Does elastic stress modify the equilibrium corner angle?

TL;DR

This study investigates how elastic stress and anisotropic surface energy influence the equilibrium shape of voids with corners, revealing that stress singularities affect apparent corner angles without changing the true energy-minimizing angles.

Contribution

The paper introduces a numerical spectral method that accounts for elastic singularities at corners, clarifying their effect on equilibrium shapes in elastic surface problems.

Findings

Corner angles minimizing total energy are unaffected by elastic singularities.

Stress singularities at corners lead to curvature singularities, altering apparent corner angles.

Results reconcile previous conflicting findings on elasticity's influence on corner angles.

Abstract

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.