Example Curvature due to thermal strain

Consider a thin film of aluminum, 1 //m in thickness, which is deposited uniformly on the 100 surface of a Si substrate, which is 500 pm thick and 200 mm in diameter, at a temperature of 50 C. The thermoelastic properties of the film are elastic modulus, Ef = i Ai = 70 GPa, Poisson s ratio, i/f = i/ ai = 0.33, and coefficient of thermal expansion, ccf = oai = 23x10 °G . The corresponding properties of the Si(lOO) substrate are Mg = Ms = 181 GPa and ag = ctsi = 3x10 °C h The fflm-substrate system is stress-free at the deposition temperature. Determine (a) the mismatch strain of the film with respect to the substrate at room temperature, that is, at 20 °G, (b) the stress in the film due to the mismatch strain, and (c) the radius of curvature of the substrate. 

Note that this is substantially smaller than a typical plastic yield stress for 

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