A Three-Dimensional Stress Analysis of Woven Fabric Composites Based on a Mixed Variational Model and Spline Approximation
Authors: Roy A. K. 
Published: January 2022

Abstract
The stiffness of fabric (woven) composites has been well studied. However, the failure mechanism of this class of composites is not well understood. The prediction of the stiffness of woven composites is relatively simple compared to the prediction of strength, and has been done extensively in the literature. It is known that the yarn crimping in woven fabric composites plays an important role in the failure initiation process. In the vicinity of yarn crimping of two-dimensional fabric (woven) architecture, two perpendicular yarns crimp over and under to each other. Due to the perpendicular yarn crimping, even under the application of simple unidirectional load, the stresses in the vicinity of the yarn crimping are three dimensional. Further, in situ experimental observation of damage initiation in textile composites reveals that the damage initiates in the form of interface cracks in the vicinity of yarn crimping, which is strongly influenced by the interlaminar stresses at the interface region. Thus an accurate prediction of the interlaminar stresses at the interface region is needed to reliably analyze damage and failure in woven composites. Most of the research work in this area, however, is based on two-dimensional stress analysis which does not reliably predict the interlaminar stresses. Further, traditional displacement-based finite-element analysis only predicts stresses accurate at the Gaussian integration points; thus, even three-dimensional finite element analysis does not yield accurate interlaminar stresses at the interface.
A mixed three-dimensional variational model has been derived for stress analysis of a representative volume element (RVE) of woven fabric composites, based on the Reissner variational principle. In this model, each yarn is modeled as a homogeneous orthotropic medium, and the matrix regions that exist around the wavy yarns are also represented as separate subregions in the model. The RVE of a two-dimensional woven fabric composite is shown in Figure 1. The representative volume element (RVE) of the model is divided into several subregions; each subregion is occupied by a characteristic fabric yarn (Figure 2). The yarns of characteristic self-similar crimping are lumped together and represented in one subregion, as shown in Figure 2. In order to accurately predict the characteristic damage (crack initiation and its propagation), the equilibrium of stresses is satisfied pointwise everywhere in the model, and the inter-yarn stress compatibility is enforced in the model. The in-plane stresses within a yarn are assumed to vary linearly in the thickness direction, and the expressions for the interlaminar stresses are obtained by satisfying the three-dimensional equilibrium equations. Further, to reduce the computational size of the problem, the thickness integration of the quantity in the variational principle is performed. Due to crimped profile of the yarns (i.e., not being parallel to RVE axes) Leibnitz’s theorem must be used to perform the thickness integration. After performing the thickness integration, the model yields a set of first order partial differential equations. Based on the linear variation of the in-plane stresses in the thickness direction, the principle yields 23 unknown variables for each subregion. The RVE, without any inter-yarn and intra-yarn cracks, contains six subregion. A finite element approach is taken to obtain an approximate solution to the set partial differential equations. For the finite element approximation, an energy norm is defined using the quantities obtained in the set of partial differential equations. A set of overlapping cubic splines is used as the shape functions for the finite element formulation. The Rayleigh-Ritz approximation technique is used to derive the stiffness matrix of the problem and to determine the unknown quantities. Further, the formulation being based on the mixed variational principle, a combination of stress and displacement can prescribed as boundary condition, which cannot be achieved in any displacement-based finite element.