Single crystals elastic constants

Single Crystal Elastic Constants and Calculated Aggregate Properties - A Handbook... 

Table 8.1 Single crystal elastic constants and lattice parameters of TiN, NbN and VN. The data are taken from Kim et al. (1992)...

However, in a cubic structure the value of G will be equal to C44 only when slip is on the 110 <001> slip system (Kelly et al 2000). In rocksalt-structured nitrides and carbides, slip in indentation at room temperature occurs on the 110 <110> slip system (Williams and Schaal, 1962 Molina-Aldareguia etal., 2002). The appropriate value of 6 is related to the different single crystal elastic constants, cy, by... 

Brown JM, Slutsky LJ, Nelson KA, Cheng L-T (1989) Single-crystal elastic constants for San Carlos Peridot An apphcation of impulsive stimulated scattering. J Geophys Res B94 9485-9492 Carpenter MA, Salje EKH, Graeme-Barber A (1998) Spontaneous strain as a determinant of thermodynamic properties for phase transitions in minerals. Em J Mineral 10 621-692... 

Consequently, if the peak shifts for one or more peaks are measured as a function of T in the range (0, ujl) at y and y + re for three fixed values of y e.g., 0, 71/4 and nj2) the stress tensor elements 5, can be determined from the intercept and the slopes of these lines. It is presumed that the single-crystal elastic constants are known and the diffraction elastic constants in Equations (109) and (110) can be calculated following one of the models presented before. This is the conventional sin T method. Alternatively Equation (107) can be used in a least-square analysis or implemented in the Rietveld codes. If diffraction patterns measured in several points (T, y) are available the stress tensor elements 5,- can be refined together with the structural and other parameters. The implementation in GSAS is the Voigt formula Equation (90) and not Equation (107). In this case refinable parameters are the strain tensor elements e,. 

Determination of the Single-crystal Elastic Constants. The dependence of the diffraction elastic constants on the Miller indices can be exploited to find the single-crystal elastic constants from powder diffraction data. Indeed,... 

FIGURE 1.10. Shear modulus G, bulk modulus K, Young s modulus E, and Poisson s ratio v of tungsten vs. temperature, as calculated fium single-crystal elastic constants (Gj, E, and v ) [1.31], and from measurements on polyciystalline tungsten (G, K, E, and Vj). [1.30] Taken from Metals Handbook [1.40]. o... 

W. Yang and R.G. Parr, Phys. Chem. Miner., 15, 191 (1987) G. Simmons and H. Wang, Single Crystal Elastic Constants, MIT Press, Cambridge, MA, 1971. 

G. Simmons and H. Wang, Single Crystal Elastic Constants, MIT Press, Cambridge, MA, 1971. 

Simmons G. and Wang H., Single Crystal Elastic Constants and Calculated Aggregate Properties A Handbook, The MIT Press, Cambridge Massachusetts, 1971. 

While techniques presently employed for quantitative description of microstructure all provide some sort of numerical representation, none are completely adequate for insertion into continuum theories that describe material behavior at the next higher length and time scales. Some success has been achieved in this regard by employing the ODF with single crystal elastic constants to obtain the bulk elastic constants [12], or upper and lower bounds thereto [13], for a textured poly crystal. 

Figure 2.28 Orientation convention of unit cells relative to orthogonal stress and strain axes for description of single-crystal elastic constants.

Name one approach for obtaining the elastic constants of a random (single-phase) polycrystal from the single-crystal elastic constants of the same phase. 

TABLE 3. Single crystal elastic constants (in GPa) of ot-quartz and os-cristobalite. 

Fig. 9. The temperature dependence of the single-crystal elastic constants Cn and C33 of terbium at zero magnetic field, and in an applied field along the easy axis. The inset shows the details of Cu at the Neel and Curie temperatures. (After Palmer et al. 1974.)...

Bel] Resonance technique Single crystal elastic constants of bee alloys... 

The room-temperature elastic constants of single crystals with the stoichiometry B5 fiC synthesized by the optical floating zone method have been measured using a resonant ultrasound spectroscopy technique [531]. The single crystal elastic constants indicate the strong anisotropy of the structure Cn = 542.81, Cg3 = 534.54, Ci3 = 63.51, C12 = 130.59, and 44= 164.79GPa, respectively. 

