Elastic constants of non-cubic crystals

Note that AR = 1 for an isotropic crystal. The physical significance of this ratio is that it represents a measure of the relative response of cubic crystals to two different types of shear strain C44 is an indicator of shear resistance in the 100 directions on 100 planes, whereas the quantity (cn — ci2)/2 is an indicator of shear resistance in the 110 directions on 110 planes. Values of anisotropy ratios for various cubic crystals are listed in Table 3.1. Among cubic metals, W is essentially isotropic and A1 is very close to being so. Many materials listed in Table 3.1 have anisotropy ratios which depart significantly from unity. 

Elastic deformation in isotropic materials is fully characterized in terms of two elastic constants, such as elastic modulus E and Poisson s ratio V, or in terms of the Lame constants, yu (the shear modulus) and A. For an isotropic material, the various elastic constants are related by 

Five elastic constants are required to characterize the elastic response of a material with hexagonal symmetry. A convenient material coordinate frame in this case is a rectangular coordinate system with the 0 3—axis aligned with the c—axis of the material, which is the axis with respect to which the material has sixfold rotational invariance. The xi—direction and a 2—direction can then be chosen to coincide with any pair of orthogonal axes in the basal plane so as to form a right-handed coordinate system. The five elements of the matrix defining elastic response can be chosen to be cn, C33, C44, C12 and C13. Several other elements can be expressed in terms of these according to 

For materials with orthorhombic symmetry (or orthotropic symmetry), there are three mutually orthogonal axes, each of two-fold rotational symmetry. Translation of the material in the direction of any of these axes leaves the material behavior unaltered, but the translation distances required to recover the same lattice in crystalline materials are distinct, in general. Nine independent constants (cn, C22, C33, C44, C55, cge, C z, C23, C12) are required to specify the elastic response of a general orthorhombic material. For a material in which one of the three symmetry axes has fourfold rotational symmetry (or tw o of the translational invariance distances are equal), the number of independent constants is reduced to six this is achieved by requiring that C22 = cn, C55 = C44 and C23 = C12, for example. 

Elastic constants of bulk crystalline materials are commonly measured by means of the ultrasound method where sound velcities in specific crystallographic orientations are monitored (( )). For example, the case of a cubic crystal for which the longitudinal wave speed along the [100] direction is a/cii/p, where p is the density of the crystal. Likewise, the shear wave speed along the [100] direction is p. The speed of the longitu- 

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Elastic constants of non cubic crystals the anisotropy ratio AR in either of the equivalent forms... 

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