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the significant localized microcracking that occurs in CC composites. This model considers a unit cell that is reinforced with three locally orthogonal fiber bundles and which contains regions of material with variable stiffness between the unit cell subregions. Actual crack geometry and crack-tip stress intensity effects are not considered. Rather, the transmission of loads across the interfaces in the unit cell is considered. This approach to the effect of microcracking leads to a definition of the efficiency of the unit cell interfaces in transmitting loads. The unit cell efficiency can be considered equivalently as either the transmission of a load, or as a measure of the uncracked contact area along the interfaces within the unit cell (see fig. 4).

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Figure 4. 3-D CC unit cell model: (a) unit cell geometry and (b) weak regions or surfaces of weakness.

The drawback with the DCAP model is that it is developed especially for three directionally reinforced materials and cannot handle multidirectional materials or thin two-dimensional material easily. Multidirectionally reinforced materials require an analysis approach similar to NDPROP, which assumes an effective matrix material. Thin-section two-dimensional laminated-type materials can be analyzed with a laminate analysis code, as described in the following sections.

Method of Analysis

The basic procedure used in both material models is the application of the stress state associated with the particular elastic constant of interest to the unit cell. This stress state results in an average strain state defined by the stresses and the unknown effective elastic constant. In a homogeneous material, this strain state would define the actual displacements throughout the material. In the composite, those displacements are taken as the displacements existing on each of the internal surfaces separating one subcell region from another. In this manner, the displacements on the entire boundary surface of each subcell region are defined. Such displacements define average strain components with respect to the principal elastic axes of each subcell region. If the elastic constants of each subcell are known (the method by which they are found is discussed subsequently), the average stresses in each subcell can be found. By proper coordinate transformations, the average stresses over the entire unit cell can be found from these subcell stresses. The unknown elastic constant is now found by equating the average stress state computed in this fashion to the applied stress state.

This procedure is equivalent to using an admissible displacement field to obtain a bound on the strain energy, from which a subsequent bound on the elastic constant can be obtained. Details of the analysis are presented in references 22 and 23.

Multidirectional Composite Model

The unit cell for these materials is regarded as an assemblage of subcells (see fig. 2), each of which contains either a unidirectional fiber composite or a matrix material (with or without dispersed voids). Utilization of the model requires definition of the effective thermoelastic properties of each subcell region as a function of the constituent properties.

When voids are present in the matrix, the effective matrix properties are computed for various combinations of spherical and cylindrical voids as appropriate. A composite sphere or a composite cylinder assemblage of voids (refs. 6 and 10) is assumed in the matrix material. The voids in a fiber bundle matrix are probably highly elongated in the fiber direction. Effective moduli for such a matrix are computed from the composite cylinder assemblage results (ref. 6) by assuming that the fibers have zero stiffness. Even an isotropic matrix will become transversely isotropic when cylindrical voids are added. The interstitial matrix is likely to have voids that are approximately spherical. For randomly dispersed spherical voids, the lower bound given in reference 10 has been utilized. This bound results in the matrix modulus being reduced by a factor that is a function only of the void volume fraction. As shown in reference 22, even small amounts of voids have a significant effect on the matrix properties.

Given the set of effective matrix properties, the one-dimensional (1-D) impregnated fiber bundle properties (shown as bundles 1, 2, or 3 in fig. 4) are computed using the composite cylinder assemblage results for composite cylinders with transversely isotropic constituents (ref. 10) in the form utilized in reference 22. In the composite cylinder assemblage model, a unidirectionally reinforced material is modeled by an assemblage of composite cylinders of variable sizes, which fill out space. Each composite cylinder consists of a fiber and concentric matrix shell. In an actual fiber-reinforced material, the fibers are more or less identical in crosssectional area and are randomly placed. Finite-element models utilizing circular fibers with equal diameters have been used to obtain exact solutions for regular fiber arrays. The composite cylinder assemblage model treats the geometric randomness that exists in real composites, but requires the fibers to have variable diameters.

The primary advantages of this approach over the regular array model stem from the fact that it can be analyzed analytically, with resulting simple closed-form expressions for detailed stresses, strains, and composite properties. This analysis is important for both researcher and designer because it allows ready observation of trends due to changes in constituent properties and volume fractions. Furthermore, the simplicity of the model makes it possible to analyze not just elastic properties, but also thermal, thermoelastic, and viscoelastic properties. The model is also readily adaptable to the important case of fiber and matrix cylindrical anisotropy. A drawback of the model is that one elastic parameter (the transverse shear modulus) cannot be calculated exactly and can be bounded only from above and below. However, comparison with experiments shows that for the usual stiff fibers, the upper bound agrees well with the experiments and can be used as an approximate result.

