An analytic solution for one-dimensional dissipational strain-gradient plasticity

Abstract

An analytic solution is developed for the one-dimensional dissipational slip gradient equation, first described by Gurtin.. [ " On the plasticity of single crystals; free energy, microforces, plastic strain-gradients", J.Mech. Phys. Solids 48 (200) 989-1036] and then investigate numerically by Annand et al, ["A one-dimensional theory of strain-gradient plasticity: formulation, analysis, numerical results". J. Mech.Phys. Solids 53 (2005) 1798-1826]. However, we find that the analytic solution is incompatible with the zero-sliprate boundary condition ("clamped boundary condition") postulated by these authors, and is in fact excluded by the theory. As a consequence, the analytic solution agrees with the numerical results except near the boundary. The equation also admits a series of higher mode solutions where the numerical result corresponds to (a particular case of) the fundamental mode. Anand et al also established that the one-dimensional dissipational gradients strengthen the material, but this proposition only holds if zero-sliprate boundary conditions can be imposed, which we have shown cannot be done. Hence the possibility remains open that dissipational gradient weakening may also occur. doi:10.1017/S1446181109000066