An asymptotic theory of sandwich plates

Elastic plates can be described by a two-dimensional theory, if the characteristic length of the stress state along the plate, l, is much larger than the plate thickness, h. If all elastic moduli of a laminated plate are of the same order, no matter how many lamina the plate has, then the normal to the mid-surface of the plane remains normal in the course of deformation, and the deformation of the plate can be described by the classical plate theory. The situation changes, when the elastic moduli are of different orders of magnitude. This occurs, in particular, for the hard-skin plates, i.e. the sandwich plates the faces of which are very hard. Due to the low deformability of the skin, normal fibers cannot remain normal to the mid-surface in the course of deformation. The deviations are characterized by transverse shear. The difference from the theory of transverse shear, introduced by Timoshenko and Reissner, is that the transverse shear effects are not the corrections to classical plate theory; they are the effects of the leading order. That is caused by the presence of an additional small parameter, the ratio of elastic moduli of the core and the skin. The additional small parameter changes the character of the asymptotics. In this paper, the governing two-dimensional equations for sandwich plates are derived by an asymptotic analysis of linear three-dimensional elasticity. We show that the classical plate theory works only within a certain range of parameters. Beyond that range the asymptotic theory differs from the classical one. We focus especially on the hard-skin plates, but obtain also the universal relations, which can be applied for any values of elastic moduli and the relative thickness of the skin and the core. As an example four-point bending problem is discussed.