9. Nonlinear Computational Homogenization

The homogenization of nonlinear heterogeneous materials is by an order of magnitude tougher than the homogenization of linear ones. The main reason is that in the linear case, the general form of the homogenized (or effective) behavior of heterogeneous materials is a priori known, and it suffices to determine a set of effective moduli by considering a finite number of macroscopic loading modes. In contrast, in the nonlinear case, the general form of the homogenized behavior of heterogeneous materials is unknown and the determination of the homogenized behavior requires solving nonlinear partial differential equations with random or periodic coefficients and entails considering, in principle, an infinite number of macroscopic loading modes. Then, the superposition principle, which was used as a basis in the previous chapters to construct the homogenized behavior no more applies. The central problem is to define the constitutive relationship to be used at the macroscale at each integration point of the structure, given an RVE and a description of the nonlinear behavior of each phase. The development of nonlinear computational homogenization methods has been an active topic of research since the end of the 90’s and many issues still remain at the time this book is written.