Metal ductility at low stress triaxiality application to sheet trimming

Abstract

The growth and coalescence of voids nucleated by decohesion or cracking of second phase particles is a common damage process for many metallic alloys. Classical damage models, based on void growth and coalescence, predict a ductility increase if the stress triaxiality is decreased. But experiments show that the material ductility decreases at very low stress triaxialities typical of sheet metal forming operations. At very low stress triaxiality no void growth is observed in metals containing second phase particles. In the present work, a new damage model for metals containing second phase particles submitted to low stress triaxiality loading is proposed.

The new model is based on the observed physical damage mechanism, i.e. strain localization by reducing the inter-particle spacing during large material rotations. A two step modelling strategy has been followed to determine the ductility at low stress triaxiality. In the first step Thomason's void coalescence model is extended to large material rotations and shearing. In the second step the principles of applying this model to damage nucleated at second phase particles are described. The large material rotations observed under low stress triaxiality loading lead to large changes in the microstructure. Thus, in the second step, first appropriate representative volume and material elements are determined and then the critical damage parameters. Finally, as an example, the trimming behaviour of two aluminium sheet alloys is analyzed by the new model and the model predictions shown to be in good agreement with the experimental data, in particular for the blade displacement to crack initiation. The main outcomes of this work are: (1) a void coalescence model valid at low stress triaxiality, (2) a damage criterion valid at small stress triaxiality and large material rotations, (3) a damage variable expressed in a simple closed form for materials containing second phase particles. The damage analysis in a small fixed volume with a representative microstructure (Eulerian approach) and the damage analysis in all the material elements of the considered structure are compared in detail. In trimming (or similar processes), the major contribution to damage the material movement bringing second phase particles closer together and the void growth may be neglected. This simplifies considerable the analysis.

Introduction

In ductile metals, voids are first nucleated by decohesion or cracking of second phase particles and then grow until they eventually coalescence to form a macroscopic crack. In the early 1970s a new branch of material science (“the local approach of ductile fracture”), was initiated by the pioneering contributions of Gurland and Plateau (1963), McClintock, 1968a, McClintock, 1968b, Rice and Tracey (1969), Rice and Johnson (1970), Needleman (1972), Brown and Embury (1973), Argon (1976), Gurson (1977), Chu and Needleman (1980), Beremin, 1981a, Beremin, 1981b, Lautridou and Pineau (1981), Tvergaard, 1981, Tvergaard, 1982, Budiansky et al. (1982), Browning et al. (1983) and Needleman and Tvergaard (1984). One outcome of all these analyses is the major role played by the stress triaxiality.

