Longitudinal elastic constant

The five distinguishable second-order elastic moduli in a hexagonal crystal are Cn, Cn, Ci3, C33 and C44. There are reports of neither measured nor calculated values but, since each depends principally on the lattice constants [21] which vary by only about 10% across the nitrides, values for AIN (q.v.) may be used as a first approximation. The comparability of bulk moduli in indium, gallium and aluminium nitrides supports this approach. Estimates of the principal transverse and longitudinal elastic constants Ct and Ci are given in TABLE 2. 

Fig. 48. Schematic temperature variations of adiabatic longitudinal elastic constants in Ce HF compounds. In the various regimes c T) is determined by anhatmonic effects (r= fl ), magnetoelastic coupling (r A = CEF gap) and Gruneisen parameter coupling (F< T ). c, Cg are extrapolations from the anharmonic and magnetoelastic regimes, respectively.

Here, the simple dispersion (oiq) v g was used for the acoustic phonons (if whole molecules vibrate with respect to each other, like the change of the stacking distance in a stack one speaks of acoustic phonons, because they have a long wavelength comparable to those of acoustic waves) and the sound velocity v, is determined by the relation o, = Ci/p), where c, is the longitudinal elastic constant and p is the mass density. One should remark that this very simple linear dispersion relation (o q) = v,g is not necessarily correct. With the help of the FG method described in Section 9.1 one can obtain more accurate dispersion curves. Equation (9.48) can now be used to calculate the charge carrier mobilities and free paths, defined in this case hy p= e xlm ) and A = (t ), respectively, where... 

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... 

In order to use this model it is necessary to estimate the ratio of the longitudinal to shear elastic constants, CjilCbb- In the original model this quantity was... 

In solids of cubic symmetry or in isotropic, homogeneous polycrystalline solids, the lateral component of stress is related to the longitudinal component of stress through appropriate elastic constants. A representation of these uniaxial strain, hydrostatic (isotropic) and shear stress states is depicted in Fig. 2.4. Such relationships are thought to apply to many solids, but exceptions are certainly possible as in the case of vitreous silica [88C02]. 

Form factor of the hat-curved model Normalized concentration of molecules Kirkwood correlation factor Steady-state energy (Hamiltonian) of a dipole Dimensionless energy of a dipole Moment of inertia of a molecule Longitudinal and transverse components of the spectral function Complex propagation constant Elasticity constant (in Section IX)... 

Longitudinal Elastic Waves on a 1-D Line of Equidistant Equal Atoms. Consider next the longitudinal motion of a one-dimensional array of E equal atoms of mass M (Fig. 5.7). These atoms at rest are equidistant—that is, spaced a (meters) apart—and can interact via Hooke s law with force constant kH (N m ), but only with their nearest neighbors. Let u be the longitudinal displacement of atom n from its equilibrium position. The net Hooke s law force on atom n, due to the displacements un, un v and un +, is... 

Figure 9.3 Elastic constants of an anisotropic fiber the longitudinal Young s modulus of fiber, or, the transverse Young s modulus or j., and the principal shear modulus, G 2 or Not shown are the two Poisson s ratios i/jj or the longitudinal Poisson s ratio of the fiber and or the transverse or inplane Poisson s ratio of the fiber cross-section.

In our discussion of elastic constants we have imagined uniform distortions of the crystal. An elastic medium can sustain vibrations, which at any instant consist of nonuniform distortions. The normal modes of an elastic continuum arc sound waves (longitudinal and transverse) propagating in the medium, and these normal modes will also exist in the crystal. Indeed, viewing the crystal as an elastic... 

The Born model [74], for example, satisfies the first condition however, it does not satisfy the second one because in it the longitudinal and transverse elastic constants of the linear chain of bonds (the lattice analog of a rod) decrease /V 1 however, the rods must behave more pliably in relation to transverse shifts (the elastic constant decreases L-3). Therefore, the Born scalar model leads to enhanced rigidity in the vicinity of pc. 

Mechanism b, responsible for the T-peak depicted in Figs. 20a and 20b, concerns elastic oscillation of two charged water molecules along the H-bond this oscillation is governed by the longitudinal force constant k. 

---