Boundary conditions in small-deformation, single-crystal plasticity that account for the Burgers vector

Abstract

This paper discusses boundary conditions appropriate to a theory of single-crystal plasticity (Gurtin, J. Mech. Phys. Solids 50 (2002) 5) that includes an accounting for the Burgers vector through energetic and dissipative dependences on the tensor G = curl H p , with H p the plastic part in the additive decomposition of the displacement gradient into elastic and plastic parts. This theory results in a flow rule in the form of N coupled second-order partial differential equations for the slip-rates γ ˙ α ( α = 1 , 2 … , N ) , and, consequently, requires higher-order boundary conditions. Motivated by the virtual-power principle in which the external power contains a boundary-integral linear in the slip-rates, hard-slip conditions in which (A) γ ˙ α = 0 on a subsurface S hard of the boundary for all slip systems α are proposed. In this paper we develop a theory that is consistent with that of (Gurtin, 2002), but that leads to an external power containing a boundary-integral linear in the tensor H ˙ ij p ɛ jrl n r , a result that motivates replacing (A) with the microhard condition (B) H ˙ ij p ɛ jrl n r = 0 on the subsurface S hard . We show that, interestingly, (B) may be interpreted as the requirement that there be no flow of the Burgers vector across S hard . What is most important, we establish uniqueness for the underlying initial/boundary-value problem associated with (B); since the conditions (A) are generally stronger than the conditions (B), this result indicates lack of existence for problems based on (A). For that reason, the hard-slip conditions (A) would seem inappropriate as boundary conditions. Finally, we discuss conditions at a grain boundary based on the flow of the Burgers vector at and across the boundary surface.