3.03-computational_aspects_of_nonlinear fracture mechanics.pdf

$: 3.03-Computational_Aspects_of_Nonlinear fracture mechanics.pdf$
3.03Computational Aspectsof NonlinearFractureMechanicsW. BROCKS,A. CORNEC, and I. SCHEIDERGKSS-Forschungzentrum, Geesthacht, Germany

3.03.1 INTRODUCTORY REMARKS ON INELASTIC MATERIAL BEHAVIOR 129

3.03.2 FE MESHES FOR STRUCTURES WITH CRACK-LIKE DEFECTS 131

3.03.2.1 General Aspects and Examples 1313.03.2.2 Singular Elements for Stationary Cracks 1333.03.2.3 Regular Element Arrangements for Growing Cracks 135

3.03.3 CHARACTERISTIC PARAMETERS OF ELASTOPLASTIC FRACTURE MECHANICS 135

3.03.3.1 The J-Integral 1353.03.3.1.1 Foundation 1353.03.3.1.2 J as stress-intensity factor: the HRR-field 1373.03.3.1.3 The domain integral or VCE method 1393.03.3.1.4 Extensions of the J-integral 1403.03.3.1.5 Path dependence of the J-integral in incremental plasticity 1413.03.3.1.6 The J-integral for growing cracks 142

3.03.3.2 The CTOD and the CTOA 1433.03.3.3 The Energy Dissipation Rate 144

3.03.4 DAMAGE MECHANICS AND ‘‘LOCAL APPROACHES’’ TO FRACTURE 146

3.03.4.1 Damage and Fracture 1463.03.4.2 Damage Indicators 147

3.03.4.2.1 Ductile tearing 1473.03.4.2.2 Cleavage fracture 148

3.03.4.3 Micromechanical Models of Ductile Tearing 150

3.03.5 THE COHESIVE MODEL 154

3.03.5.1 Introduction 1543.03.5.2 Fundamentals 155

3.03.5.2.1 Barenblatt’s model 1553.03.5.2.2 Advanced cohesive zone models 1553.03.5.2.3 Cohesive laws 1573.03.5.2.4 Cohesive models for mixed-mode loading 1613.03.5.2.5 Unloading and reverse loading of cohesive elements 1623.03.5.2.6 Rate-dependent cohesive laws 1643.03.5.2.7 Work of separation and remote plastic work 1653.03.5.2.8 Implementation of cohesive elements in FE codes 167

3.03.5.3 Relation to Micromechanical Phenomena of Damage and Fracture 1683.03.5.3.1 Ductile crack growth in metals 1683.03.5.3.2 Quasi-brittle fracture of concrete 1723.03.5.3.3 Crazing in amorphous polymers 173

3.03.5.4 Applications to Ductile Fracture of Metals 1733.03.5.4.1 Simulation of ductile resistance curves by the cohesive model 1733.03.5.4.2 Embedded ductile layer with center crack under mode I 1763.03.5.4.3 Interface cracking of dissimilar materials 1803.03.5.4.4 Verification of the cohesive model on homogeneous elastic–plastic structures 188

127

3.03.5.5 Applications to Other Materials and Phenomena 1953.03.5.5.1 Quasi-brittle materials such as concrete 1953.03.5.5.2 Heterogeneous compounds 1963.03.5.5.3 Dynamic fracture 1983.03.5.5.4 Rate- and time-dependent fracture 1993.03.5.5.5 Fatigue crack growth 200

3.03.5.6 Summary and Outlook 2013.03.5.6.1 Specific advantages 2013.03.5.6.2 Future research issues 2023.03.5.6.3 Challenges 202

3.03.6 REFERENCES 203

NOMENCLATURE

a crack lengtha0 initial crack lengthA crack areaB specimen thicknessBN specimen net thicknessC* C* integral in creep fractureCMOD crack mouth opening displacementD damage variableD, Dij strain rate tensor, components of

strain rate tensorDin, Dij

in inelastic strain rate tensor, compo-nents of inelastic strain rate tensor

Dp plastic strain rate tensorE Young’s modulusEt tangent modulusEve relaxation modulus for viscoelastic

cohesive lawsE1, E2 elastic moduli for dissimilar material

jointsf void volume fraction in Gurson and

Rousselier modelf* damage parameter in GTN modelfc critical void volume fraction at begin-

ning void coalescenceff void volume fraction at final failuref0 initial void volume fractionF applied forceG shear modulusG Griffith’s energy release rateGss Griffith’s energy release rate for stea-

dy-state conditionh triaxiality, ratio of hydrostatic, and

effective stressH constraint layer heightJ J-integral of Cherepanov and RiceJi J-integral at crack initiationJss J-integral for steady-state conditionsK stress-intensity factorK stiffness matrixKI, KII, KIII

stress-intensity factors for modes I, II,III

Kss stress-intensity factor at steady stateK0 stress-intensity factor at crack initiationm Weibull modulus

M mismatch ratio of yield stresses fordissimilar joints

Mb applied bending momentn hardening exponent of Ramberg–

Osgood power law (n¼ 1/N)ni ith component of the normal unit

vectorN work-hardening exponent (N r1)p norm of inelastic or plastic strainPf failure probabilityq interaction parameter for normal and

tangential separationq1, q2, q3

parameters in GTN modelQ Q-stress of O’Dowd and Shihr radial coordinate, distance from crack

tipR dissipation rateR(ep) yield curve of a materialR0 plastic zone size estimation at K0

s arc lengthS surface, areaS Cauchy stress tensorS0 stress deviatort plate or sheet thicknessT tractionT 3D traction vectorTN, TT normal and tangential direction of the

traction, respectivelyTN,0, TT,0

normal and tangential direction of themaximum traction, respectively

T0 maximum traction, separation strength[u] vector of the displacement jump at the

cohesive element (equivalent to theseparation vector)

ui components of displacement vectorUdis (total) dissipated (nonrecoverable) me-

chanical workUpl (global) work of plastic deformationUsep (local) work of separationV volumeVLL load-line displacement for C(T) speci-

mensVu matrix of shape functions for finite

elementsw strain energy density

128 Computational Aspects of Nonlinear Fracture Mechanics

3.03.1 INTRODUCTORY REMARKS ONINELASTIC MATERIALBEHAVIOR

In a general sense, nonlinear fracture me-chanics (for details, see Chapter 2.03) can beunderstood as mechanics of fracture formaterials with inelastic stress–strain relations,where inelastic behavior is any kind ofirreversible response to thermomechanicalloading including phenomena like

(i) temperature and rate dependence,(ii) time dependence: creep and relaxation,

Bauschinger effect,(iii) cyclic hardening and softening, and(iv) effects of multiaxial and nonpropor-

tional loading.A unified approach to describing inelastic

deformations which covers classical elastoplas-ticity as well as viscoplasticity is based on anincremental formulation (for details, see Chap-ter 2.01)

’S ¼ C/4S

�ðD�DinÞ ð1Þ

with ’S and D being an objective derivative ofthe Cauchy stress tensor and the deformationrate tensor, respectively. The inelastic part of D

is described by a ‘‘flow rule’’

Din ¼ffiffiffiffi3

2

r’pN ð2Þ

where N denotes the direction of inelasticdeformation and

’p ¼ Din�� ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Dinij Din

ij

q¼

ffiffiffi3

2

r’eine ð3Þ

represents the uniaxial effective inelastic strainrate—despite the factor

ffiffiffiffiffiffiffiffi3=2

p: A number of

scalar and tensorial variables, kn and Xm;capture the load history.

Elastoplastic constitutive theories, i.e., Din ¼Dp; like those of von Mises (1923), Prandtl(1924), Reuss (1930), Prager (1959), Ziegler(1959), Mroz (1967), and Chaboche and Rous-selier (1983), introduce a yield condition whichlimits the set of physically admissible stress states

FðS;X; pÞ ¼ S0 � X 0j jj j � k ¼ 0 ð4Þ

with evolution equations for the two internalvariables,

’k ¼ f ðS;Dp; ’p;X ; k; yÞ’X ¼@ðS;Dp; ’p;X ; k; yÞ

ð5Þ

W specimen widthxi Cartesian coordinatesX back stress tensorb second Dundurs’ parameterg Griffith’s surface energyG integration contour in the definition of

the J-integralG0 total cohesive energy, separation en-

ergy at failured separationdN, dT normal and tangential direction of the

separation, respectivelydN,0, dT,0

normal and tangential direction of thecritical separation, respectively

dt, d5 CTODdvp viscoplastic portion of the separationd0 separation at failure (critical separa-

tion)d1, d2 shape parameters for several cohesive

lawsd5 CTOD defined at 5mm gauge length

across the crack tip, mounted on thespecimen surface

Da crack extensionDb diameter reduction in round barsDl global elongation of specimense oscillation index for mode mixity

eein accumulated inelastic effective straineij strain componentsep plastic straineY yield straine0 strain at s0, used for Ramberg–

Osgood hardening lawZ viscosity coefficienty temperature coordinateW angular coordinatek acceleration factor for the GTN modelm friction coefficientn Poisson’s ratiosc critical stress, cleavage stressse von Mises effective stresssij components of Cauchy stress tensorsij0 components of stress deviator

su reference stress of Weibull distributionsw Weibull stresss0 yield stressSij mesoscopic strain components, used

for the unit cellt normalized T-stress parameterF yield function, flow potentialcL mode-mixity parameter, defined at

r¼Lc0 mode-mixity parameter, defined at

r¼R0

Introductory Remarks on Inelastic Material Behavior 129

which have the physical meaning of rules forisotropic and kinematic hardening, respectively,and may depend on temperature, y, and strainrate, ’p: S0 ¼ S � ð1=3Þskk I is the stress devia-tor, X is called ‘‘back stress’’ tensor, and

S0 � X 0j jj j ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis0ij � x0ij� �

s0ij � x0ij� �r

¼ffiffiffi2

3

rse ð6Þ

is—despite the factorffiffiffiffiffiffiffiffi2=3

p—the uniaxial ‘‘ef-

fective’’ stress which becomes the well-knownvon Mises effective stress in the case of pureisotropic hardening, where k ¼

ffiffiffiffiffiffiffiffi2=3

pRðepÞ is

the uniaxial flow stress as obtained in a tensiletest. Plastic deformations occur if the loadingcondition

’S � � S0 � X 0ð ÞZ0 ð7Þ

is fulfilled in the time increment Dt. Thedirection of plastic flow is commonly given bythe ‘‘normality rule’’

N ¼ S0 � X 0

S0 � X 0ð Þj jj j ð8Þ

and the consistency condition

’F ¼ @F@S

’Sþ @F@X

’Xþ @F@k

’k ¼ 0 ð9Þ

allows for determining ’p:Viscoplastic constitutive theories basing on

the ‘‘over-stress model,’’ like those of Bodnerand Partom (1975), Robinson (1978), or Cha-boche (1993), do not include a yield condition(Equation (4)) and hence no consistencycondition (Equation (9)) either. They intro-duce a special constitutive law for Din ¼ Dvp

instead, which commonly also postulates thenormality rule, Equation (8) and some powerlaw,

’p ¼ AS0 � X 0ð Þj jj j � k

K

� �n

ð10Þ

with A, K, n as material parameters.The above constitutive equations of

continuum mechanics describe deformations(Equations (2), (8), and (10)) under thermo-mechanical loading. The evolution equationsfor the internal variables (Equation (5)) arerestricted to material hardening, in general,so that the material behavior is ‘‘stable’’(Drucker, 1964). The corresponding boundaryvalue problem is elliptic, and finite element(FE) simulations will yield a unique solutionwhich is convergent with refining the mesh.The physical reality, however, is more complex:materials ‘‘soften’’ due to formation of micro-cracks, initiation, growth and coalescence ofvoids, etc., generally summarized as ‘‘damage.’’

This damage may lead to the initiation andgrowth of macrocracks in a structure and tofinal failure in the end. The ‘‘crack tip’’ asaddressed in fracture mechanics is a mathe-matical idealization. In reality, a region ofmaterial degradation exists in some processzone ahead of a macrocrack, where finally newsurfaces will be created. In this process zone,the microbehavior becomes important forconstitutive modeling, and this will raisequestions on materials length scales (Sun andHonig, 1994; Siegmund and Brocks, 1998b)which cannot be answered within the limita-tions of the theory of ‘‘simple materials.’’ Theboundary value problem for softening materi-als may lose ellipticity, and FE simulations willyield mesh-dependent results as the elementsize affects the separation energy (Siegmundand Brocks, 1998a).

Three different approaches exist to modeldamage, material separation, and fracturephenomena:

(i) no damage evolution is modeled andconventional material models, e.g., elastic–plastic constitutive equations, are applied, theprocess zone is assumed as infinitesimallysmall, and special fracture criteria, e.g., basedon K, J, C*, for crack extension are required;

(ii) separation of surfaces is admitted, ifsome critical value is reached locally, whereasthe material outside behaves conventionally,the process zone is some surface region,and the fracture criterion is a cohesive law; and

(iii) softening behavior is introduced intothe constitutive model, e.g., accumulation ofdamage, described by additional internal vari-ables, the process zone is a volume, and theevolution equation for damage yields a fracturecriterion.Classical elastic–plastic fracture mechanics(EPFM) (for details, see Chapter 2.03) coversa comparably small part of these constitutivetheories and phenomena of inelastic deforma-tion. It has been established under the assump-tions of the theory of finite plasticity or‘‘deformation theory’’ of plasticity by Hencky(1924) and the kinematics of small deforma-tions. It has nevertheless become the mostimportant field of fracture research besideslinear-elastic fracture mechanics (LEFM) as itallowed for analytical descriptions and solu-tions and has been successfully applied todescribe crack-growth initiation in ductilematerials like metals at low and moderatetemperatures. The plastic deformation behaviorof metals, however, is more realistically de-scribed by the theory of incremental plasticityby von Mises, Prandtl, and Reuss (Equations(1)–(9)), which accounts for effects of loadhistory, unloading, and local rearrangement of


stresses. As no analytical solutions are possiblein this case, numerical methods have won greatimportance. With the increasing capacities ofcomputers, the constitutive relations used instructural analyses have become more ad-vanced and sophisticated, including more andmore of the above mentioned phenomena.

But fracture parameters, and criteria forfracture and crack growth, which are used inpractice for engineering assessment methodsare still the same as in the early times of EPFM,namely:

(i) the J-integral of Cherepanov (1967) andRice (1968) (or its analog, C*, for creep crackgrowth (Landes and Begley, 1976)) and

(ii) crack-tip opening displacement (CTOD),d (Burdekin and Stone, 1966; Dawes, 1985).

For theoretical details of LEFM and EPFM,see Chapter 2.03. Numerical analyses andsimulations applying incremental plasticity ormore advanced constitutive theories can beused in this context for:

(i) determination of classical fracture para-meters for complex configurations and bound-ary conditions including thermomechanicalloading, if no appropriate analytical solutionsare available,

(ii) investigations of the applicability and thelimitations of these parameters and of engineer-ing assessment methods in general, and

(iii) application of local criteria of fracture,like those by Beremin (1981, 1983), if theclassical parameters fail in giving reliablepredictions.The following contribution will essentially

restrict to the application of the von Misestheory of incremental plasticity to crackedspecimens and components. In particular, theclassical parameters of EPFM, J and CTOD,as well as subsequently proposed parameterssuch as energy dissipation rate and crack-tipopening angle (CTOA) and the related compu-tational aspects will be discussed. Some re-marks follow on the ‘‘local approach tofracture’’ which is based on continuum fieldquantities, namely stresses and strains, and thedamage models of Gurson (1977) and Rousse-lier (1987), which have now found increasingapplication, will be briefly addressed in Section3.03.4. The numerical modeling of decohesionand separation phenomena by ‘‘cohesive ele-ments’’ will be presented in Section 3.03.5.

3.03.2 FE MESHES FOR STRUCTURESWITH CRACK-LIKE DEFECTS

3.03.2.1 General Aspects and Examples

Cracks and crack-like defects induce highstress and strain gradients which require a fine

discretization resulting in large numbers ofelements and degrees of freedom. Nonlinearsimulations of components with stress concen-trators are therefore expensive with respect tocomputation time and memory. All possibili-ties to reduce the number of degrees of freedomshould hence be utilized, e.g., restricting totwo-dimensional (2D) models of the structureif physically meaningful, coarsening the meshaway from the defect, introducing symmetryconditions, and applying singular elementswith special shape functions.

Modeling does always mean reduction ofcomplexity and simplification. Models shouldbe as simple as possible and only as complex asunavoidable to cover the interesting effects.For plane specimen geometries the possibilityof 2D models should always be considered, atleast for pre-analyses of a new problem. Thinspecimens and sheet metal components arecommonly adequately represented by a plane-stress model, and thick or side-grooved speci-mens by a plane-strain one. 3D analyses arenecessary, of course, if the geometry is notplane or if 3D effects through the thickness andalong the crack front are studied. Someexamples of typical meshes will be given inthe following.

Figure 1 shows the 2D model of a compactspecimen, C(T), accounting for the symmetrywith respect to the ligament. Normal displace-ments are constrained along the symmetry line.The load pin is modeled by truss elementstransferring compression, only. As the pin holeis quite remote from the crack tip, a ‘‘correct’’modeling of the load pin is not very relevantfor the quality of the results as long as no localplasticity is induced by it, which would changethe external work and, hence, the J-value.Collapsed elements are applied at the crack tip(see Figure 2 and Section 3.03.2.2).

Twofold symmetries exist for center-crackedor double edge-cracked tensile panels, M(T) orDE(T). Figure 3 shows the FE mesh of acenter-cracked cruciform specimen which isbiaxially loaded to study effects of biaxiality onfracture parameters and crack propagation.The ligament is modeled using a regulararrangement of rectangular elements as crackpropagation is simulated (see Figure 4 andSection 3.03.3.2). It demonstrates the commonstrategy used for coarsening the mesh awayfrom the crack.

Finally, an example of a 3D model is given inFigure 5. It was generated for a tubular jointunder three-point bending having a singularsymmetry plane. This is a rather complexgeometry as the two pipes are welded and asemi-elliptical crack exists in the weldment.Thus, the model has not only to account for a

FE Meshes for Structures with Crack-like Defects 131

3D curved crack but also for the weld whichhas different material properties than the restof the structure. It is a convincing example ofthe necessity to use a coarse mesh for the global

structure. The fine mesh in the crack vicinity isdisplayed in Figure 6.

Numerous further examples of structureswith surface flaws and material gradients can

Figure 1 2D FE mesh of a C(T) specimen (W¼ 50mm, a/W¼ 0.5), half model accounting for symmetry.

Figure 2 Detail of the FE mesh of the C(T) specimen at the crack tip with collapsed elements.

Figure 3 2D FE mesh of a biaxially loaded cruciform specimen with twofold symmetry.


be found in the literature, e.g., Brocks et al.(1989b, 1993) and Schmitt et al. (1997).

3.03.2.2 Singular Elements for StationaryCracks

Singular elements have been developed fornumerical analyses of fracture problems toincrease the accuracy of stress calculations andK-factors at a time when computer capacitieswere still rather limited. Barsoum (1977) foundthat triangular or prismatic isoparametricelements which were produced by collapsingone side or plane and shifting the respectivemidside nodes to a quarter position (double

distorted elements), see Figure 7, included the(1=

ffiffir

p)-singularity of strains in LEFM as well

as the 1/r-singularity of EPFM for perfectlyplastic material:

eijðr; WÞ ¼að0Þij ðWÞffiffi

rp þ

að1Þij ðWÞr

þ að2Þij ðWÞ ð11Þ

If the tip nodes undergo the same displace-

ment, i.e., u½1�x ¼ u

½4�x ¼ u

½8�x und u

½1�y ¼ u

½4�y ¼ u

½8�y ;

the second term in Equation (11) vanishes,að1Þij ¼ 0; and a pure (1=

ffiffir

p)-singularity of strain

and stress fields as in LEFM is realized. If themidside nodes are not shifted to the quarter-point positions and the displacements of the tip

Figure 4 Detail of the FE mesh of the cruciform specimen at the crack tip with a regular arrangement ofelements.

Figure 5 3D FE mesh of a tubular joint, having one symmetry plane (source Cornec et al., 1999).

FE Meshes for Structures with Crack-like Defects 133

nodes are not constrained, the first termvanishes, að0Þij ¼ 0; and the 1/r-singularity ofstrain fields at cracks in perfectly plasticmaterials is obtained. The strain energy

%w ¼Z 2p

W¼0

Z r

%r¼0

w%r d%r dW ¼Z 2p

W¼0

Z r

%r¼0

sijeij %r d%r dW ð12Þ

however remains finite for r-0 in linearelasticity as well as for HRR-like fields

(Hutchinson, 1968a, 1968b; Rice and Rosen-gren, 1968; see Section 3.03.3.1.2), because whas a singularity of the order of r–1 in bothcases. This is an important attribute for thephysical significance of the J-integral (seeSection 3.03.3.1).

Numerical studies—see, e.g., McMeekingand Rice (1975) and Brocks et al. (1985)—haveshown that triangular or prismatic collapsedeight- or 20-node elements, respectively, with a

VIEWSYMMETRYPLANE

crack

(a)

(b)

Figure 6 Detail of the FE mesh of the semi-elliptical surface flaw in the weldment of the tubular joint.


1/r-singularity, are well suited for elastic–plasticcalculations. Together with a large-strain ana-lysis, commonly performed by an updatedLagrangean formulation (Gadala et al., 1980),crack-tip blunting can be simulated (see Figure8) and principal stresses show their typicalshape exhibiting a maximum ahead of the cracktip (Rice and Johnson, 1970), see Figure 9.

Singular elements have now become lessimportant, mainly because J-integral calcula-tions by the virtual crack extension method (seeSection 3.03.3.1.3) yield reliable and accurateresults even for very coarse meshes, and singular

elements cannot be applied for crack-growthsimulations which require a regular arrange-ment of elements in the ligament as shown inFigure 4; these meshes, however, will notprovide sufficiently accurate results of CTODor stresses at the crack tip for stationary cracks.

3.03.2.3 Regular Element Arrangements forGrowing Cracks

If a critical initiation value of some fractureparameter is exceeded, a crack starts to grow.Different from crack growth in elastic materials,where crack initiation always leads to cata-strophic failure of the structure, ductile tearingmay occur in a stable manner, i.e., under stillgrowing external forces, or at least deformationcontrolled even beyond maximum load. Crackgrowth can be simulated by the following:

(i) node release techniques, controlled byany fracture mechanics parameter as J, CTOD,CTOA (see Section 3.03.3), e.g., Siegele andSchmitt (1983), Brocks et al. (1994), Brocksand Yuan (1989), and Gullerud et al. (1999);

(ii) cohesive elements (see Section 3.03.5),e.g., Needleman (1990a), Yuan et al. (1996), Linet al. (1997), and Siegmund et al. (1998); and

(iii) constitutive equations based on damagemechanics concepts (see Section 3.03.4), e.g.,Needleman and Tvergaard (1987), Rousselieret al. (1989), Sun et al. (1988), Brocks et al.(1995a), Xia et al. (1995), and Schmitt et al.(1997).

Figure 10 illustrates crack growth by noderelease which is controlled by a criterionassuming constant CTOA (see Section3.03.3.2). The simulation has to be performedunder prescribed displacements in order toproceed beyond maximum load. The effect ofcrack growth and load biaxiality, l¼Fx/Fy, onthe plastic zone is displayed in Figure 11:plastic deformation is retarded under a tensileload in x-direction (l¼ þ 0.5) and enlargedunder a compressive x-load (l¼ –0.5). Theupper row shows the plastic strain at initiationand the lower at maximum load for a crackgrowth of Da¼ 12mm (l¼ þ 0.5) andDa¼ 10mm (l¼ –0.5), respectively.

3.03.3 CHARACTERISTICPARAMETERS OFELASTOPLASTIC FRACTUREMECHANICS

3.03.3.1 The J-Integral

3.03.3.1.1 Foundation

Path-independent integrals are used in phy-sics to calculate the intensity of a singularity

Figure 7 Collapsed quarter-point element.

1

2

3

Figure 8 Deformed mesh at a crack tip showingblunting.

0 1 2 3 40

1

2

3

4 σxx

σyy

σzz

σ ij /

σ Y

J = 247 N/mm

x [mm]

Figure 9 Variation of principal stresses in theligament ahead of a crack tip for an elastic–plasticlarge-strain analysis.

Characteristic Parameters of Elastoplastic Fracture Mechanics 135

of a field quantity without knowing theexact shape of this field in the vicinity ofthe singularity. They are derived from con-

servation laws. They have been introducedinto fracture mechanics by Cherepanov(1967) and Rice (1968). Budiansky and Rice

Figure 10 Deformed mesh at the crack tip of the cruciform specimen with CTOA-controlled crack growthusing node release technique.

Figure 11 Plastic deformation, ep, in the vicinity of the crack tip at crack-growth initiation and maximumload for two biaxiality ratios, l¼Fx/Fy.


(1973) also showed that this ‘‘J-integral’’ isidentical with the energy release rate

J ¼ G ¼ �ð@U=@AÞ ð13Þ

for a plane crack extension, DA. For linear-elastic material, J is hence related to the stress-intensity factors by

J ¼ GI þ GII þ GIII ¼1

E0 K2I þ K2

II

� �þ 1

2GK2

III ð14Þ

where I, II, III denote the three fracture modes(Irwin, 1957). This relation has become acommon technique to calculate stress-intensityfactors in LEFM.

The J-integral of elastostatics can be de-duced from the equations governing the staticboundary value problem for a material body,B, namely, equilibrium and boundary condi-tions and constitutive equations. The compo-nents of the material force per thickness actingon the boundary, @B, of a plane domain B

Fi ¼I@B

wðemnÞni � sjknkuj;i

ds ð15Þ

are nonzero if and only if B contains asingularity. The closed contour G ¼G1,Gþ,G2,G� in Figure 12 does not includea singularity and hence Fi¼ 0. Assumingfurther that the crack faces are straight andstress free, the first components of the integralsalong the respective contours, Gþ , G�, vanish,and the path independence of the first compo-nent of the ‘‘J-vector,’’

J1 ¼I ’

G1

w dx2 � sjknkuj;1 ds

¼ �I -

G2

½ � ¼I ’

G2

½ �

ð16Þresults. This holds for the other two compo-nents, J2, J3, if and only if the contours aroundthe crack tip and the loading are symmetric tothe x1-axis.

This first component of the J-vector is the‘‘J-integral’’ used in fracture mechanics, defin-ing that the integration contour runs anti-clockwise, i.e., mathematically positive, aroundthe crack tip,

J ¼IG

w dx2 � sjknkuj;1 ds� �

ð17Þ

Because of its path independence, it can becalculated in the remote field and characterizesalso the near-tip situation, which establishes itsrole as a fracture parameter. But note that thepath independence does only hold if theconditions—(i) time-independent processes,no body forces, sij,j ¼ 0, (ii) small strains,eijð1=2Þðui;j þ uj;iÞ; (iii) homogeneous hyper-elastic material, sij ¼ @w=@eij; (iv) plane-stressand displacement fields, i.e., no dependence onx3, and (v) straight and stress-free crackborders parallel to x1 (see Figure 12)—aremet. Extensions of the J-integral will bediscussed in Section 3.03.3.1.4.

Analogous considerations as made for thederivation of the J-integral, Equations (15)–(17), yield the C*-integral

C� ¼IG

’w dx2 � sjknk ’uj;1 ds� �

ð18Þ

for viscoplastic material behavior (Landes andBegley, 1976) if a power (work rate) density, ’w;exists so that stresses derive from sij ¼ @ ’w=@’eij :This analogy implies that C* is path indepen-dent under the same conditions which hold forthe path independence of J.

3.03.3.1.2 J as stress-intensity factor: theHRR-field

Besides its identification as energy releaserate in hyperelastic materials (Equation (13)), Jalso plays the role of an intensity factor like Kin the case of linear-elastic materials. Hutch-inson (1968a, 1968b) and Rice and Rosengren(1968) derived the singular stress and strainfields at a crack tip in a power-law hardeningmaterial (called the HRR-field) (for details, seeChapter 2.03)

sijðr; WÞs0

¼ 1

ae0In

� �1=ðnþ1ÞJ

s0r

� �1=ðnþ1Þ*sijðWÞ ð19Þ

The parameters s0, e0, a, n characterize thematerial’s stress–strain curve according to thepower law of Ramberg and Osgood (1945):

ee0

¼ ss0

þ ass0

� �n

ð20Þ

where s0 is commonly identified as the initialyield strength of the material and e0¼ s0/E, so

Figure 12 Definition of contours for J-integralevaluation.


that the hardening is characterized by twoparameters: a and n. Figure 13 gives anexample for a Ramberg–Osgood fit of a truestress–strain curve of a ferritic steel.

The parameter In and the angular functions,*sijðWÞ; in Equation (19) result from the solutionof the respective fourth-order ordinary differ-ential equation (see tables given by Shih(1983) and Brocks et al. (1990)) and dependon the hardening exponent, n. Equation (19)yields unique stress curves independent ofthe external loading, if the abscissa is normal-ized by J. The ratio J/s0 is proportionalto the CTOD, dt, in the HRR-theory (Shih,1981):

dt ¼ dnJ

s0ð21Þ

where dn is again a parameter resulting fromthe solution of the HRR-equations which

depends on the hardening exponent of thematerial. Hence, unique stress curves indepen-dent of the external loading are also obtained,if the abscissa is normalized by dt. Figure 14shows the respective normalization for theprincipal stresses ahead of the crack front ofthe C(T) specimen (see Figures 1 and 9) for twodifferent J-values. A J-dominated stress fieldcan indeed be found by this normalization in acertain region ahead of the crack tip, thoughthe stresses, of course, cannot be described bythe singular HRR-field correctly in the regionof large strains at the blunted crack tip (Riceand Johnson, 1970; McMeeking and Rice,1975; Brocks and Olschewski, 1986). Themaximum principal stress, syy, shows a suffi-ciently good agreement between FE and HRRresults in this example, whereas the quantita-tive correspondence is worse for sxx and,hence, also for szz ¼ ð1=2Þðsxx þ syyÞ:

0 100 200 300 400 5000

1

2

3

ε / ε0

σ / σ

0

FE input

α=10.7, n=4.5

Figure 13 True stress–strain curve of a ferritic steel and Ramberg–Osgood power-law fit.

0 1 2 3 4 50

1

2

3

4

5

σ ij / σ Y

x σY / J

FE: σxx

HRR: σxx

FE: σyy

HRR: σyy

FE: σ

σzz

HRR: zz

HRR:α=10.7, n=4.5

FE: large strain

Figure 14 Variation of principal stresses in the ligament ahead of a crack tip for an elastic–plastic large-strainanalysis (open symbols: J¼ 247Nmm–1; solid symbols: J¼ 377Nmm–1) in comparison to the HRR-field.


Again, an analogy to the J-dominatedsingularity in EPFM (Equation (19)) holdsfor the C*-integral (Equation (18)) in visco-plastic materials: stress and strain fields havean HRR-like singularity if secondary creepfollows a power law (Riedel and Rice, 1980).

The HRR-field has been derived under theassumptions of ‘‘deformation theory’’ of plas-ticity, which actually describes hyperelasticbehavior and small strains. The actual stressfield as calculated in an elastic–plastic FEsimulation will, therefore, more or less deviatefrom the HRR-field (see Figure 14). Theamount of deviation indicates the ‘‘validity’’of J as an intensity parameter of the crack-tipfield. The comparison with the HRR-field is,however, significantly affected by the quality ofthe power-law fit. In particular, materialsexhibiting a Luders plateau in their stress–strain curve cannot be fitted uniquely by apower law (see Figure 13), so that it isimpossible to distinguish whether a deviationof the FE stresses from the HRR-field indicatesa loss of ‘‘J-dominance’’ (McMeeking andParks, 1979; Shih and German, 1981; Shih,1985) or just a poor fit of the material’s stress–strain curve.

Sharma and Aravas (1991) solved the planeboundary value problem with a two-termpower expansion:

sijðr; WÞ ¼ sHRRij ðr; WÞ þ Q

J

s0r

� ��r

%sð1Þij ðWÞ ð22Þ

The idea of a second nonsingular term of thestress field—as the T-stress in LEFM—wasintroduced in the discussion on ‘‘constrainteffects’’ on ductile tearing (Garwood, 1979;Brocks et al., 1989a; Link et al., 1991; Sommerand Aurich, 1991; Shih et al., 1993) andresulted in the introduction of a ‘‘Q-stress’’ inEPFM by O’Dowd and Shih (1991, 1994),based on detailed FE analyses,

sijðr; WÞ ¼ srefij ðr; WÞ þ Qs0dij ð23Þ

The second crack-tip field parameter, Q, isobtained as the difference between the ‘‘full’’stress field, i.e., the FE results, and any‘‘reference’’ (HRR or small-scale yielding(SSY)) solution:

Q ¼ sWW � srefWW

s0at

r

J=s0¼ 2; W ¼ 0 ð24Þ

Figure 14 shows that the location of r¼ 2J/s0is beyond the stress maximum and henceoutside the region of large plastic strain. Yuanand Brocks (1998) found from FE analyses offracture mechanics specimens that a linearrelation exists between Q-stress and stress

triaxiality:

h ¼ shse

¼ 2

3ffiffiffi3

p skkffiffiffiffiffiffiffiffiffiffis0ijs

0ij

q ð25Þ

which is hardly affected by the load amplitude,though the stress field is, in general, not fullydescribed by Equation (23) as the differencesbetween the stresses from the FE analysis andthe respective stresses of the HRR-field are notthe same for all components (see Figure 14again). Besides T- and Q-stresses, stress triaxi-ality, h, has also been used directly as a secondparameter to quantify crack-tip constraint(Brocks et al., 1989a; Sommer and Aurich,1991; Brocks and Schmitt, 1993, 1995) asmicromechanical considerations have shownits physical significance for ductile void growth(Rice and Tracey, 1969). If Q has to bedetermined by an FE analysis for a specificstructure, there seems to be no necessity ofdetermining the stress triaxiality via Q anyway,as the hydrostatic stress is a direct outcome ofthe FE calculation.

