Influence of film thickness on bilayer curvature

The radius of curvature is p = 120 m. The positive curvature implies that the substrate is concave on the surface on which the film is deposited, which is consistent with a state of residual tension in the film. 

Values of linear thermal expansion coefficients of commonly used non-metallic thin film, interlayer or substrate materials are given in Table 2.1 over a broad range of temperatures of practical interest. Table 2.2 provides corresponding values of linear thermal expansion coefficients for polycrys-talline metals. Table 2.3 lists the room temperature values of elastic modulus and Poisson s ratio for a wide variety of polycrystalline and amorphous materials with isotropic elastic properties, which are commonly used as thin films, interlayers or substrates. The anisotropic elastic properties of cubic and hexagonal single crystals are given in Tables 3.1 and 3.2, respectively, in the next chapter. 

In the preceding section, an estimate was made of the curvature caused by the mismatch strain when a very thin film is bonded to the surface of a substrate. It was assumed that the change in film stress due to substrate deformation was negligible, and that the stiffness of the system depended only on the properties of the substrate. These assumptions led to the Stoney formula (2.7) relating the membrane force in the film to the curvature of the midplane of the substrate. The value of film thickness /if entered the derivation only peripherally. The issue is re-examined in this section for cases where the film thickness /if is not necessarily small compared to the substrate thickness /ig. 

Detailed analyses of the effects of film thickness on substrate curvature in bimaterials date back to the early twentieth century, when interest in the use of thermostatic bimetals began to expand rapidly, as described in the historical note on thermostatic bimetals in Section 2.2.3. Timoshenko (1925) and Rich (1934) derived thermoelastic solutions for curvature and stress evolution in a bimetallic strip as a function of temperature change, for arbitrary variations in the relative thickness and elastic properties of the 

Material Linear thermal expansion coefficient a in units of (10-6 °c-l) T in units of°C Temp, range (°C) 

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