Elastic tensor

The mechanical behavior of TiB2 is characterized by its lattice parameters, valence electron density, elasticity tensor, plasmon tensor, and its heat of... 

Chemical hardness is an energy parameter that measures the stabilities of molecules—atoms (Pearson, 1997).This is fine for measuring molecular stability, but energy alone is inadequate for solids because they have two types of stability size and shape. The elastic bulk modulus measures the size stability, while the elastic shear modulus measures the shape stability. The less symmetric solids require the full set of elastic tensor coefficients to describe their stabilities. Therefore, solid structures of high symmetry require at least two parameters to describe their stability. 

Fig. 11. Tensor-valued elasticity parameters in a human breast in vivo. A dotted circle symbolizes a carcinoma previously localized using gadolinium-enhanced Ti-weighted imaging. Eigenvalues Ei, E2, and E3 of the elasticity tensor are shown in (a), (b), and (c) respectively. Also shown in (d) is the isotropic elasticity...

Using relations (2.5) and (2.6) we can determine the elasticity tensor which describes the linear relation between components of the stress and strain tensors. 2 slr.ss = CEstta n is therefore an expression of Hooke s law for anisotropic crystals... 

Example 15.2-2 Determine the non-zero elements of the elasticity tensor ciJki for a crystal of D4 symmetry. The generalized form of Hooke s law is... 

Cowin, S.C. 1985. The relationship between the elasticity tensor and the fabric tensor. Mech. Mater. 4, 137-147. 

Here, V denotes specific volume, K denotes bulk modulus, subscripts P,V, S and T denote isobaric, isochoric, isentopic and isothermal conditions, respectively s is the second-rank strain tensor, and C is the fourth-rank elastic tensor. 

The isotropic moduli, particularly the initial bulk modulus and its pressure derivative, are key ingredients in specifying the mechanical equation of state. As noted above, determination of these properties from experimental hydrostatic compression data is difficult due to issues with acquisition of high precision at low pressures and particular sensitivity in the choice of equation of state fitting form to data below about one GPa. Alternative routes to this information at low pressures included impulsive stimulated light scattering (ISLS) and resonant ultrasound spectroscopy (RUS), which can in principle provide the complete elastic tensor (ISLS) and isotropic bulk and shear moduli (RUS). 

Elastic Tensor, Volume Fluctuations, and Isotropic Moduli. Rahman and Parrinello [83] showed that the fourth-rank elastic tensor for an anisotropic crystalline solid can be calculated using fluctuations of the microscopic strain tensor ... 

Given the elastic tensor we can obtain Reuss average, isotropic bulk and shear moduli... 

The calculated isothermal elastic tensor for yS-HMX is compared in Table 8 to the one reported by Zaug (isentropic conditions). Uncertainties in the calculated elastic coefficients represent one standard deviation in values predicted from five contiguous two nanosecond simulation sequences from the overall ten nanosecond simulation. As mentioned above, Zaug s experiments sufficed to determine uniquely five of the thirteen elastic constants (modulo the... 

Room temperature elastic tensors for a- and [Pg.318]

These simplifications reduce the size of the elasticity tensors from [9 x 9] to [6 x 6], with 36 elastic coefficients. The shorthand notation normally used for the elasticity tensors are now introduced, namely, that the subscripts become 1 11 2 22 3 33 4 23, 32 5 31, 13 and 6 12, 21. With this change, the elastic stiffness tensor may be written in matrix form as ... 

It is important to use the exact strain tensor definition, Eq. (6), to achieve rotational invariance with respect to lattice rotation the conventional linear strain tensor only provides differential rotational invariance of u in Eq. (7).hierarchy of approximations may be used for the elastic tensor 7. The most rigorous approach is to transform the bulk elastic tensor c according to... 

PC Chou, NJ Pagano. Elasticity Tensor, Dyadic, and Engineering Approaches. New York Dover, 1992, Chap 4. 

We have learned already that any elastic oscillation can be represented as a superposition of the compressional and shear waves, which correspond to the potential and solenoidal parts of the elastic displacement field. Therefore, it is clear that the elastic tensor G can also be represented as the sum of the potential and solenoidal components, described by tensor functions G W and G respectively ... 

In general, in order to obtain the full elasticity tensor from Brillouin measurements in a DAC, several incident and scattered beams must be used, and proper access ports must be machined into the diamond seats. In addition, the equation of state and the variation of the refractive index with pressure are necessary. 

A more elaborate method allows the full elasticity tensor of cubic crystals to be determined, with arbitrary orientation, in a DAC. This requires measurements at several angles from the cell s axis over a large angular span (180°) and it therefore necessitates large (>90°) optical apertures. A fit of the observed velocities allows both the single-crystal orientations and the full set of elastic coefficients to be determined. 

The coupling of the mechanical field with the chemical field is realized as follows As there are bound charges present in the gel, a jump in the concentrations of the mobile ions at the interface between the gel and the solution is obtained. This difference in the concentrations leads to an osmotic pressure difference Ati between gel and solution. As a consequence of this pressure difference, the gel takes up solvent, which leads to a change of the swelling of the gel. This deformation is described by the prescribed strain e. This means that the mechanical stress is obtained by the product of the elasticity tensor C and the difference of the total (geometrical) strain e and the prescribed strain ... 

---