The direct approach to define effective properties of this material is to describe quantitatively the geometry, to obtain an exact solid mechanics boundary value solution, and to integrate the state variables to obtain the constraints that relate the average values of these state variables. However, to define every geometrical detail of filament cross section and spacing geometries in a deterministic fashion is clearly a task of undesirable complexity. Even if such a feat could be performed for the specific configuration, the actual geometry will vary from yarn bundle to yarn bundle and also along any single yarn bundle. Thus, the problem is to find a realistic yet practical alternative to the exact solution. Several alternatives to obtaining exact solid mechanics boundary value solutions to the exact geometry are available. These alternatives are presented schematically in figure 1.

Figure 1. Alternatives to solving exact fiber bundle/unit cell geometry.

The most common approach is either to obtain deterministic solutions to approximate geometries or to obtain statistical continuum mechanics solutions to a statistically defined geometry. In certain very special cases, it is possible to obtain exact solid mechanics solutions regardless of the geometry. In general, these special cases are not applicable to the total problem of composite behavior under general load states; the exact solutions obtained for the special cases can provide a good means of evaluating solutions to approximate geometries or statistical solutions. Of particular importance are those exact relations among different effective properties of a given material. Hill (ref. 1) has developed relations among three of the effective moduli of a unidirectional composite. Levin (ref. 2) has developed exact

relations between the thermal expansion coefficients of a two-phase material and other known properties, including effective elastic moduli of the composite material and constituent properties (see also refs. 3 and 4).

Approximate geometries appear to be the most fruitful approach to modeling CC composites. Several models have been used to predict the behavior of approximate composite geometries. These analytical models include exact solutions to boundary value problems, rigorous bounding approaches (whereby effective properties can be bounded from above and below), and approximate solutions such as first-order strength-of-materials analyses or the self-consistent scheme. The most important exact solution to approximate geometries is represented by several finite-element solutions (e.g., ref. 5). These results are for regular geometrical arrays (rectangular or hexagonal) of unidirectional composites (figs. 2(a) and (b)) and the composite cylinder assemblage (ref. 6 and fig. 2(c)). Both geometrical idealizations have been used extensively and successfully by many researchers (e.g., refs. 6-9). Solutions to both of these approximate geometries have been demonstrated to give results that are close to each other and to actual experimental data. Also, both approaches can be successfully applied to the modeling of CC materials. A more detailed discussion of the different methods in use for the evaluation of elastic constants of unidirectional fiber composites can be found in reference 10.

Figure 2. Representations of fibers in fiber bundle: (a) rectangular array; (b) hexagonal array; and (c) composite cylinder assemblage.

For multidirectional composites, the literature is more limited. A comparison of theoretical models of CC materials has been performed by Jerina (ref. 11). Various approaches were compared with data on a laboratory material system, which was a nylon-fiber-reinforced rubber. Because this field is relatively new, the comparison does not treat many of the state-of-the-art models that have been recently developed. This review did treat the commonly used constant strain and constant stress models for composite properties, which are the Voigt (ref. 12) and Reuss (ref. 13) models. Paul (ref. 14) showed that these models are actually upper and lower bounds on the actual stiffnesses. Jerina showed that an approximate model developed by Pagano (ref. 15), based on a laminate approach, yielded good agreement for many composite elastic properties.

Among the other approaches to unit cell modeling, which were not treated in the review of reference 11, are the mechanics of materials approaches of Greszczuk (e.g., ref. 16) and of Zimmer et al. (ref. 17), in which stiffnesses in series and parallel are used as models. More complex models, using combinations of closedform approximate solutions and finite-element models, were formulated by Crose (ref. 18).

