Ductile failure and rupture mechanisms
in combined tension and shear
Imad Barsoum
Licentiate thesis no. 96, 2006 TRITA, HFL – 0407 KTH Solid Mechanics SE-100 44 Stockholm, Sweden
Preface
The work in this licentiate thesis was carried out at the Department of Solid Mechanics at the Royal Institute of Technology (KTH) between September 2002 and March 2006. The work was 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, guidance and support during the span of this work. Thank you for giving me the opportunity to work with you.
I would also like to thank the head of the laboratory Mr. Hans ¨Oberg for helping out with the experimental part and the design of the fixture. Thanks go to Messrs. Bertil Dolk, Bengt M¨ollerberg, Martin ¨Oberg and Kurt Lindqvist for manufacturing the specimens. Dr. Torbj¨orn Narstr¨om at SSAB Oxel¨osund is acknowledged for supplying the material.
I am also thankful to my colleagues that have contributed to a stimulating environment at the 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 Fadi for their endless support and limitless love. Thank you for beeing with me along the road of life all these years. God bless each one of you.
Finally, I would like to express my profound gratitude to my fianc´e Gorgina Zeitoun. It feels great to have you in my life!
Stockholm, May 2006
List of Appended Papers
Paper A: Rupture mechanisms in combined tension and shear—experiments Imad Barsoum and Jonas Faleskog
Submitted to International Journal of Solids and Structures
Paper B: Rupture mechanisms in combined tension and shear—micromechanics Imad Barsoum and Jonas Faleskog
Ductile failure and rupture mechanisms in combined tension and shear
Introduction
The industrial application of high strength steels and high performance aluminum alloys in structural components have increased the demand on understanding the ductile failure behavior 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, resulting in 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 well known (Van Stone et al. (1985), Garrison Jr and Moody (1987)). The modeling of this failure mechanism 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 stress state and is often referred to as flat dimple rupture, which is shown in Figure 1. Here the final link up of the enlarged voids take place by necking of the intervoid ligaments. In a mode II or III loading situation however, the stress state ahead of the crack is altered resulting in 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 hydrostatic stress state in such a case impedes the growth of voids ahead of the crack tip resulting in small elongated dimples at fracture as shown in Figure 2. This mechanism is often referred to as 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 favorable near the mode I loading, whereas the shear localization mechanism leading to shear dimple rupture is favorable near mode II or III loading. Clearly there are two governing ductile rupture mechanisms, which will either compete or co-operate under mixed mode loading situation leading to ductile failure.
Mixed mode loading conditions
Previous studies by Ghosal and Narasimhan (1996), Laukkanen (2002) have attempted to relate the transition in micromechanics to the altering of the continuum fields, such as stress triaxiality T and effective plastic strain ¯εp, which depend on the mode mixity. The stress
triaxiality 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 for the symmetric mode I to 1 for the antisymmetric mode II or III. Laukkanen (2002), among others, 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 where
the maximum values of the triaxiality and the effective plastic strain are also a function of the mode mixity and hence the preferred macroscopic crack growth direction will depend
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.
Ductile failure and rupture mechanisms in combined tension and shear
upon the mode mixity (Hallb¨ack (1996), Narasimhan et al. (1999), Barsoum (2003)). In a mixed mode I/III loading however T and ¯εp will operate in the same plane ahead of the
crack and thus enhance each other.
In near mode II or III loading, the region ahead of the crack tip will experience extensive shearing and decreased triaxiality, whereas in mode I the region ahead of the crack tip will experience extensive tension and increased triaxiality. The rupture mechanisms are however distinctively 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 consequently not the sole parameter governing ductile rupture in mixed mode loading as will be shown in 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 ahead of the crack tip.
