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Materials Science and Engineering A 462 (2007) 174–177

Propagation mechanisms of microstructurally short cracks—Factorsgoverning the transition from short- to long-crack behavior

U. Krupp a,∗, O. Duber a, H.-J. Christ a, B. Kunkler b, P. Koster b, C.-P. Fritzen b

a Institut fur Werkstofftechnik, Universitat Siegen, 57068 Siegen, Germanyb Institut fur Mechanik und Regelungstechnik, Universitat Siegen, 57068 Siegen, Germany

Received 30 August 2005; received in revised form 10 March 2006; accepted 17 March 2006

bstract

Microstructural short fatigue cracks are known to exhibit an abnormal propagation behavior as compared to long cracks, which grow by a rate thatan be described by, e.g., the Klesnil–Lukas relationship. By means of carefully recording crack length versus number of cycles in the high-cycleatigue regime in combination with a microtexture analysis using automated electron back-scattered diffraction, the parameters determining thecatter in microcrack propagation rates and the transition from short- to long-crack behavior were identified for an austenitic–ferritic duplex steel.he three-dimensional orientation relationship of the slip planes in grains involved in the crack propagation process turned out to be most significant.his relationship determines the barrier effect of grain and phase boundaries as well as the local crack propagation mechanisms, either operating

rystallographically by single slip or perpendicularly to the applied load axis operating by double/multiple slip. To predict the propagation behaviorf microstructurally short cracks in a mechanism-based way, a numerical model has been developed that accounts for local interactions betweenhe crack tip and the microstructure as the current driving force for crack advance.

2006 Elsevier B.V. All rights reserved.

steel;

atTtbetltbobssi

eywords: Short cracks; Numerical modeling; Fatigue-life prediction; Duplex

. Introduction

Particularly in the case of high-cycle-fatigue-loaded smoothomponents, the major part of the service life is determinedy crack initiation and early crack growth. The propagationehavior of such microstructurally short fatigue cracks devi-tes considerably from that of long fatigue cracks. For the latter,he size of the plastic zone in the vicinity of the crack tip isegligibly small as compared to the crack length a, and theorresponding crack growth rate da/dN follows a characteristicelationship with the applied stress intensity factor �K, beingescribed by the well-known Paris law [1], or more generally byhe Klesnil–Lukas relationship [2]:

da = C(�Km − �Km), (1)

dN (1 − R)m th

ccounting for the stress ratio R and the threshold value �Kth fornitiation of “technical” cracks. The factor C and the exponent m

∗ Corresponding author. Tel.: +49 271 7402184; fax: +49 271 7402545.E-mail address: ulrich.krupp@uni-siegen.de (U. Krupp).

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921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved.oi:10.1016/j.msea.2006.03.159

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re material constants. Microstructurally short cracks may ini-iate and grow well below the threshold for long cracks �Kth.hey are of the same order of magnitude than the characteris-

ic microstructure features [3] and exhibit an oscillating growthehavior [4]. Once a microstructural crack has been initiated,.g., by accumulated irreversible slip or elastic anisotropy ofhe microstructure constituents, it usually grows along crystal-ographic slip bands driven by the cyclic shear displacement athe crack tip [5]. Depending on the crystallographic relationshipetween neighboring grains of the same or different phases andn the geometry and particular strength of the grain or phaseoundaries, various propagation scenarios of microstructurallyhort cracks have been observed: (i) slip bands may cause slipteps in the boundary planes followed by intercrystalline crack-ng [6], (ii) the grain and phase boundaries may act as barriersgainst slip transmission and reduce the crack propagation rate7], and (iii) depending on the orientation of the slip planes withespect to the local stress state and the lattice structure of the

espective grains, the microcrack propagation mechanism canhange from single slip to alternating multiple slip operating athe crack tip [8]. The latter mechanism is often referred to astage II crack propagation driven by the nominal normal stress

U. Krupp et al. / Materials Science and Engineering A 462 (2007) 174–177 175

Fi

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Fig. 2. Geometry of shallow-notched fatigue specimens.

