Mechanical metamaterials, distinguished by their extraordinary mechanical properties absent in natural structures, derive their unique attributes from intricately designed microstructures. In contrast to conventional materials relying solely on inherent properties, these metamaterials attain specific characteristics through the intricate geometric configurations of their unit cells. This study is driven by the aspiration to investigate the intricate design of three-dimensional (3D) metamaterials using topology optimization, with a particular emphasis on stress-constraint topology optimization. The motivation arises from the recognition that integrating stress constraints into the optimization process holds the potential to address critical challenges associated with singularity phenomena, providing a more robust and realistic representation of designed metamaterials. By employing a density-based approach that incorporates the Solid Isotropic Material with Penalization (SIMP) model and stress penalization, along with the adoption of the P-norm stress measure and the Method of Moving Asymptotes (MMA) as the optimization solver, the study provides a systematic means to optimize the mechanical performance of 3D metamaterials. The results demonstrate that topology optimization can significantly enhance the properties of the microstructure. These findings offer valuable insights, not only advancing the theoretical understanding of metamaterial engineering but also presenting practical implications for diverse industries, including aerospace and structural engineering. Furthermore, these insights hold the potential to contribute to the design of earthquake- resistant structures, thereby mitigating deformation during seismic events.