End effects for a generalized biharmonic equation with applications to functionally graded materials

In this paper we consider a generalized biharmonic equation modeling two-dimensional inhomogeneous elastic state in the curvilinear rectangle a<r<b, 0<θ<α, where (r,θ) denote plane polar coordinates. Such an arch-like region is maintained in equilibrium under self-equilibrated traction applied on one of the edges, while the other three edges are traction free. Our aim is to derive some explicit spatial estimates describing how some appropriate measures concerning the specific Airy stress function evolve with respect to the distance to the loaded edge. Two types of smoothly varying inhomogeneity are considered: (i) the elastic moduli vary smoothly with the polar distance, (ii) they vary smoothly with the polar angle. Such types of smoothly varying inhomogeneous elastic materials provide a model for technological important functionally graded materials. The results of the present paper prove how the spatial decay rate varies with the constitutive profile.