failure of metal

2.06Failure of Metals

A. PINEAU

Centre des Materiaux – Ecole des Mines de Paris, Evry Cedex, France

T. PARDOEN

Universite Catholique de Louvain, Louvain-la-Neuve, Belgium

2.06.1 INTRODUCTION 686

2.06.2 CLEAVAGE IN METALS 6882.06.2.1 Introduction 6882.06.2.2 Theories of Cleavage 689

2.06.2.2.1 Theoretical cleavage stress 6892.06.2.2.2 Dislocation-based theories 690

2.06.2.3 Transgranular Cleavage of Ferritic Steels 6922.06.2.3.1 Introduction 6922.06.2.3.2 Multiple barrier models 6932.06.2.3.3 Statistical aspects of cleavage fracture in steels 695

2.06.2.4 Transgranular Cleavage of Other Metals 7002.06.2.4.1 Welds in HSLA steels: Influence of MA constituents 7002.06.2.4.2 Cleavage fracture in other BCC metals 7042.06.2.4.3 Cleavage fracture in HCP metals 7042.06.2.4.4 Irradiation-induced embrittlement in ferritic steels 706

2.06.2.5 Intergranular Brittle Fracture in Ferritic Steels 708

2.06.3 DUCTILE FRACTURE IN METALS 7092.06.3.1 Introduction: Two Classes of Failure Mechanisms 7092.06.3.2 Plastic Localization Mechanisms in Homogeneous Medium 710

2.06.3.2.1 Necking under uniaxial tension 7102.06.3.2.2 Plastic localization under biaxial loading conditions 713

2.06.3.3 Void Nucleation 7152.06.3.3.1 Macroscopic evidences 7152.06.3.3.2 Microscopic observations 7152.06.3.3.3 Computational cell simulations 7172.06.3.3.4 Void nucleation models 718

2.06.3.4 Void Growth 7202.06.3.4.1 Macroscopic evidences 7202.06.3.4.2 Microscopic observations 7212.06.3.4.3 Void cell simulations 7232.06.3.4.4 Void growth models 727

2.06.3.5 Void Coalescence 7322.06.3.5.1 Macroscopic evidences 7332.06.3.5.2 Microscopic observations 7332.06.3.5.3 Void cell simulations 7352.06.3.5.4 Models for the onset of void coalescence 7372.06.3.5.5 Models for the coalescence process 740

2.06.3.6 Fracture Strain of Metals 7432.06.3.6.1 Simple closed-form estimates of the fracture strain 7432.06.3.6.2 More advanced predictions of the fracture strain 745

2.06.3.7 Fracture Toughness of Thick Ductile Metallic Components 7512.06.3.7.1 Basics 7522.06.3.7.2 Fracture initiation toughness 7572.06.3.7.3 Ductile tearing resistance 759

684

2.06.3.8 Fracture Resistance of Thin Metallic Sheets 7612.06.3.8.1 Introduction to the fracture mechanics of thin metallic sheets 7612.06.3.8.2 The EWF method 7632.06.3.8.3 Crack-tip necking work 7642.06.3.8.4 Flat mode I fracture in thin plates 7652.06.3.8.5 Competition between flat and slant fracture 7682.06.3.8.6 General views about thickness dependence of fracture resistance 769

2.06.4 DBT IN FERRITIC STEELS 7712.06.4.1 Introduction 7712.06.4.2 DBT in fracture toughness tests 772

2.06.4.2.1 Introduction 7722.06.4.2.2 A simplified approach 7722.06.4.2.3 Advanced models 773

2.06.4.3 DBT under Charpy V impact testing 7792.06.4.3.1 Introduction 7792.06.4.3.2 Modeling Charpy V-notched impact test – salient features 7802.06.4.3.3 Other applications 781

2.06.5 CONCLUSIONS 782

2.06.6 REFERENCES 783

Nomenclature 685

NOMENCLATURE

a crack length (defect size)acs critical largest crack arrest lengthb equilibrium atomic spacing and

norm of the Burger vectorC cleavage crack lengthC0 thickness of the cementite plateletCVN energy from Charpy V-notched

testsd grain sizeE Young’s modulusf void volume fractionf0 initial void volume fractionfc critical void volume fraction at

coalescenceff critical void volume fraction at

final fracturefp particle volume fractionJ J-integralJc J-integral at cracking initiationJlc J-integral at cracking initiation

under mode I plane strainconditions

k elastic bulk modulusky Hall and Petch constantk0y cleavage stress grain size-depen-

dent constantKc/fIa critical stress intensity factor for

crack arrest at carbide/ferriteinterface

Kf/fIa critical stress intensity factor for

crack arrest at ferrite/ferriteinterface

KImin threshold stress intensity factorfor cleavage

KJc critical stress intensity factordetermined from Jlc under condi-tions of large-scale yielding

L length of a pile-up

L0x initial void spacing along xL0z initial void spacing along zLx void spacing along xLz void spacing along zm Weibull shape factorm9 rate sensitivity exponentn;N strain-hardening exponent

(depends on the context of thehardening law)

P(�) failure probability functionRCl index for transition between

intergranular and cleavage fractureR0x initial void radius along xR0z initial void radius along zRx void radius along xRz void radius along zsij components of the deviatoric

stress tensorS void shape parameter (¼lnW)S0 initial void shape parameter

(¼lnW0)T stress triaxialityU bonding energyVu representative volume elementwe essential work of fractureW void aspect ratioW0 initial void aspect ratioWp particle aspect ratioX0 average distance between a crack

tip and the closest void�a crack advance�T56J transition temperature shift for a

56 Joules Charpy energy�TKlc;100

transition temperature shift for afracture toughness equal to100MPam1/2

� relative void spacing�0 initial relative void spacing�p initial relative particle spacingd crack-tip opening displacement

686 Failure of Metals

dc critical crack-tip opening displa-cement at cracking initiation

ec void nucleation straineep, epe effective (or equivalent) plastic

straineen effective (equivalent) strain in the

minimum section of a neckef fracture strainep1 plastic strain in the direction of

the highest principal stressepy average accumulated effective

plastic strain of the matrixgbas plastic slip along basal planegj surface energy of a grain boundary

gs surface energygints

breaking energy

G fracture energy per unit area ofcrack advance

G0 fracture energy per unit area ofcrack advance associated to thedamage and material separationin the fracture process zone

G0init fracture energy per unit area of

crack advance at cracking initia-tion associated to the damage andmaterial separation in the fractureprocess zone

G0ss steady-state fracture energy per

unit area of crack advance asso-ciated to the damage and materialseparation in the fracture processzone

Gn fracture energy per unit area ofcrack advance associated tocrack-tip necking

Gninit fracture energy per unit area of

crack advance at cracking initia-tion associated to crack-tipnecking

2.06.1 INTRODUCTION

The study of the micromechanisms of failureplays a key role in the development of engineer-ing metallic alloys, in manufacturing, and in theassessment of the mechanical integrity of struc-tures. For example, in the steel industry, thedevelopment of new alloys occurs rapidlysince about one half of existing compositionsare replaced by new compositions every 5 years.In today’s automotive industry, dual-phase andother multiphase steels are quickly becomingone of the most popular and versatile materials.Currently, these steels are most commonly usedin structural applications where they havereplaced more conventional high-strength low-alloy (HSLA) steels. They offer a great

GnSS steady-state fracture energy per

unit area of crack advance asso-ciated to crack-tip necking

Gp fracture energy per unit area ofcrack advance associated to grossplasticity

GpSS steady-state fracture energy per

unit area of crack advance asso-ciated to gross plasticity

m elastic shear modulusZ geometric imperfectionl void distribution parameterl0 initial void distribution parameterlp initial particle distribution

parameterðm=kÞCD ratio for the transition between

ductile and cleavage fracture� Poisson’s ratios0 yield stresss1 maximum principal stresssc theoretical cleavage stress�d fracture stress of a particlese equivalent (effective) von Mises

stresssf cleavage stresssG Griffith fracture stresssh hydrostatic stresssij component of the Cauchy stress

tensorsth threshold stresssu cleavage stress parameter in

Weibull distribution analysissw Weibull stresssw min minimum value of Weibull stress

for cleavagesy current yield or flow stress� shear stress�i friction stressz void packing geometric parameter

opportunity for weight reduction. The develop-ment of these new compositions requires aperfect knowledge of their deformability, inparticular their forming limit diagrams(FLDs) and cracking resistance essential forcontrolling crash worthiness. Another areawhere the study of the micromechanisms offailure is essential is the assessment of themechanical integrity of structures, in particularflawed mechanical structures. These flawsappear either during manufacturing or duringservice conditions. Developing damage-tolerant microstructures is thus essential inmany fields of engineering. For a long time,these developments have remained essentiallyempirical. However, more recently, new meth-odologies have been introduced.

Introduction 687

The ‘first objective’ of this chapter is to presentan overview of the new methodologies which arebased on the investigation of the micromechan-isms at a local scale and, through a multiscaleapproach, on the transfer of these local informa-tion to the macroscale. Indeed, nowadays, thefinal goal is to develop predictive approacheswhich can be used in finite element codes forstructural analysis or for simulation of formingoperations. Several excellent reviews and bookshave already been written on the micromechan-isms of failure in metals (see, e.g., Knott, 1973;Besson 2004), but very few of them haveattempted to provide a comprehensive synthesisof the state-of-the-art predictive approaches.This is one of the goals of this chapter. In parti-cular, in the present study, a special effort wasmade to incorporate the most recent develop-ments in the theoretical and numericalmodeling of both ductile and brittle fracture.

The ‘second objective’ of this chapter is tointroduce the four tools available for microstruc-tural/micromechanical investigations and toshow their complementarity. These tools include:(1) macromechanical tests under various loadingconditions (iso- and non-isothermal tests, multi-axial tests, etc.); (2) advanced characterizationmethods (scanning and transmission electronmicroscopy, electron back-scattered diffraction,X-ray tomography, with an emphasis on in situtesting where mechanical load is combined withthe characterization); (3) computational unit cellcalculations used to investigate the mechanicalresponse of elementary volume elements(RVEs) or to simulate numerically crack initia-tion and crack growth; and (4) theoretical modelswhich remain essential tools in structural analysisand to formulate analytical expression, the bestfor revealing the essence of the physics.

The ‘third objective’ is to address the mainmechanisms of fracture based on the use of thesefour tools. In this chapter we deal only with duc-tile and brittle (cleavage and intergranular)fracture. These two modes of failure are analyzedseparately. An attempt is made to cover a widerange of materials including steels and other BCCmaterials (Mo, Nb), HCP metals (Zn, Mg), alu-minum alloys, titanium alloys, etc. Ductilefracture by nucleation, growth, and coalescenceof voids, and by plastic localization are addressed.The necessity of introducing a characteristiclength or volume when dealing with crack-tipsingularities is evident. Then the ductile-to-brittletransition (DBT) in ferritic steels is reviewed. Aspecial emphasis is also laid on the effect of irra-diation embrittlement in ferritic steels.

A ‘fourth objective’ of this chapter is to covera wide range of metals and metallic alloys. Insteels, both ferritic and austenitic microstruc-tures are considered. Multiphased and

quenched-tempered steels are also included.An attempt has also been made to underline anumber of specific aspects encountered withwelds, in particular the mismatching effect andthe problems related to the variety of micro-structures encountered in these welds. Manyexamples dealing with cast and wrought alumi-num alloys are given. A number of other metalsare also considered.

Many other topics related to the mechanismsof failure in metals, such as fracture at hightemperature and environmentally assistedcracking or most generally most of the couplingeffects with chemistry (see Volume 6), are notcovered in this chapter. It should also be addedthat while an attempt has been made to givecredit through extensive references, many ofthem have been omitted. Moreover, for thesake of simplicity, many examples are oftenextracted from our own researches.

These four objectives are those of what is nowcalled the local approach to fracture. The influ-ence of crack-tip constraint and stress triaxialityon ductile and brittle fracture is of major impor-tance for the assessment of structural integrity.This assessment is usually made by means oflinear and nonlinear fracture mechanics con-cepts (for a review, see Chapter 2.03).Constraint is a structural feature which inhibitsplastic flow and causes a higher triaxiality ofstresses. Local stress triaxiality promotes voidgrowth on the micromechanical level and thuscauses ‘damage’ in the ‘process’ zone located atthe crack tip. Constitutive equations thataccount for damage as, for example, theGurson potential (Gurson, 1977), must hencebe able to describe the physical effect of cons-traint on the tearing resistance in a naturalway. Compared with conventional fracturemechanics concepts, micromechanical modelsdeveloped in the frame of the local approach tofracture have the advantage that the correspond-ing material parameters for ductile fracture canbe transferred in a more general way betweendifferent specimen geometries. It is not evennecessary to regard specimens with an initialcrack as, of course, initially uncracked structuresalso will break if the local degradation condi-tions of the material have exceeded some criticalstate. In the Gurson model, initiation and pro-pagation of a crack are a natural result of thelocal softening of the material due to the voidcoalescence which starts when a critical voidvolume fraction, fc, is reached over a character-istic distance lc. The parameter fc can thereforebe determined from rather simple tests, forexample, tensile tests of smooth and notchedround bars in combination with numerical ana-lyses of these steels, or from micromechanicalmodels. Similarly, the Weibull stress model

Table 1 Cleavage planes

StructureCleavageplane

Somematerials

BCC {100} Ferritic steels, Mo; Nb, WFCC {111} Very rarely observedHCP {0002} Be, Mg, ZnDiamond {111} Diamond, Si, GeNaCl {100} NaCl, LiF, MgO, AgClZnS {110} ZnS, BeOCaF2 {111} CaF2, UO2, ThO2

Francois, D., Pineau, A., and Zaoui, A. 1998. Mechanical

Behaviour of Materials. Kluwer, Dordrecht.


originally proposed by Beremin (1983) providesa framework to quantify their complex interac-tion among specimen size, deformation level,and material flow properties when dealing withbrittle cleavage (or intergranular) fracture. Theoriginal Beremin model includes only two para-meters which can be determined by testingsmooth or notched bars.

The identification and determination of thedamage parameters in the Gurson or in theBeremin model require a hybrid methodology ofcombined testing and numerical simulation. Thedescription of this methodology is out of thescope of this chapter. More details are given inChapter 7.05. Here, it is enough to say that,different from classical fracture mechanics, thelocal approach procedure is not subject to anysize requirements for the specimens as long as thesame fracture phenomena occur, that is, cleavagefracture or the mechanisms of void nucleation,and growth and coalescence.

This chapter is organized according to fail-ure modes: cleavage, ductile fracture, andDBT. In Section 2.06.2, the early theoriesfor this mode of failure are presented first.Then these theories and more recent theoreti-cal developments are applied to cleavagefracture in ferritic steels and other metalswith either a BCC or an HCP microstructure.Intergranular brittle fracture in steels is alsoreviewed in this section. Two modes of ductilefracture are distinguished. The first oneoccurs in metal forming and is associatedwith plastic localization mechanisms whichare briefly reviewed. The second one involvesvoid nucleation, growth, and coalescence.This mode of ductile fracture is reviewed indetails. After presenting recent results on themodeling of these three steps involved in duc-tile fracture, an attempt is made to show howit is possible to predict general trends in thevariation of fracture strain with mechanicalloading conditions, in particular stress triaxi-ality, and with the parameters representingthe second-phase particles (shape, spacing,volume fraction, etc.) responsible for cavityinitiation. The application of these theoriesto fracture toughness of thick componentsand fracture resistance of thin sheets is thenpresented. Finally, Section 2.06.4 is devotedto the study of the DBT observed in ferriticsteels. Simplified and more advanced modelsaccounting for this transition in fracturemodes are presented. Two extreme loadingconditions are considered: quasi-static similarto those met during conventional fracturetoughness tests and Charpy V impact testing.A number of applications of the methodologypresented in this chapter to the DBT, includ-ing welds and irradiation embrittlement, are

given. However, most applications are givenin Chapter 7.05.

2.06.2 CLEAVAGE IN METALS

2.06.2.1 Introduction

Cleavage and intergranular cracking are themost detrimental modes of fracture. Cleavagefracture occurs preferentially over dense atomicplanes. Table 1 lists some cleavage planes thathave been observed experimentally. Two exam-ples of cleavage fracture observed in a high-strength ferritic steel (a,b) and in pure zinc (c)are shown in Figure 1. These figures reveal thatthe orientation of cleavage plane changes whenit crosses sub-boundaries, twin boundaries, orgrain boundaries, and steps appear on the frac-ture surface to compensate for the localmisorientation. In the case of mechanical twins,these steps look like indentation marks called‘tongue’ (Figure 2). In order to maintain theequilibrium of the crack front, the nearest stepsgather to form a single step of larger heightleading to the formation of ‘rivers’, as observedin Figure 1b. These rivers align with the directionof the local propagation of the cleavage cracks.On a macroscopic scale the surfaces of the clea-vage facets are normal to the maximumprincipal stress. In fracture mechanics terminol-ogy, this is called mode I fracture.

Another brittle mode of fracture observed inpolycrystalline metals corresponds to intergra-nular fracture. If a crack forms along a grainboundary having a surface energy, �j, thebreaking energy, 2�s

int, of the atomic bondsmust be reduced by �j. Hence, one could thinkthat intergranular fracture will be easier thantransgranular cleavage. However, the aniso-tropy in the surface energy within the crystals,�s, must also be taken into account. The surfaceenergy �s for a crystallographic cleavage planeis always less than the surface energy of anintergranular surface by typically a factor of1.20. In order to characterize the transition

between intergranular and cleavage fracture,the parameter RCI is defined as

RCI ¼2gints � gj

2gs¼ 1:20�

gj2gs

½1�

Intergranular fracture is favored when RCI<1.Cottrell (1989, 1990a, 1990b) has shown that

�j depends mainly on the shear modulus, �,

10 μm 5 μm

20 μm

Figure 1 a, Cleavage microcracks observed on longitudinal sections in a low-alloy steel (Lambert-Perlade,2001); b, fracture surface in a low-alloy steel (Lambert-Perlade, 2001) showing the presence of rivers; c, fracturein polycrystalline zinc. Rivers originating from a grain boundary.

Figure 2 SEMmicrograph of a fracture surface of alow-alloy steel. The arrows indicate the presence of‘tongues’ corresponding to the intersection of themain (001) fracture plane with mechanical twins.

Cleavage in Metals 689

while �s depends on the bulk modulus, k. Thismeans that the parameter RCI is a function ofthe ratio �/k and can thus be written as

RCI ¼ 1:20� am=k ½2�

where a is a numerical constant close to 1. Innickel, for instance, a was found to be equal to0.95. Table 2 gives the values of RCI for a num-ber of metals. The table shows that, in a largenumber of pure metals including Fe, intergra-nular fracture should be the preferential modeof fracture. However, in many cases, the segre-gation of impurities like carbon in iron tends tosuppress intergranular brittleness.

2.06.2.2 Theories of Cleavage

2.06.2.2.1 Theoretical cleavage stress

The normal stress, �c, theoretically needed tofracture a crystal by cleavage can be easilydetermined provided that the bonding energy,U, between the atoms located across the clea-vage plane is known. The force required toseparate cleavage planes is the derivative ofthis energy with respect to distance. As the dis-tance between the lattice planes increases, the

Table 2 Transition parameters for fracture: �/k is the ratio of the shear modulus to bulk modulus, RCI

quantifies the risk of intergranular fracture (if smaller than 1) vs cleavage; (�/k)CD is the ratio required to forthe transition between cleavage and ductile fracture

Metal Au Ag Cu Pt Ni Rh Ir Nb Ta V Fe Mo W Cr

�/k 0.11 0.19 0.22 0.24 0.34 0.52 0.52 0.25 0.31 0.32 0.33 0.48 0.52 0.82RCI 1.09 1.02 0.99 0.97 0.87 0.71 0.70 0.97 0.91 0.89 0.88 0.75 0.71 0.42(�/k)CD 0.32 0.43 0.57 0.38 0.49 0.39 0.32 0.59 0.55 0.65 0.56 0.35 0.45 0.68

Francois, D., Pineau, A., and Zaoui, A. 1998. Mechanical Behaviour of Materials. Kluwer, Dordrecht.


stress, which is zero at the distance b corre-sponding to interatomic equilibrium, goesthrough a maximum which corresponds to thevalue of the cleavage stress, �c. Cleavage frac-ture requires the energy of two new surfacesassociated with the formation of a pair of newsurfaces, 2�s, per unit area of new surfaces.Assuming that the variation of the force withdisplacement of the crystallographic planes issinusoidal leads to (Francois et al., 1998)

sc ¼ ðEgs=bÞ1=2 ½3�

where E is Young’s modulus. With the typicalvalues E¼ 200GPa, b¼ 0.3 nm, �s� 0.1�b� 1 Jm�2, eqn [3] leads to �c¼ 26GPa�E/10.This theoretical value of �c�E/10 for the clea-vage stress is much higher than the experimentalvalues found for classical metallic materials (typi-cally 1GPa in steels). However, for a number ofwhiskers (i.e., small filaments free of disloca-tions), the measured values are of the sameorder as the theoretical value, providing a quali-tative validation of theoretical calculation.

The reasons for the large difference betweenthe observed and the calculated values for �care twofold. In crystalline ceramic materials inwhich brittle fracture occurs under purely elas-tic conditions, that is, without the nucleationand the propagation of dislocations, the clea-vage stress is related to the existence of defectswhich are inherently present. In this case, thefracture stress is given by the Griffith stress

sG ¼ ðEgs=paÞ1=2 ½4�

where a is the size of the large defects.In crystalline metallic materials, the reason

for the difference between the theoretical valuefor �c and the experimental cleavage stress isdifferent. Cleavage in these materials is alwaysaccompanied by plastic deformation. In otherwords, plastic deformation is a prerequisite toinitiate cleavage fracture. The dislocations pro-duce stress concentrations which are sufficientto reach locally the theoretical cleavage stress.This is the basis of the theories which are brieflypresented in the following.

Before introducing these theories it is worthnoting that, in crystalline solids, cleavage

cracks blunt by the emission of dislocations.Rice and Thomson (1974) investigated the con-ditions under which this mechanism operates.They derived a criterion for the transitionbetween pure and blunted cleavage. This transi-tion occurs when the ratio between the shearmodulus, �, and the bulk modulus, k, reaches acritical value given by

ðm=kÞCD ¼ 10gs=bk ½5�The propensity for blunted cleavage increases

with increasing ratio (m=k)CD. A number ofvalues for this ratio are given in Table 2, whichshows that, in almost all metallic materials,blunted cleavage is the rule. More recently, thistheoretical transition between blunted cleavageand pure cleavage has been reanalyzed by Riceet al. (1992). These authors used the Peierls con-cept to analyze dislocation nucleation from acrack tip. They showed that in most FCC metals,except iridium, blunted cleavage should always beobserved. Conversely, in most BCC metals, purecleavage should be observed before the nucleationof dislocations ahead of a crack tip. These theo-retical calculations are useful to explain theDBT in materials like silicon or pure chromiumwhich contain initially a very low density of dis-locations but they do not apply to engineeringmaterials. In these materials, cleavage fracture isexplained by the existence of cleavage initiationsites and the stress intensification produced byplastic deformation, as detailed below.

A closing remark must be made in this intro-duction devoted to the concept of cleavagestress. The results of brittle fracture are inher-ently scattered, and Section 2.06.2.3.3 focuseson the statistical aspects of cleavage fracture. Inparticular, the Weibull stress concept, which isdifferent from the cleavage stress concept, iscentral to address this stochastic behavior.

2.06.2.2.2 Dislocation-based theories

(i) Initiation-controlled cleavage

The formation of slip bands and, under givencircumstances, of mechanical twins during defor-mation are the sources of stress concentration.This is illustrated in Figure 3 where a pile-up of

τ

τ

σ

2Lr

n

Dislocation source

θ

Figure 3 Sketch showing stress concentration at thehead of a dislocation pileup generating cleavage in aneighboring grain.

(101)

(001)

(101)

2 [111]

a √3

a √3

2 [111]

¯

¯

Figure 4 Cottrell’s mechanism: (001) cleavageinitiated in a BCC metal at the intersection of two{110} slip systems.


n dislocations is blocked by a grain boundary.Many variants have been proposed for this ele-mentary mechanism (see, e.g., Zener, 1949;Stroh, 1954). For a pile-up of length 2L, thenormal stress at a distance r from the grainboundary in a direction W is given by

s ¼ ð� � �iÞðL=2rÞ1=2fðWÞ ½6�

where � and � i are the applied resolved shearstress and the lattice friction stress, respectively.It is assumed that cleavage is initiated when theapplied stress and thus the resolved shear stress,� , reaches the theoretical critical value, �c, overa sufficiently long distance r¼Xc. Equation [6]leads to

� � �i ¼scðXc=dÞ1=2

fðWÞ ,ðE=10ÞðXc=dÞ1=2

fðWÞ ½7�

Since the length,L, which characterizes the sizeof the dislocation pile-up, is a linear function ofthe grain size, d, that is, L , d/2, eqn [7] predictsthat the stress necessary to initiate cleavage frac-ture varies like 1=

ffiffiffidp

. Moreover, as the frictionstress, � i, is strongly temperature dependent, thisequation also shows that the stress for cleavageinitiation strongly increaseswhen the temperatureis decreased. Inmany cases, in particular in steels,it has been shown that, within a first approxima-tion, the cleavage stress does not depend ontemperature. This strongly suggests that cleavageis not initiation controlled, otherwise a tempera-ture dependence should be observed.

(ii) Growth-controlled cleavage

The above calculation of the stress necessaryto initiate a cleavage crack has not addressed

the question of whether the process is possibleon energy grounds. This problem was studiedby Cottrell (1958), who assumed that a {100}cleavage was initiated in a BCC material by the‘so-called’ self-blocking mechanism of two{110} slip systems (Figure 4). When a cleavagecrack of length C appears, the dislocations inthe two pile-ups climb rapidly in the crackwhich can thus be considered as a sessile dis-location with a Burgers vector nb and a corewhose size, according to Cottrell, is C/2.Simultaneously, two surfaces are created, anda potential energy equal to �(1� �2)�2C2/2E isreleased (� is the Poisson ratio). Thus, thechange in energy is given by

Ut ¼mðnbÞ2

4pð1� �Þ log2R

C

� �þ 2gsC�

s2pC2

2E1� �2� �

½8�

Writing the conditions of instability(Ut/C<0), it is found that the critical valuefor the growth of a cleavage crack is given by

s ¼ sf ¼2gsnb

½9�

As nb¼ �(1� �)L(� � � i)/� (Friedel, 1964),eqn [9] leads to

sfð� � �iÞ ¼2mgs

pð1� �Þd ½10�

The effective stress (� � � i) acting on the dis-locations in the pile-ups is a function of grainsize according to Hall and Petch (Petch, 1953):

� � �i ¼ kyd�1=2 ½11�


Combining [10] and [11] leads to

sf ¼ k0yd�1=2 with k0y ¼

2mgspð1� �Þky

½12�

Experimental results show that the cleavagestress, �f, is proportional to d�a, with a close to1/2 but the constant k0y that is found experi-mentally leads to surface energy values muchgreater than 2�s. These large values are due todissipative mechanisms which add to the work,2�s, required to break the atomic bonds.

(iii) Cleavage initiated from intergranularcarbides in mild steels

In mild steels it is assumed that cleavagefracture initiates from very brittle platelets ofcementite located along grain boundaries.According to Smith (1966), the correspondingnecessary condition for cleavage fracture isexpressed by

ðC0=dÞs2

þ ð� � �iÞ 1þ ðC0=dÞ1=24�i=pð� � �iÞh i2

� 4Egs=p 1� �2� �

d ½13�

whereC0 is the thickness of the platelets and d isthe ferrite grain size. The terms on the left-handside of eqn [13] relate to the applied stress andexpress first the direct effect of the appliedstress (Griffith stress, eqn [4]), and then theindirect effect of the stress concentration result-ing from the dislocation pileup on the particles(second term). The term on the right-hand sideof eqn [13] represents the resistance of the fer-rite to the propagation of a cleavage crackinitiated from the carbide. At the yield strength,the Petch relationship given by eqn [11] can beapplied in eqn [13], leading to

C0s2f þ k2y 1þ 4�i=pky� �

C1=20

h i2¼ 4Egs=pð1� �2Þ

½14�In this equation, the ferrite grain size does

not appear anymore contradicting experimen-tal evidences. This apparent discrepancy isoften explained by the existence for metallurgi-cal correlation between the grain size, d, and theplatelet thickness, C0. It should also be addedthat the Smith mechanism predicts that thecleavage stress �f is an increasing function oftemperature since the lattice friction stress � i isstrongly temperature dependent.

2.06.2.3 Transgranular Cleavage ofFerritic Steels

2.06.2.3.1 Introduction

Cleavage fracture in ferritic steels is ofteninitiated from brittle second-phase particles, forexample, carbides (McMahon and Cohen, 1965;Gurland, 1972; Lee et al., 2002; Hahn, 1984; Yuet al., 2006). Carbide particles can be spherical aswell as oblong. As a result of a fiber loadingmechanism, oblong carbides experience veryhigh stresses as the surrounding ferrite matrix isplastically deformed (Lindley et al., 1970;Echeverria and Rodriguez-Ibade, 1999). Oblongcarbides are thus more prone to the initiation ofcleavage fracture. Nonmetallic inclusions, suchas manganese sulfides, MnS (Tweed and Knott,1987; Alexander and Bernstein, 1989; Neville andKnott, 1986; Carassou et al., 1998), or titaniumnitrides, TiN (see, e.g., Fairchild et al., 2000a,2000b), also act as initiation sites for cleavagefracture in ferritic steels.

A cleavage crack initiated from the fracture of abrittle particle can propagate within the adjacentferrite with a rapidly advancingmicrocrack, and ifthe arrest fracture toughness of the ferrite is toolow, the crack will penetrate into the neighboringferrite grains. The growing cleavage microcrackwill encounter grain boundaries which will forcethe crack to change its propagation direction. Thegrain boundaries are also important obstacles forcontinued crack growth as discussed by a numberof authors (e.g., Qiao and Argon, 2003a, 2003b;Crocker et al., 2005). Propermodeling of cleavagein ferritic steels requires thus to account for multi-ple barriers to the propagation of cleavage cracks.The simplest models are deterministic. However,more sophisticatedmodels including the statisticalaspects of the problem have also been proposed,as discussed later.

Over the past few decades, there has been asteady decrease in many structural steels of thecarbon content and of the impurity (P, S) level,and, as a result, typical cleavage initiators likecementite particles and nonmetallic inclusionshave been largely reduced in number and size.This has contributed to the improvement of thebrittle cleavage fracture resistance. However,despite these advances, three factors virtuallyguarantee that cleavage fracture in steel willunfortunately always remain a concern. First,because of continuing improvements in struc-tural steels, users are selecting these materialsfor more severe service conditions. Second,cleavage will always remain the intrinsic brittlemode of failure in BCC materials. Third, struc-tural steels are usually fusion welded and thisleads to the presence of microstructures in theweld metal and in the heat-affected zone (HAZ)


which are typically inferior to the highly pro-cessed base metal. In the following, a specialsection is devoted to the fracture micromechan-isms in welds.

2.06.2.3.2 Multiple barrier models

In many ferritic steels, it has been foundthat the cleavage stress, �f, is independent oftemperature. This strongly suggests that inthese materials, the mechanism of cleavagefracture is growth controlled, as indicatedpreviously (see, e.g., Curry and Knott, 1979;Pineau, 1981, 1992). Cleavage microcracks areprogressively nucleated under the influence ofplastic strain. These microcracks are arrestedat microstructural barriers and fracture occurswhen the longest crack reaches the Griffithstress, given by eqn [4]. In this equation, allterms are almost independent of temperature,except the term �s which is much higher thanthe true surface energy because of the dissi-pated energy due to plastic deformation.However, this theory is too simple since itdoes not recognize the different steps encoun-tered during microcrack initiation andmicrocrack propagation (see, e.g., Martin-Meizoso et al., 1994).

Schematically, fracture of ferritic steels mostfrequently results from the successive occurrenceof three elementary events illustrated in Figure 5:

1

2

M–A

Effective bainitic packet

K Ia

K Ia

3

C

D

f/f

c/f

Figure 5 Initiation of a cleavage microcrack from aparticle (martensite/austenite (MA) constituent).The crack may eventually arrest at the interface c/f;then propagates through the matrix and is arrested atthe grain boundaries.

� slip-induced cracking of a brittle particle;� propagation of the microcrack under the

local stress state across the particle/matrixinterface and then along a cleavage plane ofthe neighboring matrix grain; and� propagation of the grain-sized crack to neigh-

boring grains across the grain boundary.

The first event which is similar to that occur-ring in ductile rupture is governed by a criticalstress, �d, when the particle size is larger than,0.1–1 mm (see, e.g., Pineau, 1992). Below thissize, a dislocation-based theory must be used,as indicated in Section 2.06.3.3. As shown later,the critical stress, �d, is related to the maximumprincipal stress, �1, the equivalent von Misesstress, �e, and the yield stress, �0, by

s1 þ kðse � s0Þ ¼ sd ½15�

where k is a function of particle shape(Beremin, 1981; Francois and Pineau, 2001).Within a first approximation, �d is independentof temperature, but the values of �d are statis-tically distributed.

The simple expression given by eqn [15] (seealso Margolin et al., 1998) shows that for agiven stress state the strain necessary to nucle-ate particle cracking strongly increases withtemperature because of the variation of theyield strength with temperature. Figure 5 repre-sents the local values of the critical stressintensity factor KIa

c/f (carbide/ferrite) and KIaf/f

(ferrite/ferrite) that must be overcome in orderfor the crack not to arrest. These values are alsostatistically distributed. Recent studies haveshown that in bainitic steels the crack arrestingboundaries are those for which the misorienta-tion between the bainite packets is large (Bouyneet al., 1998; Gourgues et al., 2000; Lambert-Perlade et al., 2004). The particle and ‘grain’size distribution functions (fc, fg) have thus tobe considered, as schematically shown inFigure 6. In the figure, the critical values of theparticle and grain size,C* andD*, correspondingto the different steps of cleavage fracture aresimply related to the local value of the maximumprincipal stress, �1, by a Griffith-like expression:

C* ¼ dKc=fIa

s1

!2

and D* ¼ dKf=fIa

s1

!2

½16�

where is a numerical factor related to theshape of the microcrack and close to 1.

There are few results in the literature to testthe validity of the above model, in spite of thelarge number of studies devoted to steels.However, a number of results are reported inTable 3. In the table, the details concerning arecent study on a bainitic steel (Lambert-Perladeet al., 2004) are given in the following. It is worth


noting that the local values of the calculatedfracture toughness, KIa

c/f, are much lower thanthe macroscopic fracture toughness, KIc.Several reasons can be invoked to explain thisdifference. The first one lies in the calculations.Applying eqn [16] necessitates, ideally, the use ofthe local values of the maximum principal stresswhich can be much larger than the macroscopicstress (used in the present calculations). The sec-ond reason could be related to the fact that thesecalculations apply to static conditions which isnot necessarily appropriate as discussed in thenext section. In Table 3, it is also worth noting

f c

f g

C

1

Particle fractureCrack arrest at c /f interface

1

C > C* Interface crossingCrack arrest at f /f interface

2

D > D* Final fracture3

2

2 3

3

C

∗ =

δ × KIa

σ1

2c/f

D

∗ =

δ × KIa

σ1

2 Df /f

Figure 6 Multiple barrier model. Three events areschematically shown with their probability ofoccurrence (Martin-Meizoso et al., 1994).

Table 3 Parameters of

Parameter Present study value Value

�d (MPa) 2112

KIac/f (MPam1/2)

7.8

2.5–5.02.51.8

KIaf/f (MPam1/2) CGHAZ-25 5.0–7.0

28 7.07.5

ICCGHAZ-25 4.818 15.2

thatKIaf/f is larger thanKIa

c/f. This conclusion com-bined with other observations obtained fromacoustic emission measurements (Lambert-Perlade et al., 2004) strongly suggests that themicromechanisms operating during fracturetoughness measurements at increasing tempera-ture are not necessarily the same. In suchconditions the existence of a constant cleavagestress over a wide temperature range couldappear as fortuitous.

The dynamic behavior of microcracksnucleated from a carbide and propagatingwithin the ferrite matrix has been studiedrecently in details by Kroon and Faleskog(2005). These authors used a unit cell-typedynamic finite element calculation (Figure 7).The initiation of cleavage fracture was modeledexplicitely by introducing a pre-existing smallcrack within the carbide. This microcrack pro-pagates through the carbide and eventually intothe surrounding ferrite. The carbide which hasa size of a few microns was modeled as anelastic cylinder (or sphere), and the ferrite asan elastic viscoplastic material with a yieldstrength at vanishing zero strain rate equal to�0. Macroscopic constitutive equations allow-ing for different strain rate sensitivity wereadopted. The crack growth was modeled usinga cohesive surface, where the tractions are gov-erned by an exponential cohesive law. Crackgrowth rates as large as the Rayleigh wavespeed and thus resulting in strain rates as largeas 104–106 s�1 were simulated.

These calculations show that the criticalstress required to propagate a microcrackinitiated from a broken carbide increases withdecreasing plastic strain rate sensitivity of thematrix. Results showing the variation of themacroscopic stress, Sz, as a function a carbidesize, C, are given in Figure 8. The results wereobtained for various values of the stress triaxi-ality measured by the ratio C¼Sr/Sz. InFigure 8 the Griffith criterion is also includedas a reference. The curve corresponding to the

multiple barrier models

Literature data

Microstructural unit Reference

Lambert-Perlade et al. (2004)

Carbides Martin-Meizoso et al. (1994)Globular carbides Hahn (1984)TiN particles Rodrigues-Ibabe (1998)

Bainite packets Martin-Meizoso et al. (1994)Bainite packets Martin-Meizoso et al. (1994)Ferrite grains Hahn (1984)Bainite packets Rodrigues-Ibabe (1998)Bainite packets Rodrigues-Ibabe (1998)

∑r ∑r

z

r

2c

2R0

2H0

2h

(a) (b)

∑z

∑z

ϕ

Figure 7 a, A cracked grain boundary carbide in ferrite. b, The axisymmetric model with a carbide embeddedin ferrite (Kroon and Faleskog, 2005).

7

6

5

4

3

20 2 4 6

c (μm)

Griffith criterion

8 10 12

∑z

/σ0

Ψ = 0.75Ψ = 0.70Ψ = 0.65Ψ = 0.60

Figure 8 Overall stress Sz/�0 as a function of criticallargest carbide size c for four levels of stresstriaxiality, C¼Sr/Sz. Comparison with Griffithcriterion (Kroon and Faleskog, 2005).


Griffith criterion is located above all the curvescorresponding to the elastic viscoplastic mate-rial. This situation may appear as being rathercounterintuitive since plastic flow is expected toincrease the resistance to crack growth.However, as stated by the authors, the Griffithcurve is valid for a stationary crack, whereas inthese numerical simulations the crack has asignificant speed when it reaches the carbide–ferrite interface. Figure 8 shows also that thestress levels required to arrest a microcrackvary at a much lower rate with decreasing car-bide size compared to the Griffith stress. Thisimplies that small carbides may play a moreprominent role in cleavage fracture of ferriticsteels than what might be expected from thestraightforward application of the staticGriffith criterion. The strain rate sensitivityand the dynamic aspect of crack growth thuscome into play in the initiation and in the con-tinued growth of a cleavage crack. Figure 8 also

suggests that if the stress triaxiality level isdecreased from 0.75 to 0.60, the critical stresslevel, required to initiate a critical microcrack,decreases by an amount approximately equal tothe initial yield stress of the ferrite matrix, �0.This effect of stress triaxiality on cleavage frac-ture initiation is opposite to what is normallyseen in cleavage fracture experiments, where adecrease in crack-tip constraint leads to anincrease of the fracture toughness. However,this last conclusion is not valid since, in thisstudy, the mechanism responsible for the initia-tion of carbide cracking was not consideredbecause of a precrack being introduced in thecarbides. The deleterious effect of the increasein stress triaxiality level on the nucleation ofmicrocracks from particles is described by eqn[15]. Further results were obtained in the studyby Kroon and Faleskog (2005), such as a rela-tionship between the applied axial stress Sz andthe critical largest crack arrest length acs, andthis relation is independent of the carbide sizeand of the level of stress triaxiality.

In spite of all the researches devoted to thestudy of cleavage micromechanisms in ferriticsteels, many questions remain open and bothexperimental and theoretical investigations arestill necessary to reach full transferability toreal life engineering problems.

2.06.2.3.3 Statistical aspects of cleavagefracture in steels

(i) Beremin model

Rather surprisingly, although the scatter incleavage stress measurements is well knownsince a long time, it was only in the 1980s thatmodels have been proposed to account for thisscatter (for review papers, see, e.g., Wallin et al.,1984, 1991a, 1991b). Nowadays, the most

Etat

wpa

w

te1g(cap

w�(1

1agaibattfc

ldt

tdq


largely used models are those derived from thework by Beremin (1983) (for a review of thesemodels, seeMudry, 1988; Pineau, 1992, 2006; seealso Evans, 1983; see Chapter 7.05). Assumingthat the material contains a population of micro-defects (particles or grain-sized microcracks)distributed according to a simple (power orexponential) law, p(a), the weakest link theorystates that the probability to failure P(�) of arepresentative volume element Vu is given by

PðsÞ ¼Z 1acðsÞ

pðaÞda ½17�

where ac is simply given by eqn [4], that is,

ac ¼2Egsas2

½18�

where a is a numerical constant. Knowing thedistribution p(a), it is therefore possible to cal-culate the associated distribution P(�).

In a volume V which is uniformly loaded andwhich contains V/Vu statistically independentelements, the probability to failure can thus beexpressed as

PR ¼ 1� exp � V

VuPðsÞ

� �½19�

As a general rule the function p(a) is notknown. However, when the critical step forcleavage fracture is the propagation of micro-cracks initiated from particles, the distributionp(a) can be determined experimentally. Twotypes of laws are usually proposed:

� a power law

pðaÞ ¼ ga�b ½20�� an exponential law including, if necessary, a

cutoff parameter (see, e.g., Carassou et al.,1998; Lee et al., 2002) such as the cumulatedprobability is given by

pðsize>aÞ ¼ exp � a� aua0

� �n� �½21�

where au and a0 are parameters of the dis-tribution. The simple power law leads to thewell-known Weibull expression


V0

ssu

� �m� �½22�

with the Weibull shape factor

m ¼ 2b� 2 ½23�

It should be noted that, within a first approx-imation, m and �u are temperatureindependent. Similarly, the exponential lawgiven by eqn [21] leads to (Tanguy et al., 2003)


V0� ð1=s

2Þ � ð1=s2uÞð1=s2c0Þ

� �n� � ½24�

sc0 ¼ ð2Egs=aa0Þ1=2 and su ¼ ð2Egs=aauÞ1=2

½25�

quation [22] is a simplified expression since nohreshold is introduced. In three dimensionsnd in the presence of smooth stress gradients,his equation can be written as

PR ¼ 1� exp �RPZ s

m1 dV

smuVu

� �½26�

here the volume integral is extended over thelastic zone (PZ). This equation can be rewrittens

sw ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZPZ

sm1 dV

Vu

m

s½27�

here �w is referred to as the ‘Weibull stress’.A number of investigators have introduced a

hreshold stress, �th, directly into eqn [27] (see,.g., Bakker and Koers, 1991; Xia and Cheng,997). One possible expression for the inte-rand of eqn [26] has the form [(�1� �th)/�u� �th)]m. However, rational calibration pro-edures for �th remain an open issue. In order tovoid these difficulties, Gao et al. (1999) pro-osed a modified form of eqn [27] given by

PR ¼ 1� exp � sw � sw min

su � sw min

� �m� �½28�

here �wmin represents the minimum value of

w at which cleavage fracture becomes possiblefor a full discussion, see Gao et al., 1998a,998b; Gao and Dodds, 2000).In the original Beremin model (Beremin,

983), an implicit threshold Weibull stress waslso included, a point which is somewhat for-otten in the literature. In this model, it isssumed that cleavage fracture cannot occurn the absence of plastic deformation, that is,elow the yield strength, �0. This means thathead of a crack tip the PZ size must be largerhan a critical size, Xc, or otherwise stated thathere exists a threshold for the stress intensityactor, KImin, below which cleavage fractureannot occur. This threshold is given by

KImin � s0ffiffiffiffiffiffiffiffiffiffiffi3pXc

p½29�

It should also be noted that the exponentialaw with a cutoff parameter (eqn [21]) used toescribe the particle distribution leads also to ahreshold stress given by eqn [24].The original Beremin model contains only

wo independent parameters, m and the pro-uct �u

mVu. A number of studies performed onuenched and tempered steels used in reactor

400 A 508

2¼ Cr300

200

100

K Ic

(M

Pa

m1/

2 )

0–200 –150 –100 –50

Temperature (°C)

0 50

Ring

CT25

CT25

CT20

CT20

CT50

CT50

CT195

CT100

Figure 9 Fracture toughness of two pressure vessel steels (A 508 and 2¼Cr–1Mo steels) as a function of testtemperature. The DBT temperature in 2¼Cr–1Mo steel with a lower bainite microstructure is much lower thanthat of A 508 steel with an upper bainite microstucture.


pressure vessels (RPVs) have shown that m �20 when no threshold is explicitly used, that is,when the Weibull stress is calculated using eqn[27] (see, e.g., Beremin, 1983). Similar valueshave been reported on structural steels with ayield strength between 490 and 685MPa(Minami et al., 2002). The strategy used forthe calibration of Weibull parameters has beendescribed in details elsewhere (Chapter 7.05).Values of m lower than 20 are found when athreshold is introduced (see, e.g., Gao andDodds, 2000; Petti and Dodds, 2005a).

