Elastic - constants

The elastic constant tensor is a 6 x 6 matrix that contains the second derivatives of the energy density with respect to external strain  

Generally speaking, the ability of shell model potentials to reproduce the elastic properties of ionic materials is much more limited, as compared to structures, with errors typically being an order of magnitude larger. This is a consequence of the fact that the perturbation of a structure about its equilibrium form is much more sensitive to higher order polarizabilities than the minimum itself, where any errors can be readily subsumed into the parameterization. A classic example is the failure of the dipolar shell model to reproduce the Cauchy violation in the elastic constants of simple cubic oxides, such as MgO (Catlow et al. 1976). 

The dielectric constants can be readily calculated both in the high frequency and low frequency, or static, limits where the deviation of the high frequency values from unity is a reflection of the shell model polarizability within the material. The elements of the 3x3 matrices are given by  

ELASTIC PROPERTIES AND PRESSURE EFFECTS Elastic constants 

Stress is defined as force per unit area. The elastic response of a solid consists of deformation which occurs in all the directions. Since force has three components in three (chosen) x, y and z directions and since the deformation also occurs in all three directions, stress, a, is a tensor of the second rank having 9 components described by a (3 x 3) matrix cry, y-i. 3. Alternately it may be easier to visualize why a is a tensor of second rank if one recalls that area itself can be described by a vector perpendicular to the surface which has three components. 

Deformation is measured by a quantity known as strain (strain is a relative extension or contraction of dimension). Strain is similarly a tensor of the second rank having nine components (3x3 matrix). The relation between stress and strain in the elastic regime is given by the classical Hooke s law. It is therefore obvious that the Hooke s proportionality constant, known as the elastic modulus, is a tensor of 4 rank and is represented by a (9 x 9) matrix. Before further discussion we note the following. The stress tensor consists of 9 elements of which stability conditions require axy=ayx, and cr,..=cr , . Therefore the number of 

Thus one would expect from a (6x6) matrix of the elastic stiffness coefficients (c,y) or compliance coefficients (sy) that there are 36 elastic constants. By the application of thermodynamic equilibrium criteria, cy (or Sjj) matrix can be shown to be symmetrical cy =cji and sy=Sji). Therefore there can be only 21 independent elastic constants for a completely anisotropic solid. These are known as first order elastic constants. For a crystalline material, periodicity brings in elements of symmetry. Therefore symmetry operation on a given crystal must be consistent with the representation of the elastic quantities. Thus for example in a cubic crystal the existence of 3C4 and 4C3 axes makes several of the elastic constants equal to each other or zero (zero when under symmetry operation cy becomes -cy,). As a result, cubic crystal has only three independent elastic constants (cu== C22=C33, C44= css= and Ci2=ci3= C2i=C23=C3i=C32). Cubic Symmetry is the highest that can be attained in a crystalline solid but a glass is even more symmetrical in the sense that it is completely isotropic. Therefore the independent elastic constants reduce further to only two, because C44=( c - C i)l2. 

Conventionally elastic properties of solids are described using moduli called Young s modulus, E, shear modulus, G, bulk modulus, K and the Poisson s ratio, v (due to the fact that mechanical properties were studied more extensively by engineers).Young s modulus is defined by the relation 

A number of experimental tools can be used to determine the elastic properties of glasses, including Brillouin scattering and ultrasonic 

Convincing evidence of the correlation of elastic transitions with the MCN comes from measurements on the GeAsSe and GeAsS systems, where the bulk moduli in these two different glass systems show the same trends [29,100]. The elastic modulus first increases with the MCN below 2.4. Beyond MCN = 2.4 there is first a saturation followed by a sharp rise beyond MCN = 2.7. The results for GeAsSe glasses are shown in Fig. 4.14, where the elastic transitions at 2.4 and 2.7 are quite evident. 

Since the elastic constants are calculated using measurements of density and acoustic wave velocity, the individual contributions of MCN dependence of these quantities on these elastic transitions should be considered. As evident from the previous section, the density itself changes with the MCN in a manner that reflects the topological order in Ge-As-Se ternary glasses. However, the fluctuation of the density is less than 2% compared with 50% 

In this chapter we consider properties of solids that are related to strength of interatomic bonding. We both bear in mind such macroscopic characteristics as elastic constants, melting temperature, and microscopic parameters as amplitudes of atom vibrations, energy of vacancy formation, the Debye temperature. All these properties can be measured experimentally with a high precision. Therefore they are factors that characterize the interatomic bonding in solids. 

The Hooke law states that the strain of a body is proportional to the applied stress. The law is applicable at the macroscopic level of strength of a soUd (see Table 1.1, the characteristic length is greater than 0.01 m). Practically, the Hooke law holds within the elastic range of a material for relative small values of strain e (e 1). 

