The unit cell for these materials is regarded as an assemblage of subcells, each of which contains either a unidirectional fiber composite or a matrix material. Utilization of the model requires definition of 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 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 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 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. Even small amounts of voids have a significant effect on the matrix properties.
Given the set of effective matrix properties, the one-dimensional impregnated fiber bundle properties are computed using the composite cylinder assemblage results for composite cylinders with transversely isotropic constituents. 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 cross-sectional 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 diamters.
The primary advantage 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 on elastic parameter can’t 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. 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 composite 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 bundles properties are available, the unit cell can be constructed. Bothe 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 directions 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.