Elastic and bulk moduli

The response of an isotropic, homogeneous solid to a force is expressed in terms of the elastic constants or elastic moduli. [Unfortunately, a standard set of symbols for these constants is not in use.] Four elastic constants are frequently defined but, as they are interrelated, the elastic properties of a solid can be defined in terms of any two. They are most conveniently defined with respect to the stress, which is the force per unit area applied to the body, and the strain, which is the deformation of the body produced by the force. 

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Table 2 Elastic constants and bulk moduli for 4d cubic elements. Comparison is made between the results of our tight-binding parametrization (TB), first-principles full potential LAP., results (LAPW), where available, and experiment (Exp.). Calculations were performed at the experimental volume.

In table 2 and 3 we present our results for the elastic constants and bulk moduli of the above metals and compare with experiment and first-principles calculations. The elastic constants are calculated by imposing an external strain on the crystal, relaxing any internal parameters (case of hep crystals) to obtain the energy as a function of the strain[8]. These calculations are also an output of onr TB approach, and especially for the hep materials, they would be very costly to be performed from first-principles. For the cubic materials the elastic constants are consistent with the LAPW values and are to within 1.5% of experiment. This is the accepted standard of comparison between first-principles calculations and experiment. An exception is Sr which has a very soft lattice and the accurate determination of elastic constants is problematic. For the hep materials our results are less accurate and specifically in Zr the is seriously underestimated. ... 

In general, there are three kinds of moduli Young s moduli E, shear moduli G, and bulk moduli K. The simplest of all materials are isotropic and homogeneous. The distinguishing feature about isotropic elastic materials is that their properties are the same in all directions. Unoriented amorphous polymers and annealed glasses are examples of such materials. They have only one of each of the three kinds of moduli, and since the moduli are interrelated, only two moduli are enough to describe the elastic behavior of isotropic substances. For isotropic materials... 

In this equation is the deviator and a is the spherical part of the stress tensor <7, eij is the strain deviator and e the volumetric part of the strain tensor ij, K = (2M + 3A) /3 is bulk modulus with M and A corresponding to the familiar Lame coefficients in the theory of elasticity, while r) and n can be termed the viscous shear and bulk moduli. 

In this Datareview, we have reviewed the mechanical properties of the group III nitrides. From this point of view, they form a very homogeneous family of compounds, with large elastic moduli, both compressional and shear, and bulk moduli of the same order of magnitude as that of diamond. 

In most papers referenced above, the standard molecular formulation of the B3LYP functional has been employed, and its results graded against a set of other Hamiltonians available in CRYSTAL. These usually include at least HP, LDA and one GGA functional (PW or PBE), and thus enable a critical appraisal of the B3LYP performance compared to other well established Hamiltonians in solid-state chemistry. Several observables have been examined, such as the equilibrium structure, elastic constants and bulk moduli, thermochemical data, electric field gradients, phonon spectra and vibrational frequencies, polarisation of the ferroelectric phases, magnetic coupling in open-shell transition metal oxides. We shall comment on each observable separately. 

For a perfectly elastic solid, or a viscoelastic solid at equilibrium, the equilibrium Young s modulus is related to the shear and bulk moduli more simply as follows ... 

In solving viscoelastic stress analysis problems, assumptions on the material properties are often essential as gathering accurate time dependent data for viscoelastic properties is difficult and time consuming. Thus, one often only has properties for shear modulus, G(t) or Young s modulus, E(t), but not both. Yet of course for even the simplest assumption of a homogeneous, isotropic viscoelastic material, two independent material properties are required for solution of two or three dimensional stress analysis problems. Consequently, three assumptions relative to material properties are frequently encountered in viscoelastic stress analysis. These are incompressibility, elastic behavior in dilatation and synchronous shear and bulk moduli. Each of the common assumptions defines a particular value for either the bulk modulus or Poisson s ratio as follows. 

J-g -K ) for alumina and 55.8 J-mof -K ( = 0.453 J-g -K" ) for zirconia. These values are in satisfactory agreement with literature values [Munro 1997, NIST 2002, Salmang Scholze 1982]. With this input information at hand. Equations (40) and (41) (the latter in connection with approximate values for the shear and bulk moduli, cf Table 8 below for the definite values) can now be used to obtain estimates for the differences that have to be expected at room temperature between adiabatic elastic constants (measured via dynamic techniques) and isothermal elastic constants (measured via static techniques). For alumina and zirconia... 

