On a tensor cross product based formulation of large strain solid mechanics

This paper describes in detail the formulation of large strain solid mechanics based on the tensor cross product, originally presented by R. de Boer (1982) and recently re-introduced by Bonet et al. (2015a) and Bonet et al. (2015b). The paper shows how the tensor cross product facilitates the algebra associated with the area and volume maps between reference and final configurations. These maps, together with the fibre map, make up the fundamental kinematic variables in polyconvex elasticity. The algebra proposed leads to novel expressions for the tangent elastic operator which neatly separates material from geometrical dependencies. The paper derives new formulas for the spatial and material stress and their corresponding elasticity tensors. These are applied to the simple case of a Mooney–Rivlin material model. The extension to transversely isotropic material models is also considered.