Third-order elastic constants

Table 2.1. Third-order elastic constants determined from Hugoniot elastic limits (after Davison and Graham [79D01]).

A number of such phenomena or materials characteristics are listed in Table 2.5. The noted effects include mechanical, physical, and chemical processes. The positive third-order elastic constants were described in Sec. 2.2. 

The measured relationships between piezoelectric polarization and strain for x-cut quartz and z-cut lithium niobate are found to be well fit by a quadratic relation as shown in Fig. 4.4. In both materials a significant nonlinear piezoelectric effect is indicated. The effect in lithium niobate is particularly notable because the measurements are limited to much smaller strains than those to which quartz can be subjected. The quadratic polynomial fits are used to determine the second- and third-order piezoelectric constants and are summarized in Table 4.1. Elastic constants determined in these investigations were shown in Chap. 2. 

It can be seen that the accuracy of the method is increased by increasing zo. The stress can then be deduced either from knowledge of the third-order elastic constants (Husson 1985), or by perturbation theory (Husson and Kino 1982), or from direct calibration. 

The fifty-six components Cpqr (which do not constitute a tensor) are the third-order elastic constants they are symmetric with respect to all interchanges ofp, q, and r. The expansion... 

Elastic constants depend on pressure and temperature because of the anharmonicity of the interatomic potentials. From the dependence of bulk and shear moduli on hydrostatic and uniaxial pressure, third order elastic constants and Griineisen parameters may be determined. Griineisen parameter shows the effect of changing volume, V, on the phonon mode frequencies, co. 

Technically, the direction of the inequality in equation (7.5) is determined by the variation of the price elasticity of demand with the accumulation constant of the chemical (which in turn is determined by the sign of a third-order cross partial derivative of the farmers production function). 

In a perfectly harmonic crystal the elastic constants would be strictly independent of temperature. However, due to the existence of third- and fourth-order anharmonic terms in the crystal potential there is a coupling between the homogeneous strains and the phonon coordinates. This will lead to a background temperature dependence of the elastic constants. It can be described within a quasiharmonic approximation (Ludwig 1967), in which the anharmonic contributions to the crystal potential are implicitly included by assuming a strain dependence of the phonon frequencies which can be characterized by the... 

For TmCd and TmZn a variety of techniques have been applied to determine the important coupling constants and gj. they are listed in table 3. In addition to the temperature dependence of the symmetry elastic constant Cj (T), the parastriction method, the third-order susceptibility and the magnetic field dependence of the structural phase transition temperature Tq B) have been used. The different experimental methods have been described in sect. 2.4.1. It is seen from table 3 that the coupling constants determined with these different methods are in good agreement with each other. 

Thurston RN, Brugger K (1964) Third-order elastic constants and the velocity of small amplitude elastic waves in homogeneously stressed media. Phys Rev 133 A1604... 

The s are then the generalized elastic constants. The third and fourth order s are directly proportional to the matrix elements for the three- and four-phonon processes respectively. 

The symbol x stands for the direction perpendicular to the surface of the polymer fibrile. The elastic constant L has a value of L = 1.1 10 J/m. The constant G, describing the strength of the surface coupling, will be our third parameter in the calculation of the average order parameter s. 

The main approximation of the FEAt coupling method is therefore given by the truncation of the Taylor series expansion [1]. Within the framework of linear local elasticity theory, it is enough to retain (and match) terms only up to the second order. However, if nonlinear elastic effects need to be included, third-order elastic constants must also be matched. Moreover, because the first-order elastic constants in the continuum are zero by definition, such a matching condition requires the interatomic potential to yield zero stress in a perfect lattice. 

It turns out that a third order process is possible which combines Vi and the k-f interaction characterized by (A,. T) = (1,1). Details can be found in Fulde (1975). The net result is that this time the quadrupole susceptibility diverges as TJ T - Tc) in molecular field approximation as the magnetic phase transition is approached from above. This would imply that an elastic constant becomes soft at a second order magnetic phase transition. In practice this requires the presence of the interaction (A, i ) = (1, 1) with reasonable strength. 

The technique of Brillouin spectroscopy (Section 6.3.3) has been applied to determine the elastic constants of oriented polymer fibres. Early studies of this nature were undertaken by Kruger et al. [46,47] on oriented polycarbonate films, also determining the third-order constants, which define the elastic non-linear behaviour. Wang, Liu and Li [48,49] have described measurements on oriented polyvinylidene fluoride and polychlorotrifluoroethy-lene films. In the latter case the results were interpreted using an aggregate model differing in detail from that of Ward discussed in Section 8.6.2. 

Third-order elastic constants of some rare earth metals (in 10 dyn/cm )... 

Cijkimn are the third-order adiabatic elastic constants (TOEC),... 

Third-order elastic constants of LaSe at 0 K were calculated using the Born-Mayer potential model. The repulsive interaction was considered up to the second nearest neighbors. The interatomic distance To = 3.030A leads to the values in lO N/m ( lO dyn/cm ) Cin = -21.439, c°i2 = Cii6=-1.860, C123 = cJse = C144 = 0.743. The temperature dependence for is given by = + where are (in lO N-m K ) am =6.837, an2 = 3.601,... 

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