Anisotropy elastic

The states of stress and strain in a deformed crystal being idealized as a continuum are characterized by symmetric second-rank tensors and Cjj, respectively, each comprising six independent components. Hooke s law of linear elasticity for the most general anisotropic solid expresses each component of the stress tensor linearly in terms of all components of the strain tensor in the form 

The tensor form of the constitutive equation in (3.1) provides a concise and effective statement of Hooke s law for use in theoretical developments. For purposes of measurement and calculation, however, it is often more convenient to adopt a matrix form of the constitutive equation. Such a form is 

For crystals with a triclinic structure, which possess only a center of symmetry and exhibit the most general elastic anisotropy, a complete characterization requires all 21 independent constants. The existence of a higher degree of crystal symmetry can further reduce the number of independent elastic constants needed for proper description. In such cases, some elastic constants may vanish and members of some subsets of constants may be related to each other in some definite way, depending on the crystal symmetry. Finally, it should be noted that, while the total number of material constants required to characterize a material of a certain class is independent of the coordinate axes used to represent components of stress or strain, the values of particular components of Qj do depend on the material reference axes chosen for their representation. 

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Clearly, the lower the density the greater is the elastic anisotropy. 

Gerk showed that Equation (2.1) is followed not only for metals, but also for ionic and covalent crystals if two adjustments are made. For covalent crystals, the temperature must be raised to a level where dislocations glide readily, but below the level where they climb readily. For ionic crystals, G (an average shear modulus) must be adjusted for elastic anisotropy. Thus it becomes ... 

It is also possible that a membrane might have an even lower symmetry than a chiral smectic-C liquid crystal in particular, it might lose the twofold rotational symmetry. This would occur if the molecular tilt defines one orientation in the membrane plane and the direction of one-dimensional chains defines another orientation. In that case, the free energy would take a form similar to Eq. (5) but with additional elastic constants favoring curvature. The argument for tubule formation presented above would still apply, but it would become more mathematically complex because of the extra elastic constants. As an approximation, we can suppose that there is one principal direction of elastic anisotropy, with some slight perturbations about the ideal twofold symmetry. In that approximation, we can use the results presented above, with 4) representing the orientation of the principal elastic anisotropy. 

All crystals are anisotropic many other structures also have elastic anisotropy. The propagation of elastic waves in anisotropic media is described by the Christoffel equation. This still depends on Newton s law and Hooke s law, but it is expressed in tensor form so that elastic anisotropy may be included. The tensor description of elastic stiffness was summarized in 6.2, especially eqns (6.23)—(6.29). The Christoffel equation is... 

Hildebrand, J. A. and Lam, L. (1983). Directional acoustic microscopy for observation of elastic anisotropy. Appl. Phys. Lett. 42,413-15. [250]... 

The technical magnetic properties such as Hc and pi are primarily a function of the magnetocrystalline anisotropy constant (K ) of the material. But once the magnetocrystalline anisotropy has been made small, the soft magnetic properties are still limited by magneto-elastic anisotropies due to internal mechanical stress. Hence, materials development has focused on compositions and microstructures where both K and the saturation... 

The condition for isotropic elasticity, as has been seen, is C44 = C55 = Cge = Cn C12). Cubic crystals, because of their high symmetry, almost satisfy this condition. Zener introduced the ratio 2C44/(cn - C12) as an elastic anisotropy factor for cubic crystals (Zener, 1948a). In a cubic crystal, if the Zener ratio is positive, the Young s modulus has a... 

Clearly shrinkage anisotropy is a eomplex issue. A number of faetors ean contribute and the relative importance of each will vary between timbers. In some cases a large microfibril angle might be significant, as in corewood and in compression wood. Ray tissue will be important in species such as beech and oak. Contrasting earlywood and latewood densities is a likely cause in Douglas fir, but would be irrelevant for a tropical hardwood. The effects of elastic anisotropy would be more apparent in low density softwoods. 

Tao, J. Feng, J.J. Effects of elastic anisotropy on the flow and orientation of sheared nematic liquid crystals. J. Rheol. 2003, 47 (4), 1051-1070. 

Elastic isotropy considerably simplifies the analyses that we are forced to undertake in our goal of characterizing the deformation fields associated with a dislocation. On the other hand, there are some instances in which it is desirable to make the extra effort to include the effects of elastic anisotropy. On the other hand, because the present work has already grown well beyond original intentions and because the addition of anisotropy is for the most part an elaboration of the physical ideas already set forth above, we refer the reader to the outstanding work of Bacon et al. (1979). 

Numerical Calculation of Two-Dimensional Equilibrium Shapes. To go beyond the relative simplicity of the set of shapes considered in the analysis presented above it is necessary to resort to numerical procedures. As a preliminary to the numerical results that will be considered below and which are required when facing the full complexity of both arbitrary shape variations and full elastic anisotropy, we note a series of finite-element calculations that have been done (Jog et al. 2000) for a wider class of geometries than those considered by Johnson and Cahn. The analysis presented above was predicated on the ability to extract analytic descriptions of both the interfacial and elastic energies for a restricted class of geometries. More general geometries resist analytic description, and thus the elastic part of the problem (at the very least) must be solved by recourse to numerical methods. 

Using Eq. (26.1) it is possible to obtain the temperature dependence of the hydrogen jump rate from the experimental data on ultrasonic loss. The Snoek relaxation measurements are especially informative if they are performed at a number of excitation frequencies. It should be noted that the Snoek effect can be observed only for sufficient elastic anisotropy, X -X2, of hydrogen sites. For hydrogen in pure b.c.c. metals, the Snoek effect has not been found [16], in spite of the uniaxial symmetry of tetrahedral sites occupied by hydrogen in these materials. It is believed that the absence of the observable Snoek effect is due to the small value of X1-X2 for hydrogen in the tetrahedral sites of b.c.c. metals. 

The driving mechanisms for the island vertical correlation have been the subject of extensive studies over the past years. Because the buried islands produce a nonuniform strain field at the surface of the spacer layer, i.e. the regions above the islands are tensely strained while the regions in between islands remain compressed, exciting models have treated the island distribution at the spacer layer surface by considering the effect of such a strain field on surface diffusion [4] or on island nucleation [3]. Recent calculations have taken into account the effect of the elastic anisotropy of the materials [16], the surface energy [18] or the elastic interaction between the buried islands with newly deposited ones [19]. However, in all of the above models it was assumed that the surface of the spacer layer becomes perfectly flat before the deposition of a new layer. From the experimental point of view, this... 

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