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

This paper discusses boundary conditions appropriate to a theory of single-crystal plasticity (Gurtin, 2002) that includes an accounting for the Burgers vector through energetic and dissipative dependences on the tensor G = curlH, with H 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 γ̇ (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) γ̇ = 0 on a subsurface Shard 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 Ḣ ijεjrlnr, a result that motivates replacing (A) with the microhard condition (B) Ḣ ijεjrlnr = 0 on the subsurface Shard. We show that, interestingly, (B) may be interpreted as the requirement that there be no flow of the Burgers vector across Shard. 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.