Elastic constants specific

Thermal Properties at Low Temperatures For sohds, the Debye model developed with the aid of statistical mechanics and quantum theoiy gives a satisfactoiy representation of the specific heat with temperature. Procedures for calculating values of d, ihe Debye characteristic temperature, using either elastic constants, the compressibility, the melting point, or the temperature dependence of the expansion coefficient are outlined by Barron (Cryogenic Systems, 2d ed., Oxford University Press, 1985, pp 24-29). 

In table 2 and 3 we present our results for the elastic constants and bulk moduli of the above metals and compare with experiment and first-principles calculations. The elastic constants are calculated by imposing an external strain on the crystal, relaxing any internal parameters (case of hep crystals) to obtain the energy as a function of the strain[8]. These calculations are also an output of onr TB approach, and especially for the hep materials, they would be very costly to be performed from first-principles. For the cubic materials the elastic constants are consistent with the LAPW values and are to within 1.5% of experiment. This is the accepted standard of comparison between first-principles calculations and experiment. An exception is Sr which has a very soft lattice and the accurate determination of elastic constants is problematic. For the hep materials our results are less accurate and specifically in Zr the is seriously underestimated. ... 

Data from ref [31] the elastic constants were measured by the pulse-echo technique j3 was determined from low-temperature specific heat data. 

Computer simulations therefore have several inter-related objectives. In the long term one would hope that molecular level simulations of structure and bonding in liquid crystal systems would become sufficiently predictive so as to remove the need for costly and time-consuming synthesis of many compounds in order to optimise certain properties. In this way, predictive simulations would become a routine tool in the design of new materials. Predictive, in this sense, refers to calculations without reference to experimental results. Such calculations are said to be from first principles or ab initio. As a step toward this goal, simulations of properties at the molecular level can be used to parametrise interaction potentials for use in the study of phase behaviour and condensed phase properties such as elastic constants, viscosities, molecular diffusion and reorientational motion with maximum specificity to real systems. Another role of ab initio computer simulation lies in its interaction... 

R y z Density [kg/m3] Specific surface area [m2/g] Elasticity constant cu [MPa]... 

Kieffer has estimated the heat capacity of a large number of minerals from readily available data [8], The model, which may be used for many kinds of materials, consists of three parts. There are three acoustic branches whose maximum cut-off frequencies are determined from speed of sound data or from elastic constants. The corresponding heat capacity contributions are calculated using a modified Debye model where dispersion is taken into account. High-frequency optic modes are determined from specific localized internal vibrations (Si-O, C-0 and O-H stretches in different groups of atoms) as observed by IR and Raman spectroscopy. The heat capacity contributions are here calculated using the Einstein model. The remaining modes are ascribed to an optic continuum, where the density of states is constant in an interval from vl to vp and where the frequency limits Vy and Vp are estimated from Raman and IR spectra. 

Specific results are calculated for SiC fiber-glass matrix composites with the elastic constants given in Table 4.1. A constant embedded fiber length L = 2.0 mm, and constant radii a = 0.2 mm and B = 2.0 mm are considered with varying matrix radius b. The stress distributions along the axial direction shown in Fig. 4.31 are predicted based on micromechanics analysis, which are essentially similar to those obtained by FE analysis for the two extremes of fiber volume fraction, V[, shown in Fig. 4.32. The corresponding FAS distribution calculated based on Eqs. (4.90) and (4.120), and IFSS at the fiber-matrix interface of Eqs. (4.93) and (4.132) are plotted along the axial direction in Fig. 4.32. 

The position of Ti and Zr is again important in this context. While the b.c.c. phase in these elements has long been known to indicate mechanical instability at 0 K, detailed calculations for Ti (Petty 1991) and Zr (Ho and Harmon 1990) show tiiat it is stabilised at high temperatures by additional entropy contributions arising from low values of the elastic constants (soft modes) in specific crystal directions. This concept had already been raised in a qualitative way by Zener (1967), but the... 

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. 

Furthermore, in gels the elastic moduli K and p treated so far, those of the so-called skeletal frame in Biot s theory, are much smaller than the bulk moduli of fluid and polymer. Note that K accompanies no changes in the total volume of network + solvent, whereas K, p, and pfcl involve them and are much larger than K. From this fact, without changing the essential physics, we assume that the solvent and polymer have the same constant specific volume, so that... 

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],... 

Melting point Specific heat Hardness Bulk modulus Elastic constants... 

Two of the more direct techniques used in the study of lattice dynamics of crystals have been the scattering of neutrons and of x-rays from crystals. In addition, the phonon vibrational spectrum can be inferred from careful analysis of measurements of specific heat and elastic constants. In studies of Bragg reflection of x-rays (which involves no loss of energy to the lattice), it was found that temperature has a strong influence on the intensity of the reflected lines. The intensity of the scattered x-rays as a function of temperature can be expressed by I (T) = IQ e"2Tr(r) where 2W(T) is called the Debye-Waller factor. Similarly in the Mossbauer effect, gamma rays are emitted or absorbed without loss of energy and without change in the quantum state of the lattice by... 

The three elastic constants are the Frank elastic constants, called after Frank, who introduced them already in 1958. They originate from the deformation of the director field as shown in Fig. 15.52. A continuous small deformation of an oriented material can be distinguished into three basis distortions splay, twist and bend distortions They are required to describe the resistance offered by the nematic phase to orientational distortions. As an example, values for Miesowicz viscosities and Frank elastic constants are presented in Table 15.10. It should be mentioned that those material constants are not known for many LCs and LCPs. Nevertheless, they have to be substituted in specific rheological constitutive equations in order to describe the rheological peculiarities of LCPs. Accordingly, the viscosity and the dynamic moduli will be functions of the Miesowicz viscosities and/or the Frank elastic constants. Several theories have been presented that are more or less able to explain the rheological peculiarities. Well-known are the Leslie-Ericksen theory and the Larson-Doi theory. It is far beyond the scope of this book to go into detail of these theories. The reader is referred to, e.g. Aciemo and Collyer (General References, 1996). 

With elastically anisotropic materials the elastic behavior varies with the crystallographic axes. The elastic properties of these materials are completely characterized only by the specification of several elastic constants. For example, it can be seen from Table 10.3 that for a cubic monocrystal, the highest symmetry class, there are three independent elastic-stiffness constants, namely, Cn, C12, and C44. By contrast, polycrystalline aggregates, with random or perfectly disordered crystallite orientation and amorphous solids, are elastically isotropic, as a whole, and only two independent elastic-stiffness coefficients, C44 and C12, need be specified to fully describe their elastic response. In other words, the fourth-order elastic modulus tensor for an isotropic body has only two independent constants. These are often referred to as the Lame constants, /r and A, named after French mathematician Gabriel Lame (1795-1870) ... 

The ionic conductivity of fluorite-structured halides such as SrF2, CaF, BaF, (3- Pb F 2, and SrCh, shows a rapid but continuous increase on heating, reaching values very dose to those shown by the liquid state [67-69]. The transition to the superionic state is also associated with peak in the specific heat Cp (the maximum value of which is generally taken to define the superionic transition temperature, f), anomalous behavior ofthe elastic constants and changes in the lattice expansivity (for details and references, see Refs. [38, 70]). 

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