Use of lithium metal electrodes in solid state batteries requires well-balanced electrochemical and mechanical properties for high performance yet safe application. We present a new methodology to account for coupling mechanisms between electrochemistry and mechanics, understand morphological stability and deduce stability maps for solid polymer electrolytes adjacent to metal electrodes. We use a rigorous electro-chemo-mechanical description of the polymer electrolyte and its reaction kinetics to predict the interface current density along a deformed electrolyte interface as a function of e.g. mechanical stiffness, transport and interface properties. With these results, we explore the stability of the electrolyte-electrode interface in regard to the tendency for protrusions to grow on the metal electrode. We find that there is a critical Young's modulus of the electrolyte above which protrusion growth is suppressed. However, the critical Young's modulus depends on the charging rate, and on the electrochemical transport properties of the electrolyte and the interface. We develop a morphological stability map in which the critical Young's modulus separating stable and unstable behavior is identified. The results agree well with experimental findings and might help in predicting new classes of polymer electrolytes which are more likely to perform well in the effort to suppress dendrite growth.