Self-consistent <i>T</i>-matrix solution for the effective elastic properties of perfectly disordered multiphase solids

Abstract

A self-consistent solution for the effective elastic properties of polycrystalline and perfectly disordered multiphase composites has been discussed by using the T-matrix method under certain suitable approximations. Compared to the existing formulas these new relations for the disordered composites are very useful in practical situations for a quick and more accurate estimate of the effective elastic properties, in particular for a case where the composite has components with widely different values of the elastic constants. For comparison we have discussed the results based on Kröner’s theory which also purports to solve the same problem. It is found that the two solutions do not agree. To resolve the difference we take help of Hill’s exact solution of the composite problem when the components have equal rigidities. It is found that while Kröner’s theory is inconsistent with the exact result the present self-consistent solution analytically reproduces it. Another interesting finding of the present investigation is that the approximations made in obtaining the self-consistent solution are exact in the limit of composites with equal-shear moduli. Finally it is indicated that although the results for composites have been derived for isotropic and cubic components it can be easily adapted for a composite with noncubic components.