Numerical homogenization with FFT method for elastic composites with spring-type interfaces

FFT based numerical methods are efficient for computing the effective properties of heterogeneous media and have encountered a great popularity in the scientific community of the homogenization theory during the ten past years. One particular challenging development concerns the extension of the approach to composite materials with imperfect interfaces and, particularly, for spring-type interfaces. From a practical point of view, such interfaces generally represent a thin interphase with very soft local elastic properties or localized damage near the interface. In this work, we provide an efficient three-dimensional numerical FFT based approach to account for displacement jump across the curved imperfect interfaces. The principle of the approach rely on three main ingredients. First, it uses the classic displacement based variational principle such as Finite Element Method and leads to a rigorous upper bound for the homogenized elastic tensor. Next, the problem discretization is based on truncated Fourier series plus an enrichment with discontinuous functions. Finally, the use of shape coefficients allows to describe rigorously the microstructure geometry. The accuracy and convergence rate of this numerical approach is assessed by comparisons with Finite Element solutions and analytical models in the case of composites with long fibers and spherical inclusions.