A Variationally Derived Quasi-3D Thickness Function for Laminated Composite Plates on Winkler, Pasternak and Kerr Foundations

This study develops a quasi-3D higher-order shear deformation theory for the static analysis of laminated composite plates on elastic foundations. The seven-field kinematic formulation employs a single through-thickness function whose successive derivatives describe higher-order in-plane deformation, transverse shear, and transverse normal deformation. The central contribution is the variational derivation of a new seventh-order variational polynomial (SVP7) thickness function. Its coefficients are determined by minimizing a kinematic functional subject to symmetry, normalization, and zero transverse shear conditions at the free surfaces. The resulting function is the unique minimizer within the adopted seventh-order polynomial space and provides a monotonic decay of the shear-related thickness mode toward the free surfaces, without internal sign changes or spurious oscillations. The Winkler, Pasternak, and Kerr foundation models are incorporated within a unified energy formulation, and the governing equations for simply supported plates are solved using the Navier procedure. The formulation is validated against available three-dimensional elasticity solutions and existing higher-order theories in terms of nondimensional deflections and stress components over a range of plate thickness ratios. The results show consistent agreement with the reference solutions throughout the investigated thickness range and demonstrate the pronounced influence of foundation characteristics on the static response.