Figure 8. Effectiveness of fiber bundle sheath on composite modulus. The determination of the necessary parameters utilized in the prediction of effective composite properties is an iterative procedure. For example, revised approximations to fiber modulus may change the estimated amount of fiber bundle sheath which may be present in the material. The degree of approximated matrix microcracking may not be compatible with the effective matrix properties. Data correlations must be performed, with options available, to continually update all necessary material model parameters. In addition to this the engineer must carefully scrutinize the processed material to determine if the specifications of weave architecture have been met because deviations in bundle orientation can result in unexpected material response. Effects of Degraded Properties The degraded property layers can be treated as additional subcell regions and the previously described procedure can be applied. However, there is some question as to the influence of using only admissible displacement fields, which are representative of upper-bound results. Hence, the approach to this problem considered both upper and lower bounds. Approximate solutions for the boundary value problem were developed using variational techniques that yield upper and lower bounds for the effective thermoelastic properties as well as approximate values of the strains and stresses in various subregions for a given applied stress Figure 10. Correlations of fiber axial modulus with tensile data. state. For upper-bound formulations, admissible displacement fields are assumed with unknown parameters, and the potential energy expression in the representative volume element is minimized to determine the values of these parameters. From the theorem of minimum potential energy, the potential energy for the assumed displacement field is greater than or equal to the actual potential energy that yields the upper bound for the elastic moduli. In the lower-bound formulation, admissible stress fields are assumed, and the complementary energy is minimized to obtain the lower bounds via the theorem of minimum complementary energy. Thus, the model provides both upper- and lower-bound solutions for the thermoelastic properties of the RVE and, hence, the composite. Each degraded property region is considered to have a reduced stiffness. This stiffness may differ for each stress component between each pair of subcell regions, and reduced stiffness may be normalized to define an efficiency for stress transfer. By assuming that the stiffnesses of the degraded regions are the same on each principal plane of the material, the unknown efficiencies can be reduced to six in number, namely, the efficiencies for tension and compression and for the three principal shear directions S1j. The DCAP computer formulation includes these six quantities. However, a single efficiency S may be utilized by simply assuming that all efficiencies are equal. A single unit cell efficiency S has been found to agree with limited experimental data on CC materials. Figure 11. Fiber axial modulus as function of fiber transverse modulus. The efficiencies are an approximate measure of the reduction of the composite Young's moduli from the value of the modulus when no weak regions are present. If one assumes that the weak regions contain flat disk-shaped (crack-like) stressfree regions, then the efficiency is also an approximate measure of the fraction of the contact area between the subregions that are considered perfect in transferring stress. Therefore, when Sij = 0, all the weak surfaces perpendicular to the coordinate direction are fully cracked. When Sij = 1, no cracks exist in the weak surfaces perpendicular to direction i. The efficiencies are similarly approximate measures of the reduction of shear modulus Gij or the fraction of the contact areas effective in transferring shear stress Tij. Details of the analytical formulations and assumed displacements within the unit cell have been given in reference 23. The upper- and lower-bound formulations were also shown to give very close results for most weave configurations; because of this, the upper-bound results are sufficient for most purposes. Concluding Remarks Both the basic model (NDPROP) and the degraded properties model (DCAP) have been shown to be capable of accurately predicting carbon-carbon (CC) composite material properties. DCAP, in particular, has been demonstrated to have the ability to model various CC materials with enough accuracy to permit reliable engineering studies of material performance with little or no experimental data. When developing analytical models for CC composites, the philosophy should be to choose an appropriate approach for each of the specific subproblems at hand. It must be emphasized that the objective of analytical modeling is to provide a cost-effective procedure for enhancing the efficiency of the materials development process. Analytical guidance of innovative material concepts will require extensive analytical screening of potential materials. Thus, the material model must be an engineering tool that treats many complex material problems in a manner that recognizes the primary objective. In the course of developing these models, various methods have been used, including composite cylinder assemblages, variational methods, and approximate solutions suited to the problem. The name materials engineering has been coined to describe this engineering approach. The materials engineering approach to the development of a material model for quantitative material synthesis utilizes a modular concept that provides a convenient capability to upgrade the quality of predictions without substantial overhaul of the total material model. Also, during the developmental stage when certain modules of the code are approximate or preliminary in nature, the code may be used for comparative rather than for quantitative predictions and/or for data enhancement. The DCAP model has been used to demonstrate the feasibility of the concept of pre- and post-processor interaction with material performance codes. The DCAP code has been written so that it can interact with such thermostructural finiteelement codes. The interaction between codes is effective; yet, the interface with the DCAP code remains accessible such that the program can be made to interface with other performance codes. Material models of these types should prove valuable to the materials manufacturer in developing current and future material systems for advanced applications. Also, the models will provide the structural analysis community with an effective material property prediction tool to be used as a preprocessor to aerospace structural analysis codes. Carbon-carbon materials are complex and possess substantial variability in properties. This complexity coupled with variability should not be looked upon as an impossible task for analytical modeling. Rather, the analysis models should be viewed as tools that can help explain the trends found in data and guide the material development process so that the next batch of material will possess those characteristics that are desirable for the structure being designed. |