3.03.3.1.3 The domain integral or VCEmethod

Calculating a contour integral, like Equation(17), is quite unfavorable in FE codes ascoordinates and displacements refer to nodalpoints, and stresses and strains to Gaussianintegration points. Stress fields are generallydiscontinuous over element boundaries, andextrapolation of stresses to nodes requiresadditional assumptions. Hence, a domainintegral method is commonly used to evaluatecontour integrals, see, e.g., ABAQUS (2000).

Applying the divergence theorem, the con-tour integral can be reformulated as an areaintegral in 2D or a volume integral in 3D overa finite domain surrounding the crack front.The method is quite robust in the sense thataccurate values are obtained even with quitecoarse meshes, because the integral is takenover a domain of elements, so that errors inlocal solution parameters have less effect. Thismethod was first suggested by Parks (1974,1977) and further worked out by deLorenzi(1982a, 1982b).

The J-integral is defined in terms of theenergy release rate (Equation (13)), associatedwith a fictitious small crack advance, Da (seeFigure 15):

J ¼ 1

DA

Z ZB0

sijuj;k � wdik

Dxk;i dS ð26Þ

where Dxk is the shift of the crack-frontcoordinates, DA is the correspondent increase


in crack area, and the integration domain is thegray area in Figure 15. Because of this physicalinterpretation, the domain integral method isalso known as ‘‘virtual crack extension’’ (VCE)method.

Equation (26) allows for an arbitrary shift ofthe crack-front coordinates, Dxk, yielding theenergy release rate, Gj, in the respectivedirection, which can be applied for investiga-tions of mixed-mode fracture problems. Thecommon J-integral, i.e., the first component ofthe J-vector, J1 ¼ Gj jj ¼0; is obtained if andonly if Dxk has the direction of x1 (or x1 in 3Dcases, see Figure 16), which means that it hasto be both, perpendicular to the crack planenormal, x2, and (in 3D cases) to the crack fronttangent, x3 (see Section 3.03.3.1.4). In casewhere the crack front intersects the externalsurface of a 3D solid, the virtual crackextension must lie in the plane of the surface.If the VCE is chosen perpendicular to the crackplane, i.e., in x2-direction, one obtains thesecond component of the J-vector, J2 ¼Gj j¼p=2j :

For 2D plane-strain or plane-stress condi-tions, the extended crack area is simplyDA¼ tDa, where t is the specimen thickness.In a 3D analysis, the VCE has to be applied toa single node on the crack front if the localvalue of the energy release rate is sought. For aconstant strain element, like the eight-noded3D isoparametric element, the interpolationfunctions are linear and a shift of a node on thecrack front will result in a triangle, DA ¼ð1=2Þðl1 þ l2ÞDa; where l1, l2 are the lengths ofthe adjacent elements. For the 20-nodedisoparametric element, the interpolation func-tions are of second order, and a node shift willproduce a crack area increase of parabolicshape which differs for corner nodes andmidside nodes. In any case, DA is linear in Daand, hence, in |Dxk|. Note also that the crack

extension is ‘‘virtual’’ in the sense that it doesnot change the stress and strain fields at thecrack tip.

3.03.3.1.4 Extensions of the J-integral

(i) The 3D J

Assuming plane crack surfaces, the J-inte-gral may be applied to 3D problems. It isdefined locally, J(sc), sc being the curved crack-front coordinate, following the concepts ofKikuchi et al. (1979), Amestoy et al. (1981),and Bakker (1984). Suppose, the crack is in the(x1, x3)-plane, then a local coordinate system(x1, x2¼ x2, x3) is introduced in any point Ptangential to the crack front (see Figure 16), sothat the (x1, x3)-plane is perpendicular to thecrack.

The domain, B0¼B�BS, is again a mate-rial sheet of constant thickness, t, with t-0,but as this is a 3D problem, its border nowalso contains the upper and lower faces,Sþ and S�, in the (x1, x2)-plane, @B0 ¼G1,Gþ,G2,G�,Sþ,S�: Equation (15)becomes

Fi ¼I ’

G1

½ � ds þIGþ½ � ds þ

I -

G2

½ � ds þIG�½ � ds

þ 1

h

Z ZSþ

½ � dS þZ Z

S�½ � dS

� �¼ 0 ð27Þ

and for an infinitesimal thickness, t, a Taylorexpansion can be applied so that under thesame assumptions and arguments as above, thefirst component of the 3D J-integral isobtained:

JðscÞ ¼I ’

G1

w dx2 � sjknkuj;ids

�Z Z

S�

@

@x3w dx2 � sjknkuj;ids

dS ð28Þ

which is a local value, i.e., it varies alongthe crack front. The second term vanishes if Jis constant with respect to the crack-front

Figure 15 Virtual crack extension.

Figure 16 Definition of the local J-integral evalua-tion for 3D problems.


coordinate; it may contribute significantly ifstrong gradients occur, e.g., at the specimensurface.

Applying the domain integral method, therespective volume integral of Equation (26)already includes 3D ‘‘effects.’’ If the wholecrack front is shifted by the same amount, Da,an average value, %J ¼ ð1=lcÞ

R lc0

JðscÞdsc; for thetotal structure is obtained as in the experi-mental procedures where J is evaluated fromthe area under the load vs. displacement curve.

(ii) Body forces, surface tractions, and thermalloading

The fundamental equation for deriving thepath independence postulates that the stresstensor is divergence free. These equilibriumconditions are restricted to static and station-ary processes without body forces or heatsources acting in B. Constant body forces,such as gravitational forces, which have apotential not explicitly depending on thecoordinates, xi, can easily be included in thew-term and do not affect path independence. Inall other cases, J becomes path dependentunless an extra term is added (deLorenzi,1982b; Siegele 1989):

J ¼ 1

DA

Z Z ZV

sijuj;k � wdik

� �Dxk;i � fiui;jDxj

dv

ð29Þ

The forces fi can be body forces such aselectromagnetic forces or ‘‘accelerationforces,’’ fi ¼ �rxi; in the case of dynamicloading.

In addition, the boundary conditions postu-late that the crack faces, Gþ;G�; are tractionfree. If this condition is not met, pathindependence has to be re-established againby a surface correction term

J ¼ 1

DA

Z Z ZV

½ � dv �Z Z

@Vc

ti ui;jDxj dS

� �ð30Þ

where the vector ti represents the surfacetractions (or pressure) acting on the crackfaces, @Vc (deLorenzi, 1982b; Siegele, 1989).

The correction term for thermal fields is

J ¼ 1

DA

Z Z ZV

½ � dv þZ Z Z

V

sij@a@y

ðy� y0Þ �

þa�@y@xk

dij Dxk dv

�ð31Þ

where y(xi) is the temperature field, y0 thereference temperature, and a the coefficient ofthermal expansion (Muscati and Lee, 1984;Siegele, 1989).

(iii) Multiphase materials

Path independence of J holds only if thematerial is homogeneous. However, the assess-ment of defects in composite or gradientmaterials or in welded structures requires anextension of J to multiphase materials. Again,correction terms have to be added to re-establish path independence. Moreover, theboundary conditions become asymmetric inthese cases (mixed-mode problem), so that asingle component of J is insufficient tocharacterize the crack field and the complete‘‘J-vector’’ has to be considered. In a 2Dproblem, this reduces to J1 and J2. If thecontour G passes a phase boundary betweentwo materials near the crack tip, it includes anadditional singularity of stresses and strains.This contribution has to be eliminated by aclosed contour integral along this interface (seeKikuchi and Miyamoto (1982) and Figure 17):

Ji ¼ZG

wni � sjknkuj;i

� �ds �

ZGpb

wni � sjknkuj;i

� �ds

ð32Þ

3.03.3.1.5 Path dependence of the J-integral inincremental plasticity

The severest restriction for J results from theassumed existence of a strain energy density, w,as a potential from which stresses can beuniquely derived. This assumption also con-ceals behind frequently used expressions suchas ‘‘deformation theory of plasticity’’ (Hutch-inson, 1968a, 1968b; Rice and Rosengren,1968) or theory of ‘‘finite plasticity’’ (Hencky,1924). However, it actually does not describeirreversible plastic deformations as in the‘‘incremental theory of plasticity’’ of vonMises, Prandtl, and Reuss, but ‘‘hyperelastic’’

Figure 17 Contours for J-integral evaluation at acrack tip located near a phase boundary.


or nonlinear elastic behavior. It does not onlyexclude any local unloading processes but alsoany local rearranging of stresses, i.e., changingof loading direction in the stress space, result-ing from the yield condition. All loading pathsin the stress space are supposed to remain‘‘radial’’ so that the ratios of principal stressesdo not change with time. The condition ofmonotonous global loading of a structure is, ofcourse, not sufficient to guarantee radial stresspaths in nonhomogeneous stress fields. Hence,the J-integral will become path dependent assoon as plasticity occurs and the contour Gpasses the plastic zone (McMeeking, 1977).

For small-scale and contained yielding, apath-independent integral can be computedoutside the plastic zone. This means that G—orthe respective evaluation domain—has to belarge enough to surround the plastic zone andpass through the elastic region only. In grossplasticity, this is not possible, and some moreor less pronounced path dependence willalways occur, so that the evaluation of a‘‘path-independent’’ integral is a question ofnumerical accuracy. Because of its relation tothe global energy release rate (Equation (13)),which is used to evaluate J from fracturemechanics test results, J has to be understoodas a ‘‘saturated’’ value reached in the ‘‘far-field’’ remote from the crack tip.

Significant stress rearrangements occur at ablunting crack tip (see Figures 9 and 14 as wellas results by McMeeking and Rice (1975) andBrocks and Olschewski (1986)), and the pathdependence increases strongly (see Figure 17).Thus, a small strain analysis is advantageous ifonly the J-integral and no stresses at the cracktip shall be calculated as only little pathdependence occurs. But note that stressesat the crack tip lack physical significance inthis case.

As the work dissipated by plastic deforma-tion always has to be positive, the calculated J-values have to increase monotonically with thesize of the domain (Yuan and Brocks, 1991),which is confirmed by Figure 18—exceptcontour #21 touching the boundary. The high-est calculated J-value with increasing domainsize is always closest to the ‘‘real’’ far-field J:

JtiprJðrÞrJfar field ð33Þ

Any different result would indicate an error inthe definition of contours or in the evaluationof J. Figure 19 illustrates the approach to asaturation value of J for two different load-linedisplacements.

Moreover, J will keep a finite value in thelimit of a vanishingly small contour if and onlyif the strain energy density, w, has a singularity

of the order of r–1:

Jtip ¼ limG-0

IG


¼ limr-0

Z þp=2

�p=2wr cosW dW ð34Þ

This holds in linear elasticity where stressesand strains have a (1=

ffiffir

p)-singularity and for

HRR-like fields. As the stress singularity at theblunting crack tip vanishes under the assump-tion of finite strains and incremental plasticity,J will not even have a finite value any more:

Jtip ¼ limG-0

IG


¼ 0 ð35Þ

3.03.3.1.6 The J-integral for growing cracks

J is also used as a fracture parameter inEPFM for growing cracks (Rice et al., 1973;Garwood et al., 1975). Resistance curvesagainst ductile crack propagation are deter-mined by plotting J vs. crack growth, Da (seeASTM E 1737, 1996). These JR-curves, how-ever, are subject to a lot of size, geometry, andother ‘‘constraint’’ effects which have filledconferences, journals, and books over the years(e.g., Garwood, 1979; Sommer and Aurich,1991; Shih et al., 1993; Brocks and Schmitt,1993, 1995; Yuan and Brocks, 1998).

A serious objection against J as fractureparameter for growing cracks comes fromcomputational mechanics. J has become afracture parameter because of its path inde-pendence stating that global and local energyrelease rates are equal. If path independence islost, an important argument in favor of J isgone, too. J is actually accumulated plasticwork in the specimen or structure during crackgrowth and this work results mostly fromglobal plastic deformation and hence dependson size, geometry, and loading configuration.But J is not the driving force for a growingcrack in the sense that it does not equal thelocal energy release rate at the tip any more.

As was shown in Figure 18 already, bluntingof a stationary crack results in a significantpath dependence, and Equation (35) states thatnot even a finite value of Jtip exists. The sameeffect occurs—even stronger—at growingcracks as was shown by Brocks and Yuan(1989) and Yuan and Brocks (1991). Thoughstresses and strains are still singular, theirsingularity is not strong enough to provide anonzero local energy release rate. This wasaddressed long ago by Rice (1965, 1979) as the‘‘paradox of EPFM,’’ stating that no ‘‘energysurplus’’ exists for crack propagation. Thus,any kind of ‘‘near-field’’ integrals as proposed


by Atluri et al. (1984) and Brust et al. (1985)are physically meaningless. What really hap-pens in the process zone, namely damageinduced strain softening, is not covered byclassical plasticity with hardening materialbehavior. Continuum damage or cohesive zonemodels have to be applied in this region tocapture the respective effects properly (seeSections 3.03.4 and 3.03.5).

3.03.3.2 The CTOD and the CTOA

The CTOD for a stationary crack can bedetermined from the numerically simulated

blunted crack tip (see Figure 8), which requiresa large-strain analysis. The mesh around thecrack tip should consist of collapsed elements(Section 3.03.2.2, Figures 2 and 7). However,no unique definition of CTOD exists, neitherfor the numerical nor for the experimentaldetermination. A comparison between experi-mental and numerical results only makes sense,of course, if a unique definition is applied toboth. Various evaluation procedures are used.

Linear extrapolation of the deformed crackfaces in the remote field to the crack tip. Thisdefinition allows one to use a rather coarsemesh at the crack tip, as CTOD is mainly

0.0 0.2 0.4 0.6 0.8 1.00

100

200

300

400

500

VLL (mm)

J in

tegr

al (

N/m

m)

J (ASTM 1737-96)

J_02

J_04

J_06

J_08

J_10

J_12

J_14

J_16

J_18

J_20

J_21 (boundary)

C(T): small strain

0.0 0.2 0.4 0.6 0.8 1.00

100

200

300

400

500

VLL (mm)

J in

tegr

al (

N/m

m)

C(T): large strain

J (ASTM 1737-96)

J_02

J_04

J_06

J_08

J_10

J_12

J_14

J_16

J_18

J_20

J_21 (boundary)

Figure 18 J-integral vs. load-line displacement curve of a C(T) specimen for small (top) and large (bottom)deformation analysis.


based on far-field displacements which arerather unaffected by the discretization in thecrack-tip vicinity. Comparison with testresults is easy because the latter are com-monly determined by similar procedures (BS7448 Part 1, 1991; ASTM E 1290, 1999).Displacement of the crack-tip node of therespective element at the free crack surface.This definition is easy to handle, but theresults depend strongly on the FE mesh atthe crack tip and no experimental equivalentexists.Displacement at the intersection of twosecants originating from the crack tip underangles of 7451 with the opening profile of theblunted crack. This definition is in accor-dance with the HRR-theory (Shih, 1981)and yields a simple analytical relation ofCTOD with the J-integral (Equation (21)).However, it cannot be realized in fracturemechanics testing.Displacement measured over a gauge lengthof 5 mm at the location of the initial crack tip.This definition was proposed as a simpleexperimental procedure which can also beapplied to growing cracks for the determi-nation of d5-based resistance curves (Hell-mann and Schwalbe, 1984; Schwalbe, 1995;Schwalbe et al., 2002). It can easily beapplied to FE data.

In the course of crack growth after initiation,no significant blunting of the crack tip occursany more (see Figure 10). Instead, the CTOAcharacterizes the near-field deformation.CTOA is applied as a crack-growth criterionespecially for thin metal sheets (Newman et al.,

1992; O’Donoghue et al., 1997; Gullerud et al.,1999). Numerical simulations of CTOA-con-trolled crack growth require a regular mesh inthe ligament (see Figures 4 and 10). As forCTOD, no unique definition exists for theopening angle neither in testing nor in numer-ical simulations. Averaging of deformationover some elements appears necessary, butthe number of elements does, of course, affectthe results. An approximate relation exists tothe derivative of the d5 R-curve (Yuan, 1990):

cEdd5da

ð36Þ

(see Figure 20).

3.03.3.3 The Energy Dissipation Rate

The cumulative quantity J, which increaseswith increasing crack length, is not the truedriving force for ductile tearing as Turner(1990) has pointed out in a basic discussion onthe necessity of defining an alternative measureof tearing toughness. He proposed to definetearing resistance in terms of energy dissipationrate (the term ‘‘dissipated energy’’ means‘‘nonrecoverable mechanical work’’):

R ¼ dUdis

dA¼ dWext

dA� dUel

dAð37Þ

where Wext is external work and Uel the(recoverable) elastic strain energy. Thisdefinition is a straight transfer of Griffith’selastic energy release rate (Griffith, 1920) toplastic processes which is consistent with the

0 5 100

100

200

300

400

500

contour number

J-In

tegr

al (

N/m

m)

JFE (VLL=0.605mm)

JASTM

JFE (VLL=0.884mm)

JASTM

Figure 19 Path dependence of J in the large deformation analysis in comparison to the respective ASTMreference values.


incremental theory of plasticity. The dissipa-tion rate has the same dimension as J andcharacterizes the increment of irreversible workper crack-growth increment. It falls withincreasing crack length in gross plasticity andconsists of two contributions, namely work ofremote plastic deformation and local work ofseparation:

R ¼ dUpl

dAþ dUsep

dAð38Þ

For specimens with constant thickness B andstraight crack front (‘‘plane’’ problems), theincrement of crack area is dA ¼ B da:

The dissipation rate, R, is more appropriatefor characterizing crack growth in plasticallydeformed structures than the conventionallyused J-integral (Memhard et al., 1993; Koled-nik et al., 1997; Atkins et al., 1998; Sumpter,1999). It is, in fact, the true ‘‘driving force’’which has to equal the structural resistance inorder to propagate the crack by some amount,Da, whereas J accumulates the plastic workdone along a given loading path. It is, how-ever, not a material but a structural propertyas it contains the work of remote plasticdeformation.

When introducing R, Turner generallydoubted that splitting it into local and globalcontributions will ever be possible. Thus, everymeasured ductile crack-growth resistance willnecessarily contain remote plastic work which,in general, is much larger than the local workof separation. In fact, only external work andelastic energy can be measured. Models ofdamage mechanics, however, provide ideashow to perform this separation.

For quasi-static processes, the dissipationrate can simply be evaluated from the areaunder the measured or calculated load vs.displacement curve:

R ¼ limDa-0

DUdis

BNDa¼ lim

Da-0

FDvpl

BNDað39Þ

if R is supposed to include the whole irrever-sible part of the work done and, hence, mayalso include dissipated energy in zones remotefrom the crack tip, e.g., around load points orsupports. As no splitting into local and globalcontributions according to Equation (38) ispossible, ‘‘dissipated’’ work equals total ‘‘plas-tic’’ work as determined from the area underthe load vs. displacement record. If therespective information is not available anymore, R can be simply re-evaluated fromexisting JR-curves by inverting the procedureof calculating J from test data. The respectiveformula is (Memhard et al., 1993)

R ¼ W � a

ZdJpl

daþ Jpl

gZ

ð40Þ

for bend-type specimens, C(T) and SE(B), withthe well-known geometry factors

Z ¼2:0þ 0:522 1� a=Wð Þ for CðTÞ

2:0 for SEðBÞ

(ð41Þ

g ¼1:0þ 0:76 1� a=Wð Þ for CðTÞ

1:0 for SEðBÞ

(ð42Þ

and

R ¼ ðW � aÞdJpl

dað43Þ

for tension-type specimens, M(T) and DE(T).R vs. Da curves are obtained as crack-growthresistance curves for the respective specimensand materials (see Figure 21). Similar to CTOAR-curves, they decrease from higher values atinitiation in a transition region to stationaryvalues.

In an elastic–plastic FE simulation of crackgrowth, the dissipation rate can be calculateddirectly from stresses and strains (Memhardet al., 1993; Siegmund and Brocks, 1999)

’Upl ¼Z

Vpl

sij ’epij dV ¼

ZVpl

se ’epe dV ð44Þ

where se and epe are von Mises effective stressand effective plastic strain (Equations (6) and(3)), respectively. The volume integral mayeither be performed over the whole body, thusyielding the result of Equation (37), or just overthe plastic zone at the crack tip, however largeit may be. If the FE simulation reflects plastic

0.3

0.2

0.1

0

dδ5/d

a; C

TO

A

CRACK EXTENSION, ∆a, mm

0 5 10 15

2024-T351

Experiments: d δ5/da

Finite Elements: d δ5/da(Plane Stress)

Finite Elements: CTOA(Plane Stress)

W = 100 mmB = 20 mmao = 71 mm

Figure 20 CTOA and dd5/da vs. Da in a thin C(T)specimen of Al 2024-T351, tests, and FE results.


processes, only, i.e., it is simply based on theMises–Prandtl–Reuss constitutive equationsand follows experimental records of eitherJ(Da) or VL(Da) (see, e.g., Siegele and Schmitt,1983), dissipated work equals total plasticwork as in the experimental procedure.

Subsequent approaches to modeling of duc-tile rupture refer to Barenblatt’s idea (Bare-nblatt, 1962) of introducing a ‘‘process zone’’ahead of the crack tip where material degrada-tion and separation occur. This approachrequires a constitutive description of the mate-rial behavior in the process zone which canmirror the local loss of stress carrying capacity.In general, two alternatives are applied:

(i) phenomenological ‘‘cohesive zone mod-els’’ describing the decohesion process by atraction strength and the work of separationper unit area, e.g., Needleman (1990a), Yuanet al. (1995), Lin et al. (1997), and Siegmundet al. (1998), see Section 3.03.5, and

(ii) models based on the micromechanismsof ductile failure, namely the nucleation,growth, and coalescence of voids, e.g., themost commonly used models of Gurson (1997),Tvergaard and Needleman (1983), the GTNmodel, or Rousselier (1987), see Section 3.03.4.

For the cohesive zone model, ’Usep iscalculated from

’Usep ¼ G0’A ¼

Z d0

0

TðdÞ dd � ’A ð45Þ

where T is the ‘‘surface traction,’’ i.e., the stressacting on the surface of a continuum element,and d is the ‘‘separation,’’ i.e., the displacementjump between adjacent continuum elements

(see Section 3.03.5.2). For the GTN or theRousselier model, the energy needed formaterial separation, ’Usep; in an incrementalcrack advance, ’A; is given by

’Usep ¼Z

Vsep

ð1� f Þse ’epe dV ð46Þ

Here, the volume integral is performed over the‘‘separation zone’’ only, i.e., a single row ofelements ahead of the crack tip being describedby the GTN model. As the damaged zone infracture specimens is generally restricted to onerow of elements in the ligament, the rest of thestructure may be described by classical vonMises plasticity, which reduces computationtime but does not affect the macroscopicbehavior. Models like this have been addressedas ‘‘computational cells’’ by Xia and Shih(1995), Xia et al. (1995), Ruggieri et al. (1996),Faleskog et al. (1998), and Gao et al. (1998).

Both approaches allow for splitting the totaldissipated work into the (local) work ofseparation in the process zone and the (global)plastic work in the embedding materialand, thus, solve a classical problem ofEPFM (Siegmund and Brocks, 2000a, 2000b).Examples will be given in Section 3.03.5.2.7.

3.03.4 DAMAGE MECHANICS AND‘‘LOCAL APPROACHES’’ TOFRACTURE

3.03.4.1 Damage and Fracture

‘‘Local approaches’’ and ‘‘micromechanicalmodeling’’ of damage and fracture (Pineau,

0 2 4 60

2.103

4.103

6.103

8.103

1.104

∆a [mm]

R [N

/mm

]

StE 460

M(T) test

M(T): FE-sim

C(T): test

C(T): FE-sim

Figure 21 Energy dissipation rate vs. crack growth for side-grooved C(T) and M(T) specimens StE 460, testresults, and FE simulation with cohesive zone model (source Siegmund and Brocks, 2000a, 2000b).


1981) have found increasing interest. Thegeneral advantage, compared with classicalfracture mechanics, is that, in principle, theparameters of the respective models dependonly on the material and not on the geometry.Thus, these concepts guarantee transferabilityfrom specimens to structures over a wide rangeof sizes and geometries and can still be appliedwhen only small pieces of material are avail-able which do not allow for standard fracturespecimens. It is not even necessary to considerspecimens with an initial crack as, of course,also initially uncracked structures, will break ifthe local degradation of material has exceededsome critical state. The identification anddetermination of the ‘‘micromechanical’’ para-meters require a hybrid methodology ofcombined testing and numerical simulation.Different from classical fracture mechanics,this procedure is not subject to any sizerequirements for the specimens as long as thesame fracture phenomena occur.

Micromechanical modeling encounters anew problem, however, namely, that—differentfrom the assumption of continuum me-chanics—the material is not uniform on themicroscale but consists of various constituentswith differing properties and shapes. A materi-al element has its own complex and evolvingmicrostructure. Micromechanics is a generalmethodology of expressing continuum quanti-ties in terms of the parameters which char-acterize the microstructure and properties ofthe microconstitutents of the material neigh-borhood (Nemat-Nasser and Hori, 1993). Tothis end, the concept of a representativevolume element (RVE) has been introducedby Hill (1963), Hashin (1964), and others. AnRVE for a material point is a material volumewhich is statistically representative of a materi-al neighborhood of that material point. By this,a length scale is introduced in the continuum.

Many constitutive models for damage evolu-tion exist for various phenomena of materialbehavior (see also Section 3.03.5.4). Fracturephenomena in ductile metals occur by either:

(i) formation of microcracks and their ex-tension with little global plastic deformation(‘‘brittle’’ or cleavage fracture), or

(ii) the nucleation, growth, and coalescenceof microvoids with significant plastic deforma-tion (ductile rupture).

Cleavage processes are stress controlled andconsume little deformation energy. Hence, thecrack grows fast, and unstable and a highamount of kinetic energy can be releasedespecially in large structures. Local fracturecriteria are based on a critical cleavage stress.Compared with this, the strain-controlledprocess of ductile rupture consumes much

more energy by plastic deformation. The crackgrows slowly and deformation-controlled oreven stable, i.e., neglecting creep effects,growth stops if the load does not increase.Local failure criteria are based on a criticalfailure strain, a critical void growth ratio, or ayield condition for porous materials.

Numerical models exist for both failurephenomena, in particular:

(i) the Beremin (1983) model which is basedon a critical fracture stress concept togetherwith the ‘‘weakest link’’ assumption andWeibull (1939a, 1939b, 1951) statistics,

(ii) the void growth law of Rice and Tracey(1969), and

(iii) the models of Gurson (1977) or Rous-selier (1987) for porous metal plasticity.

Beremin’s model for cleavage fracture andthe Rice and Tracey model for crack initiationdue to ductile tearing yield ‘‘damage indica-tors,’’ only, by an a posteriori evaluation ofstress and strain fields obtained in a conven-tional elastic–plastic FE analysis. They are thuseasy to handle, but they do not account for theeffect of damage on plastic deformation, andno crack growth can be simulated. Theconstitutive models of Gurson and Rousselier,alternatively, allow for both but requirespecific material subroutines for performingthe FE analysis as the evolution of damageaffects the yield behavior. In most of thecommercial FE codes, these subroutines haveto be provided by the user.

3.03.4.2 Damage Indicators

3.03.4.2.1 Ductile tearing

Damage indicators are obtained by a purepostprocessing of stress and strain data from aconventional elastic–plastic analysis. Damagedoes not change the constitutive equations.According to the analytical solution of Riceand Tracey (1969), void growth follows theequation

’D ¼ ’r

r¼ 0:283 ’epexp

3sh2se

� �ð47Þ

for high triaxialities. The rate of damage, ’D; isdefined as the growth rate of an averagespherical void of radius, r, with increasingplastic strain, where sh and se are the hydro-static stress and the von Mises effective stress,respectively. This evolution law can be inte-grated as

r

r0¼ exp D � D0ð Þ ¼ exp 0:283

Z ep

0

exp3sh2se

� �dep

�ð48Þ

Damage Mechanics and ‘‘Local Approaches’’ to Fracture 147

The initial void radius, r0, is not explicitlyneeded. Void coalescence and ductile crackextension is supposed to start, as soon as r=r0reaches its critical value, ðr=r0Þc: This value iscalibrated by testing of notched tensile barsand a simple numerical simulation of anaxisymmetric model with elastic–plastic mate-rial behavior (see Figure 22). The respectiveprocedure is described in an official ESISprocedure (ESIS P6, 1998). Equation (48) isevaluated for all elements which have plasti-cally deformed, and the highest value obtainedin any element at crack initiation in the testyields ðr=r0Þc; which is supposed to be amaterial parameter which can be used to assesscrack initiation in any structure.

The assumption of a single scalar damageparameter is equivalent to that of sphericalvoids. Hence, the relation

f ¼ r

r0

� �3

f0 ð49Þ

holds between the void volume fraction used inthe Gurson or Rousselier model and the

damage parameter of the Rice and Traceymodel.

3.03.4.2.2 Cleavage fracture

The critical fracture stress concept states thatfracture occurs for a critical value, sc, of themaximum principal stress, sI. This criterion isapplicable to any notched structure as long asthe notch root radius remains finite. It has tobe modified for structures with macroscopiccracks which induce singular stress fields byintroducing a characteristic distance, xc, fromthe crack tip. The critical stress is related toGriffith’s surface energy by

sc ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4Egpð1� n2Þl

sð50Þ

where E is Young’s modulus, n is Poisson’sratio, g is the surface energy, and l is the cracklength. The cleavage fracture strength, sc, canbe determined by elastic–plastic FE analyses ofnotched bend or tensile bar tests (ESIS P6,1998). In body centered cubic (b.c.c.) materials,a transition region from cleavage to ductiletearing exists where fracture toughness in-creases and becomes subject to scatter. Inorder to describe the latter phenomenon,Beremin (1983) and Mudry (1987) applied theconcept of the failure probability for brittlematerials by Weibull (1939a, 1939b, 1951) toferritic steels. The basic idea is the so-called‘‘weakest link’’ assumption that the probabilityof having cleavage fracture of a structure atany given load is equal to the probability thatits weakest element (‘‘link’’) fails at this load.The failure criterion (Equation (49)) is estab-lished on a ‘‘mesoscopic’’ level for everymaterial element, (i), which is subject to

a stress sðiÞI : The ‘‘critical length’’, lðcÞi ; of an

assumed microcrack in this element derivesfrom Griffith’s criterion as

lðcÞi ¼ 4Eg

pð1� n2Þ1

sðiÞc� �2 ð51Þ

Now assume that the probability of having acrack of length between li and li þ dli in thiselement is

PðliÞ dli ¼a

lbi

dli ð52Þ

where a and b are parameters dependingon the material’s microstructure and mechan-ism of microcrack formation. Then its failure

Figure 22 FE mesh of a notched tensile bar used tocalibrate the parameters of the local approach toductile and cleavage fracture.


probability becomes

Pfi sðiÞI� �

¼Z

N

lðcÞi

P lið Þ dli ¼sðiÞIsu

!m

ð53Þ

with the two ‘‘Weibull parameters’’

m ¼ 2b� 2 ð54Þ

and

su ¼ b� 1

a

� �1=mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4Egp 1� n2ð Þ

sð55Þ

According to the ‘‘weakest link’’ assumption,the failure probability of the whole structure(‘‘chain’’) with n elements (‘‘links’’) is theproduct

Pf ¼ 1�Yn

i¼1

1� Pfi ðs

ðiÞI Þ

h ið56Þ

and its ‘‘survival probability’’ is (1�Pf). Sincethe failure probability is supposed to be small,Pf

i{1; Equation (55) may be rewritten as

ln 1� Pfð Þ ¼Yn

i¼1

ln 1� Pfi sðiÞI� �h i

EXn

i¼1

Pfi sðiÞI� �

ð57Þ

Based on these assumptions, the fractureprobability of the entire structure follows atwo-parameter distribution function:

Pf swð Þ ¼ 1� exp � swsu

� �m �ð58Þ

where su is the scaling factor which describesthe point of the distribution function on thestress axis at ln 1/(1–Pf)¼ 0 or Pf¼ 0.632, i.e.,63.2% failure probability; m is the Weibullexponent or Weibull modulus which describesthe scatter of the distribution.

