Fundamental Solutions and Functionally Graded Materials

A fundamental solution (or Green’s function) is a singular solution of a governing partial differential equation (PDE). They can be constructed easily when the PDE has constant coefficients. They are useful for reducing boundary-value problems to boundary integral equations (BIEs). We begin by describing simple properties of fundamental solutions, and then comment on the use and construction of half-space Green’s functions. We then move on to consider functionally graded materials (FGMs). These are inhomogeneous materials: their properties vary with position. Modelling FGMs leads to PDEs with variable coefficients, and this makes the construction of fundamental solutions more difficult. In this paper, we consider FGMs where the properties vary exponentially in one prescribed direction; such ‘exponentially graded’ materials provide a reasonable model of certain real situations. We discuss the construction of fundamental solutions for steady-state heat conduction and for threedimensional elasticity. These solutions should be useful in the development of boundary integral methods for FGMs.