Combining FFT methods and standard variational principles to compute bounds and estimates for the properties of elastic composites

In this paper, we develop a computational approach based on variational principles combined with FFT technique to determine the effective elastic properties of composite materials. The unit cell problem of the composite is recast in a weak and integral form by making use of classic variational approaches, based on the strain and the stress potential, and the Hashin-Shtrikman variational principles. The problem is discretized with Fourier series and the stationary point is computed numerically by means of an iterative scheme. These algorithms use a representation of the local elastic tensor on the double grid (twice the size of the discretization) which is introduced by the integral formulation; it advantageously accounts for the exact geometry of the microstructure and allows the derivation of rigorous bounds of the effective elasticity coefficients. In the second part of the paper, we establish a hierarchy between the different FFT solutions: the strain, the stress based solutions coming from the classic variational formulations and the polarization based solution derived from the Hashin-Shtrikman principle, which depends on the choice of the elastic moduli of the reference material. It is proved that the strain and the stress based solutions deliver optimal bounds since they are always better than those obtained from the Hashin-Shtrikman variational principle when the same discretization is used. Alternatively, when the elastic moduli of the reference material is negative, the polarization based solution provides an estimate of the effective properties which is comprised between the strain based upper bound and the stress based lower bound.