Elasto-plastic materials with lattice defects modeled by second order deformations with non-zero curvature

Abstract

The paper deals with elasto-plastic materials, whose structural damage is produced by lattice defects, within the constitutive framework of finite elasto-plasticity with second order deformations. For second order deformations, which are represented by distortions and connections, we adopt the decomposition rule into second order elastic and plastic components. The split of the plastic connection into independent components, which involve Bilby’s connection type and a second order tensor, called disclination, plays a fundamental role in defining kinematical variables. The lattice defects are modeled by dislocations and disclinations, which are introduced within the finite deformation formalism and based on the differential geometry arguments. The constitutive framework is compatible with the free energy imbalance, postulated in the configuration with torsion, in such a form to include the presence of the disclinations. The model includes macro forces, which are power conjugated with the appropriate rate of the second order elastic deformations, and micro forces which are related with the plastic behaviour, as well as with the disclinations. The micro balance equations which are associated with the presence of disclinations are derived from a virtual power principle relative to the disclination mechanism. As a consequence of the principle of the free energy imbalance, the free energy function becomes potential for the macro forces. The viscoplastic type equations for micro forces are proposed in such a way to be compatible with the resulting dissipation inequality. Finally, we determine within the proposed constitutive framework, the disclinations as a solution of the micro balance equation associated with the disclination and satisfying the appropriate evolution equation. We suppose that the plastic simple shear deformation is the source for the evolution of disclination. After our knowledge this is a new framework for the mathematical description of disclinations.