Anisotropic solid

A.J. Niklasson, Ultrasonic 3D Probe Modeling in Anisotropic Solids, Tech. Report 1995 8, Div. Mech., Chalmers University of Technology (1995). 

Gengembre N. and Lhemery A., Transient ultrasonic fields radiated by water-coupled transducers into anisotropic solids. Review of Progress in Quantitative Non Destructive Evaluation (Plenum, New-York, 1998), to appear. 

Shuttleworth [26] (see also Ref. 27) gives a relation between surface free energy and stretching tension as follows. For an anisotropic solid, if the area is increased in two directions by dAi and dA2, as illustrated in Fig. VII-1, then the total increase in free energy is given by the reversible work against the surface stresses, that is. 

For a calculation of d. see R- H. Fowler. Statistical Thermodynamics. Second Edition, Cambridge University Press. 1956. p. 127. In Section 1.5a of Chapter 1 we defined the compressibility and cautioned that this compressibility can be applied rigorously only for gases, liquids, and isotropic solids. For anisotropic solids where the effect of pressure on the volume would not be the same in the three perpendicular directions, more sophisticated relationships are required. Poisson s ratio is the ratio of the strain of the transverse contraction to the strain of the parallel elongation when a rod is stretched by forces applied at the end of the rod in parallel with its axis. 

As in the case of and cv, ap does not differ significantly from av at low temperatures. For anisotropic solids, there are two or three (depending on the symmetry of the crystal) principal linear coefficients. For isotropic solids, the volumetric thermal expansion is j8 = 3a. 

All these advances have resulted not only in increases in resolution but have also alleviated the detection problems to a considerable extent. As a result, the last decade has seen a dramatic growth in 15N- and 170-NMR spectroscopy as a versatile method for studying molecular structure, both in isotropic (liquid) and anisotropic (solid) phases. Studies at a natural abundance level of the nucleides are now commonplace. The scope of chemical applications extends from inorganic, organometallic and organic chemistry to biochemistry and molecular biology, and includes the study of reactive intermediates, biopolymers and enzyme-inhibitor complexes. 

Common liquids are optically isotropic, and the solids that physicists seem to like most are cubic and therefore isotropic. As a consequence, treatments of optical properties, particularly from a microscopic point of view, usually favor isotropic matter. Among the host of naturally occurring sohds, however, most are not isotropic. This somewhat complicates both theory and experiment for example, measurements of optical constants must be made with oriented crystals and polarized light. But because of the prevalence of optically anisotropic solids, we are compelled to extend the classical models to embrace this added complexity. 

More problems must be faced when trying to extract optical constants from measurements on particles of anisotropic solids. Random orientation of the particles averages somehow the two or three sets of optical constants. We... 

In the surface of an anisotropic solid the situation is more complicated. Pure Rayleigh waves can exist only along certain symmetry directions in which pure SV waves exist. Away from these directions, however, the two quasi-shear polarizations are not pure SV and SH therefore, although the particle motions are orthogonal, at the surface they can be weakly coupled. If the SH mode has a higher velocity than the SV, then there can be no real solution to Snell s law... 

Liquid drops on anisotropic solid surfaces will tend to elongate in the higher surface energy direction and the contact angle will, therefore, vary with position. 

Biot, M.A. (1955) Theory of Elasticity and consolidation of a porous anisotropic solid. J. Appl. Phys. 26, 182-185... 

The majority of pure compounds can be obtained in crystalline form, although the individual specimens may often be very small or imperfectly formed. A well-developed crystal takes the shape of a polyhedron with planar faces, linear edges, and sharp vertices. In the simplest terms, a crystal may be defined as a homogeneous, anisotropic solid having the natural shape of a polyhedron. 

Nuclear Magnetic Resonance Imaging of Anisotropic Solid-State Chemical Reactions... 