In the Beremin model, when the value ofVu isfixed, the value of �u reflects the resistance ofthe material to brittle cleavage fracture. This isillustrated in Figure 9 where the fracture tough-ness temperature dependence of twolow-alloyed steels used in the fabrication ofpressurized water reactors was studied, that is,A 508 steel and a modern 2¼Cr–1Mo steel(Bouyne et al., 2001). These materials havesimilar yield strength at room temperature butdifferent microstructures.

As observed in Figure 10, the A 508 steel hasan upper bainite microstructure, while 2¼Cr1% steel shows a lower bainite microstructure.Figure 9 shows that the ductile-to-brittle transi-tion temperature (DBTT) is lower in the2¼ Cr–1Mo steel than in the A 508 steel.Tests on notched specimens showed that thesame ranking was observed when the values of�u were measured adopting the same value forVu (50 mm)3. In A 508 steel, it was found thatm¼ 22 and �u¼ 2600MPa (Beremin, 1983),while in 2 1/4 Cr–1Mo steel, it was found thatm¼ 20 and �u¼ 3500MPa (Bouyne, 1999).These values for �u and m are typically thoseencountered in quenched and tempered steels.

The Beremin model has also been applied tomultiple barrier models. As stated previously,this model is essentially based on the descrip-tion of the propagation of an existing criticaldefect belonging to a single population. This isa simplification which might explain why in theapplication of the model over a wide range oftemperatures, a number of investigators havereported that it was necessary to assume thatthe normalizing stress, �u, was an increasingfunction of temperature (see, e.g., Tanguyet al., 2005a, 2005b). This might simply reflectthe existence of different critical steps depend-ing on temperature, as indicated earlier.

Multiple barrier models would thereforeappear more satisfactory to account for thevariation of cleavage fracture toughness overa wide temperature range. In particular, mul-tiple barrier models addressing the fractureprocess schematically shown in Figure 11have been proposed (Martin-Meizoso et al.,1994; Lambert-Perlade et al., 2004). Thesemodels are also based on the weakest linktheory. The nature of these barriers dependson temperature. Application of these modelsrequires the knowledge of a certain number ofmetallurgical factors including the nucleatingparticle size distribution and the grain(packet) size distribution. These factors weremeasured in one specific steel in which thebrittle particles were formed by MA constitu-ents (Lambert-Perlade et al., 2004), that is,mixed MA particles. The application ofthese models also necessitates the knowledgeof the local fracture toughness, KIa

c/f andKIaf/f, and that of the cleavage fracture stress

of particles, �d. There are very few results inthe literature. However, a number of results

(c)

25 μm

50 μm

(a) (b)

(d)

Figure 10 Microstructures of pressure vessel steels: a, b, A 508 steel; c, d, 2¼Cr–1Mo steel. A 508 steelexhibits an upper bainite microstructure containing relatively coarse carbide particles, while 2¼Cr–1Mo steelhas a lower bainite microsctructure with smaller carbides.

(a) (b)

(d)(c)

Figure 11 Schematic representation of the role of microstructural barriers on fracture micromechanisms. Thecrack is assumed to nucleate from an intragranular particle: a, undamaged material; b, microcrack initiationand propagation in the particle; c, microcrack propagation across the particle/matrix interface; andd, microcrack propagation across a grain boundary or a bainite high-angle packet boundary leading to finalfracture (Lambert-Perlade et al., 2004).



are reported in Table 3. In the study devotedto a bainitic steel containing MA particles, itwas assumed that the local values of fracturetoughness, KIa

c/f and KIaf/f, were not temperature

dependent, which is a crude approximation.In spite of this approximation, a good agree-ment was found between the experimentalvalues of the fracture toughness and thoseinferred from this multiple barrier model, asshown later.

(ii) Effect of plastic strain

The Weibull expression in eqn [27], which isthe product of a stress function and a volume(PZ), is based on the assumption that thereexists a population of microcracks nucleatedat the onset of plastic deformation. Thesemicrocracks remain active during the entireloading history. This is an oversimplificationwhich requires further discussion.

Detailed investigations on a number of steelshave shown that the number of microcracksnucleated from carbides was an increasing func-tion of plastic strain and was increasing withdecreasing temperature (see, e.g., Kaechele andTetelman, 1969; Mc Mahon and Cohen, 1965).More recently in a study devoted to the initiationof microcracks from niobium carbides and tita-nium nitrides in an Ni-based superalloy (In 718),it was shown that the probability to failure ofthese particles could be written as

Pnucl ¼ 1� exp� s1 þ lðse � s0ÞsuN

� �Na

½30�

where �uN is a normalizing stress and Na is amaterial constant (Alexandre et al., 2005).Similar studies are lacking for a proper descrip-tion of the nucleation of microcracks fromcarbides and nonmetallic inclusions in ferriticsteels. This is why the recent modifications to theBeremin model presented below are essentiallybased on phenomenological considerations.

The original Beremin model has beenrecently modified by Bordet et al. (2005a,2005b) in order to include the effect of plasticstrain on the nucleation of microcracks and thedeactivation of the latter if they are not imme-diately propagated. At a material point locatedwithin the PZ, the probability of cleavage frac-ture is expressed as

PcleavðtÞ ¼Z t

0

PpropdPnucl ½31�

where t represents the ‘loading time’, whilePprop and Pnucl represent the probability ofcrack propagation and the increment of prob-ability to nucleate a microcrack. In Bordet’smodel, Pnucl is written as

dPnuclaNsy0ðT; _eepÞdeep ¼ N0ð1�PnuclÞs0deep ½32�

where eep is the equivalent plastic strain, N andN0 are respectively the remaining and the initialnumber of cleavage initiation sites. Faleskoget al. (2004) have similarly assumed that theincrement of the probability of microcracknucleation was linearly related to plastic strain.However, these authors did not include theeffect of yield strength on microcrack nuclea-tion. Equation [32] can be easily integrated.Normalizing the yield strength by a stress �y0and the plastic strain by eep0 which are depen-dent on the material, eqn [32] leads to

Pnucl ¼ 1� exp� s0sy0

eepeep0

� �½33�

which bears some similarity with eqn [30].However, there exists a significant differencebetween those two expressions since eqn [33]does not include the effect of stress state whichis well known to affect the nucleation of micro-cracks from particles (see Section 2.06.3.3).

As for the Beremin model, assuming that thenucleated microcracks can be treated asGriffith flaws, which are assumed to be distrib-uted according to a power law, and invokingthe weakest link principle over the PZ volume,the overall fracture probability can then beexpressed as

PR ¼ 1� exp �ZPZ

Z t

0

PpropdNnucl dV

� �

¼ 1� exp � s*w

s*u

� �m� �½34�

where �u* is a scaling parameter and �w

* is amodified Weibull stress defined as

s*w

� �m¼ZPZ

ðs1>sthÞZ eep

0

s0sy0

sm1 �smth� �

ð1�PnuclðtÞÞdeepeep0

� �dV

Vu

½35�

where Vu is a reference volume as in eqn [19].The similarities between the models of

Bordet and Faleskog for the nucleation ofmicrocracks have already been underlined.There exists, however, a difference in theexpression for the probability to propagatemicrocracks, since in Faleskog’s model it wasimplicitly assumed that these microcracks weredistributed according to an exponential lawsimilar to the expression introduced previously(see eqns [21] and [24]). A modification of theoriginal Beremin model has been also proposedby Bernauer et al. (1999) to account for clea-vage fracture in the transition region of aferritic steel. These authors have emphasizedthat in the transition region the number


of available microcracks is reduced by the num-ber of particles around which a void has beenformed. They assumed that the number of‘omitted’ carbides was proportional to thetotal number of voids formed at second-phaseparticles. The nucleation rate of cavities wasassumed to follow the law proposed by Chuand Needleman (1980) (see Section 2.06.3.3).Stockl et al. (2000) have applied theBernauer’s model to interpret results on warmprestress effect.

In these modifications or enhancements ofthe original Beremin model, plastic strainappears as detrimental since the microcracksare nucleated from carbide particles or non-metallic inclusions due to plastic deformation.On the contrary, a number of other observa-tions have also shown that the effect of apredeformation can be beneficial (see, e.g.,Groom and Knott, 1975; Knott, 1966, 1967;Beremin, 1981). This is why in the Bereminmodel a ‘strain correction’ factor was intro-duced to account for the effect of plasticstrain. The probability to failure was written as

PR ¼ 1� exp �RPZ s

m1 expð�mep1=aÞdV

smuVu

� �½36�

where ep1 is the plastic strain in the direction ofthe highest principal stress and a is a constantclose to 2.

To conclude this part devoted to ferritic steels,it appears that, in spite of its simplifications, theBeremin model has largely contributed to a bet-ter description of the brittle cleavage fracture inferritic steels. A full account of the application ofthis model to predict the fracture toughness of anumber of ferritic steels is given in Chapter 7.05.The main criticism to this model, which is theabsence of a threshold, is not really acceptablesince this threshold is present although it doesnot explicitly appear in the expression giving theprobability to fracture. The original Bereminmodel applied to the prediction of fracturetoughness tends to underestimate the variationof fracture toughness with temperature, in parti-cular in the transition regime. This has led anumber of authors to assume that the scalingstress, �u, was an increasing function oftemperature.

2.06.2.4 Transgranular Cleavage ofOther Metals

In this section, cleavage fracture in weldsmade of HSLA steels is reviewed because ofits technological importance. The emphasis islaid on the influence of MA constituents whichmay be formed in those welds. Then cleavagefracture in other BCC metals, especially Moand Nb, is briefly considered. Finally, cleavage

fracture in two HCP metals (Zn and Mg) isexamined.

2.06.2.4.1 Welds in HSLA steels: Influence ofMA constituents

HSLA steels are now widely used for struc-tural applications. These materials combineexcellent tensile strength and DBT properties.However, this combination of high-strengthand high-fracture toughness usually deterio-rates after welding thermal cycles. Thedegradation of the fracture toughness ofHSLA steels after welding is attributed to theformation of ‘local brittle zones’ in the weldedjoint (see, e.g., Davis and King, 1994).Significant embrittlement can be encounteredin the coarse-grained heat-affected zone(CGHAZ) and, in particular, in the intercriti-cally reheated CGHAZ (ICCGHAZ) ofmultipass welded joints (see, e.g., Toyoda,1988; Kenney et al., 1997; Zhou and Liu, 1998).

An example of such embrittlement effects isshown in Figure 12which refers to a recent studyperformed on the micromechanisms and model-ing of cleavage fracture in simulated HAZmicrostructures obtained in an HSLA steel(�0¼ 430MPa at room temperature) (Lambert-Perlade et al., 2004). In this study, a Gleeblesimulator was used to apply thermal cyclesrepresentative of those encountered during mul-tipass welding. The maximum temperature ofthe first cycle was Tp1¼ 1250 �C. Cooling timesfrom 800 to 500 �C (Dt8/5) were chosen to be100 s (CGHAZ-100) and 500 s (CGHAZ-500)corresponding to a medium and high-inputwelding energy, respectively. Intercritical heat-ing of the CGHAZ-100 microstructure(ICCGHAZ) at maximum temperature(Tp2¼ 775 �C) with the same cooling conditionsas CGHAZ-500 induced partial austenitizationof the bainitic microstructure. Upon furthercooling, austenite was partially transformedinto martensite leading to the formation of MAconstituents (Figure 13). Figure 12 shows thatthe ICCGHAZ microstructure produces a shiftof the DBTT by about 80 �C. Tensile tests onnotched specimens showed that the cleavagestress was reduced as compared to the ferrite–perlite microstructure (base metal), in particularin the ICCGHAZ-100 and CGHAZ-500 micro-structures. This clearly illustrates the deleteriouseffect of the presence of MA constituents oncleavage fracture in welds.

Fracture toughness tests were carried out onspecimens which were simulated with other(Dt8/5) cooling conditions. Figure 14 showsthat the transition temperature measured atKJ¼ 100MPam1/2 is much higher in the

0

20

40

60

80

–200 –150 –100 –50

CGHAZ-100 s

ICCGHAZ-100 s

CGHAZ-500 s

Temperature (°C)

Base metal

Str

ain

to fa

ilure

0 50

Figure 12 Variation of the strain to failure measured on notched specimens as a function of test temperature ina low-alloy steel with four microstructures (Lambert-Perlade, 2001). Reproduced with permission fromLambert-Perlade, A., Gourgues, A. F., Besson, J., Sturel, T., and Pineau, A. 2004. Mechanisms andmodeling of cleavage fracture in simulated heat-affected zone microstructure of a high-strength low alloysteel Metall. Mater. Trans. A 35, 1039–1053.

3

21

Figure 13 SEM micrograph of the microstructureof an HAZ in a weld. (1) Residual austenite locatedat lath grain boundaries; (2) MA mixed constituentat a former austenite grain boundary; and (3) bainiticpacket boundary (Lambert-Perlade, 2001).


simulated microstructures than in the basemetal since shifts as large as 150 �C are mea-sured. Cleavage crack initiation observed on afracture toughness specimen tested at 20 �C isshown in Figure 15: cleavage was initiated froman MA constituent (arrow), revealed after che-mical etching of the fracture surface.

An attempt was made to model the observedvariations of the fracture toughness with tem-perature shown in Figure 14 using the Bereminmodel. The results are reported on the samegraphs while the values of the Beremin parameter,m and �u, are given in Table 4. These values werefitted by using elastic–plastic finite element calcu-lations on notched specimens tested in the brittle

temperature range, followed by post-processingcalculation of both theWeibull stress, �w, and theprobability to fracture, PR. Figure 14 shows agood agreement with the experimental resultsfor the simulated HAZmicrostructures. A similarconclusion was reached by Tagawa et al. (1993).In Figure 14, we have also included the predic-tions obtained from the ‘master curve’ approach(Wallin, 1991a, 1991b, 1993; ASTM E 1921-02,2002; for further details, see Lambert-Perladeet al., 2004). Tagawa et al. (1993) also usedthe Beremin model for low carbon steel HAZmicrostructures. They found smaller values form (10–18) instead of 20–27. From a theoreticalpoint of view, the values of the parameter mshould be related to the defect size power-lawdistribution (p(a)) by eqn [20]. The values of theb exponent in eqn [20] were determined experi-mentally. Using these values of b and therelationship m¼ 2b� 2 leads to m� 5 which ismuch smaller than the values (m¼ 20–27) usedto draw the curves shown in Figure 14. They areeven smaller than those reported by Tagawaet al. (1993). This probably arises from the factthat cleavage fracture occurs in several stepswhich theoretically implies the use of a multiplebarrier model as stated previously.

The multiple barrier model was also appliedto the results of fracture toughness measure-ments obtained in various simulated HAZmicrostructures. The details are given elsewhere(Lambert-Perlade et al., 2004). The effect oftemperature on fracture toughness can bedescribed by dividing the problem into threetemperature ranges:

1. At very low temperature, the critical stepin cleavage fracture was assumed to correspondto the nucleation of cleavage microcracks fromMA particles.

2. At somewhat higher temperatures, micro-cracks initiate at particles and stop at theparticle/matrix interface. The critical step isthen the propagation of these particle-sizedmicrocracks.

300

(a) (b)

(d)(c)

250

200

150

100

50

0–200 –150 –100

Temperature (°C)

K J

c(M

Pa

m1/

2 )

Experiments

Pr = 10%

Pr = 90%

–50 0 50

300

250

200

150

100

50

0–200 –150 –100

Temperature (°C)

K J

c(M

Pa

m1/

2 )

–50 0 50

Experiments

Pr = 10%

Pr = 90%

300

250

200

150

100

50

0–200 –150 –100

Temperature (°C)

K J

c(M

Pa

m1/

2 )

–50 0 50

Experiments

Pr = 10%

Pr = 90%

300

250

200

150

100

50

0–200 –150 –100

Temperature (°C)

K J

c(M

Pa

m1/

2 )

Experiments

Pr = 10%

Pr = 90%

–50 0 50

Figure 14 Brittle-to-ductile toughness transition curves in a low-alloy steel and in simulated HAZmicrostructures. a, Base metal; b, CGHAZ-25; c, ICCGHAZ-25; d, CGHAZ-120 microstructures. Solid lines(respectively, dotted lines) show fracture probabilities of 10% and 90% given by the Beremin model and the‘master curve’ approach. Numerical values used in the Beremin model are given in Table 4 (Lambert-Perlade,2001; Lambert-Perlade et al., 2004). Reproduced with permission from Lambert-Perlade, A., Gourgues, A. F.,Besson, J., Sturel, T., and Pineau, A. 2004. Mechanisms and modeling of cleavage fracture in simulated heat-affected zone microstructure of a high-strength low alloy steel. Metall. Mater. Trans. A 35, 1039–1053.

Fatigue precrack

First cleavage facet

10 μm

Figure 15 Cleavage crack initiation afterinterrupted test of a fracture mechanics specimen.ICCGHAZ-25 microstructure. The cleavage crackis initiated from an MA constituent (indicated byan arrow) (Lambert-Perlade et al., 2004).Reproduced with permission from Lambert-Perlade, A., Gourgues, A. F., Besson, J., Sturel, T.,and Pineau, A. 2004. Mechanisms and modeling ofcleavage fracture in simulated heat-affected zonemicrostructure of a high-strength low alloy steel.Metall. Mater. Trans. A 35, 1039–1053.

Table 4 HSLA steel. Parameters of the Bereminmodel (unit volume Vu¼ (100mm)3)

Microstructure Bereminmodel

Experimental valuesof T*100(�C)

�u(MPa) m

Base metal 2158 27 �140CGHAZ-25 2670 20 �55ICCGHAZ-25 2351 20 �20CGHAZ-120 2085 20 �10

The parameter T*100 provides the value of temperature for

which KJc=100MPam1/2.



3. At higher temperatures, microcracks arearrested at high-angle bainitic packet bound-aries. The critical step is then the propagationof these arrested grain-sized microcracks.

The input parameters of the multiple barriermodel are, therefore, as follows:

1. The fracture probability p(c) of an MAparticle of size c. In the absence of statisticalmeasurements it was assumed that this initia-tion process occurred for a single value of thecritical stress (see eqn [15]). The value of�d¼ 2100MPa was assumed using the inclusiontheory (Eshelby, 1957) (see Table 3).

2. The distribution functions fc(C) and fg(D)for MA particles and bainitic packet size wereexperimentally determined and represented bylog–normal functions.

3. The critical size for cracked MA particlesand cracked bainitic packets were calculatedusing eqn [16]. For the sake of simplicity, itwas assumed that the values of the local frac-ture toughness, KIa

c/f and KIaf/f, were independent

of temperature, as indicated previously. Thismeans that, as in the Beremin model, the tem-perature dependence of the fracture toughnessarises mainly from the variation of the yieldstrength with temperature.

The numerical values of these input para-meters are given in Table 3. The resultsshowing the application of the multiple barriermodel to one specific condition (ICCGHAZ-25)are reported in Figure 16. A good agreement isobserved between the theory (solid lines) and theexperiments. In particular the model is able toreproduce the dispersion, which is not trivialsince the calculated scatter derives directly fromthe experimental size distribution of second-

Tempe

Tou

ghne

ss (

MP

a. m

1/2 )

200ICCGHAZ-25

150

100

–100–150–200

50

0

Figure 16 Results of the multiple barrier model. Openemission; solid circles correspond to final fracture. Solid90% probabilities for the specimen to fracture (respectivparticle/matrix boundary) as given by the multiple barriwith permission from Lambert-Perlade, A., Gourgues,Mechanisms and modeling of cleavage fracture in simstrength low alloy steel. Metall. Mater. Trans. A 35, 103

phase particles and bainitic packets. In thesetests, acoustic emission was also used to detectthe number of events occurring before fracture.Interrupted tests showed that one event corre-sponded to the nucleation and crack growth of amicrocrack which was arrested at high-anglegrain boundaries (Lambert-Perlade et al.,2004). In Figure 16, it is observed that the lowestvalue of fracture toughness (open symbols) cor-responding to the first detection of microcrackevents detected by acoustic emission occurs forstress intensity factors equal to about30–40MPam1/2. These values agree with thecalculated probability for a cleavage microcrackto propagate across the particle/matrix bound-ary which is shown by dotted lines. These lineswere drawn using the values of KIa

c/f and KIaf/f

given in Table 3 and the results of finite elementcalculations, assuming that the fracture tough-ness specimens were tested under plane strainconditions. Figure 16 successfully demonstratesthat the multiple barrier model is able to predictnot only the evolution of the fracture toughnesswith temperature but also the value of the criti-cal stress intensity factor associated to thedevelopment of temporarily stable grain sizedmicrocracks. However, the difficulty with themultiple model is that it requires the knowledgeof many microstructural parameters.

The main advantage of this type of model isthat it captures the fracture toughness, in theupper part of the DBT curve, is related to thepropagation of microcracks which are tempora-rily arrested at grain boundaries in ferritic steelsor at packet boundaries in bainitic steels. Recentstudies have shown that in bainitic steels theeffective packet boundaries are those with ahigh-angle misorientation (see, e.g., Bouyneet al., 1998; Gourgues et al., 2000; Lambert-

rature (°C)

–50 0 50

circles denote microcrack events detected by acoustic(respectively, dotted) lines represent 10%, 50%, andely, for a cleavage microcrack to propagate across aer model (Lambert-Perlade et al., 2004). ReproducedA. F., Besson, J., Sturel, T., and Pineau, A. 2004.ulated heat-affected zone microstructure of a high-9–1053.

Figure 17 SEM micrograph of a polycrystallinezinc specimen fractured at �196 �C.


Perlade et al., 2004). This suggests that the frac-ture toughness of these materials can beimproved by the development of such favorableboundaries. This links with a new research fieldsometimes called ‘interface engineering’ or ‘grainboundary engineering’. Fundamental studies inthis field must therefore be strongly encouraged.

2.06.2.4.2 Cleavage fracture in otherBCC metals

While the plastic deformation and brittle clea-vage fracture behavior of iron and ferritic steelshas, for technological reasons, received consider-able attention, other BCC metals have beenrelatively neglected. This is partly due to theexpected similarity in behavior with a iron, andpartly due to the small use in key engineeringapplications of the highmelting refractory metals(V, Nb, Ta, Cr, Mo, W) in industry. However,the increased importance of these metals, in par-ticular in gas turbines and nuclear energy sectors,pushes for more detailed investigation of theirmechanical properties. Commercially availablegroup V-A refractory metals (V, Nb, Ta) areconsiderably more ductile and have considerablylower transition temperature than commerciallyavailable group VI-A refractory metals (Cr, Mo,W). This mainly results from that the solubilityof impurities in the V-A metals is much higherthan in the VI-A metals.

Molybdenum, which is representative of themetals of group VI-A, has been investigated(see, e.g., Briggs and Campbell, 1972; Kovalet al., 1997). Niobium, which is representativeof themetals in groupV-A, has been investigatedinmuchmore detail (Briggs and Campbell, 1972;Samant and Lewandowski, 1997a, 1997b; Pahdiand Lewandowski, 2004). Lewandowski and hisco-workers have studied pure Nb and Nb–Zrsolid solutions. These authors have investigatedthe effect of grain size (,60 and 165mm). Thecleavage stress, �f, was measured using eitherblunt notched specimens or fatigue precrackedfracture toughness specimens. In blunt notchedspecimens, the Griffith and Owen solution wasused to calculate the cleavage stress (Griffithsand Owen, 1971). In fracture toughness speci-mens, the Ritchie, Knott, and Rice (RKR)model (Ritchie et al., 1973) was used in conjunc-tion with either the HRR field (Hutchinson,1968; Rice and Rosengren, 1968; see Section2.06.3.7.1) or the Tracey solution (Tracey,1976). The cleavage stress was found to be adecreasing function of grain size. The identifiedvalue for �f appeared to be larger for fracturetoughness specimens than when identified basedon blunt notched specimens (typically 1700MPacompared to 1400/1500MPa in 60mm grain size

Nb). This suggests that the sampling volumeswhich are quite different in these two geometriesmay also affect the values of �f. Additional testsand analyses are required to show if this sizeeffect can be interpreted by using the Beremintheory.

2.06.2.4.3 Cleavage fracture in HCP metals

When compared to crystal systems like BCC,HCP metals exhibit a wider variety of deforma-tion modes, including slip and twinning systems.Historically, HCPmetals have been categorized interms of c/a ratio. For metals with c=a<

ffiffiffi3p

(e.g.,beryllium, titanium, zirconium, and magnesium),the {1012} twinning mode is activated by com-pression along the c-axis and <cþa> slip isactive. All these metals have a preferential clea-vage plane which is the basal plane {0002}.However, in metals like titanium and zirconium,pure cleavage fracture along the {0002} plane hasnot been observed, except under stress corrosionconditions which are outside the scope of this (see,e.g., for Zr alloys: Kubo et al., 1985; Schuster andLemaignan, 1989a, 1989b; Cox, 1990). This iswhy in this section only cleavage fracture of zincand magnesium are considered.

(i) Cleavage fracture of zinc

Zinc shows brittle cleavage fracture along thebasal plane and along the prismatic planes. Anexample of the fracture surface of a pure poly-crystalline specimen broken at�196 �C is shownin Figure 17. The importance of prismatic clea-vage in zinc and that of the accommodationrequired at a grain boundary as a crack propa-gates from grain to grain has been underlinedrecently (Hughes et al., 2005). Cleavage on basalplane has been studied more thoroughly, in par-ticular on zinc single crystals (Gilman, 1958;Deruyttere and Greenough, 1956). More

45

40

35

30

25

20

15

10

5

00 10 20

γ (%) at fracture

σ n (M

Pa)

at f

ract

ure

χ = 89°

χ = 82°χ = 75°

χ = 60°

χ = 45°

χ = 30° χ =

30

Figure 18 Fracture in zinc. Normal stress to the basalplane at fracture as a functionof basal glide for differentmisorientations � between the c-axis and the tensiledirection (Deruyttere and Greenough, 1956; Gilman,1958). Reproduced with permission of Parisot, R.,Forest, S., Pineau, A., Nguyen, F., Demonet, X., andMataigne, J. M. 2004b. Deformation and damagemechanisms of zinc coatings on hot-dip galvanizedsteel. Part II: Damage modes. Metall. Mater. Trans.A 35, 813–823.


recently, cleavage fracture on basal plane hasalso been studied in detail on hot dip galvanizedsteel sheets (Parisot et al., 2004a, 2004b). In stu-dies devoted to the analysis of single crystals, itwas shown that the critical cleavage stress mea-sured by the stress normal to the {0002} cleavageplane is largely reduced when the amount ofplastic strain in the basal plane, �bas, is increased(see Figure 18). Similar observations have beenmade on crack propagation behavior in stronglytextured zinc sheets (Lemant and Pineau, 1981).The cleavage criterion proposed by Gilman(1958) involved the product �n�i

bas where �ibas is

the amount of plastic slip on the basal slip sys-tem, i. The reason for the decrease of the cleavagestress when basal plastic strain is increased islikely related to the existence of local stress con-centrations associated with basal dislocationpile-ups that can locally reach the theoreticalcleavage stress (see Section 2.06.2.2). A similarconclusion was reached in the study devoted to

zinc coatings on hot-dip galvanized steel sheets,except that thresholds for the cleavage stress andthe basal plastic strain were introduced, that is,

sf ¼ sth þk

g0 þ gbasep

½37�

where �th, k, and �0 are material parameters. Ineqn [37] the theoretical cleavage stress is, in theabsence of basal slip, equal to �thþ k/�0.A measure of the total basal slip activity of thethree basal slip systems was defined as

gbasep ¼ jg1jbas þ jg2jbas þ jg3jbas ½38�

Equation [37] introduces an asymptotic value�th for zinc crystals formed at the free surface ofgalvanized steel sheets and undergoing a verylarge amount of basal slip, which is not the casein the original Gilman’s criterion. This may berelated to the purity of the materials used inthose studies.

(ii) Cleavage fracture of magnesium

A major problem facing HCP metals, suchas magnesium and zinc, is their limited low-temperature ductility. The dominant slipmode in all HCP metals involves the Burgersvector a¼ 1/3<1120>whether the primaryslip plane is basal (e.g., cadmium, zinc, or mag-nesium) or prismatic (e.g., titanium orzirconium). Even if both slip planes operate,there is still no way to accommodate strainsalong the c-axis. Deformation twinning canhelp alleviating this problem but it is ofteninsufficient to provide a large ductility.

This problem is encountered not only in zincbut also in high-purity magnesium. Magnesiumalso cleaves easily along the basal plane. Thecleavage fracture stress of high-purity magnesiumis independent of temperature, but highly depen-dent upon grain size. The addition of lithium tomagnesium alloy has an interesting effect on low-temperature fracture behavior in decreasing theDBTT. Several reasons have been given toexplain this effect which is known for quite along time (Hauser et al., 1956). Many reportshave been published regarding the beneficialeffect of lithium on the ductility of magnesium(see, e.g., Raynor, 1960; Saito et al., 1997). Inmany of these studies, the lithium additionsresulted in a substantial volume fraction of thesoft Li-rich BCC b-phase which is probably onepart of the explanation for the better ductility.The increase in ductility has also been observedeven in Mg–Li a-solid-solution alloys. Anotherreason for the ductility improvement might berelated to the reduction of the stress for prismaticslip relative to that for basal slip. More recently,Agnew et al. (2001, 2002) have shown the


existence of 1/3<1123>{1122} pyramidal slip inMg–Li solid-solution alloys, while pure magne-sium exhibits only basal and pyramidal slip.Nevertheless, the structure of<cþ a>dislocations, in particular their dissociationand decomposition, remains an open issue. Verymuch remains thus to be done before the theoriesof cleavage fracture in HCP metals reach a devel-opment similar to those used for BCC metals.

2.06.2.4.4 Irradiation-induced embrittlementin ferritic steels

Although it is out of the scope to review indetailthe micromechanisms of embrittlement producedby irradiation, it is worthmentioning a number ofrelevantmechanisms causing thismode of embrit-tlement (for a review, see for instance Gurovichet al., 1997; Odette et al., 2003).

RPVs of commercial nuclear power plants aresubjected to embrittlement due to the exposure tohigh-energy neutrons from the core. The currentmethods used to determine the effect of the degra-dation by irradiation on the mechanical behaviorof RPVs rely on tensile tests and impact Charpytests. For a given fluence (f (#/cm2)) of neutrons(energy>1MeV) and irradiation temperature,irradiation-induced embrittlement is stronglydependent on material chemical composition,especially on Cu, Ni, and P contents (Haggag,1993). The Cu content plays an important role inthe hardening-induced embrittlement due to theirradiation-induced precipitation of Cu-richnanoparticles (see, e.g., Buswell et al., 1995;Auger et al., 1995; Akamatsu et al., 1995). Otherfine-scale microstructural modifications influencethe macroscopic behavior of the irradiated mate-rial. Two types of influence can be distinguished(Nikolaev et al., 2002; Wagenhofer et al., 2001):(1) modification of the plastic properties and

Tempe

Cha

rpy

ener

gy; f

ract

ure

toug

hnes

s

Figure 19 Schematic effect of irradiation e

(2) embrittlement. One of the most well-knownand well-described embrittlement effects is asso-ciated with phosphorus segregation at grainboundaries (Miller et al., 1995; Faulkner et al.,1996). This embrittlement effect is considered inthe following section devoted to intergranularfracture. Hardening mechanisms include matrixand precipitation hardening. Matrix hardening isdue to irradiation-produced point defects anddislocation loops. The deformation is thenconcentrated along channels, producing localstress concentrationswhich facilitate the initiationof cleavage fracture, as described previously.Precipitation hardening, as already stated, is asso-ciated with the irradiation-enhanced formation ofCu-rich precipitates. These two hardeningmechanisms cause an increase of the yield strengthand, usually, a decrease of the work-hardeningexponent.

The effect of irradiation embrittlement mea-sured with Charpy V-notch tests is depictedschematically in Figure 19. A shift of theDBTT, DT, is observed. Similar effects on frac-ture toughness,KIc, are observed. In many cases,the upper shelf energy (USE) determined fromCharpy specimens is lowered, as shown schema-tically in Figure 19. The yield strength measuredin RPV steels with a low to medium content inimpurities (Cu, P) increases monotonically withthe fluence, , while the work-hardening cap-ability measured by (UTS� �0) tends todecrease (Tanguy et al., 2006; EDF, 2003). Thevariation in yield strength can be represented bythe following empirical expression:

Ds0 ¼ 32:46f0:51 ½39�

where D�0 is expressed in MPa and in #/cm2.The shift of the transition temperature, DT,

can be qualitatively explained using the theoriesfor cleavage fracture which have been presented

rature

ΔTIrradiated

Non-irradiated

(USE)

ffect on the DBT curves in ferritic steels.

0


in the previous section. More quantitativedetails can be found elsewhere (Al Mundheriet al., 1989; Tanguy et al., 2006). Assuming thatthe cleavage stress is not affected by irradiation,the increase in yield strength �0 provides a sim-ple explanation for the shift of the DBTT. TheBeremin theory (Section 2.06.3.3) applied to thecrack-tip region predicts that, when small-scaleyielding (SSY) conditions are fulfilled, theprobability to failure, PR, can be written as

PR ¼ 1� exp� K4IcBs

m�40 CmðnÞVusmu

� �½40�

where B is the specimen thickness and Cm(n) is anumerical factor which is an increasing functionof the work-hardening exponent �¼ ken of thematerial; for further details, see, for example,Chapter 7.05. An increase of the yield strengthdue to irradiation will thus produce a reductionof the fracture toughness, KIc, for a given testtemperature and a given specimen thickness.Very recently, a similar approach has been usedto predict the Charpy V temperature shift inA508PVR steel by Tanguy et al. (2006). Theseauthors have simulated Charpy V specimens bythe finite element method using a material modelintegrating a description of viscoplasticity, duc-tile damage, and cleavage fracture. They alsoassumed that irradiation affects only the yieldstrength and/or the work-hardening coefficientbut not the ‘cleavage stress’, �u. Using empiricalcorrelation relating the increases in yieldstrength D�0 (in MPa) to the irradiation fluence given by [39], Tanguy et al. (2006) showed thatthe increase in the DBTT with can be reason-ably well predicted with the Beremin theory. Inparticular, the low values of the Charpy energyare properly captured. These authors alsoshowed that the shift in fracture toughness mea-surements, KIc, could be predicted using theBeremin theory. Their results are provided inTable 5, which reports the values of the DBTTcorresponding to a fracture toughness equal to100MPam1/2, the value used in manyapproaches based on the ASTM E 1921 MasterCurve Standard for measuring the fracturetoughness in the cleavage transition (ASTM E1921-02, 2002; Markle et al., 1998). In Table 5we have also included the values of the tempera-ture shift, DT56J, for a Charpy energy of 56 J,

Table 5 Comparison between

(1019#/cm2)

D�0(MPa)

DT56J

(�C)

1.90 45 307.07 88 5620.10 150 95

which is a typical value used in many standards.These values of DT56J were also predicted usingsophisticated numerical finite element calcula-tions in combination with the Beremin theory.

These variations in DT can also be deter-mined using simpler analytical calculationsprovided that the temperature dependence ofthe yield strength is known. Several expressionshave been proposed for this dependence.A polynominal function given by Rathbunet al. (2006) can be used:

s0ðTÞ ¼ 0:0085T2 � 0:4402Tþ 481:51 ½41�

where �0 is expressed in MPa while T isexpressed in �C. It is assumed that this function,valid for non-irradiated materials, can also beapplied to the irradiated material. It is alsoassumed that the slight variation in the work-hardening capability of the material with irra-diation does not influence the value of thecoefficient Cm appearing in the theoretical eqn[40]. It might be useful to remember that thisequation is strictly valid for a stationary crackunder SSY conditions (Chapter 7.05). This the-oretical expression indicates that for a givenprobability to failure, the product KIc �0

((m/4)�1)

should remain constant and is independent ofthe irradiation conditions. Differentiating thisexpression leads to

DKIc

KIc¼ m

4� 1

� � Ds0syððm=4Þ � 1Þ ½42�

This expression relates the variation in fracturetoughness, DKIc, with the increase in yieldstrength, D�0, produced by neutron irradiation.The increase of the flow strength D�0 is given byeqn [39]. The variation of yield-strength equa-tion [39] can also be differentiated to predict thetemperature shift necessary to obtain, after givenirradiation conditions, the same value of thefracture toughness, for instance, 100MPam1/2.One easily obtains that the variation in yieldstrength, D�0, is simply related to the tempera-ture shift, DT, by the following expression:

Ds0ðMPaÞ ¼ ½0:017T0ð�CÞ � 0:4402�DTð�CÞ ½43�

Assuming that, for instance, T0¼�60 �C, eqn[41] predicts that, in the non-irradiated condi-tion, � (�60 �C)¼ 538MPa, which is a typical

DT56J and predicted DTKIc,100

DTKIc,100numerical(�C)

DTKIc,100analytical(�C)

49 3173 60.3104 103


value for A508 steel. The above expression andeqn [39] simply leads to

Ds0ðMPaÞ ¼ �1:46DTð�CÞ ¼ 32:46f0:51 ½44�

where is expressed in 1019 #/cm2. This expres-sion can thus be used to assess the values of theDBTT shift, DT, corresponding to a referencefracture toughness equal to 100MPam1/2. Theresults are reported inTable 5where a comparisonwith those obtained by numerical finite elementcalculations (Tanguy et al., 2006) can be made.These results show that the simplified analyticalapproach leads to values for the temperature shiftclose to those inferred from sophisticated numer-ical computations. A better agreement is observedfor large values of the fluence. This is likelybecause the basic assumption of SSY conditionsbehind the theoretical eqn [40] is much bettersatisfied for large increases in the yield strength,D�0, due to very large values for the fluence. Thevalues of the DT shift predicted from the analyti-cal approach are closer to those numericallycalculated for the shift in Charpy tests. No simpleexplanation can be given to this observationwhich is likely purely fortuitous.

This brief overview of the irradiation-inducedembrittlement suggests that the main source ofembrittlement is related to the irradiation-hard-ening effect. As a matter of fact, the problem islikely far more complex. In particular, as alreadystated, a modification in the failure mode fromtransgranular cleavage to intergranular has beenreported (Miller et al., 1995; Faulkner et al.,1996; Gurovich et al., 1999, 2000). Moreover,the irradiation-hardening effect cannot simplyexplain the decrease of the USE depicted inFigure 19. Many theories for ductile fracturedepict that an increase in yield strength shouldproduce an elevation of the USE (see Section2.06.3.7.2), when other factors are maintained.The observed decrease in the USE may be dueto the reduction in the work hardenability of thematerial which is evidenced at high fluences. Areduction in the work-hardening exponent leadsto strain localization which is detrimental to theductility of the material (see Section 2.06.3.2) andto the fracture toughness (see Section 2.06.3.7.2).Irradiation-induced segregation of impurities likeP at the matrix/precipitate interface may alsocontribute to the reduction in the ductility ofthe material. Such segregation effects seem tohave been observed by Gurovich et al. (2000).

These difficulties related to impurities contri-bute to the fact that the predictions ofirradiation-induced embrittlement remainlargely empirical. However, in this section, anattempt has been made to show that, at least inrelatively clean steels, in which the segregationphenomena are limited, the shift in the DBTT

due to irradiation effects can reasonably be wellpredicted using the theoretical local approachto cleavage fracture. Clearly, this is anotherresearch area which should deserve moreresearch effort.

2.06.2.5 Intergranular Brittle Fracturein Ferritic Steels

As stated previously theoretical calculationssuggest that intergranular fracture should beobserved preferentially in many polycrystallinemetals instead of transgranular cleavage fracture(see Table 2). However in ferritic steels, brittlefracture occurs at low temperature by transgra-nular cleavage. This is usually attributed to thereinforcement effect of a number of elementssegregated along the grain boundaries, in parti-cular carbon. Conversely, the segregation ofother impurities, for example, phosphorus,along the grain boundaries can change the brittlefracture mode from cleavage to intergranular.This is the situation observed for instance inlow-alloy steels, such as A508 Cl3 steel used forthe fabrication of pressurized water reactors(PWRs). In these thick components the presenceof small areas of low toughness, referred to as‘ghost lines’, may be an important source ofscatter in fracture toughness values (see, e.g.,Tavassoli et al., 1983, 1989; Kantidis et al.,1994). These ghost lines can initiate intergranu-lar brittle fracture due to temper-embrittlementeffect. These lines were identified as segregatedzones containing increased amounts of impuri-ties (P, S, etc.) and alloying elements (C, Mn, Ni,Mo, etc.), as compared to the base material.Intergranular brittle fracture is therefore ofgreat practical importance.

Very few detailed studies have been made todetermine quantitatively the variation of thecritical intergranular fracture stress �CI withtest parameters using procedures similar tothose used for the measurement of the cleavagestress. Most often, the effect of temper embrit-tlement has been investigated by determiningthe shift in the DBTT, using Charpy V-notchedspecimens. However, the interesting work byKameda andMcMahon (1980) should be men-tioned. These authors showed that the criticalintergranular fracture stress was directly relatedto the amount of impurity (Sb) segregated ongrain boundaries. Similarly in an RPV steelbased on A508 steel composition, it wasshown that �CI was decreasing linearly withthe amount of phosphorus segregated alonggrain boundaries (Naudin, 1999; Naudinet al., 1999). In this material the critical stress(,2300MPa for cleavage fracture) was reducedby almost 30% when the phosphorus

σz mean

εz mean

or

Figure 20 Schematic representation of the processof nucleation, growth, and coalescence of voidsnucleated on second-phase particles inside anidealized representative volume element of themicrostructure, and the relationship with themacroscopic loading evolution.

Ductile Fracture in Metals 709

monolayer grain-boundary coverage reachedapproximately 40%. In another recent studydevoted to the statistical aspect of intergranularfracture in a low-alloy steel, it was shown thatafter a heat treatment leading to temper embrit-tlement, the intergranular fracture stress was ofthe order of 1400MPa while before applyingthis heat treatment the cleavage stress wasclose to 1560MPa (Wu and Knott, 2004).These authors showed also that �CI was inde-pendent of temperature within a firstapproximation and was distributed accordingeither a normal or a Weibull law.