The Weibull stress, sw, is defined by asummation of the maximum principal stress, sI:

sw ¼Xnpli¼1

sðiÞI� �m Vi

V0

" #1=m

ð59Þ

where V0 is some reference volume. Sinceplastic deformations are a prerequisite forcleavage fracture in a metal, the summation isperformed over the plastically deformed part ofthe volume, only, i.e., the npl elements whichhave already experienced plastic deformations.Using a summation over ‘‘elements’’ instead ofan integration over the volume does not onlyrefer to its application in FE calculations, butalso accounts for the necessity of averaging the

stress values over some characteristic volume,especially in regions of steep stress gradients,e.g., at a crack tip. For a given microstructureand a given mechanism of microcrack forma-tion, there is some minimum volume, V0, forwhich statistical independence can be assumed.Mean stress values over volumes much smallerthan V0 are not reasonable. If a finer FE meshis used, stresses have to be averaged over thischaracteristic volume. But it is a much simplerprocedure to introduce V0 as the minimummesh size, Vmin (Mudry, 1987). For low stressgradients the mesh size is not significant and thereference volume, V0, in Equation (59) may betaken arbitrarily (Beremin, 1983; Mudry, 1987).As it affects the parameter su by sm

u V0 ¼constant; a transfer of Weibull distributionsfrom specimens to components is only admis-sible for a fixed reference volume. In addition,no physical meaning should be assigned to su

in the case of an arbitrarily chosen V0.The application of the local approach to

cleavage fracture does not require a measure-ment of the basic quantities which have beenused for the derivation of the concept, such ascleavage fracture strength, sc, or Griffith’ssurface energy, g, in Equations (50) and (51)and the parameters of the microcrack distribu-tion, a, b, in Equation (52), but only of thederived quantities, m and su, in Equations (58)and (59). Nevertheless, it will appear necessaryto keep in mind the basic assumptions of theconcept and discuss their physical significancefor metals.

The Weibull parameters, m and su, aresupposed to be independent of the specimengeometry and can therefore be determined bysimple tests of notched tensile bars and acorresponding FE analysis (see, e.g., Figure22). The respective procedure is described in anofficial ESIS procedure (ESIS P6, 1998).Additional tests on cracked specimens arenecessary to fit V0, if required. A sample of Nspecimens is tested and the specimens areordered in ascending sequence with respect toa monotonically increasing loading parameter,generally a prescribed external displacement,which uniquely characterizes the event ofcleavage failure. A relative failure probabilityis assigned to the jth specimen, e.g., by

Pj ¼j � 0:5

N; j ¼ 1;y;N ð60Þ

and the fracture event is characterized by therespective Weibull stress, sðjÞw ; which is deter-mined by an elastoplastic FE analysis independence on the loading parameter. Theevaluation of Equation (59) can be done in apostprocessor program. The summation overthe Gauss points of an element may be


performed before or after putting sI to thepower m, which will result in different values,of course. In the first case, stresses are linearlyaveraged within each element. When perform-ing the summation over the volume of thespecimen, one has to pay attention that the FEmodel may have unit thickness, e.g., 1 rad inthe axisymmetric case, and may be symmetricto the center plane. Hence, a respective volumefactor has to be applied for the calculation ofVi. sw has to be calculated for every fracturedspecimen, i.e., at the time step which corre-sponds to the fracture event of the respectivespecimen. An interpolation between time stepsmight be necessary. The correlation betweenthe experimental fracture event and the timestep in the FE analysis has to be realized with amonotonically increasing parameter, e.g., theelongation, DL, or diameter reduction, DD.

As m is unknown in the beginning, theevaluation of Equation (59) starts with anassumed value which has to be corrected in aniterative procedure. The data are plotted as

yj ¼ ln ln1

1� Pf

� �vs: xj ¼ ln

sðjÞwMPa

!ð61Þ

in order to assure that they follow a Weibulldistribution with sufficient accuracy. TheWeibull parameters, su and m, are assessedby the maximum likelihood method (Khaliliand Kromp, 1991). If the calculated value of mdeviates from that used in the previous step tocalculate sw, the procedure is repeated until thedifference of two iteration steps, Dm, is lessthan a defined accuracy. Beremin (1983)applied this approach to the statistical analysisof fracture toughness data of ferritic pressurevessel steels and found a value of m¼ 22 and subetween 1,970MPa and 2,800MPa forV0¼ 1.25� 10–4 mm3.

Since the estimation is only based on asample of size N, the parameters of the entirepopulation of all possible specimens from thematerial cannot be determined exactly. Onlyconfidence intervals can be evaluated. Aconfidence level (1�a) is introduced, which isthe required probability that any one estimatewill fall within the confidence interval. If theresults of the parameter estimation procedureare called su0 and m0, and (1�a) is the desiredconfidence level, the following statementsabout m and su can be made:

m0

llrmr

m0

luð62Þ

and

su0 exp � tl

m0

� �rsursu0 exp � tu

m0

� �ð63Þ

with a probability of at least (1�a)� 100%. ll,lu, tl, and tu are numbers which depend on Nand a only (Khalili and Kromp, 1991). If m isnot expressed by a confidence interval, anotherprocedure has to be followed: when going fromthe sample to the entire population, theparameter m has to be bias corrected. It willthen be denoted mcor:

mcor ¼ m0b ð64Þ

where b depends on N only. This bias correc-tion is important as soon as the Weibullparameters shall be applied to other specimensor structures and predictions will be made.

The Weibull parameters can now be used topredict the probability of cleavage failure of afracture mechanics specimen or an entirestructure by an FE analysis. Figure 23 showsthe FE mesh at the crack tip of a C(T)specimen consisting of square elements asproposed for a European round robin onthe local approach. Since the crack tip is aregion of high stress concentration, the resultsof the calculation generally depend on themesh size. Figure 24 shows the influence of themesh size, varying from 0.1mm� 0.1mm to0.0125mm� 0.0125mm, on the predicted clea-vage-failure probability of a C(T) specimen asa function of the stress-intensity factor. TheWeibull parameters, m¼ 22 and su¼ 1,898MPa, have been determined from notchedtensile bars. The approach allows for predict-ing size effects on cleavage-failure probability(see Figure 25).

Commonly, it is assumed that the Weibullparameters do not depend on the temperature.Once determined for one temperature they can,hence, be used to predict the temperaturedependence of fracture toughness. Investiga-tions of various specimens at different tem-peratures and loading rates have raised doubts,however, and several modifications have beenproposed (see, e.g., Bernauer et al., 1999).Different from ceramics, where the defectdistribution (Equation (51)) is a materialproperty, which is determined by the manu-facturing process, defects in a ductile metalform with plastic deformation and the defectdistribution will therefore depend on the loadhistory. In addition, nucleating defects mayjust enlarge in a ductile manner and notbecome the origin of cleavage.

3.03.4.3 Micromechanical Models of DuctileTearing

The approach of continuum damage me-chanics is a promising way to overcome thenumerous problems of size and geometry


dependence of the characteristic parametersused in conventional fracture mechanics. Thereare also problems and open questions withrespect to this approach. There is no known (asof early 2000s) general proof under which

conditions a unique solution for the set ofmaterial data exists. This may look like amathematically interesting problem rather thanan engineering one. But it obviously affects thequestion whether or not the set of data may be

(a)

(b)

Figure 23 FE mesh of a C(T) specimen used for the application of the local approach to cleavage fracture: (a)total mesh and (b) detail of the crack-tip region.

Figure 24 Influence of the crack-tip element size on the predicted cleavage-failure probability of a C(T)specimen in dependence on the stress intensity factor, m¼ 22 and su¼ 1,898MPa.


transferred to any structural component inorder to assess its structural integrity underservice load conditions. A related question ofmore practical importance is how ‘‘user depen-dent’’ the results are (Bernauer and Brocks,2002). In addition, localization effects arise forany softening materials which lead to meshdependence of the results in FE calculations.This becomes especially significant in crackproblems. The necessity of introducing acharacteristic length or volume is evident(e.g., Sun and Honig, 1994; Siegmund andBrocks, 1998a; Xia and Shih, 1995).

The constitutive equations which are used todescribe ductile fracture processes are based onrelatively simple models of the growth andcoalescence of microvoids (Rice and Tracey,1969; Koplik and Needleman, 1988). Theapplication of these simple micromechanicalmodels is justified by a statistical averagingeffect over a large number of material elementscontaining microdefects. This averaging resultsin the definition of a RVE or ‘‘unit cell’’ whichrepresents the essential micromechanical phe-nomena. On the macroscale, a homogenizationprocess is applied which relates the micro-structural properties in a systematic way tofield quantities of continuum mechanics. Thevoid volume fraction, f, which is defined as theratio of the total volume of all cavities to thevolume of the body, is introduced as aninternal variable to characterize the damage.Its evolution equation consists of two termsdue to nucleation and growth, in general:

’f ¼ ’fgrowth þ ’fnucl with f ðt0Þ ¼ f0 ð65Þ

with f0 as the initial void volume fraction. Thevoid growth rate is proportional to the plasticvolume dilatation rate:

’fgrowth ¼ ð1� f Þ trDp ð66Þ

The most difficult modeling problems in thetheory of ductile fracture are concerned withthe nucleation of microvoids at the sites ofinclusions and second-phase particles in aplastically deforming matrix. An empiricalapproach for the nucleating part of voidevolution was proposed by Needleman andRice (1978):

’fnucl ¼ A ’sþ B

3tr ’S ð67Þ

Chu and Needleman (1980) suggested a normaldistribution for void nucleation where A and Bare given by

A ¼ 1

Et

fn

snffiffiffiffiffiffi2p

p exp �1

2

ep � ensn

� �2" #

; B ¼ 0 ð68Þ

for strain-controlled nucleation. Here, fn is thevolume fraction of void nucleating particles, enis the mean strain for nucleation, and sn is itsstandard deviation. The tangent modulus ofthe true stress–strain curve is called Et, and ep isthe accumulated effective plastic strain of thematrix material.

After a microvoid has nucleated in aplastically deforming matrix, it undergoes avolumetric growth and a shape change. Gurson(1977) has derived the yield function of aporous plastic continuum. Plastic flow does notonly depend on ep but, according to this model,

Figure 25 Prediction of size effects on the cleavage-failure probability of C(T) specimens by the localapproach, m¼ 22 and su¼ 1,898MPa.


on a second internal variable, the void volumefraction, ff, as well. Tvergaard and Needleman(1983) introduced some empirical modifica-tions of this yield function (for details, seeChapter 2.03)

F ðS0; tr S; f ; epÞ ¼ 3S0 � �S0

2R2ðepÞ

þ 2 q1 f �cosh q2tr S

2RðepÞ

� �� 1þ q3 f �2� �

¼ 0 ð69Þ

with three additional material parameters, q1,q2 , q3, and a damage variable, f*,

f � ¼f for frfc

fc þ k ðf � fcÞ for f4fc with k ¼ f �u � fc

ff � fc

8<:

ð70Þ

accounting for accelerated damage occurringwith beginning coalescence of voids when acritical void volume fraction, fc, is reached. Therespective modification of the Gurson modelwill be referred to as GTN model in thefollowing. The crack appears if the final voidvolume fraction, ff, is reached, where thematerial loses its stress carrying capacity andwhere the damage variable achieves its ultimatevalue, f �

u ¼ 1=q1: For this state, k can becalculated if the void volume fraction at finalfracture is known from experiments.

In rate-independent plasticity, the currentflow stress of the matrix, R(ep), depends on theaccumulated plastic strain alone. This describesstrain hardening for slow deformation pro-cesses. If the Gurson model shall be extendedto dynamic processes, the effect of strain rateon plastic hardening and the coupled effect ofthermal softening have to be considered andRðep; y; ’epÞ is introduced, instead (Sun et al.,1994).

An alternative formulation of the yieldcondition was derived by Rousselier (1987)based on thermodynamical considerations

FðS0; tr S; f ; epÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3=2ð ÞS0 � �S0

pð1� f ÞRðepÞ

þ s1RðepÞD exp

tr S

s1ð1� f Þ

� �� 1 ¼ 0 ð71Þ

with two material parameters, D and s1.The GTN model described above includes a

total number of nine parameters: three para-meters (en, fn, sn) are used to model voidnucleation; three (f0, fc, ff,) describe theevolution of void growth up to coalescenceand final failure; and the three remaining (q1,

q2, q3) characterize the yield behavior of thematerial. In particular, q1 may depend on thehardening of the matrix material and iscommonly set to 1.5. Furthermore, it isassumed that q2¼ 1 and q3 ¼ q2

1; so that thenumber of parameters reduces to six. It maybecome necessary to increase the value of q2 instructures with low triaxiality close to plane-stress-like sheet metal. As the application ofmicromechanical models to ductile fracture isstill a rather new approach, no generallyaccepted recommendations exist how to iden-tify and determine these parameters. They areobviously not independent of each other. Someexperience exists (see Tvergaard and Needle-man, 1983; Needleman and Tvergaard, 1987;Sun et al., 1988; Brocks et al., 1995; Xia et al.,1995; Ruggieri et al., 1996; Bernauer andBrocks, 2002), but further detailed studies arenecessary to analyze the interaction of theparameters as well as the sensitivity of themodel. Numerical screening tests with a tensilebar and a C(T) specimen revealed that fourparameters, namely fn, f0, fc, and q1, influenceboth, the reduction of area at fracture, DSf,and J at initiation. The volume fraction of voidnucleating particles, fn, affects DSf directly,whereas the interaction of fn and en issignificant for Ji. The ‘‘damage accelerationfactor’’ after coalescence, k, in Equation (70)—or the void volume fraction at final fracture,f—is of more numerical significance for crack-growth simulations and should be taken ashigh as convergence can be assured.

The initial void volume fraction, f0, is amicrostructural feature and can, hence, inprinciple be determined from metallurgicalobservations. This relation could be verifiedmost easily for nodular cast iron, for instance,where void nucleation occurs by decohesion ofthe graphite particles from the ferritic matrix(Steglich and Brocks, 1997). The relationbetween f0 and the volume fraction of sec-ond-phase particles is much more complex formaterials having more than one population ofinclusions and for void nucleation by breakingof particles. Ruggieri et al. (1996) state that forall ferritic steels studied thus far, f0 ranges from10–4 to 4� 10–3. The determination of thedamage parameters is still a mostly phenom-enological fitting procedure which requires ahybrid methodology of combined testing andnumerical simulation. In general, as manyparameters as possible should be taken fromliterature on similar materials, and kept con-stant while the remaining are varied in thenumerical simulations to fit experimental datawhich significantly reveal failure events.

The first question which arises is whether theanalysis should account for void nucleation or,


just for simplicity, start with an initial voidvolume fraction without accounting for nuclea-tion, or both. The critical void volume fractionat initial coalescence, fc, interacts with theinitial value, f0, and the volume fraction of voidnucleating particles, fn. Hence, the effects aredifficult to separate in the numerical simula-tions, which suggests neglecting one of the twocontributions if this is consistent with micro-structural observations. For a given f0 withoutconsidering nucleation, fc can be determined bycell model calculations (Koplik and Needle-man, 1988; Brocks et al., 1995b).

Tvergaard and Needleman (1983) were thefirst who analyzed the fracture of a roundtensile bar with the Gurson model. Theyreferred to the phenomenon that the onset ofmacroscopic fracture is associated with asudden drop of the load. Fitting the numericalresults to the experimental data at this pointhas therefore become a common technique todetermine fc. It is important to notice that fc isalmost independent of the triaxiality (seeKoplik and Needleman, 1988; Brocks et al.,1995b). An important question for the applic-ability of the model is whether or not fcdepends on temperature and loading rate.The cell model calculations of Brocks et al.(1995b) have shown that the latter has littleeffect on fc. But the experiments indicated aslight temperature dependence since the reduc-tion of diameter decreases with temperature, asis well known, whereas the numerical resultswith a temperature-independent fc showed theopposite trend.

As has been mentioned above, the resultsobtained with damage models are mesh sizedependent and, hence, the element size be-comes an additional material parameter. De-spite the general agreement that the elementsize has to be related to the microstructure ofthe material, namely the average spacing ofvoid nucleating particles, the actual physicsof the void nucleation process does not allowfor a simple and direct correlation of materialand model length scales. Several authors haveintroduced the term ‘‘computational cells’’ forthe elements in the ligament (see, e.g., Xia andShih, 1995; Xia et al., 1995; Ruggieri et al.,1996; Faleskog et al., 1998; Gao et al., 1998).The descriptive ‘‘cell’’ points at the fundamen-tal difference to the common idea of the FEmethod according to which elements aresupposed to be numerical entities solving aboundary value problem but having no physi-cal significance determining their size. Thedominant effect of the ‘‘cell size’’ on the JR-curve comes from its height, as a simpleconsideration shows. The mechanical workdissipated in one element by inelastic deforma-

tion results from

DUdis ¼ZDV

udis dV ¼Z

V

Z t

t¼0

sij ’epij dt

� �dV

¼ %udis DV ð72Þ

where the element volume is DV ¼ welhelbel

with wel¼width, hel¼ height, and bel¼thickness. The energy release rate per crack-growth increment, i.e., one element width,Da ¼ wel; is

G0 ¼DUdis

DA¼ DUdis

bel wel¼ %udis helEJi ð73Þ

which is the specific ‘‘work of separation’’ (seeSiegmund and Brocks, 1999). By increasing theelement height, the JR-curve is raised, as moremechanical work is dissipated—or energy‘‘released’’—per crack-growth increment. Thisis confirmed by numerical studies and iscommon interpretation in the literature (e.g.,Ruggieri et al., 1996). The element width, wel,affects the averaging of stress and straingradients at the crack tip but plays a minorrole for the JR-curve. The correlation betweenlength scales of material and model is estab-lished via G0 and, hence, depends not only onthe hardening behavior, but also on the type ofFEs used, linear or quadratic and integrationorder.

3.03.5 THE COHESIVE MODEL

3.03.5.1 Introduction

As mentioned in Section 3.03.2.3, the crackpropagation within a structure can be simu-lated using several different methods, i.e., thenode release technique controlled by anyfracture mechanics parameter, the constitutiveequations including damage (e.g., the Gursonmodel), continuum damage concepts based ona theory of Kachanov (1993), Lemaitre (1985),or cohesive elements. The main disadvantageof the node release technique using a fractureparameter, e.g., KIc or Ji, is that only structureswith an initial crack can be modeled. Inaddition, all drawbacks of the fracture me-chanics parameters (e.g., geometry depen-dence, validity limits) apply accordingly tothis technique.

The Gurson model, briefly described inSection 3.03.4.3, is a very powerful model forthe simulation of ductile damage and crackgrowth. Since it tries to simulate the ductilefracture mechanism by a modified plasticpotential containing the material softeningdue to void nucleation, growth, and coales-cence, it can only be used for ductile damage.Alternatively, the fracture can be simulated for


any structure with or without a crack. This isalso valid for the continuum damage modeland the cohesive model. In addition, the lattertwo are not confined to a class of materials, butcan be used for arbitrary materials, not onlyfor those with ductile behavior. In the follow-ing, the cohesive model will be described as amodel which has its main advantages in that itcontains few parameters, its universality ofapplicability, and the wide experience of morethan 20 years of use.

3.03.5.2 Fundamentals

3.03.5.2.1 Barenblatt’s model

The idea for the cohesive model is based onthe consideration that infinite stresses at thecrack tip are not realistic. Models to overcomethis drawback have been introduced by Dug-dale (1960) and by Barenblatt (1962), calledstrip-yield models. Both authors divided thecrack into two parts: one part of the cracksurfaces, Figure 26, is stress free, and the otherpart is loaded by cohesive stresses. Dugdaleintroduced the finite stress to be the yieldstress, s0, which holds only for plane stress, butthe crack-opening stresses can be much higherthan the equivalent stress in a multiaxial stressstate. Barenblatt, who investigated the fractureof brittle materials, made several assumptionsabout the cohesive stresses: The extension ofthe cohesive zone d is constant for a givenmaterial (independent from global load) andsmall compared to other dimensions. Thestresses in the cohesive zone follow a prescribeddistribution s(x), where x is the ligamentcoordinate, which is specific for a givenmaterial but independent of the global loadingconditions.

3.03.5.2.2 Advanced cohesive zone models

Most of the newer models developed andproposed are different from Barenblatt’s modelin that they define the traction acting on theligament as a function of the opening and noton the crack-tip distance as Barenblatt did.

For practical applications the model becamemore interesting when numerical methods,

mostly the FE method, were applicable tononlinear problems. Needleman (1987) was thefirst, who used the model for crack propaga-tion analyses of ductile materials. Hillerborget al. (1976) applied the cohesive zone model tobrittle fracture using the FE method for thefirst time, followed by Petersson (1981) andCarpinteri (1986) amongst others.

The material separation and, thus, damage ofthe structure is classically described by interfaceelements—no continuum elements are damagedin the cohesive model. Using this technique, thebehavior of the material is split into two parts:the damage-free continuum with an arbitrarymaterial law, and the cohesive interfacesbetween the continuum elements, which specifyonly the damage of the material. A differenttechnique to define the cohesive elements assolid elements, which contain not only thedamage of the structure but also its continuumproperties, is the so-called strong discontinuityapproach (e.g., Simo et al., 1993; Larsson et al.,1996; Moes et al., 1999). This method is notconsidered here. The interface elements openwhen damage occurs and lose their stiffness atfailure so that the continuum elements aredisconnected. For this reason the crack canpropagate only along the element boundaries.If the crack propagation direction is not knownin advance, the mesh generation has to makedifferent crack paths possible. Applications tothe crack path prediction in combination withthe cohesive model will be given in Section3.03.5.4.4.

The separation of the cohesive interfaces iscalculated from the displacement jump [u], i.e.,the difference of the displacements of theadjacent continuum elements,

d ¼ ½u� ¼ uþ � u� ð74Þ

More common than the definition of theseparation vector in global coordinates is thedescription in a local coordinate system,namely the distinction between normal separa-tion, dN, and tangential separation, dT asshown in Figure 27. The separation dependson the normal and the shear stress, respec-tively, acting on the surface of the interface.When the normal or tangential component ofthe separation reaches a critical value, dN,0 or

σ0

x

σ(x)crack length a

Figure 26 Dugdale (left) and Barenblatt (right) crack models.

The Cohesive Model 155

dT,0, respectively, the continuum elementsinitially connected by this cohesive elementare disconnected, which means that the materi-al at this point has failed.

For 3D problems, two tangential separationdirections exist, which will be denoted as dT,1and dT,2. If isotropic materials are considered,a distinction between these two directions isnot necessary. For the sake of simplicity, onemay define a resultant tangential separation by

dT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2T;1 þ d2T;2

qrdT;0 ð75Þ

which is physically meaningful if both compo-nents are monotonously increasing, at least.Generally, it can happen, however, that thecohesive surfaces slide without changing theabsolute tangential separation defined byEquation (75), which is a physically question-able assumption. For further considerations onunloading of cohesive elements, see Section3.03.5.2.5.

Besides the critical separation, d0, the max-imum traction (stress at the surface of thecontinuum element), T0, is used as a fractureparameter, also denoted as ‘‘cohesivestrength.’’ The value of T0 only describes themaximum value of a traction–separation curveT(d), in the following denoted as cohesive law.Like the separations, the stresses T can also actin normal or in tangential direction, leading tonormal or shear fracture, respectively. Theshape of the curve, T(d), which is assumed tobe a material-independent cohesive law, isdefined differently by various authors. Com-mon to all cohesive laws is that

(i) they contain the two material para-meters, T0 and d0, mentioned above, and

(ii) for total failure the stresses become zero,T(d 4d0)� 0, for both normal and tangentialseparation (this condition is not fulfilledexactly for all cohesive laws as will be shownlater).

A literature review over the most commoncohesive laws (also called traction–separationlaws, or decohesion laws) will be given inSection 3.03.5.2.3.

The integration of the traction over separa-tion, either in normal or in tangential direction,gives the energy dissipated by the cohesiveelements, G0. This third parameter can bedetermined from the former two by

G0 ¼Z d0

0

TðdÞ dd ð76Þ

and used alternatively to d0. One can showthat the cohesive energy at failure G0 equals the(Rice, 1968) integral, Equation (17), of crackdriving force J at initiation in mode I under theassumptions made for the validity of the HRR-field. If a mode-I loading is assumed and thecrack propagation direction is the x-direction,a near-field J-integral can be calculated byplacing the contour path along the border ofthe cohesive zone. Doing so, the first part inEquation (17) vanishes so that only the work ofthe normal traction component Ty has to beconsidered:

Jnf ¼IGcz

�Tyuy;x dx� �

¼Z lcz

x¼0

TNðdÞ½u�;x dx� �

ð77Þ

Figure 27 Separation of 2D continuum elements connected by a cohesive interface.


where Equation (74) has been applied and lcz isthe length of the cohesive zone. Initiationoccurs if d¼ d0 at the crack tip, so that

Jnf ;i ¼Z d0

0

TNðdÞ dd � G0 ð78Þ

If path independence of J is satisfied (seeSection 3.03.3.1.5), we thus have G0 ¼ Ji;which can easily be determined by experiments;see also the considerations in Section 3.03.5.4.4for SSY. In gross section yielding, whensignificant path dependence of J may occur,Ji can at least be taken as a first estimate for G0.The critical separation d0 can be understood asa CTOD value (Xia and Shih, 1995) as acomparison between Equations (21) and (76)shows. The result of the integral in Equation(76) may be written as

G0 ¼ aT0d0 ð79Þ

with a being in the range of 0.5oao1.0 formost of the cohesive laws used for ductilematerials (see also Section 3.03.5.2.3). As T0

can be roughly estimated as T0¼ 3s0 for hightriaxialities like in plane strain, d0 becomes

d0 ¼1

3aJi

sYð80Þ

with 1=3o1=3ao2=3; which is comparable tothe values of the factor dn in Equation (21)(Shih, 1981).

If both separation modes—the tangentialand the normal separation—occur simulta-neously, there is an influence of the normalseparation on the tangential tractions and viceversa. The description for this case of ‘‘mixedmode’’ and the basic assumptions made in theliterature are given in Section 3.03.5.2.4. Otherspecial issues are the unloading behavior of thecohesive zone and the sliding of a failedcohesive element under negative normal se-paration, which involves contact of the fracturesurfaces, described in Section 3.03.5.2.5.

3.03.5.2.3 Cohesive laws

Since the cohesive model is a phenomenolo-gical model, there is no evidence which form totake for the cohesive law, T(d). Thus, thecohesive law has to be assumed independentlyof a specific material as a model of theseparation process. Most authors take theirown formulation for the dependence of thetraction on the separation. For ductile materi-als a polynomial function of third degree, firstused by Needleman (1987) for the pure normalseparation, and some years later extended byTvergaard (1990) for mixed-mode loading, is

one of the most popular cohesive laws and usedby many authors (e.g., Chaboche et al., 1997).The 1D formulation of the function T(d) is

TðdÞ ¼ 27

4T0

dd0

1� dd0

� �2

ð81Þ

The curve described by Equation (81) is shownin Figure 28(a). The total dissipation energy ofthe cohesive element, calculated by Equation(81), is

G0 ¼9

16T0d0 ð82Þ

Another function, which is based on theuniversal atomistic binding energy functionproposed by Rose et al. (1981), is an exponen-tial function, which is used as a cohesive lawsince Needleman (1990b):

TðdÞ ¼ T0ezdd0

exp �zdd0

� �ð83Þ

with e ¼ expð1Þ and z ¼ 16e=9: The form ofthis function is shown in Figure 28(b). Acharacteristic feature of this model is that thetraction does not approach zero at d¼ d0. Infact, the traction at this point is stillT(d¼ d0)¼ 0.105T0. The factor z is chosen sothat the cohesive energy G0 at d¼ d0 is the sameas for the model characterized by Equation(81).

The exponential model is also used by anumber of authors, both for ductile (Siegmundand Brocks, 1998a) and brittle metals (Xu andNeedleman, 1993).

Mainly for brittle materials such as concreteand rocks, another cohesive law has beenestablished with a purely decreasing form.The linear decreasing function proposed byHillerborg et al. (1976) is written as

TðdÞ ¼ T0 1� dd0

� �ð84Þ

Contrary to Equations (81) and (83), thisfunction, which is shown in Figure 28(c) andalso used by Camacho and Ortiz (1996), startswith an infinite stiffness until the maximumstress T0 is reached. There are several similarseparation laws with infinite initial stiffness asshown in Figure 28(d), e.g., with a constantstress in the beginning of the separation(Guinea et al., 1994), with a bilinear descrip-tion, or with a nonlinear softening law (Bazant,1993).

A more versatile cohesive law is proposed inScheider (2000), which fulfills the followingrequirements:


(i) the initial stiffness of the cohesive ele-ment can be varied;

(ii) a region can be defined, where thetraction in the cohesive element is keptconstant; and

(iii) the curve must be continuously differ-entiable for numerical reasons.

This has been achieved by using two addi-tional parameters, d1 and d2 (see Figure 28(e)),leading to the following formulation for thefunction T(d):

T ¼ T0

2dd1

� �� d

d1

� �3

; dod1

1; d1odod2

2d� d2d0 � d2

� �3

�3d� d2d0 � d2

� �2

þ1; d2odod0

8>>>>>><>>>>>>:

ð85Þ

Using d1¼ d2¼ 0.33d0, the curve is very similarto Equation (81) and the softening branch isactually identical. This law is similar to amultilinear cohesive law proposed by Tver-gaard and Hutchinson (1992), who also intro-duced two additional parameters (see Figure28(f)), but without the requirement that thecurve is continuously differentiable. The func-tion for the tractions is given by

T ¼ T0

dd1

� �; dod1

1; d1odod2d0 � dd0 � d2

� �; d2odod0

8>>>>><>>>>>:

ð86Þ

This cohesive law has also been used, e.g., byRoy et al. (2000). A special case of Equations(85) and (86) is obtained for d1-0, d2-d0,

Figure 28 Various cohesive laws used by several authors: (a) polynomial law, Equation (81); (b) exponentiallaw, Equation (83); (c) linear decreasing law, Equation (84); (d) two different modifications of Equation (84);(e) multipolynomial law, Equation (85); and (f) trapezoidal law, Equation (86).


which gives a constant stress from the begin-ning to final failure of the element, and is oftencalled rectangular cohesive law (see Yuan andCornec, 1991; Lin et al., 1998a).

Elices et al. stated that the form of thecohesive law depends on the class of materialsunder consideration. In Elices et al. (2002) theyinvestigated the fracture behavior of concreteusing a bilinear softening function, polymethyl-methacrylate (PMMA) using a truncated linearfunction and a pearlitic steel with a constantstress function. The authors also stated that thecohesive law should not have an initial hard-ening branch, as only the continuum elementsand not the cohesive elements are supposed toaffect the global behavior of the structure.

Initially, all models were based on a puremode-I crack under monotonic loading, only.Improvements have been developed for theapplication to mixed-mode loading, time de-pendence, interaction of combined normal andtangential loading, and unloading of thecohesive element.

The influence of the shape of the cohesivelaw on the crack propagation has not yet beenstudied extensively. Yuan and Cornec (1991)have investigated crack growth in ductilemetals using a rectangular shape and a mono-tonically decreasing function of the form T ¼T0 1� d=d0ð Þ0:5: They report that load–VLL

curves and CTOD R-curves are slightlyaffected by the shape of the cohesive law.Tvergaard and Hutchinson (1992) used themultilinear function of Equation (86) andstudied the influence of the parameters d1 andd2 by changing them between (d1¼ 0.15d0,d2¼ 0.5d0) and (d1¼ 0.125d0, d2¼ 0.25d0) withthe result that these ‘‘shape’’ parameters havelittle influence on the steady-state toughness ofa material. Though this investigation coversonly a small class of cohesive laws, it is oftenreferenced to state that the shape of thecohesive law has little influence on the results.Guinea et al. (1997) have studied a lineardecreasing function and a function withinitially constant traction equal to the cohesivestrength, T0, applied to concrete. As they triedto achieve similar load–CMOD curves with allmodels by varying the parameters T0 and G0,their results do not contribute to the question

how much the results of a crack-growthsimulation are affected by the shape of T(d)for fixed cohesive strength and energy.

The main question is not whether a load–displacement curve can be predicted with allcohesive laws, but whether it is possible totransfer the material parameters from onecohesive law to another. The maximum trac-tion which has to be overcome for failure tooccur should be a material parameter at leastfor brittle materials, for which the damage isinduced by normal stresses. For ductile materi-als, the maximum traction may depend on thestress triaxiality (Siegmund and Brocks, 1999),which will be discussed in Section 3.03.5.3. In aphenomenological approach, however, T0 willalso be taken as a material parameter forductile tearing. Whether the critical separation,d0, or the fracture energy, G0, is the secondconstitutive parameter depends on the pre-ferred local failure criterion and cannot bedecided a priori. The factor a in Equation (79),which relates the two quantities, depends onthe specific cohesive law.

The influence of the shape of the cohesivelaw on the results of crack propagationanalyses has been investigated for the para-meter values T0¼ 3.3s0 and G0¼ 100Nmm–1,applying the functions of Equations (81), (83),and (85). The corresponding critical separationvalues, d0, are given in Table 1. These cohesiveparameters are reasonable for ductile metals.

C(T) and M(T) specimens with elastic–plastic material behavior (J2-theory of plasti-city) have been simulated. Details of modelingcrack growth with cohesive elements are givenin Section 3.03.5.2.7 (see also Figure 35). Theresults for the load–displacement curves aredisplayed in Figure 29 for the C(T) and inFigure 30 for the M(T) specimen.

The first and most important conclusion isthat large differences between the variouscurves exist for both specimens, with respectto the overall behavior as well as to the valuesof maximum load or the respective load-linedisplacement. The nearly rectangular cohesivelaw (#3) yields the highest crack resistanceresulting in the highest load–displacementcurve compared to all other laws. For both,C(T) and M(T), the four cohesive laws rank as

Table 1 Critical separation values for fixed cohesive energy, G0¼ 100 N mm–1, and cohesive strength,T0¼ 3.3s0, with different cohesive laws.

No. Cohesive lawd0

(mm)

1 Exponential, Equation (81) 0.1782 Cubic polynomial, Equation (83) 0.1783 Nearly rectangular (multipolynomial, Equation (85), with d1¼ 0.01 d0, d2¼ 0.75d0) 0.1154 Purely decreasing (multipolynomial, Equation (85), with d1¼ d2¼ 0.01d0) 0.200


#3 (nearly rectangular), #4 (purely decreasing),#2 (cubic polynomial), and #1 (exponential),with respect to decreasing specimen stiffnessand hence decreasing structural crack growthresistance, though the local energy release rate,G0, is the same in all cases. This ranking doesnot correlate with d0 (see Table 1).

