A laminar stagnation flow flame stabilized at a plane wall is theoretically analyzed, coupled with a thermo-elasticity model in the wall. Mathematical models for both the flame and the wall will be proposed, and corresponding analytical solutions based on their dimensionless forms will be obtained. The mathematical analysis of this flame-solid interaction primarily focuses on the effect of the flame on combustion-induced thermo-mechanical stresses and the influence of wall material on flame properties such as flame temperature and stability against extinction. A sensitivity analysis is further performed to examine how thermo-mechanical stress inside the wall is affected by changes in other combustion conditions (e.g., mixture composition, flame stretch rate). This study offers valuable insights into understanding the interaction between flames and solids, a relationship crucial in engineering applications such as the interaction between combustion processes and blades within gas turbine machinery.