Table 1Nominal chemical composition of the stainless steels used in this study (wt.%)

Fe Cr Ni Mo Mn Si Nb N C

111

(g��wissTi

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ig. 1. Schematic representation of the different crack propagation regimes dur-ng fatigue damage of metals and alloys.

erpendicular to the direction of crack advance. However, evennder these conditions the crack can still be very short, and thelastic zone is not necessary small compared to the crack length.dditionally, the microstructure may have an influence on the

rack propagation, so the crack growth rate cannot directly beescribed by continuum mechanics, certainly not by linear elas-ic fracture mechanics. Hence, one should define a transitiontage between stage I microcrack propagation along crystallo-raphic slip bands and stage II crack propagation resulting inatigue striations. This is shown schematically in Fig. 1, beingalid for both, crack propagation along the specimen surface asell as into the bulk.Prediction of the crack propagation rate during the early stage

f fatigue damage requires a model that accounts for alterationsn the local resistance to crack advance. Navarro and de los Rios9] and Taira et al. [10] developed analytical models where suchlterations are attributed to a dislocation pile up between therack tip and the grain boundary. Once the pile-up stress actingn a dislocation source in the adjacent grain exceeds a criticaltress, plastic slip sets in giving rise to an intermittent increasen the crack propagation rate.

While the analytic model of Navarro and de los Rios consid-rs the relationship between local crystallographic orientationnd the variation in crack propagation only in one dimensiony an average orientation factor, Section 4 of the present paperntroduces a numerical two-dimensional short crack model thats capable to treat single- and two-phase microstructures with

easured variation in the grain size and crystallographic orien-ation distribution.

. Experimental details

Microcrack propagation was studied on electro-polished,hallow-notched specimens (Fig. 2) of an austenitic–ferritic

sdof

.4462 Bal. 21.9 5.6 3.1 1.8 0.5 – 0.1871 0.020

.4404 Bal. 16.6 11.1 – 1.3 0.6 0.01 0.0296 0.018

.4511 Bal. 16.3 – – 0.7 0.5 0.25 – 0.012

�–�) duplex steel 1.4462 (ASTM A182 F51) with an averagerain size of d(�) = 46 �m and d(�) = 33 �m, respectively, and an/�-volume ratio of approximately 0.5. The barrier strength of� and �� grain boundaries as well as of the �� phase boundariesas quantified by means of a cyclic Hall–Petch analysis, apply-

ng incremental step tests on ferritic steel 1.4511 and austeniticteel 1.4404 (AISI 316L) reference materials with various grainizes. Details on this investigation can be found elsewhere [11].he chemical compositions of all the materials studied are given

n Table 1.Fatigue experiments were carried out in a MTS810 ser-

ohydraulic testing machine under fully reversed (R = −1),tress-controlled push-pull loading conditions. The specimensere periodically removed from the testing machine to eval-ate microstructurally short cracks within the electropolishedhallow-notched gauge length by means of scanning elec-ron microscopy (SEM). The crystallographic orientations ofhe grains and phase patches involved in the crack propaga-ion process were determined by means of automated electronack-scattered diffraction (TSL OIMTM, orientation imagingicroscopy).Information about three-dimensional effects of microcrack

ropagation was obtained: (i) by stepwise polishing the sur-ace in the vicinity of selected microcracks and (ii) by applyingSDG (interferometric strain/displacement gage) measurementso monitor the development of crack-mouth opening displace-

ent (CMOD).

. Results and discussion

In most cases, cracks in the �� duplex steel are initiated in theicinity of grain or phase boundaries. This might be attributed tohe interdependence between the elastic constants and the crys-allographic orientation of the two-phase microstructure con-tituents (elastic anisotropy). As an example, Fig. 3 shows aurface microcrack initiated at a �-austenite twin-boundary, andurther growing from (1) by alternate operating (1 1 1) 〈1 1 0〉-

lip systems in a similar way as it was recently observed andiscussed by Blochwitz and Tirschler [12]. At the left-hand sidef the cracked twin boundary, microcrack propagation resumesrom a triple point (2) into an adjacent bcc �-ferrite grain gov-

176 U. Krupp et al. / Materials Science and Engineering A 462 (2007) 174–177

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ig. 3. Microcrack propagation in �–� duplex steel at a stress amplitude ofσ/2 = 350 MPa after N = 160,000 cycles, the vertical lines correspond to inter-

als of 10,000 cycles.

rned by cyclic crack tip slip displacement �CTSD along a1 1 0) 〈1 1 1〉-type slip plane.

