A historical overview on static and dynamic analyses of sandwich or partially composite beams and plates

This paper gives a brief overview of the static and the dynamic analyses of sandwich beams and sandwich plates composed of two elastic layers connected by a soft core. We show the strict correspondence between the sandwich beam, sandwich plate, and the analogous composite beam and composite plate problem with partial interaction. The beam problem is governed by a sixth-order differential equation in space for the deflection, whereas the plate problem is governed by a tri-Laplacian equation for the deflection, as first shown by Hoff in 1950 for symmetrical three-layer component (sandwich plate or composite plate theory with partial interaction). It is shown herein that the tri-Laplacian partial differential equation is also valid for general unsymmetrical sandwich plate or partially composite plate. In the limit cases, i.e., for non-composite action and for full-composite action, the beam model reduces into the Euler–Bernoulli beam model and the plate model into the Kirchhoff–Love model. Boundary layer phenomena may be predominant in the asymptotic stiff problem. The nonlocal bending moment–curvature of the sandwich beam (or composite beam with partial interaction) is analysed, and extended to a nonlocal Marcus bending moment–curvature for the composite plate theory. The consistency of the tri-Laplacian plate theory is commented, regarding the bi-Laplacian Reissner sandwich plate theory, which neglects the bending stiffnesses of each face layer. The paper finally presents results for the static bending and vibrations of sandwich beams and plates on simply supported boundary conditions, with a discussion on asymptotic limit cases.