There has been considerable effort devoted towards predicting the effective properties of composite materials directly from the properties of their constituent phases and from their microgeometry. An impressive body of literature exists when the constitutive behavior of the phases that make up the composite is linear elastic. By comparison, much less is known about composites exhibiting a nonlinear behavior such as plasticity or creep. This has motivated the development of direct numerical simulations of composites with periodic microstructures mostly for "simple" microstructures. More recently efforts have been directed towards modeling composites with "complex" microstructures and/or containing many inclusions which are a real challenge for conventional numerical methods.
The study of media with complex microstructure can be reduced to the study of periodic media with an adequately large unit cell, which is representative of the aggregate. Classical numerical techniques (FEM) are prohibitively time consuming for the calculations required. This presentation focuses on a method based on Fast Fourier Transforms which makes direct use of digital images of the "real" microstructure, therefore avoiding the difficulty due to meshing. It is based on the exact expression of the Green function for a linear elastic and homogeneous comparison material and makes use of Fast Fourier Transforms (FFT) to solve the unit cell problem even when the constituents are nonlinear, coupled with a fixed-point method which avoids the formation and inversion of a stiffness matrix. A computational advantage of the method is that it is easy to parallelize. The comparison with the Finite Element Method shows that, in many instances, the method based on FFT is faster and more flexible.
However the performances of the method in its basic form deteriorate when the contrast between the phases increases. Alternate methods, based on augmented Lagrangians, are used to handle composites with high or infinite contrast (voided or rigidly reinforced materials).
This method has been applied to simulate the response of nonlinear materials with various behaviours such as elastoplasticity of composites or polycrystals and compared with experimental results. The results of the numerical simulations shed light on the accuracy of theoretical estimates for nonlinear composites.