Variational solution for a cracked mosaic model of woven fabric composites

Abstract

A variational solution for a cracked mosaic laminate model of woven fabric composites is presented using the principle of minimum complementary energy. The solution is derived for the woven laminate in either the plane strain or the plane stress state, with the warp/fill yarn materials being either orthotropic or transversely isotropic, unlike other existing solutions in the literature of laminate elasticity. The stress components are given in closed-form expressions in terms of a perturbation function, which is governed by two (uncoupled) fourth-order inhomogeneous ordinary differential equations (i.e., Euler–Lagrange equations) when the thermal effects are included. All possible expressions of this perturbation function are obtained in closed forms. Young’s modulus of the cracked laminate is calculated using the determined minimum complementary energy. The present closed-form solution can account for different yarn materials, applied loads (crack densities), geometrical dimensions, or their combinations. To demonstrate the solution, a total of 60 sample cases are analyzed using three different composite systems (i.e., glass fiber/epoxy, graphite fiber/epoxy and ceramic fiber/ceramic) and ten different crack densities. The obtained numerical results are also compared to two existing elasticity solutions for cross-ply laminates.