Analytical Solution for Bending of Laminated Composites with Matrix Cracks

Abstract

An analytical, closed form solution is developed for balanced (not necessarily symmetric) laminates subjected to flexural deformation. The analytical solution provides spatial distribution of displacements and curvature, from which in-plane and intralaminar strains and stresses are obtained through differentiation and constitutive equations. The deformation is shown to consist of a homogeneous deformation plus perturbations near the crack tip. A methodology is proposed to separate the perturbation from the homogeneous deformation, to eliminate ill-conditioning of the eigenvalue/eigenvector problems that occurs otherwise. It is shown that, while the homogeneous deformation provides a macroscopic measure of damage in terms of reduced flexural stiffness of the laminate, the perturbation solution provides a detailed account of the intralaminar shear induced near the crack, which is used to calculate the extent of shear lag and the maximum intralaminar shear stress. The intact portion of the laminate is modeled without lumping it into a single equivalent lamina. Furthermore, laminas can be subdivided into multiple sub-laminates to increase the accuracy of the representation of intralaminar/interlaminar shear, which is shown to improve the predicted value of maximum interlaminar shear stress, which in turn is important for the prediction of matrix-crack induced delamination.