A further aspect regards the slope of thesimulated load–displacement curve of the C(T)specimen in the elastic regime, which is lowerfor the cubic polynomial (#1), and the ex-ponential (#2) cohesive law due to the com-pliance of the traction–separation relationbefore the maximum traction is reached. Thisis an artifact, since ductile damage should haveno influence on the elastic compliance of aspecimen.

The results show that the crack-growthsimulations applying the cohesive law #3

having the lowest d0-value (see Table 1) predictthe highest crack resistance. If d0 is takenas a material constant instead of G0, e.g.,d0¼ 0.15mm, then the respective cohesiveenergy depends on the applied cohesive law(see Table 2).

It is obvious that the specimen stiffnessobtained from a simulation applying the nearlyrectangular cohesive law (#3) would be evenhigher, since the cohesive energy required forcrack propagation is higher than in thesimulation above, whereas the predicted stiff-ness corresponding to the exponential (#1) andthe cubic polynomial (#2) cohesive law wouldbe lower. This means that the differences in themechanical response of the specimens assum-ing constant critical separation d0 would beeven more pronounced than the differences forconstant cohesive energy G0.

exponentialcubic polynomial

nearly rectangularpurely decreasing

0

500

1000

1500

2000

2500

3000

0 431 52 6

vLL (mm)

load

(N

)

Figure 29 Load vs. load-line-displacement curve of a C(T) specimen showing the influence of the cohesivelaw.

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

∆l (mm)

Load

(N

)

exponential

cubic polynomial

nearly rectangular

purely decreasing

0 431 52 6

Figure 30 Load–elongation curve of an M(T) specimen showing the influence of the cohesive law.


3.03.5.2.4 Cohesive models for mixed-modeloading

All cohesive models can be used for normaland tangential separation as well as forcombined loading. Some models use threeparameters (TN,0, dN,0, and an interactionparameter called a or q), some use two pairsof independent parameters for normal andtangential material separation (TN,0, dN,0, TT,0,dT,0). At combined normal and shear fracturethe shear damage will reduce the ductility innormal direction and vice versa:

TN ¼ fN dN; dTð Þ; TT ¼ fT dN; dTð Þ ð87Þ

The interaction of shear and normal separationcan be described by a damage variable, D,which is defined as

D ¼ dNh idN;0

� �r

þ dTdT;0

� �r �1=rð88Þ

with an additional interaction parameter r (seeScheider, 2000). Macauley brackets are used toindicate that the effect of dN vanishes undercompression. For r¼ 2 and dN;0 ¼ dT;0 ¼ d0;the damage variable in Equation (88) is equalto the normalized absolute value of separation,

d ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2N þ d2T

q; and r-N defines a vanishing

interaction of the separations.Introducing D, the functions in Equation

(87) can be written as

TN ¼ fN dN;Dð Þ; TT ¼ fT dT;Dð Þ ð89Þ

The polynomial cohesive law (Equation (81))has also been extended to combined normaland shear separation by Tvergaard (1990).Four independent parameters for the materialseparation and r¼ 2 are assumed:

TN=TðdÞ ¼27

4TN=T;0

dN=T

dN=T;0

1� Dð Þ2 ð90Þ

A similar approach is used for the cohesive lawproposed by Tvergaard and Hutchinson(1993), which is an extension of Equation (86)using the damage definition of Equation (88)with r¼ 2 and three parameters—dN,0, dT,0,

and T0—plus two shape parameters, D1 andD2, describing the form of the cohesive law.The tractions are calculated by

T ¼ T0

D

D1

� �; DoD1

1; D1oDoD2

1� D

1� D2

� �; D2oDo1

8>>>>><>>>>>:

ð91Þ

which can be split into a normal and tangentialpart by

TN ¼ T

D

dNdN;0

; TT ¼ T

D

dTdT;0

dN;0

dT;0ð92Þ

The extension of the exponential cohesive law(Equation (83)) for combined normal andshear separation uses a modified formulation,in which d0 is the separation at maximumnormal traction, TNðd0Þ ¼ T0:

TN ¼T0e exp �dNd0

� �dNd0

exp �dTd0

� �2"

þð1� qÞ 1� exp �dTd0

� �2" #

dNd0

#

TT ¼ 2T0eqdTd0

� �1þ dN

d0

� �

� exp �dNd0

� �exp �dT

d0

� �2

ð93Þ

(see Xu and Needleman, 1993). The exponen-tial cohesive law has only three parameters—TN, dN, and q (0rqr1). For q¼ 1, theseparation energies of pure normal and pureshear fracture are equal, and q¼ 0.4289 givesthe same maximum stress for both fracturemodes. The maximum tangential tractionunder pure shear (dN¼ 0), TT,0, is reached atdT ¼ d0=

ffiffiffi2

p; independent of q. The value of

TT,0 itself depends on q. It is TT,0¼ 2.33T0q.Another coupling of normal and tangential

separation is introduced by Camacho andOrtiz (1996) through effective values forseparation and traction, in which differentweight is assigned to the components of theseparation by a weighting factor b:

d ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2d2T þ d2N

qð94Þ

The effective traction t is defined in a similarway. With these effective values d and t, allcohesive laws presented in Section 3.03.5.2.3can be applied to mixed-mode loadings, lead-ing to a three-parameter formulation with T0,d0, and b.

Slant fracture in thin sheets is a specialapplication of the combination of normal and

Table 2 Cohesive energy values for fixed criticalseparation, d0¼ 0.15 mm, and cohesive strength,

T0¼ 3.3s0, with different cohesive laws.

No. Cohesive lawG0

(Nmm–1)

1 Exponential 84.42 Cubic polynomial 84.43 Nearly rectangular 131.04 Purely decreasing 75.0


tangential separation (see Siegmund et al.,1999). In these structures the crack turns intoa 451 plane to the loading direction andpropagates mainly under tangential separation.This behavior is often modeled using 2D planestress or shell elements with lines of cohesiveinterfaces, where the shear fracture is treated asnormal separation, but with parameters, whichare different from those determined for normalfracture. The simulation of fracture in thinsheets is treated in Section 3.03.5.4.4.

3.03.5.2.5 Unloading and reverse loading ofcohesive elements

Local unloading effects can occur in thecases of global unloading of a structure, crackbranching, or multiple cracks. One has, there-fore, to define the behavior of the cohesiveelements under decreasing separation account-ing for the irreversibility of the damageprocess. Since damage evolution is a nonlinearprocess as inelastic deformation, the cohesivemodels are established in analogy to theprinciples of plasticity, but allowing for strainsoftening. The terms ‘‘loading’’ and ‘‘unload-ing’’ will be used in the sense of increasing ordecreasing separation, respectively, as thetractions decrease also under increasing separa-tion beyond maximum stress, T0. More gen-erally, ‘‘unloading’’ is any change of thedeformation direction by which the stress statemoves from the limiting traction–separationcurve. The latter definition also applies forshear separation.

Two principal mechanisms depending on thematerial behavior have to be distinguished:

(i) in ductile materials, the mechanical workfor producing damage is totally dissipated, thevoid growth and hence the inelastic separationare irreversible, and any reduction of separa-tion occurs purely elastically with unchangedelastic stiffness, as shown in the upper row ofFigure 31 (see, e.g., Roe, 2001; Scheider, 2000);and

(ii) in brittle materials, the elastic stiffness ofthe material is reduced by damage, but theseparation vanishes when the stresses decreaseto zero, as shown in the lower row of Figure 31.

The effect of the two unloading mechanismson normal separation is shown in the leftcolumn of Figure 31. Unloading starts at pointA where the stresses are released elastically witheither unchanged or reduced stiffness andpermanent or vanishing separation at zerotraction. If the separation increases again, theseparation stress increases linearly up to point Aand then follows the original cohesive lawafterwards. Negative normal separation indi-cates penetration of the adjacent continuumelements, which is physically not admissible. Thestiffness of the cohesive element should hence beas high as possible, at least as high as the initialelastic stiffness. This is automatically realizedfor the ductile damage mechanism, whereas forthe brittle damage mechanism the tensile stiff-ness is reduced by damage and the compressivestiffness equals the initial elastic stiffness as allmicrocracks have completely closed.

Unidirectional shear separation can betreated in the same way as normal separation.Reverse shear, however, requires a differentmodel as damage may increase in both direc-tions. In analogy to the isotropic hardening inthe theory of plasticity, it is assumed that

Figure 31 Cohesive laws at unloading: normal separation (left) and shear separation (right).


damage which has been activated by separationin one direction becomes also active in theopposite direction, if the same absolute valueof shear stresses is reached, i.e., in point B. Theshear stress will then follow the cohesive lawagain, whereas in between it varies linearlywith the separation (see the right column inFigure 31). Figure 32 illustrates how severalcycles of loading and unloading are handledfor the ductile damage mechanism. The re-spective handling in the case of the brittledamage mechanism is obvious.

A second important problem concerns load-ing and unloading under ‘‘mixed-mode’’ con-ditions, i.e., combined normal and tangentialseparation. Again, two different models can beconsidered:

(i) unloading is handled separately for nor-mal and tangential separation, i.e., a cohesiveelement can be in an unloading state fornormal, but in a loading state for tangentialseparation and vice versa; and

(ii) unloading depends on the total separa-

tion d ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2N þ d2T

q: In this case, unloading in

both modes is assumed if the total separation

decreases, even if the separation in one of thetwo modes might increase.

The contact condition, i.e., prevention ofpenetration of adjacent continuum elements,has to be ensured also after total failure of thecohesive element. The behavior after failurecan be described as a ‘‘contact cohesive law,’’depending on the mechanisms of ductile orbrittle damage, respectively. This is illustratedin Figure 33 for the normal separation mode;further details can be found in Chaboche et al.(1997) or Roe (2001).

If a structure fails under pure shear, fric-tional sliding of the fracture surfaces must betaken into account. The frictional forces can beincluded in the cohesive law by assumingCoulomb’s law of friction (Tvergaard, 1990):

TT ¼ �sgnð’dTÞmTN for TNo0 ð95Þ

If TTomTN; there is no sliding but thebehavior is represented by elastic springs inthe tangential direction

TT ¼ kdT ð96Þ

Figure 32 Cohesive laws for shear separation with repeatedly changing loading direction.

Figure 33 Cohesive law (contact law) after total failure for: (a) brittle behavior and (b) ductile behavior.


with a stiffness k chosen appropriately. Anapproach for the consideration of frictionalsliding using Coulomb friction and an addi-tional extension of the tangential tractionbefore total failure can be found in Chabocheet al. (1997). Fiber pullout in a compositematerial is an application where the effect issignificant and must be regarded. The fracturemechanism in this case is pure shear (seeSection 3.03.5.5.2).

3.03.5.2.6 Rate-dependent cohesive laws

Rate- and time-dependent fracture phenom-ena require respective cohesive models. Formost metals the rate effect is minor except forhigh-temperature or high-velocity (impact)testing. Alternatively, these effects are of greatinterest for the fracture behavior of polymersand the simulation of adhesives using thecohesive model. Therefore, most of the modelsdescribed below are developed for the applica-tion to these classes of materials.

In general, a rate formulation for thetractions has to be used which has the form

’T ¼ f d; ’d;T ; ki

� �ð97Þ

that means the rate of the traction may dependon the separation and its rate and on thetraction itself. The optional variables ki areadditional state variables (e.g., time, tempera-ture, etc.).

The rate dependency can be introduced intocohesive elements in several ways depending onthe material behaviors: explicit rate depen-dency, viscoplastic behavior, and viscoelasticbehavior.

In the first case, the tractions dependexplicitly on the separation rate:

T ¼ f ðd; ’dÞ ð98Þ

Several authors (e.g., Liechti and Wu, 2001;Costanzo and Walton, 1997) apply rheologicalmodels of the Kelvin–Voigt type, i.e., a (non-linear) spring and a (nonlinear) dashpot inparallel as shown in Figure 34, to describe thisbehavior. The cohesive traction is the sum ofthe contributions of the spring and the dash-pot:

T ¼ f1ðdÞ þ f2ð’dÞ ð99Þ

A modified simple form of this model assumesa frictional block instead of a spring and alinear dashpot. The cohesive law can then bewritten as proposed by Costanzo and Walton(1997):

Tðd; ’dÞ ¼ T0 þ Z’d for dod0 ð100Þ

A form proposed by Xu et al. (1991) forcrazing in homopolymers, which contains themixed term ð’ddÞ and thus cannot be modeled asa spring–dashpot combination according toEquation (99), is based on the linear decreasingmodel described in Equation (84) with anembedded rate dependency:

Tðd; ’dÞ ¼ T0 þ Z’d� �

1� dd0

� �ð101Þ

The implementation of such models is verysimilar to the rate-independent cohesive laws,since only the time derivative of the separationbut not of the traction is needed in thisformulation.

In viscoplastic cohesive laws, the separationis split in an elastic and a viscoplastic part as inviscoplastic constitutive relations of continuummechanics (see Section 3.03.1):

d ¼ del þ dvp ð102Þ

where the elastic part is defined by any of thecohesive laws given in Section 3.03.5.2.3. Ingeneral, we can write (see Equation (1))

T ¼ f ðd� dvpÞ ð103Þ

The viscoplastic part, dvp, is given by aviscoplastic law, which has been proposed byseveral authors in different ways. Estevez et al.(2000) gave an evolution law for the timederivative of the viscoplastic separation

’dvp ¼ ’d0 exp�Acsc

y1� T

T0

� � �ð104Þ

depending on the temperature, y, with Ac, sc,and ’d0 being model parameters.

Another evolution law for the separationvector dvp ¼ dvpN ; dvpT

� �is given by Corigliano

et al. (1997) and Corigliano and Ricci (2001),who proposed a viscoplastic relation of

Figure 34 Schematic of a rate-dependent cohesiveelement with a spring and a dashpot in parallel.


Perzyna kind, which is written as

’dvpN ¼ f ðTN;TT; dvp;accÞh in @f ðTN;TT; dvp;accÞ

@TN

’dvpT ¼ f ðTN;TT; dvp;accÞh in @f ðTN;TT; d

vp;accÞ@TT

ð105Þ

with

f ðTN;TT; dvp;accÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaN TNh i2þaTT2

T

q� 1þ hdvp;acc ð106Þ

The accumulated viscoplastic separation,dvp;acc; is defined as

dvp;acc ¼Z t

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi’dvpN� �2þ ’dvpN

� �2qdt ð107Þ

and aN, aT, n, g, and h are model parameters.In Corigliano et al. the evolution law is givenfor the 3D case, i.e., for dvp ¼ ðdvp1 ; dvp2 ; dvp3 Þand T ¼ ðT1;T2;T3Þ; respectively.

Viscoelastic cohesive laws are of increasinginterest mainly for elastomers and fiber–matrixcomposites. The nonlinear Kelvin–Voigt modelof Equation (99) describes viscoelastic beha-vior. Alternatively, a functional formulationcan be chosen, where the time-dependenttractions, Tðd; tÞ; result from a time integralover the separation history, multiplied by sometime-independent function, TstatðdÞ; which canbe any of the cohesive laws discussed in Section3.03.5.2.3 (Rahul-Kumar et al., 1999):

Tðd; tÞ ¼ TstatðdÞZ t

0

Gðt � tÞ ’D dt ð108Þ

or using a damage law, (1�a(t)), with anadditional threshold value T0 (Allen andSearcy, 2001),

Tðd; tÞ ¼ ð1� aðtÞÞ T0 þZ t

0

Eveðt � tÞ ’D dt �

ð109Þ

A rather simple standard viscoelastic linearmodel is used for the relaxation moduleEve(t�t0), which can be written as

Eveðt � t0Þ ¼ Eve;N þ Eve;0expð�t=t0Þ ð110Þ

A similar viscoelastic cohesive zone model isproposed earlier by Knauss (1993). Othermodels, which use a viscoelastic term depen-dent on the traction, are proposed, e.g., byTandon et al. (1995).

3.03.5.2.7 Work of separation and remoteplastic work

Cohesive as well as damage models allow forsplitting the total dissipated work in the course

of ductile crack growth into the work ofseparation in the process zone and the remoteplastic work in the embedding material and,thus, solve a classical problem of EPFMfracture mechanics (Siegmund and Brocks,2000a, 2000b). A process zone ahead of thecrack tip is introduced where material degra-dation and separation may occur. This zone ismodeled either by cohesive elements or by asingle row of continuum elements with incor-porated GTN equations, whereas the rest ofthe structure consists of continuum elementswith classical elastic–plastic constitutive beha-vior (see Figure 35 for a C(T) and an M(T)specimen).

A uniformly spaced mesh region with quad-rilateral elements is positioned in the ligament,i.e., along the symmetry line. A graduallycoarser mesh is used elsewhere up to thespecimen boundary. Failure within a cohesiveelement is taken to have occurred if the averagenormal opening in an element has reached thecritical separation, d0. If the GTN equationsare used, a single row of void containing cells islocated in the ligament, instead. Accountingfor the symmetry conditions only one-half of avoid containing cell is modeled. Thus, the cellshave a height of D/2. Failure has occurred ifthe average value of f* over an element in theprocess zone has reached fu.

Two different approaches will be comparedin the following example of the cohesive zonemodeling. In the ‘‘classical’’ approach, T0 andG0 are taken to be constant. However, ingeneral, the triaxiality within a cracked speci-men changes in the course of crack growth andalso from specimen to specimen. For thetriaxiality-dependent cohesive zone elements,the relationships, T0(h) and G0(h), have to beinserted as material data into the FE formula-tion of the cohesive zone elements (Siegmundand Brocks, 1999, 2000a).

The remote plastic work in the specimen andthe work of separation in the process zone inthe course of ductile crack growth can beevaluated according to Equations (44)–(46).The results for the M(T) and the C(T) specimenof Figure 35 are plotted in dependence oncrack growth, Da, in Figure 36 for bothapproaches—GTN cells and cohesive elements.The sum of the two quantities is the dissipationrate displayed in Figure 21, which allows for acomparison with experiments. The right dia-gram in Figure 36 shows the difference of theseparation energy, G0, between the two speci-mens and its evolution during crack growth;the dashed line indicates the value ofG0¼ 53.3Nmm–1 taken in the simulation withconstant cohesive parameters. Finally, the ratioof ’Upl and ’Usep is plotted in Figure 37. The


conclusions which can be drawn from thesediagrams are as follows.

(i) Ductile crack growth can be simulated byusing either GTN cells or cohesive elements inthe ligament, and geometry effects for differentspecimen types are well captured by bothapproaches.

(ii) The plastic work per crack incrementdiffers up to four times between an M(T) and aC(T) specimen, thus explaining the significantlydifferent R-curve behavior of the two speci-mens.

(iii) The actual separation energy is only1–5% of the plastic work, or in other words,95–99% of what is measured in R-curve testingis dissipated plastic work but no ‘‘fractureenergy,’’ at least for ductile materials wherecrack-growth initiation occurs in large-scaleyielding (LSY). Investigations on large andthin center-cracked panels of aluminum,where crack-growth initiation occurs in SSY,however, have revealed similar relations (Sieg-mund and Brocks, 2000c). Because of thelatter, the assumption of constant, i.e., not

0 2 4 60

50

100

150

Γ 0 [

kJ/m

2]

Γ0 (const)

M(T) - CZM

M(T) - GTN

C(T) - CZM

C(T) - GTN

∆a [mm]0 2 4 6

0

2.103

4.103

6.103

8.103

1.104

dUpl

/ Bda

[kJ

/m2]

∆a [mm]

M(T) - CZM

M(T) - GTN

C(T) - CZM

C(T) - GTN

StE 460 - num.sim.

Figure 36 Splitting of the total dissipated work into global plastic work (left) and local work of separation(right) (source Siegmund and Brocks, 2000a, 2000b).

M(T)-Specimen C(T)-Specimen

Crack tip meshes

Gurson hmin = D/2 = 0.1 mm

Cohesive zone model

Gurson model:UMAT in ABAQUS

Cohesive zone model:UEL in ABAQUS

W = 50 mma0/W = 0.59

a0

a0

2a0

2W

W = 50 mma0 /W = 0.5

2L

a0

Figure 35 Modeling of fracture mechanics specimens with a process zone of Gurson cells or cohesiveelements in the ligament.


triaxiality-dependent cohesive parameters, T0

and G0, does not have a significant effect on theglobal behavior. Hence, the two cohesiveparameters, T0 and G0, allow for a simple,mechanically based and predominantly geome-try-independent characterization of ductiletearing resistance.

The above study is a good example of howmodeling and numerical simulation help inidentifying material parameters and thus sup-port material characterization.

3.03.5.2.8 Implementation of cohesiveelements in FE codes

The principle of virtual work reads

dPi � dPe ¼ dPa ð111Þ

i.e., the internal virtual work minus the virtualwork of external loads is equal to the virtualwork of inertia. The cohesive elements do notcontribute to dPa; since they do not have amass. In the following, terms associated tomass accelerations are neglected. Since thesymbol d denotes the variation of a quantityin variational mechanics, the displacementjump [u] as defined in Equation (74) is usedfor the separations in the following to avoidconfusion.

The internal virtual work with presentcohesive surfaces is then defined by

dPi ¼Z

B

S � �dE dV þZ@B

T � d u½ � dS ð112Þ

with S being the stress tensor, E the energyconjugated tensor of strains, and T the vectorof the tractions. The cohesive elements con-tribute only to the second term of Equation

(112), which can be rewritten as

dDPi ¼Z@B

@T

@½u� � D½u� � d½u� dS ð113Þ

for an incremental analysis. Since the separa-tions are given at the nodes only, the fieldvariable vector D[u] is replaced by the productof the matrix of shape functions Vu and theseparations at the nodes [u]e:

u ¼ Vu � ue ð114Þ

The rank of Vu depends on the degree of thepolynomials which describe the element geo-metry and displacements. Details on the shapefunctions can be found in Bathe (1996). WithEquation (114), the increment of the energy ofthe cohesive element (Equation (113)) becomes

dDPi ¼ d½u� �Z@B

VTu � @T

@½u� � VudS � D½u�e

¼ d½u� � K � D½u�e ð115Þ

with K being the stiffness matrix of thecohesive element.

The derivative of the tractions with respectto the separations can be calculated analyti-cally from the given cohesive law. If thetraction is given as a function of the damagevariable D, the chain rule has to be applied.The integration over the surface of the cohesiveelement is done numerically in local coordi-nates and transformed into global ones after-wards.

Another implementation aspect concerns thecoordinate system used for the deformedcohesive element. Contrary to a conventionalcontact algorithm, there is no master and slavesurface for the cohesive element, from which

0 2 4 60

20

40

60

80

100

120

dUp

l/Γ

0B

da [

-]

∆a [mm]

M(T) - CZM (h-dep)

M(T) - CZM (const)

M(T) - GTN

C(T) - CZM (h-dep)

C(T) - CZM (const)

C(T) - GTN

StE 460 - num.sim.

Figure 37 Ratio of global plastic work and local work of separation during crack growth.


the coordinate system and reference plane canbe defined. Instead, there are two possibilitiesto define the reference plane:

(i) if the original position of the cohesiveelement is taken as the reference for thecoordinate system during the entire analysis,this means the assumption of small deforma-tions; and

(ii) according to the theory of large defor-mations in an updated Lagrangean formula-tion, the coordinate system can be defined asmoving with the element using a midsectionface, i.e., the bisector between upper and lowersurface.The choice of the coordinate system is

important, since it affects the normal and thetangential portion of the separation of thecohesive element. In Figure 38 the difference isvisible with regard to a rotation of 2Dcontinuum elements: while definition (i) withits coordinate system (B1, x1) leads to a partlytangential opening in direction x1, the elementis separated under pure mode I (direction B2)using the definition (ii) with its coordinatesystem (B2, x2).

3.03.5.3 Relation to MicromechanicalPhenomena of Damage and Fracture

The cohesive laws referenced in Section3.03.5.2.3, which describe the material decohe-sion in the process zone, are purely phenom-enological. The underlying physical processesare various, depending on the material. Ingeneral, damage evolves in a material by theevolution of areal or volumetric microdefects,i.e., microcracks, microvoids, and microcav-ities, respectively. Micromechanical models ofdamage aim at describing these phenomena ona local scale (see also Section 3.03.4.3). Theyutilize the concept of an ‘‘RVE’’ or a ‘‘cell,’’which is regarded as the smallest material unitthat contains reasonably sufficient informationabout crack growth in the material (Broberg,1997). These models can be taken to establish,verify, and calibrate cohesive laws. A few

considerations of the microstructural processesleading to either ductile or brittle damage andfracture are presented in the following in orderto give a better physical understanding of thevarious cohesive laws.

3.03.5.3.1 Ductile crack growth in metals

Ductile rupture of metals is governed by thenucleation, growth, and coalescence of micro-voids in combination with significant plasticdeformation. This process can be observed postmortem on the fracture surfaces of brokenspecimens. Figure 39 shows the fracture sur-face of a ferritic steel specimen after failureunder monotonic loading. Large dimples ofthe size of 20–30 mm are regularly distributedand surrounded by small dimples with a size ofB3 mm. These dimples represent voids whichhave coalesced and thus formed a macroscopicfracture surface. The overall mechanical beha-vior of a representative volume, including allphases from void nucleation until final separa-tion (see Figure 40), has to be captured by thecohesive law. The RVE can be either anelastic–plastic unit cell with void (Koplik andNeedleman, 1988; Brocks et al., 1995b) or ahomogeneous volume element described by themicromechanically based constitutive equa-tions of Gurson or Rousselier. Thus, a micro-mechanical material model can be used tocalibrate the cohesive law. An example is givenin the following (Siegmund and Brocks, 1998a,1998b).

An elastic–plastic cylindrical unit cell ofdiameter D0 and height H0¼D0 with aspherical void and a single FE consisting ofGTN material under plane-strain condition ofsize wel� hel¼H0�H0, and an initial voidvolume fraction of f0¼ 0.005 is analyzed. Thetwo parameters—cohesive strength and energy,T0, G0—of the cohesive law are identified bythe maximum overall (‘‘mesoscopic’’) tensilestress and the dissipated work until failure ofthe cell. The load is applied under displacementcontrol and constant overall triaxiality, defined

Figure 38 Definition of the local coordinate system within the cohesive line element.


according to Equation (25). Figure 41 showsthe comparison of the mechanical behaviorof the unit cell and the plane-strain ele-ment. The stress–strain curves of the unit cellare well matched by the GTN model overa wide range of triaxialities. The capitalletters used for the cell model indicate overallor ‘‘mesoscopic’’ values of stresses and str-ains, namely the tensile stresses on thecell surface, Szz, and the relative elongations,

Ezz ¼ log 1þ DH=H0ð Þ; respectively. Some dif-ferences, however, can be observed for thedissipation energy density

g ¼Szz dEzz for cell

szz dezz for GTN

(ð116Þ

at low triaxialities. The limiting value ofG¼ gH0 after final separation, szz¼ 0 orSzz¼ 0, respectively, is identified with the

Figure 39 Fracture surface of a ductile structural steel exhibiting several large and small voids.

Figure 40 Void growth and coalescence ahead of a crack tip and modeling by a cohesive law.


cohesive energy, G0. Despite some differences,the overall agreement between the mechanicalbehavior of the unit cell and the GTN model isgenerally satisfactory.

The plots show a significant influence of thetriaxiality, h. If triaxiality increases, the cohe-sive strength increases, but the cohesive energy(or ‘‘ductility’’) decreases, which is well knownfor ductile tearing phenomena. The CZM in itsclassical form, i.e., with constant cohesiveparameters, however, cannot account for thiseffect. In other words, the assumption of thecohesive model that T0 and G0 are uniquematerial parameters is not verified for ductiletearing (see also Tvergaard and Hutchinson,1996a; Broberg, 1997). A calibration of T0 andG0 is, hence, only possible for a giventriaxiality. This has been done for a triaxialityof h¼ 3.0, which is typical for a fracturespecimen at the crack tip, and the result isplotted in Figure 42 as normalized tractions orstresses, T/s0, Szz/s0, szz/s0, vs. relative separa-

tion, d/d0, or elongation, DH/H0, respectively.For the cell and the GTN model, the elonga-tion values on the abscissa have been addi-tionally scaled by (DHf/d0), where DHf is theelongation at final failure, szzE0, to allowfor a comparison with the cohesive curve. Theagreement between the mechanical responsesof the three models is acceptable. Thecompliance before maximum stress as de-scribed by the cohesive law of Equation (83)is too high, however, or, in other words, thisCZM is too soft in the beginning. Figure 42(a)also demonstrates the sharp discontinuityinduced by the f*(f) function in the GTNmodel at fc.

The dissipated energy plotted in Figure 42(b)is the area under the stress–elongation curves:

G ¼H0

RSzz dEzz for cell

helRszz dezz for GTNRT dd for CZM

8><>: ð117Þ

0.0 0.2 0.4 0.6 0.80.00

0.50

1.00

1.50

εzz ; Εzz

γ / σ

0

h = 1.0h = 1.5h = 2.0h = 3.0

f0 = 0.5%

εzz ; Εzz

0.0 0.2 0.4 0.6 0.80.0

1.0

2.0

3.0

(σzz

/ σ0)

; (Σ

zz / σ

0)

h = 1.0h = 1.5h = 2.0 h = 3.0

f0 = 0.5%

GTN CPE

axisymm. cell

(a)

(b)

Figure 41 Comparison of the mechanical behavior of an elastic–plastic axisymmetric unit cell with void(dashed) and a single plane-strain GTN element (lines) for various stress triaxialities: (a) tensile stress vs.tensile strain and (b) dissipated energy density vs. tensile strain.


The limiting value of G after final separation isagain identified with the cohesive energy G0. Itis obvious that the work required for the finalplastic collapse of the unit cell, i.e., G0, dependson the cell size. Equation (117) illustrates oncemore that the GTN model also requires theintroduction of an additional internal length,namely the element height hel¼H0 (Siegmundand Brocks, 1998b), as was also shown inEquation (72). In the cohesive model, thislength is introduced by d0 already, whichmakes the simulations with cohesive elementsmuch less mesh sensitive.

The triaxiality dependence of cohesivestrength and energy has been taken intoconsideration by Siegmund and Brocks(1999, 2000a) to establish a triaxiality-depen-dent CZM in which the local triaxiality of theadjacent continuum element is taken tocontrol the cohesive strength and energyvalues of the cohesive element. The dependence

of T0 and G0 on the stress triaxiality, h,has to be known, of course. They can beobtained by numerical studies as shown inFigure 41. The effect on crack-growth resis-tance curves appeared not to be very pro-nounced, as the dissipated energy in a fracturemechanics specimen consists to a great part ofplastic strain energy remote from the crack(see Section 3.03.5.2.7 and Figure 37). Tver-gaard and Hutchinson (1996b) introduced astrain-dependent cohesive zone model in whichthe peak stress for separation is reduced byplastic straining. As the plastic strain issingular at the crack tip, the results aresignificantly affected by the element width,however.

The involvement of several smaller voids inan RVE instead of just one large void makesthe decohesion part of the curve more steep,see Figure 43 (Faleskog et al., 1998; Broberg,1997).

0.0 0.5 1.0 1.50.0

0.5

1.0

1.5

2.0

2.5

3.0

T /

σ0 ;

σzz

/ σ

0 ; Σ

zz /

σ0

f0 = 0.5%

CZM

GTN CPE

axi cell

σh / σe = 3

0.0 0.5 1.0 1.50.0.100

2.5.10-3

5.0.10-3

7.5.10-3

1.0.10-2

δ / δ0 ; (∆H / H0) (δ0 / ∆H f)

δ / δ0 ; (∆H / H0) (δ0 / ∆H f)

Γ / σ

0

CZM

GTN CPE

axi cell

σh / σe = 3

f0 = 0.5%

(b)

(a)

Figure 42 Comparison of the mechanical behavior of a cohesive zone model, an elastic–plastic axisymmetricunit cell with void, and a single plane-strain GTN element: (a) overall tensile stress vs. elongation and (b)dissipated energy vs. elongation.


3.03.5.3.2 Quasi-brittle fracture of concrete

Materials like ceramics, bricks, rocks, ice, orconcrete fail in a quasi-brittle way, but someductility due to bridging and roughness effectsof particles or aggregates at the fracturesurfaces can be observed. Concrete, in parti-cular, is a highly heterogeneous material withseveral microstructural length scales. On themacroscale level large aggregates are observed,bonded by a matrix of mortar. This mortarphase consists of cement grains and possiblylarger voids and small cracks, and the hydratedcement itself shows a microstructure, too.Fracture in concrete is a continuous processof nucleating and linking up of microcracks.Due to the simultaneous creation of micro-cracks at different locations, bridges formbetween cracks. After an initially steep soft-ening response these bridges result in a long tailin the load–displacement curve (van Mier andvan Vliet, 2002).

