Three-dimensional tow inclination model for calculating elastic constants of three-dimensional triaxial braided composites

A simple three-dimensional tow inclination model was developed to calculate elastic constants of two and three-dimensional woven and braided composites. The analysis was based on the idea that textile composite is an assemblage of specially oriented inclined tows and interstitial matrix. The repeating unit cell and the representative tows for 2-D textile and 3-D triaxial four-step circular braids were defined. The elastic constitutive matrix of the composite was derived from the 3-D stress transformation and iso-strain condition. Based on this analysis a PC and Macintosh based FORTRAN computer code TowInc-3D was developed. The code was verified for 2-D woven and braided composites with the TexCAD code and test data in the literature. The E& from the code was verified for 3-D triaxial braided composite by test. However, additional verification is needed. A parametric study in variation of braid angle, fiber volume fraction and number of moving rows on elastic constants of 3-D braided composites was conducted. Although the braiding is 3-D, the transverse (through-the-thickness) orientation of braid angle was only about 2° to 7° for the planar braid angle (a/) of 15° to 75°. With increase in braid angle, as expected, £„ decreased whereas £„ increased. The shear modulus Gxy was largest at a/=45°, and v^ was largest (about 0.9) and v« is smallest at 00=30°. Introduction Textile composites offer several advantages over laminated composites. Textile preforms are relatively easy to handle and cut into required shapes and sizes. Fabrication costs for textile composites is low [1]. The through-the-thickness reinforcement in three-dimensional (3-D) textile preforms yield a more delamination resistant and damage tolerant material. The ability to braid near net shape preforms should reduce machining, joining, and scrap costs for structural components. The technology, however, is still in the development stage. Studies to develop fabrication *Graduate Student 'Research Professor, Associate Fellow AIAA ** Professor, Member ASME "Copyright ©1996 by Kevin L. Branch and Kunigal N. Shivakumar. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission." processes, mechanics models which predict stiffness and strength, and test methodologies are being continued. A number of analytical and numerical techniques have been used to calculate elastic constants of woven and braided composites. All of the proposed models exploit the periodicity of the textile architecture to isolate a representative unit cell (RUC). The RUC is assumed to lie in the interior of the composite away from edge effects. A review of analytical models used in the past was presented in references [2] and [3]. Some of the earlier models for analyzing woven and braided composites are highlighted below. Chou and Ishikawa [4] presented three different models for 2-D woven composites based on classical lamination theory (CLT) the mosaic model, the fiber undulation model, and the bridging model. The mosaic model neglected the undulation of the yarns and idealized the RUC as an assemblage of asymmetric cross-ply laminates. The fiber undulation model considered the undulation in the fill yarn direction but neglected the undulation in the warp yarns. The bridging model extended the applicability of the fiber undulation model to the analysis of satin weave composites by separately modeling the straight portions of the yarns. The fabric geometry model (FGM) [5] was another model based on CLT. It combined textile engineering methodology and a modified laminate theory. The geometric model consisted of straight yarns oriented in six different directions. Such a unit cell representation neglected the crimp hi the yarns. The FGM was also used together with an energy analysis [6] to predict stiffness properties of three-dimensional (3-D) braided composites. The FGM model was further modified by Pastore and Gowayed [7]. The fiber inclination model (FIM) [8] was also based on CLT. It treated the fabric unit cell as an assemblage of inclined unidirectional laminae. The FIM also neglected crimp in the yarns. Ramnath [9] presented an approach that used a variable yarn crosssection geometry model along with admissible displacement and stress fields to determine bounds on the elastic properties of 2-D woven composites. 1788 American Institute of Aeronautics and Astronautics Naik and Ganesh [10] and Raju and Wang [11] developed models for woven composites that accounted for undulations in both the fill and warp yarns. They computed overall stiffness properties by assuming either iso-strain or iso-stress or a combination of these and analytically integrating through the volume of the RUC. Due to the tedious analytical description of yarn cross-sectional shapes and yarn paths and the analytical volume integrations, this approach is not well suited for the analysis of more complicated architectures such as 2-D and 3-D triaxial braids. Three-dimensional finite element models (FEM) [12-16] have also been developed for the analysis of plain weave composites. The 3-D FEM are highly computer intensive and also require considerable time and effort for model generation. Mohajerjasbi [17] developed micromechanics models for 3-D Cartesian braided composites. Foye [14] developed inhomogeneous finite elements and analyzed sub-cells within the RUC to overcome these limitations. He analyzed many textile architectures including plain and satin weaves, 2-D braids, and 2-D triaxial braids. However, he had to manually calculate the stiffness properties, after estimating the material directions and volume fractions, for each element in the 3-D model. Glaessgen and Griffin [18] modeled the complete plane weave unit cell by 3-D solid elements and calculated the elastic constants. The model was complex and the size of the problem was very large. One can easily conclude from these studies that the FEM is not suitable for modeling complex 3-D fiber architectures or for performing parametric studies to investigate the sensitivity of overall mechanical properties to architecture parameters. Naik [9,19] presented a simple but accurate geometric modeling and analysis technique for 2-D triaxially braided composites. They discretely modeled the yarn centerline paths within the RUC by assuming sinusoidal undulations and used a yarn discretization scheme together with a 3-D stress averaging procedure to compute overall stiffness properties which correlated well with test data over a wide range of architecture parameters. Naik [2] generalized his 2-D micromechanics analysis for calculating the three-dimensional mechanical and thermal properties of 2-D wovens and braided composites. He also developed a PC based code called TexCAD. Shivakumar et al [20] presented a simple 3-D tow inclination model to calculate elastic constants of 3-D braided composites. In this paper, the 3-D tow inclination of Shivakumar et al is generalized for 2-D and 3-D braids and weaves. The concept of tow inclination, architecture of 2-D and 3-D braids and weaves, and the micromechanics analysis are presented. Tow Inclination Model The Concept of Tow Inclination Analysis The tow inclination model is based on the simple concept that the curvilinear profile of tows within a representative unit cell of a textile composite can be approximated by straight line segments. The spatial orientation of the yarn segment was defined by the weaving or braiding process used in textile machines. Fiber volume fraction was used to constrain the total length of the tow segments and their cross-sectional areas. The elastic constitutive matrix in the tow coordinate system was transformed to the reference or global coordinate system. The products of the transformed constitutive matrix and volume fraction for each of the tow segments and matrix were summed to obtain the global composite constitutive matrix. Then, all six elastic moduli and three Poisson's ratios were generated from the global constitutive matrix. Details of this method for satin woven and braided composites and 3-D braided composites are presented in the following sections. A number of improved geometrical models have been reported in the literature for 2-D preforms [Example 3-6, 10]. These 2-D textile models serve to verify the accuracy of the simple tow inclination model. Geometric Definition for 2-D Preforms (a) Satin Weaves The idealized tow inclination models for two of the three cases studied are shown in Figures 1 and 2. The length Lp of the square shaped repeating unit cell (RUC) is twice the width w of a tow for the plain weave. The length of the RUC is five and eight times the width of a tow for the five and eight harness weaves, respectively, and this RUC is also square. A generalization of woven fabric to m-harness satin weave results in a square RUC of length mw with 2m participating tows. The tow cross section is assumed to be of a diamond shape. The width is w and thickness is t. Assuming the tows cover the complete width of the RUC, the tow spacing a is equal to tow width w. Hereafter a will refer to both tow spacing and width. Notice in Figure 1 that for a plane weave, this tow profile is approximated by five straight line segments. There are two horizontal segments and three inclined segments. The angle of inclination of inclined segments is called the crossover angle ft.