Basic Strain Gradient Plasticity Theories with Application to Constrained Film Deformation

A family of basic rate-independent strain gradient plasticity theories is considered that generalize conventional J(2) deformation and flow theories of plasticity to include a dependence on strain gradients in a simple way. The theory builds on three recent developments: the work of Gudmundson (J. Mech. Phys. Solids 52 (2004), 1379-1406) and Gurtin and Anand (J. Mech. Phys. Solids 57 (2009), 405-421), proposing constitutive relations for flow theories consistent with requirements of positive plastic dissipation; the work of Fleck and Willis (J. Mech. Phys. Solids 57 (2009), 161-177 and 1045-1057), who clarified the structure of the new flow theories and presented the underlying variational formulation; and observations of Evans and Hutchinson (Acta Mater. 57 (2009), 1675-1688) related to preferences for specific functional compositions of strains and strain gradients. The starting point in this paper is the deformation theory formulation of Fleck and Hutchinson (J. Mech. Phys. Solids 49 (2001), 2245-2271) which provides the clearest insights into the role of strain gradients and serves as a template for the flow (incremental) theory. The flow theory is constructed such that it coincides with the deformation theory under proportional straining, analogous to the corresponding coincidence in the conventional J(2) theories. The generality of proportional straining is demonstrated for pure power-law materials, and the utility of power-law solutions is illustrated for the constrained deformation of thin films: the compression or extension of a finite layer joining rigid platens. Full elastic-plastic solutions are obtained for the same problem based on a finite element method devised for the new class of flow theories. Potential difficulties and open issues associated with the new class of flow theories are identified and discussed.