Substrate curvature for arbitrary film thickness

The issue of film thickness effects on substrate curvature evolution is pursued now by recourse to the energy minimization method which was introduced in Section 2.1 for the derivation of the Stoney formula. All other features of the system introduced in that section are retained in this discussion, which follows the work of Freund et al. (1999). It is assumed that the film material carries an elastic mismatch strain in the form of an isotropic extension em (or contraction if Cm is negative) in the plane of the interface the physical origin of the mismatch strain is immaterial. The mismatch strain is spatially uniform throughout the film material. In this case, em is 

The description of deformation in which the development in Section 2.1 is based is retained. However, if the thickness of the film is to be taken into account, then the strain energy of the film material must be included in the calculation of total potential energy. This is accomplished by adopting the strain expression (2.2) for the film as well as the substrate, but augmenting it by the elastic mismatch strain Cm in the former case. The strain energy density throughout the system is then 

The total potential energy of the film-substrate system is 

As in Section 2.1, the actual midplane deformation within the class of admissible deformations is that which renders the total potential energy V stationary with respect to variations in and k. It follows from the requirements that dVjdto = 0 and dV/dK = 0 that the curvature is given by 

The influence of film thickness or, more precisely, the influence of the ratio /if//is on substrate curvature, is considered here in two ways. First, the leading two terms in a series expansion of curvature k in powers of /if//is 

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