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.

我们考虑了弹性和各向异性表面能对双轴载荷下二维空隙的能量最小化形状的影响。特别是，我们考虑具有角的空隙形状，其应变能密度在角处是奇异的。弹性问题被表述为使用复势的边界积分方程。通过结合空隙角落处奇异弹性场的渐近行为，我们开发了一种数值谱方法来确定一类任意空隙形状和角角的应力。我们使用变分法最小化表面能和弹性势能的总能量，以获得与弹性场耦合的边界上的欧拉-拉格朗日方程。空隙边界的形状是使用数值光谱法确定的，该方法同时确定平衡空隙形状和奇异弹性场。我们的结果表明，最小化总能量的精确角角不受奇异弹性场的影响。然而，拐角处空隙表面上的应力奇异性必须通过拐角处曲率的奇异性来平衡，该奇异性有效地改变了形状的宏观几何形状并有效地改变了表观拐角角度。这些结果调和了 Srolovitz 和 Davis (2001) 以及 Siegel、Miksis 和 Voorhees (2004) 的结果中关于弹性对平衡角角的影响的明显矛盾，