The distance between the vertical lines in Fig. 3 correspondso 10,000 fatigue cycles. Hence, local microcrack propagationates seem not to depend only on the barrier effect of phasend grain boundaries but also on the slip behavior at the crackip. Crack advance driven by operation of multiple slip systemsduring the early crack propagation stage mainly in fcc � grains)s substantially slower than that driven by �CTSD on a singlelip band. In the case of the two-phase duplex steel, transitiono multiple slip crack propagation can be attributed to both, theosition of the slip systems with respect to the stress axis, and theritical shear stress to activate dislocation glide on a slip plane,hich is lower for the fcc � phase [11]. Regarding the transition

o long crack propagation being always governed by multiplelip, it has been observed that due to the two-phase microstruc-ure, microcrack propagation in the �–� duplex steel exhibits aransition stage (compare Fig. 1) during which a change from

ultiple-slip back to single-slip crack advance is possible (seeosition (3) in Fig. 3).

. Numerical modeling of short cracks

Analogously to the model of Navarro and de los Rios [9],numerical short crack model has been developed by Schick

13], which accounts for microstructural features, and which isriefly introduced in the following (for more details see Ref. [8]).he model describes a propagating microcrack and its adjacentlastic zones as yield strips, consisting of an array of mathemat-cal dislocation dipole elements (boundary elements) accordingo Fig. 4a. By means of the algebraic expression of the stresseld in the vicinity of a dislocation [14], the displacements andtresses of all the elements can be calculated based on the actualominal stress of the applied loading and the local positions ofhe slip planes, taking the following boundary conditions intoccount: (i) neither tensile stresses nor shear stresses may occurithin the crack area, (ii) the normal displacement in the crack

rea must always be positive, and (iii) the shear stress in the slipands is limited by the cyclic critical shear stress that was esti-ated for the � phase as to be τc,� = 137 MPa and for the � phase

s to be τc,� = 198 MPa, respectively [11]. Hence, one obtains a

msbt

ig. 4. Schematic sketch of (a) the dislocation-dipole representation of a micro-rack with adjacent slip bands and sensor elements for the numerical short crackodel, and (b) transition from single to multiple slip.

ystem of linear in equations (see Refs. [8,14] for details) thatields the whole set of displacements within the yield-strip area.rack advance is calculated by the respective displacement at

he crack tip �CTSD using a simple crack-propagation law thatccounts with the material constant C′ for the fraction of planarlip irreversibility during cyclic loading:

da

dN= C′�CTSDm. (2)

In the case of the duplex steel, the constant C′ was foundo be of a value of C′ = 0.005, which is in accordance withodeling parameters reported by Chan [15]. The exponent m

s material-specific and is in most cases of values between m = 1nd 1.05. When approaching a grain boundary, �CTSD andonsequently da/dN decreases. At the same time, the stressescting on so-called sensor elements (see Fig. 4a), which rep-esent the slip systems in the adjacent grain, increase. Once aritical stress τ* on one of the sensor elements is exceeded, slipn the respective slip system is activated and the plastic zone isxtended to the next grain boundary, leading again to an increasef �CTSD.

Alternatively to the single-slip mechanism, the model canccount for multiple slip by means of additional sensor ele-

ents representing secondary slip systems. Once the critical

tress on one of these additional slip bands is exceeded, itecomes “activated” and meshed by boundary elements, i.e., fur-her crack propagation is governed by alternate slip on two slip

U. Krupp et al. / Materials Science and E

Fig. 5. Crack propagation in a statistical bcc microstructure generated by theVls

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oronoi algorithm: (a) microcrack paths, and (b) corresponding microcrackength vs. number of cycles (arbitrary calculation cycles) for different criticaltresses required to activate multiple slip systems.

lanes. This is shown schematically in Fig. 4b (see Ref. [8] foretails).

