Constant material properties. Cylindrical coordinates

For a cylinder with A = const we obtain, from (2.277) and (2.279), the expression 

A further simplification is possible if the grid spacing Ar is assumed to be constant. With 

If an explicit difference equation is desired, then the time based derivative in 

The boundary conditions are treated in the same manner as in section 2.4.2. 

To transfer the Crank-Nicolson [2.65] implicit difference method, which is always stable, over to cylindrical coordinates requires the discretisation of the equation 

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In this section, the effect on substrate curvature of the variation of mismatch strain and material properties through the thickness of layered films is analyzed. The derivation of the Stoney formula (2.7) in Section 2.1 refers only to the resultant membrane force in the film any through-the-thickness variation of mismatch strain in the film is considered only peripherally. Film thickness was taken into account explicitly in Section 2.2, but it was assumed there that mismatch strain and elastic properties of the material were uniform throughout the film. However, there are situations of practical significance for which this is not the case. Two of the most common cases are compositionally graded films in which the mismatch strain and the elastic properties vary continuously through the thickness of the film, and multi-layered films for which the mismatch strain and the elastic properties are discontinuous, but piecewise constant, from layer to layer throughout the thickness of the film. In both cases, the mismatch strain and the material properties are assumed to be uniform in the plane of the interface. With reference to the cylindrical r, 6,. z—coordinate system introduced in Section 2.1, the mismatch strain and film properties are now assumed to vary with z for fixed r and 6, but both are invariant with respect to r and 9, for fixed z. 

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