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

Morton E. Gurtin $^{\scriptscriptstyle\ddagger}$ and Alan Needleman $^{\scriptscriptstyle\dagger}$

$^{\scriptscriptstyle\ddagger}\mskip-6mu$ Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213, USA

$^{\scriptscriptstyle\dagger}\mskip-6mu$ Division of Engineering Brown University Providence, RI 02912, USA

Abstract: 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 $\mbox{\boldmath$G$} = {\rm curl} \mbox{\boldmath$H$}^p$, with $\mbox{\boldmath$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 $\dot{\gamma}^\alpha$ $(\alpha=1,2\dots, N)$, 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)
$\dot{\gamma}^\alpha=0$ on a subsurface ${\cal S}_{\rm hard}$ of the boundary
for all slip systems $\alpha$ 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 $\dot{H}^p_{ij}\varepsilon_{jrl}n_r$, a result that motivates replacing (A) with the microhard condition
(B)
$\dot{H}^p_{ij}\varepsilon_{jrl}n_r=0$ on the subsurface ${\cal S}_{\rm hard}$.
We show that, interestingly, (B) may be interpreted as the requirement that there be no flow of the Burgers vector across ${\cal S}_{\rm hard}$.

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.

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