More detailed studies have been devoted tothe effect of intergranular fracture on the frac-ture toughness of A508 steel due to theimportance of this mode of failure on the frac-ture assessment of RPV components (Yahyaet al., 1998; Kantidis et al., 1994; Raoul et al.,1999). In these studies, intergranular fracturewas favored by applying a step-cooling heattreatment after tempering. It was shown thatthe Weibull statistics (eqn [22]) was able todescribe the scatter in test results on notchedbars and fracture mechanics specimens. It wasalso shown that the Beremin theory for clea-vage fracture should be slightly modified toaccount for the effect of test temperature,since �CI was found to be an increasing functionof temperature. Kantidis et al. (1994) showedthat the variation of fracture toughness withtemperature could be well represented usingthe original Beremin theory, provided that theWeibull stress included a temperature depen-dence, that is,

smWW ¼ZPZ

sm1 ½1þ lðT� T0Þ�mdV

V0½45�

where �WW is the modifiedWeibull stress, l>0is a material parameter, and T0 a referencetemperature. Equation [45] is very similar tothe original definition of the Weibull stressgiven by eqn [27]. The expression of the mod-ified Weibull stress indicates that when thetemperature is lower than T0, intergranularfracture occurs at lower stresses, as comparedto the Weibull stress calculated without anytemperature correction. In these studiesdevoted to A508 steel, it was also shown thatthe value of the shape factor, m, was muchlower (,10) for intergranular fracture than forcleavage where typically m, 20. The reasonsfor this difference in m values, which havestrong practical implications since the value ofm controls the scatter in the results, have not yetbeen discussed in detail. Clearly brittle intergra-nular fracture also requires further detailedinvestigations.

2.06.3 DUCTILE FRACTURE IN METALS

2.06.3.1 Introduction: Two Classesof Failure Mechanisms

Ductile fracture is the most common room-temperature mechanism of failure in metals.We will reserve the term ‘ductile fracture’ forthe process of damage nucleation followed by aphase of damage growth and coalescence drivenby plastic deformation. A good understandingof ductile fracture relies thus first on properappraisal of the mechanisms and theory ofplasticity, that is, physics of dislocations, ofhardening and strain-hardening mechanisms,crystal plasticity and plastic anisotropy con-cepts. Note also that, in some high-strengthalloys, dirty metals, or metal matrix compo-sites, the process of nucleation, growth, andcoalescence of voids can take place very rapidlyand lead to very low ductility, sometimes assmall as 1%. Nevertheless, in our terminology,these materials fail by a ductile fracturemechanism to contrast with a cleavage-typemechanism.

Even if ductile fracture is defined as the resultof a damage process by plasticity-controlledvoid nucleation, growth, and coalescence, it isimportant, for practical purposes, to distin-guish between two ‘modes’ of ductile fracture.In the ‘first mode’, damage occurs more or lesshomogenously in homogenously deformedregions up to the final fracture point (seeFigure 20). Only at the very end of the process,when void coalescence takes place and a crackstarts propagating in the solids, the deforma-tion can become highly heterogenous. In the‘second mode’, plastic localization occursbefore or early in the damaging process through


the development of localized necks or shearbands, usually with no or only limited couplingwith the damage evolution. In that case, thepractical failure condition is the onset of plasticlocalization. This is the common situation inlow stress triaxiality metal-forming applica-tions in which the FLD concept has beendeveloped in order to provide guidelines foravoiding plastic localization. Of course, the‘regular’ damage process still takes place withinthe localization band which involves large plas-tic strains, and is ultimately responsible for thefracture. The increasing stress constraint in theband accelerates the damage process. In termsof the macroscopic loading, the final fracturefollows rapidly after the onset of localization.The details of the failure process within theband and the prediction or measurement ofthe fracture strain locus are of limited practicalinterest. This division between plastic localiza-tion-controlled failure and damage-controlledfailure motivates the organization of the sectionin two parts. The first part consists in an intro-duction to plastic localization. This part, madeof a single subsection, is kept relatively shortbecause it deviates from the main message ofthis chapter which focuses on the fracturemicromechanisms in metals. The second partis made of several subsections devoted to thenucleation, growth, and coalescence of voidsand to the ductility and fracture resistance ofthick and thin metallic plates.

2.06.3.2 Plastic Localization Mechanismsin Homogeneous Medium

Plastic deformation can be localized becausethe material is not homogeneous at the level ofthe microstructure. Such type of localizations,when present, is usually a primary factor con-trolling the ductility and fracture toughness ofmetallic alloys. The most typical situation is amicrostructure or mesostructure that involveshard and soft regions or phases. Examples ofsuch type of microstructure-induced localiza-tions will be repetitively illustrated whendealing with damage mechanisms.

The second type of plastic localizations takesplace in homogeneous materials under loading.Advanced solid mechanics is necessary in orderto formulate a sound framework for addressingplastic localization from general perspective.Excellent reviews have been written on thattopic which goes outside the scope of this sec-tion (e.g., several papers in Koistinen andWang, 1978; Rice, 1976; Hutchinson, 1979;Semiatin and Jonas, 1984; Needleman andTvergaard, 1992; Perzyna, 1998; Forest andLorentz, 2004). We will start here from the

simple problem of necking under uniaxial ten-sion and then move to the formulation ofplastic localization conditions in 2-D planestress.

2.06.3.2.1 Necking under uniaxial tension

Let us start by considering a long perfect barof a material whose plastic behavior is charac-terized by the evolution of the flow stress as afunction of strain and strain rate. A genericbehavior is assumed for which the variationof the strain-hardening rate as a function ofstrain e is expressed by the function n(e)defined as

n ¼ @lns@lne

¼ es@s@e

½46�

Similarly, the variation of the strain rate sensi-tivity as a function of strain can be expressed bythe function m9(e) defined as

m9 ¼ @lns@ln_e

¼ _es@s@ _e

½47�

The specimen is loaded under uniaxial ten-sion along direction z. Let us assume that thisspecimen presents a infinitesimally small pertur-bation of the cross-sectional areaA(z) in the closeneighborhood of a point of coordinate z. Byimposing force equilibrium along the specimenlength and neglecting elastic strains, one candemonstrate that at the beginning of straining, aregion of the specimen that would locally presenta smaller cross-sectional area (with for instancedA/dz<0 at a particular point) will undergo alower rate of cross-sectional area reduction thanthe other regions (i.e., d/dz(dA/dt)>0 at thatparticular point). This means that the tensiledeformation of the specimen is then stable:deformation tends to distribute uniformlyalong the specimen. Stability toward sectionreduction is allowed by the strain-hardeningcapacity of the material. At larger strains, asituation is attained where the strain-hardeningcapacity is not sufficient anymore to compen-sate for the section reduction and theperturbation then amplifies while the rest ofthe specimen starts to unload elastically. Thecondition for dA/dz<0! d/dz(dA/dt)>0 ordA/dz>0 ! d/dz(dA/dt)<0 can be written,after some algebra, in the form

e � nðeÞ1�m9ðeÞ ½48�

In principle, beyond this point, plastic strain-ing in tension becomes unstable: a region withlower cross-sectional area undergoes a largerrate of contraction than the rest of the speci-men, which is the phenomenon of necking. Asshown in Figure 21, the role of the rate


sensitivity to help stabilizing plastic localizationis in fact much stronger than what is predictedby eqn [48]. A good example is provided inFigure 22 showing the engineering stress straincurves of a fully b titanium alloy at room tem-perature. Although the strain rate sensitivityseems quite low, m9� 0.03, the effect on stabi-lizing the necking process is already quitesignificant.

Conversely, the presence of material or geo-metrical imperfections leads to a significantdrop of the resistance to plastic localizationwith respect to what is predicted by eqn [48]

0.001

0.01

0.1

1

Total elongation (%)

Data collected by Woodford (1969) :- Fe base + Cr and Mo- Ni base- Mg base + Zr- Pu base- Pb –Sn- Ti base + Al and Sn- Ti base + Al and V- Zircaloy

Str

ain

rate

sen

sitiv

ity, m

′

1 10 100 1000 104

Trend lines for thecondition ε > n /(1–m ′)

Figure 21 Variation of the total elongation as afunction of strain rate sensitivity (from datacollected by Woodford, 1969; see also Hutchinsonand Neale, 1977).

1200

1000

800

600

4000 0.05 0.1

Δl

σ eng

(M

Pa)

ε =

ε = 10–2 s–1

LCB Ti all

First obs

Maximum

ε.

.

.

Figure 22 Engineering stress strain curves for a fullCourtesy of N. Clement, Universite Catholique de Lou

(e.g., Hosford and Caddell, 1993). It is worthnoting that accounting for imperfections in themodeling of the necking condition leads to amuch better description of the effect of the ratesensitivity. Considering an initial imperfection� defined as a fractional reduction of the initialcross-sectional area, the following expressionprovides a good estimate of the necking strainwhenm9 is close to 0 and when the imperfectionis small (Hutchinson and Neale, 1977):

e ¼ n 1�ffiffiffiffiffi2Zn

r !½49�

while a good estimate when n is close to 0 andwhen the imperfection is small (but not equal to 0)is given by

e ¼ �m9 ln 1� ð1� ZÞ1=m9� �

½50�

These two expressions provide a fairly goodindication of the effect of both m9 and � onthe onset of necking. For instance, an imperfec-tion of 1% in amaterial sample with n¼ 0.2 andm9¼ 0 will cause a 30% drop of the neckingstrain. Unfortunately, it is difficult to provideanalytical expressions for the onset of neckinginvolving simultaneously the effect of the ratesensitivity and the effect of the presence ofimperfection except in some asymptotic limits.More general parameter analysis is provided inHutchinson and Neale (1977).

After the onset of localization, the neckingregion develops with the height scaling with thewidth of the specimen (or the diameter for acylindrical specimen). Up to that point theshape of the section is unimportant. The shapeof the section, that is, rectangular section versuscylindrical, and the aspect ratio of the section,

0.15 0.2 0.25 /l0

10–2 s–1

oy ‘low cost beta’

ervable necking

load (ε approximately equal to n /(1–m ′))

= 1 s–1

beta titanium alloy for different applied strain rates.vain.

ht

(a)

n = 0.2n = 0.3 n = 0.4

n = 0.5n = 0.1

n = 0.01

0.500

1

2α

3

4

1 1.5εen

(c)

Axisymmetric conditionsn = 0.2

n = 0.3

n = 0.1

4

3

2

1

00 0.5 1 1.5

εen

n = 0.4 n = 0.5α

Plane strain conditions

(b)

Figure 23 Variation of the shape factor of the active necking region a as a function of straining for differentstrain-hardening exponents: a, definition of height and thickness of the neck; b, for plane strain tension ofrectangular beam; c, pure tension of cylindrical bars. From Pardoen, T., Hachez, F., Marchioni, B., Blyth, H.,and Atkins, A. G. 2004. Mode I fracture of sheet metal. J. Mech. Phys. Solids 52, 423–452.

σh/σe

f0 = 0.2%, W0 = 1, σ0

/E = 0.001

n = 0.05

n = 0.2

n = 0.1

1/3 = pure uniaxial tension

1.2

0.9

0.6

0.3

00 0.5 1 1.5εe

Figure 24 Variation of the stress triaxiality in thecenter of the minimum cross-sectional area as afunction of the average effective strain in theminimum section.


that is, small versus large ratio of width overthickness, affects the geometrical evolution ofthe neck (Zhang et al., 1999), the evolving stressstate within the neck, the possibility for shearbanding to take place within the neckingregion, and the evolution of damage inside theneck. This aspect is discussed in Section2.06.3.6.1.

Figure 23 shows the evolution of the ratio a ofthe height of the neck divided by the minimumthickness (for a plane strain tension specimen)or divided by the diameter of the minimum crosssection (for cylindrical specimens) as a functionof the effective strain within the minimum sec-tion of the neck een, for various strain-hardeningexponents n. These results were obtained byfinite element simulations assuming the isotro-pic J2 flow theory for the response of thematerial (see Pardoen et al., 2004). The strain-hardening exponent was defined here through aSwift-type representation:

ses0¼ ð1þ keepÞn ½51�

where �e is the effective stress, eep is the effectiveplastic strain, �0 is the yield stress, and k is aparameter that is usually much larger than 1(here, it is equal to E/�0). The ratio a¼ h/t (seeFigure 23) is equal to 1 for perfectly plasticmaterials (Onat and Prager, 1954; Hosfordand Atkins, 1990). A good empirical fit forthese results is given by

aaxisymmetric ¼ 1:105þ 1:422n

e1:725en

½52�

aplane strain tension ¼ 0:012þ 4:353n

e1:37en

½53�

All a values are larger than 1 (the rigid-perfectlyplastic value), and decrease with straining.The development of the neck is not affected by

the value of the ratio �0/E but the shape of thenecking zone is very much influenced by thestrain-hardening capacity. A smaller hardeningexponent leads to a more localized neck andthus smaller a. The fact that the neck is sharperwhen the strain-hardening exponent is smallerhas a direct effect on the evolution of the stresstriaxiality (see definition in Section 2.06.3.4.1)inside the neck, as shown in Figure 24, whichraises more slowly with shallower necks. Thevariation of a with straining is important andcannot be neglected in the analysis of necking.

These types of numerical simulations ofnecking (as initiated by Needleman (1972a),using the finite element method, or Norriset al. (1978), using the finite difference method)are very useful for the determination of the truestress strain response of the material after theonset of necking, through an inverse modelingprocedure (see also Pardoen and Delannay,1998a; Zhang et al., 1999). In the same vein,Zhang et al. (2001) have carefully analyzed the

in Metals 713

effect of the anisotropy of the material on theevolution of the necking process.

Note finally about necking under uniaxialtension conditions that:

1. a short specimen length delays the onset ofthe localization process when compared to con-dition [48] (Hutchinson, 1979);

2. other localizationmodes involvingmultiplenecks or surface instabilities are also predicted bythe theory (e.g., Hutchinson, 1979) but theyrequire very specific boundary conditions rarelyobserved in practice; and

3. shear band formation is unlikely to occurunder uniaxial loading conditions with smoothyield surfaces but is predicted with corneredyield surfaces (Needleman and Rice, 1978) orafter some amount of necking (then the stressstate is not uniaxial anymore).

2.06.3.2.2 Plastic localization under biaxialloading conditions

The localization of plasticity, that is, a transi-tion from a uniform to a nonuniform mode ofdeformation while the loading remains uniform,also occurs under a multiaxial state of strain andstress. As illustrated in Figure 25, one then fre-quently observes so-called ‘shear bands’, that is,bands oriented in the direction of maximumshear. The width of a shear band varies depend-ing on the material microstructure, for example,on the grain size or on the dislocation cell size. Ingeneral, the orientation of the band depends onthe stress state and the geometry of the speci-men. Necking is said to be a ‘geometric diffusemode’ of plastic localization. (‘Diffuse’ means

Ductile Fracture

that there is a progressive transition between

Diffuse neck Localized shear band

Figure 25 Diffuse neck and localized band.

the necking zone and the rest of specimen,while ‘geometric’ means that the geometry ofthe neck is imposed only by the geometry ofthe specimen and not bymicrostructural featuresof the material.) When referring to shear bands,one usually refers to bands showing a very sharptransition between the ‘localized band’ (or ‘loca-lized neck’) and the rest of the material. Assuggested in Figure 25, it is common, in homo-genous thin plates, to observe first the onset ofnecking and later the emergence of shear bandswithin the diffuse neck (e.g., the detailed experi-mental study by Carlson and Bird (1987)). Incomplex thin plate geometry, non-homogeneousdeformation conditions can prevent the appear-ance of necking but cannot generally prevent theoccurrence of shear bands.

Plastic localization is an important limitingfactor in plate-forming operations, such as indeep drawing. It is common to map the condi-tions of occurrence of plastic localization usingso-called FLDs. The FLD is intensively used inthe metal-forming industry. A schematic exam-ple of an FLD is presented in Figure 26. Thecurve represents the locus of the in-plane prin-cipal strains, e1 and e2, corresponding to theonset of a plastic instability (by construction,the FLD is symmetrical with respect to themirror line e1¼ e2). Figure 26 shows that thevalue of the ‘biaxiality ratio’ �¼ e2/e1, signifi-cantly affects the onset of plastic instability.A forming process is ‘safe’ if the state of defor-mation in the plate never reaches the forminglimit during the loading history.

In the case e2<0 and e1>0 (the left-handpart of the FLD), it can be shown that, assum-ing the response of the material under uniaxialstress is isotropic and can be described by aHollomon representation (�¼Ken) and thatthe strain biaxiality ratio �¼ e2/e1 keeps a con-stant value during loading, the condition for theformation of these bands is

Major strain ε1

Minor strain ε2

Compression

Slope –1

Slope –2Wrinkling

n

Slope –1/2

2n

Diffuse necking

Shear fracture

Shear bands

She

ar b

ands

Shear bands

Fracture

Fracture

Figure 26 Failure locus, i.e., ‘FLDs’, for thin sheetsunder biaxial loading conditions.

Noprestrain

ε = 0.07

ε = 0.12ε = 0.17

ε = 0.04

ε2

ε1

0.30

0.20

0.10

0.000.30–0.10 0.00 0.10 0.20

Figure 27 Example of nonradial loading effects onFLDs for aluminum alloy 2008-T6 (based on Grafand Hosford, 1993; see also Hosford and Duncan,1999).


e1 þ e2 ¼ n ½54�

while the orientation of the band with respectto the x2 axis is given by

tan y ¼ � e2e1

� ��1=2½55�

In the case of tensile testing (e2¼�e1/2), a loca-lized band of deformation would thus appear atan angle ¼ 54.7� when e1¼ 2n. The localizedband will thus develop inside the diffuse neckwhich started to form at e1¼ n.

In anisotropic materials characterized by aLankford coefficient l¼ ewidth/ethickness when load-ing in the long direction, the orientation of theband depends on the degree of anisotropy. Inuniaxial tension, the orientationof the bandwrites

tan y ¼ffiffiffiffiffiffiffiffiffiffil

1þ l

r½56�

which generalizes eqn [55].A condition for diffuse necking in isotropic

materials under biaxial straining condition hasalso been worked out by Swift (1952):

e1 ¼2nð1þ rþ r2Þ

ðrþ 1Þð2r2 � rþ 2Þ ½57�

The prediction of plastic localization fore2>0 and e1>0 (right-hand side of the FLD)is much more complex and requires either animperfection-type of analysis (see Marciniakand Kuszinsky, 1967) or a bifurcation analysis(e.g., Rudnicki and Rice, 1975). In bothapproaches numerical analysis is necessary.These analyses show specific features of theplastic flow response such as vertices on theyield surfaces. The presence of porosity canalso have serious effect on the onset of shearbanding (e.g., Rice, 1976; Needleman and Rice,1978; Needleman and Tvergaard, 1992;Hosford and Caddell, 1993). When the strainpath is highly nonradial, significant departuresfrom the localization strains predicted by theFLDs are observed, as shown in Figure 27 foran aluminum 2008-T6 strained first under equi-biaxial tension and then under plane straintension.

An arresting example of nonradial loadingeffects affecting plastic localization is providedby systems made of thin metal layers depositedon elastomers (see Lacour et al., 2003; Li andSuo, 2005; Li et al., 2005). If the Young’s mod-ulus of the elastomer, E, is noted and if aHollomon fit (�¼Ken) is used for the stress–strain curve, three specific mechanisms can bepredicted theoretically for plane strainconditions:

1. If the elastomer is very compliant, that is,E/K is small (e.g., E/K<0.2), then the metal

film forms a neck at small strain as it was afreestanding film.

2. If the elastomer has an intermediate com-pliance (e.g., E/K¼ 1), then the metal filmforms multiple necks and deforms very muchbeyond the bifurcation point.

3. If the elastomer is stiff, that is, E/K is large(e.g., E/K>2), then the metal film deformsuniformly to large strains.

As a matter of fact, the substrate stabilizes thelocalization process as it does not want toundergo large local elongation. These phenom-ena have been observed experimentally.

Nowadays, advanced multiscale physics-based constitutive models coupled to numer-ical simulation tools allow predicting quiteaccurately the localization locus of metallicalloys, for radial or nonradial loadings, byincorporating plastic anisotropy effectsphenomenologically or through crystal orpolycrystal plasticity theory, kinematic hard-ening, phase transformation or second phases,information about the dislocation cell struc-ture evolution, etc. (e.g., Peirce et al., 1982;Hiwatashi et al., 1998; Hill, 2001; Inal et al.,2002; Knockaert et al., 2002; Yao and Cao,2002; Chien et al., 2004; Wu et al., 2005; Heet al., 2005). The capacity of constitutivemodels to properly predict localization is anexcellent way to assess their validity. Thesemodels constitute very important tools foraccelerating the development and optimiza-tion of new forming operations. The currentchallenge with these models is to incorporateinternal lengths in computationally efficientways while keeping the physics right. At thistime, the lack of internal length gives rise tostrong mesh dependency effects when simulat-ing the development of plastic localizations(see, for a detailed discussion, Forest andLorentz, 2004; Niordson and Redanz, 2004).


As a matter of fact, in metals, the width ofthe localization band is set by the microstruc-ture (typically the grain size). Localizationresults thus from a complex competitionbetween material hardening versus materialsoftening, geometry and loading configurationeffects.

2.06.3.3 Void Nucleation

2.06.3.3.1 Macroscopic evidences

Void nucleation is usually not detectedfrom the overall mechanical response ofmetals. In most metals, the initial second-phase content is indeed (and fortunately) toosmall to bring about significant amount ofinitial porosity: the initial porosity in metalsranges typically between 10�5 and 10�2.Furthermore, voids do not nucleate at thesame time on all particles. A smart method,but not applicable to all metals, to determinethe average overall nucleation strain consistsof prestraining samples to various levels ofdeformation, heat-treating them to restorethe strain-hardening capacity and loadingthem again up to fracture. The idea is thatthe fracture strain will be independent of thelevel of prestraining as long as it does notlead to void nucleation. The level of pre-straining which affects the fracture strainafter heat treatment is an indication of thenucleation strain. This method has been usedon aluminum (Le Roy et al., 1981) and oncopper (Pardoen and Delannay, 1998b).Ultrasonic detection of the nucleation eventscan be used to estimate the nucleation strain(e.g., Montheillet and Moussy, 1986). Notealso that all the techniques that are pres-ented in the next section for measuringdamage growth macroscopically can be usedto quantify an overall nucleation strain orstress by extrapolating the data to zerodamage.

(a)

7 μm

Figure 28 SEM micrographs of damage nucleationa, particle/matrix decohesion and b, particle fracture, as

2.06.3.3.2 Microscopic observations

Void nucleation is usually associated to thepresence of second-phase particles and inclu-sions, located either within the grains or alongthe grain boundaries (see the ‘classical’ papersby Puttick, 1959; Argon and Im, 1975; Argonet al., 1975; Argon, 1976; Goods and Brown,1979; Fischer and Gurland, 1981; Beremin,1981; Wilsdorf, 1983; Van Stone et al., 1985).For illustration, the two micrographs ofFigure 28 have been obtained during in situtensile tests on 6XXX aluminum alloys.Similar observations have already beendescribed in Section 2.06.2.3.3 when dealingwith Inco 718 alloy (see Alexandre et al.,2005). The AlFeSi particles, located along thegrain boundaries, present an elongated plateletshape. These particles break into several frag-ments when aligned with respect to the mainloading axis while interface separation occurswhen their long axis is orthogonal to the mainloading direction (Lassance et al., 2006a). Thisdependence of the void nucleation mode on theorientation of the second phases with the load-ing configuration has been studied in details insteels involving long MnS inclusions (seeMontheillet and Moussy, 1986, for references).Babout et al. (2004a, 2004b) have used 3-Dtomography to determine the mode of voidnucleation in ‘ideal’ composites made of elasticceramic particles in an aluminum matrix. Asshown in Figure 29, a soft matrix (pure Al)favors particle decohesion while a hard matrix(precipitate hardening 2124 Al) leads to particlecracking. A soft matrix prevents the stress in theparticle to attain the critical stress required forthe particle to crack, while the accumulation ofplastic strain on the interface between the par-ticle and the matrix allows the progressiveopening of the submicron interface defects.Void nucleation in a perfect lattice by localcleavage or specific dislocation accumulationis not common. However, these micromechan-isms of void nucleation have been observed in

(b)

10 μm

in a 6060Al alloys during in situ uniaxial testing:indicated by an arrow (Lassance et al., 2006a).

Matrix: 2124 alloyMatrix: pure aluminum

100 μm 100 μm(a) (b)

Figure 29 Reconstructed images from in situ 3-D X-ray tomography for metal matrix composites involving4% of ZrO2/SiO2 spherical particles embedded inside: a, a pure aluminum matrix (the strain level is equal to0.27); b, a 2124T6 aluminum matrix (the strain level is equal to 0.09). From Babout, L., Brechet, Y., Maire, E.,and Fougeres, R. 2004a. On the competition between particle fracture and particle decohesion in metal matrixcomposites. Acta Mater. 52, 4517–4525.

15 μm

25 m

m

Figure 30 Cast duplex stainless steel. a, Initiation ofcavities produced by the formation of cleavagemicrocracks in the ferrite phase; b, Voronoı cellsillustrating the heterogeneity in the distribution ofcavities initiated from cleavage microcracks.Source: Devillers-Guerville, L., Besson, J., andPineau, A., 1997. Notch fracture toughness of acast duplex stainless steel: Modelling ofexperimental scatter and size effects. Nucl. Eng.Des. 168, 211–225.


very pure single-phase metals, like Ti alloys(e.g., Thompson and Williams, 1977).

The recognition of the heterogeneous natureof void nucleation is essential for understandingmany ductile fracture problems. Void nuclea-tion is inherently a discontinuous process madeof a succession of discrete nucleation events.Several studies have reported that voids nucle-ate first on the largest inclusions, which involveprobably the largest of internal or interfacialdefects, and that void nucleation becomesincreasingly difficult with decreasing particlesizes (Gurland, 1972; Garrison et al., 1997;Lewandowski et al., 1989; Dighe et al., 2002).Some materials involve different families of sec-ond phases and inclusions. Among others, theexistence of two populations of particles, onewith limited resistance to void nucleation andanother, usually of a much smaller size, invol-ving a much better resistance to voidnucleation, has been repetitively illustrated inthe literature (Cox and Low, 1974; Hahn andRosenfield, 1965; Marini et al., 1985; Li et al.,1989; Haynes and Gangloff, 1997; Bron et al.,2004; Asserin-Lebert et al., 2005). The effect ofthis second population of voids can, in somecircumstances, be a dominant feature of thedamage process and will be discussed inSection 2.06.3.5. The inhomogeneity can alsobe due to a statistical distribution of thematrix/particle cohesion stress about a meaninterfacial stress (see, e.g., Kwon and Asaro,1990) or to local microplasticity effects in rela-tion with crystallographic details (see, e.g.,Bugat et al., 1999). The existence of differentmodes of void nucleation (interface vs particlefracture) also participates to the inherently het-erogenous nature of the nucleation process. Theinhomogeneity in the particle distributionscauses local stress concentrations and is cer-tainly an important reason of heterogenous

void nucleation (see, Lewandowski et al.,1989; Dighe et al., 2002; Gammage et al.,2004). For instance, the inhomogeneity inlocal nucleation rate was thoroughly investi-gated in duplex stainless steels (Pineau andJoly, 1991; Devillers-Guerville et al., 1997;Joly et al., 1990) using interrupted tests, asshown in Figure 30 where clusters of cavitiesrepresented by Voronoı cells are clearlyobserved. The histogram of the cell sizesshows that a very small fraction of the surfacearea in Figure 30 is leading to large local nuclea-tion rates compared to the mean nucleationrate. This clustering effect, particularly pro-nounced in this material, is another feature ofcavity nucleation which should be kept in mindwhen modeling ductile rupture.


The interfacial strength between second-phase particles and the matrix is dependent onthe local chemical composition. The segrega-tion of impurity elements similar to thosewhich induce intergranular embrittlement (seeSection 2.06.2.4.4) can reduce the interfacialresistance. Hydrogen-induced ductility lossesin low-strength steels could also be at leastpartly explained in this way (Cialone andAsaro, 1979); for recent modeling of hydro-gen-induced decohesion at particle/matrixinterfaces, also see Liang and Sofronis (2003).Impurity segregation at particles interface dueto irradiation effects has also been mentionedearlier (see Section 2.06.2.4.4).

Quantifying experimentally the local mechan-ical condition for void nucleation is anexperimental challenge. Two-dimensional digitalcorrelation methods can be used to determinethe local strain field corresponding to the onsetof void nucleation but the artifact of a surfacemeasurement has to be taken into account. Inthe method proposed by Beremin (1981),notched round bars were strained up to differentamount of deformation. After unloading, a sec-tion on the specimen parallel to the loading axisand comprising the axis of the specimen wasmetallographically prepared in order to deter-mine the boundaries of the region inside whichvoid nucleation had taken place. By combiningthe experimental determination of the locus ofvoid nucleation with finite element simulationsof the specimen, it is possible to determine thelocal mechanical conditions for nucleation. The3-D in situ tomography study reported above(see Figure 29), coupled with finite elementsimulations in order to calculate the local stress

0.010.001.0

1.5

2.0

2.0

0.02 0.03 0.04 0.05

K

I

Ed

0

N = 5N = 10

(a)

Figure 31 Stress concentration factor: a, at particle–maof remote axial deviatoric strain. The particle aspect ratito 1 for the lower pair of curves, equal to 2 for the middleshown for two different strain-hardening exponents N¼representation). Source: Lee, B. J. and Mear, M. E. 1999particle in a plastically deforming solid. J. Mech. Phys. S

and strain fields, allows a more rigorous deter-mination of the void nucleation condition(Babout et al., 2004a, 2004b; Maire et al., 2005).

2.06.3.3.3 Computational cell simulations

Computational cell simulations are very use-ful for capturing quantitatively the localtransfer between a matrix and second phasesas a function of the shape and mechanical prop-erties of the particles and of the mechanicalproperties of the matrix.

Lee and Mear (1992, 1999) have performed acomprehensive set of calculations on ellipsoidalinclusions embedded in viscous or elastoplasticsolids. Stress concentration factors were deter-mined for both interface decohesion andparticle cracking. Selected results from theirstudies are given in Figure 31 showing the evo-lution of the stress concentration factors (KI forthe interface stress concentration factor and Kp

for the particle stress intensity factor) as a func-tion of the applied remote strain Ed for twodifferent strain-hardening exponents and mod-ulus contrasts, under uniaxial tensionconditions. The stress concentration factors sig-nificantly evolve only during the beginning ofthe plastic deformation (during the first 2% ofstrains) and only if the particle has a stiffnesssimilar to that of the matrix. The effect of thestrain-hardening exponent is relatively small.Figure 32 shows the ratio of the stress concen-tration factors Kp/KI as a function of remoteaxial deviatoric strain for different particleaspect ratio Wp. In agreement with the experi-mental observations of Figure 28, particlefracture is favored when the particle is

0.010.001.0

2.0

1.5

2.5

3.0

0.02 0.03 0.04 0.05

K p

Ed

0

N = 5N = 10

(b)

trix interface KI and b, within particle Kp as a functiono is equal to 2 and the modulus contrast Ep/E is equalpair, and equal to 4 for the upper pair. The results are5 and N¼ 10 (defined through a Ramberg–Osgood

. Stress concentration induced by an elastic spheroidalolids 47, 1301–1336.

Wp = 1Wp = 2Wp = 3Wp = 4

3.0

2.5

2.0

1.5

1.00.00 0.01 0.02 0.03 0.04 0.05

Ed0

K p /

K I

Figure 32 Ratio of the stress concentration factorsKp/KI as a function of remote axial deviatoric strainfor different particle aspect ratio Wp and a strain-hardening exponent n¼ 5. Source: Lee, B. J. andMear, M. E. 1999. Stress concentration induced byan elastic spheroidal particle in a plastically deformingsolid. J. Mech. Phys. Solids 47, 1301–1336.


elongated in the direction of loading. Otherimportant information in the studies by Leeand Mear (1992, 1999) concern the effect ofthe stress state and of the location of the inter-face decohesion.

Christman et al. (1989), Llorca et al. (1991),and Tvergaard (1993) conducted a series of unitcell simulations of the interface separationbetween short elastic fibers and a plasticallydeforming matrix. The system underlying thisset of studies was metal matrix composites butmost of the results are generic. The interfacebehavior is modeled using a traction separationlaw (see seminal papers by Needleman, 1987,1990). The precise location of the beginning ofthe decohesion process was found to depend onmany factors such as the stress state and parti-cle aspect ratio. Partial interface decohesionwas frequently reported. The effect of shear onvoid nucleation has been investigated by Flecket al. (1989). More elaborated representativevolume elements involving particle clusteringhave been addressed recently by Shabrov andNeedleman (2002). The fracture of brittle sec-ond-phase particles (carbides in steels) has beeninvestigated by Kroon and Faleskog (2005) inthe framework of a study of cleavage in steels asindicated earlier in Section 2.06.2.3.2. Straingradient plasticity based analysis (e.g., Fleckand Hutchinson, 1997; Xue et al., 2002;Niordson and Tvergaard, 2002; Niordson,2003) or advanced dislocation dynamics cellcalculations (e.g., Cleveringa et al., 1999) per-mit the analysis of RVE with particle sizes inthe micron or submicron size range, where

classical plasticity theory fails to capture prop-erly the extra hardening contribution providedby a large density of geometrically necessarydislocations.

2.06.3.3.4 Void nucleation models

(i) Nucleation on a single particle

Analytic or closed-form void nucleation cri-teria constitute the first essential ingredient in aconstitutive models involving damage evolu-tion. Various void nucleation criteria havebeen proposed based either on dislocation the-ory (for crystalline materials) or on purecontinuum mechanics (e.g., based on Eshelby,1957) theory to evaluate the load transfer.Dislocation-based analysis is necessary whenthe particle size is smaller than typically 1 mmto properly account for the large density ofgeometrically necessary dislocations that con-trols the hardening at that scale due to the verylarge plastic strain gradients. A good review ofthese different criteria has been made by Berdin(2004); see alsoMontheillet andMoussy (1986).We limit the presentation here to generalaspects and to one specific void nucleationcriterion.

If the particle is brittle and deforms elasti-cally, a simple one-parameter condition can bemotivated from linear-elastic fracturemechanics arguments. Second-phase particlesalways contain tiny submicron defects (e.g.,Kroon and Faleskog, 2005; see Section2.06.2.3.2). Considering a given size for theinternal cracks, particle fracture takes placewhen the energy release rate becomes largerthan the particle fracture toughness, which canbe translated also into an effective critical stresscondition within the particle (e.g., see Ghoshet al., 1997; Horstemeyer et al., 2003; Huberet al., 2005). Now, in very small and clean brit-tle particles, without defects, a critical stresscondition based on the theoretical strength ofthe material is also valid. One thus needs properhomogenization method to evaluate the stressin the particle as a function of the overall stress.

In the case of interface fracture, a one-para-meter linear-elastic fracture mechanicsapproach is, in principle, not relevant if thematrix surrounding the particle is plasticallydeforming. Both the separation energy andinterface strength play a role in the problem.In many instances, the energy condition caneasily be met while enough plastic deformationmust still be accumulated on the interface toraise the stress above the critical strength level.This is why both critical-strain-based models(e.g., see Walsh et al. (1989) for Al alloys andJoly and Pineau (1991) or Bugat et al. (1999) for


cast duplex stainless steel) have been used aswell as a critical stress based models (e.g., seeKwon and Asaro (1990) for spheroidized steel)but none of them are totally satisfactory. Theformulation of an adequate condition for voidnucleation by interface fracture is a very diffi-cult problem that many researchers try to solvethrough ‘phenomenological’ models involvingboth strain- and stress-controlled nucleationconditions with parameters tuned on experi-mental data or RVE simulations (see Chu andNeedleman, 1980; Tvergaard, 1990). Usually,the critical stress and critical strains are overallvalues and not local values in the particle oralong the interface.

One relatively advanced void nucleationmodel has already been introduced in Section2.06.2.3.2. The fracture of the particle or of theinterface is assumed to occur when the maxi-mum principal stress in the particle or at theinterface reaches a critical value

sparticle max1 ¼ sbulkd or sinterfd ½58�

which is different for each mechanism. Basedon the Eshelby theory (Eshelby, 1957) and the‘secant modulus’ extension to plasticallydeforming matrix proposed by Berveiller andZaoui (1979), the maximum principal stressin an elastic inclusion (and at the interface)�1particlemax can be related to the overall stress

state by using

sparticle max1 ¼ smax

1 þ ksðse � s0Þ ½59�

Table 6 Void nucleation stress reported in the litemechan

Particles Matrix

Particle fractureElongated MnS A508 steelCuboidal TiN 4330 steelTiN Inconel 718

4% Spherical ZrO2–SiO2 (ZS) balls Al2124 (T6)20% Spherical ZrO2–SiO2 (ZS)balls

Al2124 (T4)

Interface fractureMnS A508 steelSi Al (cast)4% Spherical ZrO2–SiO2 (ZS) balls Al2124 (T6)4% Spherical ZrO2–SiO2 (ZS) balls Pure Al20% Spherical ZrO2–SiO2 (ZS)balls

Pure Al

Rounded Fe3C Spheroidized 104steel

Cu–Cr particles Cu alloyTiC Maraging steelC nodules Cast iron

where �1max is the maximum overall principal

stress and ks is a parameter of order unitywhich is a function of the inclusion aspectratio Wp and of the loading direction (seeLeRoy et al. (1981) for a model in the samevein). An approximate explicit form for thefunction ks(Wp) can be given for long spheroi-dal particles:

ksðWpÞ ¼4

9ð2La � 1Þ 2� 3La þW2p

� �½60�

where La¼ ln(2Wp� 1/2Wp)¼ liml!1cosh�1(Wp). The higher the aspect ratio, thelarger is ks and the sooner void nucleationtakes place. This void nucleation criterion wasinitially proposed by the Beremin group(Beremin, 1981), who also identified ks fromexperimental data on steels with MnS inclu-sions. They found that the values of kspredicted by the theory were about 2 timeshigher than the experimental values because ofan overstiff response of the homogenizationscheme.

Typical mean values for particle fracturestress or interface fracture stress are gathered inTable 6. A very interesting set of results wasobtained by Babout et al. (2004a, 2004b) on‘ideal’ composites Al matrixþZS sphericalballs using 3-D tomography to detect the nuclea-tion event and finite element cell calculations toestimate the local stress in the particle and alongthe interface. As explained above, the criticalstress for interface fracture is definitely not

rature for particle fracture or particle decohesionisms

Critical stress(MPa) Reference

1100 Beremin (1981)2300 Shabrov et al. (2004)

1280–1540 Alexandre et al.(2005)

700 Babout et al. (2004b)700 Babout et al. (2004b)

800 Beremin (1981)550 Huber et al. (2005)1060 Babout et al. (2004b)250 Babout et al. (2004b)320 Babout et al. (2004b)

5 1650 Argon and Im (1975)

1000 Argon and Im (1975)1820 Argon and Im (1975)80 Dong et al. (1997)

00 0.2 0.4 0.6 0.8 1 1.2 1.4

0.002

0.004

0.006

0.008

0.01

0.012

0.014

f –f 0

εe

CuAR

CuA

CuAR

εefεefCuA

Figure 33 Variation of the porosity estimated fromdensity measurements as a function of straining fromsamples cut in the minimum cross-sectional area of anecking region, in as-received (CuAR) and annealedcopper bars (CuA) (see Pardoen and Delannay,1998a).


intrinsic to the particle but depends on the hard-ening of the matrix. Note that there are manyimportant informationmissing in this table, suchas the mean particle size and particle size distri-bution, the flow properties of each phase. Theseinformation are necessary to allow cross-com-parison between the various systems.

The model [59] remains rather qualitativewith respect to more accurate finite elementcell calculations presented in Figure 31, forinstance, and which could be used to providecloser estimate of the load transfer between thematrix and particle, or with respect to sizeeffects for small particles. These is definitelyroom for improving such models based onrecent developments in homogenization theoryfor nonlinear solids (e.g., Doghri and Ouaar,2003). Nevertheless, the complexity of the frac-ture process at that scale is such that criticalstress and particle fracture toughness will haveto be identified from experiments.

(ii) Void nucleation rate function

For all the reasons discussed at the beginningof this section, the heterogeneity of the nuclea-tion process must be introduced through adistribution function. Following the spirit ofthe model proposed by Needleman, Tvergaard,and co-workers (see, e.g., review Tvergaard,1990), a general expression for the void nuclea-tion rate, based on [59], can be written as

_fnucl ¼ An _smax1 þ ks _se

� �½61�

where

An ¼f0

sffiffiffiffiffiffi2pp exp � 1

2

smax1 þ ksðse � s0Þ � hsdi

s

� �2" #

½62�

for �1maxþ ks (se� �0)¼ �1maxþ ks (se� �0)jmax

and _smax1 þ ks _se>0. In [62], f0 is the initial void

volume fraction, h�di is the average void nuclea-tion stress, and s is the standard deviation to bedetermined experimentally or by inverse identi-fication. Note that strain-controlled voidnucleation laws are frequently preferred becausethey are more simple to implement numerically.

The initial void volume fraction f0 is equal tothe volume fraction of particles giving rise tovoids if the nucleation mechanism is by com-plete interface decohesion. For the case ofpartial decohesion or particle fracture, theinitial void shape is very flat. The easiest optionis to take the volume fraction of spherical voidshaving the same projected area. A moreadvanced procedure to identify f0 for initiallypenny-shaped voids has been recentlyaddressed by Lassance et al. (2006a); see next

section. Different populations of voids withdifferent associated shapes of the particles, cri-tical nucleation stress, and standard deviationscan be taken into account by adding their con-tribution to the void nucleation rate.

2.06.3.4 Void Growth


The macroscopic softening effect induced bythe growth of voids in a plastically deformingmatrix is relatively weak. The presence of voidscan be measured macroscopically from densitymeasurements with accurate scales or from otherindirect methods, such as the change of the elasticstiffness or change of resistivity (see Montheilletand Moussy, 1986). Figure 33 shows a typicalevolution of the density observed in samples cutin the neck of cylindrical bars pulled in tensionand made of a copper containing initially about0.3% of copper oxide particles.

The most important macroscopic observa-tion about ductile fracture is that the fracturestrain decays exponentially with increasingstress triaxiality, as first reported by Hancockand McKenzie (1976), followed by many otherteams (e.g., Marini et al., 1985; Devaux et al.,1985; Becker et al., 1988; Bauvineau, 1996;Decamp et al., 1997; Pardoen et al., 1998;Jablokov et al., 2001; Huber et al., 2005). Thiseffect is directly related to a significant increaseof the void growth rate with increasing stresstriaxiality. The fracture strain ef is defined fromthe reduction of cross-sectional area measuredon broken samples:

ef ¼ lnA0

Af

� �½63�


where A0 is the initial cross-sectional area andAf is the final cross-sectional area. The stresstriaxiality T is defined as

T ¼ sii3se¼ sh

se½64�

where �h is the hydrostatic stress. Figure 34shows the schematic evolution of the fracturestrain as a function of the stress triaxiality aver-aged along the deformation path. Asschematically represented in Figure 34, thestress triaxiality is equal to the following:

� 0 under pure shear.� 1/3 under single tension. Note that in a single

tension specimen, T is equal to 1/3 up to theonset of necking and then steadily increasesinside the neck with increasing plastic defor-mation (see Section 2.06.3.2.1).� ffiffiffi

3p

=3 under plane strain tension.� 0.6–1.8 in the center of the minimum cross

section of a notched round bar, dependingon the radius of curvature of the notch. Thecylindrical notched round bar geometry isprobably the best suited to study ductile frac-ture. It allows probing a wide range of stresstriaxiality by changing the radius of curva-ture of the notch. Moreover, the stresstriaxiality remains relatively constant allover the deformation process in the centerof the minimum cross section. A standardprocedure to test cylindrical notched roundbars and interpret the results has been devel-oped by the European Structural IntegritySociety (ESIS P6-98, 1998). Finite elementsimulations are necessary to accurately eval-uate the stress triaxiality.� 2.75–5 inside the fracture process zone

(FPZ) in front of a crack tip. The valuedepends on the strain-hardening exponent.The specific and very important problem ofvoid growth at the tip of crack, controlling

Fra

ctur

e st

rain

Stress triaxiality0.3–0.5

1

0.1

1.0 1.5 2.5 3.0 3.5

Figure 34 Typical variation of the fracture strain as afunction of the stress triaxiality in metallic materials.

the fracture toughness of ductile materials,is discussed in Section 2.06.3.6.� 1 under purely hydrostatic stress (�e¼ 0).

