The Geometry of Discombinations and its Applications to Semi-Inverse Problems in Anelasticity

The geometric formulation of continuum mechanics provides a powerful approach to understand and solve problems in anelasticity where an elastic deformation is combined with a non-elastic component arising from defects, thermal stresses, growth effects, or other effects leading to residual stresses. The central idea is to assume that the material manifold, prescribing the reference configuration for a body, has an intrinsic, non-Euclidean, geometric structure. Residual stresses then naturally arise when this configuration is mapped into Euclidean space. Here, we consider the problem of  discombinations (a new term that we introduce in this paper), that is a combined distribution of  fields of dislocations, disclinations, and point defects. Given a discombination, we compute the geometric characteristics of the material manifold (curvature, torsion, non-metricity), its  Cartan's moving frames and structural equations. This identification provides a powerful algorithm to solve semi-inverse problems with non-elastic components. As an example,  we calculate the residual stress field of a cylindrically-symmetric distribution of discombinations in an infinite circular cylindrical bar made of an incompressible hyperelastic isotropic elastic solid.