Anisotropic thin film on an isotropic substrate

Certain key features of the analysis of curvature in Section 2.1 leading to the classic Stoney formula (2.7) when hi C hs can be used to great advantage in the case of an anisotropic stress film bonded to a substrate. Among these features is that it is only the resultant membrane force in the film prior to substrate deformation which contributes to substrate curvature. Any through-the-thickness variation of the film stress giving rise to this force 

If the substrate is to be maintained in a stress-free state, the components /i and /2 must act on film edges defined by planes normal to the x and X2 directions, respectively, while the component /e represents the shear force inducing shear deformation between these two directions. If the film material is homogeneous and if the mismatch strain is uniform through the thickness then the integrands in (3.72) are independent of xs and the force expressions reduce to fi = Furthermore, if the film material is isotropic and the 

kq represents the twist of the substrate midplane, which was introduced in Section 2.6.2. These three parameters characterize the transverse midplane deflection of the substrate, which has the form U3(xi,X2,0) = Kix + kqx X2 + f 2X2 with respect to a plane tangent to the curved substrate at the point x = X2 = 0. Although the configuration is illustrated for 

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The chapter begins with an overview of elastic anisotropy in crystalline materials. Anisotropy of elastic properties in materials with cubic symmetry, as well as other classes of material symmetry, are described first. Also included here are tabulated values of typical elastic properties for a variety of useful crystals. Examples of stress measurements in anisotropic thin films of different crystallographic orientation and texture by recourse to x-ray diffraction measurements are then considered. Next, the evolution of internal stress as a consequence of epitaxial mismatch in thin films and periodic multilayers is discussed. Attention is then directed to deformation of anisotropic film-substrate systems where connections among film stress, mismatch strain and substrate curvature are presented. A Stoney-type formula is derived for an anisotropic thin film on an isotropic substrate. Anisotropic curvature due to mismatch strain induced by a piezoelectric film on a substrate is also analyzed. 

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