This moment of inertia is essential for the analysis of rotational spectra of molecules. For anisotropic solids or for molecules, the moment of inertia I is a second-rank tensor, with three principal-axis components I, I2, and f3. This moment of inertia is important when the body rotates with angular frequency co radians per second (co FIz), or with v revolutions per second. 

In practice, this integral is performed on a digital computer using a measured spectrum of R versus o>, the frequency of electromagnetic radiation. The index of refraction of anisotropic solids is described by a biaxial in-dicatrix. In triclinic crystals the indicatrix is a triaxial ellipsoid formed by three principal axes of the length 2na, 2np, and 2ny which lie along the X, Y, and Z optical directions, respectively. na, np, and ny are the three principal indices of refraction (na < < ny). For absorbing crystals the... 

This tensor is symmetric and requires a maximum of six independent parameters for low symmetry samples. The number of parameters is reduced for more symmetric samples. In general, polymeric samples will either be isotropic, with one resistivity parameter, or have axial symmetry generated by mechanical processes, see Section 1.3.6, with two resistivity parameters. Montgomery (1971) introduced a more general 4-point probe method suitable for the measurement of anisotropic solids. This method utilises a sample in the form of a rectangular prism with electrodes attached at the corners, illustrated in Fig. 5.19(c). Measurement of the voltage/current ratios for opposite pairs of electrodes,... 

Butler, L. G., Cory, D. G., Dooley, K. M., Miller, J. B. and Garroway, A. N. (1992). NMR imaging of anisotropic solid-state chemical-reactions using multiple-pulse line-narrowing techniques and H-1 T1-weighting. J. Am. Chem. Soc., 114, 125-35. [238]... 

Table 1.13 Maxwell relations for anisotropic solids constant V...

We wish to study the effects of planar Couette flow on a system that is in the NPT (fully flexible box) ensemble. In this section, we consider the effects of the external field alone on the dynamics of the cell. The intrinsic cell dynamics arising out of the internal stress is assumed implicitly. The constant NPT ensemble can be employed in simulations of crystalline materials, so as to perform dynamics consistent with the cell geometry. In this section, we assume that the shear field is applied to anisotropic systems such as liquid crystals, or crystalline polytetrafluoroethylene. For an anisotropic solid, we assume that the shear field is oriented in such a way that different weakly interacting planes of atoms in the solid slide past each other. The methodology presented is quite general hence it is straightforward to apply for simulations of shear flow in liquids in a cubic box, as well. 

Thus one would expect from a (6x6) matrix of the elastic stiffness coefficients (c,y) or compliance coefficients (sy) that there are 36 elastic constants. By the application of thermodynamic equilibrium criteria, cy (or Sjj) matrix can be shown to be symmetrical cy =cji and sy=Sji). Therefore there can be only 21 independent elastic constants for a completely anisotropic solid. These are known as first order elastic constants. For a crystalline material, periodicity brings in elements of symmetry. Therefore symmetry operation on a given crystal must be consistent with the representation of the elastic quantities. Thus for example in a cubic crystal the existence of 3C4 and 4C3 axes makes several of the elastic constants equal to each other or zero (zero when under symmetry operation cy becomes -cy,). As a result, cubic crystal has only three independent elastic constants (cu== C22=C33, C44= css= and Ci2=ci3= C2i=C23=C3i=C32). Cubic Symmetry is the highest that can be attained in a crystalline solid but a glass is even more symmetrical in the sense that it is completely isotropic. Therefore the independent elastic constants reduce further to only two, because C44=( c - C i)l2. 

The thermal expansion of solids depends on their structure symmetry, and may be either isotropic or anisotropic. For example, graphite has a layered structure, and its expansion in the direction perpendicular to the layers is quite different from that in the layers. For isotropic materials, ay w 3 a . However, in anisotropic solid materials the total volume expansion is distributed unequally among the three crystallographic axes and, as a rule, cannot be correctly measured by most dilatometric techniques. The true thermal expansion in such case should be studied using in situ X-ray diffraction (XRD) to determine any temperature dependence of the lattice parameters. 

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