The Ball-on-Ring-test: Enhancing an analytical solution by numerical analysis for elastic deformation and small displacements

The Ball-on-Ring-test is a biaxial strength testing method utilized to test brittle materials such as ceramics, glass, or semiconductor wafers. In this work, an analytical solution for the BoR-test is derived using plate theory. In contrast to previous work, a Hertzian load distribution beneath the loading ball is considered. The solution provided in this work is extensively analyzed and validated by two Finite-Element-Analysis (FEA) models. For thin specimens, excellent agreement between FEA and the analytical solution was found. For many specimen geometries and loading configurations, it is shown that plate theory generally fails to accurately describe the maximum stress. Therefore, a simple correction for these cases is proposed. With this correction, an error < 2 % to the FEA-results is achieved. By combining these methods, accurate functional expressions for the displacement field, its derivatives, and the shear force-, the bending moment- and stress distributions are provided for the entire disc.