ELASTIC CONSTANT VARIATIONS

Table 1. Phase transitions in minerals for which elastic constant variations should conform to solutions of a Landau free-energy expansion (after Carpenter and Salje 1998).

Here Qm and Q can be separate order parameters or order parameter components. In combination, Equations (4) and (5) provide a means of predicting the elastic constant variations associated with any phase transition in which the relaxation of Q in response to an applied stress is rapid relative to the time scale of the experimental measurement. A classic example of the success that this approach can have for describing the elastic behaviour of real materials is provided by the work of Errandonea (1980) for the orthorhombic monoclinic transition in LaPsOi4 (Fig. 4). 

Expressions for the elastic constant variations due to the tetragonal orthorhombic transition in stishovite derived by applying Equation (5) to Equation (23) are listed in Table 4. The most important factors in determining the form of evolution of the elastic constants are, self-evidently, the strength of coupling between the order parameter and strain ( ii, ) the order parameter susceptibility, x given by... 

Values for most of the coefficients in Equation (23) were extracted from experimental data. Values were assigned to 4 and ke arbitrarily in the absence of the relevant experimental observations. The bare elastic constants were given a linear pressure dependence based on the variations calculated by Karki et al. (1997a). When experimental data become available, a comparison between observed and predicted elastic constant variations will provide a stringent test for the model of this phase transition as represented by Equation (23). 

The form of the elastic constant variations of quartz can be predicted from Equation (28) in the same way, and the resulting expressions are listed in Table 5 (from Carpenter et al. 1998b). An important difference between stishovite and quartz is that, in the latter, there is no bilinear coupling of the form keQ. As a consequence, all the deviations of the elastic constants from their bare values depend explicitly on Q. This means that there should be no deviations associated with the phase transition as the transition point is... 

Table 5. Expressions for the elastic constant variations in quartz due to the P <-> a transition (from Carpenter et al. 1998b).

Carpenter MA, Salje EKH, Graeme-Barber A (1998a) Spontaneous strain as a determinant of thermodynamic properties for phase transitions in minerals. Eur J Mineral 10 621-691 Carpenter MA, Salje EKH, Graeme-Barber A, Wrack B, Dove MT, Knight KS (1998b) Calibration of excess thermodynamic properties and elastic constant variations due to the a p phase transition in quartz. Am Mineral 83 2-22... 

Anisotropy (elastic-constant variations with direction) can be determined on a single specimen with the pulse-echo method. 

The liquid crystal parameters K2, K3/Kx> na and cell design parameters 3g OCg, d and the orientation of polarizers were varied one at a time. The ratio K3/KX was chosen as a parameter in order to keep the Freedericksz transition threshold voltage constant while varying the elastic constants. Variations in the threshold voltage due to the natural cholesteric pitch were not compensated for. Several calculations were also conducted for the case of infinite cholesteric pitch (no cholesteric compound in the liquid crystal mixture). 

Fig. 3.24 shows the variation of these elastic constants for all values of 6 between 0 and 90°. 

The variation in wall thickness and the development of cell wall rigidity (stiffness) with time have significant consequences when considering the flow sensitivity of biomaterials in suspension. For an elastic material, stiffness can be characterised by an elastic constant, for example, by Young s modulus of elasticity (E) or shear modulus of elasticity (G). For a material that obeys Hooke s law,for example, a simple linear relationship exists between stress, , and strain, a, and the ratio of the two uniquely determines the value of the Young s modulus of the material. Furthermore, the (strain) energy associated with elastic de-... 

Sumino Y, Nishizawa O., Goto T, Ohno I., and Ozima I. (1977). Temperatnre variation of elastic constants of single crystal forsterite between 190 and 400°C. J. Phys. Earth, 28 273-280. 

However, if there are large variations in the elastic constants with pressure, this can seriously affect Pent by as much as an order of magnitude (Lam et al. 1984). 

The problem of definition of modulus applies to all tests. However there is a second problem which applies to those tests where the state of stress (or strain) is not uniform across the material cross-section during the test (i.e. to all beam tests and all torsion tests - except those for thin walled cylinders). In the derivation of the equations to determine moduli it is assumed that the relation between stress and strain is the same everywhere, this is no longer true for a non-linear material. In the beam test one half of the beam is in tension and one half in compression with maximum strains on the surfaces, so that there will be different relations between stress and strain depending on the distance from the neutral plane. For the torsion experiments the strain is zero at the centre of the specimen and increases toward the outside, thus there will be different torque-shear modulus relations for each thin cylindrical shell. Unless the precise variation of all the elastic constants with strain is known it will not be possible to obtain reliable values from beam tests or torsion tests (except for thin walled cylinders). 

For an approximate quantitative comparison of our theoretical results with the experiments on lyotropic liquid crystals we make a number of assumptions about the material parameters. As we have shown in Sect. 3.2 the different approaches cause only small variations in the critical wave number. For this estimate it suffices to use the critical wave number obtained in our earlier work [42], For lyotropics it is known [56, 57], that the elastic constants can be expressed as... 

Now we consider the valence angle bending force field as it appears from the DMM picture. For this end the geometry variation given by the vectors eq. (3.139) must be inserted in eq. (3.72) and the required elasticity constant can be obtained by extracting the second order contribution in vectors 6[Pg.260]

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