Usually, ductile fracture in industrial metal-lic alloys only takes place for stress triaxialitylarger than 0.3–0.5. This is why most formingoperations are conducted under compressiveloadings, at least in the heavily plasticallydeformed regions of materials. The effect ofthe stress triaxiality has also been widelystudied by conducting tensile tests with super-imposed hydrostatic pressure (e.g., Brownrigget al., 1983; Vasudevan, et al., 1989; andreview paper by Lowhaphandu andLewandowski, 1999). Ductile fracture caneven be suppressed if the pressure is highenough leading to a full necking process upto a final material point.

The fracture strain of metallic samples afterplastic forming operations is usually anisotro-pic due to particle morphology effects andplastic texture (e.g., Becker et al., 1989b;Achon, 1994; Bauvineau, 1996; Benzergaet al., 2004a).


After nucleation, the voids grow by plasticdeformation. The void growth process can beobserved during in situ testing inside a scan-ning electron microscope. As shown inFigure 35a, voids nucleated by particle frac-ture open and become more rounded withplastic deformation. Voids nucleated by par-ticle decohesion are initially rounded and tendto elongate in the principal loading directionunder low stress triaxiality (see Figure 35b). InFigure 35b, the particle prevents the void tocontract in the direction transverse to themain loading direction. If the void axis isnot aligned with the principal loading direc-tion or if it undergoes shear deformation,void rotation is observed (see Figure 35d).Recently, in situ 3-D tomography experimentshave provided more complete informationabout the complete 3-D morphology of thevoids, while avoiding the artifact of surfaceobservation (Babout et al., 2001, 2004a,2004b; Maire et al., 2005; see also X-raymicrotomography experiments by Everettet al. (2001)). An example of a 3-D recon-structed image obtained on aluminumcontaining ZS particles (see also Table 6) isshown in Figure 35c.

Goto et al. (1999), Jablokov et al. (2001),and Chae and Koss (2004) have prepared andanalyzed metallographic sections of samplesdeformed up to various levels of straining.

(a) (b)

(d)(c)

Loadaxis

Voidaxis

θ

Figure 35 Micrographs of voids growing by plastic yielding of the surrounding matrix: a, voids originatingfrom particle fracture (void growth in cast Al alloys nucleated by the fracture of Fe-rich particles or Si particles;b, voids originating from the decohesion of the particle interface (void growth around a copper oxide inclusionin copper; c, 3-D tomography reconstructed image where dark cavities can be seen around the large grayparticles; d, evidence of void rotation. a, from Huber, G., Brechet, Y., and Pardoen, T. 2005. Void growth andvoid nucleation controlled ductility in quasi eutectic cast aluminium alloys. Acta Mater. 53, 2739–2749.;b, from Pardoen, T., Doghri I., and Delannay, F. 1998. Experimental and numerical comparison of voidgrowth models and void coalescence criteria for the prediction of ductile fracture in copper bars. Acta Mater.46, 541–552; c, source: Maire, E., Bordreuil, C., Babout, L., and Boyer, J. C. 2005. Damage initiation andgrowth in metals. Comparison between modeling and tomography experiments. J. Mech. Phys. Solids 53,2411–2434; d, source: Benzerga, A. A., Besson, J., and Pineau, A. 2004a. Anisotropic ductile fracture. Part I:Experiments. Acta Mater. 52, 4623–4638.


The relevance of these time-consuming experi-ments very much depends on the quality ofthe polishing. Polishing can indeed signifi-cantly smear out the voids. By measuring alarge number of void sizes, they were able togenerate plots of the evolution of the voidvolume fraction as a function of the straining

in different steels. An example is given inFigure 36 (similar measurements were alsoperformed by other teams, e.g., Marini et al.(1985)).

There has been only a very limited number ofexperimental studies devoted to the couplingbetween void growth and crystal orientation

0.02

0.015

= 1.4

(σm/σeq)ave

(Af)crit

= 1.1 = 0.9 = 0.8

0.005

0.01

00

(a)

0.2 0.4 0.6

Equivalent plastic strainV

oid

area

frac

tion

0.8 1

Void–voidinteraction

300 μm(b)

Figure 36 a, Void volume fraction measurements made of multiple micrographs taken frommetallographically polished section of broken notched round bars made of HSLA-100 steel. Finite elementsimulations were used to estimate the stress triaxiality and effective plastic strain fields. b, The variation of thevoid volume fraction as a function of straining was obtained by relating the local void measurements to thelocal mechanical conditions. The method was repeated for different notch radii. Source: Chae, D. and Koss, D. A.2004. Damage accumulation and failure in HSLA-100 steel. Mater. Sci. Eng. A 366, 299–309.

L0z

L0x

(a)

(b)R0x

R0z

Rpz

FE unit cell

Figure 37 a, Principle of void cell simulation basedon cylindrical geometry subjected to axisymmetricloading conditions and involving a spheroidal voidand a particle. b, Finite element mesh.


effects (e.g., Crepin et al., 1996; Gan et al.,2006) or to the analysis of the effect of the initialvoid size on the void growth rate which tends todecrease when the voids get smaller (e.g.,Schlueter et al., 1996; Khraishi et al., 2001).

2.06.3.4.3 Void cell simulations

The principle of a void cell simulation is toidealize the microstructure by considering simplearrangements of voids and to use the finite ele-ment method with proper boundary conditions.The most simple arrangement is to consider per-iodic distribution of voids. More sophisticatedrepresentative elements can of course be con-structed based on elementary patterns thatcontain several voids. A representative set ofreferences dealing with void cell calculations isgiven by the following list: Needleman (1972a,1972b), Tvergaard (1981, 1982, 1990), Koplikand Needleman (1988), Becker et al. (1989a),Hom and McMeeking (1989a), Needleman andKushner (1990), Worswick and Pick (1990),Needleman et al. (1992), Huang (1993), Beckerand Smelser (1994), Richelsen and Tvergaard(1994), Brocks et al. (1995a), Kuna and Sun(1996), Faleskog and Shih (1997), Steglich andBrocks (1997), Steenbrink et al. (1997), Sovikand Thaulow (1997), Faleskog et al. (1998),Thomson et al. (1999), Pardoen and Hutchinson(2000), Socrate andBoyce (2000), Pijnenburg andVan derGiessen (2001), andKim et al. (2004). Allthese studies have contributed to improving our

understanding of all the possible factors that gov-ern the growth of voids in elastoplastic or elasto-viscoplastic materials.

The key information that can be extractedfrom a simple cell made of a cylinder containinga spheroidal void and loaded axisymmetrically(see Figure 37) is presented in the following.Periodic boundary conditions are enforced byimposing the external faces to remain straightand parallel to their initial orientation.Although cylinders leave open space whenpacked next to each other, this 2-D representa-tion is known (Kuna and Sun, 1996; Worswickand Pick, 1990) to give results very similar tothe 3-D unit cell based on hexagonal-type dis-tribution of voids.

0

0.5

1

1.5

2

2.5

0

σ e /σ

0

εe

n = 0.1, E /σ0 = 500, triaxiality = 1

f = 0

f = 10–2

f = 10–4

f = 10–3

0.5 1 1.5 2

Figure 38 Variation of the overall effective stress asa function of the overall effective strain predicted bya 2-D axisymmetric unit cell simulation with aninitially spherical void for three different initial voidvolume fractions. The comparison is made with theresponse of the nonporous material.


Motivated by the recognition that the stresstriaxiality is the main parameter controlling thevoid growth rate (see Figure 34), the cell calcu-lations are ideally performed at a constantimposed stress triaxiality. Figure 38 shows theoverall effective stress–effective strain curvespredicted by void cell calculations performedat a constant stress triaxiality T¼ 1 for threeinitial porosities: 10�2, 10�3, and 10�4. Thevoids are initially spherical and the void distri-bution parameter, noted l0 and defined asl0¼L0z/L0x, is equal to 1.A J2 isotropic hard-ening elastic–plastic response is assumed for thematrix, characterized by the following represen-tation in uniaxial tension:

ss0¼ E

s0e for s<s0 ½65�

ss0¼ 1þ E

s0ep

� �n

for s>s0 ½66�

where E is the Young’s modulus, �0 is the initialyield stress, and n is the strain-hardening expo-nent. The Poisson ratio of the matrix n is alwaystaken equal to 0.3. For the sake of comparison,the stress–strain curve of the reference nonpor-ous material has also been added. The softeningdue to the presence of the porosity only showsup close to the onset of void coalescence. Untilthat point, the effect of the porosity on thestress–strain curve is small for porosity smallerthan 10�2.

The various effects of the stress triaxiality aresummarized in Figure 39. Figure 39a shows theaxial stress versus axial strain curves correspond-ing to different stress triaxiality equal to 1/3, 1,

and 3. In agreement with Figure 34, the fracturestrain significantly drops with increasing stresstriaxiality. As shown in Figure 39b, the drop offracture strain is directly related to an increasingrate of porosity growth with increasing stresstriaxiality.

Figure 39 addresses also the effect of the initialvoid aspect ratio of spheroidal void and theevolution of the void aspect ratio. The aspectratio is designated by S or W, with S¼ ln(W)and W¼Rz/Rx (see Figure 37). Three differentinitial void aspect ratios are considered: elon-gated void with an aspect ratio 6; sphericalvoid with an aspect ratios of 1; and flat voidwith an aspect ratio 1/6 (see Pardoen andHutchinson, 2000; see also analyses by Beckeret al., 1989b, 1989c). The effect of void shape isvery marked at low stress triaxiality. It is inter-esting to note the saturation of the porosity atlarge strains in uniaxial tension for the voids thatare initially spherical or elongated. As shown inFigure 39c, the aspect ratio of the voids increasesa lot with plastic straining. As a matter of fact,the voids both elongate in the loading directionand contract in the transverse direction whilepreserving the volume of the void, leading atvery large strain to a needle-like shape. Thistransverse contraction is prevented if a particleis present in the voids (see Figure 35b). The voidshape effect is still pronounced at T¼ 1 butdecreases with increasing stress triaxiality. Theaspect ratio of voids that are initially spherical orelongated tends to increase at low or intermedi-ate stress triaxiality but to decrease at large stresstriaxiality (see Figure 39c; see Budiansky et al.,1982). Voids that are initially flat always tend toopen first. At the intermediate stress triaxiality,the porosity growth rate is significantly largerfor flat voids. Hence, the behavior of flat voids(also called ‘oblate’) is different than the beha-vior of more rounded or elongated voids (alsocalled ‘prolate’).

Proper understanding of the response ofmaterials involving flat voids is important asmany materials involve void nucleation by par-ticle cracking or by partial interface decohesionleading to initially penny-shaped voids (detailedanalysis can be found in Lassance et al.(2006b)). Let us consider a constant relativevoid spacing while changing only the initialvoid aspect ratio and thus the initial porosity.The initial relative void spacing �0¼R0x/L0x isrelated to the initial porosity and initial voidshape through the following expression:

�0 ¼R0x

L0x¼ R3

0x

L30x

� �1=3

¼ R20xR0zR0xL0z

L20xL0zR0zL0x

� �1=3

¼ f0zl0W0

� �1=3½67�

0

0.5

1

1.5

2

2.5

3

3.5

4

0

W 0 = 6W 0 = 1W 0 = 1/6

σ z /σ

0

εz

f0 = 10–2, λ 0 = 1, n = 0.1T = 3

T = 1

T = 1/3

Onset of coalescence

(a)

0

0.02

0.04

0.06

0.08

0.1

0

W 0 = 6W 0 = 1W 0 = 1/6

f

εz

T = 3

T = 1

T = 1/3

–2

–1

0

1

2

3

0

T = 3T = 1T = 1/3

S =

ln(W

)

εz

W 0 = 1/6

W 0 = 1

W 0 = 6

(b)

0.2 0.4 0.6 0.8 1 1.2 1.4

(c)0.2 0.4 0.6 0.8 1 1.2 1.4 0.2 0.4 0.6 0.8 1 1.2 1.4

Figure 39 Void cell results for f0¼ 10�2, l0¼ 1, �0/E¼ 0.002, n¼ 0.1, and W0¼ 1/6, 1, 6, at T¼ 1/3, 1, 3.a, (True) axial stress vs (true) axial strain; b, porosity vs axial strain; c, void shape vs axial strain.From Pardoen, T. and Hutchinson, J. 2000. An extended model for void growth and coalescence. J. Mech.Phys. Solids 48, 2467–2512.


where � is a geometric factor which depends onthe arrangement of voids: �¼ �/6¼ 0.523 for aperiodic simple cubic array; z ¼

ffiffiffi3p

p=9 ¼ 0:605for a periodic hexagonal distribution; and �¼ 2/3¼ 0.666 for a void surrounded by a cylindricalmatrix, which is the case under discussion here.Prescribing the initial void spacing is similar toconsidering voids originating from particlecracking with a fixed particle spacing. Here theparticles are virtual: they have the same proper-ties than the matrix material. The initial voidaspect ratio W0 ranges between 0.001 and 2.A‘virtual’ volume fraction of spherical particlefp¼ 1% is considered which is equivalent to pre-scribing the relative void spacing �0¼ 0.247. Theinitial void volume fraction is given by

f0 ¼W0

Wpfp ½68�

where Wp is the shape of the particle, equal to 1here. Figure 40 gathers the results obtained at aprescribed stress triaxiality equal to 1, in terms of(a) the overall stress–overall strain curves corre-sponding to the z direction; (b) the evolution ofthe porosity; and (c) the evolution of the voidaspect ratio. Themost important result emergingfrom Figure 40 is that the evolution with strain-ing of the overall stress, porosity, and void shapeis independent of the initial void aspect ratioW0

ifW0 is typically lower than 0.03–0.1. This showsthat for flat voids, the key parameter controllingthe damage evolution process is the relative voidspacing �0. The assumption of identifying aneffective initial porosity as equal to the porosityassociated to an equivalent spherical void isacceptable as long as the particle volume frac-tion is sufficiently low (fp¼ 1%) (see Figure 40)but not when the volume fraction is large

1

1.5

2

2.5

3

0 0.1 0.2 0.3 0.4 0.5 0.6

f p = 1%

T = 1 n = 0.1

W 0 =

σ z /σ

0 Onset of void coalescence

(a)

1 0.5 0.2 0.1 0.01–0.001– – – –

εz

0 0.1 0.2 0.3 0.4 0.5 0.6εz

f p = 1%

T = 1 n = 0.1

0

0.02

0.04

0.06

0.08

0.1

f

Onset of void coalescence

(b)

W 0 = 1 0.5 0.2 0.1 0.01–0.001– – – –

εz(c)

0

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.5 0.6

W

f p = 1%

T = 1 n = 0.1

Onset of void coalescence

W 0 = 1 0.5 0.2 0.1 0.01–0.001– – – –

Figure 40 Results of finite element unit cellcalculations for various initial void aspect ratios W0

ranging between 0.001 and 1, under a constant stresstriaxiality T¼ 1, with n¼ 0.1 and E/�0¼ 500, forfp¼ 1%: a, overall axial stress vs axial strain curves;b, evolution of the porosity as a function of the axialstrain; c, evolution of the void aspect ratio as afunction of the axial strain. The onsetof coalescence is indicated by a bullet. FromLassance, D., Scheyvaerts, F., and Pardoen, T.2006b. Growth and coalescence of penny-shapedvoids in metallic alloys, Eng. Fract. Mech. 73,1009–1034.


(fp¼ 10%) (see Lassance et al. (2006b): theresponse of the material becomes much moredependent on the initial void shape (whenW0>0.1) when fp¼ 10% than when fp¼ 1%.

Figure 41 shows that the effect of the strain-hardening exponent on the void growth rate isnot very significant. Note however, as discussedlater in Section 2.06.3.2, that the strain-hard-ening exponent will have an indirect effect onthe damage accumulation as the prime para-meter controlling plastic localization, and onthe strain distribution and stress triaxialityinside the necking regions.

Figure 42 provides interesting informationabout the rotation of elongated voids undercombined shear and normal loading. Two dif-ferent orientations of the voids with respect tothe loading configurations are chosen andembedded with a 3-D unit cell under fully per-iodic boundary conditions. In both cases, thevoids rotate toward the direction of the max-imum principal stress (Scheyvaerts et al., 2005;see also Bordreuil et al. (2003) for void cellsimulations under shear conditions).

Other features of the void growth processhave been addressed using void cell simulations.Several authors have shown that heterogeneitiesin the void distributions, for instance clusteringeffects, do not significantly affect the voidgrowth rates (Needleman and Kushner, 1990;Huang, 1993; Thomson et al., 2003). The pre-sence of small secondary voids in betweenprimary big voids has also been investigatedusing void cell simulations, as discussed in thenext section on void coalescence. Benzerga(2000) analyzed the effect of anisotropic materialbehavior on the void growth rate and the cou-plings with the morphological anisotropy causedby nonspherical void shapes. Void cell calcula-tions have also been performed with a crystalplasticity constitutive response for the matrixmaterial (O’Regan et al., 1997; Orsini andZikry, 2001; Kysar et al., 2005; Potirnicheet al., 2006; Gan et al., 2006) under differentloading conditions. Among others, Potirnicheet al. (2006) have shown that different crystalorientations can lead up to a factor of two dif-ference in the void growth rates. The effect of thestrain rate sensitivity on the void growth rate hasbeen addressed by Klocker and Tvergaard(2003). The void growth rate decreases andvoid aspect ratio increases faster when increasingthe rate sensitivity. Higher-order constitutivetheories including strain gradient plasticityeffects and internal length scales associated tothe accumulation of geometrically necessary dis-locations have also been used to study the effectof the void size on its growth rate (e.g., Fleck andHutchinson, 1997; Shu, 1998; Li et al., 2003; Liuet al., 2003). A decrease of the void growth rate

0

0.02

0.04

0.06

0.08

0.1

0 0.1 0.2 0.3 0.4 0.5 0.6

f

εe

FE unit cell calculationcylinder with a single void 2-D axisymmetric – T = 1

f 0 = 1%, W 0 = 1, λ0 = 1E /σ0 = 500

n = 0.1n = 0.1

n = 0.1n = 0.3

Figure 41 Variation of the void volume fraction as a function of the overall strain for strain-hardeningexponents n¼ 0.1 and 0.3 under a constant stress triaxiality T¼ 1, with E/�0¼ 500, for f0¼ 0.1% andinitially flat voids.

Figure 42 3-D void cell calculations with elongated spheroidal voids under combined plane strain shear/tension conditions, and constant stress triaxiality. Periodic boundary conditions are enforced. Two differentorientations of the voids are chosen to analyze their rotation. Source: Scheyvaerts, F., Onck, P., Brechet, Y.,and Pardoen, T. 2005. Multiscale simulation of the competition between intergranular and transgranularfracture. In: Proceedings of ICF11 – 11th International Conference on Fracture (ed. A. Carpinteri), 20–25March 2005, Turin, Italy, CD-ROM – 5331.


with decreasing void size is confirmed by allthese works. Liu et al. (2003) found that theamplitude of this size effect significantlyincreases with increasing stress triaxiality.

2.06.3.4.4 Void growth models

(i) Models for isolated voids

Although the state-of-the-art in void growthmodeling is connected to the advances in thedevelopment of constitutive models for porousmedium, simple analytical models for isolatedvoids remain very useful and powerful to per-form first-order analysis, to guide intuition, andto sustain pedagogical approaches of ductilefracture. Models for isolated spherical voids

have been proposed by McClintock (1968) andRice and Tracey (1969). The Rice and Traceymodel (RT model) (Rice and Tracey, 1969)evaluates the growth of a initially sphericalvoid in an infinite, rigid, perfectly plastic mate-rial subjected to a uniform remote strain field.Using well-chosen velocity fields, the varia-tional analysis of Rice and Tracey (seeChapter 2.03) leads, for the assumption of sphe-rical void growth, to the following expressionfor the average rate of growth:

dR

R¼ 1

3

df

f¼ a exp

3

2T

� �deep ½69�

where R is the actual radius of the cavity, eep isthe equivalent plastic strain, and a is a constant.


A value a¼ 0.283 was computed by Rice andTracey (1969) and re-evaluated by Huang(1991) using additional velocity fields. Huangretrieved the same result but with a¼ 0.427 forT>1 and a¼ 0.427T1/4 for T>1. In practice,the porosity is finite and the stress and strainfields around the voids interact. This explainswhy some authors (Marini, 1984; Marini et al.,1985) found higher values for a when calibrat-ing the model toward experimentalmeasurements. This effect of a finite porosityis addressed in the next subsection.

The general analysis of Rice and Tracey wasalso taking into account the change of voidshape, assuming ellipsoidal voids and plasticflow conditions in which the directions of theprincipal axes of the strain rates remain fixedthroughout the strain path. The rates of changeof the radii of the void, in the principal direc-tions, write (Rice and Tracey, 1969; Thomason,1990)

dRk

R¼ dR

Rþ ð1þ EvÞde pk ½70�

where Ev is a void shape parameter. From voidcell simulations, Worswick and Pick (1990) haveshown that the model would be more accurate if(1þEv) in expression [70] is not taken as a con-stant equal to 5/3, but involves a dependence onstress triaxialities, on f0, on the strain-hardeningexponent, n, and on eep at low stress triaxialites(Le Roy et al., 1981; Worswick and Pick, 1990).For instance, for f0¼ 0.01 and n¼ 0.2 and forlow stress triaxialities (<1.2), the results ofWorswick and Pick (1990) lead to (see Pardoenand Delannay, 1998b)

Ev ¼ð1:9� 0:72TÞexp ð3=2Þeep

� �2exp ð3=2Þeep

� �� 1

� 1 ½71�

It is worth mentioning the recent extension ofthe Rice and Tracey model to the so-calledTaylor-type dislocation-based strain gradientplasticity material response (Gao et al., 1999;Huang et al., 1999; Qiu et al., 2001, 2003) byLiu et al. (2003) in order to account for size effectswhen the void size is in the submicron range.

(ii) Constitutive models for porous media

As shown in Figure 38, the softening inducedby the nucleation and growth of the porosity inrealistic alloys is usually quite weak.Nevertheless, there are several circumstanceswhere a full constitutive model involving thecoupling with the porosity on the materialresponse is needed. Small amount of softeninginduced by the porosity is sometimes sufficientto significantly affect plastic localization (e.g.,Yamamoto, 1978; Ohno and Hutchinson,

1984), a critical point of investigation for sev-eral types of forming operations. The drop ofthe stress-carrying capacity near fracture initia-tion and during cracking must be accounted forwhen modeling ductile tearing problems.Constitutive models for porous media alsooffer a natural framework for the introductionof void interaction effects.

Constitutive models for porous elastoplasticor viscoplastic media (also called dilatant plas-ticity models) have been proposed based eitheron micromechanics or on thermodynamics.The micromechanical model developed byGurson (1977) is probably the model that hasreceived the largest attention in the literature.Many efforts have been devoted to extend itsrange of validity and of application since the1970s. A first subsection will thus focus on thismodel and its extensions (see also Chapter2.03). Note that advanced model for porouselastoplastic or viscoplastic media has alsobeen formulated based on variational methods(Zaidman and Ponte Castaneda, 1996; PonteCastaneda and Zaidman, 1994, 1996;Kaisalam and Ponte Castaneda, 1998; Garajeuet al., 2000). A second shorter subsection will bedevoted to a presentation of the Rousseliermodel which has been developed in the frame-work of a thermodynamical approach. Thismodel has several merits which are worthmentioning.

(iii) Gurson model and extensions

TheGursonmodel is the firstmicromechanicalmodel for ductile fracture which introduces astrong coupling effect between deformation anddamage. The model is representative of the voidgrowth stage only. It is derived from an analysissimilar to the one performed by Rice and Traceyfor an isolated void (see previous subsection). Themodel is based on a simplified representation ofthe voided material which consists in an hollowsphere. The only nondimensionalmicrostructuralfeature in themodel is the void volume fraction orporosity, f. Thematerial of the hollow sphere (i.e.,the matrix) is assumed to be rigid, isotropic, per-fectly plastic with a yield limit. Thematrix obeys astandard vonMises yield criterion and associatedflow rule. The analysis is based on an upper-bound method consisting in two steps: (1) find afamily of velocity fields compatible with theboundary conditions (here homogeneous) and(2) minimize the plastic dissipation within theproposed family (see for details Chapter 2.03).In the work of Gurson, only one field was con-sidered so that no minimization procedure wasneeded. Themain result is an estimate of the yieldfunction for the porousmetal which, applying thenormality rule, can be used to derive the plastic


flow direction. The yield surface is given by thefollowing equation:

� Xs2es2yþ 2f cosh

1

2

skksy

� �� 1� f2 ¼ 0 ½72�

This yield surface is identical to the von Mises(or J2) yield surface when f¼ 0. It is very similarto the von Mises yield surface for low hydro-static stresses (i.e., �h¼ �kk/3<<�y). The maindifference is that, owing to the presence of avoid, plastic yielding takes place under largehydrostatic stress whatever the value of thedeviatoric stress (effective stress) while a vonMises material is incompressible and does notyield plastically under pure hydrostatic stresses.As soon as voids have started to nucleate, theaccumulation of plastic deformation causes theenlargement of the voids and an increase of thevoid volume fraction which, by stating volumeconservation, writes

_fgrowth ¼ ð1� fÞ_epii ½73�

where e_ijp are the ij components of the overall

plastic strain rate tensor. The full evolution lawfor the void volume fraction is written as

_f ¼ _fgrowth þ _fnucl ½74�

Expressions for the void nucleation rate f_nuclhave been discussed in the previous section.The hardening behavior of the matrix materialis related to the overall stress and plastic strainrate through the energy balance initially pro-posed by Gurson (1977):

sy _epyð1� fÞ ¼ s_ep ½75�

where the mean yield stress of the matrix mate-rial �y is a function of ey

p:�y X h(eyp).

In order to calculate the evolution of the twovariables f and �y as well as the stress state, useis made of an associated flow rule:

_epij ¼ _g@�

@sij½76�

The stresses are calculated from

_sij ¼ Cijkl _ekl � _epkl� �

½77�

where the Cijkl are the elastic moduli. The deri-vation of an analytical expression for thetangent moduli Lijkl relating the stress rates �ijto the strain rates e_kl

_sij ¼ Lijkl _ekl ½78�

can be found by imposing the consistencycondition

_� ¼ 0 ½79�which leads to

sij ¼1

Lijkl

Lelijkl � Lel

ijmn

�@�=@smn

�Lelklmn

�@�=@smn

�� @�=@smnð ÞLel

mnrs

�@�=@srs

��@�=@f

��1� f

��@�=@stt

�� smn

�@�=@smn

��@�=@sy

��@�y=@Sp

y

��1�� _"kl ½80�

where the derivatives of the flow potential canbe easily determined analytically and Lijkl

el arethe elastic constants. In this tensorial form, theelastic moduli write for an elastic material

Lelijkl ¼

E

1þ v

1

2dikdjl þ dildjk� �

þ v

1� 2vdijdkl

� �½81�

where is the Kronecker operator.Many extensions of the Gurson model have

been proposed in the literature. The format ofthe model remains basically the same. Thevolume conservation of the matrix material[73] is almost always invoked as well as theassociated flow rule [76]. The extensionsmainly differ by the formulation of moresophisticated yield surfaces, extra evolutionlaws for the new variables introduced in themodel, and/or new hardening laws for theoverall material. A relatively comprehensivesurvey of these extensions is provided inTable 7 (see also Chapter 2.03 for detailsabout some of the extensions). Note alsothat extensions of the Gurson model to poly-meric materials have been developed(Steenbrink et al., 1997; Pijnenburg and VanDer Giessen, 2001; Challier et al., 2006).

(iv) A brief on Rousselier model

Although the Rousselier model (Rousselier,1987) has been developed from general thermo-dynamical principles, its formulation remainsclose to the Gurson model. The yield surfaceof the Rousselier model writes

� Xse

1� fþDrs1 f exp

jshj3ð1� fÞs1

� �� sy ¼ 0 ½92�

where �1 and Dr are adjustable parameters.The normality rule (76) is also assumed. It isinteresting to outline some differencesbetween the Gurson and Rousselier models.Under pure shear conditions, that is, �h¼ 0,the Rousselier model predicts a damage evo-lution (as the normal to the yield surface in a�h� �e plane does not coincide with the �eaxis), whereas, in the absence of nucleation,the Gurson model does not lead to damagechange. Under pure hydrostatic stress states,that is, �e¼ 0, the Rousselier yield surface hasa vertex which implies that at high stress

Table 7 Survey of the extensions of the Gurson model

Purpose of the extensionand main references Brief description

Improve global accuracy Better comparison toward unit cell calculations can be achieved by addingadjusting parameters q1 and q2 into the expression of the yield surface:

Tvergaard (1981) � X�2es2yþ 2q1fcosh

q22

skksy

� �� 1� ðq1fÞ2 ¼ 0 ½82�

These q factors can be used to correct for the effect of void shape changes, of largestrain-hardening exponent, and of void interaction. However, the use of thisversion of the Gurson model should preferably be restricted to the modeling ofvoids that are initially more or less spherical

Perrin and Leblond (1990) The theoretical value of q1¼ 1.47 for spherical voidsKoplik and Needleman(1988)

Calibrations of the q factors toward unit cell calculations have been performed byseveral authors. Typically, q1� 1.5 and q2� 1.More detailed calibration is given inthe following table, providing the best set (q1,q2) for different �0/E and n

Faleskog et al. (1998) n �0/E¼ 0.001 �0/E¼ 0.002 �0/E¼ 0.004

0.025 (1.88,0.956) (1.84,0.977) (1.74,1.013)0.05 (1.63,0.95) (1.57,0.974) (1.48,1.013)0.1 (1.58,0.902) (1.46,0.931) (1.29,0.982)0.15 (1.78,0.833) (1.68,0.856) (1.49,0.901)0.2 (1.96,0.781) (1.87,0.8) (1.71,0.836)

Improved modeling ofstrain hardening

The validity of the yield surface [72] and of the energy balance [75] deteriorates whenthe strain-hardening exponent becomes large (typically when n>0.2, see thechange of the q factors in the table above when n¼ 0.2). An improved yieldfunction has been worked out, involving two yield strength indices �y1 and �y2:

Leblond et al. (1995) � X�2

�2y1þ 2q1fcosh

q22

skksy2

� �� 1� ðq1fÞ2 ¼ 0 ½83�

supplemented by closed-form evolution laws for �y1 and �y2

Kinematic hardening Various extensions of the Gurson model to kinematic hardening have beenformulated. Typical yield surface writesMear and Hutchinson

(1985); Leblond et al.(1995); Arndt et al.(1997); Besson andGuillemer-Neel (2003);Muhlich and Brocks(2003)

� X3

2

ðsij � aijÞðsij � aijÞ�2y

þ 2fcosh1

2

ðskk � akkÞsy

� �� 1� ðq1fÞ2 ¼ 0 ½84�

where sij is the stress deviator and aij is the deviator of the symmetric tensorspecifying the center of the yield surface. The model must be supplemented byan evolution law for aij, see the references

Plastic anisotropy The first approximate way to extend the Gurson model to plastic anisotropy is bycalculating the effective stress based on an anisotropic yield criterion (e.g., Hill,1948, 1950; or more advanced models by Barlat et al., 1991, 1997; Karafillisand Boyce, 1993; Li et al., 2003; Bron and Besson, 2004; Van Houte and VanBaele, 2004) while preserving the expression of the yield surface [72]. For theHill model, the effective stress writes

Doege et al. (1995); Liaoet al. (1997); Rivalin et al.(2001); Grange et al.(2000); Benzerga et al.(2002); Wang et al. (2004)

�e ¼3

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih1s

211 þ h2s

222 þ h3s

233 þ 2h4s

212 þ 2h5s

223 þ 2h6s

231

2

q½85�

where the hi’s are the anisotropy coefficients and s is the deviatoric stress tensorexpressed in the orthotropic reference frame

As a matter of fact, the expression of the yield surface also changes whenreworking completely the Gurson model with an anisotropic plastic theory. Forthe Hill model, it becomes

(Continued )


Table 7 (Continued )


Benzerga and Besson(2001); Benzerga et al.(2004b)

� X�2es2yþ 2fcosh

1

h

skksy

� �� 1� f2 ¼ 0 ½86�

where h is equal to

h ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8

5

h1 þ h2 þ h3h1h2 þ h2h3 þ h3h1

þ 4

5

1

h4þ 1

h5þ 1

h6

� �2

s½87�

The error made by keeping the form [72] rather than using [86] could also becorrected by adjusting the q2 factor as a function of the anisotropy coefficients

Void shape effect A micromechanical analysis of a spheroidal void embedded in a confocalspheroidal cell can be performed along the lines of the Gurson analysis leading,after complex mathematical developments, to the following yield surface

Gologanu et al. (1993,1994, 1997)

� XC

�2ys9þ ZsghX�� 2þ 2qðgþ 1Þðgþ fÞcosh k

sghsy

� �� ðgþ 1Þ2 � q2ðgþ fÞ2 ¼ 0

½88�

where �9 is the deviatoric part of the Cauchy stress tensor; �hg is a generalized

hydrostatic stress defined by �hg¼� : J; X is a tensor associated to the void axis

and defined by 2/3 ez ez� 1/3 ex ex� 1/3 ey ey; J is a tensor associated tothe void axis and defined by (1� 2a2)ez ezþ a2ex exþ a2ey ey; k k is thevon Mises norm; C, �, g, k, a2 are analytical functions of the state variablesS¼ ln W¼ ln(Rz/Rx) and f, q is a parameter that has been calibrated as afunction of f0, W0, and n (see references for complete expressions). Forspherical voids (i.e., S¼ 0), the yield surface is identical to the initial Gursonsurface [72].

Pardoen and Hutchinson(2000)

Benzerga et al. (2004) The evolution law for S writes

_S ¼ 3

2ð1þ h1Þ _ep �

_epkk3

d� �

: Pþ h2 _epkk ½89�

Gologanu et al. (1997) where analytical expressions for h1, h2 as a function of S and f can be found in thereferences, P is a projector tensor, defined by ezez and ez is a unit vectorparallel to the main cavity axis; is the Kronecker tensor

Scheyvaerts et al. (2006a);Kaisalam and PonteCastaneda (1998)

The last new variable which enters the model is the orientation of the main voidaxis ez. The initial proposal was to rotate the void axis with the material axis.More correct rotation laws based on homogenization theory have beenformulated and assessed toward unit cell calculations (e.g., Figure 42).

Rate dependency andviscoplasticity

The most simple extension of the Gurson model to rate-sensitive behaviors of thetype

�

�0¼ 1þ E

�0"p

� �n

1þ "p

"0

� �1=m9

½90�

Peirce et al. (1982, 1984);Tvergaard (1990);Moran et al. (1991);Haghi and Anand (1992)

is to retain the expression of the yield surface [72] (or of one of its extension), and useit as a flow potential. Expression [75] allows to directly calculate the effectiveplastic strain rate. This type of extension is frequently used for numerical reasons,while keeping m9 very small (<0.01). When the rate sensitivity exponent m9departs significantly from 0, the expression of the flow potential is not relevantanymore. The flow potential for spherical voids which writes

(Continued )


Table 7 (Continued )


Leblond et al. (1994)

� X�2e þ f Fskk2

� �þm9� 1

m9þ 1

1

F skk2

� �" #

� 1�m9� 1

m9þ 1f2 FðxÞ ¼ 1þ jxj

m9þ1m9

m9

!m9

½91�

Flandi and Leblond (2005);Klocker and Tvergaard(2003)

was developed for arbitrary nonlinear viscous materials (without hardening, i.e.,n¼ 0) and is identical to the Gurson model when m9¼1. Extensions tospheroidal voids have also been proposed by combining [88] and [91]

Two populations Leblond and co-workers have developed models that account for the presence ofa second population of voids around large primary voids which lead to largervoid growth rates due to the presence of ruined materials around the largeprimary voids or to the successive coalescence of these small voids with thelarge void

Perrin and Leblond (2000);Enakoutsa et al. (2005)

Presence of the particleinside the void

The presence of a hard particle within the void prevents the void to contract in thedirection transverse to the main loading direction under low stress triaxialityconditions. This effect can only be accounted for with a model which alreadyincorporates void shape changes

Siruguet and Leblond(2004a, 2004b); Maireet al. (2005)

Effect of the void size The effect of the initial void size has been accounted for by reworking the analysisof Gurson in the case of a so-called Taylor-type dislocation-based straingradient plasticity model (Gao et al., 1999; Huang et al. 1999; Qiu et al., 2001,2003)

Wen et al. (2005)


triaxiality ratios the plastic deformation ten-sor retains a non-zero shear component.

It has been shown that the use of theRousselier model together with a rate-depen-dent flow law for the matrix does not lead torealistic results in terms of void growth rates. A‘modified’ Rousselier model has been proposedby Tanguy and Besson (2002) which keeps thespecific shape of the Rousselier yield surface(i.e., damage growth under pure shear, vertexunder pure pressure). The yield surface is nowexpressed as

� Xse

ð1� fÞsyþ 2

3Drf exp

qR2

jshjð1� fÞsy

� �� 1 ¼ 0

½93�

where DR and qR are the damage parametersused instead of �1 and Dr. Another modifica-tion has been proposed by Sainte Catherineet al. (2002) where �1 is now a function of theeffective plastic strain.

When properly calibrated, the Rousseliermodel provides predictions as good as the

regular version of the Gurson model with theadvantage of better capturing localization.

2.06.3.5 Void Coalescence

During the void growthphase, plastic deforma-tion is homogeneous at a scale of a representativevolume element, that is, at the scale of a volumecontaining one or a few voids depending on theheterogeneity of the microstructure. Except forvoids clusters present in highly heterogenousmaterials (e.g., Devillers-Guerville et al. (1997)and Achon (1994)), the interaction during thevoid growth phase between voids in industrialalloys is usually weak (e.g., Needleman andKushner (1990) and Huang (1993)) due to thelow initial porosity, and the effect of damage onthe global behavior can be taken into accountwith a single porosity parameter as explained inthe previous section about void growth in metals.Conversely, the coalescence of voids leads to atransition from a stable phase of diffuse plasticdeformation to a localized mode of plastic defor-mation within the ligament between the voids


located in the most critical region, with materialoff the localization plane usually undergoing elas-tic unloading. Void coalescence leads to theformation of a macroscopic crack that can thenpropagate through the material with limitedamount of additional external work. This localplastic localization phenomenon called void coa-lescence is the physical transition to fracturecontrolling the resistance of a metallic alloy toductile fracture. To establish precise terminology,‘void coalescence’ has been reserved for the partof the void enlargement evolution after the transi-tion to the localized mode of plasticity, and ‘voidgrowth’ is used to characterize void enlargementbefore localization.The void coalescencemechan-ism is a localization mechanism at the scale of thevoid size that must thus be distinguished from thelocalization in a shear band at the ‘mesoscopic’scale, as discussed in Section 2.06.3.2, whosewidth is, in general, not controlled by the voidspacing. The confusion can arise because whensuch a mesoscopic-localized band develops, thevoid growth rate inside the band increases andfracture occurs rapidly with small additionalincrease of remote displacements.

In ductile materials, void coalescence occursafter significant amount of void growth, typi-cally when the diameter of the void hasdoubled. In less ductile materials, such as inhigh-strength aluminum, coalescence can startright after the nucleation of the voids(Worswick et al., 2001) and the mechanism issaid to be nucleation controlled.


The schematic of Figure 20 shows a clearchange of slope after the maximum load. Thissharp slope change can be associated to the initia-tion of fracture. The initiation of fracture consistsin the linking of two or a few voids in the mostdamaged region of the material. In an overallload/displacement plot such as shown inFigure 20, fracture initiation occurs soon afterthe onset of coalescence due to the highly localizedcharacter of the process. It is thus possible todetect quite accurately the occurrence of the firstcoalescence events on the global stress–straincurve. This point has been confirmed experimen-tally by several authors (e.g., Benzerga, 2000). Inductile alloys, fracture initiation occurs after themaximum load while in less ductile alloys, frac-ture initiates during the rising part of the load.


The most interesting experimental observa-tions of the void coalescence mechanisms areobtained by interrupting mechanical tests just

after cracking initiation and by preparingmetallographical sections. This is a tediouswork which requires trial and errors and repe-titive polishing sequences to reveal the region ofthe material involving the coalescing cavities.The ductile fracture process can also be investi-gated using ‘academic’ systems in which holesare drilled into plates (see Dubensky and Koss,1987; Magnusen et al., 1988, 1990; Becker andSmelser, 1994; Weck et al., 2006). An example isprovided in Figures 43a–43f for differentarrangements of cavities. The cavities arecylindrical. These two types of experimentsallow the study of the two main modes of coa-lescence in an ideal situation.

The first mode of coalescence, shown in thesequence a–c–e, is the ‘internal necking’ modeof coalescence, where the ligament between thetwo voids shrinks with a shape typical of anecking process. During the process of coales-cence, the voids evolve toward a diamondshape. The fracture profile will be flat orientedat 90� with respect to the main loading direc-tion. Another example of internal necking,now for initially rounded voids, is shown inFigure 43g. This micrograph has beenobtained by carefully polishing prestrainedsteel samples.

The second mode of void coalescence shownin the sequence b–d–f consists in a ‘shear loca-lization’ between large primary voids. It isfrequently observed in high-strength materialswith low or moderate strain-hardening capa-city. The transition from shear localization tointernal necking has been frequently observedin aluminum alloys when moving from a low-constrained to highly constrained geometry (seeAsserin-Lebert et al., 2005; Bron et al., 2002),when increasing the strain-hardening capacityby specific heat treatment (Asserin-Lebertet al., 2005; Bron et al., 2002) or when decreas-ing the loading rate (Rivalin et al., 2001a). Thetransition between shear localization and inter-nal necking can be related to the transitionbetween slant fracture to flat fracture mode.This aspect will be discussed in more details inSection 2.06.3.8. Another micrograph showingthe shear localization mechanism is provided inFigure 43h, again on prepared metallographicsections of pre-deformed samples. The micro-graph shows inside the shear localization bandthe presence of secondary small voids. Thismode of coalescence is frequently called ‘voidsheeting’.

Nowadays, 3-D tomography is becoming themost versatile method to investigate in details theprocess of void coalescence (e.g., Babout et al.,2004a, 2004b; Maire et al., 2005). Nevertheless, atthis time, the resolution of the method is not yetsufficient to analyze in details the submicron

(e)

(c)

(f)(d)

(a)(b)

(g) (h)

5 μm

(i)

10 μm

450 μm 35 μm 140 μm

100 μm35 μm

450 μm

Figure 43 Micrographs demonstrating the different void coalescence mechanisms in metals. The loadingdirection is vertical except for (i), from (a)–(f) the experiments have been made on perforated aluminumsheets (courtesy of A. Weck and D. S. Wilkinson) with different arrangements of voids: the arrangement(a) leads to internal necking (see (b) and (c)) and the arrangement (d) leads to coalescence in shear (see(e) and (f)); g, an internal necking process in steel showing the presence of one or two secondary voids inthe ligament; h, a classic micrograph of void sheet mechanisms by Cox and Low (1974) withmany secondary voids along the microshear band; i, an example of void coalescence in column or‘necklace’ in copper. a–f, see Weck, A., Wilkinson, D. S., Maire, E., and Toda, H. 2006. 3Dvisualization of ductile fracture. Proceedings of the Euromat Conference. Adv. Eng. Mater. 8, 469–472;g, source: Benzerga, A. 2000. Rupture ductile des toles anisatropes: Simulation de la propagationlongitudinale dans un tube pressurise. Ph.D. thesis, Ecole Nationale Superieure des Mines de Pariso; h,source: Cox, T. B. and Low, J. R. 1974. An investigation of the plastic fracture of AISI 4340 and 18nickel-200 grade maraging steels. Metall. Trans. A, 5, 1457–1470; i, source: Pardoen, T. 1998. DuctileFracture of Cold-Drawn Copper Bars: Experimental Investigation and Micromechanical Modeling.Ph.D. thesis, Universite Catholique de Louvain, Belgium.



mechanisms taking place in industrial alloyswithin the ligament between primary voids.

