Example Temperature change for buckling of a debond

A 1 jim thick diamond-like carbon film is deposited at 500 °C on a Ti alloy substrate. The film with elastic modulus Ef = 500 GPa and Poisson ratio Ui = 0.2, is essentially free of any internal stress at the deposition temperature. When cooled to the temperature 20 °C, however, an equibiaxial compressive mismatch stress of 5 GPa is expected to exist in the film as a consequence of thermal mismatch with the substrate. An unbonded circular patch, 30 gm in diameter, developed at the film-substrate interface during film deposition. Determine whether the film buckles upon cooling to 20 °C If so, determine the temperature at which buckling begins. 

Using the notation introduced in Section 5.2.1, the radius of the circular region of debond at the interface a = 15 m, the film thickness hf = 1 m, and the film biaxial modulus E = f/(l — vf) = 625 GPa. A temperature change AT = 480 ° G from the stress-free deposition temperature to room temperature, produces a compressive mismatch stress (Tm = 5 GPa. The mismatch stress necessary to 

Since this compressive mismatch stress is smaller than that in the film upon cool 

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