The Lode parameter (Lode (1925)) µ, which is related to the third invariant of the deviatoric stress 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 corresponds
to generalized tension, µ = 0 generalized shear and µ = 1 generalized compression. The stress state is now characterized by T and µ, which describe the stress state ahead of the crack 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 function of the mode mixity βI−II and βI−III defined in Eq. (2). KI, KII and KIII are the stress
intensity 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 modified
boundary layer analysis are shown in Figure 3, where solid lines corresponds to mixed mode I/II and dot-dashed line to mixed mode I/III loading. Here, T and µ are determined near the 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 in
Figure 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. In 3(b), where µ vs. the mode mixity is plotted, it is seen that the stress state goes toward generalized shear, µ = 0, for increased mode II or III loading in the direction where T is maximum. For the elastic-ideally plastic material and mixed mode I/II, µ does not vary markedly with the mode mixity. It can also be noted that for both materials the stress state at pure mode I loading is not generalized tension, µ = −1, as might be expected. In the direction 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 in the direction where the effective plastic strain is maximum. Obviously, the changes in stress
Imad Barsoum – KTH Solid Mechanics
state near the crack tip with respect to the mode mixity, Figure 3, have a strong influence on the characteristics in the rupture mechanisms as seen in Figure 1 and 2.
βI−II = 2 π arctan(KII/KI), βI−III = 2 πarctan(KIII/KI) (2) 0 0.5 1 0 1 2 3 max T max ¯εp βI −I I, βI −I I I T N= 0 N= 0.1 N= 0 N = 0.1 (a) 0 0.5 1 −1 −0.5 0 βI −I I, βI −I I I µ N = 0, N = 0.1 max ¯εp max T N = 0 N= 0.1 (b)
Figure 3. The stress state ahead of a crack tip obtained from an elasto-plastic modified boundary layer 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/II loading 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 in ductile mixed mode fracture discussed above. Of special interest is the transition between the different failure mechanisms with respect to the stress state, here characterized by the stress triaxiality T and the Lode parameter µ. For this reason, in Paper A a double notched tube specimen is used and tested in combined tension and torsion, giving rise to variations in the Lode parameter. The triaxiality is controlled and kept constant throughout the test by keeping the tension to torsion ratio fixed. A decrease in the torsion portion of the loading gives an decrease in the triaxiality, and vice versa. Two different materials are tested, a high strength steel Weldox 960 and a medium strength steel Weldox 420. The average effective plastic strain over the notch at failure is determined from the experiments. All the tests where analyzed by means of finite elements and the effective plastic strain in the centre of the notch at failure was determined for each test. The stress state at failure in the notch region was also carefully analyzed. Failure loci for the two materials are constructed, where the strains at failure are plotted vs. T at failure and µ vs. T at failure. A abrupt change
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 the transition between the two rupture mechanisms, necking of intervoid ligament and shearing of intervoid ligament, could be mapped.
In Paper B, a micromechanics analysis of the observed mechanisms from the experimental work is performed. The micromechanics model employed consists of an array of equally sized cells located within a planar band, where each cell contains a spherical void located at its centre. The periodic arrangement of the cells allows the study of a single unit cell. The unit cell is loaded in a way such that it resembles the stress state, T and µ, in the centre of the notch at failure for each test, obtained from Paper A. It was found that at high triaxiality the dominating rupture mechanism is growth and internal necking of the ligaments between voids. However, at low triaxiailty and near a generalized shear state of stress the presence and growth of voids does not play a significant role. Here rupture occurs by internal shearing between voids and seems to be governed by a simple shear deformation criteria postulated in 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 rupfrac-ture criteria and develop a nonlinear computational fracture mechanics tool. In order to do so there are supplementary issues that need to be addressed and additional work remain to be done. The following items serve as a good guideline for future work:
• Perform a comprehensive parametric cellmodel study based on the stress state ahead of a cracktip in a mixed mode loading situation, with the initial void size, the stress triaxiality and the Lode parameter as model parameters.
• Build a continuum model based on the rupture mechanisms in combined tension and shear, where void coalescence accounts for T and µ.
• Implementation of the continuum model in a material subroutine and carry out mixed mode experiments to validate the model.
Imad Barsoum – KTH Solid Mechanics
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