Addressing the non-linearity and singularity phenomena of stress-based optimal design of material microstructures

Material design is an active research field since composites have met increasingly interest, for instance, in lightweight construction as it happens in aerospace industry. One assumes in the present work a given macroscopic stress or strain field (one that may occur at a certain point of a macro-structure) and computes through homogenization the micro-stress distribution across the two (weaker and stronger) composite constituents mixed in a unit-cell domain which is representative of a periodic heterogeneous material. Stress gradients depend a lot of design details but typically the stress field is highly non-linear. In the frame of finite element models for material microstructures one pursues here an investigation about mesh convergence. Since stress distribution is strongly design dependent, that motivates one to pursue optimal design of the material microstructure to comply with admissible stress criteria. The inverse homogenization method using density-based topology optimization is applied here for such purpose. This is quite a challenge not only because of the aforementioned non-linearity of the stress field but also due to the singularity phenomena which one overcomes using standard relaxation techniques. Some preliminary results are obtained in order to get some insight into the fine structure of composite materials and the influence of the stresses therein.