On asymptotically correct linear laminated plate theory

Abstract The focus of this paper is the development of asymptotically correct theories for laminated plates, the material properties of which vary through the thickness and for which each lamina is orthotropic. This work is based on the variational-asymptotical method, a mathematical technique by which the three-dimensional analysis of plate deformation can be split into two separate analyses: a one-dimensional through-the-thickness analysis and a two-dimensional “plate” analysis. The through-the-thickness analysis includes elastic constants for use in the plate theory and approximate closed-form recovering relations for all three-dimensional field variables expressed in terms of plate variables. In general, the specific type of plate theory that results from this procedure is determined by the procedure itself. However, in this paper only “Reissner-like” plate theories are considered, often called first-order shear deformation theories. This paper makes three main contributions: first it is shown that construction of an asymptotically correct Reissner-like theory for laminated plates of the type considered is not possible in general. Second, a new point of view on the variational-asymptotical method is presented, leading to an optimization procedure that permits a derived theory to be as close to asymptotical correctness as possible. Third, numerical results from such an optimum Reissner-like theory are presented. These results include comparisons of plate displacement as well as of three-dimensional field variables and are the best of all extant Reissner-like theories. Indeed, they even surpass results from theories that carry many more generalized displacement variables.