A constitutive framework for finite viscoelasticity and damage based on the Gram–Schmidt decomposition

A novel thermodynamic framework for the continuum mechanical response of nonlinear solids is described. Large deformations, nonlinear hyperelasticity, viscoelasticity, and property changes due to evolution of damage in the material are all encompassed by the general theory. The deformation gradient is decomposed in Gram–Schmidt fashion into the product of an orthogonal matrix and an upper triangular matrix, where the latter can be populated by six independent strain attributes. Strain attributes, in turn, are used as fundamental independent variables in the thermodynamic potentials, rather than the usual scalar invariants of deformation tensors as invoked in more conventional approaches. A complementary set of internal variables also enters the thermodynamic potentials to enable history and rate dependence, i.e., viscoelasticity, and irreversible stiffness degradation, i.e., damage. Governing equations and thermodynamic restrictions imposed by the entropy production inequality are derived. Mechanical, thermodynamic, and kinetic relations are presented for material symmetries that reduce to cubic or isotropic thermoelasticity in the small strain limit, restricted to isotropic damage. Representative models and example problems demonstrate utility and flexibility of this theory for depicting nonlinear hyperelasticity, viscoelasticity, and/or damage from cracks or voids, with physically measurable parameters.