A one-dimensional theory of strain-gradient plasticity : Formulation, analysis, numerical results

This study develops a one-dimensional theory of strain-gradient plasticity based on: (i) a system of microstresses consistent with a microforce balance; (ii) a mechanical version of the second law that includes, via microstresses, work performed during viscoplastic flow; (iii) a constitutive theory that allows • the free-energy to depend on the gradient of the plastic strain, and • the microstresses to depend on the gradient of the plastic strain-rate. The constitutive equations, whose rate-dependence is of power-law form, are endowed with energetic and dissipative gradient length-scales L and l, respectively, and allow for a gradient-dependent generalization of standard internal-variable hardening. The microforce balance when augmented by the constitutive relations for the microstresses results in a nonlocal flow rule in the form of a partial differential equation for the plastic strain. Typical macroscopic boundary conditions are supplemented by nonstandard microscopic boundary conditions associated with flow, and properties of the resulting boundary-value problem are studied both analytically and numerically. The resulting solutions are shown to exhibit three distinct physical phenomena: (i) standard (isotropic) internal-variable hardening; (ii) energetic hardening, with concomitant back stress, associated with plastic-strain gradients and resulting in boundary layer effects; (iii) dissipative strengthening associated with plastic strain-rate gradients and resulting in a size-dependent increase in yield strength.