The model described in this paper resulted from a series of developments by Materials Sciences Corporation over a number of years. The general methodology for evaluating unit cell properties stems from an approach method originally presented by Dow and Rosen (ref. 19). The approach utilized geometry models that are extensions of those presented in references 20 and 21. In a series of unpublished company papers, the authors and their colleagues at the Materials Sciences Corporation developed a methodology to treat multidirectional, threedimensional composite materials reinforced in any number of spatial directions. In recent developments, the effects of localized regions of high microcracking were incorporated via the concept of a degraded property region. These regions of high microcracking are of particular importance to CC materials, which develop severe microcracking between constituents during the fabrication process. These developments have been reported in reference 22.

Characteristics of CC Materials

One of the principal difficulties in predicting properties of CC materials is in defining the constituents to be used as input to the models. This difficulty is a problem because the constituents that one starts with are modified by the high temperatures during processing. Consequently, the process history becomes a major influence upon the final in situ constituent properties.

Previous chapters described processing approaches and indicated how processing can affect the constituents. The analyst must use previous experience or similar processed material properties to define a starting point for constituents. The following points should be considered when beginning to define input properties:

1. High graphitization temperatures plus tensile stresses during processing will generally increase the in situ fiber modulus and lower the fiber expansion characteristics. These properties are highly dependent upon the fiber type used in the material.

2. First cycle pitch impregnation will promote high alignment of the matrix material surrounding the fibers, leading to an increasingly effective composite modulus.

3. Pyrolytic graphite will result locally in very anisotropic matrix properties, which can influence the matrix-dominated composite properties.

4. Cooldown from graphitization cycles will result in cracks developing between subcell regions, thus reducing the composite shear modulus and expansion coefficients. Whether a final graphitization cycle is or is not used will have a large effect on the properties during the first subsequent heat-up of the material. If multiple temperature cycles must be sustained by the composite, then final graphitization should be taken into account when deciding on the final processing conditions.

Description of Model

The basic mini-mechanical material model proposed for multidirectional reinforced composite materials is based upon the concept that a composite material system can be regarded as an assemblage of unit cells. This fact is illustrated in figure 3. The basic unit cell is defined by fiber and matrix properties and the phase geometry. The effective property analysis is performed on the representative volume element (RVE) which contains a large number of unit cells. The material variability can be incorporated by treating an assemblage of RVE's having some statistical variability of, for example, fiber volume fraction.

The unit cell models are constructed from impregnated yarn bundles oriented in space and from interstitial matrix regions as shown schematically in figure 3. Several basic versions of this unit cell model have been analyzed. To be effective, a model of the material must include the following material characteristics:

1. Anisotropic constituents, including the effects of oriented matrix material such as in a fiber "sheath"

2. Definition of fiber content within each fiber bundle, as well as definition of total fiber content within a repeating element

3. Definition of orientation of each fiber bundle within the repeating element

4. Definition of the interface properties between subcell regions to model the subcell cracking that exists in most CC materials

5. Dispersed voids, of both random and elongated shapes

6. Temperature-dependent constituent properties

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Figure 3. Section of 3-D orthogonally reinforced material showing assembly of unit cells.

The MSC NDPROP (N-Directional Composite Analysis Program) model has been developed for the analysis of very general composite unit cells. The unit cell is composed of fiber bundles oriented in space by defining direction numbers that determine the orientation of each fiber bundle. The cell can contain any number of bundles, provided that the total bundle and interstitial matrix volume fractions do not exceed unity. The material does not need to be orthotropic, but can be fully anisotropic, if necessary. Each of the fiber materials can be different, thus allowing treatment of hybrid composites. The constituent fiber and matrix materials may have transversely isotropic elastic constants and thermal expansion coefficients. Material porosity can be treated and matrix properties in each fiber bundle and in the interstitial matrix regions can differ.

The deficiency in the NDPROP model is the lack of ability to model explicitly the interface regions between bundles. Rather, an effective matrix is defined which includes the effects of cracks between bundles.

The DCAP (Directional Composite Analysis Program) model, which was developed under U.S. Navy funding and is available to qualified government contractors through the Naval Surface Weapons Center, focuses on the effects of