Compatibility conditions for a general problem were formulated within the constitutive framework of finite elasto-plasticity. No potentiality condition for an invertible second-order tensor field, F, due to the significance of F being a plastic distortion, was assumed. The affine connection, F, with vanishing Riemann curvature, was thought to be a plastic connection. In order to ensure the existence of a continuously distributed dislocation, the Cartan torsion attached to the connection was supposed to be non-zero. The principal result concerned the compatibility conditions which were viewed, for a given symmetrical and positive definite tensor (the metric tensor), as being partial differential equations for the torsion (defined in terms of the second-order torsion tensor). The non-zero torsion was essential here while, in finite elasticity, the compatibility conditions were formulated in terms of zero torsion. Various implications of the theorem could be deduced relative to the torsion, concerning the evolution equations for the pair of plastic distortion and plastic connections. That is, only the evolution equation for the plastic metric had to be defined because the torsion could be defined as being a solution of the appropriate partial differential equations within the approach to finite elastoplasticity with a plastic connection having a zero fourth-order curvature tensor, Also, if no relationships between the plastic metric and (plastic) torsion were imposed, then the evolution equations could be defined for plastic distortion as well as for plastic connection.

Torsion Equation in Anisotropic Elasto-Plastic Materials with Continuously Distributed Dislocations. S.Cleja-Tigoiu, D.Fortune, C.Vallee: Mathematics and Mechanics of Solids, 2008, 13[8], 667-89