The third mode of void coalescence, called‘necklace coalescence’, is more anecdotic. It hasbeen observed in between row of voids gatheredin elongated clusters. It consists in a localizationprocess in a direction parallel to the main load-ing axis. An example is provided in Figure 43iwith a column of voids resulting from a longcluster of copper oxide particles within a coppermatrix and giving rise to multiple void coales-cence in a direction transverse to the mainloading direction.

Hence, plastic localization at the scale of thevoid spacing can occur at any orientation rela-tive to the principal straining axis depending onthe orientation of the ligament between the twocoalescing voids: tensile (i.e., normal separa-tions) or shear localizations are possible to themain direction (see also Figure 35d).

Fractographic analysis can also provideinteresting information about the coalescenceprocess. The internal necking coalescencebrings about the flat dimpled fracture morphol-ogy widely observed in a large range of ductilematerials under a wide range of stress states.Fractography also provides information aboutthe final linking process. For instance,Figure 44 shows the presence of secondaryvoids in between large primary voids whichhave contributed to the linking process. Thepresence of a second population of smallervoids inside the intervoid ligament (seeFigure 43h) has been observed in several steelsand aluminum alloys (Cox and Low, 1974;Hancock and Mackenzie, 1976; Achon, 1994;Asserin-Lebert et al., 2005; Gallais et al., 2006)and precipitate ligament failure before impinge-ment of the large voids.

Figure 44 Fractography of a 6060-T4 Al alloyshowing large dimples originating from theintermetallic particles and small dimples resultingfrom the decohesion of Mn dispersoıds.

2.06.3.5.3 Void cell simulations

The different modes of void coalescence havebeen simulated numerically using FE void cellsimulations.

Coalescence by internal necking is observedin axisymmetric void cell simulations (due to akinematical constraint which prevents shearlocalization) as well as in 3-D cell calculations,when the strain-hardening exponent is not toolow, when the stress triaxiality is moderate orhigh, and when the periodic void packing doesnot involve closest neighbors near to 45� fromthe main loading direction (e.g., Richelsen andTvergaard, 1994). The main characteristic ofthis tensile mode of coalescence is a sharp tran-sition into an uniaxial straining mode of thevolume element imposed by the material out-side the localization band which unloadselastically and behaves as a rigid block (seeKoplik and Needleman, 1988; Becker et al.,1989a; Brocks et al., 1995a, and Richelsen andTvergaard (1994)).

Figure 45 shows thevariationof the radial strainer as a function of the axial strain ez predicted byaxisymmetric void cell calculations performedwith three different initial void aspect ratios andunder three different applied stress triaxialities.The transition to an uniaxial straining mode isalmost instantaneous (see also Koplik andNeedleman (1988) for initially oblate voids, andBrocks et al. (1995b) andRichelsenandTvergaard(1994) for 3-Dcomputations). This transition con-stitutes a direct indicator of localization and isused for quantifying the strain at the onset ofvoid coalescence. The onset of coalescence canalso be detected by a sharp change in the overallstress–strain curves (see Figures 38 and 39a).

Void coalescence induces an increase of thevoid growth rate and a transition in the void

–0.6

–0.5

–0.4

–0.3

–0.2

–0.1

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4

W 0 = 6

W 0 = 1

W 0 = 1/6

ε r

εz

T = 3

T = 1

T = 1/3

Figure 45 Void cell results for f0¼ 10�2, l0¼ 1,�0/E¼ 0.002, n¼ 0.1, and W0¼ 1/6, 1, 6 at T¼ 1/3, 1,3 showing the variation of the radial strain as a functionof axial strain (Pardoen and Hutchinson, 2000).


shape evolution. After the onset of void coales-cence, the radial growth is significantly largerthan the axial growth. The end of the coalescenceprocess in a real material usually consists in thefailure of the remaining ligament (by microclea-vage, crystallographic shearing, or with the helpof a second population of smaller voids) ratherthan radial void growth until impingement. Thus,after the onset of void coalescence, the voidexpands rapidly in the radial direction until thefinal failure of the ligament. During this process,axial void growth remains small. Consequently,the mean depth of the void (i.e., Rz) measured onthe fracture profile is a good approximation ofthe void half-height at the onset of void coales-cence. Hence, if the mean distance betweendimple centers (ideally corrected by the value ofthe overall transverse strain) provides informa-tion about the spacing between particles givingrise to void nucleation, the depth of the voids isthe most pertinent dimension to measure on afracture profile to gain information about thevoid coalescence conditions.

Void cell simulations provide important infor-mation about the state of the damage at theonset of void coalescence. Figure 46 shows thatthe value of the porosity at the onset of voidcoalescence varies with the stress triaxiality, theporosity, and initial void shape (Pardoen andHutchinson, 2000). If the initial porosity issmall enough (typically, f0<0.001), the criticalporosity can reasonably be considered as inde-pendent of the stress triaxiality for mostpractical purposes. The value of the strain-hard-ening exponent also affects the critical porosity.

0

0.02

0.04

0.06

0.08

0.1

0.12

stress triaxiality

f c

0 1 2 3 4 5 6

W 0 = 1

W 0 = 1/6

W 0 = 6

λ 0 = 1, n = 0.1

f 0 = 10–4

f 0 = 10–2

Figure 46 Stress triaxiality dependence of the criticalporosity at the onset of coalescence, for differentinitial porosity and void shapes, with n¼ 0.1 andl0¼ 1. From Pardoen, T. and Hutchinson, J. 2000.An extended model for void growth and coalescence.J. Mech. Phys. Solids 48, 2467–2512.

As shown in Figure 47, by elongating the cell(i.e., by increasing l0 while keeping the voidspacing constant), the zone undergoing elasticunloading becomes larger and, in terms of theoverall strain, the unloading slope during coales-cence becomes more and more steep. It isremarkable to note the straight unloading slopeduring coalescence, which has been verified byTvergaard (1997) up to very large strains usingremeshing techniques. The reason for the coales-cence to take place earlier when the cell lengthincreases, although the initial porosity decreasesproportionally, is caused by the elevation of thestress in the ligament which favors the localiza-tion of the deformation in the ligament (see laterthe section on coalescence models).

The effect of the presence of a second popula-tion of voids has been investigated using cellcalculations by different authors. Tvergaard(1998) and Faleskog and Shih (1997) treatedthe secondary voids explicitly in the finite ele-ment mesh using 2-D plane strain conditions.Several authors have modeled the second popu-lation using a Gurson-type model for the matrixsurrounding the primary voids (e.g., Brockset al., 1995a, 1995b; Fabregue and Pardoen,2006). Figure 48 shows the overall stress–straincurve obtained with a fixed total initial voidvolume fraction f10þ f20¼ 1.5� 10�3, oncewithout secondary voids (f10¼ 1.5� 10�3,f20¼ 0) and once with secondary voids(f10¼ 1� 10�3, f20¼ 0.5� 10�3). The influenceof the presence of a second population(nucleated here immediately) on the onset ofcoalescence is very clear as well as the negligibleeffect on the overall stress evolution before coa-lescence (note that the void growth rate of theprimary void is also not affected by the presenceof the second population).

0

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

σ e/σ

0

εe

Lr0 / Rr0 = 3.22 W 0 = 1 T = 1 n = 0.1

f 0 =

0

f 0 =

1.25 × 10–3

λ0 =

16

f 0 =

2.5 × 10–3

λ 0 =

8

f 0 =

5 × 10–3

λ 0 =

4

f 0 =

2 × 10–2

λ 0 =

1

f 0 =

4 × 10–2

λ 0 =

1/2

f 0 =

10–2

λ 0 =

2

Onset of coalescence

Figure 47 Effective stress vs effective strain curvesfor n¼ 0.1, W0¼ 1, a constant Lr0/Rr0 ratio equal to3.22, and T¼ 1. From Pardoen, T. and Hutchinson, J.2000. An extended model for void growth andcoalescence. J. Mech. Phys. Solids 48, 2467–2512.


‘Coalescence in shear’ was first investigatednumerically by Yamamoto (1978) andTvergaard (1981, 1982) using finite elementcell calculations involving a cylindrical voidunder plane strain conditions in the directionof the void axis and periodic boundary condi-tions. The kinematics associated to this loadingconfiguration favors the development of shearlocalization. Coalescence by shear localizationis also observed in 3-D cell calculations invol-ving spherical voids for low strain-hardeningexponents and low stress triaxiality (Richelsenand Tvergaard, 1994). Specific packing of voidswhere the closest neighbors are located at 45�

from the main loading direction also favors ashear coalescence process (Figure 49).

0

1

2

3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

σ Ζ/σ

0

εz

f10 = 1 × 10–3

f20 = 5 × 10–4

1.5 × 10–3

0

εn2 = 0, T = 1, W0 = 1, n = 0.1, σ0/E = 500

Figure 48 Overall stress–strain curve obtained withcylindrical void cell calculations and showing theinfluence of the presence of a second population on theonset of coalescence for a fixed total initial void volumefraction f10þ f20¼ 1.5� 10�3. The stress triaxiality T isequal to 1, the initial void aspect ratio W0¼ 1, and theinitial void distribution parameter l0¼ 1.

ε

p

0.50

0.40

0.30

0.20

0.10

–0.00

Figure 49 Void sheet mode of coalescence involvinglarge primary voids and a population of secondarysmall voids for low strain biaxiality and low strainhardening exponent. From J. Faleskog and C. Shih,1997, Micromechanics of coalescence – I. Synergisticeffects of elasticity, plastic yielding and multi-size-scale voids. J. Mech. Phys. Solids, 45, 21–45.

Compared to the tensile necking mode whichis diffuse by nature and has a thickness scalingwith the ligament length, the thickness of theshear band in the shear coalescence mode is notso well defined. Tvergaard and Needleman(1997) have shown that the length scale asso-ciated to the microshear band scales with thevoid size. Nevertheless, these calculations sufferfrom mesh dependency effects. In reality, theshear localization band thickness is probablycontrolled by a material length scale related todislocation accumulation mechanisms (Fleckand Hutchinson, 1997) rather than by the voidgeometry. Faleskog and Shih (1997) have shownthat when a second population of cavities ispresent in the microshear band joining largeprimary voids, the thickness of the shear bandis controlled by the size of the secondary voids(see Figure 49).

2.06.3.5.4 Models for the onset of voidcoalescence

(i) Generic phenomenological models

When using constitutive models couplingdamage and plasticity, for instance the Gurson(1977) model, there is a strain at which the truestress reaches a maximum value. This maxi-mum corresponds to the point where strainhardening does not compensate anymore forthe damage-induced softening. According toMudry (1982), Koplik and Needleman (1988),or Becker (1987), fracture initiates rapidly afterthis maximum has been attained. This criterionbased on the attainment of a maximum effec-tive stress is valid at low stress triaxiality butnot at large stress triaxiality (Pardoen andHutchinson, 2000) where the coalescence pro-cess, which takes place after the peak stress,requires a large additional strain increment tobe completed. Also, this condition is limited toradial loading conditions.

The most widely employed criterion for theonset of void coalescence states that voidcoalescence starts at a critical porosity(McClintock, 1968, 1971; d’Escatha andDevaux, 1979). Several numerical (e.g., Koplikand Needleman, 1988; Tvergaard, 1990; Brockset al., 1995a, 1995b) and experimental/numer-ical works (e.g., Marini et al., 1985; Becker,1987; Pardoen et al., 1998) have assessed thevalidity of this attractive fracture criterion(see, for instance, Figure 46). For a well-definedmaterial (and microstructure) and range ofstress states, a constant critical porosity isacceptable from a practical standpoint, espe-cially in low-porosity alloys, such as mostmodern steels. However, as discussed next,any general void coalescence model requires

dsplo

ð

wipanPsncvocmilsm(dwal

afmasaGct

εz

Exact solution

Approximate solution for localized flow (e.g., Thomason)

σ z /σ

0

Approximate solution for diffuse flow (e.g., Gurson)

Figure 50 Schematic description of the competitionbetween the two modes of plasticity reflected by thepredictions of the two types of theories.


the introduction of at least some microstruc-tural information related to the void/ligamentdimensions and geometry.

(ii) Micromechanical model for voidcoalescence by internal necking

Coalescence by internal necking is the mostdocumented mode of coalescence and has, up tonow, received the largest attention. The ‘tensilevoid coalescence’ or ‘internal necking’ mode is adiffuse localization at the microscale. Differentcriteria have been proposed in the literature topredict the onset of void coalescence.

The criterion of Brown and Embury (1973) isbased on an explicit micromechanical point ofview: in a perfectly plastic material, internalnecking is considered to start when it is possiblefor 45� microshear bands to connect two neigh-boring voids. The original criterion of Brownand Embury states that, for elongated voids,this condition is met when the radius R isequal to half the distance, L, between the cen-ters of the two voids. From simple geometricalarguments, the criterion can be generalized forany pair of ellipsoidal voids: Lmust be equal to

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2

x þ R2z

pto allow mutual 45� tangents. In

terms of nondimensional parameters, the con-dition of Brown and Embury rewrites:

2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiW2 þ 1

p¼ 1 ½94�

The evolution of the relative void spacing in thecase of spheroidal voids, that is, �¼Rx/Lx, canbe written as

_�

�¼ 1

3_ez � _ex þ

_f

f�

_W

W

!½95�

A more realistic void coalescence model canbe obtained by directly treating the mechanismof tensile plastic localization in the intervoidligaments: diffuse plasticity throughout void-containing cells gives way to localized deforma-tion within the ligament with the materialoutside the ligament unloading elastically.Consider a thin annular cylindrical disk of elas-tic–perfectly plastic material welded to rigidplates and constrained against flow at the outerradius. An approximate analysis for the limitload of this configuration, with associated aver-age true stress, can be carried out along the linesof the Hill (1950) plane strain analysis of a thinplastic layer welded to and squeezed by two rigidplates. The analysis assumes that the material inthe disk moves outward flowing in shear andotherwise supporting only hydrostatic tensionsuch that the three normal stresses are approxi-mately equal (see Pardoen and Hutchinson,2000). More accurate representations have been

eveloped by Thomason (1990), who extensivelytudied the transition to localization for elastic–erfectly plastic solids using slip-line solutionseading to the following condition for the onsetf void coalescence:

sn1� f2Þsy

1

ð1� �2Þ ¼2

3

"a

1� ��W

� �2

þ1:24

ffiffiffi1

�

s #½96�

here the volume fraction of secondary voids f2s equal to 0 in the original criterion and thearameter a has been fitted as a function of theverage value of the strain-hardening exponent: a(n)¼ 0.1þ 0.22nþ 4.8n2 (0 n 0.3); seeardoen and Hutchinson (2000). Criterion [96]tates that coalescence occurs when the stressormal to the localization plane �n reaches aritical value. This critical value decreases as theoids open (W increases) and get closer to eachther (� increases). The dominant parameterontrolling the transition to the coalescenceode is the relative void spacing �. The poros-

ty affects the coalescence indirectly through theink with the void spacing � and through itsoftening effect on the applied stress �z. In aore advanced application of this modelScheyvaerts et al., 2006), the localization con-ition [96] is tested in all possible directionshich requires to generalize the definition of �nd W as a function of the orientation of theocalization plane.The relationship between �z/�0 and the over-

ll strain ez is sketched qualitatively in Figure 50or a cell of material containing a finite porosityade of voids that grow and coalesce throughn internal necking process. At low overalltrain, �z/�0 from [96] is far greater than thectual value, which is better predicted by aurson-type model. This last model was indeedonstructed with velocity fields capturing plas-icity all around the void. However, the actual


solution peaks and falls until localization sets in,and then the actual solution merges with theartificially constrained localized solution. Thisis the transition point, and from this point on,plasticity is localized within the ligament. Thevelocity fields used to build the Gurson modelare not realistic anymore. Internal necking is nota bifurcation phenomenon. Nevertheless, thetransition to localization occurs sharply, andthe competition depicted in Figure 50 is an aidto thinking about the transition condition.

Based on experimental and theoretical stu-dies described previously (e.g., Cox and Low,1974; Hahn and Rosenfield, 1965;, Mariniet al., 1985; Faleskog and Shih, 1997; Haynesand Gangloff, 1997; Tvergaard (1998), Perrinand Leblond, 1990, 2000; Bron et al., 2002;Asserin-Lebert et al., 2005), the nucleationand growth of a second population of voidshas been shown to significantly accelerate thedamage process and cut down the ductility.Motivated by unit cell calculations involvingboth large primary voids and small secondaryvoids (e.g., Tvergaard, 1998; Fabregue andPardoen, 2006), the second population ofvoids is assumed to affect only the coalescencebetween the primary voids and not the growthof the primary voids. If we consider that thepresence of secondary voids within the ligamentbetween two large primary voids does not influ-ence the overall response of the material, theonly possible effect of these small voids isthrough a local softening of the matrix inbetween the primary voids. Hence, the effectof secondary voids on the onset of coalescenceby internal necking can be introduced heuristi-cally into eqn [96] by multiplying the currentmean yield stress of the matrix material by(1� f2). Based on finite element void cell simu-lations treating the matrix as a Gurson material(see Figure 48 and Fabregue and Pardoen,2006), it was found that the localization processis triggered by the local value of f2 in the mostdamaged region of the ligament between twoprimary voids. In other words, if the cell calcu-lations are used to evaluate both f2 in the mostdamaged part of the ligament and �z, �y, W,and �, then eqn [96] provides relatively accuratepredictions of the onset of void coalescence.The most damaged region was always foundto be located next to the surface of the primaryvoids in the minimum section of the ligament(see Fabregue and Pardoen, 2006).Consequently, f2 in eqn [96] should be seen asthe local value of f2 next to the surface of theprimary voids. The condition [96] was validatedtoward the finite element calculations andturned out to accurately capture the onset ofvoid coalescence for a wide range of initialparameters (see Fabregue and Pardoen, 2006).

An estimate of the evolution of f2 in theregion next to the surface of the primary voidis required in order to couple the void coales-cence condition [96] to a void growth model.The state of deformation adjacent to the surfaceof the primary voids can be described in thefollowing way. The local circumferential strainrate on the equatorial section of the void sur-face (z¼ 0) is directly related to the transverserate of the void growth through

_esyy ¼_Rx

Rx¼ 1

3_ez þ 2_ex þ

_f

f�

_W

W

!½97�

The estimation of the local strain rate ezzs is

more complex and requires some approxima-tion to relate it to the evolution of the cavity:

_eszz� _ez þ 2_ex þ_f

fþ 2

_W

W

!p2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2

1

W2þ 1

� �s !½98�

The derivation of eqn [98] and its validation isdiscussed in Fabregue and Pardoen (2006). Atthe surface of the voids, we also have that�rrs ¼ 0. By applying these loading conditions to

a constitutive model involving a void nucleationlaw (e.g., Beremin criterion) and a void growthlaw (e.g., Gurson), one can estimate the evolu-tion of f2 and use the extended coalescencecondition [96] to predict the onset of fracture inthe presence of two populations of voids.

(iii) Void coalescence in shear

The first option for predicting coalescence byshear localization is to analyze the process atthe scale of a row of discrete voids similar to theThomason model for internal necking devel-oped in the previous section. There are only avery limited number of attempts in the litera-ture to formulate such a closed-form condition.McClintock (1968) stated that a shear localiza-tion band sets in between two ellipsoidal voidswhen

1

se

dsedee¼

ffiffiffi3

8

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þW2

p 1þW0

1þW

� �2

� f

f0

1� f01� f

� �2=3

�0 expðeIÞ ½99�

where eI is the maximum principal strain. Thiscoalescence model has been recently used byRagab (2004) for analyzing various experimen-tal data. Note that a simple upper-boundcondition has also been proposed by Richelsenand Tvergaard (1994) and compared to finiteelement cell calculations.

The second option is to analyze the problemfrom a slightly different perspective by addressingthe prediction of a shear localization band within


the framework of constitutive model for porousmaterials. Such kinds of calculations have beeninitially proposed by Yamamoto (1978) and thenby several authors (e.g., Saje et al., 1982; Panet al., 1983; Ohno and Hutchinson, 1984;Mear and Hutchinson, 1985; Tvergaard,1982, 1987; Tvergaard and Van der Giessen,1991). Early approaches involved the incor-poration of heterogenous zones of higherinitial porosity to trigger the shear localiza-tion. Following the works of Rudnicki andRice (1975), Doghri and Billardon (1995)have proposed to compute the bifurcationcondition to detect cracking initiation andthe direction of the localization band (seeBesson et al., 2003). Theoretically, a shearband analysis in a continuum described witha damage model is different than solving alocalization problem with the exact geometryand constraint resulting from the presence ofdiscrete voids. Nevertheless, for practical pur-poses, the two approaches seem to lead tocomparable predictions. Proper modeling ofshear coalescence is still a matter of opendebate. Ideally, a micromechanics-based ana-lysis of ducticle fracture should rest onmultiple coalescence criteria: one for internalnecking and one for void sheeting.

(iv) Coalescence in columns

Note finally that Gologanu et al. (2001) haveworked out theoretically the third mode of coa-lescence called ‘coalescence in columns’ inwhich voids concentrate along vertical columnsinstead of horizontal layers. Evidences of coa-lescence in columns can be found in the thesis ofBenzerga (2000). This mode of coalescence canbe of importance in the transverse delaminationof plates having elongated bands of closelyspaced particles.

2.06.3.5.5 Models for the coalescence process

For low-stress triaxiality loadings, the energyassociated to the final enlargement of the voidduring the coalescence stage is small withrespect to the energy associated to the voidgrowth stage. The onset of void coalescence isan excellent indicator of the fracture strain ofthe material. At large stress triaxiality, typicalof the state of loading in the region located infront of a static or growing crack, coalescencestarts at small strains and then the amount ofenergy spent during coalescence can be signifi-cant. Hence, a full constitutive model for theresponse of the material during coalescence isnecessary for simulating crack propagation.Four types of approaches can be found in theliterature. The first two have been developed in

order to provide simple numerical implementa-tion, the third one is specific for capturing voidsheeting mechanisms, and the fourth is specificto the internal necking process.

(i) Phenomenological accelerationof the void growth rate

Tvergaard and Needleman (1984) (see alsoNeedleman and Tvergaard, 1984) have pro-posed to simulate the coalescence process byartificially accelerating the rate of increase ofthe porosity in the following way:

f* ¼f if f<fc

fc þ dðf� fcÞ if f � fc

(½100�

where fc is the porosity at the onset of coales-cence which is either prescribed as a ‘materialparameter’ or predicted from a void coales-cence condition, for instance [94] or [96] (e.g.,Zhang and Niemi, 1995). The factor is anadjusting parameter which varies typicallybetween 3 and 8 and which can be fitted onvoid cell calculations or on experimental data.The idea is to preserve the format of the Gursonmodel after the onset of coalescence which isnumerically very convenient. This model doesnot distinguish between different modes of coa-lescence. Note that Benzerga (2002) proposed atheoretical analysis for estimating .

(ii) Numerical reduction of the load-carryingcapacity

Several authors (Xia et al., 1995) proposed,in the framework of a finite element implemen-tation, to ramp down the stress in a givennumber of time increments after a critical por-osity fc is attained. This method is based on theassumption that an accurate description of thecoalescence process is unimportant, which isprobably a good approximation for many pro-blems. There is no control on the energydissipated during coalescence. Note that theseminal simulations of crack propagationusing a ‘noncoupled’ damage model (e.g., theRice and Tracey model) had the same spirit,i.e., neglecting the coalescence phase by simplyreleasing nodes when a critical damage valuewas reached (d’Escatha and Devaux, 1979).

A more advanced implementation has beendeveloped by Shih and co-workers (e.g., Xiaand Shih, 1995a, 1995b, 1996; Ruggieri et al.,1996; Gao et al., 1998a; Gullerud et al.,2000). Basically, after the critical porosity isreached, a cohesive zone representation (seeSection 2.06.3.7.1) is used to ramp down thestress to zero (see Figure 51). In this

(b) (a)

(c)

γ = fraction of extinct cell forces remaining on nodes after f – fg

00

1

D – D0λ D0

γ

D0D

Figure 51 Linear traction–separation model withrelease fraction l. a, Cell at extinction value of theporosity height is D0; b, force release completed atD¼ (1þ l)D0; c, linear traction–separation model.Source: Gullerud, A. S., Gao, X., Dodds, R. H., Jr.,and Haj-Ali, R. 2000. Simulation of ductile crackgrowth using computational cells: Numericalaspects. Eng. Fract. Mech. 66, 65–92.


approach, the stress decreases with applieddisplacement and there is thus a control ofthe energy dissipated during coalescence. Theadjusting parameter is the slope of theunloading ramp.

(iii) Model for coalescence in shear

Besson et al. (2003) have proposed to specifi-cally model the void sheet mechanism in theframework of the finite element technique byprescribing a critical porosity (condition for theonset of coalescence) above which secondaryvoids are nucleated. This nucleation of a secondpopulation naturally accelerates the rate ofincrease of the overall porosity and triggers theoccurrence of a localization band. Thisapproach, which does not really impose anyextra coalescence condition, allows, for instance,reproducing quite well the cup-and-cone frac-ture profile. The localization orientation agreeswith the prediction of a bifurcation analysisusing the Rudnicki and Rice (1975) condition.

(iv) A full constitutive model for the internalnecking process

As a matter of fact, the solution [96] is notonly a condition for the onset of coalescencebut also provides the evolution of the stressduring the full coalescence process as a function

of the evolution of the shape and spacing of thevoids. It constitutes thus the basis for elaborat-ing a constitutive model for the full coalescenceprocess. This idea has been followed byPardoen and Hutchinson (2003) and Benzergaet al. (2004b). In order to use solution [96] as afull constitutive model, it was transformedheuristically into a yield surface (Pardoen andHutchinson, 2003):

�coalescence Xk s9 ksyþ 3

2

jshjsy� FðW; �Þ ¼ 0 ½101�

with

FðW; �Þ X 3

2ð1� �2Þ a

1� ��W

� �2

þ b

ffiffiffi1

�

s" #½102�

A more physical generalization of [96] instress space to occur in a well-defined orienta-tion and to lead to a large constraint in thetransverse direction; see Scheyvaerts et al.(2005). The yield surface is supplemented bythe normality rule to evaluate the plasticstrains:

_epij ¼ gd�

dsij½103�

Evolution laws for the geometrical variables,that is, the void aspect ratio, W, the porosity,f, and the relative void spacing, �, as well as forthe current mean yield stress of the matrixmaterial have been formulated based on simplegeometrical rules:

_W ¼ 3Wð2�2 � 1Þ4f

_epe ½104�

_� ¼ �ð3� 2�2Þ4f

_epe ½105�

_f ¼ ð1� fÞ_epkk ½106�

_sy ¼@sy@epy

�2

f_epe ½107�

(see Benzerga et al. (2004b) for more advancedvoid shape law during coalescence, accountingfor the evolution toward a diamond shape).Figure 52 illustrates the competition betweenthe two modes of plastic deformation (diffuseplasticity during void growth vs localized plas-ticity during void coalescence), generalizing themessage of Figure 50 for a general stress state.The yield surfaces and loading history corre-spond to a high constraint situation (aconstant stress triaxiality of 3) relevant toa material element in front of a crack tip.Figure 52a corresponds to elastic loading: thestress state lies within the elastic domain of thetwo yield surfaces. As shown in Figure 52b,

1

0

1

0

1

0

1

0

1

0

Φcoalescence

Φgrowth

Φvon Mises

Currentstressstate

Slope–3/2

∑h/σf

∑eq

/σf

∑h/σf

∑eq

/σf

∑h/σf∑h/σf

∑eq

/σf

∑eq

/σf

∑h/σf

∑eq

/σf

(a) (b)

(d)(c)

(e)

Figure 52 Transition from: (a) elastic behavior, to (b), (c) plasticity/void growth, to (d), (e) plasticity/voidcoalescence in terms of the variation of the yield surfaces and loading point as a function of increasing straining.Calculations performed for a constant stress triaxiality equal to 3.


the first yield surface to be reached is Fgrowth. Inthe beginning of the deformation, the voids aresmall and the spacing is relatively large resultingin a diffuse plastic deformation. With increasingdeformation,Fgrowth first tends to expand due tohardening and then contract due to void growthsoftening. Void growth and ligament reductionalso induces a contraction of Fcoalescence.Figure 52c corresponds to an overall strain justbefore the onset of coalescence and Figure 52d toan overall strain just after the onset of coales-cence. When the two yield surfaces intersect atthe current loading point, the transition to

coalescence occurs. With increasing deforma-tion, the coalescence yield surface tends tocontract very rapidly toward the zero stressstate. Figure 52e shows the two yield surfacesfar in the coalescence stage.

This coalescence model has been shown tocapture quite well the drop of the load-carryingcapacity after the onset of coalescence (seePardoen and Hutchinson, 2000). However,numerical problems related to the jump fromone yield surface to another make the implemen-tation within robust finite element proceduresvery complex. Furthermore, it is difficult to


formulate rules for the update of the quantitiesentering the void growth model during the coa-lescence stage, in case the coalescence processstops and the void growth resumes.

2.06.3.6 Fracture Strain of Metals

The understanding and modeling of the dif-ferent steps of the process of nucleation,growth, and coalescence of voids have reacheda level of maturity that makes possible the for-mulation of fairly generic views about ductility.The ductility is defined here as the fracturestrain of a representative volume element ofmaterial subjected to homogeneous loadingconditions. The fracture strain can be measuredexperimentally directly on the fracture surfacebased on the reduction of the cross-sectionalarea. In the ‘first subsection’, a closed-formmodel for the fracture strain will be developedbased on the analytical elementary models pro-posed in the last three sections. In the ‘secondsubsection’, more complex effects or morequantitative predictions will be highlighted byproviding results obtained indifferently eitherwith FE void cell calculations or with anadvanced damage model (the idea is that theseadvanced damage models have been success-fully assessed by comparison with thenumerical unit cell simulations). These genericpredictions will be quantitatively or sometimesonly qualitatively assessed toward selectedexperimental data. For the sake of simplicity,we will always assimilate the void coalescencestrain as the fracture strain which is an approx-imation because the material keeps deforming,but only very moderately, after the onset ofcoalescence up to final material separation.

2.06.3.6.1 Simple closed-form estimatesof the fracture strain

A first approximate expression for the frac-ture strain can be formulated by assuming thatvoids nucleate at ee¼ ec with a spherical shape,grow spherically, and coalesce following theBrown and Embury condition [94]. Let us con-sider an average stress triaxiality T along adeformation history which is characterized bya strain ez in the main loading direction, andey¼ bex in the two transverse directions withjexj � jeyj. The principal axis remains fixed dur-ing deformation. Elasticity is neglected. Theinitial porosity (f0) is small, allowing the use ofthe isolated voids assumption. Using volumeconservation, the effective strain writes

ee ¼2

3ð1þ bÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ð1þ bþ b2Þ

qez

¼ � 2

3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ð1þ bþ b2Þ

qex ½108�

For instance, ee¼ ez for axisymmetric condi-tions and ee ¼ 2ez=

ffiffiffi3p

for plane strainconditions. The Brown and Embury void coa-lescence condition [94] writes for sphericalvoids

� ¼ 1

2ffiffiffi2p ½109�

Note that one could also use a coalescence cri-terion based on a critical void radius whichwould lead essentially to the same conclusions.Coalescence will take place in the x-directionbecause jexj�jeyj. Hence, the relevant void spa-cing measure is given by

� ¼ R

Lx¼

R0 exp 0:427ðee � ecÞ exp 32T� ��

L�0 expðe�Þ½110�

in which the void radius has been evaluatedusing the Rice and Tracey relation integratedfrom the nucleation strain ec. By combining[108]–[110], the effective fracture strain writes

ef ¼ln 1=2

ffiffiffi2p

�0

� �þ 0:427exp ð3=2ÞTð Þec

0:427exp ð3=2ÞTð Þ þ ð3=2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ð1þ bþ b2Þ

qÞ½111�

The initial relative void spacing can be relatedto the initial porosity using eqn [67]. Assumingimmediate void nucleation (ec¼ 0) and axisym-metric loading conditions, eqn [111] rewrites

ef ¼�1:04� ð2=3Þln f0=zð Þ

0:85 exp ð3=2ÞTð Þ ½112�

The model can be qualitatively assessed bycomparing the predictions of [112] with a wideset of results collected for various copper sys-tems by Edelson and Baldwin (1962), as shownin Figure 53. The copper bars have been loadedup to fracture under uniaxial tension condition.As explained in Section 2.06.3.2.1, the difficultywith the tensile test is to estimate the stresstriaxiality evolution in the necking region.A good approximation of the mean value ofthe stress triaxiality during a tensile test on acylindrical bar can be given as a function of thefracture strain based on the results of Figure 24:

T ¼ 1

3if ef<n

T ¼ 1

3þ 1

3

ðef � nÞ2

efif ef>n

½113�

The two eqns [112] and [113] were combinedin order to generate the curves drawn inFigure 53 for different geometrical factors �,

wRet[e

R

wt

ctt

mtr(

0

0.5

1

1.5

0 0.05 0.1 0.15 0.2

ζ = 0.3, n = 0.1ζ = 0.4, n = 0.1ζ = 0.5 (simple cubic), n = 0.1ζ = 0.6 (hexagonal), n = 0.1ζ = 0.6 (hexagonal), n = 0.2Experimental mean line

ε f

Volume fraction of second phases

Present workCopper–iron–molybdenumCopper holesCopper chromiumCopper aluminaCopper ironCopper molybdenum

Copper aluminaCopper silica

Zwilsky and Grant

Data collected byEdelson and Baldwin (1962)

1.5

1.0

0.5

0 0.1 0.2Volume fraction, f

Duc

tility

, In

A0 /A

f

0.3

Figure 53 Variation of the fracture strain as a function of the volume fraction of second-phase particles.Comparison between a large set of experimental results collected on copper by Edelson and Baldwin (1962) andthe predictions of the simple analytical model [112] using different value for the void arrangement factor � anddifferent strain-hardening exponents n.


for two strain-hardening exponents n¼ 0.1 and0.2. The model captures quite well the experi-mental trends. It underestimates the fracturestrain even for the highest geometrical factor�. In real engineering materials, the effectivegeometrical factor � should probably be smallerthan the factor resulting from periodic voiddistribution in order to account for heteroge-neous distribution effects. The underestimationin the predictions mainly comes from thatthe change of the void aspect ratio was notaccounted for in the model. Indeed, asexplained earlier, the elongation of the void atsmall stress triaxiality leads to larger fracturestrains. Furthermore, this very simple modelneglects possible delayed void nucleation andis not valid for porosity larger than about 5%.

This first simple model can be improved inmany different ways, for instance by introdu-cing the Beremin condition for void nucleation[59], by using the version of the Rice and Traceymodel accounting for void shape changes [70](still for initially spherical voids), and/or byusing the more accurate coalescence model byThomason [96]. Here, we provide one variantdestinated specifically to materials involving‘initially penny-shaped voids’. Using the rela-tionship between the stress triaxiality and theaxial stress in axisymmetric conditions

szsy¼ Tþ 2

3½114�

the Thomason condition [96] writes

ð3=2ÞTþ 1ð Þð1� �2Þ ¼ a

1� ��W

� �2

þ 1:24

ffiffiffi1

�

s" #½115�

ith a(n)¼ 0.1þ 0.22nþ 4.8n2 (0 n 0.3). Theice and Tracey evolution law for the radialvolution of the void radius under constant stressriaxiality writes (note an error in eqns [23] and28] of Lassance et al. (2006b) now corrected inqn [116]: the last ‘�’ sign must read ‘þ’)

Rx

x0¼�

1

0:427 exp ð3=2ÞTð Þ

�� 0:427 exp ð3=2ÞTð Þ � 1ð Þ½� exp 0:427 exp ð3=2ÞTð Þðee � ecÞð Þ þ 1� ½116�

hich allows the estimation of the current rela-ive void spacing �:

� ¼ �0R�

R�0

expez2

� �½117�

The Beremin model for void nucleation [60]an also be rewritten in terms of the stressriaxiality, providing the following estimate ofhe nucleation strain:

ec ¼s0E

ðsc=s0Þ þ 1� T� 23

ks

� �1=n

½118�

In the case of initially flat voids and axisym-etric loading conditions, one can show thathe following relationship for the void aspectatio (note an error in eqn [29] of Lassance et al.2006b) now corrected in eqn [119])

0.01

0.1

1

0 1 2 3 4

W0 = 1/6

f0 = 1%, n = 0.1, σ0 /E = 0.001

ε f

Stress triaxiality

W0 = 1

W0 = 6

Figure 54 Variation of the ductility as a function ofthe stress triaxiality for various initial void shapes,with an initial porosity equal to 1%.

0.01

0.1

1

0 0.1 0.2 0.3 0.4

n = 0.1n = 0.3

ε f

f p

σh /σe = 1/3

E/σ0 = 500 σd/σ0 = 0

λ0 = 1W0 = 0.01

W0 = 0.05

W0 = 0.2

W0 = 0.5

fp*

Necking for n = 0.1

Necking for n = 0.3

Figure 55 Variation of the ductility as a function ofthe particle volume fraction for different initial voidaspect ratios W0¼ 0.01, 0.05, 0.2, 0.5, strain-hardening exponent n¼ 0.1 or 0.3, and a stresstriaxiality equal to 1/3. The necking conditionpredicted by the Considere criterion is indicated toshow the range of particle volume fraction for whichfracture can be expected before the occurrence ofnecking under uniaxial tension conditions (seeLassance et al., 2006b).


W ¼ l0expðez � ecÞ � 1

expððec � ezÞ=2Þ½119�

is relatively accurate (see Lassance et al.,2006b). The coalescence condition can besolved by combining [114] to [119] in order toestimate the fracture strain for a given averagestress triaxiality level. The only parameters ofthe model are the critical fracture stress of theparticle �c and the particle volume fraction andshape (from which �0 can be calculated).

2.06.3.6.2 More advanced predictionsof the fracture strain

Finite element unit cell calculations involvingone or a few voids under well-chosen boundaryconditions, as described in the three sectionsdevoted to the nucleation, growth, and coales-cence of voids, enlighten how and to whatextent changing the loading or microstructuralparameters affects the fracture strain. Theseresults can be supplemented by results obtainedwith an advanced constitutive damage model,relying on the Beremin model for void nuclea-tion [60], on the extension of the Gurson modelby Gologanu [88], and on the extension of theThomason model for void coalescence [96]. Thepredictions of these models will be presentedand compared to experimental data in orderto discuss successively the effects of

1. the initial void volume fraction,2. the initial void shape,3. the void distribution,4. the presence of secondary small voids,5. the flow properties (yield strength and

strain hardening exponent),6. the resistance to void nucleation, and7. the microstructural heterogeneities

on the ductility, considering always the fullpractical range of stress triaxiality.

The effect of the ‘initial volume fraction ofvoids’ (or of particles giving rise to voids) aswell as of the ‘stress triaxiality’ has been dis-cussed in details above. Once again, the fracturestrain is a strong function of the stress triaxial-ity and the ductility is thus in no way a materialproperty.

The ‘effect of the void shape’ is difficult tocapture with simple models because the voidgrowth rate depends on the evolution of thevoid shape which depends itself on the stresstriaxiality and initial void shape. Figure 54shows the variation of the ductility as a func-tion of the stress triaxiality for three differentinitial void shapes. At low stress triaxiality, theeffect is such that coalescence might never occurfor elongated voids and might occur for flatvoids due to the important difference in the

relative void spacing �¼R�/L� (L� is thesame but R� is different).

Specific results related to flat initial voids areshown in Figure 55 providing the variation ofthe ductility as a function of the particle volumefraction for different initial void shapes, con-sidering uniaxial tension conditions (T¼ 1/3)(see Lassance et al., 2006b). The initial voidvolume fraction is prescribed by [68]. Thethreshold porosity fp

*under which coalescence

never occurs is indicated. This figure clearlyshows that, for initially flat voids, a widerange of particle volume fractions gives rise toa process of stable void growth followed byvoid coalescence, at this low stress triaxiality.


In contrast, more rounded voids involve amuch more abrupt transition: under fp

* no coa-lescence takes place while above fp

* coalescenceis almost immediate without any stable voidgrowth stage. Another interesting outcome ofthe calculations presented in Figure 55 is thatvoid coalescence, and thus ductile fracture, ispossible, if the volume fraction of particles issufficiently large, before the onset of necking,that is, under purely uniaxial tension conditions(e.g., Miserez et al., 2006). As explained inSection 2.06.3.2.1, necking starts in the absenceof geometric or material imperfection (and rateinsensitive materials) when e¼ n. Hence, forn¼ 0.1 and n¼ 0.3, fracture takes place beforenecking if fp>23% and fp>17%,respectively.

At larger stress triaxiality, typically largerthan 2, the effect of the shape becomes negligi-ble (see Figure 39a) which validates afterwardmany studies in the literature on crack propa-gation (high stress triaxiality) performed withthe Gurson model without void shape effects(e.g., Xia and Shih, 1995a, 1995b; Xia et al.,1995; Ruggieri et al., 1996).

The analysis of the room-temperature ducti-lity of 6xxx aluminum alloys provides apractical illustration of the importance of theinitial void shape in industrial applications (seeLassance et al., 2006a, for details). The materialconsists of industrial direct-chill casts ofAA6060 aluminum alloys. The main micro-structural feature regarding the damage andfracture process consists of b-type elongatedintermetallic Al–Fe–Si particles. A heat treat-ment allows the transformation of theb-particles into rounded a-intermetallics. Theamount of a-particles depends on the tempera-ture and duration of the heat treatment. In situtensile tests within an SEM have shown that theb-particles when oriented parallel to the mainloading direction break into several fragmentsand when oriented perpendicular to the mainloading direction give rise to decohesion (seeFigure 28). The a-particles always give rise todecohesion. This system provides a basis forlooking at the effect of the initial void shapeas it allows changing the initial shape of theprimary voids (by increasing the conversion ofb-particles into a-particles). Smooth round barswith a diameter of 9mm and a gage length of40mm were machined from the homogenizedlogs parallel to the casting direction. Notcheswere machined in some of the specimens, withnotch radii equal to 2 and 5mm. These speci-mens were loaded in tension. The hardeninglaws were identified using an inverse procedurefrom the tensile tests on smooth bars. Figure 56shows the variation of the fracture strain (mea-sured on the fracture surface) as a function of

the volume fraction of a-particles for variousstress triaxialities. As expected, the ductilityincreases when increasing the volume fractionof a-particles (involving the decrease of thevolume fraction of b-particles) due to theirmore rounded shape. The predictions of theconstitutive model, assuming immediatenucleation of all the voids, are also providedin the figure. The agreement between theexperimental results and predictions is excellent(see details in Lassance et al., 2006a). A keypoint here is that all the parameters of themodel, that is, hardening law, initial porosities,and initial void shape, were identified experi-mentally without any adjustment or fittingprocedures.

The effect of the ‘void distribution’ para-meter l0 is relatively difficult to grasp.Figure 57 shows the variation of the ductilitywith stress triaxiality for different l0. Here, dif-ferent l0, for the same fp, mean differentrelative void spacings �0. This plot is interestingbecause it approximately quantifies the maxi-mum level of anisotropy in the ductility that canbe expected for a material with l0¼ a in onesymmetry direction, which, loaded in the ortho-gonal direction, would be characterized byl0¼ 1/a. For instance, a metal with 1% ofparticles exhibiting an anisotropic void distri-bution characterized by a¼ 2 will present aductility that might change by more than afactor of 2 when tested in the two orthogonaldirections. Examples of anisotropic particle dis-tributions can be found after severe plasticdeformation typical of many forming opera-tions: oxide columns in extruded copper bars,potassium bubble columns in tungsten wires,and columns of broken intermetallic particlesin rolled aluminum sheets. Note that the effectof particle clustering can be approximately cap-tured by prescribing higher local values of l0.

