In this work, a NURBS-based time integration method is proposed for the temporal discretization and numerical solution of the effects of stress concentration in solid–solid phase transformation (PT). The performance and efficiency of the method are assessed through this multiphysics problem, discretized by standard nonlinear finite elements. Due to the presence of strong stress concentration effects and nonlinear multiphysics equations, a highly stable numerical scheme in time is required, which is presented in this work to address this necessity. Validation is conducted through comparisons with analytical solutions and simulations obtained using COMSOL Multiphysics, demonstrating the accuracy and robustness of the introduced NURBS method. The computational efficiency of this method shows improvements of up to 44.5% and 8.4% under identical conditions compared to the Adams–Bashforth and Bézier multistep schemes, respectively. The results further reveal that stress concentration induces significant stretching and curvature of the interface toward regions of elevated elastic energy. Size-effect analysis indicates that increasing the void diameter enhances the driving force, interface curvature, and interface velocity. These findings highlight two key aspects: first, the robustness and computational efficiency of the proposed NURBS-based time method in addressing nonlinear multiphysics equations; second, the critical role of stress concentration in PT, which significantly influences the mechanical response of smart materials such as shape memory alloys.