Exact solutions for functionally graded and laminated elastic materials

We consider isotropic linearly elastic materials in which, referred to a rectangular Cartesian coordinate system Oxyz, the Lame elastic moduli lambda and mu depend in an arbitrary specified manner on the coordinate z. If this dependence is continuous the material may be regarded as a functionally graded elastic material; the case in which it is discontinuous represents a laminate. A large class of exact solutions of the three-dimensional elasticity equations for materials of this type is established. It is shown that exact three-dimensional solutions for a thick plate are generated, in a simple manner, by solutions of the two-dimensional classical equations for stretching and bending of an "equivalent plate". This is a hypothetical homogeneous plate with elastic moduli that are appropriate weighted averages of the moduli of the inhomogeneous plate. The formulation in cylindrical polar coordinates is also given, and the theory is illustrated by examining solutions with radial symmetry about the z axis. (C) 1998 Elsevier Science Ltd. All rights reserved.