Correlation functions of the elastic field of multiphase polycrystals

Abstract

An approximation of the homogeneity of a linear combination of the stresses and strains σ + bε = const is proposed to evaluate the correlation functions of the elastic field of micro-inhomogeneous media. This approximation is a generalisation of the Voigt and Reuss hypotheses according to which the strains ε and the stresses a are considered homogeneous, respectively. Independence of the spatial fluctuations of the volume and shear components of the elastic field holds within the scope of the approximation made. It is shown that the proposed relationship is satisfied exactly for laminar materials, but approximately for fibrous and granular materials. An explicit form is found for the tensor b in the singular approximation of random function theory under the assumption of isotropy of the properties of each of the fibrous and granular material phases and the correlation functions and stress and strain fields dispersions are calculated. It is shown that in this approximation the coordinate and tensor dependences of the correlation functions of the stress and strain fields are separated. An analogous computation is performed for multiphase polycrystals in the correlation approximation according to which correlation functions of elastic moduli of not higher than the second order are taken into account. In this approximation, the coordinate and tensor dependences of the correlation functions of the elastic field do not separate. Conditions are found under which the correlation approximation results in independence of the volume and shear components of the elastic field fluctuations. The exact computation of the stress and strain fields is a complex problem for the deformation of micro-inhomogeneous media (composite materials, singleand multiphase polycrystals, etc.). The difficulty is due to the need to take account of the interaction between all the inhomogeneity elements, which reduces to evaluating multiple integrals of certain functions including multipoint correlation or structural functions. To overcome this difficulty we can use different approximate methods. The simplest were proposed by Voigt [1] and Reuss [2] in connection with the problem of evaluating the effective elastic moduli of polycrystals. According to these hypotheses, it is assumed that homogeneity of the stresses (Reuss approximation) σ = 〈a〉 = const, or the strains (Voigt approximation) ε = 〈ε〉 = const holds. The angular brackets here denote the statistical average over the ensemble of monotypic situations, and σ and ε, are second rank tensors. The authors [3] proposed a singular approximation of the renormalization method equivalent to the hypothesis [4] of strong isotropy. In this approximation only the singular components of the second derivatives of the Green's tensor of the equilibrium equation are taken into account, which means “spreading” of the elastic field over the grain of the inhomogeneity. On the other hand, the approximation of the homogeneous field within the phase limits is fundamental in the variational method of evaluating the effective elastic moduli. Such an approach is taken implicitly in the self-consistency method [5].