A theory of strain-gradient plasticity with effect of internal microforce

This paper develops a theory of strain gradient plasticity for isotropic bodies undergoing small deformation in the absence of plastic spin. The proposed theory is based on a system of microstresses which include a microstress vector consistent with microforce balance; the mechanical form of the second law of thermodynamics which includes work performed by the microstresses during plastic flow; and a constitutive theory that allows the free energy to depend on the elastic strain $\mathbf{E}^{e}$, divergence of plastic strain $\operatorname{div}\mathbf{E}^{p}$ and the Burgers tensor $\mathbf{G}$. Substitution of the constitutive relations into the microforce balance leads to a nonlinear partial differential equation in the plastic strain known as flow rule which captures the presence of an additional energetic length scale arising from the accounting of microstress vector. In addition to the flow rule, nonstandard boundary conditions are obtained, and as an aid to finite element solution a variational formulation of the flow rule is deduced. Finite element solution is obtained of one-dimensional problem of viscoplastic simple shearing under gravity force, where it is shown that for a fixed dissipative length scale, increase in the energetic length scales will result in decrease in the plastic strain.