A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. Part I: Small deformations

Abstract

This study develops a small-deformation theory of strain-gradient plasticity for isotropic materials in the absence of plastic rotation. The theory is based on a system of microstresses consistent with a microforce balance; a mechanical version of the second law that includes, via microstresses, work performed during viscoplastic flow; a constitutive theory that allows: • the microstresses to depend on $\nabla {\dot{\mathbf{E}}}^{\mathrm{p}}$, the gradient of the plastic strain-rate, and • the free energy $\psi$ to depend on the Burgers tensor $\mathbf{G}=\mathrm{curl}{\mathbf{E}}^{\mathrm{p}}$. The microforce balance when augmented by constitutive relations for the microstresses results in a nonlocal flow rule in the form of a tensorial second-order partial differential equation for the plastic strain. The microstresses are strictly dissipative when $\psi$ is independent of the Burgers tensor, but when $\psi$ depends on $\mathbf{G}$ the gradient microstress is partially energetic, and this, in turn, leads to a back stress and (hence) to Bauschinger-effects in the flow rule. It is further shown that dependencies of the microstresses on $\nabla {\dot{\mathbf{E}}}^{\mathrm{p}}$ lead to strengthening and weakening effects in the flow rule.Typical macroscopic boundary conditions are supplemented by nonstandard microscopic boundary conditions associated with flow, and, as an aid to numerical solutions, a weak (virtual power) formulation of the nonlocal flow rule is derived.