  We consider the influence of elasticity and anisotropic surface energy on the
energy-minimizing shape of a two-dimensional void under biaxial loading. In
particular, we consider void shapes with corners for which the strain energy
density is singular at the corner. The elasticity problem is formulated as a
boundary integral equation using complex potentials. By incorporating the
asymptotic behavior of the singular elastic fields at corners of the void, we
develop a numerical spectral method for determining the stress for a class of
arbitrary void shapes and corner angles. We minimize the total energy of
surface energy and elastic potential energy using calculus of variations to
obtain an Euler-Lagrange equation on the boundary that is coupled to the
elastic field. The shape of the void boundary is determined using a numerical
spectral method that simultaneously determines the equilibrium void shape and
singular elastic fields. Our results show that the precise corner angles that
minimize the total energy are not affected by the presence of the singular
elastic fields. However, the stress singularity on the void surface at the
corner must be balanced by a singularity in the curvature at the corner that
effectively changes the macroscopic geometry of the shape and effectively
changes the apparent corner angle. These results reconcile the apparent
contradiction regarding the effect of elasticity on equilibrium corner angles
in the results of Srolovitz and Davis (2001) and Siegel, Miksis and Voorhees
(2004), and identify an important nontrivial singular behavior associated with
corners on free-boundary elasticity problems.
