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

In this paper,we provide exact fast Fourier transform(FFT)-based numerical bounds for the elastic prop-erties of composites having arbitrary microstructures.Two bounds,an upper and a lower,are derivedby considering usual variational principles based on the strain and the stress potentials.The bounds arecomputed by solving the Lippmann-Schwinger equation together with the shape coefficients which al-low an exact description of the microstructure of the composite.These coefficients are the exact Fouriertransform of the characteristic functions of the phases.In this study,the geometry of the microstructureis approximated by polygonals(two-dimensional,2D objects)and by polyhedrons(three-dimensional,3Dobjects)for which exact expressions of the shape coefficients are available.Various applications are pre-sented in the paper showing the relevance of the approach.In the first benchmark example,we considerthe case of a composite with fibers.The effective elastic coefficients ares derived and compared,consider-ing the exact shape coefficient of the circular inclusion and its approximation with a polygonal.Next,thehomogenized elastic coefficients are derived for a composite reinforced by 2D flower-shaped inclusionsand with 3D toroidal-shaped inclusions.Finally,the method is applied to polycristals considering Voronoitessellations for which the description with polygonals and polyhedrons becomes exact.The comparisonwith the original FFT method of Moulinec and Suquet is provided in order to show the relevance of thesenumerical bounds.