An exact analytical solution to the three-dimensional elasticity problem for a transversely-isotropic composite layer is constructed by making use of the direct integration method along with the Fourier double-integral transform. The original problem is reduced to a system of governing partial-differential equations for separate stress-tensor components. The governing equations are accompanied with corresponding local and integral boundary conditions, obtained on the basis of the original local boundary conditions imposing the normal and shearing forces on the limiting planes of the layer. The numerical analysis of the obtained solution is presented for certain transversely-isotropic composite materials.

In this paper, we present an analytical solution to the general three-dimensional elasticity problem in a layer, whose Young’s modulus varies exponentially within the thickness coordinate and the Poisson’s ratio is constant. By making use of the direct integration method, the complete set of the governing equations in terms of stresses has been formed. The latter equations were reduced to separate equations for each stress-tensor component and then solved by means of the Fourier double-integral transformation with respect to the planar coordinates.