Example Maximum thermal stress in a bilayer

Consider a film-substrate bilayer system of circular geometry, where the film and the substrate have the same thicknesses and biaxial moduli h /hs = 1 and Mf/Mg = 1, and the bilayer diameter, d (hg + hf). Let the mismatch strain in this case be a consequence of a temperature change from an initial, stress-free temperature To to another temperature T, and let the thermal expansion coefficients of the film and the substrate be denoted by Of and Og, respectively, (a) Determine the variation of the radial stress and circumferential stress across the thickness of the film and the substrate, (b) Find the magnitude and sign of the thermal mismatch stress as the interface is approached from the film and from the substrate. Show that the magnitude of the stress at the interface is independent of the thickness of the film or the substrate for a fixed thickness ratio. 

Since the in-plane dimension of the film-substrate system is much larger than its total thickness, an equi-biaxial state of stress can be assumed in the system (ignoring edge effects). Therefore, ase z) = arr (z)-Prom (2.14), note that Cm = (os 0 ) (T — To). From (2.19)-(2.21), it is seen that Cq = — Cm/S and k = 3etn/(4fis)- At the interface between the film and the substrate at z = fig/2, the stress values for Mi = Ms = M are 

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