The stress minimization multi-material topology optimization (MMTO) approach has recently attracted significant attention because of its applications in aerospace and mechanical engineering. Nonetheless, the stress minimization MMTO approach may result in stress surpassing the material’s tolerance limit, potentially culminating in failure. This research proposes a novel way for imposing stress constraints on each material to regulate their respective stress levels. The fundamental concept is that each material possesses its own interpolation function for the stress model. The maximum von Mises stress for each material can be established with the definition of an upper limit, ensuring that the materials will perform safely and effectively. This aids topological structures in resisting failure and augmenting strength. A multi-physics system including thermoelastic and self-weight loads is concurrently examined alongside stress limitations. The global stress constraint utilizes the p-norm function, and the adjoint method is used to derive sensitivity. This work employs a three-field strategy utilizing density filtering and Heaviside projection functions to mitigate the artificial stress in low density. The technique is assessed through two-dimensional (2D) and three-dimensional (3D) examples, illustrating the influence of stress limits on the compliance minimization under heat and self-weight loads. The optimized results indicate a substantial decrease in the stress levels accompanied by a minor gain in compliance, while maintaining the stress within the specified range for all materials.