The geometric paradigm in computational elasto-plasticity

Computational methods, for large displacements of continua in the elasto-plastic range, rely on the mathematical modeling of the nonlinear constitutive behavior. In last decades an increasing favor has been deserved to nonlinear models based on a chain decomposition of the deformation gradient. The troubles involved in a structural analysis based on this model are well-known and have not been overcome although many efforts devoted to this end. Our investigation towards a more satisfactory model starts from the new analysis of the rate elastic behavior performed in [1,2] since the difficulties faced by previous formulations were the very motivation for the discard of rate constitutive models in elasto-plasticity [3]. The new definition of hypo-elasticity, the detection of simple integrability conditions and a new formulation of conservativeness, lead to a definition of rate elasticity suitable for an effective modeling of rate elasto-plastic constitutive behaviors [4]. The treatment is based on a geometric definition of spatial and material fields and on the statement of a geometric paradigm assessing the rules for comparison of material fields naturally provided by pushpull according to the relevant transformation. The rates involved in constitutive relations are Liederivatives of stress field and constitutive parameters. Geometric compatibility requires that elastic and plastic stretchings additively give the Lie-derivative of metric field. No privileged reference configuration is involved and no consequent multiplicative decomposition of deformation gradient is assumed. Computational methods are shown to be based on the pull-back of constitutive relations to a straightened out trajectory segment which plays the role of computation chamber wherein linear operations of differentiation and integration may be performed. Accordingly, finite elastic and plastic stretches are considered as purely computational tools with no physical interpretation in constitutive relations. Both 3D and lower dimensional structural models, such as wires and membranes, may be analysed by a direct application of the theory. The outcome is a significant improvement of physical insight and computational effectiveness with respect to previous treatments of finite elasto-plasticity.