Detailed analyses of a cementitious compo-site have been carried out by Tijssens (2000)and Tijssens et al. (2001). The RVE typicallyhas a size of a few millimeters. The structure issimplified to a three-phase material consistingof aggregates embedded in a cementitiousmatrix and weaker interface regions (see Figure44). Cohesive laws are applied for both, thematrix and the interfaces to the aggregates,whereas the aggregates behave purely elastic.Interfacial cracks can only link up when thecement ligament between the aggregates fails.The overall stress–strain response of the RVEis shown in Figure 45. Damage evolves veryslowly in the beginning, which results in a

nearly linear elastic response up to point 1.Since the interfaces are weaker than the matrixand due to the stress concentrating effects ofthe stiffer aggregates, damage starts at thepoles of the aggregates. Localization of defor-mation and damage starts beyond maximumstress (point 2) and causes a steep softening.Bridging effects which start at point 3 result in

multiplevoid failure

singlevoid failure

elongation

mac

rosc

opic

cel

l str

ess

Figure 43 Decohesion behavior of a cell with asingle large void (dashed line) and several smallvoids (solid line) with same void volume fraction(source Broberg, 1997). Figure 44 Representative volume element of con-

crete on a mesolevel consisting of aggregates in acement paste and detail of the FE mesh withcohesive surfaces between the continuum elementsof the cement paste and at the interfaces to theaggregates (source Tijssens, 2000).

0 1 2 3 4 5 6 70

1

2

3

ε = 10-7 sec-1

0

1

T T0

δ/δ00 1

cohesive law of cementand aggregate interfaces

1: onset of non-linearity2: localization and peak stress3: onset of bridging resistance

Σ (M

Pa)

E (x10-3)

1

2

3

Figure 45 Overall stress–strain response of thecementitious composite, points denoting 1: onset ofnonlinearity; 2: beginning localization of deforma-tion; and 3: start of resistance due to bridging(source Tijssens, 2000).


a long tail of the curve. The overall response ofthe cell can be approximated qualitatively by acohesive law as shown in Figure 28(d), whichcharacterizes the average behavior of the three-phase composite. The actual values of thecohesive parameters—T0, d0, and G0—dependon the volume fraction and the arrangement ofaggregates, the properties of the cement andthe interfaces, and are hence subject tostochastic variations.

3.03.5.3.3 Crazing in amorphous polymers

Polymers exhibit two types of inelasticbehavior: shear yielding and crazing. The latteris often the precursor to brittle fracture undertensile loading. Crazes in amorphous polymersreach lengths of the order of tenths ofmillimeters, whereas the width of the crazeremains of the order of several micrometers. Acraze can be modeled by a cohesive surface.The separation between two initially adjacentmaterial points is described by a separationvector d with a normal component, dN, andtangential components, dT1, dT2, with respectto the midplane of the cohesive surface (see

Section 3.03.5.2.4). The cohesive law forcrazing should incorporate the initiation, thewidening, and the breakdown of the crazes (seeFigure 46(a)).

No generally accepted criterion for crazeinitiation exists currently. Once a craze hasinitiated, widening by drawing in new polymermaterial from the craze–bulk interface occurs(see Figure 46(b)). A highly stretched networkof molecules in the fibrils resists furtherelongation and instead pulls new amorphousmaterial into the fibril. The cavities in betweentake a prolate toroidal shape. This process isstrongly rate dependent (see Section3.03.5.2.6). Tijssens (2000), therefore, adoptedthe separation law of Equation (104) to modelthis process. Craze material is a complexstructure in which long cylindrical fibrils areinterconnected by cross-tie fibrils which alsogive the craze some tangential stiffness whichhas to be accounted for in a cohesive law interms of ’dT TTð Þ: In addition, a rule for theinteraction of shear and normal separation isneeded (see Section 3.03.5.2.4). Final break-down of the craze can be assumed to occur atsome critical value of the normal separation,though this process—like initiation—is farfrom being understood theoretically.

3.03.5.4 Applications to Ductile Fracture ofMetals

3.03.5.4.1 Simulation of ductile resistancecurves by the cohesive model

R-curves in terms of the J-integral play a keyrole in the structural integrity assessmentconcept. They express the fracture resistanceand generally depend on individual structures.The advantage of R-curves is their simpledetermination from experimental test data.The disadvantage of this concept is the transfer-ability problem from specimens to structures.Thus, it is important to know more precisely theeffects contributing to geometry dependenceand the quantitative margins for transferability.The cohesive model provides a valuable tool tosimulate the resistance behavior of structures(see Section 3.03.5.4.4). In the following, atten-tion is focused on parameter studies providing acomprehensive overview of several relevanteffects. Only 2D cases are considered. Isotropicelastic–plastic bulk material and quasi-staticloading are assumed and no effects of tempera-ture or loading rate are considered.

(i) Small-scale yielding

In SSY conditions the plastic zone isrestricted to a sufficiently small extension in

4

32 1

1: single voids, precursor of craze nucleation2: craze initiation3: formation of primitive fibrils4: saturated fibril elongation

process stages

fibril between cavities

(a)

open crack

TN

TNrate-dependent

bulk

cavity

fibrils with highlystretched molucules

idealized RVE

(b)

Figure 46 Cohesive model for crazing in amor-phous polymers: (a) stages of separation ahead of acrack tip and (b) craze widening by drawing in ofpolymer material from the craze–bulk interface(source Tijssens, 2000).


relation to the dimensions of the crack orstructure; it is not valid any more if plasticzones reach any outer surfaces of the structure.It is not necessary to consider a specificgeometry in SSY, but just the elastic K-dominated stress field around the crack tip.

Initial stimulating work with the cohesivemodel was performed by Needleman (1987).Comprehensive parameter studies on R-curvebehavior for ductile materials under SSYconditions were continued by Tvergaard andHutchinson (1992, 1994a) and Lin and Cornec(1998). Tvergaard and Hutchinson (1992)introduced a cohesive law described in Section3.03.5.2.3 (see Equation (86)). Fixing the twoparameters, d1¼ 0.15d0 and d2¼ 0.5d0, thecohesive energy G0 then results fromG0¼ 0.675T0d0. Lin and Cornec (1998) applieda cohesive law with rectangular shape as thesimplest case for d1¼ 0 and d2¼ 1 withG0¼T0d0. Effects of the shape of the cohesivelaw are discussed in Section 3.03.5.2.3; theywill, however, not affect the qualitativeconclusions of the results presented in thefollowing.

The configuration is the well-known bound-ary layer model (see Figure 47). A mode-I K-dominated displacement field is applied at theouter boundary, and, in addition, an indepen-dent stress field is acting parallel to the crackplane, the so-called T-stresses defined by t � s0,ranging from compression to tension. Cohesiveelements are inserted along the ligament in theFE mesh. The plastic behavior of the bulkmaterial is described by a power law, e/e0¼(s/s0)

N, with the parameters: yield stress s0,yield strain e0, elastic modulus E¼ s0/e0, andhardening exponent N (0rNr1).

The crack-growth resistance is expressed interms of stress-intensity factor, K, which isrelated to the energy release rate, G, byEquation (14). Crack initiation occurs atG0¼G0 or K0 ¼

ffiffiffiffiffiffiffiffiffiffiG0E0p

: The R-curves underSSY conditions are, in general, dependent onthe following parameters:

KR

K0¼ F

Da

R0;

E

s0; N;

T0

s0; t

� �ð118Þ

Figures 48–51 present R-curves in dimension-less axes as obtained by Lin and Cornec (1998)based on a rectangular cohesive law. The KR-value on the ordinate is normalized by K0, andthe crack extension on the abscissa is normal-ized by an analytical approximation of theplastic zone size

R0 ¼1

3pK0

s0

� �2

ð119Þ

The R-curves are presented in dependence onthe parameters T0/s0, N, and t for plane strainand plane stress, respectively. Crack initiationin all the simulations occurs at K0, whichverifies the assumption made above thatG0¼G0. All parameters of Equation (118)except e0 ð¼ s0=EÞ have an effect on the R-curves, so that they need careful consideration.Toughness increases with T0/s0, in both—planestress (Figure 48) and plane strain (Figure49)—but decreases with increasing hardening,N (Figure 50) and increasing T-stress (con-straint), ts0 (Figure 51).

After some crack extension, some of the R-curves approach a steady-state value, Kss. Thisplateau is of particular interest as it providesthe maximum achievable toughening effect.Figure 52 summarizes the dependence of Kss onthe transverse T-stress, ts0, for varying cohe-sive strength, T0/s0, and for two hardeningexponents (N¼ 0.1 and 0.2): for tensile T-stresses, t40, the steady-state value of tough-ness does not change significantly, whereas forcompressive T-stresses, to0, a steep increaseof Kss occurs. This effect is due to the influenceof the T-stress on the plastic zone developmentwhich is responsible for the apparent macro-scopic toughening enhancement.

The cohesive strength, T0, is a very impor-tant parameter. Figure 53 shows the depen-dence of Kss in the steady-state condition onthe cohesive strength, T0/s0, for three differenthardening exponents (N¼ 0, 0.1, 0.2) (seeTvergaard and Hutchinson, 1992). No R-curveeffects exist at all, i.e., Kss¼K0, for T0/s0r2independent of the hardening. In this case noremote plasticity is involved and only localiza-tion in the cohesive zone takes place. Upper

r

x2

x1

crack θ

cohesive zonewith T0, Γ0

τσ0

σij =K

2πrfij (ϑ ) +τσ0δ1jδ1j

Figure 47 Boundary layer model for SSY ofhomogeneous structures with embedded cohesivezone.


limits of T0/s0 exist for which only remoteplastic deformation with increasing crack-tipblunting but without crack extension takesplace. In particular for N¼ 0, this limit occursat T0/s0¼ 2.97. Increasing hardening shiftsthat limit to higher values.

A further application of the modifiedboundary layer model in SSY conditions forstudying the R-curve behavior of metallicfoams is presented by Chen et al. (2001) andFleck et al. (2001). A linearly decreasingtraction is used as cohesive law.

(ii) Large-scale yielding

LSY goes beyond the limits of SSY up tofully plastic conditions. For such conditionsthe linear elastic energy release rate, G, has to

be replaced by the J-integral which includes thenonlinear effects. The R-curves are normalizedby the cohesive energy, G0. For LSY theindividual geometry and loading type have tobe introduced as further parameters:

JR

G0¼ F

Da

R0;

E

s0; N;

T0

s0; geometry; loading

� �ð120Þ

The R-curve behavior can be studied forstandard test specimens used in fracturemechanics. Three specimen geometries, namelySE(B), SE(T) and M(T), under plane-strainconditions have been investigated (Cornecet al., 1998) by numerical simulations apply-ing the cohesive model. Specimen width andbulk material properties are kept constant.The results for fixed initial crack length,a0/W¼ 0.5, and fixed cohesive parameters are

∆a/R0

T0/σ0

1.5

1.65

1.8

initiation

0 21 3 0 21 30

4

6

8

10

2

N = 0.1, τ = 02.0

∆a/R0

0

4

6

8

2

10

N = 0.2, τ = 0

1

T0/σ0

11.5

1.65

1.8

2.0

K/K

0

K/K

0

(a) (b)

Figure 48 R-curves for plane stress and SSY: effect of cohesive strength—(a) low hardening and (b)moderate hardening.

∆a/R0

T0 /σ0

2.5

3.0

3.2

initiation

0 21 3∆a /R0

0 21 30

2

3

4

5

1

N = 0.1, τ = 03.5

2.53.03.23.5

T0 /σ0

N = 0.2, τ = 0

0

2

3

4

5

1

(a) (b)

K/K

0

K/K

0

Figure 49 R-curves for plane strain and SSY: effect of cohesive strength—(a) low hardening and (b)moderate hardening.


presented in Figure 54, showing the well-known trends of R-curves between bend andtensile-type loading. Bend loading alwayscauses lower R-curves. Higher hardening re-duces toughness.

Further effects of parameter variations onthe R-curves are displayed in the followingthree figures—the effects of: (i) the cohesivestrength, T0/s0, in Figure 55, (ii) the relativecrack length, a0/W, for an SE(B) specimen inFigure 56, and (iii) the relative crack length,a0/W, for the M(T) in comparison to the SE(B)specimen in Figure 57. Crack initiation tough-

ness is always equal to the cohesive energy, G0,which confirms experimental findings that theinitiation toughness is not significantly affectedby the specimen constraint. This may becomedifferent for very large plastic deformationprior to crack initiation when the J-integraldoes not dominate the crack-tip fields anymore.

The significant increase of the global tough-ness of structures can be taken into account infracture mechanics assessments, which can beevaluated with high reliability by computa-tional methods using the cohesive model.

3.03.5.4.2 Embedded ductile layer with centercrack under mode I

A thin ductile layer embedded between twoelastic materials represents an idealization fordifferent engineering applications: mulitlayeredmaterials as in welded or brazed joints, e.g., inmicroelectronics. The following case study forstrength mismatch situations in structurespresents some interesting characteristics de-monstrating the differences to homogeneouscases. A plastic zone is confined to the layerand grows rapidly through the height andsubsequently along the layer direction. Thelayer contains a center crack.

(i) Ductile layer under SSY

Conventional 2D FE analyses without usingthe cohesive model have been carried out byVarias et al. (1991) for a thin elastic–plasticisotropic layer embedded in an elastic infinitemedium under SSY conditions. Besides the

∆a/R0

N

0.30.2

0.1

initiation

0 1 2 30

2

3

4

5

1

0.05

T0/σ0 = 3, τ = 0

K/K

0

Figure 50 R-curves for plane strain and SSY: effectof hardening exponent.

K/K

0

K/K

0

∆a/R0

τ

-0.250.000.50initiation

0 21 3 0 21 30

2

3

4

5

1

N = 0.1, T0 /σ0= 2.5

∆a/R0

0

2

3

4

5

1

-0.50

-0.75

τ

-0.250.000.50

-0.50-0.75

N = 0.2, T0/σ0= 2.5

(a) (b)

Figure 51 R-curves for plane strain and SSY: effect of T-stress—(a) low hardening and (b) moderatehardening.


strength mismatch, a mismatch of elasticproperties has been assumed. Hydrostaticstress peaks up to six times the yield strengthwere observed, depending inversely on thelayer thickness.

Lin et al. (1997) extended these investiga-tions to simulations with a cohesive model. R-curves are presented in Figure 58(a) showingthe increasing toughness with layer thickness,H, normalized by the plastic zone radius,R0, of Equation (119). For moderate cohesive

strength values, T0/s0r3.5, crack initiationoccurs always at the initial crack tip. All R-curves lie below that of a homogeneous systemconsisting of the layer material only, H-N.The constraint conditions along the layerinterfaces due to the hard elastic materialgenerate high stresses even at low remoteloading. For very thin layer heights, H, noincrease of toughening is possible and sponta-neous unstable crack extension takes place justafter reaching crack initiation at K¼K0.

1 0.10.003 0.15 0.52 0.10.006 0.15 0.53 0.120.003 0.125 0.25

No ε0 δ1/δ0 δ0/∆0δ2/δ0

0

3

4

5

6

1

2

T0/σ0

K

/Kss

0

2 30 4 51 6

2.97 (rigid plastic limit)

N

0.2

0.10

11

13

2

cohesive zone onlywithout plasticityin the bulk material

(∆0 : smallest finite element size)

δ

TT0

δ1 δ2 δ0

Figure 53 Steady-state toughness in dependence on increasing cohesive strength and hardening (sourceTvergaard and Hutchinson, 1992).

3.5

0

2

4

6

8N = 0.1 N = 0.2

3.02.5

T0 /σ0

3.5

0

2

4

6

8

3.02.5

T0 /σ0

K

/ss

K0

K

/ss

K0

0-1.0 0.5 1.0-0.5τ

0-1.0 0.5 1.0-0.5τ(a) (b)

Figure 52 Steady-state toughness in dependence on the T-stress and the effect of cohesive strength: (a) lowhardening and (b) moderate hardening.


Hence, thin layers, even though they areductile, are sensitive to unstable failure. Figure58(b) presents the steady-state value of the R-curves in dependence on the cohesive strengthfor varying layer height.

For high cohesive strength, i.e., T0/s0Z4,crack extension develops differently from thecase of medium cohesive strength (see Figure59). Crack initiation starts at the existing cracktip, but the crack does not propagate further.Instead, a first cavitation away from the cracktip initiates in the layer followed by a secondone after a certain growth of the first. Thisprocess repeats. The bridges between thecavitations can sustain further loading, as thehigh peak stress is released after cavitation. Asthe elastic boundaries allow only uniform

deformation, these bridges will fail like in auniaxially tested tensile bar. Failure of the firstbridge initiates a zip-fastener process andresults in unstable crack propagation.

(ii) Weld joint with strength mismatch

Real welded joints are complex materialcompounds with high gradients of materialproperties. The principal mechanical behaviorof a weld joint can be understood by consider-ing an idealized model of a cracked specimencontaining a material layer with differentelastoplastic properties. The respective simula-tions with the cohesive model were performedby Lin et al. (1999). Figure 60 shows an SE(B)specimen with a center-cracked layer, B and W

0

5

10

15

20

J/

3 41 520 3 41 520∆a (mm)(a) (b)

1

SE(B)

Γ0

J/Γ

0

∆a (mm)

1

N = 0.1T0 /σ0

3.0

3.5

4.0

0

5

10

15

20

SE(B)N = 0.2

3.0

T0 /σ0

3.54.0

5.0

Figure 55 R-curves for bend specimen SE(B) in LLY: effect of cohesive strength—(a) low hardening and (b)moderate hardening.

0

5

10

15

20

N = 0.1, T0/σ0= 30

2

4

6

8

10

J/Γ 0

J/Γ 0

3 41 520 3 41 520(a) (b)

1

M(T) SE(T)

Tension

SE(B)Bending

N = 0.2, T0/σ0= 3

M(T)

SE(T)

SE(B)

1

�a (mm)�a (mm)

Figure 54 R-curves for different fracture mechanics specimens under tension and bending and LSY: (a) lowhardening and (b) moderate hardening.


designating base or weld metal, respectively.Both materials are elastic–plastic. The strengthmismatch is characterized by the parameterM¼ s0W/s0B. Undermatching, where the weldis softer than the base metal, is present for0oMo1, and overmatching, where the weld isharder than the base metal, for 1oMoN,respectively.

Figure 61(a) presents R-curves in terms of theJ-integral for a fixed ligament length to weldthickness ratio, (W�a0)/H¼ 10, and M in therange 0.5rMr2. M¼ 1 denotes the homo-geneous case. The R-curves for the case ofundermatching lie below that of the homoge-neous structure. Fracture toughness decreaseswith decreasing M, and no R-curve exists for

the limiting case M-0. For strong under-matching, Mo0.5, the simulations indicatesudden failure of the specimen by breakdownof several cohesive elements at the same time,characterized as ‘‘pop-in’’ of the crack in Figure61. The simulated curves are in accordance withthe trends of experimental observations.

Normalized load–displacement curves, i.e.,bending moment, Mb, vs. crack opening,CMOD, are shown in Figure 61(b). Thebending moment is normalized by the lowerbound limit load and CMOD by the ligamentlength. Overmatching reduces the normalizedmaximum bending moment compared to thehomogeneous case but the structure behavespredominantly stable, whereas undermatching

M(T)

T0 /σ0 = 3N = 0.1

initiation

0 1 2 3 4 50

10

20

30

J/Γ 0

J/Γ 0

∆ a (mm)

0.5

0 1 2 3 4 50

2

4

6

8

10

1

∆ a (mm)(a) (b)

SE(B)

M(T)

T0 /σ0 = 3N = 0.2

a0 /W = 0.1

SE(B)

1

0.5

a0 /W = 0.1

Figure 57 R-curves for tension and bend specimens, M(T) and SE(B), in LLY: effect of initial crack length—(a) low hardening and (b) moderate hardening.

SE(B)T0/σ0 = 3

N = 0.1

a/W

0.5

0.3

0.2

0.1

initiation

0 1 2 3 4 50

2

4

6

8

1

J/Γ 0

∆ a (mm)

a/W0.5

0.1

SE(B)T0/σ0 = 3

N = 0.2

0 1 2 3 4 50

2

4

6

8

1

∆ a (mm)

J/Γ 0

(a) (b)

0.30.2

Figure 56 R-curves for bend specimen SE(B) in LLY: effect of initial crack length—(a) low hardening and (b)moderate hardening.


increases the normalized maximum bendingmoment but reduces ductility significantly.

3.03.5.4.3 Interface cracking of dissimilarmaterials

Some applications demonstrate the abilitiesand the potential of the cohesive model.

(i) Type A: debonding of a bimaterial interfaceunder remote mixed-mode loading

A compound of two semi-infinite media withdistinct properties is considered in plane-strainand SSY conditions. It is assumed that theinterface exhibits the weakest strength so thatthe crack will propagate along the interface.For higher interface strength, the crack maykink into the softer material. Figure 62 showsthe boundary value problem for remote mixed-

mode loading. Again, the boundary layermodel is applied for SSY. The plastic zonemust be small compared to the crack lengthwhich is satisfied due to the low toughness ofthe interface.

Basic investigations using the cohesive modelto simulate fracture of interfaces were per-formed by Needleman (1990a, 1990b) and Xuand Needleman (1995). Systematic parameterstudies for mixed-mode loading of interfaceswere presented by Tvergaard and Hutchinson(1993), Tvergaard (2001), and Wei and Hutch-inson (1998). Tvergaard and Hutchinson(1993) analyzed the limit case of a rigidsubstrate at first and Tvergaard and Hutch-inson (2001) extended this to the more generalcase of elastic as well as elastic–plastic sub-strates. The layer material on the substratebehaves always elastic plastically.

Mixed-mode loading causes normal andshear tractions along the interface, Figure63(c)which have to be considered in thesimulations. Figure 63 presents schematicdrawings of the mixed-mode situation: theplastic zone development in Figure 63(a), andthe deformation of the cohesive zone undermixed mode in Figure 63(b). The cohesive lawsfor normal and shear separation, Figure 63(c)which were used by Tvergaard and Hutchinson(1993) and Tvergaard (2001), have beendescribed in Section 3.03.5.2.4 and in Equation(86). The authors used d1¼ 0.15d0, d2¼ 0.5d0,and dN;0=dT;0 ¼ 1 for their simulations, andonly in some cases also considered dN;0=dT;0 ¼0:5 and dN;0=dT;0 ¼ 100: The FE mesh used byTvergaard and Hutchinson (2001) is shown inFigure 64. A large area with a regular mesh isplaced in the center of the specimen to allowfor crack extension. The smallest element size isD0, which is used as a reference length.

The number of parameters affecting the R-curves, which have been introduced in Equa-tions (118) and (120), has to be furtherextended by the additional mismatch andmixed-mode parameters:

K

K0¼ F

Da

R0;

E1

E2;n1n2;s01s02

;N1

N2;

T0

s01;dN;0

dT;0

� �ð121Þ

The indices 1, 2 denote the softer elastic–plasticmaterial and the substrate medium, respec-tively. The dimensionless Dundurs factors

b ¼ 1

2

m1 1� 2n2ð Þ � m2 1� 2n1ð Þm1 1� 2n2ð Þ þ m2 1� 2n1ð Þ

�ð122Þ

and

mi ¼Ei

1þ ni

; i ¼ 1; 2 ð123Þ

KI

elastic

elastic

el-pl

5

4

3

2

1

0

N1 = 0.1E2/E1 = 1

ν1 = ν2 = 0.3

H/R0

0.1360.40824

∞

Kss

/K0

3.53.02.52.0Τ0 /σ0

5

4

3

2

1

03210

∆a/R0

N = 0.1, T0/σ0 = 3.5

0.136

H/R0

0.408

2

4

˚

cavity growth

K/K

0

(a)

(b)

Figure 58 R-curves for a constrained ductile layercontaining a center crack under SSY: (a) dependenceof the R-curves on the layer height and (b) thesteady-state toughness on cohesive strength andlayer height.


are commonly introduced to characterizeelastic mismatch. A ‘‘mode-mixity’’ parameteris defined by

tan cL ¼Im KI þ iKIIð ÞLie

Re KI þ iKIIð ÞLie½ � ð124Þ

Figure 59 Formation of cavitations in a constrained ductile layer: (a) crack initiation at existing center crackfollowed by the initiation of the first cavitation; (b and c) initiation of the second cavitation ahead of the firstone, the first one stops growing; and (d) initiation of the third cavitation ahead of the second one, the secondone stops growing.

Mb

BB W

a0

W

CMOD

el-pl e l-plel-pl

Mb

2H

Figure 60 Bend specimen with mismatched weld: B—base metal; W—weld metal; and CMOD—crack mouthopening displacement.


with i ¼ffiffiffiffiffiffiffi�1

pand

e ¼ 1

2pln

1� b1þ b

� �ð125Þ

where L is a reference distance ahead of thecrack tip on the interface, for which tan cL ¼s12=s22 is given. For elastic mismatch this ratiodepends on the chosen value L. For thelimiting case b¼ 0, i.e., without mismatch ofelastic moduli, Equation (124) results in thecommon definition of tan c ¼ KII=KI (inde-pendent of the distance from the crack tip).The plastic zone size for mixed-mode load-ing at initiation can analytically be estima-ted by

R0 ¼1

3pKj j0s01

� �2

ð126Þ

where s01 is the reference yield stress of theelastic–plastic material and Kj j0 the effectivestress-intensity factor at initiation, i.e., when G0

is achieved. This is at

Kj j0¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2G0=ð1� b2Þ

ð1� n21Þ=E1 þ ð1� n22Þ=E2

sð127Þ

The mode-mixity parameter for any otherdistance r ahead of the crack tip can becalculated by

cr ¼ cL þ e ln r=Lð Þ ð128Þ

The mode mixity at r ¼ R0 is used in thefollowing figures with c0 ¼ crðr ¼ R0Þ:

Figure 65 displays the R-curve of a bimater-ial system consisting of a rigid substrate and anelastic–plastic material connected by an inter-face, compared to the R-curve of a homo-geneous elastic–plastic material of the samecohesive strength, T0/s0¼ 3. Both axes arenormalized by the reference values definedabove. The mode mixity for this example iscLE0 which corresponds to c0¼ –251 forthe chosen parameter value of b¼ –1/4. A

pop-ininitiation

3

2

1

00 0.1 0.2 0.3 0.4

CMOD/(W-a0)

SE(B)

0.5

0.75 = M

2.01.0

1.25

elastic slope

homogeneous

30

20

30

0

J/J i

1

2.01.51.00.50

0.75

1.0

1.25

2.0

M

∆a (mm)

SE(B)a0 /W = 0.5W-a0

H= 10

homogeneous

overmatching

undermatching

Mb/(

σ 0W

W2 /4

)

a0/W = 0.5W-a0

H= 10

0.5

(a)

(b)

Figure 61 Effects of strength mismatching on abend specimen SE(B) with the parameters: EW¼EB,

vW¼ vB, NW¼NB, T0W/s0W¼ 3, G0W¼ 90Nmm–1,a0/W¼ 0.5—(a) R-curves for various mismatchratios 0.5rMr2 and (b) load–displacement curvesfor various mismatch ratios 0.5rMr2.

L

x2

x1

KI, KII

1

2

elastic-plastic

rigid or elastic or elastic-plastic

structural mixed-mode loading

2D, plane strain

Figure 62 Boundary layer model of a bimaterialjoint with a cohesive interface under SSY andmixed-mode loading.


significant increase of the resistance is foundfor the bimaterial system, and the steady-statecondition is reached later compared to thehomogeneous reference case. The comparisonwith results by Lin and Cornec (1998) for thehomogeneous case demonstrates the influenceof the shape of the cohesive law as well as ofthe absolute values of the cohesive parameterson the steady-state toughness.

The effect of modulus mismatch, E2/E1Z1,on R-curves is shown in Figure 66 for an elasticsubstrate material for a mode mixity of cLE0,again. The stiff elastic substrate constrains thedeformation at the interface and thus imposesa shielding effect which retards the activationof the cohesive elements. At moderate elasticmismatch this effect is less pronounced.

Figure 67 presents the steady-state toughnessfor a rigid substrate as a function of the mode-mixity parameter, c0 (see Equation (128)).

Mixed-mode loading increases the structuraltoughness significantly, while varying valuesfor the critical shear and normal separationhave only a minor effect.

Tvergaard (2001) also extended the para-meter study to the case where both materialsbehave elastic plastically. In addition to b, astrength mismatch ratio is introduced, M ¼s02=s0141; with s01, s02 as yield stresses of thematerials #1 and #2, respectively. M¼ 1 andE2¼E1 represent the homogenous case andM¼N characterizes an elastic substrate. Fig-ure 68 displays steady-state toughness valuesfor some strength mismatch ratios and twocohesive strength values, T0/s0¼ 3 and 4.25,respectively, in dependence on the mode-mixityparameter, c0. The trends are similar as for thecase of a rigid substrate. The strength mismatchincreases the toughness, and this effect issignificantly higher for increasing cohesivestrength. However, the steady-state level islower than for the case of a rigid substrate.

(ii) Type B: constrained ductile layer withcracked interface under remote mixed-modeloading

A thin ductile layer of thickness H is embed-ded in an elastic material (see Figure 69).Remote loading is applied within the frame ofSSY conditions. The interface contains a crack.

1

2

Rout

el-pl

el-pl

KI, KII

Rout = 8000 ∆0

∆0

linearisoparametricelements

crack

Figure 64 FE mesh for the boundary layer modelof interfaces under mixed-mode loading (sourceTvergaard, 2001).

T(D)

D =

1

break of the cohesive element at D = 1(b)

1 (soft)

2 (hard)

initialcrack tip

unloading wake

cohesive zone

active plastic zone

cohesive elementswith T0, Γ0

x1

x2

L

∆a

wake

10.50.250

1

0

TN,0

TTN,0

TT,0

pure N direction (normal fracture)

pure T direction (shear fracture)N+T direction

δN

δN,0

δT

δT,0;

(a)

(c)

interface elementswith collapsing nodesD

<1

Figure 63 Bimaterial with cohesive interface: (a)plastic zone in soft and hard material (sourceTvergaard and Hutchinson, 1993), (b) interactionof normal and shear traction in the cohesive zone,and (c) cohesive laws for pure normal and shearfracture modes.


Basic investigations of this case under mode-Iloading are published by Tvergaard andHutchinson (1994b, 1996a). If the elasticproperties differ, mixed mode occurs even forremote mode-I loading. The formalisms fortype A interfaces described above can also beused here. The limiting case of H¼N isequivalent to a type A interface in a bimaterial.

Parameter variations of the cohesivestrength and layer height are displayed. The

steady-state condition depends strongly of thelayer height, H. Figure 70 shows the maximumtoughness enhancement in dependence of thelayer height, H, normalized by the plastic zoneextension, R0. The larger the height H, thehigher the toughness increase. The toughnessreduces significantly if the layer is fully plastic,i.e., when H is smaller than R0. For the samecohesive strength, the respective R-curves liebelow the homogeneous reference case. This

el-pl

homogen

KI

1

2rigid

el-pl

KI, KII

4

3

2

1

0

5

6

∆a/R0

±ψ0

T0/σ0 = 3 (all)

N1 = 0.1

ε01 = 0.003ν = 1/3β = -1/4

δN0 = 0.005 ∆0

L = 2000 ∆0

δδ

N,0/ T,0 = 1

N = 0.1ν = 0.3ε0 = 0.003

43210 6

K/K

0 ψ0 = -26°

ψ0 = 0°

ψ0 = +20°

(1)

(2)homogen

(1) Tvergaard and Hutchinson (1992)

(2) Lin and Cornec (1998)

(1) (2)

T

δ

min

5

Figure 65 R-curves for interfaces between dissimilar materials (source Tvergaard and Hutchinson, 1993), thebulk material #1 is elastic–plastic.

KI, KII

1el-pl

2elastic

1

2rigid

el-pl

KI, KII4

3

2

1

0

T0/σ0 = 3

4 53210∆a/R0

|K|/K

0

E2/E1

1

2

6

1000

K ss

ψ0 = 0

8

Figure 66 R-curve behavior for interfaces in dissimilar joint with elastic modulus mismatch for a specificmixed-model loading (c0E01) (source Tvergaard and Hutchinson, 2001).


1

2rigid

el-pl

KI, KII

-60 -30 30 60

KK0

ss

0

0

4

3

2

1

T0/σ0 3.02.0

1.4

δN,0

δT,0= 100

δN,0

δT,0= 1

ψ0 (˚)

Figure 67 Steady-state toughness for interfaces in dissimilar rigid/elastic–plastic joints under variable mixed-mode loading (source Tvergaard and Hutchinson, 2001).

KI , KII

1

2

el-pl

el-pl

E2 = E1ν1 = ν2 = 1/3

M = σ02/σ01

N2 = N1 =0.1

KI , KII

rigid

el-pl

T0/σ0 = 3.0

-10 30

KK0

ss

0

0

4

3

2

1

5

1.25

1.35

1.5

δN,0δT,0

= 1

ψ0 (˚)10 20 40-20-30-40

M

T0/σ0 = 3.5

T0/σ0 = 4.251.50

M (σ02= )

(σ02= )

ν = 1/3N = 0.1

8 8

8 8

Figure 68 Steady-state toughness for interfaces in dissimilar joints with strength mismatch materials underremote mixed-mode loading in dependence on mode mixity (source Tvergaard, 2001).


decrease of toughness is different from thebimaterial case.

An overview of only the steady-state tough-ness is presented in Figure 71 taking threelimiting cases (Tvergaard and Hutchinson,1994b). The rigid substrates of a bimaterialinterface provide the highest apparent struc-tural toughness, while the thin embedded layerrepresents the weakest case, when both elasticmoduli are equal.

For high cohesive strengths, T0/s0Z4, thethin layer with an interface crack behaveslike a layer with a center crack, which has

been described in Section 3.03.5.4.2. Furtherinvestigations of thin layers on a substrateare presented by Wei and Hutchinson(1997) and Abdul-Baqi and van der Giessen(2002).

