It is interesting to note that the reduction in compacting frequency from weave pattern A to B resulted in a substantial increase in strength and modulus without a significant reduction in shear strength.

Modeling of Textile Structural Composites

The mechanical properties of textile-reinforced composites can be predicted with a knowledge of the fiber properties, matrix properties, and fiber architecture through a modified laminate theory approach. Geometric unit cells defining the fabric structure (or fiber architecture) can be identified and quantified to form a basis for the analysis. For 2-D woven fabric-reinforced composites, Dow et al. (ref. 7) and Yang et al. (ref. 43) have developed models for the thermomechanical properties of plain-, twill-, and satin-reinforced composites. Examples of these include the mosaic, crimp, and bridging models developed by Yang et al. (ref. 43). In the mosaic model, fiber continuity is ignored and the composite is treated as an assembly of cross-ply elements. With the crimp model, the nonlinear crimp geometry and the yarn continuity are considered. Based on the geometric repeating unit cell, each yarn segment is treated as a laminae. Although the crimp model was found to be suitable for plain weave composites, the bridging model was found to be best for satin weave composites because it takes the relative stiffness contribution of the linear and nonlinear yarn segments into consideration.

The modeling of 3-D fiber-reinforced composites also begins with the establishment of geometric unit cells. A summary of the four major classes of 3-D fiber architecture is given in figure 25. Three basic yarn components make up the unit cell for a 3-D orthogonal nonwoven fabric, defined according to the yarn orientation: 0° (warp), 90° (weft), and through-the-thickness. Five basic yarn components make up the unit cell for a general MWK fabric, defined according to the yarn orientation: 0° (warp), 90° (weft), 45° (bias), -45° (bias), and the stitching yarn (through thickness). The fractional volume of fiber in each of the directions can be calculated geometrically on the basis of yarn size, yarn spacing, and stitch construction that should reflect a combination of the orthogonal unit cell and crimp geometry. Several yarns running parallel to the body diagonal of the cell represent the unit cell for the 3-D braid. However, in some instances, yarns are placed in longitudinal (0°) and transverse (90°) directions of the fabric and are referred to respectively as longitudinal and transverse reinforcements (or lay-ins).

Among the 3-D composites, the 3-D braided composites have received much attention because of their improved stiffness and strength in the thickness direction, their delamination-free characteristics, and their near-net shape manufacturing capabilities. Several analytic models have been developed to characterize the elastic moduli and structural behavior of 3-D braided composites.

Table V. Effect of Fiber Architecture on Mechanical Properties
of 3-D Braided CCC

Figure 25. Unit cell geometry of 3-D fabrics: (a) 3-D braid. (b) MWK, (c) orthogonal nonwoven, and (d) multiwarp weave (angle interlock).

Ma et al. (ref. 44) assumed the yarns in a unit cell of a 3-D braided composite as composite rods forming a parallel pipe. Strain energies (because of yarn axial tension, bending and lateral compression) are considered and formulated within the unit cell. By Castigliano's theorem, closed-form expressions for axial elastic moduli and Poisson's ratios have been derived as functions of fiber volume fractions and fiber orientations.

Yang et al. (see ref. 43) also developed a Fiber Inclination Model according to the idealized, zig-zagging yarn arrangement in the braided preform (ref. 43). They assumed an inclined lamina as a representation of one set of diagonal yarns in a unit cell. In this way, four inclined, unidirectional laminae formed a unit cell. Using classical laminate theory, the elastic moduli then can be expressed in terms of the laminae properties.

From the preform processing science aspect, Ko et al. (ref. 45) developed a fabric geometric model (FGM) based on the unit cell geometry shown in figure 26. The stiffness of a 3-D braided composite was considered to be the sum of stiffnesses of all its laminae. The unique feature of the FGM is its ability to handle 3-D braid and other multidirectional reinforcements including 5-D, 6-D, and 7-D fabrics with straight or curvilinear yarns. The product of the FGM is a stiffness matrix that provides a link between applied strains and the corresponding stress responses. With a properly selected failure criterion, the stress-strain relationship of the fiberreinforced composite can be predicted. The development of the FGM has been detailed by Ko (refs. 6, 16, and 45); only a brief introduction is presented here.

The first step in the modeling process is defining fiber orientation and fiber volume fraction Vf in terms of the preform processing parameters U, V, W, Ny, and Dy. The preform processing parameters U, V, and W are track displacement, column displacement, and combing frequency, respectively; Ny is the number of yarns in the cross section and Dy is the yarn linear density (denier)* from which the total fiber cross-sectional density can be computed. For a given fiber with density p and a give composite cross-sectional area Ac, the fiber orientation and fiber volume fraction Vƒ for a (U × V × W) 3-D braid composite structure are shown (ref. 44):

*1 denier = 1 gm/9000 meters. Specific cross-sectional area of the fiber As

(1)

(2)

With the knowledge of fiber orientation 9, the stiffness matrix [C] for each composite yarn segment in the unit cell can be transformed from the stiffness matrix [C] of equivalent unidirectional composites, as follows:

The systems stiffness matrix [C] for the composite subsequently can be determined by the summation of the [C] according to their fractional volume fraction contribution K:

With the material properties and geometry of the 3-D braided composite defined piecewise and linearly. the stress response of the composite system can be determined for each incremental strain {A} imposed to produce the stressstrain relationship:

where {A}, is the incremental stress response of the composite system and {o}.i is the total stress on the composite system.

For a given failure criteria based on maximum strain energy (Umax), contributions are made by tension (Ut) and bending (U) as well as other sources (Uq),

Ut - Ub + Uq> Umax

(6)

The stress-strain relationship for the composite system can be determined according to the computational flow diagram as shown in figure 27.

With the FGM, one can also examine the elastic response of various three dimensionally reinforced composites. For a carbon-carbon system with 50-percent fiber volume fraction, the uniaxial tensile modulus Er, through-the-thickness modulus Ez, and in-plane shear modulus Gry of an XYZ orthogonal fabric, and a 3-D braid reinforced composite are examined. These two reinforcement systems represent the extreme case of linear and curvilinear reinforcement systems as illustrated in the unit cell structures (figs. 25(a) and 25(c)). Assuming equal distribution of fibers in the X and Y directions for the orthogonal fabrics and a 1- by 1-braided structure, figure 28 shows that E, increases as Ex decreases for both composites. The curvilinear system has high tensile modulus at low values of Ez. At higher tensile modulus values, however, the behavior of both reinforcement systems is similar. On the other hand, figure 29 shows opposite trends of the