A nanosized rigid spherical inclusion with interfacial diffusion and sliding

Abstract

We examine the time-dependent deformations around a nanosized rigid spherical inclusion in an infinite elastic matrix under uniaxial tension at infinity. The elastic matrix is first endowed with separate Gurtin–Murdoch surface elasticity. Furthermore, interfacial diffusion and sliding both occur on the inclusion–matrix interface. Closed-form expressions of the time-dependent displacements and stresses in the matrix are derived by using Papkovich–Neuber displacement potentials. A concise and elegant expression of the steady-state normal stress on the surface of the inclusion is also obtained. It is seen that the displacements and stresses in the matrix evolve with two relaxation times which are reliant on three size-dependent parameters, one from surface elasticity and the other two from interfacial diffusion and sliding. Numerical results are presented to demonstrate the influence of surface elasticity on the relaxation times and on the stress distribution near the inclusion. It is observed that the surface elasticity can alter the nature of the steady state normal stress on the surface of the inclusion from tension to compression. When the radius of the inclusion is not greater than the ratio of residual surface tension to remote tension, the steady state normal stress on the surface of the inclusion is always compressive. The related problem of a nanosized rigid spherical inclusion with a spring-type imperfect interface is also solved. We find that it is feasible to design a neutral spherical inclusion that does not disturb a prescribed uniform uniaxial stress field or even any uniform stress field outside the inclusion through a judicious choice of the four imperfect interface parameters.