On Dual Extremal Variational Principles for Geometrically Nonlinear Composite Plates

Thin composite geometrically nonlinear plates are under consideration. The plates are linearly elastic, obeying the Kirchhoff straight normal hypothesis and the classical theory of laminated plates. The layer fibers may have different orientation angles depending on the local position of a point in the midplane. The deformation of the plates obeys the von Karman approximation. The kinematic and static variational principles for the plates are considered. The proven statement is: for everywhere positive principal in-plane (membrane) force resultants, the principles lead to a minimum and a maximum, respectively. The duality gap for the considered variational principles is absent. The sum of the total plate strain potential energy and the total complementary energy is equal to the potential of external forces. For the composite plates, a generalization of the Clapeyron theorem of the linear theory of elasticity is proven. A comparison of the behavior of geometrically linear and geometrically nonlinear composite plates (under the same loading) suggests greater generalized stiffness and lower generalized compliance of the latter plate. An example of a composite plate illustrates the variational principles discussed and their error bounds. The results of the paper are applicable to the analysis and design of composite structures, such as lower panels of large-span composite wings.