Elastic constant measurement temperature dependence

Instead of measuring the force-temperature dependence at constant volume and length, one can measure this dependence at constant pressure and length but in this case it is necessary to introduce the corresponding corrections. The corrections include such thermomechanical coefficients as iso-baric volumetric expansion coefficient, the thermal pressure coefficient or the pressure coefficient of elastic force at constant length 22,23,42). 

The Debye temperature can also be obtained from the elastic constants. The measurement of the elastic constants of polycrystalline AIN was used by Slack et al [8] to derive the Debye temperature, giving 0D = 950 K. Therefore, Slack et al have criticised the value of the AIN Debye temperature 0D = 800 + 2 K, derived from the heat capacity measurements by Koshchenko et al [6], as too low. Also, Slack s value differs considerably from Meng s result [7]. Since the cubic dependence T3 approximates the Debye specific heat well in the temperature range below T = 0d/5O [9], it is likely that the upper temperature limit used by Meng is too high and led to error and the difference from the results of Slack et al [8],... 

As we see from these formulae the elasticity constant Cee (the reciprocal of elastic susceptibility) tends to zero at T = Tc. The experimental dependence of the modulus of elasticity upon temperature is measured at T > Tc only because of the strong domain wall related ultrasound attenuation. The agreement between the MFA theory (the continuous line in the Fig. 2) and experiment is very good. 

Fig. 3 Lowest electronic states (a) for DyV04 crystal, temperature dependence of the energy gap between ground and excited doublets (b) (for comparison Raman scattering results are shown for the DyAs04 crystal with similar electronic and crystal structures), and the ultrasonic measurements (c) of the elastic constant Ci = l/2(Cn-Ci2) for DyV04 crystal...

Elastic Properties [1.30,1.31,1.35]. In regard to elasticity, at least below room temperature, tungsten behaves nearly isotropically the anisotropy coefficient at 24 °C is = 1.010 [1.35]. The elastic constants for polycrystalline tungsten at 20 °C are given below. Their temperature dependence as well as the respective values for singlecrystal elastic constants are shown in Fig. 1.10 [1.40], based on ultrasonic measurements [1.30,1.31]. 

The order parameter S is a very important quantity in a partially ordered system. It is the measure of the extent of the anisotropy of the liquid crystal physical properties, e.g., elastic constants, viscosity coefficients, dielectric anisotropy, birefringence, and so on. S is temperature dependent and decreases as the temperature increases. The typical temperature dependence of S is shown in Figure 1.16. 

Fig. 5.3.10. Temperature dependence of the elastic constants of the smectic A phase of diethyl 4,4 -azoxydibenzoate determined by ultrasonic velocity measurements open circles, 2 MHz filled circles, 5 MHz triangles, 12 MHz crosses, 20 MHz. (After Miyano and Ketterson. )...

Interpretation of mechanical measurements in terms of molecular structure was until fairly recently confined essentially to identification of the temperatures of the major viscoelastic relaxations through extensional or torsional dynamic mechanical studies. Now, however, investigations of the elastic constants and their temperature dependence—allied with dynamic mechanical, creep and both wide and small angle X-ray diffraction— are yielding fairly detailed pictures of the interrelation of the crystalline and less well ordered regions of some oriented solid polymers. 

The perfect-crystal model of Karasawa et al. considers a basic force-field approach for the analysis of crystal properties. The model contains covalent-bonded interactions along the polymer chain as well as non-bonded van der Waals interactions between molecules and Coulombic interactions when relevant, all with appropriate temperature dependences. Table 4.3 lists some of the temperature-dependent elastic moduli and some elastic constants cy of ideal polyethylene, determined by Karasawa et al. (1991), which will be of interest to us in later chapters. These are shown also in Fig. 4.1. Of these Ec (= l/ssj) gives directly the main-chain Young s modulus of polyethylene. Also listed is the transverse shear elastic constant c e, which can be considered to be a good measure of... 

Equation (102) shows that MAQO can provide important information about the electronic parameters (extremal Fermi surface cross-sectional area, effective masses, electronic relaxation times) and about the electron-phonon interaction (strain derivatives of the cross-sectional area for different symmetry strains). With the help of this technique, combined with de Haas-van Alphen susceptibility measurements, one can put the deformation potential interaction and the temperature dependence of the elastic constants, discussed above in sect. 3.2, on a solid basis. In the following we discuss some compounds. 

Elastic anomalies in actinide-based heavy-fermion systems are most pronounced for UPt3. The temperature dependences of two representative elastic modes are shown in flg. 36 (Yoshizawa et al. 1985). Unlike the case of UPdj, CF effects as a possible source do not seem very likely. The phonon dispersions of UBOjj, as measured by neutron scattering (Robinson et al. 1986), do not show strong anomalies. The elastic constant Cj2, however, is negative at low temperatures, similar to some IV compounds. Remarkably, UBejj possesses a low Debye... 

The oscillations of I (U) are well seen in the experimental plot. Fig. 11.21. The measurements were made at 27°C on 55 nm thick cell filled with a mixture having ta = 22. From the I (U) curve, the field dependence of the phase retardation 8(17) and the Frederiks transition threshold Uc were obtained. In mm, from Ec = UJd and Fq. (11.56) the splay elastic constant Ku was found. The bend modulus "33 was calculated from the derivative dbldU. The same material parameters may be found for the whole temperature range of the nematic phase. 

Strain dependence should also be considered when comparing acoustically measured elastic constants with statically measured values. As an example, for polyethylene at room temperature, the modulus is independent of strain up to a strain of about 10 (7). Beyond this point, the modulus decreases as the strain increases. Typically, acoustic measurements are made in the strain range 10 where the moduli are strain-independent, but static measin-ements... 

---