Mechanical responses of elastic layers with through-the-thickness variation in eigenstrain

A general procedure is developed for the three-dimensional analysis of homogeneous isotropic elastic layers with eigenstrain varying arbitrarily through the thickness. By using the state-space method coupled with asymptotic expansion, the displacement and stress fields are derived and expressed in terms of the displacement components on the mid-plane. The governing equations for these displacement components are formally the same as those in the classic plate theory. Obtaining the displacement of the mid-plane immediately provides a three-dimensional solution to the elastic layer. The solution is exact in Saint Venant's sense that the boundary conditions at the edge of the layer are satisfied in the average manner as in the classic plate theory. As application of the present method, some examples are given and numerical results are shown.