On uniqueness, characteristic topology, and singularities present in hyperbolic equations which determine stress from full-field strain

A system of hyperbolic partial differential equations (PDEs) can be used to determine stress fields that are consistent with observed kinematic fields, such as those obtained from digital image correlation, when the material properties are unknown. The PDEs enforce force balance, require traction boundary conditions, and make a material isotropy assumption. We show that singularities commonly arise in these equations at isolated points. These singularities globally modify the characteristic topology, raising questions about the uniqueness of solutions to the PDEs, and what numerical algorithms might be applied to solve problems. By studying the topology of the characteristic curves in domains with and without singularities, we make four contributions. First, we prove that the equations admit unique solutions in bounded, simply connected domains with no singularities present. Second, we show that the characteristic topologies associated with generic singularities can be classified into the three classical Darbouxian structures. Third, we conjecture, with supporting numerical evidence, that a single generic singularity still permits a unique solution. Fourth, we provide a practical numerical technique that enables us to solve the equations for all three singularity types, including a case with two singularities, where standard approaches become numerically unstable.