Ductile failure and rupture mechanisms

in combined tension and shear

Imad Barsoum

Licentiate thesis no. 96, 2006TRITA, HFL – 0407KTH Solid Mechanics

SE-100 44 Stockholm, Sweden

Preface

The work in this licentiate thesis was carried out at the Department of Solid Mechanics at theRoyal Institute of Technology (KTH) between September 2002 and March 2006. The workwas financially supported by the Swedish Research Council which is sincerely acknowledged.

I would like to thank my advisor Dr. Jonas Faleskog for his excellent supervision, guidanceand support during the span of this work. Thank you for giving me the opportunity to workwith you.

I would also like to thank the head of the laboratory Mr. Hans Oberg for helping out withthe experimental part and the design of the fixture. Thanks go to Messrs. Bertil Dolk,Bengt Mollerberg, Martin Oberg and Kurt Lindqvist for manufacturing the specimens. Dr.Torbjorn Narstrom at SSAB Oxelosund is acknowledged for supplying the material.

I am also thankful to my colleagues that have contributed to a stimulating environment atthe department, especially my officemates Mr. Mateusz Stec and Dr. Kaj Pettersson.

I am immensely grateful to my parents Nadhira and Samir and my brothers Zuheir and Fadifor their endless support and limitless love. Thank you for beeing with me along the roadof life all these years. God bless each one of you.

Finally, I would like to express my profound gratitude to my fiance Gorgina Zeitoun. It feelsgreat to have you in my life!

Stockholm, May 2006

Imad Barsoum

List of Appended Papers

Paper A: Rupture mechanisms in combined tension and shear—experimentsImad Barsoum and Jonas FaleskogSubmitted to International Journal of Solids and Structures

Paper B: Rupture mechanisms in combined tension and shear—micromechanicsImad Barsoum and Jonas FaleskogTo be submitted for international publication

Ductile failure and rupture mechanisms in combined tension and shear

Introduction

The industrial application of high strength steels and high performance aluminum alloysin structural components have increased the demand on understanding the ductile failurebehavior of this type of materials. In practical situations the loading experienced in compo-nents made out of these materials could be very complex when a crack is present, resultingin mixed mode ductile failure involving combinations of mode I, II and III.

The understanding of the governing ductile failure mechanisms under mode I loading is wellknown (Van Stone et al. (1985), Garrison Jr and Moody (1987)). The modeling of this failuremechanism is also rather established (McClintock. (1968), Rice and Tracey (1969), Gurson(1977)) and has advanced rapidly during the recent years (Tvergaard and Needleman (1984),Gao et al. (1998), Pardoen and Hutchinson (2000), Benzerga (2002)) involving nucleation,growth and coalescence of voids. This mechanism is promoted by a high hydrostatic stressstate and is often referred to as flat dimple rupture, which is shown in Figure 1. Here thefinal link up of the enlarged voids take place by necking of the intervoid ligaments. In a modeII or III loading situation however, the stress state ahead of the crack is altered resultingin a different failure mechanism. Here the mechanism is shear localization of plastic flow,which is promoted by the shear stress state ahead of the crack tip. The low hydrostaticstress state in such a case impedes the growth of voids ahead of the crack tip resulting insmall elongated dimples at fracture as shown in Figure 2. This mechanism is often referred toas shear dimple rupture, where final failure take place by shearing of the intervoid ligaments.

Hence, the void growth and coalescence mechanism leading to flat dimple rupture is favorablenear the mode I loading, whereas the shear localization mechanism leading to shear dimplerupture is favorable near mode II or III loading. Clearly there are two governing ductilerupture mechanisms, which will either compete or co-operate under mixed mode loadingsituation leading to ductile failure.

Mixed mode loading conditions

Previous studies by Ghosal and Narasimhan (1996), Laukkanen (2002) have attempted torelate the transition in micromechanics to the altering of the continuum fields, such as stresstriaxiality T and effective plastic strain εp, which depend on the mode mixity. The stresstriaxiality T is defined as the ratio between the hydrostatic and the Mises effective stress,respectively and the mode mixity is commonly defined as a parameter ranging from 0 forthe symmetric mode I to 1 for the antisymmetric mode II or III. Laukkanen (2002), amongothers, showed that T ahead of the crack tip in a mixed mode I/II situation decreases,whereas εp increases, as the portion of the mode II loading increases. The location of wherethe maximum values of the triaxiality and the effective plastic strain are also a function ofthe mode mixity and hence the preferred macroscopic crack growth direction will depend

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Imad Barsoum – KTH Solid Mechanics

Figure 1. Flat dimple rupture where final failure takes place by necking of the intervoid ligaments.Fractograph obtained from the scanning electron microscopy investigation made in Paper A.

Figure 2. Shear dimple rupture where final failure takes place by shearing of the intervoid ligaments.Fractograph obtained from the scanning electron microscopy investigation made in Paper A.

