Optimal shape and topology of multi-material microstructures in min-max stress design problems

The present dissertation seeks to optimize the unit cell of a two-dimensional cellular material, pursuing the minimization of the peak equivalent stress in the microstructure. This class of materials is particularly relevant to the design of lightweight structures. By minimizing the peak stress in the microstructure, it is possible to use material in a more rational way. Given the periodic nature of the problem, asymptotic homogenization is employed to compute the stress distribution in the microstructure when a macroscopic load is applied, since periodicity boundary conditions are imposed. With this being a purely conceptual study, only three macroscopic loads are considered: the hydrostatic, biaxial, and pure shear ones. Initially, the single-material problem is solved through shape optimization. Then, the potential to reduce the peak equivalent stress through the introduction of additional material phases is explored. Also with shape optimization, the in uence of one additional material phase is studied. Additionally, topology optimization is used to discover the functionally graded material that minimizes the peak stress in the microstructure. The obtained results show that an increased design exibility always leads to milder stress states. The known theoretical results were successfully replicated, with minimal error measures associated. By increasing the number of material phases in the microstructure, peak stress reduction are attainable. A uniformly stressed microstructure is possible to obtain, by means of a functionally graded material