On a generalized biharmonic equation in plane polars with applications to functionally graded material

In this paper we consider a generalized biharmonic equation modelling a 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 the edge θ = 0, while the other three edges r = a, r = b and θ = α are traction free. Our aim is to derive some explicit spatial exponential decay bounds for the specific Airy stress function and its derivatives. Two types of smoothly varying inhomogeneity are considered: (i) the elastic moduli vary smoothly with the polar angle, (ii) they vary smoothly with the polar distance. 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.