Contact Analysis Using Surface Green’s Functions for Isotropic Materials With Surface Stress and Surface Elasticity

Surface stress and surface elasticity are related to an organization of surface pattern and reconstruction of surface atoms. When the size of material reduces to a nanometer level, a ratio of surface to volume increases. Then, surface stress and surface elasticity influence on mechanical response near surface for an external force on the surface. Stroh formalism is very useful for analyzing the stress and displacement in anisotropic materials. When the Stroh’s formalism is applied to isotropic materials, the eigen matrix derived from equilibrium equation yields a triple root of i (i: imaginary unit), and then an independent eigen vector corresponding to the eigen value can not be determined. In this paper, surface Green function for isotropic materials is derived using Stroh’s formalism. The derived Green function considering neither surface stress nor surface elasticity agrees with the solution of Boussinesq. The surface Green’s function considering surface stress and surface elasticity is used for analyzing the displacement fields in amorphous silicon. It was found that the displacements obtained from the Green’s function were less than those from Boussinesq’s solution. Furthermore, the derived surface Green’s function is applied to a contact analysis for isotropic materials such as amorphous silicon. It is found that an apparent Young’s modulus determined from a force-indentation depth curve increases when surface stress and elasticity is taken into account in the analysis.

The Stroh Formalism

In this chapter we study Stroh's sextic formalism for two-dimensional deformations of an anisotropic elastic body. The Stroh formalism can be traced to the work of Eshelby, Read, and Shockley (1953). We therefore present the latter first. Not all results presented in this chapter are due to Stroh (1958, 1962). Nevertheless we name the sextic formalism after Stroh because he laid the foundations for researchers who followed him. The derivation of Stroh's formalism is rather simple and straightforward. The general solution resembles that obtained by the Lekhnitskii formalism. However, the resemblance between the two formalisms stops there. As we will see in the rest of the book, the Stroh formalism is indeed mathematically elegant and technically powerful in solving two-dimensional anisotropic elasticity problems. The possibility of extending the formalism to three-dimensional deformations is explored in Chapter 15.

Punch Problems for an Anisotropic Elastic Half-Plane

By combining Stroh’s formalism and the method of analytical continuation, several mixed-typed boundary value problems of an anisotropic elastic half-plane are studied in this paper. First, we consider a set of rigid punches of arbitrary profiles indenting into the surface of an anisotropic elastic half-plane with no slip occurring. Illustrations are presented for the normal and rotary indentation by a flat-ended punch. A sliding punch with or without friction is then considered under the complete or incomplete indentation condition.