A local viewpoint for evaluating the influence of stress triaxiality and Lode angle on ductile failure and hardening
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 σEq(εEq) 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 σEq 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.
Section snippets
Experiments and procedure for material characterization
According to the classical theory of elastoplasticity, the stress--plastic strain incremental relationship is approximated as pressure-independent so that the isotropic hardening σEq(εEq), based on the von Mises stress, is sufficient to describe the behavior of a ductile metal when kinematic hardening is negligible.
Recent studies by Wierzbicki et al., 2005, Barsoum and Faleksog, 2007, starting from concepts applying to soils and granular materials, postulated that the evolution of the yield
Discussion of the failure models
Damage models as well as failure criteria allow to predict failure of structures undergoing different load histories, usually by expressing the failure configuration in terms of equivalent plastic strain at the material point where failure firstly occurs in the structure.
Damage models also describe the progressive deterioration of the material performance through dissipative functions which influence the stress--strain relationship; on the contrary, failure criteria do not affect the
Finite elements and failure calculations
Finite elements simulations of experimental tests are performed with the commercial code MSC-MARC. Four-noded axisymmetric elements are used for modeling the round specimens, while the notched flat samples are modeled by 8-noded hexahedral elements; displacements are imposed at the prescribed nodes for simulating the motion of the test machine crosshead or that of the loading pins. All analyses are based on the updated Lagrangian formulation for large displacements and finite strains, additive
Main results and considerations
The first finding which, according to the authors’ knowledge, is proposed in this work for the first time, is that the intuitive relationship between fracture and notch sharpness of round specimens, expressing that failure strains are as lower as the notches are sharper, has not a general validity. In fact, when the notch is very acute and failure initiates at the outer surface of round bars, the local strain may be much higher than it is for less severe notches failing at the neck center,
Conclusions
The prediction of ductile failure according to different theories is performed in this paper with reference to round smooth bars, round notched bars and notched flat specimens.
Non-linear finite elements analyses are ran with and without a failure modeling subroutine for deleting elements when they reach a critical condition according to the Wierzbicki W1 failure model. Also failure predictions according to the Tresca model and to the Wierzbicki W2 model are performed by using results from the
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