Gurtin and Murdoch's surface effect on the elastic behavior of an elastic half space subjected to body forces

Abstract

Surface effect have significant influence on mechanical behavior of material at nano scale. Classical solutions have limitations in predicting the micromechanics of materials due to the neglect of surface effect. This paper examines a generalized Mindlin's problem for an elastic half space with surface effect and subjected to arbitrarily orientated body forces. It employs the Gurtin and Murdoch's theory of surface mechanics to describe the surface effect. It fully considers both the residual stress and surface elastic constants. It utilizes the GKS based method to solve the boundary value problem. Analytical solutions of the full elastic field due to uniform circular loading and point loading are derived and expressed in terms of Hankel transform. These new solutions are general and can be reduced to several existing solutions for the surface loading problems and those derived based on simplified version of surface elasticity. Additionally, exact closed-form solutions for the half space with limiting boundary conditions, including traction free, rigidly fixed, inextensible and rolling surfaces can also be obtained from the present solutions. The new solutions have revealed some unique physical phenomena for incompressible solid, which are more general than existing findings. Numerical studies are performed using the solutions to investigate the elastic behavior of the half space with the presence of the surface effect. The numerical results show the substantial differences between the elastic fields predicted by present solutions and those by classical solutions without the consideration of surface effect. The new solutions for point loading can serve as Green's function to address related dislocation problems and other mixed boundary value problems in nano-scale materials, where the surface effect can play significant role on the mechanical behavior of material.