At higher temperatures the same sample (Lounasmaa and Roach 1962, Lounasmaa and Sundstrom 1966) showed an anomalous hysteresis effect, centered around 16 K and an impurity related peak at 4.5 K. As neither of these phenomena have been satisfactorily explained, it is gratifying that Hill et al. (1974) and Wells et al. (1976) found no low temperature impurity anomaly in heat capacity measurements on S.S.E. purified terbium between 1.5 and 16 K. Thus their best results (Wells et al. 1976) indicate for Tb that y = 4.4 0.1 mJ/mole-K and 0d(O) = 178 3 K. The same arguments as above for Gd can be put forward to explain the unexpectedly low y value. The limiting 6d(0) value is in reasonable agreement with the elastic constant result of 177 K by Rosen (1968), but is considerably lower than the 187 K found from single crystal elastic constant data by Palmer (1973).t... 

Due mainly to impurity oxide, the low temperature heat capacity of dysprosium is strongly sample dependent, especially between about 0.5 and 0.8 K (see fig. 1 of Flotow and Osborne 1964). Thus any analysis based on this temperature range alone cannot be relied upon. The only safe criterion is that the purest samples presumably give the lowest heat capacity from which we can obtain an upper limit for y and d(0). On this basis y 9.0 nJ/mole K for Dy. To obtain any further information from the experimental results, some assumptions must be made about Ce and Cl- Single crystal elastic constant data for Dy (Palmer 1970) give d(0)= 183 K as against polycrystalline results (Rosen 1968) where d(4.2) = 178 K. Flotow and Osborne (1%3) assumed Ce = 10.5 T mJ/mole-K and du(0) = 186 K and found a fit to Cooper s (1962) expression, eq. (5.7), with EJka 50 K. Lounasmaa and Sundstrdm (1966), on the other hand, using Ce+Cl= Cp(Lu) proposed the behaviour Cm= 107 T exp( —31/T) mJ/moIe-K. 

For single crystals with transverse dimensions large enough to permit a plane wave condition to be attained, the results are unambiguous and virtually free from theoretical assumptions. Five independent elastic constants (stiffnesses or compliances) are required to describe the linear elastic stress-strain relations for hexagonal materials. Only three independent constants are required for cubic (y-Ce, Eu, Yb) materials. Since there are no single crystal elastic constant data for the cubic rare earth metals, this discussion will concentrate on the relationships for hexagonal symmetry. 

No single crystal elastic constant data for cerium were found by the writer This probably reflects the difficulty of obtaining suitable single phase crystals. The most complete and systematic investigation of the polycrystalline elastic properties of cerium was conducted by Rosen (1969a). He used an ultrasonic pulse technique employing a frequency of 10 MHz to measure sound velocities in spectrographically pure (99.9 + %) metal and corrected the acoustic path... 

Greiner et al. (1976) also have determined the single crystal elastic constants from 6 to 300 K using the ultrasonic pulse-echo-overlap (10 MHz) method. No thermal expansion corrections for density and acoustic path length were made because no thermal expansion data were available in the experimental temperature range. In addition to the four constants measured by Lenkkeri and Palmer (1977), Greiner et al. measured C13 consequently, polycrystalline elastic... 

Single crystal elastic constants of dysprosium were determined by Rosen and Klimker (1970), Palmer and Lee (1972), and Fisher et al. (1973). Palmer (1975) investigated the effect of cooling and heating on just one constant, C33. The results of Palmer and Lee (1972) and Fisher et al. (1973) showed qualitative and... 

The compressibility of the rare earth metals was first investigated up to 40 kbar by Bridgman (see Lawson s review, 1956) for most of the members of the series, and in some cases to about 90 kbar. Since Bridgman s earlier measurements, Stephens (1964) has published compression data on Pr, Eu, Tb, Yb and Sc to 45 kbar and Perez-Albuerne et al. (1966) have reported on the compressibility of Ho, Er and Tm to about 200 kbar (see Drickamer et al., 1966). More recently Liu et al. (1973) and Liu (1975) have measured the compressibility of Tm and Lu respectively in a diamond anvil high pressure X-ray apparatus to several hundred kbar. Syassen and Holzapfel (1975) have reported on the compression of La to 120 kbar. Anomalies corresponding to the phase transitions discussed earlier were noted in some cases in these static compression measurements at the y to a-Ce transition near 7 kbar (ch. 4, section 2.1) and in La at the dhep-fee transition near 25 kbar (Bridgman, see Lawson s review, 1956), in Yb at the fcc-bcc transition near 40 kbar (Stephens, 1964) and in Lu from the hep to Sm-type transition near 230 kbar (Liu, 1975). Bulk moduli evaluated from these data are plotted in fig. 9.11. For Gd, Dy, and Er single crystal elastic constant data and their pressure variation have been obtained (Fisher et al., 1973). 

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