  In this paper, we investigate a variational polycrystalline model in finite
crystal plasticity with one active slip system and rigid elasticity. The task
is to determine inner and outer bounds on the domain of the constrained
macroscopic elastoplastic energy density, or equivalently, the affine boundary
values of a related inhomogeneous differential inclusion problem. A
geometry-independent Taylor inner bound, which we calculate directly, follows
from considering constant-strain solutions to a relaxed problem in combination
with well-known relaxation and convex integration results. On the other hand,
we deduce outer bounds from a rank-one compatibility condition between the
affine boundary data and the microscopic strain at the boundary grains. While
there are examples of polycrystals for which the two above-mentioned bounds
coincide, we present an explicit construction to prove that the Taylor bound is
non-optimal in general.
