Cracked composite laminates least prone to delamination

Abstract

In [($\pm \theta $)$_{n_{2}}$/(90 degrees)$_{n_{1}}$/($\mp \theta $)$_{n_{2}}$] fibre-reinforced composites, the outer ($\pm \theta $)$_{n_{2}}$ sublaminates are known to act as crack arrestors, i.e. to reduce the stress intensity factors at the tips of a crack in the central (90 degrees)$_{n_{1}}$ layer in all three modes of loading. The degree of reduction depends on the stiffness of the plies, the ply angle $\theta $ and the thickness of the outer sublaminates. However, although the stress intensity factor decreases, the crack-induced largest interfacial principal tensile stress increases. The situation is particularly severe under transverse mode II loading, inevitably resulting in interfacial delamination. The aim of this paper is to choose the design variables of the laminate, namely the ply angle $\theta $, relative ply stiffness and thickness, in such a way as to minimize the stress intensity factor at a crack tip in the (90 degrees)$_{n_{1}}$ layer without exceeding the interfacial bond strength. A constraint is also placed on the minimum flexural stiffness of the laminate. An alternative optimization problem in which the largest interfacial principal tensile stress is minimized subject to a limit on the stress intensity factor is also formulated and solved. The above optimization problem is solved in two stages. First, an analysis of the non-homogeneous, anisotropic medium containing a flaw is performed within the bounds of the classical lamination theory and fracture mechanics. This allows development of mathematical expressions relating the stress intensity factor in the central (90 degrees)$_{n_{1}}$ layer and the interfacial tensile stress to the sublaminate thickness and stiffness and the ply angle $\theta $. These expressions are highly complicated, precluding a completely analytical approach to the calculation of design sensitivities. In the next stage the optimization problem is reduced to a nonlinear mathematical programming one, whose solution is attempted by several techniques. The sensitivities with respect to the design variables required in these techniques are calculated by a mixed analytical/numerical approach.