Plane wave solutions and modal analysis in higher order shear and normal deformable plate theories

We use the three-dimensional Hellinger-Reissner mixed variational principle to derive a Kth order (K = 0, 1, 2,...) shear and normal deformable plate theory. The balance laws, the constitutive relations and the boundary conditions for the plate theory are deduced. The constitutive relations incorporate the shear and the normal tractions applied on the top and the bottom surfaces of the plate. For a Kth order plate theory with displacements expressed as a power series in the thickness co-ordinate z with terms up to z(K), the transverse shear and the transverse normal stresses involve terms up to z(K+2) while in-plane stress components have terms up to z(K). The equations for the plate theory are expressed in a compact form by taking Legendre polynomials as the basis functions. The plate theory is used to study plane travelling waves and in particular the lengths of decay of the displacement components; this allows for a rigorous ordering of the importance of the displacement descriptors in terms of decaying properties. Finally, we study the free vibrations of a simply supported rectangular orthotropic thick plate; results from the present theory are compared with an exact three-dimensional solution and with other plate theories. To this end, a Kth order compatible plate theory is also deduced; the term compatible alludes to the fact that the reduction map for the stress fields is induced by the kinematical reduction map, whilst in the mixed models it is postulated independently. It is found that the frequencies up to the fifth mode of vibration computed with the fifth order theory and without introducing any shear correction factors match very well with the corresponding analytical solution. Also, through-the-thickness distribution of all of the stress components is found to agree well with the three-dimensional elasticity solution, while the stress distribution obtained from the compatible plate theories deviates considerably, especially for the higher modes. (C) 2002 Elsevier Science Ltd. All rights reserved.