Use of Thermodynamic Formalism in Generalized Continuum Theories and a Model for Damage Evolution

A technique for setting up generalized continuum theories based on a balance law and nonlocal thermodynamics is suggested. The methodology does not require the introduction of gradients of the internal variable in the free energy, while allowing for its possibility. Elements of a generalized (brittle) damage model with porosity as the internal variable are developed as an example. The notion of a flux of porosity arises, and we distinguish between the physical notion of a flux of voids (with underpinnings of corpuscular transport) and a flux of void volume that can arise merely due to void expansion. A hypothetical, local free energy function with classical limits for the damaged stress and modulus is constructed to show that the model admits a nonlinear diffusion-advection equation with positive diffusivity for the porosity as a governing equation. This equation is shown to be intimately related to Burgers equation of fluid dynamics, and an analytical solution of the corresponding constant-coefficient, semilinear equation without source term is solved by the Hopf–Cole transformation, that admits the Hopf–Lax entropy weak solution for the corresponding Hamilton–Jacobi equation in the limit of vanishing diffusion. Constraints on the class of admissible porosity and strain-dependent free energy functions arising from the mathematical structure of the theory are deduced. This work may be thought of as providing a continuum thermodynamic formalism for the internal variable gradient models proposed by Aifantis in 1984 in the context of local stress and free-energy functions. However, the degree of diffusive smoothing is not found to be arbitrarily specifiable as mechanical coupling produces an “antidiffusion” effect, and the model also inextricably links propagation of regions of high gradients with their diffusive smoothing.