Analytical solutions of heterogeneous rectangular plates with transverse small periodicity

摘要

We consider the cylindrical bending of a simply-supported orthotropic rectangular plate, which is small periodically heterogeneous in the thickness direction. A homogenized plate model is first established by using the two-scale asymptotic expansion method. The state-space method is then adopted to analyze the homogenized plate exactly. Analytical expressions for two sets of approximate stresses, i.e. the homogenized model stresses and the zeroth-order two-scale model stresses, are presented. To check the accuracy of the approximate stresses, the state-space method is also applied to the original laminate plate or the approximate laminate model of the original heterogeneous plate with continuously varying material properties to obtain an analytical solution. In the latter case, the analytical solution is approx imate but approaches the exact solution gradually when the number of layers increases. In order to avoid numerical instability, the joint coupling matrix is utilized. Numerical results illustrate that the two-scale model can predict accurately the realistic stress field in the original plate if it contains enough repeated units along the thickness.