A formulation for the micromorphic continuum at finite inelastic strains

The paper presents a generalized theory of deformation which can capture scale effects also in a homogenously deforming body. Scale effects are relevant when the dimensions of the specimen or structure themselves are in the micron and submicron scale, but also when it comes to high strain concentrations as in the case of localised shear bands or at crack tips, etc. In this context, so-called generalized continuum formulations have been proven to provide remedy as they allow for the incorporation of internal length-scale parameters which reflect the micro-structural influence on the macroscopic material response. Here, we want to adopt a generalized continuum framework which is based on the mathematical description of a combined macro- and micro-space (Sansour, 1998b). The approach introduces additional degrees of freedom which constitute a so-called micromorphic deformation. First the treatment presented is general in nature but will be specified for the sake of an example and the number of extra degrees of freedom will be reduced to four. Based on the generalized deformation description new strain and stress measures are defined which lead to the formulation of a corresponding generalized variational principle. The theory is completed by Dirichlet boundary conditions for the displacement field. Of great advantage is the fact that the constitutive law is defined in the generalized space but can be classical otherwise. This limits the number of the extra material parameters necessary to those needed for the specification of the micro-space, in the example presented to only one. An example of scale effects in a homogenously deforming specimen and a further example of shearband formation are presented where the constitutive law is a classical viscoplastic one.