Nonuniform mismatch strain and elastic properties

All features of the film-substrate system introduced in Section 2.2 are retained. In addition, it is assumed that the film material carries a mismatch strain in the form of an isotropic equibiaxial extensional strain em( ) parallel to the interface. This strain may depend on distance from the interface in an arbitrary way, as suggested by writing the mismatch as a function of z] this function need not be continuous in 2. Similarly, the biaxial elastic modulus of the film Mf z) may vary through the thickness of the film. 

Cm(-z) = M (z) z), and its effect is to render the film compatible with respect to the stress-free substrate. Upon release of this artificial externally applied traction, the substrate takes on a curvature which is to be estimated. This is first pursued on the basis of the principle of minimum potential energy, followed by a discussion of an equilibrium approach leading to the formula relating curvature and mismatch strain. 

The strain energy density of the system can be written just as in (2.17), if it is understood that Cm and M are now functions of 2. To make the results useful for a broader class of systems, it is convenient to adopt a slightly modified notation. First, suppose that the definition of mismatch strain is modified so that em( ) = 0 for — hs z In addition, 

If the total potential energy is minimized with respect to variations in Co and K, it is found that, at the minimum value, 

While these results appear to be simple in form, they are not sufficiently transparent to permit easy interpretation for the general case. Thus, several particular cases are pursued in detail. First, the result for curvature obtained in Section 2.2 is recovered. To do so, it is noted that if Mf z) and (2) have the constant values Mf and Cm throughout the film, the constants in (2.51) are 

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