Bending of crystalline plates under the influence of surface stress — a finite element analysis

Changes in the surface stress Δτ(s) at the solid–vacuum and the solid–liquid interface are typically measured via the bending of a (thin) crystalline plate. The bending is converted into Δτ(s) with the help of elasticity theory. Although the conversion is trivial for the unconstrained bending of an elastic isotropic substrate, it becomes less straightforward for single crystals. In this paper we firstly consider the analytical solution for the free two-dimensional bending of a cubic substrate under the influence of a homogeneous, but anisotropic surface stress. Secondly, the bending is calculated numerically for the experimentally mostly realized case of a sample with (100) or (111) surfaces clamped along one edge. In principle, separate numerical calculations are required for each individual material and surface orientation. It is shown that a parameterized form of the analytical solution for the unconstrained bending can still be used in most cases. The appropriate parameters are calculated for various Poisson ratios and crystal anisotropies.