On spatial and material settings of hyperelastostatic crystal defects

Based on the spatial and material settings of hyperelastostatics the appropriate volume force densities associated to either spatial or material dislocation densities are derived. Thereby spatial and material dislocation densities are defined corresponding to incompatibilities of either the spatial or the material configuration, respectively. All derivations are performed within the geometrically nonlinear framework of continuum mechanics in order to emphasize the intriguing duality of the spatial and the material motion problem. For the particular cases of a single dislocation with either spatial or material Burgers vector the corresponding single forces in material or spatial description, respectively, take the format and interpretation of the well-known Peach-Koehler force (Peach and Koehler, Phys. Rev. 80 (1950) 436). As a result, an attempt is made to clarify some possible irritations documented in the literature (Ericksen, Int. J. Plasticity 14 (1998) 9) of how to set up the Peach-Koehler force within the framework of geometrically nonlinear hyperelastostatics