Elastic constants complex

It is also possible that a membrane might have an even lower symmetry than a chiral smectic-C liquid crystal in particular, it might lose the twofold rotational symmetry. This would occur if the molecular tilt defines one orientation in the membrane plane and the direction of one-dimensional chains defines another orientation. In that case, the free energy would take a form similar to Eq. (5) but with additional elastic constants favoring curvature. The argument for tubule formation presented above would still apply, but it would become more mathematically complex because of the extra elastic constants. As an approximation, we can suppose that there is one principal direction of elastic anisotropy, with some slight perturbations about the ideal twofold symmetry. In that approximation, we can use the results presented above, with 4) representing the orientation of the principal elastic anisotropy. 

Most of the experimental results on CJTE can be explained on the basis of molecular field theory. This is because the interaction between the electron strain and elastic strain is fairly long-range. Employing simple molecular field theory, expressions have been derived for the order parameter, transverse susceptibility, vibronic states, specific heat, and elastic constants. A detailed discussion of the theory and its applications may be found in the excellent review by Gehring Gehring (1975). In Fig. 4.23 various possible situations of different degrees of complexity that can arise in JT systems are presented. 

Mobile H centres in alkali halides are known to aggregate in a form of complex hole centres [64] this process is stimulated by elastic attraction. It was estimated [65, 66] that for such similar defect attraction the elastic constant A is larger for a factor of 5 than that for dissimilar defects - F, H centres. Therefore, elastic interaction has to play a considerable role in the colloid formation in alkali halides observed at high temperatures [67]. In this Section following [68] we study effects of the elastic interaction in the kinetics of concentration decay whereas in Chapter 7 the concentration accumulation kinetics under permanent particle source will be discussed in detail. 

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)... 

Thus it may produce the transitional densities of the alg, egc, and tluz symmetries. At this point selection rules pertinent to the frontier orbitals approximation enter for the 12-electron complexes the symmetries of the frontier orbitals are Th = eg and Tl = ai3, the tensor product Th <8> TL = eg aig = eg contains only the irreducible representation eg so that the selection rules allow only the density component of the egc symmetry to appear. In its turn this density induces additional deformation of the same symmetry. That means that in the frontier orbitals approximation, only the elastic constant for the vibration modes of the symmetry eg is renormalized. This result is to be understood in terms of individual nuclear shifts of the ligands in the trans- and cis-positions relative to the apical one. They, respectively, are ... 

By contrast, for the 14-electron complexes (nontransition nonmetals) the symmetries of the frontier orbitals are I n = aig and rL = l u and the tensor product Th rL = a g 11 = 11 so that only the transition density corresponding to the representation t survive. Analogous moves allow us to conclude that the off-diagonal elastic constant for stretching the /.ran.s-bonds has the form ... 

The complexity of N205 and the absence of compressibility or elastic constant data preclude any reliable calculation of the nonelectrostatic... 

Solution In the case of harmonic motion, for which Sj= joiSj, Equation 2.17 implies that attenuation may be accounted for by representing the elastic constants cjj by complex elastic constants cu + jmr u. (This is analogous to accounting for dielectric loss in electromagnetic and optical waveguides by the well-known method of postulating a complex dielectric constant or a complex index of refraction.) Equation 2.13, the lossless wave equation for this shear wave, becomes... 

The prediction of the elastic constants of an oriented polymer provides a further, more stringent, test for the deformation models because these properties depend in a more complex way on the details of the distribution... 

Elastic constants and strength properties needed for the characterization of directional behavior can be used. The number of constants needed depends primarily on the complexity of the construction, that is, whether it is isotropic, planar isotropic, or orthotropic. With these stiffness constants known, the directional stiffness properties can be calculated using textbook equations. 

As q " is strictly independent of the temperature, equation (2) gives in the fast motion (27cfx 1) as well as in the slow motion case (27ifx 1) the refractive index n(T) at the laser wavelength Xq (c.f (9)). In the acoustic relaxation regime D(q ", T) exeeds n(T). In (35) we present different theoretical curves of D(q ", q , T) calculated under the assumption, that the real part of the complex elastic constant c (q, T) can be written in the form c (q, T) = c (T)-Ac/ 1 + 47i (q,T)x (T). For the exponent P<1 this formular describes a Cole davidson function. The relaxation time x was assumed to follow a VFT law. Under these conditions the OADF deviates from n(T) only well above the TGT and... 

Expressions are available for the HS bounds for the other crystal classes but the mathematics increases in complexity and, thus, the VRH averages are often used. The data in Table 2.1 were calculated using this procedure and Table 3.4 compares some experimental values with the theoretical values. It appears from the data that these constitutive equations are extremely successful in estimating the elastic constants of the polycrystalline material. 

For both magnesite and calcite, the elastic bulk modulus Bq was computed straightforwardly by the Murnaghan interpolation formula, while of the elasticity tensor only the C33 component and the C + C 2 linear combination could be calculated in a simple way. The relations used are C = (l/Vo)c (d L /crystal structure. To derive other elastic constants, the symmetry must be lowered with a consequent need of complex calculations for structural relaxation. A detailed account of how to compute the Ml tensor of crystal elasticity by use of simple lattice strains and structure relaxation was given previously[10, 11]. For the present deformations only the c-o ( ) relaxation need be considered. The results are reported in Table 6, together with the corresponding values extrapolated to 0 K from experimental data (Table 2). For calcite, the mea-... 

Characteristic material properties for fibre reinforced composites cannot be so rigorously defined as for most other structural materials. Standard data are either not generally available or cannot be relied upon except for preliminary design. Because of its nonhomogeneity and anisotropy, up to 21 elastic constants may, in theory, be required for a full analysis of a complex laminate. In practice though, due to symmetry, these reduce to ... 

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