A hyperelastodynamic ALE formulation based on referential, spatial and material settings of continuum mechanics

The present contribution aims at providing a variational arbitrary Lagrangian Eulerian (ALE) framework for hyperelastostatic as well as hyperelastodynamic problem classes. To this end, a fixed third configuration, the ALE reference or intrinsic configuration, has to be introduced next to the moving spatial and material configuration. The essential idea of the present work is the reformulation of the total variation as the sum of a variation with respect to the spatial coordinates at fixed material and referential placements plus a variation with respect to the material coordinates at fixed spatial and referential placements. For the hyperelastostatic case, the appropriate variational setting is defined through the ALE Dirichlet principle. For the corresponding hyperelastodynamic problem, the ALE Hamilton principle introduces the governing equations. A general frame is set up for both, the static and the dynamic case. Various existing ALE formulations are identified as special cases embedded in this generalized overall framework.