Stiffness coefficients for inhomogeneous elastic plates

In a series of papers commencing in 1988, Rogers and Spencer and their co-authors developed a procedure for deriving exact solutions to the equations of elasticity for materials that are isotropic but are inhomogeneous along a specified direction, so that the elastic constants may be taken as functions of a single space variable. In the case of a thick plate we will suppose the elastic moduli are known functions of the coordinate normal to the plane of the plate, so that laminated plates and functionally-graded plates are covered by this analysis. In the case of a plate with traction-free upper and lower surfaces, England and Spencer [A.H. England, A.J.M. Spencer, Complex variable solutions for inhomogeneous and laminated elastic plates, Math. Mech. Solids 10 (2005) 503–539] have derived general solutions which may be expressed in terms of four analytic functions of the complex variable ζ=x+iy in the mid-plane of the plate. These solutions are generalisations of the Kolosov–Muskhelishvili solutions for plane-strain elasticity. This analysis has been extended to cover the case of a pressure field applied to one face of the plate. In general the bending and extensional behaviour of the plate is coupled. The intention in this paper is to illustrate these solutions and to derive some stiffness coefficients for general inhomogeneous circular plates of this type.