Kelvin decomposition for nonlinear hyperelastic modeling in large deformation (Q2107569)

The constitutive assumption of linear elasticity is a linear relation between the infinitesimal strain and the stress, both symmetric second-order tensors. It is well-known that for linear isotropic materials the Hooke tensor has two distinct eigenvalues: \(2\mu\) (multiplicity 5) and \(3\lambda+2\mu\) (multiplicity 1). This provides a direct sum decomposition (the Kelvin decomposition) of the vector space of Hooke tensors as a sum of two projections. Obviously, this algebraic concept can be generalized to the geometrically nonlinear setting as long as the second Piola-Kirchhoff stress depends linearly on a Lagrangian finite measure of strain, as for instance the Green-Lagrange strains, \(E.\) The authors extend the Kelvin decomposition to cover also both anisotropic linear elasticity and geometrically nonlinear but quadratic with respect to finite strains) elastic energy densities and introduce the so-called generalized Saint Venant-Kirchhoff and generalized Odgen materials.