Elastic anisotropy factor

The condition for isotropic elasticity, as has been seen, is C44 = C55 = Cge = Cn C12). Cubic crystals, because of their high symmetry, almost satisfy this condition. Zener introduced the ratio 2C44/(cn - C12) as an elastic anisotropy factor for cubic crystals (Zener, 1948a). In a cubic crystal, if the Zener ratio is positive, the Young s modulus has a... 

The parameter Z is known as the Zener ratio or the elastic anisotropy factor. For the Young s modulus representational surface in cubic crystals, this anisotropy... 

In crystals with the LI2 structure (the fcc-based ordered structure), there exist three independent elastic constants-in the contracted notation, Cn, C12 and 044. A set of three independent ab initio total-energy calculations (i.e. total energy as a function of strain) is required to determine these elastic constants. We have determined the bulk modulus, Cii, and C44 from distortion energies associated with uniform hydrostatic pressure, uniaxial strain and pure shear strain, respectively. The shear moduli for the 001 plane along the [100] direction and for the 110 plane along the [110] direction, are G ooi = G44 and G no = (Cu — G12), respectively. The shear anisotropy factor, A = provides a measure of the degree of anisotropy of the electronic charge... 

Table I. Experimental and calculated lattice constants a (in A), elastic constants, bulk and shear moduli (in units of 10 ) for the M3X (X = Mn, Al, Ga, Ge, Si) intermetallic series. Also listed are values of the anisotropy factor A and Poisson s ratio V. The experimental data for a are from Ref. . The experimental data for B, the elastic constants, A and v are taken from Ref. . The theoretical values for NiaSi are from Ref.. Also listed in the table are values of the polycrystalline elastic quantities-shear moduli G, Yoimg moduli (in units of and the ratio The experimental data for these quantities are from Ref. ...

It is interesting to note that haversian bones, whether human or bovine, have both their compressive and shear anisotropy factors considerably lower than the respective values for plexiform bone. Thus, not only is plexiform bone both stiffer and more rigid than haversian bone, it is also more anisotropic. These two scalar anisotropy quantities also provide a means of assessing whether there is the possibility either of systematic errors in the measurements or artifacts in the modeling of the elastic properties of hard tissues. This is determined when the values of Ac (%) and/or As (%) are much greater than the close range of lower values obtained by calculations on a variety of different ultrasonic measurements (Table 47.5). A possible example of this is the value of As (%) = 7.88 calculated from the mechanical testing data of Knets [1978], Table 47.2. 

Table 2.3 contains an overview of the elastic constants for some metals and ceramics. As can be seen, the anisotropy factor of tungsten is 1.0, so it is (almost) isotropic even as a single crystal. For most other materials, almost isotropic properties can only be found in a polycrystalline state. The direction dependence of Young s modulus for selected materials is plotted in figure 2.10. 

The fourth factor is an element of the structure tensor and depends on the internal structure (expressed by the angle a) and the parameter/, which is controlled by the bonding properties of the contact. Compressional and shear waves with propagation in vertical and horizontal directions differ only in this last term. It follows immediately that for a dry rock the velocity ratios (e.g. VpIV, elastic anisotropy parameters) depend only on structural and bonding properties. 

Composite materials have many distinctive characteristics reiative to isotropic materials that render application of linear elastic fracture mechanics difficult. The anisotropy and heterogeneity, both from the standpoint of the fibers versus the matrix, and from the standpoint of multiple laminae of different orientations, are the principal problems. The extension to homogeneous anisotropic materials should be straightfor-wrard because none of the basic principles used in fracture mechanics is then changed. Thus, the approximation of composite materials by homogeneous anisotropic materials is often made. Then, stress-intensity factors for anisotropic materials are calculated by use of complex variable mapping techniques. 

Ki = a y/a is the stress intensity factor, and F, the material constant, both of which depend on the degree of anisotropy of the composite controlled by the composite elastic moduli in the longitudinal and transverse directions, El and Ej, in-plane Poisson ratio, vlt, and Glj. For a perfectly isotropic material, jr/8(l + Vlt) 0.3. Also, the material parameters, i and < 2 are given by ... 

Table 3.10 Transition temperatures (°C). elastic constant ratio kjj/kj ). birefringence (A j, dipole moment (pD), Kirkwood-Froehlich factor (g) and dielectric anisotropy (As) for the compounds (106-109)... 

Blackman, D. K., Wenk, H.-R. Kendall, J.-M. 2002. Seismic anisotropy of the upper mantle 1. Factors that affect mineral texture and effective elastic constants. Geochemistry, Geophysics, Geosystems, (in press). 

Both pure elastic and elastoplastic models show similar response until the k value reaches approximately 2.5. With the increase of horizontal stress, the anisotropy in the permeability becomes significant. This is because the sub-vertical fractures are more vulnerable to the horizontal stress and its effect on y-directional permeability is larger. The difference between x- and y-directional permeability is about a factor of 2 for k ratios of 0.5 and 2. 

In this paper the compressive strength/elastic modulus of the jointed rock mass was estimated as a function of intact rock strength/modulus and joint factor. The joint factor reflects the combined effect of joint frequency, joint inclination and joint strength. Therefore, having known the intact rock properties and the joint factor, jointed rock properties can be estimated. The test results indicated that the rock mass strength decreases with an increase in the joint frequency and a sharp transition was observed from brittle to ductile behaviour with an increase in the number of joints. It was also found that the rocks with planar anisotropy exhibit the highest strength in the direction perpendicular to the anisotropy and the lowest at an inclination of 30o-45o in jointed samples. The anisotropy of the specimen influences the dynamic elastic modulus more than the static elastic modulus. The results were also compared well with the published works of different authors for different type of rocks. 

Temperature dependence of the optical threshold voltage Vth is a dominant factor in the figure of merit (M), It is worth analyzing the relations between AV, or and LC material properties, or molecular structures. Concerning physical constants, viz, dielectric anisotropy and elastic constant, both of which govern the threshold voltage of LC, we made some investigation of temperature dependence of V h for each LC in relation to that of ECH ... 

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