Variational analysis of cracked general cross-ply laminates under bending and biaxial extension

In the current study, a variational model is developed for predicting stress transfer due to ply cracking in general cross-ply laminates subject to out-of-plane bending and biaxial in-plane loading. The model is valid for multiple ply laminates that can be nonsymmetric. Using the principle of minimum complementary energy, an optimal admissible stress field is derived that satisfies equilibrium, boundary, and traction continuity conditions. Natural boundary conditions are derived from the variational principle to overcome the limitations of the existing variational methods on the analysis of cracked general cross-ply laminates. Comparing laminates of glass/epoxy and graphite/epoxy with the available finite element results shows that stress components are in very good accordance with the analytical results. It is also shown that the existing variational models are specific cases of the current formulation. In the next step, the obtained stress field is used in conjunction with the principle of minimum complementary energy to get the effective stiffness modulus of a cracked general cross-ply laminate. It is indicated that the method provides a lower bound for the stiffness modulus of a cracked laminate.