On the effects of microstress on macroscopic diffusion processes

Summary

The effects of internal microstress fields are neglected in a usual simulation of the diffusion of a small solute trough solid heterogeneous media. However, when a heterogeneous material is used in a structure undergoing external mechanical loading, highly nonuniform stress fields can arise locally due to the variable microstructure. In this paper a simple quasi-Fickian model is studied which employs a spatially variable stress-dependent diffusivity,D ^σ. The structure ofD ^σ stems from an assumption that the stresses nonuniformly open and close the pores of the material microstructure, thus providing preferential sites for accumulation of the diffusing solute. When σ=0, the usual stress-free Fickian diffusivity,D ⁰, is recovered. Because of the highly oscillatory stress fields on the micro level, when employing numerical methods, such as the finite element or finite difference method, the distance between discretization nodes must be far smaller than the microstructural oscillations to obtain accurate simulations. This fact makes direct numerical simulations involvingD ^σ virtually impossible without computationally intensive, and complicated, special techniques. In this paper upper bounds are developed for the difference between solutions produced when usingD ^σ and alternativelyD ⁰ in the body under analysis. The general case, whenD ⁰, σ and consequentlyD ^σ, are spatially variable, is considered. The bounds are a function of onlyD ⁰ and σand do not require any knowledge of the stress-dependent solution, and can thus be used as an a-priori check to determine whether potentially expensive computations are necessary.