Keywords: plasticity, finite deformations, integration algorithm, anisotropy, exponential algorithm.

The object of this paper is the integration algorithm for nonlinear constitutive models in the context of finite deformation plasticity. The main motivation for the research is in the fact that, at the current state of the art, the majority of the numerical applications present in the literature employ the so-called "exponential integration algorithm", systemized by [1]. The standard model for isotropic elastoplastic materials in finite deformations is revised in the paper and the fundamental assumptions on which it is based are shown. In particular it is stressed that this algorithm can be used only for materials that are isotropic in their elastic and inelastic behavior [2,3,4]. In this case it can be proved that the stress, the elastic deformations and the inelastic deformations are coaxial, so that it is possible a common spectral decomposition. Whenever anisotropic material behaviors are considered, thus, the exponential algorithm fails. This is of special interest in damaging materials, that develop preferential directions according to the level of damage.

In this work non isotropic elasto-plastic materials are considered and. For this kind of materials a full set of constitutive equation is developed on the basis of thermodynamic principles.

The classical kinematic multiplicative decomposition of the deformation gradient tensor in its elastic and inelastic parts is used. The elastic part of the process is assumed to be ruled by an hyperelastic potential fulfilling the objectivity requirements (but not the isotropy restrictions). An explicit dependence of the hyperelastic potential from the elastic volumetric deformation is considered (that induces additional complications to the numerical solution).

The inelastic evolution laws have been obtained consistently with the maximum dissipation principle, in order to develop a sound integration algorithm.

The constitutive equations are integrated in full tensorial form, after introducing the hypothesis that either the covariant anelastic velocity gradient, that gives the instantaneous rate of change of the intermediate configuration, or the material derivative of the anelastic deformation gradient be constant in the step. This differs from the exponential algorithm, that assumes constant the spatial anelastic gradient of velocity.

The algorithm proposed is compared with standard exponential algorithm analyzing isotropic problems. The numerical efficiency appears to be as good as that of standard algorithms.

1

J.C. Simo. Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory. Comp. Meth. Appl. Mech. Engrg., 99:61-112, 1992. doi:10.1016/0045-7825(92)90123-2

2

A. Ibrahimbegovic and L. Chorfi. Covariant principal axis formulation of associated coupled thermoplasticity at finite strains and its numerical implementation. Int. J. Solids Struct., 39:499-528, 2002. doi:10.1016/S0020-7683(01)00221-9

3

C. Miehe. A formulation of finite elastoplasticity based on dual co- and contravariant eigenvector triads normalized with respect to a plastic metric. Comp. Meth. Appl. Mech. Engrg., 159:223-260, 1998. doi:10.1016/S0045-7825(97)00273-9

4

G. Meschke and W.N. Liu. A re-formulation of the exponential algorithm for finite strain plasticity in terms of cauchy stresses. Comp. Meth. Appl. Mech. Engrg., 173:167-187, 1999. doi:10.1016/S0045-7825(98)00267-9

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