Variational arguments and Noether’s theorem on the nonlinear continuum theory of dislocations

A formulation concerning the hyperelastic simple bodies with a continuous distribution of dislocations at finite strain is proposed. The presence of dislocations within the body renders the stored energy function non-homogeneous. Based on the notion of uniform reference, as developed by Noll, the stored energy density per unit uniform reference configuration is introduced. The main property of the latter is that it receives all the explicit dependence of the standard stored energy function on the material variables. Variational formulations for the direct and inverse deformation descriptions are established and it is shown that the Euler–Lagrange equation for the direct deformation description provides the standard balance of physical forces as in the classical case. Moreover, the lack of invariance under translations in material space results in a non-conservation law describing the balance of material forces. The additional source term appearing in it, i.e., an additional material force, is due to the presence of dislocations. Thus, one may conclude that the force acting on dislocations is a material one. Repeating the aforementioned procedure for the inverse deformation description, one obtains again the same two equations in a different order, that is, the balance of material forces is derived from the Euler–Lagrange equation, whereas the balance of physical forces is obtained from the invariance of the energy functional under translations in the physical space.