Homogenization theory of regular cross-ply laminates
Tadeusz Niezgoda, Marian Klasztorny
Quarterly No. 2, 2009 pages 154-158
DOI:
keywords: regular cross-ply laminate, homogenization, constitutive equations of elasticity, computer-aided algorithm
abstract The paper concerns regular cross-plyfibre-reinforced-plastic (CP xFRP) laminates, i.e. a stack of plies of [0/90]nS, n 4 configuration. Each ply is a UD xFRP composite, i.e., an isotropic hardening plastic reinforced with long monotropic fibres packed unidirectionally in a hexagonal scheme. The plies are identical with respect to their thickness and microstructure. The considerations are limited to stress levels protecting geometrically and physically linear elastic behaviour of the material. The study presents the homogenization theory of a regular CP xFRP laminates. The theory employs respective boundary-value problems put on the representative volume element of the laminate: uniform tension in the x direction, uniform tension in the z direction, pure shear in the xz plane, pure shear in the xy plane. The representative volume element is considered in these problems under the following requirements: 1) elastic behaviour of the unhomogenized and homogenized representative volume element must be compatible with behaviour of the whole laminate, 2) the unhomogenized and homogenized representative volume element satisfy the compatibility conditions put on the total stress and displacement states. A concept of homogenization in each boundary-value problem contains the following steps: a) formulation of stress and strain components of the homogenized representative volume element, b) formulation of constitutive equations of elasticity, c) formulation of stress and strain components in each ply of 0° orientation before homogenization, d) formulation of stress and strain components in each ply of 90° orientation before homogenization. The final analytic formulae for effective elasticity constants of the laminate, describing an orthotropic model of the homogenized material, are presented. Based on the exact homogenization theory of UD xFRP composites and the exact stiffness theory of regular CP xFRP laminates presented in this study, the authors have written a computer programme in PASCAL for predicting the effective elasticity constants of these materials. As an example, a regular CP U/E53 laminate of [0/90]nS, n 4 configuration is considered. The matrix (E53 hardening plastic) is made of Epidian 53 epoxide resin, reinforced with UTS 5631 carbon fibres produced by Tenax Fibers.