Tensor of elasticity

In nematic liquid crystals we see a novel feamre. There is no shear modulus as in isotropic liquids, but the orientational, for example, torsional elasticity appears. Such elasticity is also characteristic of crystals but, in that case, the corresponding moduli are much smaller than the other moduli. The orientational elasticity determines almost all fascinating properties and applications of nematics. 

Other liquid crystal phases combine many types of elasticity of solid crystals with orientational elasticity and we encounter enormous variations of the mag-nimde of elasticity moduli from almost zero to that typical of three-dimensional crystals. 

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This result was obtained by Persson [266] before the more general equation for nonsteady state was obtained in Ref. 267. An example for the connection between the loss part of the elastic modulus and the friction coefficient is shown in Fig. 22. We also want to reemphasize that (q, to) should not be taken literally as e bulk modulus or a coefficient of the tensor of elastic constants but as a more generalized expression, which is discussed in detail in Ref. 266. For a more detailed presentation that also includes contact mechanics and that allows one to calculate the friction coefficient, we refer to the original literature [267]. 

The veracity of this expression can be shown by running the divergence theorem in reverse, basically turning the argument given above to prove the principle of minimum potential energy around. This result in conjimction with the application of symmetries of the tensor of elastic moduli allows us to note... 

The coefficent array C,)H relates the stress tensor to the strain tensor and is itself a tensor (called the tensor of elasticity). Fortunately for a homogeneous material the 81 terms in the tensor of elasticity reduce through symmetry considerations to only three,... 

Glinchuk, M.D., Skorokhod, V.V., Eliseev, E.A., Khist, V, Zaulychny, B.Y. Surface influence on the tensor of elasticity moduli. Reports of National Academy of Sciences of Ukraine N 12, 72-78 (2011) (in Russian)... 

The tensor of elastic compliancy S T) = C T), and die elastic constant matrix (ignoring the terms due to electron-rotation interaction) is equal to ... 

Hruska K (1962) An attempt at a phenomenological interpretation of the influence of a polarizing field on piezoelectric resonators. Czech J Phys B 12 338 Hruska K, Kazda V (1968) The polarizing tensor of elastic coefficients and moduli of alpha-quartz. Czech J Phys B18 500... 

Deformation of a solid object is given by the well-known strain tensor of elasticity theory. Let the point originally at location x be displaced by a small amount The strain tensor uy is then given by... 

For an isotropic material, the number of independent constants reduces to two and the tensor of elasticity has the form... 

The tensor of elasticity for a transversely isotropic (or hexagonal) medium with the z-axis as the axis of symmetry has five independent elements ... 

FIGURE 6. 35 Hudson s model Calculated elements of the tensor of elasticity as a function of porosity (water-saturated rock). Input parameters are fm the solid material (calcite) As=54 GPa, /ts = 31 GPa, and for the fluid kg = 2.2 GPa the curve parameter is the aspect ratio a. The upper figure shows cu and C33 controlling compressional wave velocities in the main axis direction the lower figure shows C44 and Cgs ermtroUing shear wave velocities in the main axis direction (note is not influenced by the low fiacture pOTosity). Also plotted is the component C13 nec-... 

Hudson s model Calculation of the components of the tensor of elasticity based on the assumption of a VTI medium (horizontal cracks). Additional input parameter is the aspect ratio. 

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