Elastic Coefficient Tensor

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. 

Generally, the elastic properties of crystals should be described by 36 elasticity constants Cit but usually a proportion of them are equal to zero or are interrelated. It follows that in crystals, the tensors (2.6) and (2.7) are symmetric tensors, owing to which the number of elastic compliance coefficients is reduced, e.g., in the triclinic configuration, from 36 to 21 (Table 2.1). With increasing symmetry, the number of independent co-... 

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]

Because stress and strain are vectors (first-rank tensors), the forms of Eqs. 10.5 and 10.6 state that the elastic constants that relate stress to strain must be fourth-rank tensors. In general, an wth-rank tensor property in p dimensional space requires p" coefficients. Thus, the elastic stiffness constant is comprised of 81 (3 ) elastic stiffness coefficients,... 

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 ... 

Unlike stress and strain, which are field tensors, elasticity is a matter tensor. It is subject to Neumann s principle. Hence, the number of independent elastic coefficients is further reduced by the crystal symmetry. The proof is beyond the scope of this book (the interested reader is referred to Nye, 1957), here the results will merely be presented. For example, even with triclinic crystals, the lowest symmetry class, there are only 21 independent elastic-stiffness coefficients ... 

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) ... 

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. 

Isotropy of piezoelectric ceramics is destroyed during poling process but remains in the direction perpendicular to the poling field direction. Stractuie of tensor material coefficients of oomm cylindrical polar symmetiy is the same as for the hexagonal 6mm symmetry for dielectric, piezoelectric and elastic tensors. 

Book content is otganized in seven chapters and one Appendix. Chapter 1 is devoted to the fnndamental principles of piezoelectricity and its application including related histoiy of phenomenon discoveiy. A brief description of crystallography and tensor analysis needed for the piezoelectricity forms the content of Chap. 2. Covariant and contravariant formulation of tensor analysis is omitted in the new edition with respect to the old one. Chapter 3 is focused on the definition and basic properties of linear elastic properties of solids. Necessary thermodynamic description of piezoelectricity, definition of coupled field material coefficients and linear constitutive equations are discussed in Chap. 4. Piezoelectricity and its properties, tensor coefficients and their difierent possibilities, ferroelectricity, ferroics and physical models of it are given in Chap. 5. Chapter 6. is substantially enlarged in this new edition and it is focused especially on non-linear phenomena in electroelasticity. Chapter 7. has been also enlarged due to mary new materials and their properties which appeared since the last book edition in 1980. This chapter includes lot of helpful tables with the material data for the most today s applied materials. Finally, Appendix contains material tensor tables for the electromechanical coefficients listed in matrix form for reader s easy use and convenience. 

Here we demonstrate in two examples how symmetry considerations help eliminate coupling constants. The first example describes Frank s considerations in determining the relevant elastic constants of nematic liquid crystals, and the second example will show us how many piezoelectric coupling constants are possible in systems with different symmetries. In both cases the nonvanishing coefficients are determined by the symmetry of the material and using the Curie principle, which states that tensor coefficients characterizing material properties should be invariant under the symmetry transformations of the substance. ... 

Ctjki is a fourth order tensor that linearly relates a and e. It is sometimes called the elastic rigidity tensor and contains 81 elements that completely describe the elastic characteristics of the medium. Because of the symmetry of a and e, only 36 elements of Cyu are independent in general cases. Moreover only 2 independent rigidity constants are present in Cyti for linear homogeneous isotropic purely elastic medium Lame coefficient A and /r have a stress dimension, A is related to longitudinal strain and n to shear strain. For the purpose of clarity, a condensed notation is often used... 

In this equation is the deviator and a is the spherical part of the stress tensor <7, eij is the strain deviator and e the volumetric part of the strain tensor ij, K = (2M + 3A) /3 is bulk modulus with M and A corresponding to the familiar Lame coefficients in the theory of elasticity, while r) and n can be termed the viscous shear and bulk moduli. 

Not all the tensor components are independent. Between Eqs (6.29a) and (6.29b) there are 45 independent tensor components, 21 for the elastic compliance sE, six for the permittivity sx and 18 for the piezoelectric coefficient d. Fortunately crystal symmetry and the choice of reference axes reduces the number even further. Here the discussion is restricted to poled polycrystalline ceramics, which have oo-fold symmetry in a plane normal to the poling direction. The symmetry of a poled ceramic is therefore described as oomm, which is equivalent to 6mm in the hexagonal symmetry system. 

S]). The direct piezoelectric effect is the production of electric displacement by the application of a mechanical stress the converse piezoelectric effect results in the production of a strain when an electric field is applied to a piezoelectric crystal. The relation between stress and strain, expressed by Equation 2.7, is indicated by the term Elasticity. Numbers in square brackets show the ranks of the crystal property tensors the piezoelectric coefficients are 3rd-rank tensors, and the elastic stiffnesses are 4th-rank tensors. Numbers in parentheses identify Ist-rank tensors (vectors, such as electric field and electric displacement), and 2nd-rai tensors (stress and strain). Note that one could expand this representation to include thermal variables (see [5]) and magnetic variables. 

Wall thickness Channel width Acoustic velocity Friction coefficient Conductance Capillary number Discharge coefficient Drag coefficient Diameter Diameter Dean number Deformation rate tensor components Elastic modulus Energy dissipation rate Eotvos number Fanning friction factor Vortex shedding frequency Force... 

The first two terms in this expression for the stress tensor are elastic terms due to Brownian motion and the nematic potential, respectively. The last term is a purely viscous term produced by the drag of solvent as it flows past the rod-like molecules [see Eq. (6-36)]. is a drag coefficient, which for modestly concentrated solutions is predicted to follow the dilnte-solution formula (Doi and Edwards 1986 see Section 6.3.1.4) ... 

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]. 

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