The geometry of discombinations and its applications to semi-inverse problems in anelasticity

Abstract

The geometrical formulation of continuum mechanics provides us with a powerful approach to understand and solve problems in anelasticity where an elastic deformation is combined with a nonelastic 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, nonEuclidean, geometrical 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 geometrical 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.

Circumferentially-symmetric finite eigenstrains in incompressible isotropic nonlinear elastic wedges

“…Similar constructions using non-trivial material geometries have been (footnote continued) introduced in thermoelasticity, growth mechanics, and the mechanics of distributed defects[21,[23][24][25][26][27][28][29].…”

Line and point defects in nonlinear anisotropic solids

“…Only a handful of exact solutions for defects in nonlinear elastic solids exist in the literature, and they are all restricted to isotropic materials. We should mention [14,76,46,11,1,13,67,47] for dislocations, [76,8,70] for disclinations, and [68,71,6] for point defects and discombinations.…”

Non-metric Connection and Metric Anomalies in Materially Uniform Elastic Solids

“…In the context of elastic solids, where the relevant geometric space is the stress-free material space, they can be identified with material inhomogeneity fields arising from a distribution of point defects (intrinsic interstitials, vacancies, point stacking faults), thermal deformation, biological growth, etc. [3,7,19,21,26,[41][42][43]. This association is made on the basis of the metrical nature of these material inhomogeneities, such as that leading to an inhomogeneous volume change due to a distribution of spherical point defects, isotropic thermal deformation or isotropic growth.…”