Weyl geometry and the nonlinear mechanics of distributed point defects

Abstract

The residual stress field of a nonlinear elastic solid with a spherically symmetric distribution of point defects is obtained explicitly using methods from differential geometry. The material manifold of a solid with distributed point defects-where the body is stress-free-is a flat Weyl manifold, i.e. a manifold with an affine connection that has non-metricity with vanishing traceless part, but both its torsion and curvature tensors vanish. Given a spherically symmetric point defect distribution, we construct its Weyl material manifold using the method of Cartan's moving frames. Having the material manifold, the anelasticity problem is transformed to a nonlinear elasticity problem and reduces the problem of computing the residual stresses to finding an embedding into the Euclidean ambient space. In the case of incompressible neo-Hookean solids, we calculate explicitly this residual stress field. We consider the example of a finite ball and a point defect distribution uniform in a smaller ball and vanishing elsewhere. We show that the residual stress field inside the smaller ball is uniform and hydrostatic. We also prove a nonlinear analogue of Eshelby's celebrated inclusion problem for a spherical inclusion in an isotropic incompressible nonlinear solid.

Nonlinear elastic inclusions in isotropic solids

“…The body being residually stressed means that this 3-manifold, which we call the material manifold, cannot be isometrically embedded in R 3 . This geometric framework is identical to that used for calculating residual stresses in the presence of non-uniform temperature distributions [15], bodies with bulk growth [21] and bodies with distributed defects [22][23][24]. It should also be noted that this approach is general, i.e.…”

“…The ambient space is a Riemannian manifold (S, g), and hence the computation of stresses requires a Riemannian material manifold (B, G) and a mapping ϕ : B → S. For example, in the case of non-uniform temperature changes and bulk growth [15,21], one starts with a material metric G that specifies the relaxed distances of the material points. In the case of distributed defects, the material metric is calculated indirectly [22][23][24]. When there are eigenstrains distributed in a body, the material manifold has a metric that explicitly depends on the eigenstrain distribution.…”

Geometry and mechanics of two-dimensional defects in amorphous materials

“…[8][9][10]. Inspired by properties of defects in crystalline materials, they suggested to describe defects in continuum by geometric fields such as curvature and torsion tensors.…”

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

“…When ∇ is the Levi-Civita connection of G, for example, (B, ∇, G) is reduced to a Riemannian manifold. We briefly review the geometrical machinery needed for the analysis of defective solids; details are given in [7,22,23]. A linear (affine) connection on a manifold B is an operation ∇ : X (B) × X (B) → X (B), where X (B) is the set of vector fields on B, such that ∀X, Y, X 1 , X 2 , Y 1 , Y 2 ∈ X (B), ∀f , f 1 , f 2 ∈ C ∞ (B), ∀a 1 , a 2 ∈ R: (i) Finally, we quantify non-metricity.…”

“…Unfortunately, these geometrical works remained mostly formal with very few stress calculations for distributed defects. Recently, we revisited the geometric theory of solids with distributed defects and showed that it is also suitable for the calculation of stress fields in nonlinear solids with distributed defects by computing explicitly solutions with either dislocations, disclinations or point defects [7,22,23]. To further emphasize the power of such an approach, here we first present the general geometric theory for combined defects and derive from it a general method for semi-inverse problems in anelasticity based on Cartan's moving frames and structural equations.…”