Void expansion as wave phenomena - might damage evolution be mathematically related to fluid dynamics and turbulence?

The main idea is the following: a most natural mathematical setup for considering the motion of the void-solid interface of an expanding void is that of the traveling wave. Thus, a theory for macroscopic damage evolution may be suspected as being a homogenized version of basic theory that has such wave phenomena as an essential ingredient. This paper is a first step in probing such questions. 

This is a paper that has been accepted for publication in a special volume on generalized and nonlocal continuum theories in the ASCE Journal of Engineering Mechanics, edited by George Voyiadjis and Rich Regueiro. I had posted an earlier version of this paper at imechanica - I have since then been able to solve an example that lends insight into some interesting qualitative features of the model. Hence the repost.

Abstract

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 associated 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 (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 ‘anti-diffusion’ effect, and the model also inextricably links propagation of regions of high gradients with their diffusive smoothing.