Topology optimization is a powerful engineering design tool that can lead to innovative layouts and significantly enhance the performance of engineered systems in various sectors. In a world where we are searching for cost reduction while being ecologically responsible, we should seek practical applications of topology optimization. Reducing weight while sustaining strength requirements is one of them. Another concern is the accurate prediction of the mechanical behavior of the wide variety of available materials, such as soft and rubber-like elastomers. To this end, incorporating nonlinearities will extend conventional topology optimization to hyperelastic structures and significantly enhance the performance at the primary design stage. We consider the density-based approach, which enforces us to properly address numerical instabilities of low-density regions through an energy interpolation scheme. An augmented Lagrangian-based formulation is used to deal with the large number of stress evaluation points, whereas polynomial vanishing constraints are employed to overturn the singularity phenomenon. We conducted a preliminary investigation under linear-elastic circumstances to explore different strategies related to stress constraints which justify implementing the augmented Lagrangian method. In addition, we extract analytical expressions for sensitivity analysis with extreme rigor and detail. Problems in plane stress scenarios requires effective computation of the out-of-plane strain component. Then, in order to do this, we deduced analytical expressions and a numerical solution based on the Newton s method. Different examples validate our method, demonstrating the significance of considering stress constraints and nonlinearities in topology optimization. We additionally point out that solutions derived from linear theory often violate stress limits under nonlinear conditions, making them unsuitable for modeling structures that undergo large deformations.