Torsion Equation in Anisotropic Elasto-Plastic Materials with Continuously Distributed Dislocations

Abstract

Within the constitutive framework of finite elasto-plasticity we formulate compatibility conditions in a general problem, in which no potentiality condition for an invertible second-order tensor field F has been assumed, due to the signification of F to be a plastic distortion. In the problem the affine connection F, with vanishing Riemann curvature, is thought to be a plastic connection. Consequently, in order to ensure the existence of the continuously distributed dislocation, the Cartan torsion attached to the connection is supposed to be non-zero. The principal result concerns the compatibility conditions, which are viewed, for a given symmetric and positive definite tensor (the metric tensor), as partial differential equations for the torsion (defined in terms of the second-order torsion tensor). In our problem, the non-zero torsion is essential, while in finite elasticity the compatibility conditions are formulated in terms of zero torsion. The following implications of the theorem relative to the torsion, concerning the evolution equations for the pair of the plastic distortion and plastic connections, can be drawn: 1. only the evolution equation for the plastic metric has to be defined, because the torsion could be defined as a solution of the appropriate partial differential equations, within the approach to finite elastoplasticity with plastic connection having a zero fourth-order curvature tensor, 2. if no relationships between the plastic metric and (plastic) torsion have been imposed then the evolution equations could be defined for plastic distortion as well as for plastic connection.