Isotropic solids

In Chapter III, surface free energy and surface stress were treated as equivalent, and both were discussed in terms of the energy to form unit additional surface. It is now desirable to consider an independent, more mechanical definition of surface stress. If a surface is cut by a plane normal to it, then, in order that the atoms on either side of the cut remain in equilibrium, it will be necessary to apply some external force to them. The total such force per unit length is the surface stress, and half the sum of the two surface stresses along mutually perpendicular cuts is equal to the surface tension. (Similarly, one-third of the sum of the three principal stresses in the body of a liquid is equal to its hydrostatic pressure.) In the case of a liquid or isotropic solid the two surface stresses are equal, but for a nonisotropic solid or crystal, this will not be true. In such a case the partial surface stresses or stretching tensions may be denoted as Ti and T2-... 

Generally, Oijki depends on x. The isotropic solid is characterized by the constant coefficients Oijki of the form... 

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. 

The mechanism of conduction is most easily understood by the study of conduction through homogeneous isotropic solids, because in this case convection is not present. As a simple illustration of heat transfer by conduction, let a flat parallel-sided plate of a uniform solid material, whose flat faces are maintained at temperatures Tt and T2 respectively (Tj > T2) be considered (Figure 3.15). Heat would be transferred from the face at the higher temperature (Tj) to that at the lower temperature (T2). Let the rate of this transfer be dQjdt, and the area of the plate perpendicular to the direction of heat flow be S. If L is the plate thickness, then it is found that dQ/dt is proportional to (Tt - T2) S/L. In other words,... 

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. 

Solid bertoid Predominantly Isotropic -solid, massive particle with no vesiculation, coal precursor material not recognisable, high reflectance solid, massive particle with no vesiculation. 

Since polymers are viscoelastic solids, combinations of these models are used to demonstrate the deformation resulting from the application of stress to an isotropic solid polymer. Maxwell joined the two models in series to explain the mechanical properties of pitch and tar (Figure 14.2a). He assumed that the contributions of both the spring and dashpot to strain were additive and that the application of stress would cause an instantaneous elongation of the spring, followed by a slow response of the piston in the dashpot. Thus, the relaxation time (t), when the stress and elongation have reached equilibrium, is equal to rj/G. 

The propagation of linear acoustic waves in solids depends on two laws discovered by two of the most illustrious physicists of the seventeenth century, one from Cambridge and the other from Oxford. Consider a volume element of an isotropic solid subjected to shear, as shown in Fig. 6.1. If the displacement in the transverse direction is , and the component of shear stress in that direction is os, then Newton s second law may be written... 

The simple method just described is applicable as it stands only to isotropic solids, that is, to glasses and amorphous solids in general, and to crystals belonging to the cubic system. In all other crystals the refractive index varies with the direction of vibration of the light in the crystal the optical phenomena are more complex, and it is necessary to disentangle them. 

If now we immerse the crystals in various liquids, and observe each crystal in light vibrating parallel to its fourfold axis, we observe consistent effects as in the case of isotropic solids in ordinary light, and we find the refractive index is 1 479. If we use light vibrating perpendicular to the fourfold axis of the crystal, we again observe consistent effects and this time find the refractive index to be 1 525 (Fig. 44 a and 6). 

According to Eq. (20) the compression of isotropic solid polymers having positive thermal expansivity must be accompanied by the internal energy inversion. AU inversion at compression has been estimated71 to occur at strains 5-15%. At compression, irreversible plastic deformations occur which prevents a correct experimental determination of AU. With inversion parameters, AU for isotropic poly-... 

The equation so far given are of very general validity. They also hold for an isotropic solid, on which a shear stress p21 is applied. 

This equation relates sM to the orientation of the stress-ellipsoid [cf. eq. (1.3)]. This result is first quoted by Lodge. It differs by a factor of one half from that for the (completely recoverable) simple shear s of a perfectly elastic isotropic solid (50) ... 

THERMAL EXPANSION COEFFICIENT. The change in volume per unit volume per degree change in temperature (cubical coefficient). For isotropic solids the expansion is equal in all directions, and the cubical coefficient is about three times the linear coefficient of expansion. These coefficients vary with temperature, but for gases at constant pressure the coefficient of volume expansion is nearly constant and equals 0.00367 for each degree Celsius at any temperature. 

In line with these solution-phase 13C NMR results, the isotropic solid-state 13C NMR chemical shifts of the cage carbons of silatranes appeared to be nearly independent on the Si — N bond length254. 

If the thermal conductivity k and the product pCp are temperature independent, Eq. 5.3-1 reduces for homogeneous and isotropic solids to a linear partial differential equation, greatly simplifying the mathematics of solving the class of heat transfer problems it describes.1... 

Elasticity of solids determines their strain response to stress. Small elastic changes produce proportional, recoverable strains. The coefficient of proportionality is the modulus of elasticity, which varies with the mode of deformation. In axial tension, E is Young s modulus for changes in shape, G is the shear modulus for changes in volume, B is the bulk modulus. For isotropic solids, the three moduli are interrelated by Poisson s ratio, the ratio of traverse to longitudinal strain under axial load. 

In Fig. XIV-2 we have a representation of the frequencies of vibration of an elastic solid, under the assumption that the waves are propagated as in an isotropic solid the velocity of propagation being independent... 

The calculation which we have carried out in this section has been limited in accuracy by our assumption that the velocity of propagation of the clastic waves was independent of direction and of wave length. Actually neither of these assumptions is correct for a crystal. Even for a cubic crystal, the elastic properties are more complicated than for an isotropic solid and the velocity of propagation depends on direction. 

In isotropic solids one can distinguish longitudinal waves and shear waves. When the lateral dimensions are much less than the wavelength (threads) an extensional wave is propagated. 

The measurement of the work needed to increase the surface area of a solid material (e.g., an electrode metal) is more difficult. The work required to form unit area of new surface by stretching under equilibrium conditions is the surface stress (g1 ) which is a tensor because it is generally anisotropic. For an isotropic solid the work, the generalized surface parameter , or specific surface energy (ys) is the sum of two contributions ... 

Figure 10 Examples of the analysis of long-range cross peaks in (A) isotropic and (B) aligned sample of Me-/ -D-xylopyranoside. Short-dashed and solids lines show anti-phase (AP) and in-phase (IP) multiplets. Long-dashed singlets were obtained by the addition and subtraction of the IP and AP multiplets using appropriate scaling factors. (C) Overlay of two one-bond cross peaks from isotropic (solid line) and aligned (dashed line) samples. The multiplets were shifted to overlay on one line of the doublet in order to accentuate the difference between J and D.

The stress tensor a in the perfectly elastic and isotropic solid phase of the porous medium is described by the constitutive equation... 

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