Our intention is to give a brief survey of advanced theoretical methods used to detennine the electronic and geometric stmcture of solids and surfaces. The electronic stmcture encompasses the energies and wavefunctions (and other properties derived from them) of the electronic states in solids, while the geometric stmcture refers to the equilibrium atomic positions. Quantities that can be derived from the electronic stmcture calculations include the electronic (electron energies, charge densities), vibrational (phonon spectra), stmctiiral (lattice constants, equilibrium stmctiires), mechanical (bulk moduli, elastic constants) and optical (absorption, transmission) properties of crystals. We will also report on teclmiques used to study solid surfaces, with particular examples drawn from chemisorption on transition metal surfaces. 

Pa, would deform appreciably under the action of loads comparable to the pull-off force given by Eq. 16. It is for this reason that the JKR type measurements are usually done on soft elastic materials such as crosslinked PI rubber [45,46] or crosslinked PDMS [42-44,47-50]. However glassy polymers such as polystyrene (PS) and PMMA are relatively hard, with bulk moduli of the order of 10 Pa. It can be seen from Eq. 11 that a varies as Thus, increasing K a factor of... 

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 shown that the stabilities of solids can be related to Parr s physical hardness parameter for solids, and that this is proportional to Pearson s chemical hardness parameter for molecules. For sp-bonded metals, the bulk moduli correlate with the chemical hardness density (CffD), and for covalently bonded crystals, the octahedral shear moduli correlate with CHD. By analogy with molecules, the chemical hardness is related to the gap in the spectrum of bonding energies. This is verified for the Group IV elements and the isoelec-tronic III-V compounds. Since polarization requires excitation of the valence electrons, polarizability is related to band-gaps, and thence to chemical hardness and elastic moduli. Another measure of stability is indentation hardness, and it is shown that this correlates linearly with reciprocal polarizability. Finally, it is shown that theoretical values of critical transformation pressures correlate linearly with indentation hardness numbers, so the latter are a good measure of phase stability. 

When we compared the viscosities of solutions of natural rubber and of guttapercha and of other elastomers and later of polyethylene vs.(poly)cis-butadiene, with such bulk properties as moduli, densities, X-ray structures, and adhesiveness, we were greatly helped in understanding these behavioral differences by the studies of Wood (6) on the temperature and stress dependent, melting and freezing,hysteresis of natural rubber, and by the work of Treloar (7) and of Flory (8) on the elasticity and crystallinity of elastomers on stretching. Molecular symmetry and stiffness among closely similar chemical structures, as they affect the enthalpy, the entropy, and phase transitions (perhaps best expressed by AHm and by Clapeyron s... 

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-... 

Furthermore, in gels the elastic moduli K and p treated so far, those of the so-called skeletal frame in Biot s theory, are much smaller than the bulk moduli of fluid and polymer. Note that K accompanies no changes in the total volume of network + solvent, whereas K, p, and pfcl involve them and are much larger than K. From this fact, without changing the essential physics, we assume that the solvent and polymer have the same constant specific volume, so that... 

The isotropic moduli, particularly the initial bulk modulus and its pressure derivative, are key ingredients in specifying the mechanical equation of state. As noted above, determination of these properties from experimental hydrostatic compression data is difficult due to issues with acquisition of high precision at low pressures and particular sensitivity in the choice of equation of state fitting form to data below about one GPa. Alternative routes to this information at low pressures included impulsive stimulated light scattering (ISLS) and resonant ultrasound spectroscopy (RUS), which can in principle provide the complete elastic tensor (ISLS) and isotropic bulk and shear moduli (RUS). 

Given the elastic tensor we can obtain Reuss average, isotropic bulk and shear moduli... 

Apart from the calculations using the Keating-Harrison model [17] the calculated bulk moduli are very near to the experimental results. The elastic moduli are less accurately reproduced, between 28% for the Cn and 2% for C33. 

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. 

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