Finite strain generalization of smallstrain constitutive relations for any finite strain tensor and additive volumetric-deviatoric split

Abstract

The paper deals with finite strain generalization of small-strain constitutive equations for isotropic materials for which the strain is split into a volumetric part and a deviatoric part (the latter characterizing the isochoric strain, i.e. a strain at constant volume). The volumetric-deviatoric split has so far been handled by a multiplicative decomposition of the transformation tensor; but the existing sophisticated complex constitutive models for small strains of cohesive pressure-sensitive dilatant materials, such as concrete and geomaterials, involve an additive decomposition and would be difficult to convert a multiplicative decomposition. It is shown that an additive decomposition of any finite strain tensor, and of the Green-Lagrange strain tensor in particular, is possible, provided that the higher-order terms of the deviatoric strain tensor are allowed to depend on the volumetric strain. This dependence is negligible for concrete and geomaterials because the volumetric strains are normally small, whether or not the deviatoric strains are large. Furthermore, the related question of the choice of the finite-strain measure to be used for the finite-strain generalization is analysed. A transformation of the Green-Lagrange finite strain tensor whose parameters approximately reflect the degrees of freedom equivalent to replacing the small strain tensor by any other possible finite strain measure is proposed. Finally a method by which the stress tensor that is workconjugate to any finite strain tensor can be converted to the Green-Lagrange strain tensor is presented.