A small-deformation strain-gradient theory for isotropic viscoplastic materials

This study develops a thermodynamically-consistent small-deformation theory of strain-gradient viscoplasticity for isotropic materials based on: (i) a scalar and a vector microstress consistent with a microforce balance; (ii) a mechanical version of the two laws of thermodynamics for isothermal conditions, that includes via the microstresses the work performed during viscoplastic flow; and (iii) a constitutive theory that allows: the free energy to depend on p, the gradient of equivalent plastic strain p, and this leads to the vector microstress having an energetic component; strain-hardening dependent on the equivalent plastic strain p, and a scalar measure p related to the accumulation of geometrically necessary dislocations; and a dissipative part of the vector microstress to depend on [image omitted], the gradient of the equivalent plastic strain rate. The microscopic force balance, when augmented by constitutive relations for the microscopic stresses, results in a non-local flow rule in the form of a second-order partial differential equation for the equivalent plastic strain p. The flow rule, being non-local, requires microscopic boundary conditions. The theory is numerically implemented by writing a user-element for a commercial finite element program. Using this numerical capability, the major characteristics of the theory are revealed by studying the standard problem of simple shear of a constrained plate. Additional boundary-value problems representing idealized two-dimensional models of grain-size strengthening and dispersion strengthening of metallic materials are also studied.