Cauchy generalized the Hooke law to a three-dimensional elastic body and stated that the six components of stress are hnearly related to the six components of strain. The law can be written in a tensor form as 

IrOenOomic Boning in Solids Fund(mienkds,Simulation,(aid plications, First Edition. Valim Levitin. 

The relationship between stress and strain written in a matrix form is given by 

The flow viscosity of a nematic phase also determines the spatial and temporal response of the director to an applied field. The bulk viscosity of a nematic phase depends on the direction of flow of each molecule with respect to the director, averaged out over the whole of the sample. Therefore, bulk viscosity is 

El = Longitudinal tensUe modulus = Transverse tensUe modulus 

Vi2 = Transverse eontraction on uniaxial extension (major Poisson s ratio) i 2i = Longitudinal eontraction in transverse extension (minor Poisson s ratio) 

The axial YM is determined by dynamic resonance methods which are at very low strain and the determined values are lower than values measured at higher levels of strain, the reason being the stress strain curve for carbon fiber is not linear. 

The representative elastic properties of carbon fiber. E-glass and aramid fibers in unidirectional fiber reinforced epoxy resins are given in Table 20.13. 

Elastic constants for carbon fiber as determined by Reynolds are given in Table 20.14, whilst values determined by Goggin are given in Table 20.15. 

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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. 

Mehl M J and Papaconstantopoulos D A 1996 Applications of a tight-binding total-energy method for transition and noble metals Elastic constants, vacancies and surfaces of monatomic metals Phys. Rev. B 54 4519... 

Allen M P, Warren M A, Wilson M R, Sauron A and Wiliam S 1996 Molecular dynamics calculation of elastic constants in Gay-Berne nematic liquid crystals J. Chem. Phys. 105 2850-8... 

Here and are elastic constants. The first, is associated with a splay defonnation, is associated with... 

Figure C2.2.11. (a) Splay, (b) twist and (c) bend defonnations in a nematic liquid crystal. The director is indicated by a dot, when nonnal to the page. The corresponding Frank elastic constants are indicated (equation(C2.2.9)).

Here Fq is tire free energy of the isotropic phase. As usual, tire z direction is nonnal to tire layers. Thus, two elastic constants, B (compression) and (splay), are necessary to describe tire elasticity of a smectic phase [20,19, 86]. 

Here B is again a compressional elastic constant, is a bend elastic constant and tire elastic constant C results from an elliptical defonnation of tire rods (tliis tenn is absent if tire column is liquid). 

The strains on the lattice are equal to the stress divided by the elastic constant matrix ... 

If all the components have the same elastic constants,the condition for reversed yielding is the same as that given by equation 20. 

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

Table 3. Measured Values of Elastic Constants at Small Extensions and 25°C...

Table 1 Hsts the properties of several semiconductors relevant to device design and epitaxy. The properties are appropriate to the 2incblende crystal stmcture in those cases where hexagonal polytypes exist, ie, ZnS and ZnSe. This first group of crystal parameters appHes to the growth of epitaxial heterostmctures the cubic lattice constant, a the elastic constants, congment sublimation temperature, T. Eor growth of defect-free...

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

Mechanical Properties. The hexagonal symmetry of a graphite crystal causes the elastic properties to be transversely isotropic ia the layer plane only five independent constants are necessary to define the complete set. The self-consistent set of elastic constants given ia Table 2 has been measured ia air at room temperature for highly ordered pyrolytic graphite (20). With the exception of these values are expected to be representative of... 

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... 

Example 3. The mean free path of electrons scattered by a crystal lattice is known to iavolve temperature 9, energy E, the elastic constant C, the Planck s constant the Boltzmann constant and the electron mass M. (see, for example, (25)). The problem is to derive a general equation among these variables. 

Thermal Properties at Low Temperatures For sohds, the Debye model developed with the aid of statistical mechanics and quantum theoiy gives a satisfactoiy representation of the specific heat with temperature. Procedures for calculating values of d, ihe Debye characteristic temperature, using either elastic constants, the compressibility, the melting point, or the temperature dependence of the expansion coefficient are outlined by Barron (Cryogenic Systems, 2d ed., Oxford University Press, 1985, pp 24-29). 

The data shown in Fig. 8.11 are for an 80 ml/kg grade oil shale obtained from a mine near central Colorado. Oil shale grades from this region vary from 40-320 ml/kg. Properties such as fracture toughness and elastic constants are found to depend on oil shale grade. For the oil shale studied in Fig. 8.11, a fracture toughness of x 0.9 MN/m, a density of p = 2000 kg/m and an elastic wave speed of c = 3000 m/s are representative. 

One final point. We earlier defined Poisson s ratio as the negative of the lateral shrinkage strain to the tensile strain. This quantity, Poisson s ratio, is also an elastic constant, so we have four elastic constants E, G, K and v. In a moment when we give data for the elastic constants we list data only for . For many materials it is useful to know that... 

Contrary to spherical contact, the minimum force, or the pull-off force under load-controlled conditions, is dependent on the elastic constant of the system... 

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