Some Geometrical Aspects of Inhomogeneous Elasticity

Abstract

The balance of pseudomomentum (canonical momentum) is a fully material equation [1]–[2] (i.e., with components on the abstract material manifold Μ 3 and not in physical space E3) which allows one to compute directly either inhomogeneity forces (also called configurational forces) which are the main ingredients in devising fracture criteria in engineering [3] or, in another context, perturbing forces on soliton solutions in nonlinear elastic systems with dispersion [4]. Here we raise the question as to whether this balance law can be rewritten in a completely covariant manner on the material manifold. The answer (geometrical structure of Μ 3) depends on the degree of fineness of the elastic description and the degree of singularity ascribed to the corresponding elastic field, i.e., the type of continuous distribution of structural defects admitted by the material. We illustrate this for nonlinear anisotropic elastostatics via the cases of simple and second-grade elastic materials which are likely to admit (translational) dislocations and/or disclinations,respectively, as elastic singularities. The results are obtained by studying the transformation properties of the elastic response on the material manifold (on the first and second tangent spaces). This follows previous works [5]–[8].