Elastic Constants. Bulk Modulus

Most polycrystalline solids are considered to be isotropic, where, by definition, the material properties are independent of direction. Such materials have only two independent variables (that is elastic constants) in matrix (7.3), as opposed to the 21 elastic constants in the general anisotropic case. The two elastic constants are the Young modulus E and the Poisson ratio v. The alternative elastic constants bulk modulus B and shear modulus /< can also be used. For isotropic materials, n and B can be found from E and t by a set of equations, and on the contrary. 

The calculated and experimental values of the equilibrium lattice constant, bulk modulus and elastic stiffness constants across the M3X series are listed in Table I. With the exception of NiaGa, the calculated values of the elastic constants agree with the experimental values to within 30 %. The calculated elastic constants of NiaGa show a large discrepancy with the experimental values. Our calculated value of 2.49 for the bulk modulus for NiaGa, which agrees well with the FLAPW result of 2.24 differs substantially from experiment. The error in C44 of NiaGe is... 

Bulk Compressibility and Bulk Modulus is one of the important constants of aa elastic solid Bulk modulus is defined as the tatio of stress to atrsin when the stress is a pressure applied equally on all surfaces of the sample and the strain is the resulting change in volume per suit volume. The reciprocal of bulk modulus Is called bulk compressibility. One apparatus for the direct exnd measurement of the dynamic bulk modulus of a solid was developed at the NOL, White Oak, Md(Ref 1). Some data obtained, on several HE a, using this apparatus are given in Refa 2, Refs l)NAVORD Kept No 1534(1950) 2)NAVORD Rept No 4380(1956) 3)PATR 1740,Rev 1(1958)... 

Fig. 7. Relations between elastic constants and ultrasonic wave velocities, (a) Young s modulus (b) shear modulus (c) Poisson s ratio and (d) bulk...

The are the elastic constants the bulk modulus of the material is computed as -B = + 2c 2 )/3- Values in parentheses are estimates. 

Another commonly used elastic constant is the Poisson s ratio V, which relates the lateral contraction to longitudinal extension in uniaxial tension. Typical Poisson s ratios are also given in Table 1. Other less commonly used elastic moduH include the shear modulus G, which describes the amount of strain induced by a shear stress, and the bulk modulus K, which is a proportionaHty constant between hydrostatic pressure and the negative of the volume... 

The elastic constants of bulk amorphous Pd-Ni-P and Pd-Cu-P alloys were determined using a resonant i rasound spectroscopy technique. The Pd-Ni-P glasses are slightly stiffer than the Pd-Cu-P glasses. Within each alloy system, the Young s modulus and the bulk modulus show little change with alloy composition. 

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... 

Poison s ratio is used by engineer s in place of the more fundamental quality desired, the bulk modulus. The latter is in fact determined by r for linearly elastic systems—h ncc the widespread use of v engineering equation for large deformations, however, where the Strain is not proportional to the stress, a single value of the hulk modulus may still suffice even when the value of y is not- constant,... 

What we would like to do is use these thermodynamic properties to calculate an equilibrium elastic moduli. The bulk modulus is by definition the constant of proportionality that links the infinitesimal pressure change resulting from a fractional change in volume (Section 2.2.1). In colloidal terms this becomes... 

In addition to the tensile and shear moduli, a compressive modulus, or modulus of compressibility, K, exists to describe the elastic response to compressive stresses (see Fignre 5.7). The compressive modulus is also sometimes called the bulk modulus. It is the proportionality constant between the compressive stress, CTc, and the bulk strain, represented by the relative change in bulk volume, AV/Vo-... 

Spiering et al. (1982) have developed a model where the high-spin and low-spin states of the complex are treated as hard spheres of volume and respectively and the crystal is taken as an isotropic elastic medium characterized by bulk modulus and Poisson constant. The complex is regarded as an inelastic inclusion embedded in spherical volume V. The decrease in the elastic self-energy of the incompressible sphere in an expanding crystal leads to a deviation of the high-spin fraction from the Boltzmann population. Pressure and temperature effects on spin-state transitions in Fe(II) complexes have been explained based on such models (Usha et al., 1985). 

Elastic constants of minerals are the key to understanding geophysical properties of the Earth s interior. Bulk modulus and rigidity parameters, for example, influence the velocities of seismic waves through the Earth. Numerous experi-... 

Nevertheless, some conclusions may be drawn from the set of results presented here. First, with the notable exception of InN, the group III nitrides form a family of hard and incompressible materials. Their elastic moduli and bulk modulus are of the same order of magnitude as those of diamond. In diamond, the elastic constants are [49] Cu = 1076 GPa, Cn = 125 GPa and Cm = 577 GPa, and therefore, B = (Cn + 2Ci2)/3 = 442 GPa. In order to make the comparison with the wurtzite structured compounds, we will use the average compressional modulus as Cp = (Cu + C33)/2 and the average shear modulus as Cs = (Cu + Ci3)/2. The result of this comparison is shown in TABLE 8. 

Melting point Specific heat Hardness Bulk modulus Elastic constants... 

In a similar fashion, the rigidity modulus, G, for an elastically isotropic solid is given by 0-4/84 = C44 = 0-5/85 = C55 = cTs/8g = cgg = i(cn - C12) = /r, or C44, which represents a shape change without a volume change. Therefore, the second Lame constant (fi) is the shear modulus for an elastically isotropic body. The Lame constants may also be related directly to the bulk modulus, B, for an elastically isotropic body, which can be obtained through the relations /r = ( )(B - A) and = B - ( )G. 

Use the following values of the elastic-stiffness constants and the elastic-comphance constants (Kisi and Howard, 1998) for tetragonal zirconia monocrystals to determine the Voigt-Reuss-HiU averages for the Young s modulus, E, the shear modulus, G, and the bulk modulus, B. 

For a crystal having the symmetry of diamond or /.incblende (thus having cubic elasticity), there are three independent clastic constants, c, t 12, and C4.4. The bulk modulus that was discussed in Chapter 7 is B = (c, + 2c,2)/3. We can discuss the bulk modulus, and the combination c, — c,2, entirely in terms of rigid hybrids, and therefore the two elastic constants c, and c,2 do not require deviations from this simple picture. This will not be true for the strain, which is relevant to c 44, and this is a complication of some importance. 

The wurtzitc structure has lower symmetry and six, rather than three, independent elastic constants. These could be estimated by obtaining Cq from the measured bulk modulus and from Eq. (8-22), if one wished them. 

A correction for TDS, that is, the inelastic phonon scattering, is a difficult problem not yet satisfactorily solved. A proper correction can only be made if the elastic constants of the crystal are known, although an empirical method has been suggested. TDS is inversely correlated with the bulk modulus, thus for soft molecular crystals at room temperature it can be as high as 20-30% of the Bragg intensity, while a much smaller percentage for hard inorganic solids. The effect of no correction is to underestimate the ADP. For neutrons, the problem is even more complicated since the TDS correction will depend on the neutron velocity relative to the velocity of sound in the crystal. ... 

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