(iii) Experimental verification I

An epoxy block is bonded on top of a bendbar made of an aluminum alloy (see Figure 72).This test was first presented by Mohammedand Liechti (1998) and Hutchinson and Evans(2000) to study the influence of the notch angle,a, of the block on crack initiation. As no initialcrack is present in this configuration, fracturemechanics cannot be applied. Cohesive-ele-ment simulations, however, allow for predict-ing crack initiation at arbitrary notches evenwithout any precrack. The applied cohesivelaw is again that of Tvergaard and Hutchinson(1992) (see Equation (86)). The separationenergy was determined from the initiationvalue obtained from a specimen with a sharpcrack, a¼ 0, and the cohesive strength bysimulations of the respective test data, yieldingG0¼ 4Nmm–1 and T0¼ 3MPa (Hutchinsonand Evans, 2000).

Figure 73 displays the results of experimentsand simulations, showing a perfect coinci-dence, though only pure mode-I separation atthe interface was considered. The notch anglesignificantly raises the apparent structuraltoughness. Neither conventional fracture me-chanics nor any critical stress concepts wouldbe able to predict this trend.

KI

2

2

1 el-pl

elastic

elastic

T0, Γ0

N = 0.1ν = 0.3ε0 = 0.003E2 = E1ν2 = ν1

crackH

Figure 69 Boundary layer model for a constrainedlayer with cracked interface under mode-I loadingand SSY (sources Tvergaard and Hutchinson,1994b; Tvergaard and Hutchinson, 1996a).

elastic

elastic

Hel-pl

KI

2

0

4

6

8

10

ss/Γ

0

T0/σ0

4.25

4.0

3.53.02.5

2 40 6H/R0

4.5

steady state becomesindependent of H(homogeneous)

Figure 70 Steady-state toughness of a thin constraint layer with cracked interface in dependence on layerheight and cohesive strength (sources Tvergaard and Hutchinson, 1994b; Tvergaard and Hutchinson, 1996a).


(iv) Experimental verification II

A successful application of the cohesivemodel to the failure analysis of adhesivelybonded sheet metal was presented by Hutch-inson and Evans (2000) and initiated by Yanget al. (1999), who tested and analyzed tearingof adhesive joints under various loadingmodes. The geometry of a mode-I peel test isshown in Figure 74. Two metal sheets of width,B, and thickness, t, are bonded by an adhesive

and then torn apart by equal and oppositeforces, F. When steady-state peeling isachieved, 2F/B is the macroscopic work percrack extension, such that Gss ¼ 2F=B: Thus,

KI

elastic

elastic

Hel-pl

2

0

4

6

8

10

ss/Γ

0

el-pl

homogen

KI

rigid

el-pl1

2

E2/E1 = 1ν2 = ν1 = 1/3

KI

21

E2/E1 = 1ν2 = ν1 = 1/3

0 21 3 4 5 6

N = 0.1 H/R0 > 6

enhancement due tocrack tip shieldingfrom the rigid body

ν = 0.2

T0/σ0

Figure 71 Steady-state toughness of dissimilar joints with cracked interfaces (sources Tvergaard andHutchinson, 1994a, 1994b; Tvergaard and Hutchinson, 1996a).

0˚ < α < 135˚

adhesive layer

T0 = 3 MPa

Γ 0 = 4 N/mm

(b)

el-pl

elastic

notch

epoxy

FF

aluminum bar

α12.7

12.7

254

(a)

177.8

Figure 72 Setup for studying crack initiation fromnotches at interfaces between bonded materials:aluminum bend bar with epoxy block bonded ontop (source Hutchinson and Evans, 2000).

Figure 73 Predicted crack initiation for the testsetup of Figure 72 in comparison to experimentaldata (source Hutchinson and Evans, 2000).


this test provides a method for measuring thesteady-state macroscopic toughness, which isthe sum of the work per unit area consumed inseparating the epoxy adhesive plus the plasticdissipation induced in the sheets (see Equation(38)). Thus, Gss will depend on the thickness, t,of the sheets, and the actual separation energy,G0, of the adhesive can be determined by thecohesive model only.

The results of the measurements fort¼ 1mm and 2mm and of the 2D simulationsfor varying thickness but identical cohesiveparameters are shown in Figure 75. Theapplied cohesive law is the same as used incase I. The parameters were fitted to one of thetwo experiments, yielding T0¼ 100MPa andG0¼ 1.4 N mm–1. The predicted trend of thesimulations, that due to plastic deformation ofthe metal sheets, the macroscopic steady-statetoughness increases with increasing sheet thick-ness, agrees with the experimental result. Inparticular, extrapolation t-0 yields Gss-G0.

More experimental investigations of inter-face problems can be found in Liang andLiechti (1995), Dauskardt et al. (1998), Qiuet al. (2001), Yang et al. (2001a), Ismar et al.(2001), Madhusudhana and Narasimhan(2002), and S�rensen (2002).

3.03.5.4.4 Verification of the cohesive modelon homogeneous elastic–plasticstructures

All simulations presented in the previoussections are restricted to 2D simulations, and afictitious material behavior has been assumed

for the parameter studies. Treating ductilefracture in real materials and structures by3D simulations is necessary for a verification ofthe cohesive model with respect to engineeringapplications. A general problem arises, how-ever, when comparing simulations and experi-ments: only measurable global quantities canbe obtained from the tests, which are often notsensitive enough to identify the local processesof damage and fracture uniquely.

A first 3D verification example was success-fully conducted for a soft aluminum alloy byLin (1998) and Lin et al. (1998b). Verificationsfor laser-welded specimens followed by Cornecand Scheider (2001) and Scheider (2000). Forhigh-strength steel some verification examplesare given by Elices et al. (2002) for uncracked,shallow, and deep notched and Charpy bendbars.

(i) Determination of cohesive parameters forductile materials

The two cohesive parameters, T0 and G0, aresupposed to describe damage and fracture of aRVE. In the following, an engineering treat-ment will be presented which was proposedand applied by Lin (1998), Lin et al. (1998b),Scheider (2000), and Cornec and Scheider(2001).

Figure 76 presents specimens for normaland shear fracture. The tensile bars aremeant for determining the cohesive strength,and the fracture mechanics specimens for thecohesive energy. In the tensile specimens,unstable failure follows immediately after

F

F

epoxy based adhesive layerH = 0.25 mm

sheet metal (elastic-plastic)

t

tH

B

Figure 74 Double cantilever beam test used foradhesively bonded thin sheet metals (source Hutch-inson and Evans, 2000).

30

20

10

0

J ss/

Γ 0

sheet thickness, t (mm)

Γ0 = 1.4 N/mm

T0 = 100 MPa

2-D CM simulation

experimental data

0 0.15 0.5 10

1

Γ0δδ0

TT0

0 1 2 3

Figure 75 Predicted steady-state toughness for thedouble cantilever beam in comparison with experi-mental data (source Hutchinson and Evans, 2000).


crack initiation. In this case the initiation pointcan be identified directly on the remote load–displacement record. Otherwise the crackinitiation has to be detected by appropriateexperimental methods like multiple specimensor potential drop technique. However, inductile materials the local principal stress atthe point of crack initiation depends on thetriaxiality and cannot be estimated fromaveraged experimentally measured data only.A conventional 3D FE analysis is required forthe identification of the cohesive parameters,which allows for an evaluation of the quantitiesat the very load level and place where fractureactually takes place.

The first example deals with normal fracture.Figure 77 demonstrates the determination ofthe cohesive strength, T0, from a test on around notched bar made of a soft aluminumalloy 2024-FC. Fracture initiates in the speci-men center after significant plastic deforma-tion. The corresponding principal stress fromthe FE analysis provides T0¼ 420MPa. Crackinitiation, characterized by the J-integral, isdetermined by fracture mechanics tests, yield-ing Ji¼ 10Nmm–1, which is identical to thecohesive energy G0.

Alternatively, the cohesive parameters canbe calibrated by simulations with cohesive

elements through fitting numerical to experi-mental records of remote quantities. Figure 78presents the results of axisymmetric simula-tions of the notched tensile bar test for varyingT0. The cohesive energy has been taken fromthe fracture mechanics test. With T0¼ 420MPathe experimentally detected unstable failurepoint is well described. Note that the differentT0 values do not change the load–displacementcurve but only the point of instability. It can beconcluded that both procedures provide thesame cohesive stress.

Similar progress has been achieved by Kolheet al. (1999), Yang et al. (2001a), and S�rensen(2002) for adhesive-bonded layers or interfaces.Under certain conditions, interface strengthand the shape of the cohesive law can directlybe determined from experimental curves asproposed by S�rensen (2002).

(ii) Verification tests for mode-I normalfracture crack extension

Verifications of the cohesive model werecarried out on the two specimens shown inFigure 79, namely a side-grooved C(T) and a

J J

initiation along45˚ slant plane

Shear FractureNormal Fracture

σmax

mode I inititationof side-grooved

specimens

thin

T0 τmax

Γ0

FfractureFfracture

Figure 76 Test specimens for determination of thecohesive parameters for normal and slant fracture.

dk

σ˚

0

100

200

300

400

0.2 0.4 0.6 0.8 1.0 1.2 1.40

σmax = T0 = 420 MPa

14.6

distance from center (mm)notchcenter

stress just beforeunstable fracture

point

loadincrease

dk = 3 mmrk = 1 mm

d0 = 8 mm

σ = 43.5

105

T0

d0

rk

axia

l str

ess

com

pone

nt in

the

notc

hed

sect

ion

(M

Pa)

8

Figure 77 Determination of the cohesive strengthfor the aluminum alloy 2024-FC by a tensile test of around notched bar based on a conventional elastic–plastic FE analysis: maximum tensile stress in thespecimen center at fracture provides T0¼ 420MPa.


tensile panel with semi-elliptical surface crack,SC(T) (Lin, 1998; Lin et al., 1998b). They wereperformed three-dimensionally as 2D casestudies by Yuan et al. (1996) under plane-stress and plane-strain conditions revealed thatsuch idealizations may lead to misinterpreta-tions. In both specimens mode-I crack exten-sion occurs in the symmetry plane which can beintroduced a priori in the FE mesh. Figure 80shows the FE mesh in the region of crackpropagation. The mesh is relatively coarse butmesh dependence is not a serious problem forsimulations with cohesive elements, fortu-nately.

Figure 81 presents the curves of load vs.CMOD of the side-grooved C(T) specimen.The results from the conventional FEcalculations without cohesive model mustcoincide with the initial part of the F–CMODcurve up to and somewhat beyond the initia-tion point before it deviates from the experi-mental records. An excellent coincidence ofnumerical and experimental curves is realizedwith the cohesive elements. The same corre-spondence is observed for the R-curves (seeFigure 82), where CMOD is chosen as the‘‘driving force’’ of crack growth. The crackextension, Da, is an averaged value over thespecimen thickness for both, numerical andexperimental curves.

Figure 83 shows the numerically simulatedcrack-front profile in the C(T) specimen andthe comparison with the final crack front afterthe test. The crack profiles are almost self-similar. The coincidence between predictionand test indicates that the cohesive model andthe respective parameters are not only able topredict the global behavior but also to simulatethe real fracture process. An equally goodcoincidence of the predicted and the measuredcrack-front curvature was obtained for the

0 0.5 1.0 1.5 2.00

100

200

300

400

ln(1+∆L/L0) (%)

F/R

k2

(N/m

m2)

round notched bar

Γ0 = 10 N/mm

Experiment

CM: T0 = 400 N/mm2

CM: T0 = 420 N/mm2

CM: T0 = 430 N/mm2

fracture

experimentwith unstable

T0

Figure 78 Determination of the cohesive strengthfor the aluminum alloy 2024-FC by a tensile test of around notched bar based on simulations with thecohesive model: parameter variation yields a best fitof the cohesive strength of T0¼ 420MPa.

crack frontligament (half width)with cohesive elements

side-groove

Figure 80 FE mesh of the C(T) specimen in thearea of crack propagation.

W

a0

BN

B

a0 = 25.42 mm

W = 50 mmB = 5 mmBN = 3 mm(a)

C(T)

2co

(b)

a0

c0

a0 = 4.66 mmc0 = 5.37 mmW = 50 mmB = 10 mm

B

SC(T)

CMOD

F

2W

Figure 79 Test configurations for 3D cohesivemodels: (a) deeply side-grooved C(T) specimen,W¼ 50mm, a0/W¼ 0.5, representing a bend-typeconfiguration and (b) SC(T) specimen, W¼ 50mm,containing a surface crack, representing a tension-type configuration.


tensile panel containing a surface crack, SC(T),which is shown in Figure 84. Similar goodresults for crack extension in laser-weldedSE(B) specimens were obtained by Cornecand Scheider (2001).

(iii) Thin sheet fracture

The second example deals with fracture ofthin sheets. This application is of basicimportance for all light-weight structures. Inthin sheets, ductile crack growth starts from afatigue crack as normal mode-I fracture in thecenter and forms a continuously curved crackfront over the thickness as in Figure 83, whichis known as ‘‘tunneling’’ effect or ‘‘thumb-nail’’shape. After some extension shear lips ofincreasing thickness are formed at the specimensurface, which finally turn into a complete slantfracture plane across the thickness. The frac-ture mode has changed from a pure mode I to acombination of modes I and III.

2D plane-stress simulations are sufficient tomodel the global mechanical behavior and topredict R-curves for large crack propagation(see Lin and Cornec, 1998; Siegmund et al.,1999). The respective cohesive parameters,however, are purely phenomenological and donot characterize the local fracture process. Thisconcerns not only the cohesive strength, whichis a tensile stress, T0, in normal (mode-I)fracture, but normal plus shear stress, TN,0,TT,0, in slant (mixed-mode) fracture also (seeSection 3.03.5.2.4). It also affects the cohesiveenergy, G0, as this quantity characterizes thelocal energy release per crack surface incrementas defined in Equation (13). Now the realcrack-surface increment, DA, equals (B �Da) innormal fracture and (O2B �Da) in 451 slantfracture. A plane model of unit thicknesscannot distinguish between the two cases, andhence the parameter G0 which gives a best fit ofthe global mechanical behavior is not related tothe local energy release rate.

Two verifications of cohesive models underplane-stress conditions by Lin and Cornec(1998) are presented in the following for C(T)specimens with different widths, W¼ 50mmand 1,000mm, made of a high-strength alumi-num Al 2024 FC sheet of 1.6mm thickness.The mode-I ‘‘cohesive strength,’’ T0, wasdetermined directly from test data at theinstability point of flat tensile specimens.Figure 85 presents the results of the R-curvesimulations in comparison to the tests. CTOD,specified as d5 measured on the surface with5mm gauge length, is used as fracture para-meter. The ‘‘cohesive energy,’’ G0, is deter-mined by calibrating the initial part of theR-curve. Note that both quantities, T0 and G0,characterize a pseudo-2D situation, only.The simulated curves are in an excellentcoincidence with the experimental curves.Some deviations occur for the large C(T)specimen of 1,000mm width, which canbe explained by the contribution of the

0 0.2 0.4 0.6 0.8 1.0 1.20.0

0.5

1.0

1.5

2.0

2.5

3.0

load

F (k

N)

CMOD (mm)

Experiment C(T)

3-D CM simulation

3-D FE el-pl (no crack growth)

initiation experiment

initiation simulation

initiation

2024-FC

C(T)

W = 50 mm

side-grooved

Figure 81 Load vs. crack opening curve for theC(T) specimen shown in Figure 79 from experimentand 3D simulations with cohesive model.

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

∆a (mm)

CMOD

(m

m)

Experiment

3-D CM simulation

2-D CM simulation, plane strain

2024-FC

C(T)

W = 50 mm

side-grooved

Γ0 = 10 N/mm

T0 = 420 MPa

Figure 82 CMOD resistance curve for the C(T)specimen shown in Figure 79 from experiment and3D simulations with cohesive model.


anti-buckling plates in the test, which enforcean in-plane deformation.

A further study on thin 1.7mm thick center-cracked panels of an aluminum alloy waspresented by Chabanet et al. (2001, 2003) usingan exponential function for the cohesive law.The cohesive parameters were determined fromtests on Kahn specimens.

Multisite cracks in a 1mm thick aluminumpanel of 2024-T3 of 1mm have been consideredby Siegmund and Brocks (2000c), and a newapplication of the cohesive model to shellelements has been presented by Li andSiegmund (2002). Full 3D cohesive simulationsof thin sheet specimens were published by Royand Dodds (2001). Interaction between normaland shear separation is considered. A curvedcrack front develops in the beginning, whichlater propagates in a self-similar steady-stateshape for larger crack extension. However,modeling of the real transition from flat toslant fracture is acknowledged to lie beyondpractical efforts.

(iv) Problem of arbitrary crack propagation inhomogeneous structures

In all problems considered so far, the crackpath was predefined by the location of cohesive

elements in the symmetry plane, which isactually the real crack path in symmetricmode-I situations. For general loading, how-ever, a crack would propagate in an arbitrarydirection. Some newer approaches for arbitrarycrack paths introduce a fine grid of interfaceelements in orthogonal and diagonal arrange-ments between all continuum elements whichhave to be triangular or tetrahedral (see Figure86). The region of the area meshed in this wayhas to be chosen large enough to cover anypossible crack path. This approach was firstapplied by Xu and Needleman (1994) for 2Dsimulations.

An example by Scheider (2001) is given forthe cup-cone fracture occurring in roundtensile bars, where normal fracture starts inthe center and shear lips develop at thecircumference. Due to its axial symmetry, thisproblem can be modeled two-dimensionally.The mesh consists of triangular solid elementswith cohesive elements at all interfaces asshown in Figure 86 across the complete sectionin the middle part of the specimen (see Figure87). Cohesive parameters are needed fornormal and shear fracture which have beenchosen as TN,0/s0¼ 3.33, TT,0/s0¼ 1.2,dN,0¼ 0.1mm, and dT,0¼ 0.15mm. Cup-conefracture could be successfully simulated:

1 mmC(T)

end

of te

st th

ickn

ess

initi

al th

ickn

ess

crack profile fitted in the center

finalcrack

real crack front

initi

al c

rack

fro

nt CM prediction

Comparison of final crack front: experiment vs. simulation

3-D CM simulation of crack propagation

finalcrack

initi

al c

rack

fro

nt

FE mesh

(a)

(b)

Figure 83 Crack propagation in the C(T) specimen shown in Figure 79: (a) simulated profiles of the growingcrack and (b) comparison of measured and predicted crack front at the end of the test.


fracture starts as normal fracture in the centerand is followed by slant fracture as the crackapproaches the free surface of the specimen,due to the competitive activation of tensile andshear decohesion modes. Branching of thecrack path is a local instability point, whichoccurs in 451 directions, actually, and due to‘‘numerical noise’’ one of the two finallydominates.

Extensive 2D cohesive simulations for het-erogeneous and quasi-brittle materials witharbitrary crack path evolution have beenpresented by Tijssens (2000) and Tijssens et al.(2001). Extremely refined triangular mesheswith interface elements have been used whichallowed for the development of crack patternswith several short secondary cracks besides themain crack.

Despite these successful modeling examplesfor the phenomena of arbitrary crack paths,many open questions remain. The influence ofthe discretization as well as a unique identifica-tion of the cohesive parameters for a givenmaterial are still uncertain. The coupling ofnormal and shear decohesion laws remains tobe related more closely to the physical process.How meshes, as shown in Figure 86, can

also be applied in 3D cases is also an openquestion.

A promising new approach for modelingarbitrary crack paths are current developmentsof new element formulations, where separationis integrated as a displacement jump. Alocalized cohesive zone can thus develop inarbitrary direction with arbitrary curvature.Examples using this new numerical methodhave been presented by Melenk and Babuska(1996), Moes et al. (1999), Sukumar et al.(2000), Belytschko and Black (2001), deBorst(2001), Wells and Sluys (2001), and Moes andBelytschko (2002).

An important issue of modeling aredamage mechanisms in heterogeneous struc-tures consisting of multiple constituents con-tributing to failure. Such situations occur on

experiment

CM prediction

3-D CM simulation of crack propagation

initialcrackfront

sym

met

ry li

ne

symmetry line

initialcrackfront

SC(T)

∆a

2024-FC

(a)

(b)

Figure 84 Crack propagation in the SC(T) speci-men shown in Figure 79: (a) simulated profiles of thegrowing crack and (b) comparison of measured andpredicted crack front at the end of the test.

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

∆a (mm)

δ 5 (m

m)

thickness 1.6 mm

C(T)

W = 50 mm

a/W = 0.5

Test 1

Test 2

CM simulation(plane stress)

δ 5 (m

m)

C(T)

W = 1000 mm

a/W = 0.5

Test 1

Test 2

CM simulation

(plane stress)

thickness 1.6 mm

∆a (mm)

(a)

(b)

Figure 85 Ductile fracture in thin metal sheets:comparison of d5-based R-curves from testing andnumerical simulations of C(T) specimens: (a)W¼ 50mm and (b) W¼ 1,000mm.


the microstructural level of a material as wellas on the macrolevel of structures consisting ofregions with varying properties, e.g., in weldedjoints. Cohesive models coupled with FE

methods provide an excellent tool for assessingfailure of heterogeneous materials and struc-tures without the limitations of classicalfracture mechanics.

possible crack path

nodes

interface elementsbetween the triangularcontinuum elements

continuum elements

detail of 8 collapsingnodes at cross points

Figure 86 2D FE mesh for zigzag crack propagation along cohesive elements located at the interfaces oftriangular continuum elements.

Figure 87 Simulation of cup-cone fracture by cohesive elements: (a) round tensile bar and FE meshaccounting for symmetry conditions and (b) local mesh exhibiting transition from normal to slant fracture.


3.03.5.5 Applications to Other Materials andPhenomena

3.03.5.5.1 Quasi-brittle materials such asconcrete

Concrete structures experience complexloading spectra and various environments overlong times. Concrete fails in a quasi-brittleway, but some ductility can be observed due tobridging and roughness effects of particles oraggregates at the fracture surfaces (see Section3.03.5.3.2 on micromechanisms). Due to itssimplicity and efficiency, the cohesive modelhas successfully been applied to concretestructures for many years. An overview onthe state of the art is given by Bazant andPlanas (1998). Among those who have pro-moted the application of the cohesive modelfor concrete are Elices et al. (1993, 2000, 2002),Planas et al. (1993), and Elices and Planas(1996). Some subsequent studies on concretefracture on a mesolevel have already beenpresented in Section 3.03.5.3.2.

Initially, Petersson (1981) proposed an ex-ponential cohesive law as shown in Figure28(b). Bilinear curves introduced by Planaset al. (1999) and Guinea et al. (2000)—seeFigure 28(d)—have been accepted to be usefulapproaches to model the traction–separationbehavior of concrete. Though the uniaxialtensile test still is the most fundamentalexperiment for material characterization, thetesting of concrete is confronted with a numberof problems (van Mier and van Vliet, 2002).Local strain gradients due to the boundaryconditions and specimen size effects may affectthe experimental strength and fracture proper-ties (van Vliet and van Mier, 1999). As of early2000s, it is a common view that cohesive lawsof disordered materials cannot be measured ina direct way. Instead of this, inverse methodsare acknowledged to be the preferred way toidentify the shape of cohesive laws and itsparameters. The latter are fitted iteratively tothe experimentally obtained macroscopic be-havior. Two specimen geometries are pre-ferred: the so-called Brazilian disk forsplitting tests and notched bend specimens. Atest guideline has been established, which isnow generally accepted to determine the tensilestrength, T0,N, of concrete (Planas et al., 1999;Guinea et al., 2000).

Some validation of the cohesive law is shownin Figure 88. The parameters for the bilinearcurve in Figure 88(a) were calibrated by fittingthe peak load from splitting tests and thedecreasing branch of the load–displacementcurve of notched bend bar tests. Figure 88(b)points to the fact that the first decreasing

branch of the cohesive law is related to thenarrow range up to and shortly beyondmaximum load (curve #2), only. However,any prediction of the complete load–displace-ment curve (curve #1) requires to take intoaccount the second part of the bilinear curvealso. That means that two specimens arenecessary to obtain a complete set of cohesiveparameters.

Another example has been presented byGalvez et al. (1998) and Cendon et al. (2000).Tests have been carried out on a beam loadedin mixed mode. During propagation the crackshowed path deviation and curving, as illu-strated in Figure 89. The test results werecompared with 2D cohesive-model simula-tions, which applied a bilinear cohesive law asshown in Figure 88(a) with T0¼ 3MPa,G0¼ 0.069Nmm–1, d1¼ 0.023mm, and d0¼0.16mm. The crack path, however, was notpredicted by the simulations. Instead of this, astraightforward LEFM analysis had beenperformed in advance. The trajectories of the

Figure 88 (a) Cohesive laws used for concrete and(b) respective simulations of a bend test compared toexperimental records (source Guinea et al., 2000).


maximum principal stress were then chosen todefine the crack propagation path in thecohesive model simulations. The cohesiveelements opened locally under pure mode-Iconditions, and thus no interaction of normaland shear separation had to be considered. Thecomplete load–displacement curves of severaltests could be predicted as shown in Figure 90.

The cohesive model was also applied to thesimulation of steel-reinforced concrete beams(e.g., Ruiz et al., 1998; Ruiz, 2001). Steelreinforcement is introduced to improve thetensile strength of concrete. That does, how-ever, not exclude concrete cracking due toshear stresses in the interface, which causes asudden load drop after reaching a first loadpeak. Mechanisms like this are well knownfrom pullout tests. Depending on the proper-ties of the reinforcement (e.g., surface rough-ness, volume fraction), the load-carryingcapacity after the load drop can remain

constant or increase before approaching asteady-state stress level.

The behavior of newer fiber-reinforced con-cretes was also investigated using the cohesivemodel. In this case, the parameters of thecohesive model depended sensitively on thearrangement of the reinforcement elements.Even small fiber volume fractions enhanced themacroscopic strength significantly (Li, 1998).More detailed microstructural considerationswould be helpful for a better understanding ofthe local failure mechanisms and thus for thefurther improvement of the materials.

Since the current cohesive models for con-crete, or more generally for quasi-brittlematerials, incorporate phenomenological de-scriptions based on experimental experience, anumber of essential aspects is still not coveredby them (Bazant, 2002), e.g.:

(i) The influence of the triaxial stress stateon the softening curve is not considered. It ishowever well known that, in particular, thetransverse stresses parallel to the crack planecan affect the process zone size.

(ii) Mixed-mode cohesive laws featuring theinteraction of normal and shear separation asneeded for predicting arbitrary crack propaga-tion have not yet been verified for concrete.

(iii) The application to 3D engineeringstructures has not yet been demonstrated.

3.03.5.5.2 Heterogeneous compounds

Compound materials consist in two or moreconstituents with different properties to com-plement each other. The degrading propertiesof one constituent are leveled off by the betterproperties of the others. The great variety ofcompound materials cannot exclusively beinvestigated by experiments. Therefore, com-putational design of materials is the futuredemand. Damage evolution, for instance, issensitive to morphological parameters of themicrostructure such as volume fraction, size,shape, and spatial distributions of reinforce-ments, interfacial strength and process-relateddefects. All these parameters will have to betaken into account in advanced assessmentmethodologies. Numerical modeling and, inparticular, the cohesive model can contributeto optimum material design.

Metal matrix composites (MMCs) are ma-terials, which combine the high strength, thehigh stiffness, and the good creep resistance ofthe quasi-brittle reinforcements with the hightoughness of the ductile metallic matrix. Forinstance, the stiffness and tensile strengths ofaluminum alloys are significantly improved byfiber reinforcements, but due to the limited

2 mm W/2W

W = 150 mm

FW

2W3W2

W2

experimentally observed crack path

Figure 89 Rectangular notched bend bar withunsymmetrical loading and resulting crack pathobtained in experiments (source Cendon et al.,2000).

0

2

4

6

8

10

12

14

0.200.150.100.050CMOD (mm)

experimentalscatter band

simulation (2-D)with a bi-linearcohesive law

F (k

N)

Figure 90 Load vs. crack-opening curve fromexperiments and comparison with CM simulations.A bilinear cohesive law is considered (sourceCendon et al., 2000).


resistance against fracture and debonding, thelatter can also yield poor ductility and lowfracture toughness of the overall structure.

For a compromise between strength andductility, the interface between fiber and matrixplays an essential role. The propagation ofcracks at the fiber–matrix interfaces has beensuccessfully modeled using the cohesive model,first by Needleman (1990a, 1992) and Tver-gaard (1990, 1993, 1995). A different approachhas been presented later by Lo and Allen(1994), Allen et al. (1994), and Lissenden andHerakovich (1995).

Most of these studies have used unit cellmodels. A unit cell represents the smallestrepresentative volume in the structure. Thewhole structure is assumed to consist of aperiodic arrangement of such unit cells. Ghoshet al. (2000) extended the model towardsnonuniform distributions and shapes of theso-called Voronoi cells including cohesive prop-erties, which are closer to realistic applications.

Figure 91 shows a periodic short-fiber(whisker) arrangement and the respective unitcell, the latter idealized by a round cylinder, aspresented by Tvergaard (1993). During me-chanical interface debonding, both—normal

and tangential separation—occur simulta-neously, and following the complete separa-tion, slip friction between fiber and matrix hasto be taken into account. The cohesive modelfor such kind of mixed-mode loading isdescribed in Section 3.03.5.2.4. A typical resultillustrating the mechanical behavior of a unitcell is presented in Figure 92. The cohesiveparameters chosen for this application werefictitious but of realistic order. The unit cellwas loaded by axial tension, s11, and radialstresses, s22¼ 0 and s22¼ 0.25s11, respectively.The diagram shows the typical drop afterinitiation of failure due to interface debonding,followed by complete separation of the fiber atits ends. Fiber pullout occurred accompaniedby reducing the load-carrying capacity. Note,however, that this capacity was still higher thanthat of the pure matrix material. A radialstress, s2240, raises the axial stress level. Whenthe interface strength, T0,I, was increased, nodamage occurred at all. Fiber breakage was notobserved, as the fiber strength, T0,F, was simplytoo high. Similar studies of interface debond-ing including Coulomb friction at the fibersurface have been presented by Ismar et al.(2001).

x2

rFrM

LFLM

x1

σ22

σ11

F

I

T0,I, Γ0,IT0,F, Γ0,F

(b)

M

SiC whisker

2aM

M

(a) aluminum matrix

r

Figure 91 Short-fiber-reinforced metal matrix and respective cell model (source Tvergaard, 1993): (a)periodically staggered SiC fibers in an aluminum matrix and (b) axisymmetric unit cell model under axial andradial loading.


Cantilever beam tests can be used for theexperimental determination of interface crack-ing, where crack propagation occurs understeady-state conditions (La Saponara et al.,2002), see Section 3.03.5.4.3. However, thisspecimen is restricted to mode-I straight crackextension. Further experimental fiber pushouttests and 2D CM simulations with varyingparameters are presented by Lin et al. (2001).Various parameters such as stiffness mismatch,friction coefficients, residual strains in theinterface, and interfacial strength have beenvaried within the investigation.

Some modern alloys, such as intermetallictitanium aluminides (TiAl), have a lamellarmicrostructure with different orientations. Thelamellae within the grains form interfaces, atwhich cracks can propagate. If a microcrackapproaches another domain of differentlyoriented lamellae, it cannot retain its path buthas to change the direction of propagation. Forinvestigating the behavior of such lamellarmicrostructures, 2D cohesive-model simula-tions have been carried out by Arata et al.(2001).

3.03.5.5.3 Dynamic fracture

The models and examples presented aboveare restricted to quasi-static loading conditionsand rate-independent material behavior.

Two different effects can be distinguishedwith respect of the treatment of dynamicfracture phenomena: material inertia effectsare treated by including the respective terms in

the balance of virtual work, and rate depen-dence of material properties. The latter is oftenrestricted to rate-dependent bulk materialproperties, whereas the cohesive propertiesare identical to those for quasi-static loading.Rate- and time-dependent effects of the cohe-sive properties are considered in more detail inSection 3.03.5.5.4. Within this section, onlyhigh-speed impact problems are discussed. Thetime, for which the structural response iscalculated, is in the range of microseconds oreven less.

Pioneering investigations on dynamic frac-ture using the cohesive model have beenpublished by Xu and Needleman (1994).Fast-growing cracks in brittle materialsshowed a rather complex pattern of crackbranching instead of cracks that grew at oneplane. To allow for arbitrary crack branching,cohesive surfaces have been introduced be-tween all continuum elements by a grid oforthogonal and diagonal interfaces in the sameway as for the meshes shown in Section3.03.5.4.4 for quasi-static loading. Using thatspecific mesh approach a variety of phenomenahas been investigated, such as crack branching,the effect of the impact velocity on the crackspeed, or crack arrest in brittle solids.

Xu and Needleman (1996) later presented aninvestigation on interfacial cracking betweenPMMA polymer and an aluminum materialblock under various impact loading. Thephenomenon of short crack branching couldbe simulated, but the effects depend quantita-tively on the assumed normal and shearseparation interaction. Further studies oninterfacial dynamic cracking were presentedby Siegmund and Needleman (1997), Arataand Needleman (1998), the latter using aviscoplastic bulk material.

A cohesive simulation of a drop-weightdynamic fracture test carried out on a notchedbend bar is presented by Ortiz and Pandolfi(1999), who used a viscoplastic bulk material.The model was extended to incorporate ther-mal effects by Pandolfi et al. (2000). Theunloading of the cohesive elements followed alinear path to the origin (see Section3.03.5.2.5). The normal and shear tractionsoperated interactively according to Equation(94). Contact and friction phenomena were notmodeled within the frame of the cohesive law.