As explained in Sections 2.06.3.5.3 and2.06.3.5.4, the presence of ‘secondary voids’which nucleate and grow in the ligamentbetween primary voids can significantly decreasethe fracture strain and lead to void sheet-typefracture mechanism. Figure 58 shows the varia-tion of the fracture strain as a function of thevolume fraction of second population, f20, fordifferent nucleation strains (the primary voidsnucleate immediately). Minute fraction of sec-ondary voids causes a serious drop of theductility. In an application on a 6056Al alloy ineither T4 or T78 state, the presence of about 1%dispersoids was shown to have a major effect onthe ductility (Gallais et al., 2006).

Figure 59 presents the effect of the ‘flowproperties’ of the materials, that is, yieldstrength and strain-hardening exponent on thefracture strain for initially penny-shape voids,

40 60 80 1002000

0.5Fra

ctur

e st

rain

, εf

Uniaxial with necking: T ~ 0.5Notch 1: T ~ 0.7Notch 2: T ~ 0.9

AA6060

ε = 0.2 s–1

T homog = 585 °C

α -AlfeMnSi content (%)

1

1.5

Model results

Exp. results

.

100% α100% β

Figure 56 Evolution of the calculated ductility (fracture strain) as a function of the a-AlFeMnSi content foran aluminum alloys AA6060 deformed at 20 �C with different tensile specimen geometries: uniaxial tensionwith necking, notch 1 with radius of 5mm, and notch 2 with radius of 2mm (see Lassance et al., 2006a).

0.01

0.1

1

0 1 2 3

f p = 10%

f p = 1%

f p = 0.1%

ε f

σh/σe

W0 = 0.01

E /σ0 = 500 n = 0.1 σd /σ0 = 0

λ 0 = 2

λ 0 = 1

λ 0 = 0.5

Figure 57 Variation of the ductility as a function ofthe stress triaxiality for penny-shaped voids of aspectratio W0¼ 0.01 and for three volume fractions ofparticles (with no effect on the strength of the mediumin the framework of the present model), fp¼ 0.1%, 1%,and 10%, considering three different anisotropydistribution parameters, l0¼ 1/2, 1, and 2, withE/�0¼ 500 and n¼ 0.1 (see Lassance et al., 2006b).

0

0.2

0.4

0.6

0.8

1

0 × 100 4 × 10–3 8 × 10–3 1.2 × 10–2

f 20

0.84

0.56

0.28

0

T = 1, W 0 = 1, n = 0.1, σ 0 /E = 500

f 10 = 1 × 10–3

No 2nd population

ε f

εc

Figure 58 Variation of the fracture strain as afunction of the initial volume fraction of secondaryporosity for different nucleation strains, ec. Thevolume fraction of the primary porosityf10¼ 1� 10�3, the stress triaxiality T¼ 1, the initialvoid aspect ratio W0¼ 1, and the initial voiddistribution parameter l0¼ 1 (see Fabregue andPardoen, 2006).


considering various initial volume fractions ofparticles (the results are similar with initiallyrounded voids). All these results consider thatthe voids are present from the beginning of theloading. The strain-hardening exponent has amoderate effect on the ductility while the yieldstrength has no effect. Changing for instancethe strain-hardening exponent n from 0.1 to 0.2by proper thermal treatment will not markedlyimprove the ductility as such. But, of course, anenhanced strain-hardening capacity will, in

practical structural loading conditions, delaynecking and shear banding which significantlycontributes to improving the ductility by post-poning the rise of the stress triaxiality(Figure 24). Those last effects are geometricand not a result of an intrinsic influence of thestrain hardening on the damage evolution.Note also that, in practice, strain hardeningcan hardly be changed without affecting the

0.1

1

0 1 2 3

f p = 10%

f p = 1%

f p = 0.1%

ε f

σ h /σ e

W 0 = 0.01 λ 0 = 1

E /σ 0 = 500 σd /σ 0 = 0

n = 0.3

n = 0.2

n = 0.1

n = 0.01

0 1 2 3

ε f

σ h /σ e

W 0 = 0.01 λ 0 = 1

σd /σ 0 = 0 n = 0.1

0.1

1

f p = 10%

f p = 1%

f p = 0.1%

E /σ 0 = 100

E /σ 0 = 500

E /σ 0 = 2500

(a)

(b)

Figure 59 Variation of the ductility as a function ofthe stress triaxiality for penny-shaped voids of aspectratio W0¼ 0.01 and for three volume fractions ofparticles (which have no effect on the strength of themedium in the framework of the present model),fp¼ 0.1%, 1%, and 10%, considering: a, differentstrain-hardening exponents n¼ 0.01, 0.1, 0.2, 0.3 withE/�0¼ 500 and l0¼ 1; b, different ratios E/�0¼ 100,500, 2500 with n¼ 0.1 and l0¼ 1 (see Lassance et al.,2006b).

0.01

0.1

1

0 1 2 3

ε f

ε c

ε f or

ε c

σ h/σ e

W 0 = 0.01 f p = 1% λ 0 = 1

E /σ 0 = 500 n = 0.1 E p /E = 10

σd /σ0 = 0

σd /σ0 = 4

σd /σ0 = 5

σd /σ0 = 6

Figure 60 Variation of the ductility and voidnucleation strain as a function of the stress triaxialityfor a material involving 1% of spherical hard particlesgiving rise to penny-shaped voids (W0¼ 0.01) when anucleation stress �d is attained (Beremin criterion [59])for various nucleation stresses �d/�0¼ 0, 4, 5, 6, andn¼ 0.1 (see Lassance et al., 2006b).


strength (one important exception is given byaluminum alloys which show, at low tempera-ture, a change of the strain-hardening capacitywithout much variation of the yield stress).

The picture about the effect of the flow prop-erties becomes very different when ‘voidnucleation’ does not occur immediately buttakes place only when a critical stress �d isreached in the particle or along the interface,requiring thus moderate to large amount ofplastic deformation accumulation prior nuclea-tion. Figure 60 shows the variations of thenucleation strain ed and of the fracture strainef as a function of the stress triaxiality for amaterial involving 1% of spherical hard parti-cles giving rise to penny-shaped voids(W0¼ 0.01). Let us define the ‘void growthstrain’ ef as the strain increment required to

bring freshly nucleated void to coalescence,that is, eg¼ ef� ec. Different void nucleationstresses �d/�0¼ 0, 4, 5, and 6 are analyzed,with n¼ 0.1. The increase of the ductility efwith increasing nucleation stress is smallerthan the increase of the void nucleation strainec. In other words, the void growth straindecreases with increasing resistance to voidnucleation. The main reason for this effect isthat the mean spacing between particles in theplane normal to the principal loading directiondecreases before void nucleation. The maineffect of the strain-hardening capacity will beto accelerate the attainment of the nucleationcondition by raising the stress in the particle.Hence, improving the strain-hardening capa-city is beneficial for the ductility only whenthe ductility is not controlled by the voidnucleation step.

The effect of the flow properties on the nuclea-tion of voids and on the resulting ductility hasbeen investigated in an application on quasi-eutectic cast aluminum alloy by varying theheat treatment without affecting the second-phase particles (15% of Si spherical particles)(see Huber et al., 2005, for more details). Theannealed state noted A has a low yield stress(�0¼ 87MPa) and the T6 state has a high yieldstress (�0¼ 234MPa) while the strain-hardeningexponents are similar. The sequence of events inthe damage accumulation process in both A andT6 materials observed during in situ tensile test-ing are gathered in Figure 61. The two materials,A and T6, present a similar behavior: the firststep is the cracking of the silicon particles, givingbirth to penny-shaped voids which then growand coalesce. The only obvious differencebetween the two materials is that in the T6

Lpz

Lpx

2Rpx

2Rpz

5 μm

Figure 61 The schematics outline the damage events sequence. The micrographs were taken during in situtensile test at the final stage of deformation (see Huber et al., 2005).

0

0.2

0.4

0.6

0.8

0.4 0.8 1.2

Experimental

Model

ε f

T

T6

A

Figure 62 Variation of the ductility as a function ofthe mean stress triaxiality, comparison ofexperiments and modeling for (1) the A heattreatment using fp¼ 15%, W0¼ 0.01, �d¼ 6.15; (2)the T6 heat treatment using fp¼ 14%, W0¼ 0.01,�d/�0¼ 6.15 (see Huber et al., 2005).


sample, particle cracking occurs right from thebeginning of plastic yielding, whereas it occursmuch later in the softer material A. Tensile testson smooth and notched round bars were per-formed on both A and T6 material samples.The results are given in Figure 62 in terms ofthe variation of the ductility as a function of thestress triaxiality.

The damage evolution during the tensile testson the notched and smooth bars was modeledusing the void nucleation condition [60], voidgrowth law [88], and void coalescence criterion[96]. The only ‘free’ parameter to be calibratedis the critical stress �d for void nucleation whichhas been adjusted based on the fracture strainof material A measured on broken smooth spe-cimens loaded in uniaxial tension. The initialporosity f0 was related to the particle contentusing f0¼W0 fp (see eqn [68] and W0 wasimposed to be equal to 0.01. The calibrationprovides a value of �d equal to 550MPa. The

response of material T6 has been simulatedusing exactly the same values, that is,�d¼ 550MPa and W0¼ 0.01, and the properflow properties. The comparison between theexperimental results and the model is given inFigure 62. For the case of the T6 treatment,�d¼ 550MPa leads to very early nucleation asalso observed experimentally. On the otherhand, modeling the A alloy without accountingfor delayed void nucleation, that is, using�d¼ 0, underestimates the ductility by morethan a factor of 2. In other words, dependingon the state of hardening in the matrix, theductility of these materials can be controlledby both the nucleation and growth stages.None of these two stages can be neglected.

In some materials, ‘heterogeneities in themicrostructure’ leading to heterogeneties inlocal mechanical properties are responsible forthe coexistence of different ductile failure modes.The effect of the presence of both soft and hardregions in a material can have a marked influ-ence on the fracture strain due to the accelerateddamage rates induced inside the soft regions dueto the constraint imposed by the surroundingharder zones. Many examples of alloys consist-ing of two ductile phases are provided byAnkemet al. (2006). In particular, in some aluminumalloys, the microstructure consists of precipitate-free zone (PFZ) along the grain boundariescovered with large inclusions and a precipita-tion-hardened state within the grain involvingalso coarse intermetallic particles. As shown inFigure 63, the failure mode of the material can beeither intragranular or intergranular ductile frac-ture, or a combination of the two (e.g., Dumont,2001; Pardoen et al., 2003).

A schematic of the microstructure is shownin Figure 64a. The grain interior, after heattreatment, has a high yield stress �0g and a lowstrain-hardening rate ng. On the other hand, the

10 μm 100 μm

(a) (b)

Figure 63 Fractography of (a) intergranular ductile fracture and (b) intragranular ductile fracture in a 7xxxalloy. From Dumont D. 2001. Relations Microstructures/Tenacite dans les alliages aeronautiques de la serie7000. Ph.D. thesis, Institut National Polytechnique Grenoble. Pardoen, T., Dumont, D., Deschamps,A., Brechet, Y. 2003. Grain boundary versus transgranular ductile failure. J. Mech. Phys. Solids 51, 637–665.

Grain particles

Grain-boundary particles

Grain damage

Grain-boundary damage

Grain boundaryPrecipitate-free zone

d

h

Lgx

Lgz

Dpx

DpzLpx

Dgx

Dgz

ng

E ν

fg0 Wg0

npE ν

σ0g

σ0p

λg0

fp0 Wp0 λ p0

(a) (b) (c)

Figure 64 Description of: a, the real microstructure and failure mechanisms; b, the idealized microstructure;and c, the continuum micromechanical model. The parameters appearing in (b) and (c) are defined in Table 1

(see Pardoen et al., 2003).


PFZ has a low yield stress �0p and a high strain-hardening rate np. The idealized microstructureis shown in Figure 64b with the various micro-structural parameters entering the problem andthe relevant dimensionless quantities.

The competition between intergranular andtransgranular failure can be qualitatively under-stood in the following way (see Figure 65). ThePFZ is soft and is thus the first to deform plas-tically. The elastic grain imposes a strongconstraint on the PFZ involving a large stresstriaxiality. The large void growth rate in the PFZleads to rapid coalescence of the voids. However,in some circumstances, the stress in the grainreaches the yield stress before the onset of coa-lescence in the PFZ. The stress triaxiality thendrops in the PFZ which, due to its higher strain-hardening capacity, now imposes a higher con-straint inside the grain. Voids then tend to growmore rapidly within the grain. Due to the lowstrain-hardening capacity of the grain, a state ofdamage-induced softening is rapidly attaineduntil voids finally coalesce within the grain.

Experimentally, by increasing the time ofheat treatment, the yield stress first increases,favoring the propensity toward intergranularfracture, and then decreases, favoring again atransgranular fracture mode. In parallel, thestrain-hardening rate decreases when precipita-tion occurs. A more quantitative analysis of thishighly nonlinear problem of failure mode tran-sition requires a detailed model for void growthand coalescence to be incorporated in eachlayer. In Pardoen et al. (2003), the PFZ andthe grain interior have been modeled using thesame void growth and coalescence constitutivelaws (Gologanu model combined by Thomasoncoalescence criterion). The material was mod-eled as a bilayer (see Figures 64b and 64c),neglecting thus the effect of inclined grainboundaries (see Scheyvaerts et al. (2006) formore advanced representation of the micro-structure). The initial relative spacing betweenparticle along the grain boundary, Lp0/Dp0, isthe most relevant parameter to interpret theresults. Figure 66 presents failure maps for

Str

ess

tria

xial

ity, T

= σ i

i /3σ e

Relative particle spacing in PFZ, L p0 /Dp0

Intergranularfracture

Transgranularfracture

Grain

PFZΕe

Σe

Grain

PFZ

Εe

Σe

Figure 65 Failure map providing a qualitativeunderstanding of the competition betweenintergranular and transgranular ductile failure (seePardoen et al., 2003).

0

1

2

3

4

4 8 12 16 20

T

Lp0 /Dp0

n p =

0.4, W p0

= 1

λ p0 =

1, R 0 = 0.2

σ 0g /σ 0p = 4

σ 0g /σ 0p = 5

σ 0g /σ 0p = 6



Figure 66 Effect of the yield stress ratio on thefailure mode in a stress triaxiality vs relative particlespacing map (see Pardoen et al., 2003).

0.5

1

1.5

2

2.5

3

4 8 12 16 20

λ p0 =

1

λ p0 =

3

λ p0 =

1/3

T

Lp0 /Dp0



σ 0g /σ 0p = 5, n p = 0.4

W p0 =

1, R 0

= 0.01

Figure 67 Effect of the PFZ thickness/void spacingratio on the failure mode in a stress triaxiality vsrelative particle spacing map (see Pardoen et al.,2003).


three different ratios of �0g/�0p. As expected, anincrease of the grain yield stress promotes grainboundary failure. An increase of the PFZstrain-hardening capacity has an effect similarto a decrease of �0g/�0p. The most importantparameter, as exhibited in Figure 67, is the spa-cing relative to the PFZ thickness, �p0.

These failure maps show that, whatever theflow properties and microstructure, lower par-ticle spacing Lp0/Dp0 and high stress triaxialityalways tend to promote intergranular fractureas expected from the qualitative description ofFigure 65. The stress triaxiality failure modedependence has been qualitatively observed byDumont (2001). Realistic values for Lp0/Dp0 arebetween 2 and 5. In that range, the stress triaxi-ality corresponding with the failure modetransition is very much dependent on particle

spacing which also complicates the optimiza-tion of the material properties.

2.06.3.7 Fracture Toughness of ThickDuctile Metallic Components

Two sections are now devoted to the fractureresistance of cracked structures. The first sec-tion is limited to ‘thick components’ or ‘thicktest specimens’ which is the classical topic ofelastoplastic fracture mechanics. By ‘thick’, itis meant that the thickness is larger than the PZsize and that plane strain or near plane strainconditions prevail, on average, along the crackfront line. We limit this presentation to mode Ifracture, which is the most important for amajority of practical applications. The secondsection is devoted to the fracture of ‘thinsheets’, a subject which has recently received arecrudescence of interest, motivated by anincreasing use in structural application of thin-ner and thinner sheets made of more and moreductile alloys. In ‘thin’ sheets, the thicknessplays a key role in controlling the dissipationof energy at the crack tip through its directeffect on the energy spent in localized neckingand an indirect effect on the rate of damageevolution in the FPZ. While in ‘thick’ planestrain samples, loss of constraint results fromin-plane effects, in ‘thin’ plate one must alsoconsider out-of-plane effects. In thin plates,mixed mode I and III fracture is also an impor-tant issue.

In this section devoted to plane strain frac-ture (thick components), we start by reviewingsome basics of elastoplastic fracture mechanicsnecessary to address the cracking mechanisms,also useful for the next section on thin sheetfracture. Then, a second subsection addressesthe resistance to fracture initiation, which is

y

x

δ

Initial crackr

ux

uy

45°

Figure 68 Definition of the CTOD.


usually (but not always) the definition of thefracture toughness of a material, while the thirdsubsection focuses on the resistance to ductiletearing (crack propagation).

2.06.3.7.1 Basics

(i) Essentials of elastoplastic fracturemechanics

The theoretical foundations of these conceptshave been presented in details in Chapter 2.03(see Broberg, 1999; Anderson, 1995). In thick-walled components or relatively thick labora-tory samples, the zone in front of the crack tipin which the fracture phenomena take placeundergoes plane strain conditions. The crack-tip loading is quantified through the value ofthe J-integral defined by (Rice, 1968)

J ¼Z

�

WVnx � nisij@uj@x

� �ds ½120�

where G is a contour surrounding the crack tipin the anticlockwise direction, WV(eij) is thestrain energy density, and ui are the displace-ments. Rice (1968) demonstrated that the valueof the integral [120] is independent of the pathof integration G. Assuming that a nonlinearelastic behavior provides an adequate descrip-tion of the mechanical response (which is trueunder radial or approximately radial loadings),J provides a unique intensity measure for char-acterizing the crack-tip loading under yieldingconditions. The analysis of the crack-tip stressfields within the context of small strain defor-mation theory of plasticity has been performedby Hutchinson (1968) and Rice and Rosengren(1968). It is known as the HRR solution. Apower law relation of the form

ee0¼ a

ss0

� �N

½121�

is assumed to represent the uniaxial flow proper-ties of the material (a is a constant andN¼ 1/n isthe inverse of the strain-hardening exponent n).The asymptotic solution for the stress and strainfields is given by

sij ¼ s0J

as0e0INr

� �1=ðNþ1Þsijðy;NÞ ½122�

se ¼ s0J

as0e0INr

� �1=ðNþ1Þseðy;NÞ ½123�

eij ¼ ae0J

as0e0INr

� �N=ðNþ1Þeijðy;NÞ ½124�

ui ¼ ae0rJ

as0e0INr

� �N=ðNþ1Þuiðy;NÞ ½125�

where IN is a dimensionless constant whichdepends onN, and �~ij, �~e, e~ij, and u~i are functionsof andN. The constant IN and functions �~ij, �~e,e~ij, and ui have been computed and tabulated byShih (1983) for both plane stress and plane strainconditions. In the elastic case, that is, N¼ 1, eqn[122] reduces to the inverse square root singular-ity of linear-elastic fracture mechanics. The levelof J determines the crack-tip stress field in thenonlinear elastic cases.

According to the displacement fields [125],the opening separation of the two crack facesvaries like r1/(Nþ1) as r! 0. The opening of thecrack at r ! 0 is thus zero. Rice (1968) hassuggested to define the opening separation ofthe two crack faces by taking the opening¼ 2uy at the intercepts of the two 45� linesdrawn back from the tip of the deformed pro-file, that is where r� ux¼ /2 (Figure 68). Thislocal opening separation, , called the crack-tipopening displacement (CTOD), can be mea-sured experimentally on a section transverse tothe crack front. The use of the HRR solution[125] implies that

d ¼ dðae0;NÞJ

s0½126�

where d has also been tabulated by Shih (1983)as a function of ae0 and N. For typical metalalloys, d ranges between 0.3 and 0.6 in planestrain and between 0.5 and 0.9 in plane stress.

The detailed analysis of the crack-tip blunt-ing process in the framework of finite strain J2plasticity has been made first by McMeeking(1977) using the finite element method(extending the slip line analysis of Rice andJohnson (1970) limited to perfectly plasticmaterials). The evolution of the key parametercontrolling the ductile damage process, that is,the stress triaxiality, is shown in Figure 69a as afunction of the distance to the crack tip normal-ized by J/�0. As shown in Figure 69b, themaximum stress triaxiality is equal to 2.75 forperfectly plastic materials (n¼ 0) and increaseswith increasing strain-hardening exponent.These values of the stress triaxiality arecorrectly predicted by Hutchinson and Riceand Rosengren (the so-called HRR theory).The stress triaxiality drops down near the

(a)

(b)

3 4 51 62

0.05–0.1

εp

0.577

Stress triaxiality

Loss of constraint

Increasingn

Constraint changes

r /δ

Fracture process zone

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2 0.25 0.3

Str

ess

tria

xial

ity

n

Plane stress

Plane strain

Figure 69 Variation of the stress triaxiality as afunction of (a), the distance to the crack tip(supplemented by the effective plastic strainevolution); (b), the strain-hardening exponent (forplane strain and plane stress conditions).

Δa0.2 mm

Ji

J Loss of constraint

JIc

Steady-state regime

Active plasticzone

Crack wake

ΓSSBlu

ntin

g lin

e

Figure 70 Schematic JR curve and PZ extensionduring crack propagation.


blunted crack tip due to the presence of thefree surface and reaches the plane strain tensionvalue T ¼

ffiffiffi3p

=3 � 0:58. The plastic strainsbecome larger than a few percent at a distanceequal to about 1–3 CTOD . The FPZ willthus extend in a zone equal to roughly 1 to3. The plastic strains up to a value equal to0.1–0.2 are correctly predicted by the HRRsolution.

The value of J corresponding to the initiationof cracking is noted in mode I JI and the corre-sponding critical CTOD is noted c. Althoughits validity is not guaranteed anymore, theJ-integral is still used to quantify the loadingduring the propagation of the crack, at least forpropagation over small distances (typically afew millimeters). The tearing resistance of thematerial is thus quantified through the evolu-tion of J with crack advance, Da. A typical JRcurve is shown in Figure 70. An engineeringfracture toughness is defined by the value of Jafter 0.2mm of crack advance, noted JIc. Insome materials, J0.2 and JIc can be very differ-ent. In this chapter, we do not distinguishbetween these two definitions. The fact that J

keeps increasing with the loading is extrinsic tothe fracture process taking place in the near-crack-tip region. It is thus not a true increase ofthe fracture resistance (see Cotterell andAtkins, 1996). J increases due to the plasticdissipation taking place in the crack wake dueto the progressive elastic unloading, due to thenonradial loadings in the active PZ of a propa-gating crack, and due to changes in the crack-tip geometry (see further). In very large sam-ples, a steady-state regime should in principlebe reached but it is almost never observed inmetallic samples because of specimen sizelimitations.

The analysis of the stress and strain conditionsin front of a growing crack in a plasticallydeforming solid has been made by Rice andSorensen (1978) (see also Hutchinson (1974) forthe case of steady-state conditions). Under SSYconditions and for an elastic–perfectly plasticmaterial, the main difference in the predictedstress–strain field between a stationary and agrowing crack lies in the strain singularity andnot the stress profile at the crack tip. This is animportant point for the analysis of the brittle-to-ductile transition which is discussed in moredetails in Section 2.06.4. Many observationshave shown a major difference in crack-tip geo-metry between initiation of crack growth andcrack propagation. This is illustrated inFigure 71. At cracking initiation, the crack tipis blunted while during propagation the unzip-ping process from one inclusion to another onegives rise to a crack tip which is much sharper.This modification in crack profile leads to anelevation of the normal stress ahead of thecrack tip when the crack propagates (this effectis not accounted for in the theoretical analysisby Rice and Sorensen, 1978): the higher thisincrease of the normal stress, the steeper theslope of the J resistance curve. This phenom-enon will also be a key in the analysis of thebrittle-to-ductile transition (see Section 2.06.4).

tddtitmrb

vd2oFbectppwua

FpuANa1G1H2aplotflmdisapfpsNuaa

gU

Δabl Δatear

Δatot

a0

∂1

∂2

50 μ

m

(b)

(a)

200 μm

Figure 71 Ductile crack initiation and crack growthfrom a fatigue precrack: a, in copper (see Pardoenand Delannay, 2000); b, A 508 RPV steel. The crackpropagates from one inclusion to another oneleading to a zigzag pattern (Lautridou, 1980).


The JR curve is very sensitive to slight changesof constraint which affect a lot the extrinsicplastic dissipation (see, e.g., Sumpter, 1993;Brocks and Schmitt, 1994; Anderson, 1995; Xiaet al., 1995; Xia and Shih, 1995a, 1995b). Theconstraint effect can be quantified in the frame-work of fracture mechanics through either theso-called T-stress (only as long as SSY condi-tions prevail) or the so-called Q-stress, whichquantifies the departure from the HR solution(O’Dowd and Shih, 1991, 1992). A good reviewabout the fracture mechanics approach of con-straint is given in the book by Anderson (1995)and in Chapter 2.03.

To sum up, the work of fracture can be writ-ten for plane strain conditions as

� ¼ �0 þ �p ½127�

where G0¼ J1c and Gp is highly dependent onthe level of constraint (the question whetherG0 is dependent on the constraint will be dis-cussed in the next subsection as it requires toaddress the fracture mechanisms inside theFPZ).

(ii) Computational strategies to simulate crackpropagation in ductile materials

There are four main types of modeling stra-tegies which have been proposed in theliterature to address the prediction of the frac-

ure resistance, that is, in order to couple theamage mechanisms and the mechanical fieldsescribed just before. In all these approaches,he key is that a characteristic length must bentroduced. The energy dissipation scales withhe height of the zone in which the fractureechanisms take place. This size is directlyelated to the void spacing or to the spacingetween void clusters.

1. The ‘first strategy’ consists in using theoid nucleation, growth, and coalescence lawsescribed in Sections 2.06.3.3, 2.06.3.4, and.06.3.5, and integrate them using the solutionsf the elastoplastic crack problems (seeigure 69a), thus neglecting the couplingetween the mechanical fields and the damagevolution. This approach allows relating theracking initiation toughness to the microstruc-ure and also allows the simulation of crackropagation using finite element methods withroper node release technique (see pioneeringork by d’Escatha and Devaux (1979), whosed a critical void radius condition to bettained at a distance X0 from the crack tip).2. The ‘second strategy’ is shown inigure 72b: finite element calculations can beerformed with the voids explicitly modeledsing a refined finite element mesh (e.g.,ravas and McMeeking, 1985a, 1985b;eedleman and Tvergaard, 1987, 1991; Homnd McMeeking, 1989a, 1989b; McMeeking,992; Tvergaard and Needleman, 1992;hosal and Narasimhan, 1996; Gao et al.,996; Yan and Mai, 1997; Tvergaard andutchinson, 2002; Kim et al., 2003; Gao et al.,005; Petti and Dodds, 2005b). These analysesccurately model the growth and coalescencerocess while properly accounting for theength scale introduced by the void spacing. Inrder to simulate the propagation of the crack,his approach still requires a criterion for theinal failure of the intervoid ligament to simu-ate crack propagation, for example, byodeling shear localization of the ligamentue to a second population of smaller voids asn Needleman and Tvergaard (1987), or by pre-cribing a critical void spacing as in Tvergaardnd Hutchinson (2002). This approach is com-utationally intensive. It is only able to accountor a few voids ahead of the crack tip and thusrobably not attractive for simulating fulltructures or test specimens, especially in 3-D.evertheless, such simulations provide veryseful results for assessing the validity of thessumptions and the predictions of the otherpproaches.3. The ‘third strategy’ pursued mainly by

roups in France, Germany, the UK, and theS employs a constitutive model, such as the

(b)

Large inclusions

Carbides

Active layer(ductile tearing)

(a)

Voids

Figure 72 (Continued)


Gurson or the Rousselier model, which accountsfor the damage-induced softening, see Figure 72c(e.g., Mudry et al., 1989; Rousselier et al., 1989;Bilby et al., 1993; Xia et al., 1995; Xia and Shih,1995a, 1995b; Brocks et al., 1995a; Ruggieriet al., 1996; Gao et al., 1998a; Koppenhoeferand Dodds, 1998; Zhang et al., 2000;Roychowdhury and Narasimhan, 2000; Rivalinet al., 2001a, 2001b; Pardoen and Hutchinson,2003; Chabanet et al., 2003; Negre et al., 2003,2004, 2005). The constitutivemodel is implemen-ted in a finite element code to simulate theinitiation and growth of the crack. A microme-chanics-based damage model, such as theGurson model (and its extensions), is derived insuch a way that it should adequately reproducethe behavior of a material cell involving a singlevoid subjected to homogeneous conditions at theboundaries. Here, however, near a crack tip,strong strain and stress gradients develop at thescale of the void cell size. These gradients areaveraged in a relatively crude way by only usinga single element to represent one void cell. Theerror coming from this approximation is difficultto evaluate a priori.

As explained above, this approach requiresthe introduction of a length scale in the modelrelated to the spacing between voids. This is

usually accomplished by tying the element sizeto the void spacing, calibrated on experimentalcrack growth data. This simple approach has thedisadvantage to artificially tie a physical lengthto a numerical parameter. Hence, teams havebeen working, motivated by seminal contribu-tions in the field of fracture in concrete (e.g.,Pijaudier-Cabot and Bazant, 1987), on formu-lating nonlocal constitutive models for ductilefracture (e.g., Leblond et al., 1994; Tvergaardand Needleman, 1995; Engelen et al., 2003;Geers et al., 2003; Reusch et al., 2003a, 2003b;Zhenhuan et al. 2003; Geers, 2004; Yuan andChen, 2004; Mediavilla et al., 2006a, 2006b,2006c). There are several methods for introdu-cing internal lengths into the model and noconsensus has emerged yet about the bestapproach.

4. A ‘fourth strategy’, schematically illu-strated in Figure 72d, initiated by Tvergaardand Hutchinson (1992), makes use of cohesivezone surfaces to simulate the fracture process inductile metals (e.g., Tvergaard and Hutchinson,1996; Keller et al., 1999; Siegmund and Brocks,1999, 2000; Li and Siegmund, 2002; Roy andDodds, 2001; Roychowdhury et al., 2002;Brocks et al., 2003; Chen et al., 2003; Cornecet al., 2003; Scheider and Brocks, 2003; Chen

aoercvp

altem7

cσσ

cδδ

1

1

σ δ

0Γ

cδδ1

cδδ 2

cσσ

cδδ

1

1

0Γ

Trilinear function

cσσ

cδδ

1

1

0Γ

Polynomial function

Exponential function

fE f0Cell element with void

Layer of void-containing cell elementsCrack

D

D D /2

(c)

(d)

Figure 72 The three ‘coupled’modeling strategies referred in the text asmethods 2, 3, and 4 described schematicallyin (b)–(d) in order to simulate cracking in ductile materials represented in (a); b, discrete voids modeling; c,computational cell model; d, cohesive zone model approach with the three typical traction separation laws used inthe literature. a, From Gullerud, A. S., Gao, X., Dodds, R. H., Jr. Haj-Ali, R. 2000. Simulation of ductile crackgrowth using computational cells: Numerical aspects Eng. Fract. Mech. 66, 65–92. b, From Tvergaard, V. andHutchinson, J.W. 1992. The relationship between crack growth resistance and fracture process parameters in elasticplastic solids. J.Mech. Phys. Solids 40, 1377–1397. c, FromGao, X., Faleskog, J., and Shih, C. F. 1998a. Cell modelfor nonlinear fracture analysis. II: Fracture-process calibration and verification. Int. J. Fract. 89, 374–386.


et al., 2005; Chen and Kolednik, 2005; Negreet al., 2005; Scheider et al., 2006). The responseof the FPZ is approximately modeled by a trac-tion–separation curve (see Figure 72d). Themain characteristics of the traction–separationcurve are the work of separation and the max-imum stress, also called cohesive stress or peakstress. The main advantage of this method isthat the characteristic length is introduced in anatural way into the model since the cohesiveproperties involve the work of separation andthe cohesive stress. However, problems areencountered when introducing dependencies ofthe cohesive zone parameters on the mechanicalfields next to the cohesive surfaces (Tvergaard

nd Hutchinson, 1996; Keller et al., 1999) inrder to artificially account for constraintffects on the damage evolution. Also, inegular finite element implementations, therack path must be prescribed in advance pre-enting thus the modeling of complex crackaths.

Even though these various modelingpproaches have all attained a relatively highevel of maturity, comparisons with experimen-al data are still extremely scarce. Nevertheless,ncouraging results have been demonstratedostly using strategies 3 and 4 (see Chapter.05).


2.06.3.7.2 Fracture initiation toughness

The resistance to crack initiation of pre-existing sharp crack provides a measure of theso-called fracture toughness. It provides, formaterial scientist, a very useful way to indexthe quality and compare materials with respectto their ability to resist cracking. The resistanceto cracking initiation constitutes also, for manyapplications, the failure conditions on whichstructural integrity assessment methods arebased. In this section, we are mainly interestedin understanding and predicting the fracturetoughness at cracking initiation in ductilemetals through its relationship with the micro-structure and flow properties.

Figure 73 depicts the model envisioned forsimulating cracking in an idealized ductilemetal. The initial geometry is a precrack ofopening 0 in an idealized material having reg-ularly distributed voids with initial spacing X0.The crack is long, and SSY is assumed to apply.The matrix is characterized by the followingmechanical properties E, v, �0, and n. Thevoids have an initial shape W0, volume fractionf0, and distribution parameter l0.

The predicted evolutions of the fracturetoughness reported hereafter have beenobtained by using the third type of modelingstrategy (computational unit cell, seeFigure 72c) using the combination of theGologanu and Thomason models (see Sections2.06.3.4.4 and 2.06.3.5.4) and under SSY con-ditions in order to avoid any complicationsrelated to constraint effects (see Pardoenand Hutchinson, 2003, for details). The size ofthe elements is directly related to void spacingin the portion of the mesh that experiencesvoid growth and coalescence, that is, in theFPZ. In all cases here, the elements in the FPZ

X0

δ0

I. Multiple void interaction

Plastic localization= coalescence= finite strain zone

Figure 73 The initial geometry of a precrack in a ideal msketch of the two ideal modes of crack initiation, that is,

are taken to be square and of dimension X0.Thus, normalization of the toughness by theonly length scale X0 obviously leads to resultsthat are independent of the degree of meshrefinement. Dimensional analysis shows indeedthat

JIcs0X0

¼ Fs0E; n; f0;W0; l0; sd

� �½128�

The validity of this relationship has beenconfirmed experimentally, for instance byLautridou and Pineau (1981) (see also therecent work by Miserez et al., 2006). The resultsof the simulations have been successfully com-pared to the results of Tvergaard and Hutchinson(2002) and Gao et al. (2005), obtained with thesecond modeling strategy. A few selected resultsare presented hereafter. Note that void nucleationis not considered (i.e., �d¼ 0).

As shown in Figure 73, two limiting situationscan be found (Tvergaard and Hutchinson, 2002;Pardoen and Hutchinson, 2003):

1. At sufficiently high porosity, the void nearthe tip is influenced by its nearest neighbor,which experiences almost the same rate ofgrowth. The interaction among the voids,including voids even farther from the tip,results in significantly higher rate of voidgrowth for all of the voids. Coalescencebetween several voids and with the crack startsearly, almost simultaneously. This is the ‘mul-tiple void interaction’ mechanism.

2. For sufficiently small void volume frac-tion, a single void process prevails, which isessentially the process envisioned by the Rice–Johnson (1970) model. The void nearest to thetip grows with little influence from its nearestneighbor further from the tip. This is the ‘void-by-void growth’ mechanism. Experimental data

Z0

2R0

2Rz 0

Finitestrainzone

II. Void by void growth

Plastic localization= coalescence

aterial with regularly distributed spheroidal inclusions;multiple void process and single void/crack process.

0

2

4

6

8

(a)

10–6 10–5 10–4 10–3 10–2 10–1

σ0 /E = 0.01

σ0 /E = 0.003

σ0 /E = 0.001

f 0

J Ic /σ

0X 0

n = 0.1, W 0 = 1, λ 0 = 1

(b)

10–5 10–4 10–3 10–2

f 0

J Ic/

σ 0X

0

0

2

4

6

8

10

0.2

n =

0.1

0.01

σ0 /E = 0.003, W 0 = 1, λ 0 = 1

(c)

10–5 10–4 10–3 10–2

f 0

J Ic/

σ 0X

0

0

2

4

6

8

1/3 1

W0 =

1/10

3

10

σ0 /E = 0.003, n = 0.1, λ 0 = 1

Figure 74 Variation of the fracture toughness(normalized by �0X0) as a function of the initialporosity, for various: a, ratios �0/E; b, strain-hardening exponents n; and c, initial void aspectratios W0 (Pardoen and Hutchinson, 2003).


on HSLA steels (Luo et al., 1989) show, fromlocal strain measurements, strains of about0.5�0.75 in front of the crack tip at crackinginitiation, a value that would never be attainedwith the multiple void interaction mechanismwhich involves typical fracture strains of about0.1. Most metallic alloys have initial voidvolume fraction smaller than 10�2 and willthus fail from void by void growth mechanism.

Whatever the mechanism, initially sphericalvoids get first oblate (see Aravas andMcMeeking, 1985a). In the single void interac-tion problem, if the initial porosity is very low,the voids enter the low stress triaxiality zone(see Figure 69) before coalescing with thecrack. In that case, the voids tend to elongatebefore cracking initiates.

Figure 74 shows the variation of the fracturetoughness (normalized by �0X0) as a function ofthe initial porosity, for various (a) ratios �0/E,(b) strain-hardening exponents n; and (c) initialvoid aspect ratios W0. The ratio �0/E has noeffect on the fracture toughness as long as voidnucleation is not considered. Figure 74b exhi-bits the effect of the strain-hardening index, n,on the fracture toughness. Strain hardening hasa major influence on fracture toughness. Thefact that the fracture toughness is linearly pro-portional to the yield stress seems to contradictmuch experimental evidences showing that thetoughness (JIc) of a family of alloys usuallydecreases with increasing yield stress (�0).Several points are relevant to this apparent con-tradiction. First, in many alloys, an increase of�0 by metallurgical intervention is usuallyaccompanied by a decrease of the strain-hard-ening index n that has the opposite effect on thetoughness. For instance, these countervailingtrends occur in precipitation hardening of alu-minum alloys, where the precipitates do not, ingeneral, take part in the failure process. Second,the present model does not incorporate void ormicrocrack nucleation criterion. In manyinstances, an increasing yield strength willaffect the nucleation stage by raising the stresson the second-phase particles or grain bound-aries. A larger yield stress may also favornucleation on smaller particles or on a secondpopulation of particles at an earlier stage of thedeformation. However, if all other parameters,including strain hardening, can be kept con-stant, a higher yield stress directly implies ahigher fracture toughness. A good example isgiven by the decrease with increasing tempera-ture of the fracture toughness of ferritic steel inthe upper shelf region (see Section 2.06.4). Forthe typical temperature range covered whenmeasuring a ductile–brittle transition curve,no modification of microstructure and

hardening mechanisms is expected, except forthe decrease of the yield stress with increasingtemperature.

Figure 74c shows that the effect of the initialvoid shape follows the intuition: at a givenporosity, prolate voids have a smaller area frac-tion projected onto the fracture plane thanspheres and conversely for oblate shapes.Thus, prolate shapes increase JIc/�0X0 whileoblate shapes reduce it relative to sphericalvoids at the same volume fraction. The results

0

0.5

1

1.5

2

0 1 2 3 4 5

Γ 0/σ

0X

0

Stress triaxiality, T

f 0 = 0.01; n = 0.1; σ0 /E = 500

Typical stress triaxiality for ‘thick’ cracked specimens with possible constraint effects

Figure 75 Variation of the work of fracture per unitarea (normalized by �0X0) as a function of the stresstriaxiality of a material element involving an initialvolume fraction f0¼ 0.01 of spherical voids.


of Figure 74c can be used to guide understand-ing of, as well as to predict the variation of, thefracture toughness as a function of the loadingdirection for rolled plates with preferentialorientation of the second phase. Clearly, voidshape has a significant effect on fracture tough-ness. The effect of anisotropic voiddistributions (l0 not equal to 1) has also beeninvestigated by Pardoen and Hutchinson (2003)and turns out be also very significant.

Finally, it is important to recognize that theprevious analysis is valid only for tensile locali-zation mode leading to internal neckingbetween voids. However, localization in shearis sometimes observed at the crack tip of high-strength/low-hardening materials. Examples ofshear coalescence in an FPZ under plane straincondition are provided by observations of‘zigzag’ cracking in low-hardening steels(Clayton and Knott, 1976; Needleman andTvergaard, 1987; Xia and Shih, 1995b). Theearly shear localization process between theblunted crack tip and the nearest void is detri-mental to the fracture toughness as it involvesmuch less plastic work than a full void growth/coalescence to final impingement.

The most simple method to estimate the frac-ture toughness at crack initiation is theaforementioned method 1, which requires tointegrate the constitutive model with themechanical fields evaluated for a crack in anondamaging elastoplastic material. An evensimpler approach is to consider only the stresstriaxiality as the dominant feature controllingthe damage process. Figure 75 shows thevariation with the imposed stress triaxiality ofthe work per unit area spent in deforming amaterial element involving an initial volumefraction f0¼ 0.01 of spherical voids up to frac-ture. For that specific f0, the high stresstriaxiality work of fracture G0 calculated thisway is about a factor of 2 smaller than thefracture toughness calculated with the full com-putational cell model (see Figure 74). Theimportant message to extract from Figure 75 isthat the work of fracture is relatively indepen-dent of the stress triaxiality as long as the stresstriaxiality is larger than typically 2–2.5. Thispoint has also been raised by Siegmund andBrocks (2000). For thick component, typicalloss of constraint encountered with small speci-men sizes or short crack lengths does notusually induce a drop of the stress triaxialityunder a value equal to 2–2.5. This explainswhy, again for plane strain conditions, con-straint effects mainly affect the tearingmodulus and not much the initiation of crack-ing. Nevertheless, under large-scale yieldingconditions, the magnitude of plastic strainsnear a crack tip can sometimes depend on the

specimen geometry, resulting in differences inthe crack initiation toughness (see Pardoenet al., 2000).

2.06.3.7.3 Ductile tearing resistance

For several applications, the presence ofcracks is tolerated, for example, in some aircraftstructural components or pipelines, and theintegrity of structure is assessed towardunstable crack propagation. It is thus essentialto develop materials with the highest possibletearing resistance as well as to develop modelsthat allow transferring resistance curvesobtained on laboratory specimens to complexstructures undergoing realistic complex loadingconditions.

The first comprehensive effort to predictcrack resistance curves based on an embeddedFPZ model reproducing the response of a rowof voids during deformation, has been pro-posed by Tvergaard and Hutchinson (1992)using the cohesive zone methodology (seeabove, method 4), SSY, and plane strainconditions. These authors have shown that thekey factor controlling the dissipation of energyduring crack propagation is the ‘cohesivestrength’. In their work, the tearing resistancewas quantified by the ratio GSS/G0 where GSS isthe steady-state work of fracture (seeFigure 70). Figure 76 shows the variation ofGSS/G0 as a function of the cohesive strengthfor different strain-hardening exponents. Ifthe peak strength is lower than about 3,GSS/G0� 1, that is, the JR curve is flat andthere is no other dissipation than the work offracture spent in the FPZ, G0. When thecohesive strength increases, plastic strainsmust accumulate next to the FPZ in order toraise the stress up to the cohesive strength.This stress increase is made possible owing to

00

1 2

2

3 4

4

5 6

6

8

10

n = 0 0.1 0.2

σ/σy

Γ ss / Γ

0

Figure 76 The extrinsic plasticity contribution toplane strain steady-state toughness as predicted bythe cohesive zone model for mode I crack growth in aductile solid with tensile yield stress �y and strain-hardening exponent n. The curves give the ratio ofthe steady-state macroscopic work of fracture to theintrinsic work of separation as a function of the ratioof peak separation stress to yield stress. FromTvergaard, V. and Hutchinson, J. W. 1992. Therelationship between crack growth resistance andfracture process parameters in elastic plastic solids.J. Mech. Phys. Solids 40, 1377–1397.

