Thermodynamic Significance and Basis of Damage Variables and Equivalences

In this article, thermodynamic significance and basis of damage variables and equivalences are addressed within the irreversible thermodynamic framework with internal variables by Rice [Rice, J.R. (1971). Inelastic Constitutive Relations for Solids: An Internal Variable Theory and its Application to Metal Plasticity. Journal of the Mechanics and Physics of Solids, 19:433—455]. It is revealed that, as soon as the multiscale kinematic relation is prescribed, the condition of free or complementary energy equivalence leads to the full equivalence of the microscale and macroscale thermodynamic formulations. In order to achieve the fully thermodynamic equivalence, the damage equivalence hypothesis should be based on free energy or (thermodynamic) complementary energy from a thermodynamic viewpoint. In this sense, the complementary energy equivalence hypothesis by Sidoroff [Sidoroff, F. (1981) Description of Anisotropic Damage Application to Elasticity, IUTAM Colloquium on Physical Nonlinearities in Structural Analysis, pp. 237—244, Springer, Berlin] is much better than the strain equivalence hypothesis by Lemaitre and Chaboche [Lemaitre, J. and Chaboche, J. L. (1978). Aspect Phenomnologique de la Rupture par Endommagement (in French). J. de Mecanique Appliquee, 2:317—365]. One single scalar damage variable plays a role of fictitious or intrinsic time. The introduction of the scalar damage variable implies that the microscale thermodynamic fluxes measured by the intrinsic time should be potential or irrotational and the corresponding flow potential function is just thermodynamic force conjugate to the damage variable.