Ruiz et al. (2001) performed 3D simulationsof tension–shear processes and mixed-modefracture in concrete under dynamic loading.Again, the normal and shear components werecoupled by Equation (94). In order to get asatisfying coupling of normal and shear se-paration, which is important with respectof a good agreement between experiment and

0 0.02 0.04 0.06 0.08 0.10 0.12

true strain (-)

0

1

2

3

EF /EM = 5.7

no damage: 10-20-0

5-10-0.25

5-10-0

pure matrix

T0,I /σ0MT0,F /σ0Mσ22 /σ0M

σ22

σ11

FM

I

norm

aliz

ed r

emot

e st

ress

, σ11

/σ0M

Figure 92 Overall (mesoscopic) stress–stain curvesof a fiber-reinforced unit cell undergoing interfacialdebonding for varying parameter combinations(source Tvergaard, 1993).


simulation, quadratic 10-noded tetrahedralelements were used. The mesh was stepwiserefined and the elements in the vicinity of theexpected crack propagation path were almostregularly distributed. They were surrounded bycohesive elements allowing for crack propaga-tion along all element boundaries. The con-sequence was crack path deviation andbranching corresponding to a rough fracturesurface. The simulated crack paths were inclose agreement with the experimentally ob-served crack trajectories. Note that the regularmesh pattern leads to preferred directions forthe crack propagation.

The cohesive model was also applied tofragmentation due to impact loading. Repettoet al. (2000) used it for simulating thefragmentation of glass rods due to an impactwith a velocity of 210m s–1, which made therod bursting after a few microseconds. Thefragmentation after 14.7 ms is shown in Figure93. A good agreement between the predictedfailure wave and the experiments was achieved.

In Zavattieri and Espinosa (2001) andZavattieri et al. (2001), the microcracking atgrain boundaries of brittle ceramics wasinvestigated using a representative unit cell of540 mm height including about 200 grains, seethe FE mesh with grains indicated by thicklines in Figure 94(a). The bulk material was ananistropic elastic solid. Several effects—namelystochastics, impact velocity, and grain size—were discussed by the authors. As an examplefrom Zavattieri et al. (2001), the fragmenta-tions after 100 ns and 1 ms are shown in Figures94(b) and (c), respectively.

In contrast to approaches based on con-tinuum damage theories, the cohesive model isable to describe cracks which form alongelement boundaries, show branching andcoalescence, and eventually lead to the frag-mentation of the specimen. The cohesivemodeling approach provides the chance of

predicting fragmentation patterns and crackpropagation velocities based on basic prin-ciples of material separation.

3.03.5.5.4 Rate- and time-dependent fracture

Both rate and time dependence are related tothe deformation and separation resistance ofthe material during either rapid loading orstress relaxation. Note that rate dependence isless important for metals, but plays a majorrole for polymers. The constitutive formula-tions for rate-dependent cohesive models arepresented in Section 3.03.5.2.6.

A thermodynamically based formulationwas presented by Costanzo and Allen (1996)and Costanzo and Walton (2002). Tijssens(2000) developed a new approach for modelingrate-dependent phenomena like craze forma-tion in polymers, based on micromechanicallybased constitutive relations (see Section3.03.5.3.3). That the rate effects may realisti-cally be predicted by this model has been

Figure 94 Fragmentation of a ceramic microstruc-ture subject to dynamic loading: (a) FE mesh andgrain boundaries of the periodic cell, (b) crackformation after 100 ns, and (c) fragmentation after1,000 ns (source Zavattieri et al., 2001).

Figure 93 Fragmentation of a glass rod subjected to impact loading. Snapshot after 14.7 ms (source Repettoet al., 2000).


confirmed by experiments. Results for adhesivepolymer layers have been presented by Rahul-Kumar et al. (1999, 2000). Several differentexperiments have been proposed for thedetermination of the respective material para-meters, for instance the peel test described inSection 3.03.5.4.3, used by Yang et al. (1999),or the compressive shear test, used by Rahul-Kumar et al. (1999). Note that such tests canonly be used for specific applications.

Li and Bazant (1997) introduced a nonlinearrate-dependent cohesive law of the Kelvin–Voigt type for concrete. The bulk material wasassumed to behave as linear viscoelasticincluding aging effects. Rate dependence wasconsidered in a simple way by varying thesoftening law over time, the latter beingcontrolled by the actual crack length recordedduring the test. The integration approach isproposed as being more accurate and numeri-cally efficient.

3.03.5.5.5 Fatigue crack growth

Current predictions of fatigue life are basedon phenomenological laws, relating the ampli-tude of the applied stress-intensity factor, DK,to the crack-growth rate, da/dN, like the well-known Paris law, da/dNp(DK)m, with m beinga material parameter. This equation describesthe fatigue crack growth under SSY conditionsand constant amplitude loading. A furtherrestriction is its applicability to so-called longcracks only. Within these limitations, it is veryuseful tool for engineering assessments. Itsapplication to life-time assessments becomes,however, questionable outside that frame.

Here, cohesive modeling can provide analternative approach, which is in general notrestricted to size and geometry requirements,crack lengths, or loading conditions. Note,however, that linear un- and reloading de-scribed in Section 3.03.5.2.5 (i.e., un- andreloading follow the same path) is not capableto model crack growth under cyclic loading.The consequence of doing so would be that fora structure, subjected to constant amplitudecyclic loading, shake down and crack arrestwould be predicted, as shown by Yang andRavi-Chandar (1998).

DeAndres et al. (1999) applied the cohesivemodel using an exponential cohesive law and alinear unloading–reloading option, see thebottom-left part of Figure 31, to predict theshape and extension of the crack front in asurface-notched round bar under high-cyclefatigue loading, modeled with 3D FEs. Thecohesive law they used for fatigue was the sameas for quasi-static monotonically increasingload. The development of separation in a

cohesive element was calculated for a fewloading cycles and then extrapolated to alarger number of cycles by

Dnþ1 ¼ Dn þ@D

@N

��n

Nnþ1 � Nnð Þ ð129Þ

where Nn is the number of cycles at the end ofthe FE calculation and Nnþ 1 is the number ofcycles, for which the damage parameter D hadto be determined. The cycle incrementNnþ 1�Nn used in Equation (129) was selectedsuch that the damage increment Dnþ 1�Dn wassufficiently small. By repeated extrapolationand determination of the damage rate @D=@N;more than 3� 105 cycles were simulated. Thesimulation was in good agreement with experi-ments carried out on the round bar. Irrespec-tive of the progress made by this cyclic model,a continuous damage evolution law was notprovided. The limits with respect of a generalpredictability of fatigue life remain uncertain.

A cohesive law with an unloading–reloadinghysteresis behavior was introduced by Yanget al. (2001b) and Nguyen et al. (2001). In bothcases the bulk material was described by the J2-theory of plasticity with rate-dependent kine-matic hardening. Linear unloading combinedwith nonlinear reloading made it possible totake dissipative mechanisms into account suchas frictional interactions between asperities aswell as crystallographic slip. As these physicaleffects are described phenomenologically, therespective parameters have to be fitted toexperimental results. Figure 95 illustrates thebehavior of the model. Material degradationcan accumulate below the limiting curve of thecohesive law for monotonic loading (the ‘‘da-mage locus’’ or ‘‘envelope’’) prior to failure dueto the fact that unloading and reloading do notfollow the same path, but show a hysteresis andthus dissipated energy. The authors combinedlinear unloading with quadratic reloading. Theapplied (global) loading range controlled the(local) upper and lower loading levels in thecohesive elements. Yang et al. (2001b) analyzedan SE(T) specimen under modes I and IIsubjected to a few thousand cycles by thedual-boundary element method. Nguyen et al.(2001) modeled an M(T) specimen under cyclicmode-I loading with up to 6,000 load cycles andcompared the results with experiments in theParis regime. In addition, they investigated theeffect of short cracks and overload.

Roe (2001) (see also Roe and Siegmund,2003) proposed a different approach, in whichthe traction–separation path within a cohesiveelement always followed the original cohesivelaw as given by Equation (83), and cyclicdamage evolution was accounted for by a


decreasing maximum cohesive traction accord-ing to

T0;fatigue ¼ T0 1� Daccð Þ ð130Þ

Here Dacc is accumulated damage, whichaccounts for the separation history. It isderived by an evolution law including athreshold value, below which no fatiguedamage can occur. Different to the unload-ing–reloading hysteresis as described above,the unloading and reloading paths wereassumed to be identical. The unloading stiff-ness was chosen to be equal to the stiffness ofthe cohesive law at zero separation, @T=@djd¼0:For decreasing maximum cohesive strength(Equation (130)), the unloading stiffness@T=@djd¼0 was also decreasing during damageaccumulation.

A new microstructurally based approach tofatigue crack propagation has been developedby Deshpande et al. (2001) using discretedislocation dynamics in the bulk material incombination with a cohesive model. Near thethreshold, the plastic deformation is confinedto a small volume around the propagatingcrack tip. Yielding is modeled by the motion ofedge dislocations in an elastic solid taking intoaccount constitutive laws for lattice resistanceto dislocation motion, dislocation nucleation,interaction with obstacles and annihilation. Aconventional cohesive law for irreversible

separation is introduced at the crack tip, whichincludes linear unloading and reloading (seeFigure 96). Thus, continuous crack growthunder cyclic loading is exclusively triggered bythe irreversible motion of dislocations, which ismodeled within the continuum elements, notwithin the cohesive elements.

Furthermore, retarded crack growth due toan overload cycle was well simulated by theapproach. The cyclic crack-growth rate, da/dN,vs. the applied stress-intensity factor range,DK, curve emerged implicitly from the bound-ary value problem. Deshpande et al. (2002)extended this approach from the near thresh-old to the Paris law regime.

3.03.5.6 Summary and Outlook

The cohesive model can be regarded as aflexible, versatile, and robust tool for computa-tional simulations of damage localization, ma-terial separation up to structural failure. Due toits phenomenological character, the model isadjustable to many different types of materialsand failure phenomena; it can be applied toanalyses of macroscopic engineering structuresas well as heterogeneous microstructures.

3.03.5.6.1 Specific advantages

The cohesive relation can be regarded as amaterial law, and the respective parameters

Figure 95 Cohesive law with accumulating damage under cyclic loading: the unloading–reloading hysteresisloops accounting for material degradation are enclosed by the limiting curve for monotonic loading.


characterize material properties with respect todamage and fracture independent of a specificgeometry.

There are no problems, in principle, withtransferring the fracture parameters from smallspecimens to large components as in theclassical macroscopic fracture mechanics ap-proach.

By entrusting the nucleation, propagation,branching, and other aspects of the fracturebehavior of materials to a master cohesive law,the amount of phenomenology is considerablyreduced compared to theories of distributeddamage.

Cohesive laws can be established for variousseparation phenomena and can also be ex-tended to time-dependent material behavior.

Separation processes, damage and fractureon different length scales can be simulated.

Cohesive models endow materials with acharacteristic length, and unlike damage the-ories, a cohesive law introduces a well-definedfracture energy, which eliminates mesh depen-dence, so that FE solutions attain properconvergence in the limit of vanishing meshsize.

3.03.5.6.2 Future research issues

The phenomenological nature of the modelis an advantage with respect to its versatilitybut may raise some uncertainties with respectto the physics of the underlying processes.Advanced test and measuring techniques arehence required for a closer comparison be-tween ‘‘reality’’ and model simulations. Somefuture research issues are as follows.

The separation laws used so far are basedupon the maximum principal stress or themaximum shear stress. Interaction of normaland shear separation is treated in a more or lessempirical way and multiaxial effects in theprocess zone affecting the cohesive law are notyet established with sufficient evidence.

Local separation processes are describedphenomenologically only; relations to micro-structural quantities need further enlighten-ment.

Simulation of arbitrary crack propagation isfar from being an established method. Re-quirements on the mesh design are of greatconcern and, in particular, more experience isneeded for 3D crack-growth simulations. Newelement formulations including displacementdiscontinuities are very promising for modelingarbitrary crack paths with reduced numericalefforts.

Modeling of fatigue crack growth withcohesive elements is a challenging applicationfield. Only few fundamental investigations areavailable at the moment.

New experimental test methods, e.g., in situtechniques, and instrumentations are requiredespecially for determining microstructuralproperties and for the calibration of numericalanalyses.

Enhanced computational equipment is re-quired, e.g., parallel processing for the largenumbers of elements and nodes necessary foradvanced investigation of micro- and macro-structures.

The present applications of cohesive modelsare still far away from practical engineeringemployment in structural integrity assessments.There is a strong need to standardize thesimulation techniques and the determination ofthe model parameters.

3.03.5.6.3 Challenges

The approach of cohesive modeling providesthe basis for both scientific investigations andthe solution of engineering problems. Amongmany possibilities, some challenges will be thefollowing.

Advanced hybrid or smart materials can bedesigned in detail by computer simulations and

KI

rigid substrate

3 slip systems

15 x 15 µm area withedge dislocations onslip lines

elastic, isotropic

crack

500 x 1000 µm

(a)

a

T, δ

cycles

KI,max

KI,min

KIcohesive lawfor monotonicloading

TN

TN,0

δΝ

00

(b)

cohesive elements

Figure 96 Modeling of fatigue crack growth bydiscrete dislocation dynamics combined with acohesive surface (source Deshpande et al., 2001):(a) boundary layer model with a crack along aninterface between a metal and a rigid substrate and(b) exponential cohesive law with linear unloading–reloading according to Equation (93).


their performance can be studied under variousoperating conditions.

Qualification of new materials and alloys forengineering applications can be supported bynumerical simulations, thus reducing time andcosts for marketing new products.

Transferability of material parameters fromsmall-sized test specimens to arbitrary struc-tures is of great benefit for safety and lifetimeassessments. The safety margins between ser-vice loading and failure scenarios can bequantified more reliably, avoiding expensivelarge component tests.

3.03.6 REFERENCES

ABAQUS, Standard Version 6.1, User’s Manual, 2000,vol. I, chap. 7.8.2.

A. Abdul-Baqi and E. van der Giessen, 2002, Numericalanalysis of indentation-induced cracking of brittle coat-ings on ductile substrates. Int. J. Solids Struct., 39, 1427–1442.

D. H. Allen and C. R. Searcy, 2001, A micromechanicalmodel for a viscoelastic cohesive zone. Int. J. Fract., 107,159–176.

D. H. Allen, R. H. Jones and J. G. Boyd, 1994, Micro-mechanical analysis of continuous fiber metal matrixcomposite including the effects of matrix visco-plasticityand evoling damage. J. Mech. Phys. Solids, 42, 505–529.

M. Amestoy, H. D. Bui and R. Labbens, 1981, On thedefinition of local path independent integrals in 3D crackproblems. Mech. Res. Commun., 8, 231–236.

J. J. M. Arata, K. S. Kumar, W. A. Curtin and A.Needleman, 2001, Crack growth in lamellar titaniumaluminide. Int. J. Fract., 111, 163–189.

J. J. M. Arata and A. Needleman, 1998, The effect ofplasticity on dynamic crack growth across an interface.Int. J. Fract., 94, 383–399.

ASTM E 1290, 1999, Standard test method for crack tipopening displacement (CTOD) fracture toughness mea-surement. In: ‘‘Annual Book of ASTM Standards,’’American Society for Testing and Materials, vol. 03.01.West Conshohocken, PA.

ASTM E 1737, 1996, Standard test method for J-integralcharacterization of fracture toughness. In: ‘‘AnnualBook of ASTM Standards,’’ American Society forTesting and Materials, vol. 03.01 West Conshohocken,PA.

A. G. Atkins, Z. Chen and B. Cotterell, 1998, The essentialwork of fracture and JR curves for the double cantileverbeam specimen: an examination of elastoplastic crackpropagation for the double cantilever beam specimen.Proc. Roy. Soc., A, 454, 815–833.

S. N. Atluri, T. Nishioka and M. Nakagaki, 1984,Incremental path-independent integrals in inelastic anddynamic fracture mechanics. Eng. Fract. Mech., 20, 209–244.

A. Bakker, 1984, ‘‘The Three-dimensional J-integral,’’Report WTHD 167, Delft University of Technology,The Netherlands.

G. I. Barenblatt, 1962, The mathematical theory ofequilibrium cracks in brittle fracture. Adv. Appl. Mech.,7, 55–129.

R. S. Barsoum, 1977, Triangular quarterpoint elements aselastic and perfectly-plastic crack tip elements. Int. J.Numer. Meth. Eng., 11, 85–98.

K.-J. Bathe, 1996, ‘‘Finite element procedures,’’ Prentice-Hall, New York.

Z. P. Bazant, 1993, Current status and advances in thetheory of creep and interaction with fracture. In: ‘‘Proc.

5th Int. RILEM Symposium on Creep and Shrinkage ofConcrete,’’ eds. Z. P. Bazant and I. Carol, E & FN Spon,London and New York, pp. 291–307.

Z. P. Bazant, 2002, Concrete fracture models: testing andpractice. Eng. Fract. Mech., 69, 165–205.

Z. P. Bazant and J. Planas, 1998, ‘‘Fracture and Size Effectin Concrete and Other Quasibrittle materials,’’ CRCPress, Boca Raton.

T. Belytschko, N. Moes, S. Usui and C. Parimi, 2001,Arbitrary discontinuities in finite elements. Int. J. Numer.Meth. Eng., 50, 993–1013.

F. M. Beremin, 1981, Study of the fracture criteria forductile rupture of A 508 steel. In: ‘‘5th Int. Conf. onFracture: Advances in Fracture Research,’’ ed. D. Fran-

-cois, Pergamon, Oxford and New York, pp. 809–816.F. M. Beremin, 1983, A local criterion for cleavage fracture

of a nuclear pressure vessel steel. Met. Trans. A, 14A,2277–2287.

G. Bernauer and W. Brocks, 2002, Micro-mechanicalmodelling of ductile damage and tearing—results of aEuropean numerical round robin. Fatigue Fract. Eng.Mater. Struct., 25, 363–384.

G. Bernauer, W. Brocks and W. O. Schmitt, 1999,Modifications of the Beremin model for cleavagefracture in the transition region of a ferritic steel. Eng.Fract. Mech., 64, 305–325.

S. R. Bodner and Y. Partom, 1975, Constitutive equationsfor elastic–viscoplastic strainhardening materials. J.Appl. Mech., 42, 385–389.

K. B. Broberg, 1997, The cell model of materials. Comput.Mech., 19, 447–452.

W. Brocks, A. Eberle, S. Fricke and H. Veith, 1994, Largestable crack growth in fracture mechanics specimens.Nucl. Eng. Des., 151, 387–400.

W. Brocks, D. Klingbeil, G. Kunecke and D.-Z. Sun,1995a, Application of the Gurson model to ductiletearing resistance. In: ‘‘Constraint Effects in FractureTheory and Applications: Second volume, SecondSymposium on Constraint Effects, ASTM STP 1244,’’eds. M. Kirk and A. Bakker, American Society forTesting and Materials, Philadelphia, pp. 232–252.

W. Brocks, D. Klingbeil and J. Olschewski, 1990, Losungder HRR-Feldgleichungen der elastisch-plastischenBruchmechanik, Forschungsbericht 137, Bundesanstaltfur Materialforschung und-prufung, Berlin.

W. Brocks, G. Kunecke, H.-D. Noack and H. Veith,1989a, On the transferability of fracture mechanicsparameters from specimens to structures using FEM.Nucl. Eng. Des., 112, 1–14.

W. Brocks, G. Kunecke and K. Wobst, 1989b, Stable crackgrowth of axial surface flaws in pressure vessels. Int. J.Press. Vessels Pip., 40, 77–90.

W. Brocks, W. Muller, H.-D. Noack and H. Veith, 1993,Fracture mechanics investigations on a pipe with acircumferential flaw supported by FEM. Nucl. Eng. Des.,143, 171–185.

W. Brocks, W. Muller and J. Olschewski, 1985, Experi-ences in applying ADINA to the analysis of crack tipfields in elastic-–plastic fracture mechanics. Comput.Struct., 21, 137–158.

W. Brocks and J. Olschewski, 1986, On J-dominance ofcrack-tip fields in largely yielded 3D structures. Int. J.Solids Struct., 22, 693–708.

W. Brocks and W. Schmitt, 1993, Quantitative assessmentof the role of crack tip constraint on ductile tearing. In:‘‘Constraint Effects in Fracture, ASTM STP 1171,’’ eds.E. M. Hackett, K.-H. Schwalbe and R. H. Dodds,American Society for Testing and Materials, Philadel-phia, pp. 64–78.

W. Brocks and W. Schmitt, 1995, The second parameter inJ–R curves, constraint or triaxiality? In: ‘‘ConstraintEffects in Fracture Theory and Applications: Secondvolume, Second Symposium on Constraint Effects,

References 203

ASTM STP 1244,’’ eds. M. Kirk and A. Bakker,American Society for Testing and Materials, Philadel-phia, pp. 209–231.

W. Brocks, D.-Z. Sun and A. Honig, 1995b, Verification ofthe transferability of micromechanical parameters by cellmodel calculations for visco-plastic materials. Int. J.Plast., 11, 971–989.

W. Brocks and H. Yuan, 1989, Numerical investigations onthe significance of J for large stable crack growth. Eng.Fract. Mech., 32, 459–468.

F. W. Brust, T. Nishioka, S. N. Atluri and M. Nakagaki,1985, Further studies on elastic-plastic stable fractureutilizing the T*-Integral. Eng. Fract. Mech., 22, 1079–1103.

BS 7448, Part 1, 1991, Fracture mechanics toughness tests.Methods for determination of KIc, critical CTOD andcritical J values of metallic materials, British StandardsInstitution, London.

E. Budiansky and J. R. Rice, 1973, Conservation laws andenergy release rates. J. Appl. Mech., 40, 201–203.

F. M. Burdekin and D. E. W. Stone, 1966, The crackopening displacement approach to fracture mechanics inyielding materials. J. Strain Anal., 1, 145–153.

G. T. Camacho and M. Ortiz, 1996, Computationalmodelling of impact damage in brittle materials. Int. J.Solids Struct., 33(20–22), 2899–2938.

A. Carpinteri, 1986, ‘‘Mechanical damage and crackgrowth in concrete,’’ Martinus Nijhoff–Kluwer, Dor-drecht–Boston.

D. A. Cendon, J. C. Galvez, M. Elices and J. Planas, 2000,Modelling the fracture of concrete under mixed loading.Int. J. Fract., 103, 293–310.

O. Chabanet, D. Steglich, J. Besson, V. Heitmann, D.Hellmann and W. Brocks, 2003, Predicting crack growthresistance of aluminium sheets. Comput. Mater. Sci., 26,1–246.

O. Chabanet, D. Steglich and W. Brocks, 2001, Beschrei-bung des Risswiderstandsverhaltens von Aluminiumble-chen durch Schadigungsmodelle. In: ‘‘DVM-Bericht233,’’ ed. H. A. Richard, Deutscher Verband furMaterialforschung und -prufung e.V, Berlin, pp. 87–95.

J. L. Chaboche, 1993, Cyclic viscoplastic constitutiveequations: Part I. Thermodynamically consistent for-mulation. J. Appl. Mech., 60, 813–821.

J. Chaboche, R. Girard and P. Levasseaur, 1997, On theinterface debonding models. Int. J. Damage Mech., 6,220–257.

J. L. Chaboche and G. Rousselier, 1983, On the plastic andviscoplastic constitutive equations: Part I. Rules devel-oped with internal variable concept. J. Press. VesselTechnol., 105, 153–158.

C. Chen, N. A. Fleck and T. J. Lu, 2001, The mode I crackgrowth resistance of metallic foams. J. Mech. Phys.Solids, 49, 231–259.

G. P. Cherepanov, 1967, Crack propagation in continuousmedia. J. Appl. Math. Mech., 31, 476–488.

C. C. Chu and A. Needleman, 1980, Void nucleation effectsin biaxially stretched sheets. J. Eng. Mater. Technol.,102, 249–256.

A. Corigliano and M. Ricci, 2001, Rate-dependent inter-face models: formulation and numerical applications.Int. J. Solids Struct., 38, 547–576.

A. Corigliano, M. Ricci and R. Contro, 1997, Rate-dependent delamination in polymer–matrix composites.In: ‘‘Proceedings of the Fifth International Conferenceon Computational Plasticity,’’ eds. D. R. J. Owen, E.Onate and E. Hinton, CIMNE, Barcelona, pp. 1168–1175.

A. Cornec, G. Lin and K.-H. Schwalbe, 1998, Simulationvon Ri�widerstandskurven mit dem Kohasivmodell:large scale yielding (Simulation of crack resistance usinga cohesive model: large scale yielding). Mat.-wiss. u.Werkstofftechnik, 29, 652–661.

A. Cornec and I. Scheider, 2001, Failure assessment oflaser weldments based on numerical modelling. Mat.-wiss. u. Werkstofftech., 32, 316–328.

A. Cornec, W. Schonfeld and U. Zerbst, 1999, FiniteElement Analyse fur ingenieurma�ige Fehlerbewertungs-verfahren: Verifizierung am Rohrknoten fur den ESISTC1.3 Round Robin. GKSS Report 99/E/SS, GKSSResearch Centre Publication, GKSS Research Centre,Geesthacht, Germany.

F. Costanzo and D. H. Allen, 1995, A continuumthermodynamic analysis of cohesive zone models. Int.J. Eng. Sci., 33, 2197–2219.

F. Costanzo and J. R. Walton, 1997, A study of dynamiccrack growth in elastic materials using a cohesive zonemodel. Int. J. Eng. Sci., 35, 1085–1114.

F. Costanzo and J. R. Walton, 2002, Steady growth of acrack with a rate and temperature sensitive cohesivezone. J. Mech. Phys. Solids, 50, 1649–1679.

R. H. Dauskardt, M. Lane, Q. Ma and N. Krishna, 1998,Adhesion and debonding of multi-layer thin filmstructures. Eng. Fract. Mech., 61, 141–162.

M. G. Dawes, 1985, ‘‘The CTOD Design Curve Approach:Limitations, Finite Size and Application,’’ TWI report278/1985, The Welding Institute, Cambridge.

A. deAndres, J. L. Perez and M. Ortiz, 1999, Elastoplasticfinite element analysis of three-dimensional fatigue crackgrowth in aluminium shafts subjected to axial loading.Int. J. Solids Struct., 36, 2231–2258.

R. deBorst, 2001, Some recent issues in computationalfailure mechanics. Int. J. Numer. Meth. Eng., 52, 63–95.

H. G. deLorenzi, 1982a, On the energy release rate and theJ-integral for 3D crack configurations. Int. J. Fract., 19,183–193.

H. G. deLorenzi, 1982b, ‘‘Energy Release Rate Calcula-tions by the Finite Element Method,’’ General ElectricTechnical Infomation Series, Report No. 82CRD205.

V. S. Deshpande, A. Needleman and E. van der Giessen,2001, A discrete dislocation analysis of near-thresholdfatigue crack growth. Acta Mater., 49, 3189–3203.

V. S. Deshpande, A. Needleman and E. van der Giessen,2002, Discrete dislocation modeling of fatigue crackpropagation. Acta Mater., 50, 831–846.

D. C. Drucker, 1964, On the postulate of stability ofmaterial in the mechanics of continua. J. de Mecanique,3, 235–249.

D. S. Dugdale, 1960, Yielding of steel sheets containingslits. J. Mech. Phys. Solids, 8, 100–104.

M. Elices and J. Planas, 1996, Fracture mechanicsparameters of concrete. Adv. Cem. Bas. Mat., 4, 116–127.

M. Elices, G. V. Guinea, J. Gomez and J. Planas, 2002, Thecohesive zone model: advantages, limitations and chal-lenges. Eng. Fract. Mech., 69, 137–163.

M. Elices, J. Planas and G. V. Guinea, 1993, Modellingcrack in rocks and cementitious materials. In: ‘‘Fractureand Damage of Concrete and Rock,’’ ed. H. P.Rossmanith, E&FN Spon, London, pp. 3–33.

M. Elices, J. Planas and G. V. Guinea, 2000, Fracturemechanics applied to concrete. In: ‘‘Fracture Mechanics,Applications and Challenges,’’ eds. M. Fuentes, M.Elices, A. Matin-Meizoso and J. M. Martinez-Esnaola,ESIS Publication 26, Elsevier, Amsterdam, pp. 183–210.

ESIS P6-98, 1998, ‘‘Procedure to Measure and CalculateMaterial Parameters for the Local Approach to Fractureusing Notch Tensile Specimens,’’ European StructuralIntegrity.

R. Estevez, M. G. A. Tijssens and E. Van der Giessen,2000, Modelling of the competition between shearyielding and crazing in glassy polymers. J. Mech. Phys.Solids, 48(12), 2585–2617.

J. Faleskog, X. Gao and C. F. Shih, 1998, Cell model fornonlinear fracture analysis: I. Micromechanics calibra-tion. Int. J. Fract., 89, 355–373.


N. A. Fleck, O. B. Olurin, C. Chen and M. F. Ashby, 2001,The effect of hole size upon the strength of metallic andpolymeric foams. J. Mech. Phys. Solids, 49, 2015–2030.

M. S. Gadala, M. A. Dokainish and G. A. Oravas, 1980,Geometric and material nonlinearity problems—Lagran-gian and updated Lagrangian formulations. In: ‘‘Proc.2nd Int. Conf. Numerical Methods in Fracture Me-chanics,’’ eds. D. R. L. Owen and A. R. Luxmoore,Pineridge Press, Swansea, pp. 277–293.

J. C. Galvez, M. Elices, G. V. Guinea and J. Planas, 1998,Mixed mode fracture of concrete under proportional andnonproportional loading. Int. J. Fract., 94, 267–284.

X. Gao, J. Faleskog and C. F. Shih, 1998, Cell model fornonlinear fracture analysis: II. Fracture-process calibra-tion and verification. Int. J. Fract., 89, 375–398.

S. J. Garwood, 1979, Effect of specimen geometry on crackgrowth resistance. In: ‘‘Fracture Mechanics, 11th Conf.,ASTM STP 677,’’ ed. C. W. Smith, American Society forTesting and Materials, Philadelphia, pp. 511–532.

S. J. Garwood, J. N. Robinson and C. E. Turner, 1975, Themeasurement of crack growth resistance curves using theJ-integral. Int. J. Fract., 11, 528–530.

S. Ghosh, Y. Ling, B. Majumdar and R. Kim, 2000,Interfacial debonding analysis in multiple fiber rein-forced composites. Mech. Mater., 32, 561–591.

A. A. Griffith, 1920, The phenomena of rupture and flow insolids. Phil. Trans. Roy. Soc. London A, 211, 163–198.

G. V. Guinea, M. Elices and J. Planas, 1997, On the initialshape of the softening function of cohesive materials. Int.J. Fract., 87, 139–149.

G. V. Guinea, M. Elices and J. Planas, 2000, Assessment ofthe tensile strength through size effect curves. Eng. Fract.Mech., 65, 189–207.

G. V. Guinea, J. Planas and M. Elices, 1994, A generalbilinear fit for the softening curve of the concrete. Mater.Struct., 27, 99–105.

A. S. Gullerud, R. H. Dodds, R. W. Hampton and D. S.Dawicke, 1999, Three-dimensional modeling of ductilecrack growth in thin sheet metals: computational aspectsand validation. Eng. Fract. Mech., 63, 347–374.

A. L. Gurson, 1977, Continuum theory of ductile ruptureby void nucleation and growth: Part I. Yield criteria andflow rules for pourous ductile media. J. Eng. Mater.Technol., 99, 2–15.

Z. Hashin, 1964, Theory of mechanical behaviour ofheterogeneous media. Appl. Mech. Rev., 17, 1–9.

D. Hellmann and K.-H. Schwalbe, 1984, Geometry andsize-effects on J–R and d–R curves under plane stressconditions. In: ‘‘Fracture Mechanics, 15th Symp.,ASTM STP 833,’’ ed. R. J. Sanford, American Societyfor Testing and Materials, Philadelphia, pp. 577–605.

H. Hencky, 1924, Zur Theorie plastischer Deformationenund der hierdurch im Material hervorgerufenen Nach-spannungen. Z. angew. Math. Mech., 4, 323–334.

R. Hill, 1963, Elastic properties of reinforced solids: sometheoretical principles. J. Mech. Phys. Solids, 11, 349–354.

A. Hillerborg, M. Modeer and P. E. Petersson, 1976,Analysis of crack formation and crack growth inconcrete by means of fracture mechanics and finiteelements. Cement Concrete Res., 6, 773–782.

J. W. Hutchinson, 1968a, Singular behaviour at the end ofa tensile crack in a hardening material. J. Mech. Phys.Solids, 16, 13–31.

J. W. Hutchinson, 1968b, Plastic stress and strain fields at acrack tip. J. Mech. Phys. Solids, 16, 337–347.

J. W. Hutchinson and A. G. Evans, 2000, Mechanics ofmaterials: top–down approaches to fracture. ActaMater., 48, 125–135.

G. R. Irwin, 1957, Analysis of stresses and strains near theend of a crack traversing a plate. J. Appl. Mech., 24,361–364.

H. Ismar, F. Schroter and F. Streicher, 2001, Effects ofinterfacial debonding on the transverse loading beha-

viour of continuous fibre-reinforced metal matrix com-posites. Comput. Struct., 79, 1713–1722.

L. M. Kachanov, 1993, ‘‘Introduction to ContinuumDamage Mechanics,’’ Kluwer Academic, Dordrecht.

A. Khalili and K. Kromp, 1991, Statistical properties ofWeibull estimators. J. Mater. Sci., 26, 6741–6752.

M. Kikuchi and H. Miyamoto, 1982, Evaluation of Jkintegrals for a crack in multiphase materials. ‘‘RecentResearch on Mechanical Behavior of Materials, Bulletinof Fracture Mechanics Laboratory,’’ Science Universityof Tokyo, Tokyo, vol. 1.

