On discontinuities when computing the stress-field from the strain: A finite volume discretization

This article focuses on developing a finite volume discretization for a recently-developed and widely-applicable system of PDEs, enabling computation of the full-field stress from a measured strain field and traction boundary conditions. This is applicable to datasets obtained via DIC, and applies when the material properties are unknown. Interestingly, the strain or strain-rate discontinuities at the interfaces between finite volumes cause reflection and transmission of the principal stresses. Hence, we analytically analyze the solution structure in this case, enabling the development of the discretization. In addition, analysis is conducted translating knowledge of the traction at the boundary into boundary conditions, so that the numerical method can be applied to a variety of loading conditions. The finite volume discretization is validated using strain fields obtained from finite element simulations. The developed discretization enables the system of equations to be applied to irregular geometries, finite deformation, general boundary conditions, and avoids instabilities that arise due to the Courant-Friedrichs-Lewy condition, significantly increasing the range of applications for which these PDEs can be applied.