Derivation of FFT numerical bounds of the effective properties of composites and polycristals

International audienceIn this paper, we provide exact FFT-based numerical bounds for the elastic properties of composites having arbitrary microstructures. Two bounds, an upper and a lower, are derived by considering usual variational principles based on the strain and the stress potentials. The bounds are computed by solving the Lippmann-Schwinger equation together with the shape coefficients which allow an exact description of the microstructure of the composite. These coefficients are the exact Fourier transform of the characteristic functions of the phases. In this study, the geometry of the microstructure is approximated by polygonals (2D objects) and by polyhedrons (3D objects) for which exact expressions of the shape coefficients are available. Various applications are presented in the paper showing the relevance of the approach. In the first benchmark example, we consider the case of a composite with fibers. The effective elastic coefficients ares derived and compared, considering the exact shape coefficient of the circular inclusion and its approximation with a polygonal. Next, the homogenized elastic coefficients are derived for a composite reinforced by 2D flower-shaped inclusions and with 3D toroidal-shaped inclusions. Finally, the method is applied to polycristals considering Voronoi tessellations for which the description with polygonals and polyhedrons becomes exact. The comparison with the original FFT method of Moulinec and Suquet is provided in order to show the relevance of these numerical bounds