We consider the problem of finding the effective stiffness tensor $\mathbb{C}^e$ of periodic heterogeneous martrix-inclusion materials. Given the distribution of the constituents, the cell problem must be solved first and the linear relation between average stress and strain is then established. Estimates can be obtained by making relevant approximation to the ingredients constituting the effective tensor. Although the present contribution concerns the theory of optimally estimating $\mathbb{C}^e$ from the microstructure, it is closely related to FFT numerical homogenization methods.

By introducing a reference material $\mathbb{C}^0$, the heterogeneity effect can be viewed as a distribution of eigenstrains within an homogeneous material. Using the related Green tensor, our problem can be formulated as a Lippmann-Schwinger equation for eigenstrain. The integral equation is the origin of resolution methods based on iteration and Fast Fourier Transform (FFT) techniques (Michel et al., 1999; Bhattacharya and Suquet, 2005). Significant progresses have been made regarding the improvement of convergence rate (Michel et al., 1999; Eyre and Milton, 1999; Milton, 2002; Monchiet and Bonnet, 2012; Brisard and Dormieux, 2010). The study of convergence rate in those works will be extended in the present contribution in the case of new integral equations.

The iteration scheme used to solve the Lippmann-Schwinger equation corresponds to the Neumann series summation. The latter can be used to derive exact theoretical relations and estimates. In this paper, we propose a new estimate based on series expansion that works at all contrast ratio, while using the matrix as a reference material. Additionally, we can control and optimize the convergence rate so that the series converges in the quickest way, and therefore produces the best estimates when using a finite sum in the series expansion. A class of integral equations for eigenstrain depending on two parameters $\alpha,\beta$ is first derived. The spectral radius and norm of the corresponding operators are bounded by analytical expressions. Different optimization methods are proposed to find the fastest series convergence and the associated estimates.

Similarly to the estimations of the effective elasticity tensor using correlation functions, the new method presented in this paper allows to estimate the effective elasticity tensor using the $n-$ order structure factors, which represent the counterpart in Fourier space of correlation functions. As an example, a direct connection of the effective elasticity tensor to $n-$th order structure factors is given in the case of randomly distributed spheres. Numerical applications for cubic arrays and random distribution of spheres yield very good results in comparison with FFT based methods and other results from the literature.