This paper develops a nonlinear observer for lithium-ion battery electrode particle stress and state-of-charge (SOC) estimation using the single particle model (SPM) coupled with mechanical stress. Particle fracture due to stress generation is a critical mechanism causing capacity fade, and thus reducing battery life. The stress sub-model captures stress developed during lithium ion intercalation and deintercalation. State estimation based on coupled SPM and mechanical stress model is particularly challenging because the coupled model is given by nonlinear partial differential equations (PDEs). We address this problem by reducing the coupled model to a system of nonlinear ordinary differential equations (ODEs), and then apply nonlinear observer design methods. The key novelty of this design is a nonlinear internal state estimation algorithm, from which the internal stress can be monitored from current and terminal voltage measurements only. Simulation studies illustrate the performance of the proposed estimation scheme.