Anisotropy elastic properties

Knowledge of the elastic constants of a single crystal is necessary to calculate or estimate various elastic properties like elastic anisotropy, elastic properties of dislocations, etc., of intermetallic compounds. Such information is also necessary to estimate or check the potential energy between different atoms, which is used for computer simulations like molecular dynamics, etc., although the elastic constants themselves can be calculated using various methods like EAM, F-LAPW, etc. The temperature dependence of the elastic moduli may also be further used for the investigation of various kinds of transformations, since they are sensitive to composition, temperature, etc. 

The hardness values of the crystals are given in Table 1. These data show that fi has a much lower hardness than a. Also a high anisotropy of hardness and elastic properties are found for (i, which is less pronounced in a. 

Figure 3. Ratio of the mean square root displacements derivatives along directions of weak and strong coupling, calculated in the model of a highly anisotropic layered crystal. Its anisotropy of interatomic interaction and elastic properties correspond to those of NbSe2. The pronounced maximum on this curve corresponds to a minimum on the temperature dependence of the thermal expansion along the layers.

The big difference between normal isotropic liquids and nematic liquids is the effect of anisotropy on the viscous and elastic properties of the material. Liquid crystals of low molecular weight can be Newtonian anisotropic fluids, whereas liquid crystalline polymers can be rate and strain dependent anisotropic non-Newtonian fluids. The anisotropy gives rise to 5 viscosities and 3 elastic constants. In addition, the effective flow properties are determined by the flow dependent and history dependent texture. This all makes the rheology of LCPs extremely complicated. 

Convergence of estimates of the melting curve of iron places a tighter constraint on the temperature at the ICB. Experimental measurements and theoretical calculations of elastic properties and plastic deformation of iron offer new interpretations for the inner core anisotropy. Prehminary results have been obtained on the properties of liquid iron, which allow a more direct comparison between laboratory measurements and seismic observations. 

Later, Polyakov and Tarakanov modified the model by presenting if of a hexahedral cell having an initial curvature (eccentricity) near the rods disposed in two perpendicular directions. This model, which makes allowance for the initial anisotropy of the plastic foam, satisfactorily describes the elastic properties of flexible foams under considerable strains, but generally predicts unduly high values of the elastic properties of foamed plastics. 

Elastic Properties [1.30,1.31,1.35]. In regard to elasticity, at least below room temperature, tungsten behaves nearly isotropically the anisotropy coefficient at 24 °C is = 1.010 [1.35]. The elastic constants for polycrystalline tungsten at 20 °C are given below. Their temperature dependence as well as the respective values for singlecrystal elastic constants are shown in Fig. 1.10 [1.40], based on ultrasonic measurements [1.30,1.31]. 

Most micromechanical theories treat composites where the thermoelastic properties of the matrix and of each filler particle are assumed to be homogeneous and isotropic within each phase domain. Under this simplifying assumption, the elastic properties of the matrix phase and of the filler particles are each described by two independent quantities, usually the Young s modulus E and Poisson s ratio v. The thermal expansion behavior of each constituent of the composite is described by its linear thermal expansion coefficient (3. It is far more complicated to treat composites where the properties of some of the individual components (such as high-modulus aromatic polyamide fibers) are themselves inhomogeneous and/or anisotropic within the individual phase domains, at a level of theory that accounts for the internal inhomogeneities and/or anisotropies of these phase domains. Consequently, there are very few analytical models that can treat such very complicated but not uncommon systems truly adequately. 

The analytical expressions of micromechanics are generally most accurate at low volume fractions of the filler phase. The details of the morphology become increasingly more important at higher volume fractions. This fact was illustrated by Bush [64] with boundary element simulations of the elastic properties of particulate-reinforced and whisker-reinforced composites. The volume fraction at which such details become more important decreases with increasing filler anisotropy, as was shown by Fredrickson and Bicerano [60] in the context of analytical models for the permeability of nanocomposites. 

For cubic symmetry materials, three independent elastic properties that are orientation dependent are required to describe the mechanical behavior of the material. This anisotropy effect increases significantly the number of the nonzero elements in the FE stiffness matrix leading to alteration in the calculated stress components and the wave speed. In order to test these anisotropy effects, we plot the wave profiles of three different orientations and compare it with the isotropic behavior with a loading axis in the [001] directions as shown in Fig 8. We observed that under the same loading condition, the peak stress of [111] and [Oil] orientations are slightly higher than those of the [001] which is lower that that of isotropic material. Furthermore, wave speed varies moderately with orientation with the fastest moving wave in the [ 111 ] followed by [011 ], isotropic medium and [001 ] respectively. 

Considerable anisotropy of properties like electrical conductivity, modulus of elasticity, etc due to the layered structure. 

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