M. Kikuchi, H. Miyamoto and Y. Sakaguchi, 1979,Evaluation of three-dimensional J-integral of semi-elliptical surface crack in pressure vessel. In: ‘‘Trans.5th Int. Conf. Structural Mechanics in Reactor Technol-ogy (5th SMiRT),’’ paper G7/2, Berlin.

W. G. Knauss, 1993, Time dependent fracture and cohesivezones. J. Eng. Mater. Technol., 115, 262–267.

O. Kolednik, G. Shan and F. D. Fischer, 1997, The energydissipation rate—a new tool to interpret geometry andsize effects. In: ‘‘Fatigue and Fracture Mechanics,ASTM STP 1296,’’ eds. R. S. Piascik, J. C. Newmanand N. E. Dowling, American Society for Testing andMaterials, Philadelphia, vol. 27, pp. 126–151.

R. Kolhe, S. Tang, C.-Y. Hui and A. T. Zehnder, 1999,Cohesive properties of nickel–alumina interfaces deter-mined via simulation of ductile bridging experiments.Int. J. Solids Struct., 36, 5573–5595.

J. Koplik and A. Needleman, 1988, Void growth andcoalescence in porous plastic solids. Int. J. Solids Struct.,24, 835–853.

J. D. Landes and J. A. Begley, 1976, A fracture mechanicsapproach to creep crack growth. In: ‘‘Mechanics ofCrack Growth, ASTM STP 590,’’ eds. Landes andBegley, J. R. Rice and P. C. Paris, American Society forTesting and Materials, Philadelphia, pp. 128–148.

R. Larsson, K. Runesson and S. Sature, 1996, Embeddedlocalization band in undrained soil based on regularizedstrong discontinuity theory and FE analysis. Int. J.Solids Struct., 33, 3081–3101.

V. La Saponara, H. Muliana, R. Haj-Ali and G. A.Kardomateas, 2002, Experimental and numerical analy-sis of delamination growth in double cantilever lami-nated beams. Eng. Fract. Mech., 69, 687–699.

J. A. Lemaitre, 1985, Continuous damage mechanicsmodel for ductile fracture. J. Eng. Mater. Technol.,107, 83–89.

V. C. Li, 1998, Engineering cementitious composites(ECC)—tailored composites through micromechanicalmodeling. In: ‘‘Fibre Reinforced Concrete: Present andFuture,’’ eds. N. Bathia, et al., Canadian Society of CivilEngineering, Montreal, pp. 64–98.

Y.-N. Li and Z. Bazant, 1997, Cohesive crack model withrate-dependent opening and viscoelasticity: II. Numer-ical algorithm, behaviour and size effect. Int. J. Fract.,86, 267–288.

W. Li and Th. Siegmund, 2002, An analysis of crackgrowth in thin-sheet metal via a cohesive zone model.Eng. Fract. Mech., 69, 2037–2093.

Y.-M. Liang and K. H. Liechti, 1995, Tougheningmechanisms in mixed-mode interfacial fracture. Int. J.Solids Struct., 32, 957–978.

K. M. Liechti and J.-D. Wu, 2001, Mixed-mode, time-dependent rubber–metal debonding. J. Mech. Phys.Solids, 49, 1039–1072.

G. Lin, 1998, Numerical investigation of crack growthbehaviour using a cohesive zone model. Ph.D. thesis,University of Hamburg-Harburg, Report GKSS 98/E/8,GKSS Publication Centre, GKSS Research Centre,Germany.

G. Lin and A. Cornec, 1998, Characterization of crackresistance: simulation with a new cohesive model(Numerische Untersuchung zum Verhalten von

References 205

Ri�widerstandskurven: Simulation mit dem kohasivmo-dell). Mat.-wiss. u. Werkstofftechnik, 27, 252–258.

G. Lin, P. H. Geubelle and N. R. Sottos, 2001, Simulationof fibre debonding with friction in a model compositepushout test. Int. J. Solids Struct., 38, 8547–8562.

G. Lin, Y.-J. Kim, A. Cornec and K.-H. Schwalbe, 1997,Fracture toughness of a constrained metal layer.Comput. Mater. Sci., 9, 36–47.

G. Lin, Y.-J. Kim, A. Cornec and K.-H. Schwalbe, 1998a,Numerical analysis of ductile failure of undermatchedinterleaf in tension. Int. J. Fract., 91, 323–347.

G. Lin, A. Cornec and K.-H. Schwalbe, 1998b, Three-dimensional finite element simulation of crack extensionin aluminium alloy 2024-FC. Fatigue Fract. Eng. Mater.Struct., 21, 1159–1173.

G. Lin, X.-G. Meng, A. Cornec and K.-H. Schwalbe, 1999,The effect of strength mis-match on mechanical perfor-mance of weld joints. Int. J. Fracture, 96, 37–54.

R. E. Link, J. D. Landes, R. Herrera and Z. Zhou, 1991,Something new on size and constraint effects for J–Rcurves. In: ‘‘Defect assessment in Components, Funda-mentals and Applications, ESIS/EGF 9,’’ eds. J. G.Blauel and K.-H. Schwalbe, Mechanical EngineeringPublications, London, pp. 707–721.

C. J. Lissenden and C. T. Herakovich, 1995, Numericalmodelling of damage development and viscoplasticity inmetal matrix composites. Comput. Meth. Appl. Mech.Eng., 126, 289–303.

D. C. Lo and D. H. Allen, 1994, Modelling of delaminationdamage evolution in laminated composites subjected tolow velocity impact. Int. J. Damage Mech., 3, 378–407.

K. S. Madhusudhana and R. Narasimhan, 2002, Experi-mental and numerical investigations of mixed modecrack growth resistance of a ductile adhesive joint. Eng.Fract. Mech., 69, 865–883.

R. M. McMeeking, 1977, Path dependence of the J-integraland the role of J as a parameter characterizing the neartip field. In: ‘‘Flaw Growth and Fracture (10th Conf.),ASTM STP 631,’’ ed. J. M. Barsom, symposiumchairman American Society for Testing and Materials,Philadelphia, pp. 28–41.

R. M. McMeeking and D. M. Parks, 1979, On criteria forJ-dominance of crack-tip fields in large scale yielding. In:‘‘Elastic–Plastic Fracture, ASTM STP 668,’’ eds. J. D.Landes, J. A. Begley and G. A. Clarke, American Societyfor Testing and Materials, Philadelphia, pp. 175–194.

R. M. McMeeking and J. R. Rice, 1975, Finite-elementformulations for problems of large elastic–plastic defor-mation. Int. J. Solids Struct., 11, 601–616.

J. M. Melenk and I. Babuska, 1996, The partition of unityfinite element method: basic theory and applications.Comput. Meth. Appl. Mech. Eng., 139, 289–314.

D. Memhard, W. Brocks and S. Fricke, 1993, Character-ization of ductile tearing resistance by energy dissipationrate. Fatigue Fract. Eng. Mater. Struct., 16, 1109–1124.

N. Moes and T. Belytschko, 2002, Extended finite elementmethod for cohesive crack growth. Eng. Fract. Mech.,69, 813–833.

N. Moes, J. Dolbow and T. Belytschko, 1999, A finiteelement method for crack growth without remeshing.Int. J. Numer. Meth. Eng., 46, 131–150.

I. Mohammed and K. M. Liechti, 1998, ‘‘Cohesive ZoneModeling of Crack Nucleation at Bimaterial Corners,’’Report EMRL 98-20, Research Centre for Mechanics ofSolids, Structures and Materials, University of Texas atAustin.

Z. Mroz, 1967, On the description of anisotropic workhardening. J. Mech. Phys. Solids, 15, 163–175.

F. Mudry, 1987, A local approach to cleavage fracture.Nucl. Eng. Des., 105, 65–76.

A. Muscati and D. J. Lee, 1984, Elastic–plastic finiteelement analysis of thermally loaded cracked structures.Int. J. Fract., 25, 227–246.

A. Needleman, 1987, A continuum model for voidnucleation by inclusion debonding. J. Appl. Mech., 54,525–531.

A. Needleman, 1990a, An analysis of tensile decohesionalong an interface. J. Mech. Phys. Solids, 38, 289–324.

A. Needleman, 1990b, An analysis of decohesion along animperfect interface. Int. J. Fract., 42, 21–40.

A. Needleman, 1992, Micromechanical modeling of inter-facial decohesion. Ultramicroscopy, 40, 203–214.

A. Needleman and J. R. Rice, 1978, Limits to ductility byplastic flow localization. In: ‘‘Mechanics of Sheet MetalForming, Proceedings of General Motors ResearchLaboratories Symposium,’’ eds. D. P. Koistinen andN.-M. Wang, Plenum Press, New York, pp. 237–267.

A. Needleman and V. Tvergaard, 1987, An analysis ofductile rupture at a crack tip. J. Mech. Phys. Solids, 35,151–183.

S. Nemat-Nasser and M. Hori, 1993, ‘‘Micromechanics:overall properties of heterogeneous materials,’’ North-Holland, Amsterdam.

J. C. Newman, C. A. Bigelow and D. S. Dawicke, 1992,Finite-element analyses and fracture simulation in thin-sheet aluminum alloy. In: ‘‘Durability of Metal AircraftStructures, Proc. Int. Workshop on Structural Integrityof Aging Airplanes,’’ ed. S. N. Atluri, Atlanta Technol-ogy Publications, Atlanta, GA, pp. 167–186.

O. Nguyen, E. A. Repetto, M. Ortiz and R. A. Radovitzky,2001, A cohesive model of fatigue crack growth. Int. J.Fract., 110, 351–369.

P. E. O’Donoghue, M. F. Kanninen, C. P. Leung, G.Demofonti and S. Venzi, 1997, The development andvalidation of a dynamic fracture propagation model forgas transmission pipelines. Int. J. Press. Vessels Pip., 70,11–25.

N. P. O’Dowd and C. F. Shih, 1991, Family of crack-tipfields characterized by a triaxiality parameter: I. Struc-ture of fields. J. Mech. Phys. Solids, 39, 989–1015.

N. P. O’Dowd and C. F. Shih, 1994, Two-parameterfracture mechanics, theory and applications. In:‘‘Fracture Mechanics, ASTM STP 1207,’’ eds. J. D.Landes, D. E. McCabe and J. M. Boulet, AmericanSociety for Testing and Materials, Philadelphia, vol. 24,pp. 21–47.

M. Ortiz and A. Pandolfi, 1999, Finite-deformationirreversible cohesive elements for three-dimensionalcrack-propagation analysis. Int. J. Numer. Meth. Eng.,44, 1267–1282.

A. Pandolfi, P. R. Guduru, M. Oritz and A. J. Rosakis,2000, Three dimensional cohesive element analysis andexperiments of dynamic fracture in C300 steel. Int. J.Solids Struct., 37, 3733–3760.

D. M. Parks, 1974, A stiffness derivative finite elementtechnique for determination of crack tip stress intensityfactors. Int. J. Fract., 10, 487–502.

D. M. Parks, 1977, The virtual crack extension method fornonlinear material behavior. Comput. Meth. Appl. Mech.Eng., 12, 353–364.

P. E. Petersson, 1981, ‘‘Crack Growth and Development ofFracture Zones in Plain Concrete and Similar Materi-als,’’ Report LUTVDG/TVBM-1006, Division of Build-ing Materials, Lund Institute of Technology, Lund,Sweden.

A. Pineau, 1981, Review of fracture micromechanisms anda local approach to predicting crack resistance. In:‘‘Advances in Fracture Research, Proc. 5th Int. Conf. onFracture,’’ ed. D. Francois, Pergamon, Oxford and NewYork, vol. 2, pp. 553–577.

J. Planas, M. Elices and G. V. Guinea, 1993, Cohesivecracks versus nonlocal models: closing the cap. Int. J.Fract., 63, 173–187.

J. Planas, G. V. Guinea and M. Elices, 1999, Size effect andinverse analysis in concrete fracture. Int. J. Fract., 95,367–378.


W. Prager, 1959, ‘‘An Introduction to Plasticity,’’ Adison-Wesley, Reading/London.

L. Prandtl, 1924, ‘‘Proc. 1st Int. Conf. App. Mech.,’’ Delft.eds C. B. Biezeno and J. M. Burgers, TechnischeBoekhandel en Drukkerij J. Waltman Jr.

Y. Qiu, M. A. Crisfield and G. Alfano, 2001, An interfaceelement formulation for the simulation of delaminationwith buckling. Eng. Fract. Mech., 68, 1755–1776.

P. Rahul-Kumar, A. Jagota, S. J. Bennison and S. Saigal,2000, Cohesive element modelling of viscoelastic frac-ture: application to peel testing of polymers. Int. J. SolidsStruct., 37, 1873–1897.

P. Rahul-Kumar, A. Jagota, S. J. Bennison, S. Saigal and S.Muralidhar, 1999, Polymer interfacial fracture simula-tions using cohesive elements. Acta Mater., 47, 4161–4169.

W. Ramberg and W. R. Osgood, 1945, Description ofstress–strain curves by three parameters. NACA Tech-nical Note no. 902.

E. A. Repetto, R. Radovitzky and M. Ortiz, 2000, Finiteelement simulation of dynamic fracture and fragmenta-tion of glass rods. Comput. Meth. Appl. Mech. Eng., 183,3–14.

A. Reuss, 1930, Berucksichtigung der elastischen Forman-derung in der Plastizitatstheorie. Z. angew. Math. Mech.,10, 266–274.

J. R. Rice, 1965, An examination of the fracture mechanicsenergy balance from the point of view of continuummechanics. In: ‘‘Proc. 1st Int. Conf. on Fracture,’’ vol.I,eds. T. Yokobori, T. Kawasaki and J. K. Swedlow,Sendai, Japan, Japanese Society for Strength andFracture of Materials, Tokyo, 1966, pp. 309–340.

J. R. Rice, 1968, A path independent integral and theapproximate analysis of strain concentration by notchesand cracks. J. Appl. Mech., 35, 379–386.

J. R. Rice, 1979, The mechanics of quasi-static crackgrowth. In: ‘‘Proc. 8th Int. Congress of AppliedMechanics,’’ ed. R. E. Kelly, Western Periodicals, NorthHollywood, California, pp. 191–216.

J. R. Rice and G. F. Johnson, 1970, The role of large cracktip geometry changes in plane strain fracture. In:‘‘Inelastic Behavior of Solids,’’ eds. M. F. Kanninen,et al., McGraw-Hill, New York, pp. 641–672.

J. R. Rice, P. C. Paris and J. G. Merkle, 1973, Some furtherresults of J-integral analysis and estimates. In: ‘‘Progressin Flaw Growth and Fracture Toughness Testing,ASTM STP 536,’’ ed. J. G. Kaufman, American Societyfor Testing and Materials, Philadelphia, pp. 231–245.

J. R. Rice and G. F. Rosengren, 1968, Plane straindeformation near a crack-tip in a power-law hardeningmaterial. J. Mech. Phys. Solids, 16, 1–12.

J. R. Rice and D. M. Tracey, 1969, On the ductileenlargement of voids in triaxial stress fields. J. Mech.Phys. Solids, 17, 201–217.

H. Riedel and J. R. Rice, 1980, Tensile cracks in creepingsolids. In: ‘‘Fracture Mechanics, 12th Conf.,’’ ASTMSTP 700, P. C. Paris, symposium chairman,’’ AmericanSociety for Testing and Materials, Philadelphia, pp. 112–130.

D. N. Robinson, 1978, ‘‘A Unified Creep-plasticity Modelfor Structural Metals at High Temperature,’’ ORNLreport TM 5969.

K. L. Roe, 2001, A cohesive zone model for fatigue crackgrowth simulation. Master Science thesis, Purdue Uni-versity, USA.

K. L. Roe and T. Siegmund, 2003, An irreversible cohesivezone model for interface fatigue crack growth simula-tion. Eng. Fract. Mech., 70, 209–232.

J. Rose, J. Ferrante and J. Smith, 1981, Universal bindingenergy curves for metals and bimetallic interfaces. Phys.Rev. Lett., 47, 675–678.

G. Rousselier, 1987, Ductile fracture models and theirpotential in local approach of fracture. Nucl. Eng. Des.,105, 97–111.

G. Rousselier, J.-C. Devaux, G. Mottet and G. Devesa,1989, A methodology for ductile fracture analysisbased on damage mechanics, an illustration of a localapproach of fracture. In: ‘‘Nonlinear Fracture Me-chanics, Volume II—Elastic–Plastic Fracture, ASTMSTP 995,’’ eds. J. D. Landes, A. Saxena and J. G.Merkle, American Society for Testing and Materials,,Philadelphia, pp. 332–354.

Y. A. Roy and R. H. Dodds, 2001, Simulation of ductilecrack growth in thin aluminium panel using 3-D surfacecohesive elements. Int. J. Fract., 110, 21–45.

S. Roy Chowdhury and R. Narasimhan, 2000, A finiteelement analysis of quasistatic crack growth in a pressuresensitive constrained ductile layer. Eng. Fract. Mech., 66,551–571.

C. Ruggieri, T. L. Panontin and R. H. Dodds, 1996,Numerical modeling of ductile crack growth in 3-D usingcomputational cell elements. Int. J. Fract., 82, 67–95.

G. Ruiz, 2001, Propagation of a cohesive crack crossing areinforced layer. Int. J. Fract., 111, 265–282.

G. Ruiz, M. Elices and J. Planas, 1998, Experimental studyof fracture of lightly reinforced concrete beams. Mater.Struct., 31, 683–691.

G. Ruiz, A. Pandolfi and M. Ortiz, 2001, Three-dimen-sional cohesive modeling of dynamic mixed-modefracture. Int. J. Numer. Meth. Eng., 52, 97–120.

I. Scheider, 2000, Bruchmechanische Bewertung vonLaserschwei�verbindungen durch numerischeRi�fortschrittsanalysen mit dem Kohasivzonenmodell.Ph.D. thesis, Technical University Hamburg-Harburg,Germany.

I. Scheider, 2001, Simulation of cup-cone fracture in roundbars using the cohesive zone model. In: ‘‘First M.I.T.Conf. on Computational Fluid and Solid Mechanics,’’ed. K. J. Bathe, Elsevier, Amsterdam, vol. 1, pp. 460–462.

W. Schmitt, D.-Z. Sun and J. G. Blauel, 1997, Damagemechanics analysis (Gurson model) and experimentalverification of the behaviour of a crack in a weld-claddedcomponent. Nucl. Eng. Des., 174, 237–246.

K.-H. Schwalbe, 1995, Introduction of d5 as an operationaldefinition of the CTOD and its practical use. In:‘‘Fracture Mechanics, ASTM STP 1256,’’ eds. W. G.Reuter, J. H. Underwood and J. C. Newman, AmericanSociety for Testing and Materials, Philadelphia, vol. 26,pp. 763–778.

K.-H. Schwalbe, J. Heerens, U. Zerbst and M. Ko-cak,2002, ‘‘The GKSS Procedure for Determining theFracture Behaviour of Materials,’’ 2nd edn., GKSSResearch Centre, Geesthacht.

S. M. Sharma and N. Aravas, 1991, Determination ofhigher-order terms in asymptotic elastoplastic crack tipsolutions. J. Mech. Phys. Solids, 39, 1043–1072.

C. F. Shih, 1981, Relationship between the J-integral andthe COD for stationary and extending cracks. J. Mech.Phys. Solids, 29, 305–326.

C. F. Shih, 1983, ‘‘Tables of the Hutchinson–Rice–Rosengren Singular Field Quantities,’’ Report MRL E-147, Brown University, Providence, RI.

C. F. Shih and M. German, 1981, Requirements for a oneparameter characterization of crack tip fields by theHRR-singularity. Int. J. Fract., 17, 27–43.

C. F. Shih, 1985, J-dominance under plane strain fullyplastic conditions: the edge crack panel subject tocombined tension and bending. Int. J. Fract., 29, 73–84.

C. F. Shih, N. P. O’Dowd and M. T. Kirk, 1993, Aframework for quantifying crack tip constraint tosurface flaws. In: ‘‘Constraint Effects in Fracture, ASTMSTP 1171,’’ eds. E. M. Hackett, K.-H. Schwalbe and R.H. Dodds, American Society for Testing and Materials,Philadelphia, pp. 2–20.

D. Siegele, 1989, 3D Crack propagation using ADINA.Comput. Struct., 32, 639–645.

References 207

D. Siegele and W. Schmitt, 1983, Determination andsimulation of stable crack growth in ADINA. Comput.Struct., 17, 697–703.

Th. Siegmund, G. W. Bernauer and W. Brocks, 1998, Twomodels of ductile fracture in contest, porous metalplasticity and cohesive elements. In: ‘‘Proc. ECF 12,Fracture from Defects,’’ eds. M. W. Brown, E. R. de losRios and K. J. Miller, Engineering Materials AdvisoryServices, Ort, vol. II, pp. 933–938.

Th. Siegmund and W. Brocks, 1998a, Tensile decohesionby local failure criteria. Technische Mechanik, 18, 261–270.

Th. Siegmund, W. Brocks, 1998b, Local fracture criteria,lengthscales and applications In: ‘‘PROC. EURO-MECH-MECAMAT Conf.,’’ J. de Physique IV, 8,347–354.

Th. Siegmund and W. Brocks, 1999, Prediction of the workof separation and implications to modeling. Int. J.Fract., 99, 97–116.

Th. Siegmund and W. Brocks, 2000a, The role of cohesivestrength and separation energy for modeling of ductilefracture. In: ‘‘Fatigue Fracture Mechanics, ASTM STP1360,’’ eds. K. L. Jerina and P. C. Paris, AmericanSociety for Testing and Materials, Philadelphia, vol. 30,pp. 139–151.

Th. Siegmund and W. Brocks, 2000b, A numerical studyon the correlation between the work of separation andthe dissipation rate in ductile fracture. Eng. Fract. Mech.,67, 139–154.

Th. Siegmund and W. Brocks, 2000c, Modeling crackgrowth in thin sheet aluminum alloys. In: ‘‘Fatigue andFracture Mechanics, ASTM STP 1389,’’ eds. G. L.Halford and J. P. Gallagher, American Society forTesting and Materials, Philadelphia, vol. 31, pp. 475–485.

Th. Siegmund, W. Brocks, J. Heerens, G. Tempus and W.Zink, 1999, Modeling of crack growth in thin sheetaluminium. In: ‘‘ASME Int. Mechanical EngineeringCongress and Exposition: Recent Advances in Solids andStructures,’’ ASME PVP 398, Nashville, pp. 15–22.

Th. Siegmund and A. Needleman, 1997, A numerical studyof dynamic crack growth in elastic–viscoplastic solids.Int. J. Solids Struct., 34, 769–787.

J. Simo, J. Oliver and F. Armero, 1993, An analysis ofstrong discontinuities induced by strain-softening in rate-independent inelastic solids. Comput. Mech., 12, 277–296.

E. Sommer and D. Aurich, 1991, On the effect ofconstraint on ductile fracture. In: ‘‘Defect Assessmentin Components, Fundamentals and Applications, ESIS/EGF9,’’ eds. F. G. Blauel and K.-H. Schwalbe,Mechanical Engineering Publications, London, pp.141–174.

B. F. S�rensen, 2002, Cohesive law and notch sensitivity ofadhesive joints. Acta Mater., 50, 1053–1061.

D. Steglich and W. Brocks, 1997, Micromechanicalmodelling of the behaviour of ductile materials includingparticles. Comput. Mater. Sci., 9, 7–17.

N. Sukumar, N. Mose, B. Moran and T. Belytschko, 2000,Extended finite element method for three-dimensionalcrack modelling. Int. J. Numer. Meth. Eng., 48, 1549–1570.

J. D. G. Sumpter, 1999, An alternative view of R-curvetesting. Eng. Fract. Mech., 64, 161–176.

D.-Z. Sun and A. Honig, 1994, Significance of thecharacteristic length for micromechanical modelling ofductile fracture. In: ‘‘Proc. Third Int. Conf. on LocalizedDamage,’’ eds. M. H. Aliabadi, A. Carpinteri, S. Kaliskyand D. J. Cartwright, Comp. Mechanics Publication,Southampton, pp. 287–296.

D.-Z. Sun, A. Honig, W. Bohme and W. Schmitt, 1994,Application of micromechanical models to the analysisof ductile fracture under dynamic loading. In: ‘‘Fracture

Mechanics, 25th Symp., ASTM STP 1220,’’ ed. F.Schmitt, Erdogan, American Society for Testing andMaterials,, Philadelphia, pp. 343–357.

D.-Z. Sun, D. Siegele, B. Voss and W. Schmitt, 1988,Application of local damage models to the numericalanalysis of ductile rupture. Fatigue Fract. Eng. Mater.Struct., 12, 201–212.

S. Tandon, K. T. Faber, Z. P. Bazant and Y. N. Li, 1995,Cohesive crack modeling of influence of sudden changesin loading rate on concrete fracture. Eng. Fract. Mech.,52(6), 987–997.

M. G. A. Tijssens, 2000, ‘‘On the cohesive surfacemethodology for fracture of brittle heterogeneoussolids,’’ Shaker Publishing B.V, ISBN 90-423-0129-5,Ph.D. at Technical University Delft, Netherlands.

M. G. A. Tijssens, E. van der Giessen and L. J. Sluys, 2001,Modeling quasi-static fracture of heterogeneous materi-als with the cohesive surface methodology. In: ‘‘Compu-tational Fluid and Solid Mechanics (First MITConference),’’ ed. K. J. Bathe, Elsevier, Amsterdamand London, vol. 1, pp. 509–512.

C. E. Turner, 1990, A re-assessment of ductile tearingresistance (Parts I and II). In: ‘‘Fracture Behaviour andDesign of Materials and Structures, Proc. ECF 8,’’ ed.D. Firrao, vol. II, pp. 933–949 and 951–968.

V. Tvergaard, 1990, Effect of fibre debonding in awhisker-reinforced metal. Mater. Sci. Eng., A125, 203–213.

V. Tvergaard, 1993, Model studies of fibre breakage anddebonding in a metal reinforced by short fibres. J. Mech.Phys. Solids, 41, 1309–1326.

V. Tvergaard, 1995, Fibre debonding and breakage in awhisker-reinforced metal. Mat. Sci. Eng. A, 190, 215–222.

V. Tvergaard, 2001, Crack growth predictions by cohesivezone model for ductile fracture. J. Mech. Phys. Solids,49, 2191–2207.

V. Tvergaard and J. W. Hutchinson, 1992, The relationbetween crack growth resistance and fracture processparameters in elastic–plastic solids. J. Mech. Phys.Solids, 40, 1377–1397.

V. Tvergaard and J. W. Hutchinson, 1993, The influence ofplasticity on mixed mode interface toughness. J. Mech.Phys. Solids, 41, 1119–1135.

V. Tvergaard and J. W. Hutchinson, 1994a, Effect of T-stress on mode I crack growth resistance in a ductilesolid. Int. J. Solids Struct., 31, 823–833.

V. Tvergaard and J. W. Hutchinson, 1994b, Toughness ofan interface along a thin ductile layer joining elasticsolids. Philos. Magazine A, 70, 641–656.

V. Tvergaard and J. W. Hutchinson, 1996a, On thetoughness of ductile adhesive joints. J. Mech. Phys.Solids, 44, 789–800.

V. Tvergaard and J. W. Hutchinson, 1996b, Effect ofstrain-dependent cohesive zone model on predictions ofcrack growth resistance. Int. J. Solids Struct., 33, 3297–3308.

V. Tvergaard and A. Needleman, 1984, Analysis of thecup-cone fracture in a round tensile bar. Acta Metall.,32, 157–169.

J. G. M. van Mier and M. R. A. van Vliet, 2002, Uniaxialtension test for the determination of fracture parametersof concrete: state of the art. Eng. Fract. Mech., 69, 235–247.

M. R. A. van Vliet and J. G. M. van Mier, 1999, Effect ofstrain gradients on the size effect of concrete in uniaxialtension. Int. J. Fract., 95, 195–219.

A. G. Varias, Z. Suo and C. F. Shih, 1991, Ductile failureof a constrained metal foil. J. Mech. Phys. Solids, 39,963–986.

R. von Mises, 1923, Die Mechanik der festen Korper implastischen deformablen Zustand. Z. angew. Math.Mechanik, 3, 406.


Y. Wei and J. W. Hutchinson, 1997, Nonlinear delamina-tion mechanics for thin films. J. Mech. Phys. Solids, 45,1137–1159.

Y. Wei, J. W. Hutchinson, 1998, ‘‘Models of InterfaceSeparation Accompanied by Plastic Dissipation,’’ Re-port MECH-341, Harvard University, Division ofEngineering and Applied Science.

W. Weibull, 1939a, A statistical theory of the strength ofmaterials. Ingeni�rsvetenskapakademiens, Nandlingerno. 151.

W. Weibull, 1939b, The phenomenon of rupture in solids.Ingeni�rsvetenskapakademiens, Nandlinger no. 153.

W. Weibull, 1951, A statistical distribution function ofwide applicability. J. Appl. Mech., 81, 293–297.

G. N. Wells and L. J. Sluys, 2001, A new method formodelling cohesive cracks using finite elements. Int. J.Numer. Meth. Eng., 50, 2667–2682.

L. Xia and F. C. Shih, 1995, Ductile crack growth: I. Anumerical study using computational cells with micro-structurally-based length scales conditions. J. Mech.Phys. Solids, 43, 233–259.

L. Xia, C. F. Shih and J. W. Hutchinson, 1995, Acomputational approach to ductile crack growthunder large scale yielding. J. Mech. Phys. Solids, 43,389–413.

D.-B. Xu, C.-Y. Hui, E. J. Kramer and C. Creton, 1991, Amicromechanical model of crack growth along polymerinterfaces. Mech. Mater., 11(3), 257–268.

X.-P. Xu and A. Needleman, 1993, Void nucleation byinclusion debonding in a crystal matrix. Modell. Simul.Sci. Eng., 1, 111–132.

X. Xu and A. Needleman, 1994, Numerical simulations offast crack growth in brittle solids. J. Mech. Phys. Solids,42, 1397–1434.

X.-P. Xu and A. Needleman, 1995, Numerical simulationsof dynamic interfacial crack growth allowing for crackgrowth away from the bond line. Int. J. Fract., 74, 253–275.

X.-P. Xu and A. Needleman, 1996, Numerical simulationsof dynamic crack growth along an interface. Int. J.Fract., 74, 289–324.

B. Yang and K. Ravi-Chandar, 1998, A single-domaindual-boundary-element formulation incorporating acohesive zone model for elastostatic cracks. Int. J.Fract., 93, 115–144.

Q. D. Yang, M. D. Thouless and S. M. Ward, 1999,Numerical simulations of adhesively-bonded beams fail-ing with extensive plastic deformation. J. Mech. Phys.Solids, 47, 1337–1353.

Q. D. Yang, M. D. Thouless and S. M. Ward, 2001a,Elastic–plastic mode II fracture of adhesive joints. Int. J.Solids Struct., 38, 3251–3262.

B. Yang, S. Mall and K. Ravi-Chandar, 2001b, A cohesivezone model for fatigue crack growth in quasibrittlematerials. Int. J. Solids Struct., 38, 3927–3944.

H. Yuan, 1990, ‘‘Untersuchung bruchmechanischer Para-meter fur elastisch-plastisches Ri�wachstum, Fortschr.-Ber. VDI. Reihe 18 Nr. 82,’’ Dusseldorf, VDI–Verlag.

H. Yuan and W. Brocks, 1991, On the J-integral conceptfor elastic–plastic crack extension. Nucl. Eng. Des., 131,157–173.

H. Yuan and W. Brocks, 1998, Quantification of constrainteffects in elastic–plastic crack front fields. J. Mech. Phys.Solids, 46, 219–241.

H. Yuan and A. Cornec, 1991, ‘‘Numerical Simulations ofDuctile Crack Growth in Thin Specimens Based on aCohesive Zone Model,’’ GKSS report 91/E/23.

H. Yuan, G. Lin and A. Cornec, 1995, Quantifications ofcrack constraint effects in an austenitic steel. Int. J.Fract., 71, 273–291.

H. Yuan, G. Lin and A. Cornec, 1996, Verification of acohesive zone model for ductile fracture. J. Eng. Mater.Technol., 118, 192–200.

P. D. Zavattieri and H. D. Espinosa, 2001, Grain levelanalysis of crack initiation and propagation in brittlematerials. Acta Mater., 49, 4291–4311.

P. D. Zavattieri, P. V. Raghuram and H. D. Espinosa,2001, A computational model of ceramic microstructuressubjected to multiaxial dynamic loading. J. Mech. Phys.Solids, 49, 27–68.

H. Ziegler, 1959, A modification of Prager’s hardeningrule. Quart. Appl. Math., 17, 55–71.

Copyright 2003, Elsevier Ltd. All Rights Reserved. Comprehensive Structural Integrity

No part of this publication may be reproduced, stored in any retrieval system or

transmitted in any form or by any means: electronic, electrostatic, magnetic tape,

mechanical, photocopying, recording or otherwise, without permission in writing

from the publishers.

ISBN (set): 0-08-043749-4

Volume 3; (ISBN: 0-08-044158-0); pp. 127–209

References 209

3.03-computational_aspects_of_nonlinear fracture mechanics.pdf

Documents

0 gw dx2 sjknkuj

van der giessen

aplastically deforming matrix

nonlinear fracture mechanics3

american society fortesting

american society fortesting

themaximum principal stress

normaland shear separation