A one-dimensional theory of solute diffusion and degradation in elastic solids
The paper presents a one-dimensional theoretical model to describe the interaction between solid-assisted diffusion and degradation in elastic solids by extending the idea of Gibbs. The solid-solute system is described by four independent fields, namely solid displacement and degradation, solute density and flow. The authors introduce balance laws, a free-energy inbalance that represents the second law of thermodynamics, and constitutive equations. First, they consider the balances of mass for the solid and solute, and the balances of forces conjugate to the selected independent kinematical fields, namely solid displacement and degradation being a measure of the degree of degradation, solute flaw and density. Then, the authors introduce a free-energy inbalance taking into account energy inflow due to the power expended by all external forces and the solute supply. The thermodynamically consistent constitutive theory is based upon the next assumptions: (i) strain decomposes additively into elastic and solute-induced parts; (ii) degradation and diffusion are the only source of dissipation. The requirement of thermodynamic consistency implies that the constitutive equations are defined in terms of response functions for the free-energy density, viscosity degradation, friction-like resistance to degradation growth, and solute mobility. The governing equations, namely the equations for the degradation and the solute flux are obtained from the corresponding conjugate force balances, and the equation for the chemical potential are found from the solute-density-conjugate force balance. It is shown how the solute-degradation interactions can be implemented into the theory. The specialization of the general theory is based on the constitutive assumptions: (i) the free-energy density is given as the sum of degradation-dependent strain energy, degradation gradient energy and entropic contribution due to the mixing of dilute ideal interstitial solid solution; (ii) the viscosity degradation and the solute mobility are constant; (iii) frictional resistance is non-increasing function of the solute constant. The equation for the degradation is rate-dependent and weakly non-local. Finally, the authors develop a numerical scheme based on the finite element method, a backward Euler time-stepping scheme, and on the operator-splitting algorithm to solve the weak forms of displacement, degradation and diffusion equations. As a numerical example, they examine the solid-assisted degradation of a cylindrical notched bar.
scientific article; zbMATH DE number 5640224