Regularization of non-convex strain energy function for non-monotonic stress–strain relation in the Hencky elastic–plastic model

Abstract

The boundary value problem of small static deformation range is formulated as the variational problem for the displacement. The general local conditions of convexity and rank-one-convexity are obtained for smooth elastic–plastic potentials depending on two first convex invariants of the Cauchy strain tensor. It is demonstrated that all types of convexity are equivalent for the Hencky elastic–plastic potential, but, for example, the strain energy function of continuum fracture can have every type of convexity for different relationships between material parameters. The classical regularization method is applied for the construction of the lower convex envelope of the Hencky strain energy potential for the experimental non-monotonic stress–strain relation with falling part. For this relation, the concept of the Maxwell line is used by analogy with the Van der Waals’ gas theory. The results of 1-D and 2-D numerical examples show that this regularization is principally necessary for the convergence of the finite-difference methods that are usually used for numerical computation within the framework of the small deformation models with non-monotonic stress–strain relations.