A comparison of finite-element analyses of regular arrays and composite cylinder assemblage equations shows that property and stress results are very close. In fact, hexagonal array results and composite cylinder assemblage results are practically indistinguishable. Both geometrical idealizations have been used extensively and successfully by many researchers (e.g., refs. 21-24). 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 modeling of CC materials. The greater flexibility of the composite cylinder assemblage model and its ability to achieve significantly lower cost results are important factors in choosing it to model unidirectional composites.

Once the matrix and impregnated fiber bundle properties are available, the unit cell can be constructed as in figure 4. Both the NDPROP and DCAP computer programs allow a large amount of freedom in constructing the unit cell analysis. This feature allows the user to supply one set of constituent properties to the program and to construct a very wide range of unit cell configurations, as desired.

The unit cell is constructed by specifying the fiber bundles in each direction and in the interstitial matrix material, along with volume fractions of each material. In the case of NDPROP, the bundle direction numbers define the orientation of each bundle relative to the global axes. Any number of bundles can be defined, each with its own fiber, matrix, and orientation, provided that the total volume fraction of all fiber bundles is less than unity.

Degraded Properties Model

Cracking between subregions within the unit cell occurs as a result of cooldown from the graphitization cycle during fabrication of CC composites. The fibers in current CC materials have a much lower axial thermal expansion coefficient than the matrix material. Because the material is graphitized at high temperature, the material becomes stress free at that temperature. Upon cooldown, the matrix is loaded in tension since it tries to contract at a much higher rate than do the fibers. This loading results in cracks occurring between the fiber bundles and the matrix material. The degraded properties model was developed specifically to include the effects of cooldown cracking upon resulting composite properties. Actual crack geometry and crack-tip stress intensity effects have not been considered. Rather, the model considers the ability of the material to transmit loads across the interface between subregions within the unit cell.

The weak regions are approximated as thin plate-like regions in which the thickness of such regions is assumed to be small in comparison to the dimensions of the subregions and the unit cell. In order to develop a simple analytical model that yields a suitable representative volume element, all parallel planes of weakness are assumed to have similar stress transfer characteristics, although each of the unit cell interconnecting surfaces may have its own unique stress transfer characteristics defined, if desired.

The inclusion of regions of degraded load transfer capability in DCAP requires some definition of the effectiveness (efficiency) of the degraded material layers in transmitting axial and shear loads between subregions. The efficiency of the degraded material may be considered equivalently as the stress transfer capability (i.e., an approximate measure of the uncracked contact area between subregions) or as a measure of the stiffness of the degraded material. The analysis of the effect of these added regions upon unit cell properties is described in reference 22.

Both the degraded property model (DCAP) and the basic multidirectional composite model (NDPROP) treat thermomechanical properties including the effects of temperature-dependent constituent properties. Thermal expansion data are defined for the constituents in the form of free thermal expansion versus temperature. The model then can compute secant thermal expansion coefficients for each subcell of the unit cell from the analysis temperature to the reference stress-free temperature. The secant thermal expansion coefficients for the subcells

are then combined to form the composite thermal expansion coefficients. The resulting values for the composite, from the stress-free temperature to the given temperature, are used to obtain a set of predicted composite thermal expansion coefficients versus temperature.

The output of the models provides all of the thermoelastic properties of the unit cells for each desired temperature. In addition, if composite stresses are applied, the unit cell strains and average strains and stresses in each subregion can be computed for each temperature. An example of property prediction and correlations with experimental data follows.

Thin CC Composites

Historically, the development of CC composites has focused on multidirectionally reinforced materials. Focusing was a result of material being used for basically nonstructural applications in which heat resistance was the primary material function. These components were thick; consequently, the poor shear and throughthe-thickness direction properties required additional reinforcement directions.

Future CC applications will continue to require multidirectionally reinforced materials, but there is also a growing interest in thin, laminate construction, CC composites for structural applications requiring light weight and good hightemperature behavior.

For these thin-section CC materials, one asks how the properties might be predicted. The author and Materials Sciences Corporation have investigated this area and found that standard laminated plate analysis codes are adequate for predicting the behavior of thin-section CC composites.

The code utilized for thin-section CC materials can be simply a standard laminate analysis code. The author has been utilizing CLASS, which is a personal computer based laminate code. The advantage of this code is that the composite cylinder assemblage model is built into the code so that the user can predict layer properties for a variety of fibers and matrix materials and then can construct the laminates from these layers. In addition, the code has the ability to combine particles such as needles, spheres, or platelets in a matrix by using upper and lower bound solutions as well as the differential scheme approach. This code allows creating matrices with various inhibitors or short fiber reinforcements and then adding long fibers to create a layer. Also, this code permits transversely isotropic constituents, all of which are temperature dependent. All of the constituent and layer properties are stored within a small data base for later access by the code. Thus, properties can be entered once and used over again.

The basic analyses of thin CC material are identical to those of standard epoxy laminates. The difference is in the difficulty of defining the constituent properties

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