Comparative study of three techniques for the computation of the macroscopic tangent moduli by periodic homogenization scheme

In numerical strategies developed for determining the effective macroscopic properties of heterogeneous media, the efficient and robust computation of macroscopic tangent moduli represents an essential step to achieve. Indeed, these tangent moduli are usually required in several numerical applications, such as the FE² method and the prediction of the onset of material and structural instabilities in heterogeneous media by loss of ellipticity approaches. This paper presents a comparative study of three numerical techniques for the computation of such tangent moduli in the context of periodic homogenization: the perturbation technique, the condensation technique and the fluctuation technique. The practical implementations of these techniques within ABAQUS/Standard finite element (FE) code are especially underlined. These implementations are based on the development of a set of Python scripts, which are connected to the finite element computations to handle the computation of the tangent moduli. The extension of these techniques to mechanical problems exhibiting symmetry properties is also detailed in this contribution. The reliability, accuracy and ease of implementation of these techniques are evaluated through some typical numerical examples. It is shown from this numerical and technical study that the condensation method reveals to be the most reliable and efficient. Also, this paper provides valuable reference guidelines to ABAQUS/Standard users for the determination of the homogenized tangent moduli of linear or nonlinear heterogeneous materials, such as composites, polycrystalline aggregates and porous solids.