High-temperature ceramic materials used in solid-oxide fuel cells (SOFCs) are subject to high thermal stresses during operation due to the unequal thermal expansion between different layers. As a result, solid oxide fuel cells are prone to mechanical failure at elevated temperatures, limiting the maximum operating temperature and, therefore, limiting the maximum power density obtained from the fuel cell. Fuel cells with graded electrodes in the thickness direction have been used and extensively investigated to reduce the effect of non-uniform thermal expansion. In this study, two dimensional functionally graded electrodes are proposed for the first time. Thus, a comprehensive theoretical model is developed for a high-temperature SOFCs that includes the charge, species, energy, and momentum transport equations. Also, the bilinear elastoplastic material model is used to calculate thermal stresses and failure in solid materials. The model is used to study two-dimensional functionally graded electrodes introduced to investigate their effect on thermal stresses. The material grading will be implemented in two directions for each layer; thickness and length. Results indicate that using the two-dimensional grading reduced thermal stresses by over 40 % for a specific grading scheme compared to the conventional case. Grading the electrodes also positively affects the electrochemical performance, as the cell’s maximum power density was increased by over 60 %. These results prove that two-dimensional graded SOFCs can achieve much higher operating temperatures with safe thermal stresses, creating a potential for compact, high-temperature SOFCs designed for high power density applications.