In-plane deformations of a nano-sized circular inhomogeneity with interface slip and diffusion

We consider time-dependent plane strain deformations of a nanosized circular elastic inhomogeneity embedded in an infinite elastic matrix subjected to uniform remote stresses. The inhomogeneity and the matrix are each endowed with separate and distinct Gurtin-Murdoch surface elasticities. In addition, both rate-dependent slip and mass transport resulting from stress-driven diffusion occur concurrently on the inhomogeneity/matrix interface. A simple yet effective method is proposed to derive a closed-form solution. The stress distributions in the composite are size-dependent and evolve with two relaxation times. Explicit expressions for the relaxation times depend on four size-dependent parameters: two arising from interface slip and diffusion and two from surface elasticities. The stress field inside the inhomogeneity is spatially non-uniform and time-dependent when the remote loading is non-hydrostatic; conversely, it is uniform, hydrostatic and time-independent when the remote loading is hydrostatic.