Relation between the elastic constants

In section 2.4.3, we introduced equation (2.23), C44 = (Cn—Ci2)/2, specifying the relation between the components Cn, C12, and C44 of the elasticity matrix (Cap) of an isotropic material. Check this equation by prescribing a strain tensor 

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In the case of FCC transition and noble metals one has to introduce non central Interactions. In fact for central forces should hold the Cauchy relation between the elastic constants c... 

As a consequence of the complexity of molecular interactions in the nematic phase, there are no general quantitative relations between the elastic constants and molecular structure. However, several general rules have been extracted empirically from a large... 

The relations between the elastic constants and the velocities of longitudinal and transverse ultrasonic waves in composites with a uni-directional alignement of the fibres and in materials having a hexagonal symmetry are given by [4,6] ... 

Table 1 shows relations between the elastic constants of materials of a hexagonal symmetry and the velocity,direction of propagation and plane of polarisation of the ultrasonic waves. The relations between the material constants (Young"s modulus, stiffness modulus and Poisson number) and the elastic constants, derived from the equations quoted in Table 1 are as follows ... 

It is possible to derive relations between the elastic constants and the atomic force constants for crystals with several atoms per unit cell. For a general crystal structure, the expression corresponding to (3.128) is then quite complicated [3.20]. However, if every atom is at a center of inversion, the relation (3.128) still applies but the quantities C, defined in (3.116) are now replaced by the more general expression... 

Hint Use the two-suffix notation and the relations between the elastic constants for cubic crystals discussed in Sect.3.6. 

In Sect.3.6, we established relations between the elastic constants and the atomic force constants (3.116,128,137). In Sect.5.2.2, we obtained the result that due to anharmonicity, the atomic force constants depend on temperature and pressure, (5.54,55). We therefore expect that the elastic con-... 

For isotropic materials, certain relations between the engineering constants must be satisfied. For example, the shear modulus is defined in terms of the elastic modulus, E, and Poisson s ratio, v, as... 

When p is in units of g cm and is in units of cm sec , the shear modulus has units of dynes cm (10 GPa= Idynecm" ). Through relations between isotropic elastic constants the longitudinal velocity (ve) gives directly the following combination of bulk modulus (K) and shear modulus (G) ... 

Anomalies in the phonon spectra of UTe have also been found (Buyers and Holden 1985) and attempts were made by the authors to use the same theories described above, which essentially relate to an intermediate-valence picture. Unfortunately, as Buyers and Holden describe (see their p. 304) this does not lead to a convincing conclusion. There is, of course, no direct electronic evidence (e.g., from photoemission) that materials such as UTe exhibit valence fluctuations. We must conclude that the electron-phonon interaction is different in detail between the 4f intermediate-valence matmals and tiie actinide compounds. In both cases, however, the result is a negative Poisson ratio (which is directly related to the elastic constant C12). 

Table 2.1 Relation between various elastic constants. X and G are often termed Lame constants and K is the bulk modulus.

Within the elastic regime, the conservation relations for shock profiles can be directly applied to the loading pulse, and for most solids, positive curvature to the stress volume will lead to the increase in shock speed required to propagate a shock. The resulting stress-volume relations determined for elastic solids can be used to determine higher-order elastic constants. The division between the elastic and elastic-plastic regimes is ideally marked by the Hugoniot elastic limit of the solid. 

The fundamental quantities in elasticity are second-order tensors, or dyadicx the deformation is represented by the strain thudte. and the internal forces are represented by Ihe stress dyadic. The physical constitution of the defurmuble body determines ihe relation between the strain dyadic and the stress dyadic, which relation is. in the infinitesimal theory, assumed lo be linear and homogeneous. While for anisotropic bodies this relation may involve as much as 21 independent constants, in the euse of isotropic bodies, the number of elastic constants is reduced lo two. 

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). 

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. 

The theoretical description in terms of spherical harmonics also yields a relation between the size polydispersity index p of the microemulsion droplets and the bending elastic constants [43]. The quantity p is accessible by SANS [51, 52, 59-61]. For polydisperse shells as obtained by using deuterated oil and heavy water for the preparation of the microemulsion (contrast variation), one can account for the droplet polydispersity by applying an appropriate form factor, e.g. containing a Gaussian function to model the size distribution [52, 59, 62]. A possible often-used choice is the following form factor... 

All these results apply to a completely general triclinic crystal system whose elastic properties are expressed by the twenty-one independent quantities Cy or Jy. For crystals of higher symmetry there are further relations between the Cy or 5y which reduce their number still further. For the hexagonal and cubic systems these relations are illustrated in fig. 8.1, together with similar relations for a completely isotropic, non-crystalline material. It can be seen that for a hexagonal crystal like ice there are only five non-zero independent elastic constants Jn, i3> % and 44 or the corresponding Cy. 

The domain flow theory of LCP assumes a balance between the alignment tendency under a velocity field and the elastic resistance to deformation of the director field [Marrucci, 1984]. The average value of the Eriksen distortion stress, as, was taken as proportional to the elastic constant, K, and inversely proportional to the domain size. The flow behavior should depend on the local orientation for high velocity in the region where the orientation director and velocity vector are parallel to each other, with low velocity for the opposite direction. As a result, the relation between the stress and the deformation rate might be scaled by the domain size ... 

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