Modeling of the response of elastic plastic materials treated as a mixture of hard and soft regions

Inelastic materials that form dislocation cells on being deformed are modeled as a constrained-mixture of plastically hard and soft regions by associating different natural states with these regions. The deformation gradient from the reference configuration to the natural configuration is identified as the plastic deformation tensor and the stress is measured from a changing set of natural configurations. Two sets of natural configurations are introduced: one for the hard phase and the other for the soft phase. The full elastic response of the body is determined by elastic responses from different natural configurations. The energy stored in the dislocation networks is explicitly accounted for in the Helmholtz potential. Within a specialized constitutive set up, the soft phase is assumed to be non-hardening while the hardening response of the hard phase is dependent upon the response of both the hard and soft phases. These special forms are used to model the response of the material that forms cellular structures when subjected to cyclic loading.