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Ductile failure and rupture mechanisms in combined tension and shear

upon the mode mixity (Hallback (1996), Narasimhan et al. (1999), Barsoum (2003)). In amixed mode I/III loading however T and εp will operate in the same plane ahead of thecrack and thus enhance each other.

In near mode II or III loading, the region ahead of the crack tip will experience extensiveshearing and decreased triaxiality, whereas in mode I the region ahead of the crack tip willexperience extensive tension and increased triaxiality. The rupture mechanisms are howeverdistinctively different in tension and shear as seen in Figure 1 and 2. The triaxiality param-eter do not completely account for these differences in the stress state and is consequentlynot the sole parameter governing ductile rupture in mixed mode loading as will be shownin the present study. In order to quantify the transition in the ductile rupture mechanisms,from flat to shear dimple rupture, the stress state must be accounted for more adequately.Hence, it is a matter of identifying whether it is a shear or tension type of stress state aheadof the crack tip.

The Lode parameter (Lode (1925)) µ, which is related to the third invariant of the deviatoricstress tensor, takes into account the shear or tension state of stress and is given by

µ =2σ2 − σ1 − σ3

σ1 − σ3

, (1)

where σ1, σ2 and σ3 are the principal stresses with σ1 ≥ σ2 ≥ σ3. Here µ = −1 correspondsto generalized tension, µ = 0 generalized shear and µ = 1 generalized compression. Thestress state is now characterized by T and µ, which describe the stress state ahead of thecrack tip in mixed mode loading more adequately.

By performing a modified boundary layer analysis proposed by Larsson and Carlsson (1973)and Rice (1973), T and µ can be determined ahead of a initially blunted crack as functionof the mode mixity βI−II and βI−III defined in Eq. (2). KI , KII and KIII are the stressintensity factors in mode I, II and III respectively. In pure mode I, βI−II = 0 and βI−III = 0,and pure mode II or III, βI−II = 1 and βI−III = 1, respectively. Result from a modifiedboundary layer analysis are shown in Figure 3, where solid lines corresponds to mixed modeI/II and dot-dashed line to mixed mode I/III loading. Here, T and µ are determined nearthe crack tip in the direction where T or εp are maximum. In the case of mixed mode I/III,the directions where T and εp are maximum coincide ahead of the crack tip. As shown inFigure 3(a), the triaxiality decreases with an increased proportion of the unsymmetric mode(II or III) for both hardening (N = 0.1) and elastic-ideally plastic (N = 0) material. In3(b), where µ vs. the mode mixity is plotted, it is seen that the stress state goes towardgeneralized shear, µ = 0, for increased mode II or III loading in the direction where T ismaximum. For the elastic-ideally plastic material and mixed mode I/II, µ does not varymarkedly with the mode mixity. It can also be noted that for both materials the stress stateat pure mode I loading is not generalized tension, µ = −1, as might be expected. In thedirection where εp is maximum the stress state is approximately generalized shear, µ ≈ 0,for the whole range of mode mixity. This is a consequence of the plane strain condition inthe direction where the effective plastic strain is maximum. Obviously, the changes in stress

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Imad Barsoum – KTH Solid Mechanics

state near the crack tip with respect to the mode mixity, Figure 3, have a strong influenceon the characteristics in the rupture mechanisms as seen in Figure 1 and 2.

βI−II =2

πarctan(KII/KI), βI−III =

2

πarctan(KIII/KI) (2)

0 0.5 10

1

2

3

max T

max εp

βI−II, βI−III

T

N = 0

N = 0.1

N = 0

N = 0.1 (a)

0 0.5 1−1

−0.5

0

βI−II, βI−III

µ

N = 0, N = 0.1 max εp

max TN = 0

N = 0.1(b)

Figure 3. The stress state ahead of a crack tip obtained from an elasto-plastic modified boundarylayer analysis subjected to mixed mode I/II (solid lines) and I/III (dot-dashed lines) loading. (a)The triaxiality and (b) the Lode parameter vs. the mode mixity. In the case of mixed mode I/IIloading T and µ are evaluated in the direction of maximum T or εp, the effective plastic strain.

Present Work

The objective of this work is to study the co-operating or competing rupture mechanisms inductile mixed mode fracture discussed above. Of special interest is the transition betweenthe different failure mechanisms with respect to the stress state, here characterized by thestress triaxiality T and the Lode parameter µ. For this reason, in Paper A a double notchedtube specimen is used and tested in combined tension and torsion, giving rise to variationsin the Lode parameter. The triaxiality is controlled and kept constant throughout the testby keeping the tension to torsion ratio fixed. A decrease in the torsion portion of the loadinggives an decrease in the triaxiality, and vice versa. Two different materials are tested, a highstrength steel Weldox 960 and a medium strength steel Weldox 420. The average effectiveplastic strain over the notch at failure is determined from the experiments. All the testswhere analyzed by means of finite elements and the effective plastic strain in the centre ofthe notch at failure was determined for each test. The stress state at failure in the notchregion was also carefully analyzed. Failure loci for the two materials are constructed, wherethe strains at failure are plotted vs. T at failure and µ vs. T at failure. A abrupt change

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Ductile failure and rupture mechanisms in combined tension and shear

in trend in the failure loci is clearly noted, indicating a transition in rupture mechanism.By examining the fracture surfaces systematically with a scanning electron microscope thetransition between the two rupture mechanisms, necking of intervoid ligament and shearingof intervoid ligament, could be mapped.

