A local viewpoint for evaluating the influence of stress triaxiality and Lode angle on ductile failure and hardening

Abstract

All damage and failure models, describing either the evolution of microvoids, the development of shear bands or local rupture, rely on the knowledge of the hardening function at large plastic strains which, then, becomes an essential prerequisite for any failure prediction.

The phenomenon of ductile failure is analyzed here by focusing on its relationship with the variables for the stress--strain characterization, and by discussing the influence of plastic strain, stress triaxiality and Lode angle parameters on both the above aspects of materials behavior.

Failure predictions are presented for different metals and different combinations of load-specimen geometry, according to three theories (the Tresca criteria and two models by Wierzbicki et al.) and to a procedure previously developed for the stress--strain characterization in the post-necking range.

Experimental tests are performed by pulling tensile specimens and notched flat samples up to failure, then finite elements simulations are used to calculate the required failure-related variables within the volume of failing specimens; the results of the failure calculations are compared each other and with experimental data, and a discussion about the peculiarities of the methods used for predicting failure is also provided.

Introduction

The fracture of ductile metals occurs after microvoids or shear bands develop and evolve within the metal matrix, usually around inclusions or other discontinuities which locally act as precursors. Early theories about failure of elastoplastic metals assumed that the equivalent plastic strain alone was capable of representing a form of damage so that the achievement of a limit strain alone was assumed as the critical condition for failure initiation. Successive and more sounding theories demonstrated that also the stress triaxiality, together with the plastic strain, was a key factor in determining how the damage evolves in metals and how its critical values may trigger local failure. Many models describing possible relationships between plastic strain, triaxiality and voids damage have been proposed since McClintock, 1968, Rice and Tracey, 1969 presented their work and considerable experimental evidence supported it, such as Hancock and Mackenzie, 1976, Mackenzie et al., 1977.

The continuous damage mechanics due to Lemaitre, 1985, Chaboche, 1988, Chaboche, 2008 is based on the principle that damage growth causes a partial release of the elastic strain energy stored within the material, and many models like Bonora, 1997, Wang, 1992, La Rosa et al., 2001, Haddag et al., 2008, have been derived and are still progressing from this conceptual root. Many CDM models, for determining material constants from tensile tests assumed that proportional loading occurs, but this often led to neglecting the unavoidable deviations from uniaxiality due to the post-necking phase of every tensile test with ductile metals.

An alternative to the CDM approach is represented by the Gurson-derived models like Gurson, 1977, Tvergaard and Needleman, 1984, Gologanu et al., 1997, Pardoen et al., 1998, Ragab, 2004, Sanchez et al., 2008, Hsu et al., 2009, in which the damage variable, obtained from variational principles applied to a spherical enlarging void within a spherical unit volume, is coupled to the yielding function. The Gurson-derived models improved the originally poor ability of the Gurson model in predicting pure shear failure, but, for identifying a material, a large number of constants is required such as nucleation strain, void volume fraction, critical void fraction, statistical parameters for triggering the nucleation of new voids, etc.

Other models relating ductile failure to stress triaxiality and plastic strain are based either on energetic considerations, Rice--Tracey variational approach like Le Roy et al., 1981, Johnson and Cook, 1985, Chaouadi et al., 1994, Schiffmann et al., 2003, Mirone, 2004a, Mirone, 2004b, or on phenomenology as for Bao and Wierzbicki (2004).

Recently, a further variable has been claimed to influence either the ductile failure by Wierzbicki et al., 2005, Barsoum and Faleksog, 2007, Brunig et al., 2008 or the stress--plastic strain relationship by Bigoni and Piccolroaz, 2004, Bai and Wierzbicki, 2007; it is the third invariant of the deviatoric stress tensor also expressed as the Lode angle parameter, well known to scientists working with granular materials.

Failure models uncoupled from the yield function require the preliminary determination of the stress--strain histories all over a mechanical component (usually by way of finite elements), while, for coupled models, the damage evolution is evaluated together with the damage-induced transformations of the yield surface and to the stress--strain histories; then, for every failure calculation, the knowledge of a reliable function relating the equivalent stress to the equivalent plastic strain for the metal at hand is mandatory. Uncoupled models may be less physically sounding and possibly also less accurate than coupled models, but they are much simpler and can give reasonable estimations so that their use is very common in engineering applications.

The effect of the Lode angle on the stress--strain behavior of metals is usually neglected, so the material stress--strain function obtained from smooth axisymmetric tensile specimens is assumed to be representative of whatever isotropic hardening for that material; this induces approximations especially when the material curve is used for simulating stress--strain histories close to plane strain.

Also the hydrostatic pressure is claimed to play a role in modifying the yield surface as in Wierzbicki, 2007, Bai and Wierzbicki, 2007, but its influence is thought to be less pronounced than that of the Lode angle and is as well neglected in usual engineering calculations.

The function σ_Eq(ε_Eq) can be easily determined by experiments under the hypotheses of uniaxiality and uniformity of stress and strain within the specimen volume, but these hypotheses are only verified for smooth specimens prior to the necking occurrence.

Well developed necking induces increasing pressure fields and triaxial non-uniform stress states whose effects on the calculation of σ_Eq have to be accounted for. This is made through a corrective function able to transform the experimental true stress into an estimation of the equivalent stress. The most known model for performing such a correction is due to Bridgman (1956) but modifications to this theory have been proposed for specific cases by Alexandrov and Goldstein, 1998, Zhang et al., 1999, and for coupling the effects of necking instability and Gurson-type microvoids enlargement as in Yaning and Dale (2008). The limits of the Bridgman approach have been outlined in Alves and Jones, 1999, La Rosa et al., 2003, and a new method for performing the post-necking correction, named MLR, was presented by Mirone, 2004a, Mirone, 2004b achieving improvements in terms of accuracy and ease of application.

This paper aims at illustrating the influence of stress triaxiality and Lode angle on metals failure and elastoplastic response, by adopting different failure models and the von Mises plasticity for simulating experiments.

A novel insight in the relationship between notch shape and ductile failure of round specimens is also proposed by focusing on the local nature of failure initiation: detailed analyses of experimental results also coming from the literature allowed to demonstrate that, in some cases, the usual considerations about the behavior of round specimens may hide important aspects of failure and may lead to wrong conclusions.

In the successive sections of this paper, the correction method for the stress--strain characterization of metals is discussed, then three failure models from the literature are analyzed by referring to various experimental data from round tensile bars and from flat samples. Then, finite elements simulations of the tests, performed with and without a failure model triggering elements deletion, are presented in order to perform a local-scale investigation about how pressure and Lode angle act on the behavior of metals.