On the strain energy of laminated composite plates

Abstract

A strain energy function is obtained for nonhomogeneous. laminated, composite plates for the case when each lamina exhibits monoclinic material symmetry about its middle surface. The starting point is the three-dimensional strain energy based on geometrically nonlinear elasticity theory. The variational-asymptotical method is used to decompose the nonlinear three-dimensional problem into two separate problems: (1) a linear, through-the-thickness. one-dimensional analysis to obtain appropriate plate elastic constants and relations between plate deformation variables and three-dimensional results: and (2) a nonlinear, two-dimensional analysis to analyse the plate deformation. Closed-form analytical expressions are derived for the plate elastic constants as well as the displacement and strain distributions through the thickness of the plate. Even with this generality, no more variables are involved than in Rcissner Mindlin plate theory. Also, in spite of the simple form for the plate strain energy, there are no restrictions on the magnitudes of displacement and rotation measures. Two approximations are obtained in the through-thc-thickness analysis, the tirst being equivalent to classical laminated plate theory, and the second incorporating shear deformation etfects. The first approximation is asymptotically correct for plates of the form considered. The second approximation is asymptotically correct for plates with certain additional material restrictions. In applying the method, one first solves the through-thc-thickness problem and then uses the resulting elastic constants to pose the nonlinear plate problem. After solving the nonlinear problem, one substitutes these results back into the linear three-dimensional relations for displacement and strain throughout the plate.