For heat fluxes ranging above 10 MW/m2 or so, solid surfaces usually experience large thermal stresses and degradation of mechanical properties. The resulting mechanical failure of such surfaces is a primary limitation to the design of thermal systems at extremely high heat flux. This investigation considers the elastic stresses in circular plates subjected to extremely high heat fluxes. A gaussian distributed heat load is applied to one surface of the plate and the heat flux at which yielding occurs is identified. Several candidate materials are examined, accounting for the temperature dependence of yield strength and other properties. The mechanical boundary conditions on the plate are varied. Figures of merit are given for the high flux performance of a number of materials.