00

2

4

6

8

10

12

14

16

18

20

5

(b)

(a)

10

Δa /D

Δa /D

15 20

Γ/(D

σ 0)

f0 = 0.005

n = 0.3

n = 0.2

n = 0.1

n = 0.05

Γ/(D

σ 0)

E/σ0 = 500

E/σ0 = 400

E/σ0 = 300

E/σ0 = 200

00

2

4

5

(c)

10 15 20

f0 = 0.005

n = 0.1

00

2

4

6

8

10

12

n = 0.1

5 10 15 20

Δa /D

ƒ0 = 0.02

ƒ0 = 0.01

ƒ0 = 0.005

ƒ0 = 0.0025

ƒ0 = 0.001

Γ/(D

σ 0)

Figure 77 Crack resistance curves for: a, differentinitial porosities f0; b, different strain-hardeningexponents n; and c, different ratios �0/E. FromXia, L. and Shih, C. F. 1995a. Ductile crack growth. I:A numerical study using computational cells withmicrostructurally based length scales. J. Mech. Phys.Solids 43, 233–259.


the strain-hardening capacity of the material.Hence, the magnitude of the plastic defor-mation as well as the size of the PZ increasewith increasing load, leading to an increasingcontribution of the plastic dissipationterms Gp (see eqn [127]). The value of thecohesive strength is representative of the valuesof the parameters controlling the damageprocess.

The first comprehensive effort to predictcrack resistance curves based on the more rea-listic ‘computational cell methodology’(referred as method 3 above) was proposed byXia and Shih (1995a, 1995b, 1996) and Xiaet al. (1995) for 2-D plane strain conditions. Intheir application of the computational cellmethod, void growth remains confined to asingle layer of material symmetrically locatedabout the crack plane and having a thicknessD,where the parameter D is identical to theparameter X0 used before. This layer consistsof cubical cell elements with dimension D oneach side; each cell contains a centered sphericalcavity of initial volume fraction, f0. Progressivevoid growth and subsequent macroscopic mate-rial softening in each cell are usually describedwith the version of the Gurson model extendedby Tvergaard (see Section 2.06.3.4). When thecalculated void volume fraction, f, in the celladjacent to the crack tip reaches a critical value,

fc, the load is ramped down linearly with theincrease of the normal displacement. This pro-duces the growth of the crack tip in discreteincrements of the cell size. Figure 77 is a selec-tion of predicted crack resistance curves for (a)different initial porosities f0, (b) different strainhardening exponents n, and (c) different ratios�0/E.

eCdt‘(nsiumftdOreiacs22

2

2

Smc‘tc

00

1

2

3

4

5

6

7

8

9

f0 = 0.005

5

Δa /D

Γ/(D

σ 0)

T/σ0 = 0.0

T/σ0 = –0.25

T/σ0 = –0.5

T/σ0 = 0.5

10 15 20

Figure 78 Crack resistance curves simulated by thecomputational cell method under SSY conditions forvarious T-stress levels. From Xia, L. and Shih C. F.1995a. Ductile crack growth. I: A numerical studyusing computational cells with microstructurallybased length scales. J. Mech. Phys. Solids 43,233–259.

Δa (mm)

J (k

J m

–2)

a/ W = 0.6W = 50 mmW = 100 mmW = 1000 mm

D = 200 μmn = 0.1f0 = 0.005

1000

800

600

400

200

01 2 3 4 50

Figure 79 Crack resistance curves simulated by thecomputational cell method for three-point bendingspecimens of different sizes (different widths W).From Xia, L. and Shih, C. F. 1995a. Ductile crackgrowth. I: A numerical study using computationalcells with microstructurally based length scales.J. Mech. Phys. Solids 43, 233–259.


As already shown in Figure 38, the maximumstress that a voided ductile solid can attainunder a given stress triaxiality increases withdecreasing initial porosity. This maximumstress is the peak stress used in a cohesive zonetype model. The result of Figure 77a can thus bedirectly related to the predictions of Figure 76by Tvergaard and Hutchinson (1992). A lowerinitial porosity has a strong effect on the tearingresistance by affecting the cohesive strength ofthe FPZ. Increasing the strain-hardening capa-city or decreasing the yield strength tends toincrease the PZ size, hence the plastic dissipa-tion. Note again in Figure 77c, that the fractureinitiation is not affected by the ratio �0/E, inagreement with Figure 74a.

Xia and Shih (1995a, 1995b) have also usedthe computational cell simulations to addressconstraint effects. Figure 78 shows crack resis-tance curves corresponding to different levelof constraints imposed through varying theT-stress values (see Chapter 2.03 for the defini-tion of the T-stress), still using SSY conditions.Loss of constraints are associated to negativeT-stress. The constraint effect on the tearingresistance is very clear, while, in agreementwith the conclusions drawn from Figure 75,cracking initiation is not much affected.Figure 79 shows crack resistance curves belong-ing now to three-point bending laboratoryspecimens. The three different specimens arehomothetic with three different width W.When decreasing the specimen size, the con-straint moves from SSY to large-scale yieldingconditions. In the small specimens, the PZinteracts with the specimen boundaries leadingto significant loss of constraint.

Although the number of comparisons withxperimental data remains limited (seehapter 7.05), these crack resistance curve pre-ictions have granted a considerable success tohe so-called ‘local approach’ (Chapter 7.05) ortop-down approach’ to ductile fractureHutchinson and Evans, 2000) in the commu-ity dealing with the fracture integrity oftructural components under large-scale yield-ng conditions. The success of this approach isnderpinned by the requirement that theicrostructural parameters (the void volumeraction, void spacing, etc.) must be set suchhat the model reproduces experimental crackata for specific specimens (see Chapter 7.05).nce calibrated, these approaches have accu-ately accounted for a wide range of constraintffects. Three-dimensional aspects of cracknitiation and growth have also been simulated,s explained in Chapter 7.05. In that case, theomparisons with experiments are even morecarce (see Gao et al., 1998a; Rivalin et al.,001b; Chabanet et al., 2003; Negre et al.,005).

.06.3.8 Fracture Resistance of Thin MetallicSheets

.06.3.8.1 Introduction to the fracturemechanics of thin metallic sheets

This section extends the presentation given inection 2.06.3.7.1 of the elastoplastic fractureechanics basics to specific aspects related toracked thin sheets. A plate is considered to bethin’ if the PZ size during cracking is largerhan the thickness, thus preventing plane strainonditions to build up next to the crack tip.

L 0

t 0

X0

Diffuse plasticzone

Localized neckingzone

Final microzone ofdamage-induced localization

Figure 80 The DENT geometry with the diffuse and localized PZs (macroscale), the localized necking zone(mesoscale), and the true fracture zone with the void spacing definition (microscale) (see Pardoen et al., 2004).

3 4 5 61 2

HRR ~ 0.6

Stress triaxiality

Increasingsheet thickness

r /δ

Fracture process zone= necking zone

0.6

1.0

MidplaneSurface

Figure 81 Variation of the stress triaxiality in thinductile sheets as a function of the distance to thecrack tip (normalized by the CTOD) for varioussheet thicknesses.


Detailed 3-D finite element simulations ofcracked sheets made of a J2 elastoplastic solidhave been performed by Hom and McMeeking(1989a, 1989b) and Nakamura and Parks(1990) within a finite strain setup in order toinvestigate the stress and strain fields at the tipof a static crack (see also Pardoen et al., 1999).The first key result of these simulations is, inagreement with many experimental observa-tions, the development of localized neckingzone in the near-crack-tip region, as schemati-cally shown in Figure 80. This localized neckingregion will be considered to be the FPZ. Asshown in Figure 80 for the specific case of aDENT (double edge notched tension) specimengeometry, the necking region is surrounded bya large zone of gross plasticity, such as in thickplates. Due to the development of this neck,moderate out-of-plane stress builds up in thenear-crack-tip region.

Figure 81 shows the variation of the stresstriaxiality as a function of the distance to thecrack tip in sheets of various thicknesses, at themidplane, and along the surface. Along the sur-face and away from the crack tip, the planestress HRR solution is recovered (seeFigure 69). But, locally, stress triaxiality risesin the FPZ. Hence, the ‘plane stress’ conditionis an idealization which is only attainedapproximately for the limit of extremely thinsheets. Note that the maximum stress triaxialityprevailing inside the FPZ typically variesbetween 0.6 and 2.0 depending on the thickness,a range in which the work of damage and mate-rial separation G0 is expected to change a lot(see Figure 75). Finally, as first predicted by Hill

(1952), plane strain or near plane strain condi-tions prevail along the neck in the ligamentdirection.

In thin sheets, the total work of fracture G isthus made of three contributions:

�ðDa; tÞ ¼ �0ðDa; tÞ þ �nðDa; tÞ þ �pðDa; tÞ ½129�

where G0 is the more ‘intrinsic’ fracture tough-ness accounting for damage and materialseparation, Gn is the work per unit crackadvance required for localized necking, and Gp

is the extrinsic contribution resulting fromgross plastic dissipation during crack propaga-tion. As is discussed in details in this section, thepresence of the necking work in [129] will be


responsible for the thickness dependence of thefracture toughness in thin sheets.

At cracking initiation (Da¼ 0), G0 XG0init,

Gn XGninit, and Gp¼ 0. The fracture toughness,

quantified by the value of the J-integral atcracking initiation, Jc, involves both damageand necking works:

Jc ¼ �init0 þ �init

n ½130�The so-called ‘essential work of fracture’

(EWF) method introduced by Cotterell andReddel (1977) provides an alternative to thefracture mechanics approach for characterizingthe tearing resistance of thin sheets. The EWFmethod is an experimental method allowing themeasurement of an index we which should, inprinciple, be equal to the steady-state workspent in the FPZ:

we ¼ �SS0 þ �SS

n ½131�This important and revealing method for

thin sheet fracture analysis will be discussed ina first specific section.

After cracking has been initiated, two possi-ble fracture modes, shown in Figure 82, areusually observed experimentally, ‘flat fracture’and ‘slant fracture’. The first mode of fracture,called here ‘flat fracture’, consists in a crackrunning in mode I, usually accompanied withsignificant amount of crack-tip necking. Twosections will be devoted to that (probably notenough studied) fracture mode: Section2.06.3.8.3 provides a specific discussion aboutthe necking contribution which is one of themain peculiarities of thin sheet fracture whileSection 2.06.3.8.4 gives other informationabout this fracture mode and focuses on theimportance of the necking contribution withrespect to the damage contribution leading tothe thickness dependency of the fracture tough-ness in many thin sheet materials. In the ‘slant

(a) (b)

Figure 82 The two usual modes of fractureobserved in thin metallic sheets: a, flat mode Ifracture; b, mixed mode I and III slant fracture.

fracture’ mode, rapidly after initiation the crackplane tilts at 45� giving rise to a mixed modeI–III process. This fracture mode is usuallyobserved in high-strength aluminum alloysand high-strength steels. Section 2.06.3.8.5 pro-vides further details about the conditionsleading to slant versus flat fracture. Finally,the thickness dependence of the fracture resis-tance in thin sheets is addressed, in Section2.06.3.8.6, in order to give general views aboutthe thin versus thick plate regimes.

2.06.3.8.2 The EWF method

The EWF concept was introduced byCotterell and Reddel (1977) as a means ofquantifying the fracture resistance of thin duc-tile metal sheets (see also Mai and Cotterell,1980). The concept is simple (see Cotterellet al. (2005) for recent revisiting of the methodand more references about the method). Thegoal of the method is to separate, based ondimensional considerations, the work per-formed within the PZ (gross plasticity) fromthe total work of fracture in order to providean estimate of the work spent per unit areawithin the FPZ to fracture the material. Theidea of separating the two fracture regions wassuggested by Broberg (1974, 1975). Rememberthat at initiation the J-integral separates thework performed in the FPZ from the plasticwork (Rice, 1968). The EWF concept was intro-duced to tackle ductile fracture not frominitiation measurements, but from the otherextreme of a completely fractured specimen. Ifthe ligament of a sheet specimen is completelyyielded before initiation, and the PZ is confinedto the notched ligament, then the plastic workperformed for total fracture is proportional tothe plastic volume at initiation and the workperformed in the FPZ is proportional to thefracture area. That is, the plastic work and theEWF scale differently. Thus, if a series of geo-metrically similar specimens of different sizesare tested, then the two works of fracture canbe separated. In principle, any specimen geome-try can be used, but, for thin sheets, the DENTgeometry (see Figure 80) is particularly suitablebecause the transverse stress between thenotches is tensile and there are no bucklingproblems. The ligament between the notchesmust completely yield before fracture initiation.In metal sheets the diffuse PZ is almost circularfor metals with reasonably high strain-harden-ing exponents (Cotterell and Reddel, 1977). Forsmaller strain-hardening exponent, the PZ isnarrower and elliptically shaped. The area ofthe PZ and the plastic work performed to com-pletely fracture the specimen is proportional to

1 × 106

8 × 105

6 × 105

4 × 105

2 × 105

0

w 0

(J m

–² )

0 0.5 1 1.5 2 2.5 3 3.5

y = 2.2123e + 05 + 2.0406e + 05x R = 0.99723

Sheet thickness

Γ0ss Γn

ss

Stainless steel AISI 316L

(b)

0

1.4 × 106

1.2 × 105

1 × 106

8 × 105

6 × 105

4 × 105

2 × 105

0 1 2 3 4

t0 = 0.65 mm t0 = 1.49 mm t0 = 3.04 mm

y = 3.4025e + 05 + 38787x R = 0.99272

y = 5.4627e + 05 + 31076x R = 0.96309

y = 8.3418e + 05 + 24556x R = 0.90604

wf (

J m

–² )

l 0 (mm)

we wp

(a)

Figure 83 Application of the essential work offracture method to AISI 316L stainless steel ofdifferent thicknesses, allowing first to: a, to separatethe EWF we from the gross plasticity contribution wp

by linear interpolation as a function of the ligamentlength, and b, separate the damage work G0 from thenecking contribution Gn (results by Marchioni,2002).


the ligament, l0, squared if the FPZ is smallenough. The work performed in the FPZ isproportional to l0. The work of fracture, Wf,can be written as the sum of the essential work,We, and the plastic work, Wp:

Wf ¼We þWp ¼ tl0we þ atl20wp ½132�

where we is the specific EWF, wp is an averageplastic work density, t the sheet thickness, and ais a shape factor. The specific work of fracture,wf, is given by

wf ¼ we þ l0wp ¼ we þ h�pi ½133�

where we¼G0SSþGn

SS. Thus, if different-sizedspecimens are tested, the specific essentialwork is the constant term in the linear evolutionof the specific work of fracture against ligamentlength.

Now, the two components of the specificessential work G0

SS and GnSS of fracture can also

be identified using the same reasoning as forinitial EWF approach (Pardoen et al., 1999,2002, 2004) by assuming that G0

SS is thicknessindependent at low thickness. Indeed, it isobserved experimentally that the fractionalreduction in sheet thickness at fracture is prac-tically independent of sheet thickness for manyductile metal sheets as long as the thickness issmall enough (see Section 2.06.3.8.4).Furthermore, the width of the FPZ is propor-tional to the sheet thickness. The extensionacross the necked FPZ, under plane strainalong the neck, is therefore also proportionalto the sheet thickness and the work of necking isproportional to the square of the sheet thick-ness. Hence, Gn is proportional to the sheetthickness and so

we ¼ �SS0 þ btwSS

n ½134�

where wn is the work of necking per unit volumeand b is a shape factor. Figure 83 shows anapplication to this extended EWF methodogyto AISI 316L stainless steel (see also Pardoenet al. (1999, 2004) and Cottrell et al. (2005) forother applications). We will come back to theanalysis of the thickness effect in Section2.06.3.8.4.

Note that the EWF method can also beapplied to other specimen geometry and othermaterials families (Atkins and Mai, 1985). Ithas for instance received a lot of attention inrecent years in the polymer community to eval-uate the fracture resistance of various polymerfilms (e.g., Mai and Powell, 1991; Chan andWilliams, 1994; Levita et al., 1996; Wu andMai, 1996; Clutton, 2001). The fact that theEWF method does not require crack detectionmethods nor complex extensometry equip-ments makes it easy to implement

experimentally and attractive when workingfor instance at high temperature (e.g., Chehabet al., 2006) or in the presence of aggressiveenvironmental conditions.

2.06.3.8.3 Crack-tip necking work

A simple model for the work spent in crack-tip necking has been worked out by Pardoenet al. (2004). The material obeys von Misesplasticity with the following hardening rule:

ss0¼ ð1þ kepÞn ½135�

where �0 is the yield stress, n is the strain-hard-ening exponent, and k is a parameter that is


usually much larger than 1. Dimensional ana-lysis shows that, for a material with a flowbehavior represented by eqn [135] and for agiven geometry and stress state, the averagework per unit volume wn spent in the neck canbe expressed as

wn

s0¼ F n; k;

s0E; �; ef

h i½136�

where ef is the strain at fracture. As shown byHill (1952), a plane strain tension stress statecan be assumed.

The model (not described here) is closed formexcept for the evaluation of the shape para-meter of the necking region (height overthickness of the active PZ) whose adjustmentrequired conventional 2-D finite element simu-lations already presented in Figure 23. Figure 84shows the variation of the ratio wn/�0k

nXGn/

�0knt0 as a function of the equivalent strain at

fracture ef minus the equivalent necking straineu (corresponding to plane strain tension con-ditions, see Section 2.06.3.2.2) for different n.Indeed, the important factor for the neckingcontribution is not the fracture strain but thedifference between the strain at necking and thefracture strain. The work of necking per unitvolume levels out at high fracture strains as theactive necking zone becomes increasingly small,involving less and less additional plastic work.

Note that plastic anisotropy can significantlyaffect the work necking. For instance, in alumi-num sheets, the Lankford coefficient is lowerthan 1, favoring the through thickness reductionand leading thus to a smaller work of neckingthan the predictions given in Figure 84.

2.06.3.8.4 Flat mode I fracture in thin plates

The perceived wisdom about thin sheet frac-ture is that (1) the crack propagates undermixed mode I and III giving rise to a slant

0

0.1

0.2

0.3

0 0.5 1 1.5 2 2.5

wn /σ

0k n

n = 0.1

n = 0.25

n = 0.4

n = 0.5

ε f – εu

Figure 84 Variation of the average work ofnecking (per unit volume) as a function of thefracture strain minus the necking strain (seePardoen et al., 2004).

through-thickness fracture profile and (2) thefracture toughness remains constant at lowthickness and eventually decreases with increas-ing thickness. In a study by Pardoen et al.(2004), fracture tests performed on thinDENT plates of various thicknesses made ofstainless steel, 6082-O and NS4 aluminumalloy, brass, bronze, lead, and zinc (seeTable 8) systematically exhibit mode I‘bathtub’, that is, ‘cup-and-cup’, fracture pro-files with limited shear lips and significantlocalized necking. Furthermore, the fractureresistance systematically increases with sheetthickness, in a linear way, as anticipated inSection 2.06.3.8.2. The discussion about flatfracture in thin sheets of the present section isessentially based on this work.

For the sake of illustration, Figure 85 showsthat the two matching fracture surfaces of analuminum alloy (NS4) are similar (many othermicrographs can be found in Pardoen et al.(2004) or Rivalin et al. (2001a)). A regular frac-ture surface with dimples is observed along thesides of the specimen without any evidence ofthe shear distortion typical of fracture surfacesresulting from shear failure. As representedschematically in Figure 86, the mode I bathtubfracture profile originates from the difference ofstress triaxiality between the center and the sur-faces of the specimens. The surface is in a pureplane stress state involving thus a fracturestrain larger than in the center where the stresstriaxiality is larger due to necking inducedstress concentration. This significant differencebetween the stress state in the center and alongthe surface leads to the tunneling effect with thecrack length in the center plane longer than thecrack length along the surface by about onethickness when the steady-state regime isattained. The steady-state regime, associatedto constant thickness reduction on the cracksurface, is attained after the crack has propa-gated on a distance equal to about 1 or 2thicknesses.

The following properties of the materialslisted in Table 8 have been systematicallymeasured:

� Flow properties using the hardening law[135]. It is worth noticing the moderate orhigh strain-hardening capacity of all thesemetallic materials.� Thickness reduction factor, rf, defined as

rf ¼t0 � tft0

½137�

where tf is the final plate thickness along thefracture plane. As no significant or systematicdependence of rf on thickness is observed, therf values are averaged over the different

rrBtatcmTtis

Table 8 Properties of ductile metallic sheets with thicknesses in the range 0.5–5mm

MaterialsE

(GPa)�0

(MPa) n k rf

X0

(mm)wn

exp

(MJm�3)wn

model

(MJm�3)G0

exp

(kJm�2)G0

model

(kJm�2)

Steel A316L 210 310 0.48 25 0.48 25–50 204 165 221 172–345Al 6082-O 70 50 0.26 265 0.6 10–20 33 32 28 8–16Brass A 110 100 0.6 33 0.78 5–10 161 195 87 108–217Al NS4 // RD 70 140 0.17 159 0.8 8–15 42 39 51 26–48Zinc // RD 61 100 0.15 118 0.59 25* 64 23 34 16.5Lead 16 7 0.25 290 1 7.3 5.3 0 0Bronze A 100 120 0.51 38 0.7 4–5 218 166 70 51–63

E is the Young’s modulus, �0 is the yield stress, k and n are the parameters in relation [135], obtained by a power law fit on the

uniaxial stress strain curve, rf is the thickness reduction factor defined by eqn [137],X0 is the mean initial void spacing (* mean

grain size for Zn), G0 is the true work of fracture, and wn is the work of necking per unit volume.

(a) (b)

200 μm10 μm

Figure 85 Transverse sections (along the thickness) of fracture surfaces of thin A1 NS4 sheets: a, the bathtubprofile; b, higher magnification showing a regular mode 1 fracture surface with dimples (the voids are notdistorted by shear type localization associated to a slant fracture mode).

Triax = T

TcenterTsides = 0.6 ε f

Figure 86 Bathtub fracture profile (or ‘cup andcup’) resulting from a higher stress triaxiality andthus lower fracture strain in the center of the plate.


thicknesses. It is interesting also to point outthe behavior of lead which fails by full neck-ing, that is, without any apparent damagemechanism. For the other materials, thereduction of thickness is always larger thanabout 50%.

� The fracture surfaces have been observed bySEM in order to estimate the dimple sizesand spacing. The spacing has been quantifiedin the direction of crack advance which,owing to the near plane strain conditions,gives a direct image of the initial defect spa-cing, X0 (see Table 8).� The two contributions to the fracture resis-

tance, G0 and wn, were evaluated using theextended EWF procedure presented in

Section 2.06.3.8.2 from tests performed onmultiple DENT specimen thicknesses andligament lengths (Figure 83 shows the resultsfor the steel A316L). The most striking fea-ture in the measured values reported inTable 8 is that the work of necking per unitarea Gn (i.e., wn multiplied by the initialthickness of the plate) is at least similar andusually larger than G0 for thicknesses in themillimeter range. This implies that the frac-ture resistance quantified by G0þGn

significantly increases (linearly) with thick-ness, in the investigated range of thicknesses.

The thickness dependence of the fractureesistance in thin sheets, although it has noteceived much attention in the literature (e.g.,luhm, 1961; Swedlow, 1965), is a very impor-ant effect that must be properly understoodnd controlled in order to design fail-safe struc-ures made of thin sheets and in order to allowomparing the fracture resistance of differentaterials processed with different thicknesses.he thickness dependence directly results fromhe necking work. A simple model for the neck-ng work has been proposed in the previousubsection. It remains now to develop a model

Γ0 /J Ic

Γ0 /σ0X 0

n = 0.1

n = 0.3

n = 0.5

n = 0.1

f 0

0.1

1

10

100

1000

10–5 10–4 10–3 10–2 10–1 100

Figure 87 Variation of the predicted fracture energyas a function of the initial porosity for differentstrain-hardening exponents; variation of the ratio ofthe plane stress fracture energy over the plane strainfracture toughness as a function of the initialporosity (see Pardoen et al., 2004).


for G0 in order to capture the relative influenceof the necking work and the importance of thethickness effect as a function of the flow prop-erties and microstructural features.

The model presented aims at calculating theenergy G0 spent for the growth and coalescenceof voids in front of a crack tip following thereasoning developed in Section 2.06.3.7.2 forthick plates, but now in the case of thin sheets.Based on the schematic drawing of Figure 75,we expect nowmuch larger values of G0 becausethe stress triaxiality is much lower (seeFigure 81). We again assume, as shown inFigure 73, a material made of regularly distrib-uted voids (the voids are supposed to be presentfrom the beginning of the loading) with initialspacing X0 and initial volume fraction f0. Onlyvoids that are initially spherical will be consid-ered. Dimensional analysis for the hardeninglaw [135] gives

�0

s0X0¼ F n;

s0E; k; �; f0

� �½138�

In order to simplify the analysis, the complex3-D stress state existing at the crack tip will beapproximated by plane strain tension allowingfor necking development (see Hill, 1952). The2-D finite strain simulations have been per-formed using the same constitutive materialresponse as for the thick plate case (i.e.,Gologanu and Thomason models as presentedin Sections 2.06.3.4.4 and 2.06.3.5.4). The impor-tance of using a damage model which properlyincorporates void shape changes is more impor-tant than for thick plates. Indeed, the low stresstriaxiality in thin sheets leads to significant voidelongation. G0 is evaluated in the most loadedelement, that is, the element located in the centerof the minimum section, from the calculatedstress displacement response.

The model has been validated in the follow-ing way. The initial void volume fraction f0 wasidentified by simulating the uniaxial tensile testsand finding the value that allows reproducingthe experimental fracture strains. The void spa-cings X0 measured experimentally (see Table 8)have been used to estimate G0. The experimen-tal and predicted G0 are compared in Table 8.These results remain semiquantitative due tothe numerous approximations in the modeland the experimental uncertainty but the maintrends are captured.

For a given geometry, loading configuration,and a constant product �0X0, the two mostimportant parameters affecting ductile fracture,that is, affecting the nondimensional function Fin [138], are the initial void volume fraction f0and the strain-hardening exponent n. The otherparameters that have been kept constant are:�0/E¼ 1/k¼ 10�3, �¼ 0.3. Figure 87 presents

the variation of G0/X0�0 as a function of f0 forn equal to 0.1, 0.3, and 0.5. The effect of both nand f0 is obvious: large n and low f0 significantlyincrease G0.

The ‘plane stress fracture energy’ G0 is alsocompared to the plane strain fracture toughnessJIc in Figure 87 for n¼ 0.1. To a first approx-imation the ratio G0/JIc is not significantlyaffected by the initial porosity, nor by thevalue of the strain-hardening exponent (resultnot shown). It ranges between 2.5 and 3.5 whichagrees very well with full 3-D finite elementsimulation of crack propagation in thin andthick HSLA steel plates reported by Rivalinet al. (2001b). The plane stress fracture energyis significantly larger than the plane strain valuebecause of the much smaller stress triaxiality,which involves smaller void growth rate. To ourknowledge, there exist no noncontroversialexperimental data in the literature to assessthis prediction. The difficulty is to find a mate-rial which will be (1) ductile enough to show thenecking mechanism at small thickness but (2)not too ductile, otherwise it is not possible tomeasure plane strain fracture toughnessvalues because the required thickness forvalid measurements would be too large. Wewill come back to the general analysis of thinversus thick plate fracture resistance inSection 2.06.3.8.6.

At this point, it is possible to come back tothe problem of the coupling between crack-tipnecking and crack-tip damage. Figure 88 gath-ers the results of Figures 84 and 87 in terms ofthe variation of the ratio G0X0k

n/Gnt0 as a func-tion of f0. The proportion of damage andnecking contributions in the work of fracture

Fatigue surface


depends very much on n and f0 and linearlyscales with X0/t0. Figure 88 exhibits two limits:one when f0 ! 0.1–0.2 and one when f0 ! 0.The first limit corresponds to highly porousmaterials where the fracture strain becomessmaller than the necking strain and thus Gn isequal to 0. This limit is attained for very largeinitial porosity (>0.1) that is not encounteredin typical industrial alloys. The other limit whenf0 ! 0 also leads to fracture toughness that ismainly controlled by the damage mechanisms,because Gn then saturates at large fracturestrains (see Figure 87). However, this limit isnot really meaningful as the model is no longervalid when f0 tends to zero and thus X0 goes toinfinity in finite specimens. Indeed, the thick-ness of the plate sets a second length scale. Thusthe real limit for f0! 0, which probably corre-sponds to the experimental results obtained forlead (see Table 8), is that G0 tends to 0 and thatfracture is then only controlled by plastic neck-ing. In between these two limits, a localminimum appears over the range of porosity.This minimum corresponds to the maximumpossible amount of necking dissipation withrespect to damage. Figure 88 also shows themarked effect of the strain-hardening capacityon the energy partitioning. Increasing the strain-hardening capacity affects much more G0 thanthe necking contribution for a given ratio X0/t0.The wide variety of behaviors that can bededuced from Figure 88 probably explains thewide variations of apparent properties encoun-tered in thin plates and the difficulty ofrationalizing experimental measurements fromonly a macroscopic point of view (e.g., Broek,1978).

The summary of this section is that the workof necking (per unit area) Gn (1) scales linearlywith thickness, (2) depends on the strain-hard-ening exponent, and (3) increases with thefracture strain to reach a constant value at

10

10–5 10–4 10–3 10–2 10–1

100

1000

10 000

Γ 0t 0

k n / Γ

nX0

f 0

n = 0.3

n = 0.5

n = 0.1

Figure 88 Variation of the ratio of the fracture andthe necking work as a function of the initial porosityfor different strain-hardening exponents.

large fracture strains. The work of fracture(per unit area) G0 scales linearly with the yieldstress and void spacing, and strongly dependson the initial porosity and strain-hardeningexponent.

2.06.3.8.5 Competition between flatand slant fracture

The most important structural metallicsheets, that is, high-strength aluminum alloysin an age-hardened state, high-strength steels,and the classical TA6V titanium alloy, exhibit aslant fracture mode (e.g., Irwin et al., 1958;Krafft et al., 1961; Zinkham, 1968; Allen,1971; Knott, 1973; Krambour and Miller,1977; Broek, 1978; Atkins and Mai, 1985;Sutton et al., 1995; Taira and Tanaka, 1979;Rivalin et al., 2001a; Mahmoud and Lease,2003; James and Newman, 2003; Chabanetet al., 2003; Asserin Lebert et al., 2005). Asdepicted in Figure 89, the crack starts propagat-ing in a flat mode I with significant tunneling.After moderate amount of crack growth, shearlocalization sets in and the crack plane tilts atan angle which depends on the plastic aniso-tropy of the material (45� in plastically isotropicmaterials). As a result, the fracture surfaceshows a first ‘triangle’ of flat fracture beforethe profile becomes slanted.

The following conditions have been shown tofavor slant fracture:

(i) Flow properties

� Low strain-hardening capacity. As explainedabove, most thin sheets made of high-strength alloys exhibit a slant fracturemode. It is a common rule in materials

Slant fracture

Tensilemode

Tunneling

Δas

a

Crackfront

B

W2

Figure 89 Typical slant fracture surface. FromJames, M. A. and Newman, J. C., Jr. 2003. The effectof crack tunneling on crack growth: Experiments andCTOA analyses. Eng. Fract. Mech. 70, 457–468.

�

(

�

�

(

�

�

(�

�

tmetHgtilcctf3tshtcMtwm(

2

biaoctitoibhtPiawitaaDiocpt


science that high strength, attained by coldrolling and/or proper aging treatment lead-ing to optimal precipitation hardening, isusually associated to low strain-hardeningcapacity. For instance, as shown byAsserin-Lebert et al. (2005), the sameAl6056 alloy exhibits a slant profile in theage-hardened heat-treated condition T751(�0¼ 300MPa, n� 0.06) but a flat fracturemode in the annealed state (�0¼ 70MPa,n� 0.2).Plastic anisotropy. Sutton et al. (1995)reported that the fracture of a 2024-T3 Alsheet was slant when loaded in a LT config-uration but flat when loaded in a TLconfiguration. The strain hardening capacitywas slightly larger in the TL orientation. It isnot clear whether it is the only reason toexplain the change of fracture mode.

ii) Microstructure

Second population of voids. Bron et al. (2004)have shown for a 2024 T4 Al alloy that theslant regions were covered by both primaryand secondary dimples whereas the initial flattriangular region had only primary dimples.The presence of secondary voids favors thetransition from internal necking coalescenceto the void sheet shear-type coalescencemechanisms (under specific loading condi-tions), which is naturally related to thetransition to a slant fracture mode (see alsoSection 2.06.3.5 on void coalescence).One grain along the thickness. Slant fractureis favored when only one grain is presentalong the thickness.

iii) Geometry

Small thickness. Increasing the thicknesschanges the stress state at the crack tip byincreasing the stress triaxiality, leading to adecrease of the propensity toward slant frac-ture (see Asserin-Lebert et al., 2005).Flat surfaces. Side grooves (rather than flatsurface) are machined in order to force thecrack to remain flat.

iv) Loading conditionsLow load biaxiality. Although no evidencehave been found in the literature, adding aload transverse to the crack plane wouldprobably decrease the propensity towardslant fracture by increasing the constraint inthe near-crack-tip region.High loading rates. As shown by Rivalinet al. (2001a) in an application on a high-strength X70 ferritic pearlitic steels, highloading rates can lead to adiabatic shear

bands and slant fracture, whereas at lowloading rates fracture remains flat.

As a matter of fact, all these conditions tendo favor plastic shear localization, with theost important factors being a low strain-hard-ning capacity and a low constraint. Althoughhe problem can be addressed theoretically (seeahn and Rosenfield, 1965), formulating aeneral predictive criterion for shear localiza-ion in cracked sheets is a formidable task thats most probably outside the scope of any ana-ytical theoretical developments. Theomplexity is related to the finite strain 3-Dharacter of the boundary value problem ando the complexity of realistic elastoviscoplasticlow laws of metallic materials. Hence, only full-D numerical simulations can provide quanti-ative insights into the competition betweenlant and flat fracture in thin sheets. Thereave been only a few number of works devotedo the simulation of the slant fracture able toapture the complex initial crack tilting; seeathur et al. (1996) for 3-D dynamic simula-

ions involving two populations of voids asell as adiabatic heating and associated ther-al softening effects; see also Rivalin et al.2001b).

.06.3.8.6 General views about thicknessdependence of fracture resistance

This section, which goes beyond the analysisy Pardoen et al. (2004), aims now at synthesiz-ng the messages conveyed in Sections 2.06.3.7nd 2.06.3.8 about the ductile tearing resistancef ductile metals. A generic crack resistanceurve of a ductile metallic plate (sufficientlyhin to exhibit some amount of crack-tip neck-ng) is shown in Figure 90 with the variation ofhe different contributions to the overall workf fracture. As explained earlier, crack-tip neck-ng increases with increasing crack advanceefore reaching a steady value after the crackas propagated for a distance equal to one orwo thicknesses (see Cotterell and Reddel, 1977;ardoen et al., 1999, 2004). The work of neck-ng thus increases with increasing crackdvance to reach a constant value noted Gn

SS

hen Da¼ 1–2 t. The fact that the fracture strainncreases with increasing crack advance meanshat the damage work also changes, as a result ofchange of stress state. Hence, G0 also moder-tely evolves and reaches a value G0

SS whena¼ 1–2 t. A change of G0 with crack advances also expected in thick plates due to the changef constraint during crack propagation whenompared to cracking initiation. This change isrobably small and could be a decrease ratherhan an increase of G0. The gross plastic


dissipation Gp keeps increasing (and could theo-retically reach a steady-state value too if largeenough samples are used). The EWFwe¼Gn

SSþG0SS is also added on Figure 90.

Now, the generic evolution of the crackingresistance characterized by the sumG0þGn¼Gc,either at cracking initiation or during propaga-tion as a function of the plate thickness, is shownin Figure 91. The thickness is normalized by thetheoretical SSY PZ size. The fracture resistanceGc is normalized by the product �0X0. Thesecurves are anticipated from the discussions ofthe last two sections (note that generic curvesshowing the toughness increasing and thendecreasing with increasing thickness are shownin several classical fracture mechanics textbooks(e.g., Broek, 1978; Atkins and Mai, 1985;Barsom and Rolfe, 1987). Nevertheless, to ourknowledge, fully valid fracture resistance curvescovering the whole range of behavior have notyet been obtained experimentally nor simulatedfor the full range of thicknesses. The reasoningwas the following.

1–2Δa/t0

we

Γpss

Γ0ss

ΓnssΓn

Γninit

Γ0init

Γp

Γ

Jc

Γ0

Figure 90 Generic crack resistance curve of aductile metallic plate.

1

planestress

A

B

C

C−Γ0

C−Γn

Γc

Γ0

σ0x0

σ0x0

t /ryssy

α1

α2

Figure 91 Tentative generic variations of the fracture toa function of the plate thickness for slant fracture (curv

First, the transition between the quasi-planestress and plane strain regime occurs when thethickness is roughly equal to PZ size ry esti-mated by

rSSYy � 1

10

KIc

s0

� �2

¼ 1

10

E

s0

JIcs0

½139�

This transition corresponds to the significantrise of the stress triaxiality in the FPZowing to the increase of the through-thicknessstress.

Then, we distinguish between slant fracture(curve A) and flat fracture (curves B and C).Slant fracture usually involves lower fracturestrain in the FPZ due to the appearance ofshear localization bands with high internal con-straint. Necking can thus not significantlydevelop and the increase of the fracture resis-tance is expected to be limited before itdecreases down to the plane strain limit. Thisis confirmed qualitatively by the results ofAsserin-Lebert et al. (2005) on as- receivedhigh-strength low hardening 6056 Al alloysshowing slant fracture with minor increase ofthe fracture resistance at small thickness(<3mm) and a significant drop for larger thick-nesses (>3mm). Note also that Zinkham (1968)reported an increase of the fracture resistancewith increasing thickness in a 7075-T6 Al alloyshowing a slant fracture mode, while thedecrease at larger thickness has been measuredby several authors (Bluhm, 1961; Taira andTanaka, 1979; Lai and Ferguson, 1986; Guoet al., 2002). Now, the level of the slant fracturecurve cannot be compared to the level of the flatfracture curve, even expressed in normalizedterms. Among others, even for the same mate-rial differing only by the yield strength andstrain-hardening exponent (e.g., heat-treated

planestrain

A Slant fracture (tentative)

B

C

Flat fracture with large σ0 small n

Flat fracture with small σ0 large n

wnE30kn1α =

Γ0σ0x0

σ0σ0kn

=JIc

ughness (either at initiation or during propagation) ase A) of for flat fracture (curve B and C).

DBT in Ferritic Steels 771

Al alloy), the representative X0 might be differ-ent due to different damage mechanisms.

More detailed considerations can be elabo-rated for the flat fracture mode. We start byexpressing the normalized fracture resistance asa function of the normalized thickness in thesmall thickness range (where G0 is almost con-stant and equal to the plane stress lowtriaxiality G0

P� limit):

�c

s0X0¼ �Ps

0

s0X0þ �n

s0X0¼ �Ps

0

s0X01þ s0

�Ps0

wn

s0t0

� �

¼ �Ps0

s0X01þ 1

30

knE

s0

wn

kns0

t0ry

� �½140�

where relationship [139] has been used as well asG0P�� 3JIc (see Section 2.06.3.8.4). The slope of

the initial linear increase of the fracture resis-tance is thus given by the product

slope X a � 1

30

knE

s0

wn

kns0½141�

This expression reveals several limiting cases.First, if the metal presents a low fracture strain,ef is not much larger than eu (a good example isprovided by some multiphase TRIP steelswhich show excellent ultimate tensile strainbut limited postnecking ductility; see Jacqueset al., 2001). In that case, the amount of neckingremains limited and the behavior will ressemblecurve B.

Secondly, if the metal is sufficiently ductile,the necking term wn/k

n�0 does not depend onthe fracture strain (see Figure 84), that is, on thedetails of the microstructure. It only dependsmoderately on the strain hardening. The domi-nant term in [141] is thus knE/�0. For instance,between a moderately strain-hardening mate-rial such as Al NS4 and a high strainhardening metal alloy such as brass (seeTable 8), this last term can increase by almosta factor of 10 (high strain hardening is usuallyassociated to low-yield-strength alloys), whilewn/k

n�0 only increases by a factor of 2. Thesetwo extreme situations correspond to curves Band C in Figure 91. Curve B belongs to high-strength low-hardening alloys (such as in mostnonannealed cold-formed metals). In that case,the necking contribution will be limited. In fact,a continuous decrease of the toughness withincreasing thickness will probably be measuredin such materials. Indeed, G0 is affected by thestress state. With increasing thickness, stresstriaxiality will tend to increase at the crack tip,to accelerate the void growth rate with respectto the plane strain tension situation, to decreasethe fracture strain, and thus to lead to adecrease of G0. This can justify results (e.g.,Broek, 1978), where the toughness is observedto always decrease with increasing thickness

even at small thicknesses. Curves C belong tolow-strength high-strain-hardening materials(such as in most FCC annealed pure metalsand alloys), like most materials of Table 8. Inthose materials, the fracture toughness mark-edly increases with thickness, while remainingin the quasi-plane stress regime (see Table 8 orFigure 83b). It is thus not surprising that the G0

contribution has not been detected in somematerials (e.g., Mai and Powell, 1991). In suchmaterials, the fracture resistance is essentiallycontrolled by the necking contribution.

Again, we have to be cautious when inter-preting Figure 91. First of all, as explainedabove, these curves have never been measuredexperimentally in a fully conclusive manner.Second, several metals can show a transitionof fracture mode (from slant to flat or viceversa) when changing the thickness (seeAsserin-Lebert et al., 2005) or when using ornot side-grooves. Finally, only full 3-D calcula-tions are capable of describing either thetransient or to encompass both thin and thicksheets, as well as to better capture the competi-tion with the slant fracture mode.

2.06.4 DBT IN FERRITIC STEELS

2.06.4.1 Introduction

This part is devoted to the analysis of themacroscopic transition in fracture mode whichis observed in ferritic steels. This transition isalso observed in other BCC metals and in someHCP metals which are susceptible to cleave atlow temperature. The situation correspondingto ferritic steels is selected because of its theore-tical and practical importance. This section doesnot deal with the microscopic transition between‘blunted’ cleavage and ‘pure’ cleavage since thistopic has already been briefly discussed whendescribing cleavage theories (see Section 2.06.2).

The competition between ductile tearing andcleavage fracture controls the macroscopic frac-ture toughness of ferritic steels in the upper partof the DBT regime. Ductile crack growth (DCG)can occur under increasing load, and the struc-ture can withstand a significant amount of stableductile tearing without substantial loss of load-bearing capacity. Cleavage fracture, on the otherhand, leads to catastrophic failure of structuralcomponents, and the onset of cleavage is thecritical mechanism limiting the load-bearingcapacity of the structure. Here it is worth notingthat, in this part, the DBT transition is onlyanalyzed when cleavage fracture is preceded bysome DCG. Cleavage fracture occurring duringblunting, that is, for a stationary crack, is notconsidered, since this has been largely discussedelsewhere (see Chapter 7.05).


It is well known that theDBTT in ferritic steelsis strongly affected by the strain rate in smoothand notched specimens or the loading rate incracked geometries. The DBTT decreases whenthe strain rate or the loading rate are increased.The transition at relatively low loading rate isusually investigated using fracture toughness spe-cimens. The transition in these specimenscorresponds to relatively simple loading condi-tions which are quasi-static and isothermal. Thetransition temperature is defined for a givenvalue of the stress intensity factor (for instance,K¼ 100MPam1/2). The DBT at high loadingrates is usually investigated by using impactCharpy V-notch specimens. This includes themeasurement of the total energy absorbed tofracture and the determination of the DBTT.The transition temperature is defined for agiven level of the Charpy (CVN), for instanceCVN¼ 57 J. Charpy impact testing has remainedfor a long time essentially a technological toolwhich has proved to be extremely useful forranking the fracture properties of materials.However the correlations between the Charpyenergy and the fracture toughness have remainedlargely empirical. It is only recently that the localapproach to fracture has been used for a betterunderstanding of the Charpy test. A recent con-ference for the centenary anniversary of theCharpy test has been devoted to this topic(Francois and Pineau, 2002).

