Variational analysis for angle-ply laminates with matrix cracks

A stress-based variational model is developed to study stiffness reduction and stress distribution in angle-ply laminates [θ2l/θ1m/90n]s with matrix cracks. The inter-laminar shear stresses between 90° and θ1°-plies and between θ1° and θ2°-plies, respectively, are assumed to be in the form of general functions. The normal stresses σx in θ1°-plies and θ2°-plies are introduced with partition coefficient λ for solving the problem of statically indeterminate boundary because the normal stresses σx cannot be obtained by using the condition of statics due to the loads at the boundary for each uncracked layer. This leads to expressions derived from equilibrium equations and boundary conditions for stress components in terms of the general functions and the partition coefficient. The governing equations for the general functions and the partition coefficient are derived by using a variational approach with the principle of minimum complementary energy. As an application, reduction of Young's modulus for different laminates is evaluated and compared with available experimental results. Distributions of in-planar and inter-laminar stresses are also presented by means of the finite difference method. The results show that the present approach is suitable to analyze stiffness reduction for multi-angle-ply laminates with transverse cracks in 90° layer.