On Recent Progress in the Theory and Numerics of Material Forces

Within the setting of analytical mechanics forces are regarded as a secondary concept, i.e. as being derived from a potential. In the spririt of virtual work we may call a force a quantity that is energetically conjugated to variations of kinematic quantities. If on the one hand these kinematic quantities are considered as the (standard) spatial deformation map in continuum mechanics of solids, i.e. relating the (initial) material placements to the (deformed) spatial coordinates, the corresponding forces are the standard once. In the slang of difierential geometry, these forces live in the tangent space to the (deformed) spatial configuration. They are usually the given Neumann data for the solution of a boundary value problem. If on the other hand the (inverse) material deformation map, i.e. the map relating the spatial placement to the material coordinates, is considered, the forces conjugated to these kinematic quantities turn out to be (configurational) material forces. Material forces act in the tangent space to the (initial) material configuration. They are results of the computation of a boundary value problem and serve to assess the tendency of defects like cracks, dislocations, interfaces etc. to change their position with respect to the ambient material. Since material forces contribute to a balance equation of material momentum, more or less standard schemes can be invoked to a-posteriori compute discrete material forces for example within a FE-setting