Stress-driven diffusion in a deforming and evolving elastic circular tube of single component solid with vacancies

The title problem is considered for an elastic circular tube of inner radius A and outer radius B. The tube is made of a single component solid with vacancies as its second component. The mole fraction of the massive species is denoted by x1, while that of the vacancies by x0 = 1 – x1. The tube is completely surrounded by vacuum, serving as a reservoir of vacancies. One of the standard elasticity boundary conditions is applied at time t = 0, when the composition is uniform. The ensuing coupled deformation and diffusion leads to the evolving of A(t), B(t) and x1(R, t) as functions of time. Since the single component solid is not in contact with its vapor or liquid, the diffusion boundary condition is always tied to the elasticity problem through a surface condition that involves the normal configurational traction. Our chemical potential has an energy density term that serves as a source in the interior and the boundary conditions for the diffusion problem are such that the time rates of boundary accretion Ȧ(t) and Ḃ(t) must simultaneously satisfy two dissipative inequalities, one governed by the gradient of the internal chemical potential and the other by the normal configurational traction.