Measurement method for axisymmetric 3D stress-tensor fields using background-oriented schlieren

Abstract

Stress fields affect various characteristics of materials. However, measurement methods for three-dimensional stress fields tend to be complicated because stress tensors generally consist of six independent components. We therefore propose a simple method for measuring axisymmetric three-dimensional stress fields by measuring deflection angles of light rays using the background-oriented schlieren technique. Assuming axial symmetry of the stress field, we use the Lagrangian optics to derive a relationship between two-dimensional deflection angle and three-dimensional stress. Using independent ray polarizations, the method can reconstruct the respective refractive indices from the deflection angle. Equilibrium equations for the stress tensor can be written under the assumption of axial symmetry. By applying appropriate approximations, we formulate normal and shear stress-tensor components, which enables reconstruction of these components from refractive indices. Stress-tensor components can be experimentally generated by applying a load to a glass plate surface on the basis of the Hertz contact theory. The proposed method reconstructs the stress-tensor components from a refractive index field that is induced by the stress in the glass. The reconstructed axial-stress tensor σzz and shear-stress tensor τrz well agree with numeric calculations using structural analysis software. Other stress-tensor components, however, slightly deviate from those of the numeric calculations.