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 carbon 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.
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