A general spectral nonlinear elastic consistent tangent modulus tensor formula for finite element software

Abstract

The consistent tangent modulus tensor requires the formulae for the derivatives, with respect to the right Cauchy–Green tensor, of invariants that described the invariant-based potential function. Currently, a cumbersome process of individually evaluating the formulae for the derivatives of tensor invariants was done and only derivative formulae for invariants that can be expressed explicitly in terms of the right Cauchy–Green tensor can be found in the literature; derivative formulae for spectral invariants that cannot be expressed explicitly in terms of the right Cauchy–Green tensor, for use in finite element software, do not exist in the literature. We note that these spectral invariants have been recently used in non-linear anisotropic elasticity. Hence, in this communication, to avoid the cumbersome process of individually evaluating the derivative-invariant formulae and to supply the currently non-existent derivative-invariant formulae for spectral invariants, we give a general spectral formula for the consistent tangent modulus tensor for an invariant-based potential function that contain invariants, which may depend explicitly or implicitly on the right Cauchy–Green tensor.