Crystallographic strain piezoelectrics

Crystallographic strain copper-modified zinc oxide, 38-39 hydrodynamic cavitation, 34-39 piezoelectrics, 37-38 titania, 35-37... 

When written in matrix form these equations relate the properties to the crystallographic directions. For ceramics and other crystals the piezoelectric constants are anisotropic. For this reason, they are expressed in tensor form. The directional properties are defined by the use of subscripts. For example, d i is the piezoelectric strain coefficient where the stress or strain direction is along the 1 axis and the dielectric displacement or electric field direction is along the 3 axis (i.e., the electrodes are perpendicular to the 3 axis). The notation can be understood by looking at Figure 31.19. 

Figure 2. Effect of piezoelectric anisotropy on the single-crystal orientational behavior. Left inset embodies the polar response of the normalized piezoelectric behavior for different anisotropy factors. Note that the optimal orientation changes as the degree of anisotropy increases. Left inset shows the optimal orientation of each single-crystal, as a function of crystallographic anisotropy. Note that contrary to what it is intuitively expected, in the limit of high anisotropy, A 2/3, the crystallographic orientation at which highest piezoelectric strains will occur will asymptotically align with the direction of the applied field. Furthermore, the optimal orientation for materials with weak anisotropy will asymptotically converge to 0=54.16°.

Generally, the piezoelectric effect could exist just in non-centrosymmetrical crystallographic symmetry classes. Mechanical stress/strain as a second-rank symmetrical tensors are basically centrosymmetrical external fields. If the materials crystallographic symmetry include cerrtre of symmetry operation, the resulting symmetry of material subjected to such field is also cerrtrosymmetrical (see Neuman s Law in Nye (1985)). Therefore, piezoelectric effect is excluded. Centrosymmetrical crystal stays centrosymmetrical even after the application of the mechanical stress and no polar direction for the polarization vector might exist in such stmcture. 

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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