Large values of the stress triaxiality (T > 0.4) lead to a decrease of the fracture strain (ductility). Thus, during the last 20 years, most efforts concerned essentially medium to high stress triaxiality loadings (Benzerga et al., 2004a, Benzerga et al., 2004b, Ragab, 2004, Besson, 2004, Huber et al., 2005, Klöcker and Tvergaard, 2003, Gologanu et al., 1995, Gologanu et al., 2001, Siruguet and Leblond, 2004, Pardoen and Hutchinson, 2000, Pardoen and Hutchinson, 2003, Shabrov and Needleman, 2002, Gammage et al., 2004, Worswick et al., 2001, Thomson et al., 2003, Kim et al., 2004, Pardoen et al., 2003, Lee and Mear, 1999, Christman et al., 1989, Joly, 1992, Koplik and Needleman, 1988, Hom and McMeeking, 1989a, Hom and McMeeking, 1989b, Worswick and Pick, 1990, Brocks et al., 1995, Steglich and Brocks, 1997, Gao et al., 1998, Faleskog and Shih, 1997, Kuna and Sun, 1996, Bordreuil et al., 2003, Kroon and Faleskog, 2005, Doghri and Ouaar, 2003, Babout et al., 2004, Leblond et al., 1994, Leblond et al., 1995, Scheyvaerts, 2006, Pineau, 1992, Huang, 1991). But, in many forming operations, the material is submitted to large compressive or small tensile loads corresponding to negative or small stress triaxiality loadings. In this context, Bao and Wierzbicki (2004) deformed an AA2024 alloy under various stress strain paths covering small and large values of the stress triaxiality; their results are summarized in Fig. 1. The high stress triaxility results are well described by several classical damage models (Gurland and Plateau, 1963, McClintock, 1968a, McClintock, 1968b, Rice and Tracey, 1969, Rice and Johnson, 1970, Needleman, 1972, Brown and Embury, 1973, Argon, 1976, Gurson, 1977, Chu and Needleman, 1980, Beremin, 1981a, Beremin, 1981b, Lautridou and Pineau, 1981, Tvergaard, 1981, Tvergaard, 1982, Budiansky et al., 1982, Browning et al., 1983, Needleman and Tvergaard, 1984, Benzerga et al., 2004a, Benzerga et al., 2004b, Ragab, 2004, Besson, 2004, Huber et al., 2005, Klöcker and Tvergaard, 2003, Gologanu et al., 1995, Gologanu et al., 2001, Siruguet and Leblond, 2004, Pardoen and Hutchinson, 2000, Pardoen and Hutchinson, 2003, Shabrov and Needleman, 2002, Gammage et al., 2004, Worswick et al., 2001, Thomson et al., 2003, Kim et al., 2004, Pardoen et al., 2003, Lee and Mear, 1999, Christman et al., 1989, Joly, 1992, Koplik and Needleman, 1988, Hom and McMeeking, 1989a, Hom and McMeeking, 1989b, Worswick and Pick, 1990, Brocks et al., 1995, Steglich and Brocks, 1997, Gao et al., 1998, Faleskog and Shih, 1997, Kuna and Sun, 1996, Bordreuil et al., 2003, Kroon and Faleskog, 2005, Doghri and Ouaar, 2003, Babout et al., 2004, Leblond et al., 1994, Leblond et al., 1995, Scheyvaerts, 2006, Pineau, 1992, Huang, 1991). However, when the stress triaxiality is small (T ∈ [−0.3,0.4]), these classical models significantly over-estimate the strain to failure. To our knowledge there is no adequate model to predict ductility at low stress triaxiality.

In the present work, a new damage model for metals containing second phase particles submitted to low stress triaxiality loading and large material rotations is proposed. During forming of heterogeneous materials, the second phase particles undergo large movements (Fig. 2). Thus, the particle–matrix interface is broken, but, since the stress triaxiality is low, no significant void growth is observed (as will be confirmed here). As the broken matrix particle interface no longer carries any load, it is proposed that failure occurs due to the reduction of interparticle spacing and subsequent strain localization.

A two step modelling strategy is followed in this paper to determine the ductility, i.e. the local strain to failure at low stress triaxiality (Fig. 2). In the first step (Section 2), strain localization between rectangular voids in a sheared microstructure is considered (Fig. 2c and d). This first step, an extension of Thomason's void coalescence model (Thomason, 1990), determines the critical void arrangement as a function of the local stress. Similar approaches were used previously to describe void coalescence under medium or high triaxiality loading (Benzerga et al., 2004a, Benzerga et al., 2004b, Pardoen and Hutchinson, 2003). But under medium or high stress triaxiality loading the material does not undergo large movements before void coalescence. This point is addressed in the second step. The second step (Section 3) explains how to determine the critical volume (material) element leading to crack initiation and final failure in a material containing second phase particles (Fig. 2a and b). First appropriate representative volume and materials element are determined and then the critical damage parameters. The strain leading to void nucleation is considered negligible compared to the failure strain. In Section 4 the new damage criterion is applied to sheet metal cutting. Metal cutting leads to very large deformations and very large rotations of the material elements and the microstructure. The new model predictions are compared to the experimental values of the blade penetration at crack initiation.