In Paper B, a micromechanics analysis of the observed mechanisms from the experimentalwork is performed. The micromechanics model employed consists of an array of equally sizedcells located within a planar band, where each cell contains a spherical void located at itscentre. The periodic arrangement of the cells allows the study of a single unit cell. The unitcell is loaded in a way such that it resembles the stress state, T and µ, in the centre of thenotch at failure for each test, obtained from Paper A. It was found that at high triaxialitythe dominating rupture mechanism is growth and internal necking of the ligaments betweenvoids. However, at low triaxiailty and near a generalized shear state of stress the presenceand growth of voids does not play a significant role. Here rupture occurs by internal shearingbetween voids and seems to be governed by a simple shear deformation criteria postulatedin Paper B.

Future Work

The main objective of this project is to micromechanically model mixed mode ductile frac-ture and the goal is to formulate mechanism based ductile rupture criteria and develop anonlinear computational fracture mechanics tool. In order to do so there are supplementaryissues that need to be addressed and additional work remain to be done. The following itemsserve as a good guideline for future work:

• Perform a comprehensive parametric cellmodel study based on the stress state ahead of acracktip in a mixed mode loading situation, with the initial void size, the stress triaxialityand the Lode parameter as model parameters.

• Build a continuum model based on the rupture mechanisms in combined tension andshear, where void coalescence accounts for T and µ.

• Implementation of the continuum model in a material subroutine and carry out mixedmode experiments to validate the model.

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Imad Barsoum – KTH Solid Mechanics

References

Barsoum, I., 2003. Ductile Mixed Mode Fracture - a litterature review. Dep. of Solid Me-chanics, Royal Institute of Technology, Stockholm, Sweden.

Benzerga, A. A., 2002. Micromechanics of coalescence in ductile fraacture. Journal of Me-chanics and Physics of Solids 50, 1331–1362.

Gao, X., Faleskog, J., Shih, C. F., Dodds, R. H., 1998. Dutcile tearing in part-through cracks:experiments and cell-model predictions. Engineering Fracture Mechanics 59, 761–777.

Garrison Jr, W. M., Moody, N. R., 1987. Ductile fracture. Journal of Physics and Chemistryof Solids 48, 1035–1074.

Ghosal, A. K., Narasimhan, R., 1996. Numerical simulations of hole growth and ductilefracture initiation under mixed-mode loading. International Journal of Fracture MechanicsA211, 117–127.

Gurson, A. L., 1977. Continuum theory of ductile rupture by void nucleation and growth:Part I - yield criteria and flow rules for porous ductile media. Journal of EngineeringMaterials and Technology, 2–17.

Hallback, N., 1996. Mixed Mode Fracture in Homogeneous Materials. PhD thesis, Dep. ofSolid Mechanics, Royal Institute of Technology, Stockholm, Sweden.

Larsson, S. G., Carlsson, A. J., 1973. Influence of non-singular stress terms and specimengeometry on small-scale yielding at crack tips in elastic-plastic materials. Journal of Me-chanics and Physics of Solids 21, 263–277.

Laukkanen, A., 2002. Applicibility of gurson-tvergaard constitutive model to characterizemixed-mode and mode II. Technical Report VALB350, VTT Manufacturing Technology.

Lode, W., 1925. The influence of the intermediate principal stress on yielding and failure ofiron, copper and nickel. Eng. Math. Mech 5, 142.

McClintock., F. A., 1968. A criterion for ductile fracture growth of holes. Journal of AppliedMechanics 35, 363.

Narasimhan, R., Roy, Y. A., Arora, P. R., 1999. An experimental investigation of constrainteffects on mixed mode fracture initiation in a ductile aluminium alloy. Acta Materialia45, 1587–1596.

Pardoen, T., Hutchinson, J. W., 2000. An extended model for void growth and coalescence.Journal of Mechanics and Physics of Solids 48, 2467–2512.

Rice, J. R., 1973. Limitations to small scake yielding approximation for crack tip plasticity.Journal of Mechanics and Physics of Solids 22, 17–26.

Rice, J. R., Tracey, D. M., 1969. On the ductile enlargement of voids in triaxial stress fields.Journal of Mechanics and Physics of Solids 17, 201–217.

Tvergaard, V., Needleman, A., 1984. Analysis of the cup-cone fracture in a round tensilebar. Acta Metallurgica 32, 157–169.

Van Stone, R. H., Cox, T. B., Low, J. R., Psioda, J. A., 1985. Microstructural aspects offracture by dimpled rupture. International Metals Reviews 30, 157–179.

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