The model has been applied to numerous microstructurallyhort cracks in the �-titanium alloy LCB [13] as well as in the–� duplex steel 1.4462 [8,11] showing an excellent agreementith the experimentally measured crack length. The presentaper highlights the transition from crystalline microcrack prop-gation to long-crack behavior that depends on the local crys-allographic orientation, grain morphology, and phase distribu-ion. Fig. 5 shows the application of the numerical short-crack

odel to a statistical microstructure generated by the Voronoilgorithm. The crack paths in Fig. 5a correspond to the crackropagation curves in Fig. 5b. The position of transition fromingle- to multiple-slip crack propagation is marked by pointsnd coincides with a strong decrease in the crack propagationate (most clearly pronounced in the propagation curve of crack). This effect can be attributed mainly to the partitioning oflastic deformation on two slip planes.

As it can also be derived from Fig. 5, crack propagation rate

ncreases with increasing crack length when growing by alter-ate operating slip systems. Eventual transition to long crackehavior occurs per definition when the plastic zone size exceedsore than one grain diameter.

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ngineering A 462 (2007) 174–177 177

. Summary

Crack initiation and propagation during high-cycle fatigue ofn austenitic-ferritic duplex steel were shown to be determinedy the local microstructural features, crystallographic orienta-ion, morphology and size of grains and phase patches, as wells the structure of the grain and phase boundaries. In the two-hase alloy microcracks grow either in a crystalline manneroverned by single-slip (mainly bcc ferrite grains) or perpen-icular to the applied normal stress axis governed by multiplelip (mainly fcc austenite grains). The microcrack propagationate is high when the crack enters a new grain by single-slipechanism and slows down either when entering a grain byultiple-slip mechanism or when approaching a grain or phase

oundary due to dislocation pile-up. The latter depends on thearrier effect of the respective boundary that is correlated withhe misorientation angle between neighboring slip planes. Theseicrostructure/crack-propagation interactions were accounted

or by a numerical short-crack model which is briefly intro-uced in the present paper. By applying the model to a statisticalicrostructure a strong decrease in the crack propagation rate

ould be correlated with the transition from single- to multiple-lip microcrack propagation.

cknowledgement

The financial support of Deutsche ForschungsgemeinschaftDFG) in the framework of the priority program SPP1036Mechanism-Based Life Prediction for Cyclically Loadedetallic Materials” is gratefully acknowledged.

eferences

[1] P.C. Paris, F. Erdogan, J. Basic Eng. 85 (1963) 528.[2] M. Klesnil, P. Lukas, Eng. Fracture Mech. 4 (1972) 77.[3] S. Suresh, R.O. Ritchie, Intern. Met. Rev. 29 (1984) 445.[4] A.F. Blom, A. Hedlund, W. Zhao, A. Fathulla, B. Weiss, R. Stickler, in:

K.J. Miller, E.R. de los Rios (Eds.), Proceedings of the Behaviour of ShortFatigue Cracks, Mechanical Engineering Publications, London, 1986, p.37.

[5] A.J. Wilkinson, S.G. Roberts, P.B. Hirsch, Acta Mater. 46 (1998) 379.[6] J.C. Figuera, C. Laird, Mater. Sci. Eng. A 60 (1983) 45.[7] Y.H. Zhang, L. Edwards, Mater. Sci. Eng. A 188 (1994) 121.[8] O. Duber, B. Kunkler, U. Krupp, H.-J. Christ, C.-P. Fritzen, Int. J. Fatigue

28 (2006) 983.[9] A. Navarro, E.R. de los Rios, Phil. Mag. A 57 (1988) 15.10] S. Taira, K. Tanaka, Y. Nakai, Mech. Res. Commun. 5/6 (1978) 375.11] U. Krupp, O. Duber, H.-J. Christ, B. Kunkler, A. Schick, C.-P. Fritzen, J.

Microsc. 213 (2004) 313.12] C. Blochwitz, W. Tirschler, Mater. Sci. Eng. A 339 (2003) 318.13] A. Schick, Ein neues Modell zur Mechanismenorientierten Simulation der

Mikrostrukturbestimmten Kurzrissausbreitung, Doctorate Thesis, Univer-sity of Siegen, Fortschritt-Berichte VDI, Reihe 18, No. 292, VDI-Verlag,

Dusseldorf, 2004.

14] D.A. Mills, P.A. Kelly, D.N. Dai, A.M. Korsunsky, Solution of CrackProblems—The Distributed Dislocation Technique, Kluwer, Dordrecht,1995.

15] K.S. Chan, Metall. Mater. Trans. A 34 (2003) 43.