| Abstract: | It is shown that the theory of random functions permits the expansion of the effective tensor X~jkl for the elastic moduli with respect to correlation functions and that it leads in the second approximation in the Voigt-Reuss scheme to values that lie to one side of the Xijkl, while in the third approximation it brackets the latter. The analysis is used to refine the Hashin limits to the elastic moduli for a mechanical mixture of isotrcpic components and polycrystalline aggregates of cubic structure.There are two methods for calculating the effective elastic moduli of heterogeneous solids: virial expansion 2] (as a power series in the concentration of one of the components) and the method of correlation functions 2] (expansion with respect to relative fluctuation of the elastic moduli). Identical results should be obtained in the two cases if all terms are incorporated, but great mathematical difficulties restrict one to the lowest approximations. The first approximation in the virial method gives better results when the concentration of one component is low, while the method of correlation functions gives better results when the fluctuations in the elastic moduli are small and the concentrations are similar.Methods have been developed for determining the upper and lower bounds in both approaches, and various schemes of averaging are used for this purpose in the correlation-function method. The upper bound is established by renormalizing the equation of equilibrium, while the lower one is found by renormalizing the equation of incompatibility. The range of the bracketing can be reduced by means of higher approximations. The range can be reduced in the limit to zero, which implies passing from an approximate effective tensor to the true one, which relates the means in stress and strain over the material. Here we show that the two methods of renormalization give identical results when all terms of the series are summed.If the tensor has a Gaussian distribution, the moment functions of odd order are zero, while the even ones are expressed via combinations of the binary functions 3]. However, a mechanical mixture of several components is not Gaussian, and the odd moments are not zero. Splitting of the higher-order correlation functions is possible also for mechanical mixtures having determinate phase interfaces, but this involves various simplifying assumptions. A derivation is given for a moment of arbitrary order, which allows one to formulate the conditions under which such splitting is possible. The results are used in calculating the exact value of the effective bulk modulus for a medium with a homogeneous shear modulus.We are indebted to V. V. Bolotin for a discussion. |
