Stresses exerted by a source of diffusion in a case of a non-parabolic diffusion equation

The theory of diffusive stresses based on the diffusion-wave equation with time-fractional derivative of fractional order α is formulated. The non-parabolic diffusion equation is a mathematical model of a wide range of important physical phenomena and can be obtained as a consequence of the non-local constitutive equation for the matter flux vector with the long-tale power time-non-local kernel. Because the considered equation in the case 1 ⩽ α ⩽ 2 interpolates the parabolic equation (α = 1) and the wave equation (α = 2), the proposed theory interpolates a classical theory of diffusive stresses and that without energy dissipation introduced by Green and Naghdi. The stresses caused by a source of diffusion in an unbounded solid are found in one-dimensional and axially symmetric cases (for plane deformation). Numerical results for the concentration and stress distributions are given and illustrated graphically.