In this section, the DBT behavior observed infracture toughness tests is described first.Simplified and more advanced models basedon the study and the numerical simulation ofthe fracture micromechanisms are presented.Then the results obtained from recent studiesdevoted to the analysis of Charpy V impacttests are presented.

2.06.4.2 DBT in fracture toughness tests


A simple explanation for the existence of atransition is that the stress–strain curvedecreases with increasing temperature, whichresults in decreasing normal stress ahead of acrack tip with increasing temperature.Therefore, in the transition regime, there existsa temperature at which the maximum normalstress ahead of the crack tip reaches therequired cleavage fracture stress. Ductile tear-ing can then be initiated which can transforminto cleavage fracture due to the increase of thenormal stress associated with crack growth.However, no single explanation can be givenfor this change in failure mechanism from duc-tile rupture to cleavage fracture. A combination

of different causes seems to be more realistic.Possible causes for this change are:

1. The stress level ahead of a growing ductilecrack is higher than the stress ahead of a staticblunted crack.

2. The volume of material, with a high max-imum normal stress, which is sampled duringthe fracture process, increases as the crackextends. This results, according to the weakestlink theory for cleavage fracture, in an increas-ing probability of cleavage fracture. However,due to the competition between ductile damageand cleavage fracture initiated from second-phase particles, the number of cleavage initia-tion sites may decrease due to ductile voidformation. This competition between ductileand cleavage fracture implies that some modi-fications to the original cleavage theories whichwere presented previously (Section 2.06.2.2)must be made.

The two possible causes for the change infracture mode in the transition region are ana-lyzed and discussed successively. A simple (andnaıve) approach to the prediction of the DBTbehavior is introduced first. Then, more sophis-ticated models accounting for the selection ofpotential cleavage initiation sites are presented.The application of these models to a number offerritic steels is illustrated.

2.06.4.2.2 A simplified approach

One simple, but likely too naıve, approach tothe prediction of the DBT behavior is to con-sider the existence of a unique ductile tearingresistance curve giving the value of J-integral(or better J/�0 to account for a slight tempera-ture dependence) versus crack length, and tocalculate the probability to fracture, using, forinstance, the Beremin model for various station-ary crack lengths, as schematically shown inFigure 92a. In this figure, it is schematicallyshown that the value of J, at which the DBToccurs (for a given probability), increases withtemperature, essentially because of the decreaseof the yield strength due to the increase in tem-perature. This approach is easy to implementsince the probability to fracture is calculatedusing a postprocessing procedure. The methodrequires to determine experimentally or to simu-late numerically the J–Da curve at a temperatureat which cleavage fracture does not occur andthen to normalize the value of the loading para-meter J by the corresponding yield strength.

This simplified approach which does notaccount for the details associated with a propa-gating crack was applied to A508 RPV steel(Amar and Pineau, 1987) using cracked round

Δa (mm)

Δa (mm)

J/σ y

(m

m)

J/σ y

(m

m)

T1 < T2 < T3

T1

T2

T3

PR = 0.10

PR = 0.90

PR = 0.10

PR = 0.10

PR = 0.90

PR = 0.90

T = –50 °C

T = –50 °C

T = –20 °C

T = –20 °C

T = –80 °C

T = –80 °CT = –100 °C

00

1

2

3

A508

0.25 0.5 0.75 1 1.25 1.5 1.75

0.90

0.90

0.10

0.10

0.10

0.90

σu = 3000 MPa

(a) (b)

Figure 92 J-resistance curve. a, Sketch showing the probabilities to cleavage fracture of 10% and 90% aftersome ductile crack extension at three increasing temperatures, T1, T2, T3; b, experimental results on a pressurevessel steel (A508) (Amar and Pineau, 1987).


bars which were tested at four temperatures:�100, �80, �50, �20 �C. Figure 92b reportsthe experimental results and the probability tofracture predicted from the Beremin model. It isobserved that this simplified approach givestheoretical predictions which are in reasonableagreement with the experiments. This situationmight be partly fortuitous because of the roughapproximations made in the analysis. A similarapproach was adopted by other authors on aC–Mn steel (Fe 510Nb), tested at �170 �C(Koers et al., 1995). These authors tested four-point single edge notched bend specimens todetermine the fracture properties of the mate-rial, and applied the Beremin theory to predictcleavage fracture occurring after some ductiletearing. They showed that, in their material,the calculated fracture probabilities weresignificantly larger than those determinedexperimentally, at least for small ductile crackextensions (Da 1.5mm). However, no expla-nation was given about how crack advance dueto crack blunting effect was taken into account.The difference in the conclusions drawn bythese two groups of researchers might be dueto the definition in crack extension and crackblunting effect or, more likely, to the differencein materials and specimen geometries. In parti-cular, as shown by Scibetta et al. (2000), thecircumferentially cracked round geometryused in the study on A508 steel maintains ahigh constraint effect, which is not the situationmet with bend specimens used in the study onFe 510Nb steel. This insight partly explains thedifferences in both studies.

As already stated, this simplified approachinvolves strong limitations since it does not

account for the modifications in the stressfield ahead of a propagating crack and doesnot rely on a detailed study of the microme-chanisms accompanying crack growth. Inparticular, this approach cannot providegood results when cleavage fracture is con-trolled by the nucleation of microcracksinitiated from particles, which was likely themechanism in Fe 510 Nb steel tested at verylow temperature. On the other hand, theassumptions behind this approach are prob-ably better verified when cleavage fracture iscontrolled by the propagation of microcracksarrested at grain (or packet) boundarieswhich is more likely when the test tempera-ture is increased.

2.06.4.2.3 Advanced models

(i) Stress profiles ahead of a growing crack

The starting point for any analysis of thebrittle-to-ductile transition is to rely on asound model for ductile tearing. These modelshave been presented in details in Section2.06.3.7. Only the points specific to the under-standing of the conditions leading to thetransition into cleavage fracture are elaboratedhere. As explained in Section 2.06.3.7.1, thetheoretical work by Rice and Sorensen (1978)has shown that, under SSY conditions and foran elastic–perfectly plastic nondamaging mate-rial, the main difference in the stress–strain fieldbetween a stationary and a growing crack lies inthe strain singularity and not the stress profileat the crack tip. The Prandtl slip-line field isthought to apply also to a propagating crack

1500

1000

500

00

(a)

(b)0

0

1

2

3

4

5

10

σ 22/

σ 0

W = 50 mma/W = 0.60a/W = 0.25a/W = 0.10

Δa = D Δa = 10D Δa = 20D

20

X1/D

30 40

1 2 3Δa (mm)

W = 50 mm (TPB)a/W = 0.1

a/W = 0.25

a/W = 0.6

J (K

J m

–2)

4 5

Figure 93 a, Distribution of tensile stress �22 aheadof a propagation crack for Da¼D, Da¼ 10D, andDa¼ 20D and for three different ratios, a/W, ofinitial crack length to specimen width; b, DCGresistance curves for three different a/W ratios inthree-point bend specimen, with W¼ 50mm;E/�0¼ 500, n¼ 0.10, v¼ 0.3; D¼ 200 mm. Source:Xia, L. and Shih, C. F. 1995a. Ductile crackgrowth. I: A numerical study using computationalcells with microstructurally based length scales.J. Mech. Phys. Solids 43, 233–259. Xia, L. andShih, C. F. 1995b. Ductile crack growth – II. Voidnucleation and geometry effects on macroscopicfracture behavior. J. Mech. Phys. Solids 43, 1953–1981. Xia, L. and Shih, C. F. 1996. Ductile crackgrowth. III: Transition to cleavage fractureincorporating statistics. J. Mech. Phys. Solids 44,603–639.


while the strain singularity is much lower, thatis, varying as ln(r), than that corresponding tothe HRR field for a stationary crack. The nor-mal stress ahead of the crack tip remains closeto 3 times the yield strength of the elastic–per-fectly plastic material with a yield strength �0.These theoretical results apply to a crack whichdoes not give rise to blunting effect. Thisassumption is far from reality. As alreadyshown in Figure 71, the crack tip is blunted atcrack initiation while during propagation theunzipping process from one inclusion toanother one gives rise to a crack-tip profilewhich is much sharper.

The detailed simulations by Xia and Shih(1995a, 1995b, 1996), already discussed inSection 2.06.3.7.3, have shown that, duringDCG, the maximum tensile stress, �22, aheadof a simulated propagating crack, increaseswith crack extension, as shown in Figure 93.These results were obtained from numericalsimulations of three-point bend specimens(W¼ 50mm) in a given material; for example,the ratio between Young’s modulus and yieldstrength is equal to 500 and the work-hardeningexponent is equal to 0.1. The mesh size, D, wasequal to 300 mm, and ductile damage was simu-lated using the version of the Gurson potentialenhanced by Tvergaard and Needleman (seeSection 2.06.3.4). Three crack depths corre-sponding to a/W¼ 0.10, 0.25, and 0.60 weresimulated. In this bend specimen geometry,the constraint effect is largely dependent onthe crack depth. Figure 93 shows that at theearly stage of crack growth, Da¼D, the max-imum tensile stress for a/W¼ 0.25 is lower thanthat for a/W¼ 0.60. When the stress level for a/W¼ 0.25 is followed for increasing cracklengths, it is observed that �22 increases quicklywith crack growth, and at Da¼ 20D¼ 6mm,the peak stress has reached the level fora/W¼ 0.60. A similar effect is observed fora/W¼ 0.1 although the effect is less pro-nounced. This steady elevation of crack-tipconstraint with ductile crack extension canthen increase the risk of cleavage fracture.

This effect of the stress elevation duringcrack growth is likely less pronounced in speci-men geometries in which the constraint is lessdependent on the crack length, such as tensilespecimens with one single edge crack, as shownby Xia and Cheng (1997). For further discus-sion on the effect of a growing crack on stressprofiles at crack tip, see also Dodds et al.(1997), Tanguy (2001), and Tanguy et al.(2002a, 2002b). It appears therefore that theDBT behavior will be strongly dependent onthe specimen geometry. Cleavage fracture willbe favored in bend specimens in which theinitial crack size is relatively small. This is

the situation which will be illustrated later inthe analyses of Charpy V-notch specimens.

The introduction of damage ahead of thecrack tip produces a reduction of the ‘local’stresses, when using for instance the GTNmodel (see Section 2.06.4.3.2) to simulateDCG. This softening effect and its consequenceon the calculation of the probability to cleavagefracture has been quantified by Busso et al.(1998). However this reduction of the ‘local’stress is macroscopic. At metallurgical scalesmuch smaller than the cell size, D, formationand growth of the macroscopic cell voids driv-ing ductile crack extension likely alter andamplify the local stress fields acting on thesmaller particles that can trigger and control

A C

y

x

aa +Δa

Figure 94 Sketch showing the PZs ahead of aninitial crack (a) and after crack growth (aþDa).The plastic wake left behind the propagating crackis shown.


cleavage fracture. This local stress amplifica-tion has been studied recently by Petti andDodds (2005b). These authors used a very sim-plified model consisting in cylindrical inclusionsparallel to the crack front and extending overall the specimen thickness. The effect of localstress intensification due to the presence of duc-tile cavities has been evidenced experimentallyby a number of authors; see, for example,Carassou (1999) and Carassou et al. (1998).These authors showed that, under given cir-cumstances, cleavage cracks in A508 RPVsteel were initiated from small carbide particleslocated around cavities initiated from largerinclusions. This effect of stress intensificationon cleavage fracture will be more pronouncedat low temperature and in materials containinga significant amount of large inclusions whencleavage is controlled by the nucleation ofmicrocracks from particles. This influence ofductile damage on cleavage fracture appearstherefore opposite to the softening effectdescribed earlier. Only detailed metallographi-cal observations on the position and the natureof the cleavage initiating sites can be used todifferentiate these two conflicting effects oflocal ductile damage on the DBT behavior.

(ii) Sampling effect due to crack growth

The potential FPZ for an extending cracknow comprises two distinct zones dependingon the material stress history. Figure 94provides a schematic illustration of thesezones, denoted as A and C. Material in theunloaded zone A behind the current physicalposition of the tip experienced severe stress andstrain fields without triggering cleavage frac-ture. Due to the reduced blunting at thelocation of the physical crack tip, the peakvalue of opening mode stress develops at asmall distance (roughly the blunted opening)ahead of the tip. Material located outside theblunting region (zone C) experiences increasedstresses involving the generation of new micro-voids and cracked carbides due to progressivedeformation. Catastrophic cleavage fracture iseventually initiated in this zone not onlybecause the stresses due to crack growth areincreased but also because of an increasingprobability to contain larger microdefectswhich were not sampled during the earliercrack extension.

The probability to cleavage fracture is anincreasing function of the volume of materialexperiencing large stresses (see Section2.06.2.4). The volume of material hatched inFigure 94 must be taken into account. Thisintroduces a correction of DCG (see Brucknerand Munz, 1984; Wallin, 1989, 1991a, 1991b,

1993). The basic assumptions for the DCG cor-rection have been presented by Wallin (1989).The volume increment due to both increase inloading parameter KI¼ f(Da) as well as crackgrowth, Da, leads to an increase of the Weibullstress (see eqn [27]). The probability to fracture,PR, can thus be written as

Ln1

1� PR

� �¼ ðfðDaÞÞ

4

K40

þ2s2fK4

0B

Z Da

0

ðfðDaÞÞ2dDa

½142�

where K0 is a normalizing value for the stressintensity factor which can easily be calculatedunder SSY conditions (Beremin, 1983), and �fis the fracture (cleavage) stress. B is the thick-ness of the specimen.

The above expression of the DCG correctionproposed by Wallin (1993) is not unique.Another DCG correction has been proposedby Bruckner and Munz (1984). Their expres-sion can be written as

Ln1

1� PR

� �¼ Ki

K0

� �4

þ 1

K40Wi

Z Da

0

ðfðDaÞÞ4dDa

½143�

where Ki is the value of the stress intensityfactor corresponding to the initiation of DCG,and Wi is a constant describing the size of theactive volume.

Equations [142] and [143] are quite similar.The difference is that Wallin assumes that theeffective volume continues to grow as a func-tion of (KI

2)2 even after the onset of ductilecrack extension, whereas Bruckner and Munzassume that the size of the active volume isconstant during crack growth. In both cases,eqns [142] and [143] require the evaluation ofthe crack growth integrals. This means that the


R-curve must be known. In order to overcomethis difficulty, a simplified expression of theDCG correction has been proposed by Wallin(1993a, 1993b). If the DCG is independent ofKI, the DCG correction can be simplified andwritten as

Ln1

1� PR

� �� 1=4

¼ Ki

K01þ

2Das2fK2

i b

!½144�

where b � x/(Ki/�f)2 defines the cleavage FPZ

size. When the crack growth is small or the R-curve is relatively flat or both, eqn [144] caneasily be used to approximate eqn [142] byreplacing KI by Ki.

(iii) Elimination of eligible particles

As already stated in Section 2.06.2.3,enhancements of the Beremin model havebeen proposed to account for continuousnucleation of cleavage microcracks from parti-cles such as carbides. The idea emerged from anumber of observations, in particular thosemade by Chen et al. (1990), who showed thatthe fracture of carbide particles is a continuousprocess; see also Chen et al. (1996, 2003), Chenand Wang (1998), and Wang and Chen (2001).The microcrack blunts and generates a micro-void if the conditions to nucleate a cleavagemicrocrack from these particles are not met(see eqn [16]). The nucleation of microvoids isalso statistically distributed. If the probabilityto nucleate a microvoid is called Pvoid

i , theWeibull stress in eqn [27] must therefore bemodified and written as

sw ¼Xn

i¼1 1� Pivoid

� �siI� �mVi

V0

� �1m

½145�

where Vi is the volume of the ith element and �Ii

is the maximum principal stress in Vi. Equation[145] means that once a void has been nucleatedfrom a given particle, this particle cannot con-tribute to cleavage fracture. This idea wasimplemented by Koers et al. (1995).Unfortunately, there are very few experimentalresults in the literature dealing with quantita-tive measurements of void nucleation rate (seeSection 2.06.3.4.2). This explains why, in mostcases, this elimination of eligible particles byvoid formation is neglected.

(iv) Applications

In the early work by D’Escatha and Devaux(1979), the simulation of DCG was simply per-formed with an uncoupled model using the Riceand Tracey expression [69]. It was assumed thatfracture occurred over a critical distance D

(equal to the mesh size) ahead of the crack tipwhen the calculated value of the void growthreached a critical value, (R/R0)c. Crack growthwas simulated using the node release technique.The value of (R/R0)c was determined from testson notched bars. It was shown later that thistechnique gave consistent results when appliedto round cracked bars in A508 RPV steel(Devaux et al., 1989). This pioneering workwas followed by many authors who used the‘computational cell methodology’ initially pro-posed by Xia and Shih (1995a, 1995b, 1996) tomodel ductile crack extension, as described inSection 2.06.3.7.3. Materials properties for thiscomputational cell methodology are limited.They include, for the base material, Young’smodulus (E), Poisson’s ratio (�), yield stress(�0), and hardening exponent (n), or the actualmeasured stress–strain curve, and for the com-putational cells, D, f0, and fc. The strategywhich is generally used to calibrate these prop-erties and these parameters, in particular D, f0,and fc, is based on a series of finite elementanalyses of conventional specimens. Very fewstudies have attempted to use these parametersidentified from volume element specimens, suchas notched specimens, although there are anumber of exceptions (see, e.g., Tanguy, 2001;Tanguy et al., 2002a; Bauvineau, 1996; Decampet al., 1997; Devillers-Guerville, et al., 1997).

The Weibull stress model is used to simulatethe initiation of cleavage fracture. The calculationof the Weibull stress is based on the Bereminmodel (see Section 2.06.2.3). The application ofthis model to the situation corresponding to sig-nificant DCG requires the definition of aneffective Weibull stress. Each part of the materialis subjected to a loading history, �(t), p(t) where tis time. The probability of survival of each pointat time, t, is determined by the maximum value ofthe loading parameter experienced by this pointduring the time interval [0, t] (Tanguy, 2001;Lefevre et al., 2002). The ‘effective’ Weibull stressis thus defined by

seffðtÞ ¼ maxsIðt9Þ t9 2 ð0; tÞ; _pðt9Þ>0 ½146�

where �0(t)9 is the maximum principal stress.The condition that the cumulative plastic strainrate is positive, p(t9)>0, means that plasticdeformation must be active to trigger cleavagefracture. In eqn [146], the correction due toplastic strain which appears in the originalBeremin model (eqn [37]) is sometimes intro-duced. The Weibull stress is thus defined as

sw ¼ZPZ

smeffdV

V0

� �1=m½147�

Three applications of these advanced modelsto predict the DBT behavior are presented below.


In the first two applications, the emphasis is laidon the comparison between experiments and cal-culations. The third application which deals withwelds has been selected in order to illustrate howthe local approach to fracture can give an insightinto complex, but representative, situations.

1. HSLA steel. This application based on com-putational cell methodology was made byRuggieri and Dodds (1996). These authors usedthe results obtained on an HSLA steel(�0¼ 663MPa) with a relatively low strain hard-ening (UTS/�0¼ 1.08). The material was testedat�120 �C, that is very near the lower shelf usingthree-point bend specimens with shallow anddeep cracks. The experiments were performedby Toyoda et al. (1991). The experimental resultsfor shallow cracks (a/W¼ 0.10) are reported inFigure 95. The predicted curve is also displayedon this figure. The predictions were obtained byusing the following set of parameters:f0¼ 0.00025, fc¼ 0.15, D¼ 0.2mm, m¼ 15.6,�u¼ 1757 MPa, Vu¼ 1mm3.

Figure 95 shows that the agreement betweenthe calculated and the observed J–Da curves isonly qualitative. More details on cleavage frac-ture behavior are given in Figure 96, where theprediction of the probability distribution for thecleavage fracture toughness data of specimenswith a/W¼ 0.10 are shown. The solid symbolsshow the experimental fracture toughness datafor those specimens. The solid line represents thepredicted Weibull distribution. The dashed linesrepresent the 90% confidence bounds generatedfrom the 90% confidence limits for the distribu-tion of the Weibull stress in the specimens. Thepredicted distribution displayed in Figure 96agrees reasonably well with the experimental

00

500

1000

1500

2000

0.25

PR = 0.50

PR = 0.10

PR = 0.90

0.5 0.75 1

Experimental data

HSLA steel

J c (

kJ m

–2)

3PB specimensa/W = 0.10

Δa (mm)

f0 = 0.000 25

Figure 95 Crack growth resistance curve in anHSLA steel. Experimental results (Toyoda et al.,1991); numerical simulation (Ruggieri and Dodds,1996). Reproduced from Ruggieri, C. and Dodds,R. H., Jr. 1996. A transferability model for brittlefracture including constraint and ductile tearingeffects: A probabilistic approach. Int. J. Fract. 79,309–340, Copyright 1996, with kind permission ofSpringer Science and Business Media.

data. In particular, all the measured Jc valueslie within the 90% confidence bounds.

It is worth noting that in these experimentsductile crack extension preceding cleavage frac-ture was rather limited (Da<0.6mm). Thismight partly explain why even after someDCG the slope of ln(ln(1/1�PR)) versus ln(Jc)curves remains almost equal to 2, which is thetheoretical value obtained from the applicationof the Beremin theory to a stationary crack.This would suggest that, in spite of their theo-retical importance, these results could beanalyzed using the simplified approach devel-oped in Section 2.06.4.2.2.

2. Low-strength 2¼Cr–1Mo steel. This studydealt with a low-strength high-hardening pres-sure vessel steel (2¼Cr–1Mo steel) taken from adecommissioned 20-year-old chemical reactor.All the experiments were performed at roomtemperature, which falls in the middle of theDBT interval suggested by the Charpy V impacttest (Nilsson et al., 1992). The 0.2% offset yieldstrength at room temperature is �0¼ 300MPa,while the ultimate tensile strength isUTS¼ 530MPa. The work hardening exponentn is equal to 0.2. The fracture toughness testswere performed on 25-mm-thick compact ten-sion (CT) specimens with 20% side grooves anddeep cracks (a/W¼ 0.60). Details on the testprocedures and the results have been reportedby Wallin (1993) who conducted the statisticalanalysis of these data. The value of the J-integralat cleavage fracture initiation as well as theamount of ductile tearing, measured by SEMobservations, were recorded. These experimentsrepresent a wide data basis obtained from 105KJc

tests with identical specimens of a single material.The results are reported in Figure 97. Seven of thespecimens failed to initiate cleavage before theend of the test. These results are included inFigure 97 with black symbols but they wereomitted from Wallin’s statistical analysis.

The numerical modeling of these tests wasperformed by Gao et al. (1999). These authorsalso used the computational cell model to simu-late DCG and cleavage fracture. The followingparameters were used: f0¼ 0.004 5, fc¼ 0.2,D¼ 0.3mm, m¼ 11.86, �u¼ 2490MPa. Thevalue of m for this material is quite low, ascompared to the values determined in RPVsteels where m, 20. Gao et al. (1999) usedalso a Weibull stress model with a threshold,�th, but they found that there is no trivial way todetermine �th. This is the reason why in thefollowing only the results obtained fromnumerical simulations with �th¼ 0 are pre-sented. In their calculation of the Weibullstress and therefore the probability to failure,the authors used only the ‘history approach’,that is, based on eqns [146] and [147].

5 5.5

90% Conf. limits

Predicted

a / W = 0.1

–3.5

–2.5

–1.5

–0.5

0.5

1.5

6 6.5

ln Jc (kJ m–2)

ln [l

n (1

/(1

– P

R))

]

7 7.5

1

2

m = 15.6 ; σ th = 0

PR (%

)

8

5

20

45

63

80

95

Figure 96 Calculated probability to fracture vs J-integral. Numerical simulation of crack growth resistance inan HSLA steel. Reproduced from Ruggieri, C. and Dodds, R. H., Jr. 1996. A transferability model for brittlefracture including constraint and ductile tearing effects: A probabilistic approach. Int. J. Fract. 79, 309–340,Copyright 1996, with kind permission of Springer Science and Business Media.

1200

1000

800

600

400

200

00 1 2 3 4 5 6

Δa (mm)

J (k

J m

2 )

2¼Cr–1MoCT (a /W = 0.6)Plane strain modelD = 300 μm, f0 = 0.004 5

PR = 0.90

PR = 0.50

PR = 0.10

No cleavage

Experiments (Wallin, 1993)

Model prediction (m = 11.86;σ = 2490 MPa; σth = 0)

Figure 97 Crack growth resistance curve in a 2¼Cr–1Mo steel. Experiments (Wallin, 1993) and numericalsimulations (Gao et al., 1999).


Figure 97 compares the experimental and thetheoretical results. The model gives a goodrepresentation of the J–Da curve, except forlow values of ductile crack extension where themodel tends to overestimate the ductile tearingresistance of the material. This might be relatedto the definition of crack growth since it is notclear how in the calculation the extension corre-sponding to crack blunting effect was taken intoaccount. Gao et al. (1999) have also calculatedthe probability to cleavage fracture during crackextension. Their results are also included inFigure 97, where it is observed that the calcu-lated values for PR are in reasonable agreementwith test results, although the calculated valuesof Jc at cleavage initiation tend to underestimatethe experimental results. This effect might be

partly reduced by using the ‘strain correction’,which has to be applied in particular for largecrack extensions, as originally proposed in theBeremin model (eqn [37]). However, due to thesimplicity of the model used by the authors, itcan be considered that these results validate theuse of this methodology to predict with a reason-able accuracy the DBT behavior even after largecrack extensions.

3. Welds. In this example, it is assumed that acrack is located in the weld metal and propagatesin this material parallel to the fusion line, but farfrom the transition between the weldmetal (WM)and the base material (BM). Both undermatched(�0

WM<�0BS) and overmatched (�0

WM> �0BS)

conditions are considered. The application ofthe local approach to welds has already been


illustrated (Chapter 7.05). The focus here is on thefracture resistance due to ductile tearing and theonset of cleavage fracture. Moran and Shih(1998) have used the computational cell modelto investigate the theoretical effect of mismatchbetween the base and the weld metal.

Overmatched welds have the advantage thatthe weld metal is stronger than the surroundingbase material, inducing thus a state of low con-straint. Consequently, crack growth within theweld is accompanied by extensive plastic defor-mation both in the weld and in the surroundingbase metal. Therefore overmatched welds arevery resistant to ductile tearing. However,because the stress required to cause ductile tear-ing is relatively high, it can reach levelssufficient to initiate unstable cleavage fracture.Therefore, cleavage fracture can occur aftervery little DCG. The resulting fracture tough-ness of the assembly can thus be low eventhough it is resistant to ductile tearing.

Lowering the strength of the weld to produceundermatching confines the plastic deforma-tion into the weld. Both the tearing resistanceand the stress ahead of the crack are relativelylow. As a result, cleavage fracture is less likelyto occur. Therefore, a crack can grow stablyover considerable distances before catastrophiccleavage occurs. Hence, the critical fracturetoughness of undermatched weld specimenscan be higher than that of overmatched compo-nents although the J–Da resistance curve islower, as schematically shown in Figure 98.

This qualitative analysis of mismatching effectwas supported by the numerical simulations per-formed byMoran and Shih (1998). These authorsalso used the computational cell methodology.As their analysis is purely parametric, it is notessential to give here the values of the parametersused in their simulations. Moran and Shih

Overmatched

Overmatched

PR = 0.50

Δa (mm)

J (k

J m

–2)

Undermatched

Undermatchedf0.2

f0.1

f0.1 < f0.2

Figure 98 Schematic variation of J-integral as afunction of crack length for two weld configurations(under- and overmatched) and two values for theinitial value of the inclusion volume fraction.

investigated the importance of the relative thick-ness of the weld metal. Here we refer only tosituations where this thickness is such that thecontour maps of the maximum principal stressahead of the growing crack located in the middleof the weld can hit the interface between the weldand the base metal (i.e., typically a few milli-meters for J values of the order of 500kJm�2).

The results of the numerical calculations areschematically shown in Figure 98 where theJ–Da resistance curves corresponding to under-matching and overmatching situations aredrawn. The effect of increasing the initialvalue of the void volume fraction, f0, is dis-played. In Figure 98, we have also reported thevalue of the cleavage fracture toughness corre-sponding to a probability of 50%. From thisfigure, we see that the overmatched cases leadto critical crack length, which is larger when theinitial volume fraction of defects increases.Both of the undermatched cases are muchmore resistant to cleavage fracture. The cleanerweld has a critical crack length which is smallerthan in the dirty weld. As stated previously, thiscan be anticipated since the crack in a materialcontaining large defects can grow by ductiletearing under a low stress, and this produces arelatively low Weibull stress. However, theobvious disadvantage is that the J–Da curve isalso lower, as shown in Figure 98.

These simulations cannot be easily comparedto experiments since there are no detailedresults in the literature providing a completeset of data for ductile tearing and the subse-quent failure by cleavage in welds. This is whythe conclusions drawn from this third applica-tion of advanced models to predict the DBTbehavior remain speculative.

2.06.4.3 DBT under Charpy V impact testing


Modeling the Charpy impact test is a challen-ging issue, since several aspects of this testrequire a detailed analysis. They include: (1) theinertial effects, (2) the complexity of the loadingat high impact rate (5m s�1) and the boundaryconditions, (3) the effect of high strain rates onconstitutive equations, (4) the nonisothermalcharacter of the test, (5) the 3-D aspect of thefracture behavior, in particular the tunnelingeffect associated with DCG preceding cleavagefracture above the lower-shelf temperature, and(6) the competition between ductile and brittlefracture. However, recent developments in theinstrumentation of the Charpy test largely facil-itates the task. Moreover, recent developmentsin the local approach to fracture have alsoevolved the Charpy V impact test from a purely


quality control test to an evaluation tool forstructural integrity assessment of materials. Asalready stated in the introduction of this part, arecent conference has been devoted to this test(Francois and Pineau, 2001).

It is out of the scope of this chapter to reviewin detail the models used to simulate the Charpytest and to calculate the Charpy energy (CVN).This is already presented in Chapter 7.05. Thisreview was largely based on the work byTanguy (2001). Further details can also befound elsewhere (Tanguy et al., 2002a, 2002b,2002c, 2005a, 2005b). Here the focus is laid onsalient features in modeling Charpy impacttests. Besides the work by Tanguy (2001),other studies should also be mentioned, in par-ticular those by Rossoll (Rossoll, 1998; Rossolet al., 1999, 2002a, 2002b) and those publishedby the Freiburg group (Bohme et al., 1992,1996; Schmitt et al., 1994, 1999, 1998; Sunet al., 1995). Other theoretical studies but with-out detailed comparisons with experimentsshould also be indicated (Mathur et al., 1993;Tvergaard and Needleman, 1988, 2000;Needleman and Tvergaard, 2000). After theanalysis of the salient features arising fromthose studies, an attempt is made to underlinehow modeling Charpy V test can be used toinvestigate the fracture properties of materialsunder specific conditions, in particular thosefound with irradiated materials.

2.06.4.3.2 Modeling Charpy V-notchedimpact test – salient features

Inertial effects have been shown to affect fail-ure only at very low temperatures in the lowershelf regime and not in the transition regionwhere plastic deformation is sufficiently largeto damp the oscillations on the load–displace-ment curve recorded in an instrumentedCharpy test (Tvergaard and Needleman, 1988;Tahar, 1998; Rossoll et al., 1999). The impactCharpy test can thus be simulated under quasi-static conditions when dealing with resultsobtained in the upper part of the DBT curveand at the upper shelf.

The Charpy test specimen is essentially 3-D.Finite element modeling must thereforeaccount for this effect. Contact between thestriker and the support must also be takeninto account using a friction coefficient.

Local strain rates as large as 103 s�1 are cal-culated during the deformation of the notchunder impact and during crack propagation.Modeling the mechanical response of the speci-men requires to properly capture the strain rateeffect in the constitutive equations of the mate-rial. Many authors have used the Cowper

Symonds law to represent the strain rate effect.However, this representation assumes that thestrain rate effect is the same over all the raterange encountered in these tests. This is thereason why in Tanguy (2001) the flow strengthof the material was expressed as a function oftemperature, T, and plastic strain, p, with twoisotropic components as:

syðp;TÞ ¼ R0 þQ1½1� expð�b1pÞ�þQ2½1� expð�b2pÞ� ½148�

where the parameters R0, Q1, and b1 are tem-perature dependent, while Q2 and b2 areconstant. The equivalent plastic strain rate, p,is given by a viscoplastic flow function_p¼F(�e� �0) expressed as

1

_p¼ 1

F9¼ 1

_e1þ 1

_e2½149�

with

_ei ¼se � s0

Ki

� �ni

i ¼ 1; 2 ½150�

where �e is the von Mises equivalent stress. Thestrain rate p, e1, and e2 are representative of oneof the following deformation micromechan-isms: (1) Peierls friction forces acting mainly atmoderate strain rates, and (2) phonon drageffect prevailing at higher strain rates(,103 s�1).

The comparison between the experimentaland the calculated load–displacement curves isgenerally used to test the quality of the simula-tions. This is however a global method whichdoes not guarantee that the local stress–strainfields are correctly calculated. In Tanguy(2001), a special effort was made to measurethe local strains around the notch of theCharpy specimen by using a recrystallizationtechnique.

In most studies, ductile damage is simulatedusing the Gurson model as extended byTvergaard and Needleman (see Section2.06.3.4), referred hereafter as the GTNmodel. However, Tanguy (2001) used also theRousselier model (see Section 2.06.3.4.4) tosimulate ductile fracture in impact Charpytests. The original Rousselier model was mod-ified to account for strain rate effect (Tanguyand Besson, 2002). The damage parametersappearing either in the GTN model or in theRousselier model were calibrated to simulatethe ductility of notched tensile bars. The initialvolume fraction of inclusions initiating cavitieswas determined from detailed metallographicalobservations and chemical analysis (Tanguy,2001).

Cleavage fracture was simulated using theBeremin model (eqn [27]). A special care must


be taken when computing the probability tofracture occurring after some crack extensionsince, for a growing crack, propagation pro-duces the unloading of the material left behindthe advancing crack tip. The equivalent stressmust be defined as in eqns [146] and [147] tocalculate the probability to fracture. Here itshould be added that interrupted tests whichare difficult to perform have proved to be extre-mely useful to compare the observed ductilecrack extension and the measured crack growth(Tanguy, 2001; Tanguy et al., 2005a).

This methodology was applied to an A508RPV steel to predict the Charpy V transitioncurve. The Charpy energy, CVN, correspond-ing to a failure probability of 10%, 50%, and90% is plotted as a function of the test tempera-ture (Figure 99). The normalizing stress �u ofthe Beremin model was assumed to remain con-stant. The predictions are satisfactory up toT¼�80 �C but, above this temperature, themodel largely underestimates the Charpyenergy. Similar results have been reported inthe literature (Rossoll et al., 2002a, 2002b;Bernauer et al., 1999). The transition curvewas adjusted using a temperature-dependent�u as already proposed by Tanguy et al.(2002b, 2005b) and Lefevre et al. (2002). Abetter description of the Charpy transitioncurve is obtained when a temperature depen-dence for �u is applied, as illustrated inFigure 99b. This effect of temperature is stilllargely debated and requires further studies. Inparticular, this effect might reflect a modifica-tion in the micromechanisms initiating cleavagefracture (see Section 2.06.2.3). Detailed metal-lographical studies would be useful for a betterunderstanding of this effect.

This analysis can be used to determine theamount of energy spent in crack initiation and

–2000

100

50

(a)

150

200

250

–150 –100 –50 0 50

10%

Temperature (°C)

σu = Cte

CV

N (

J)

50%

90%

Figure 99 A508 RPV steel. Prediction of the Charpyparameter �u or a temperature-dependent �u (USE: upp

crack propagation in a Charpy test. Thesevalues are strongly dependent upon tempera-ture, as shown in Figure 100 where we haveincluded the calculated ‘crack resistance’ curve(Charpy energy, CVN, vs crack growth, Da).This curve is almost independent on tempera-ture between �80 �C (lower part of the DBTcurve) and 20 �C (USE). This situation is simi-lar to what is found for J–Da ductile tearingresistance curves. Figure 100 shows that thedata points obtained with interrupted tests at�60 �C are in very good agreement with thecalculated curve. The results of the calculationsfor the probability to failure are also reportedfor two temperatures: �60 and �40 �C. Theenergy spent in crack propagation largelyincreases when the temperature is increased.

2.06.4.3.3 Other applications

Similar models can be applied to multiple-material components, such as those found inwelds. The notch of the Charpy specimens canbe machined at different positions from thefusion line to test the various regions of thematerial, that is, the weld metal, the HAZwhich is often more brittle than the weldmetal, or the base material. The simulationsmust therefore incorporate the constitutiveequations for at least three materials. The lim-ited dimensions of the weld metal and the HAZmakes difficult to extract specimens from thesezones in order to determine the correspondingmechanical properties. Thermal welding cyclesare generally applied to bulk specimens byusing Gleeble-like thermomechanical simula-tors (see, e.g., Bilat et al., 2006) beforemechanical testing. The constitutive equationsand fracture criteria are introduced in the finiteelement modeling of multiple-material Charpy

–2000

100

(b)

50

150

200

250

–150 –100 –50 0 50

10%

USE

Temperature (°C)

σu (T )

CV

N (

J)

90%

V transition curve assuming a constant value for theer shelf energy) (Tanguy et al., 2005a, 2005b).

1

Exp. tests –60 °CCalculated PR

Δa max (mm)

CV

N (

J)

(–60 °C)

(–40 °C)0.50

0.50

0.10

0.10

0.90

0.90

A508 C1.3

00

20

40

60

80

100

120

140

160

180

200

220

2 3 4 5

Figure 100 A508 RPV steel. Variation of theCharpy energy (CVN) with crack extension Damax

from the notch. Experimental results at �60 �C.Calculated probabilities to cleavage fracture(PR¼ 0.10, 0.50, 0.90) for two temperatures (�60and �40 �C) (Tanguy et al., 2005a, 2005b).


specimen to simulate the mismatch effects pre-viously discussed.

Charpy V test results are also widely used inthe nuclear industry for surveillance program ofRPV embrittlement by neutron irradiation.Service life extension of nuclear power plantsand more stringent safety requirements increasethe request for smaller test specimens than thetypical 10� 10mm2 Charpy test pieces, such assubsized Charpy specimens. This rises the pro-blem of the transferability of fracture criteria.In a recent study, it was shown that reasonablepredictions of the Charpy energy measured onsubsized Charpy specimens could be obtainedusing the Beremin cleavage model at low tem-perature and the Rousselier model at the uppershelf (Poussard et al., 2002). See also the studyby Schmitt et al. (1998).

Charpy V impact testing is also largely used todetect and to monitor the effect of irradiationembrittlement in RPV steels. A shift of theDBTT, DT, is observed, as indicated earlier, andas schematically shown in Figure 19. Veryrecently, an approach based on the simulation ofCharpyV-notch specimens has been developed topredict the temperature shift in A508 RPV steel(Tanguy et al., 2006). These authors showed thatthe increase in theDBTTwith the neutron fluencecan be well predicted using the Beremin theory.The results have already been reported inTable 5.

All these examples show that it appears nowpossible to adapt the Charpy impact test from apurely technological test to a more quantitativetool for the evaluation of the fracture propertiesof materials and components.

2.06.5 CONCLUSIONS

A unifying approach of fracture must startwith a physical model, not just a phenomeno-logical model. In this chapter, an attempt hasbeen made to illustrate the benefits of themicromechanical modeling of fracture. In par-ticular, it has been shown that sophisticatedmodels have now been developed which capturethe influence of a large number of physicalparameters describing the detailed microstruc-ture of materials, under complex loadingconditions.

The micromechanical approach to fracture isfar more complex than the global approachwhich assumes that fracture can be describedby a single (eventually two) loading parameter,such as K or J. In particular, it requires formetallic alloys detailed metallographical mea-surements (grain size, grain or packetorientation, second-phase volume fraction, par-ticle shape, etc.), and also advanced FEMcalculations. Contrary to the global approachin which the material is considered as a ‘blackbox’ to which macroscopic, statistically basedfailure criteria apply, the local approach tofracture has largely contributed to the issuesrelated to the transferability of laboratory testresults to components in case of size or con-straint effects. It has also allowed to modelcomplex macroscopic phenomena, such as thenonexistence of a unique crack growth resis-tance curve for ductile rupture or thebeneficial warm-prestressing effect observedwhen a material is prestressed above the DBTcurve and then loaded below this transitioncurve.

The micromechanical approach to fracture isnot really new. At the turn of the millennium,one can wonder how much progress has beenmade over the past 10 or 20 years. It is clear thatthe modeling of cavity formation, cavitygrowth, and coalescence in the frame of conti-nuum solid mechanics has been tremendouslyimproved. Similarly, it is now well accepted thatcleavage fracture toughness is largely scatteredand is specimen size dependent. These pro-gresses have been possible thanks to theprodigious development of numerical methods.The introduction of new experimental techni-ques, such as in situ mechanical tests, in situobservations, and tomography has also largelycontributed to these improvements.

Many results derived from the micromecha-nical approach to fracture still remainspeculative in the absence of a sufficientlylarge basis of experimental verifications. It iswell to remember that 20 years ago the unique-ness of the ductile crack growth resistance curvewas considered as acquired in spite of all the

References 783

micromechanical studies showing that cavitygrowth was largely sensitive to stress triaxialityratio, which is largely dependent on specimengeometry. Similarly, for cleavage fracture, itwas believed that the fracture toughness wasan intrinsic property for a given material.Micromechanical approaches to fracture haveclearly shown that, due to statistical effects, thecleavage fracture toughness was specimen sizedependent. This property is now introduced inthe ASTM E 1921-03 standards.

The development of the simulation tools willcertainly continue at an increasing speed. Asituation is reached where one can figure outthe development of a complete chain of predic-tion for the final mechanical properties startingfrom the fabrication processes, including solidi-fication, solid-state transformations, heattreatments, aging, etc. However, the retroactivefit of the mechanical properties with processingmodels is still too limited and requires furtherdevelopment. It has been shown that the rela-tive simplicity of the micromechanical modelsproposed for brittle cleavage fracture, either theconcept of a critical stress over a critical dis-tance or the concept of the Weibull stress, havelargely contributed to the modeling of thismode of failure. Improvements remain to bemade to include in more details the varioussteps in cleavage fracture, in particular todevelop what we have called the multiple bar-rier models. Similarly, brittle intergranularfracture has not yet received enough attention.The use of the continuous mechanics for porousplastic materials has largely contributed to abetter understanding of the issues involved inthe study of ductile fracture. A significant efforthas to be made to incorporate in more detail thestatistical aspects of this mode of failure and toreinforce the modeling of final stage of ductilefracture, that of coalescence. Crystalline plasti-city will also be required when the cavity size issmaller than the grain size, which is more andmore typical in microsystems and with thedevelopments of nanostructured alloys.Finally, the quest for a physically relevant andcomputationally robust method to introduceintrinsic length(s) related to the fracture processinto the models and numerical schemes remainsa matter of open debate.

ACKNOWLEDGMENTS

This chapter results from an accumulation ofstudies since the early 1980s supported by indus-try and government for Andre Pineau and sincethe mid-1990s for Thomas Pardoen. The authorswould like to acknowledge all the former Ph.D.students and post-docs of their research groups,

as well as the numerous and fruitful collabora-tions with many colleagues all over the world.T. Pardoen specifically acknowledges the contin-uous support of the University Attraction Poles(IAP) Programme, financed by the BelgianState, Federal Office for Scientific, Technicaland Cultural Affaires, under contract PAI 41(1997–2001) and then P8/05 (2002–2006), aswell as of the Fonds National de la RechercheScientifique, FNRS, Belgium.

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