Isotropic, defined

A thin isotropic homogeneous plate is assumed to occupy a bounded domain C with the smooth boundary T. The crack Tc inside 0 is described by a sufficiently smooth function. The chosen direction of the normal n = to Tc defines positive T+ and negative T crack faces. 

A similar effect occurs in highly chiral nematic Hquid crystals. In a narrow temperature range (seldom wider than 1°C) between the chiral nematic phase and the isotropic Hquid phase, up to three phases are stable in which a cubic lattice of defects (where the director is not defined) exist in a compHcated, orientationaHy ordered twisted stmcture (11). Again, the introduction of these defects allows the bulk of the Hquid crystal to adopt a chiral stmcture which is energetically more favorable than both the chiral nematic and isotropic phases. The distance between defects is hundreds of nanometers, so these phases reflect light just as crystals reflect x-rays. They are called the blue phases because the first phases of this type observed reflected light in the blue part of the spectmm. The arrangement of defects possesses body-centered cubic symmetry for one blue phase, simple cubic symmetry for another blue phase, and seems to be amorphous for a third blue phase. 

It would be incomplete for any discussion of soap crystal phase properties to ignore the colloidal aspects of soap and its impact. At room temperature, the soap—water phase diagram suggests that the soap crystals should be surrounded by an isotropic Hquid phase. The colloidal properties are defined by the size, geometry, and interconnectiviness of the soap crystals. Correlations between the coUoid stmcture of the soap bar and the performance of the product are somewhat quaUtative, as there is tittle hard data presented in the literature. However, it might be anticipated that smaller crystals would lead to a softer product. Furthermore, these smaller crystals might also be expected to dissolve more readily, leading to more lather. Translucent and transparent products rely on the formation of extremely small crystals to impart optical clarity. 

Mechanical Properties. The hexagonal symmetry of a graphite crystal causes the elastic properties to be transversely isotropic ia the layer plane only five independent constants are necessary to define the complete set. The self-consistent set of elastic constants given ia Table 2 has been measured ia air at room temperature for highly ordered pyrolytic graphite (20). With the exception of these values are expected to be representative of... 

Quasi-isotropic laminates have the same ia-plane stiffness properties ia all directions (1), which are defined ia terms of the [A] matrix of the laminate. For the laminate to be quasi-isotropic. 

In the case of most nonporous minerals at sufficiently low-shock stresses, two shock fronts form. The first wave is the elastic shock, a finite-amplitude essentially elastic wave as indicated in Fig. 4.11. The amplitude of this shock is often called the Hugoniot elastic limit Phel- This would correspond to state 1 of Fig. 4.10(a). The Hugoniot elastic limit is defined as the maximum stress sustainable by a solid in one-dimensional shock compression without irreversible deformation taking place at the shock front. The particle velocity associated with a Hugoniot elastic limit shock is often measured by observing the free-surface velocity profile as, for example, in Fig. 4.16. In the case of a polycrystalline and/or isotropic material at shock stresses at or below HEL> the lateral compressive stress in a plane perpendicular to the shock front... 

In Raman spectroscopy the intensity of scattered radiation depends not only on the polarizability and concentration of the analyte molecules, but also on the optical properties of the sample and the adjustment of the instrument. Absolute Raman intensities are not, therefore, inherently a very accurate measure of concentration. These intensities are, of course, useful for quantification under well-defined experimental conditions and for well characterized samples otherwise relative intensities should be used instead. Raman bands of the major component, the solvent, or another component of known concentration can be used as internal standards. For isotropic phases, intensity ratios of Raman bands of the analyte and the reference compound depend linearly on the concentration ratio over a wide concentration range and are, therefore, very well-suited for quantification. Changes of temperature and the refractive index of the sample can, however, influence Raman intensities, and the band positions can be shifted by different solvation at higher concentrations or... 

For isotropic materials, certain relations between the engineering constants must be satisfied. For example, the shear modulus is defined in terms of the elastic modulus, E, and Poisson s ratio, v, as... 

In Section 2.2, the stress-strain relations (generalized Hooke s law) for anisotropic and orthotropic as well as isotropic materials are discussed. These relations have two commonly accepted manners of expression compliances and stiffnesses as coefficients (elastic constants) of the stress-strain relations. The most attractive form of the stress-strain relations for orthotropic materials involves the engineering constants described in Section 2.3. The engineering constants are particularly helpful in describing composite material behavior because they are defined by the use of very obvious and simple physical measurements. Restrictions in the form of bounds are derived for the elastic constants in Section 2.4. These restrictions are useful in understanding the unusual behavior of composite materials relative to conventional isotropic materials. Attention is focused in Section 2.5 on stress-strain relations for an orthotropic material under plane stress conditions, the most common use of a composite lamina. These stress-strain relations are transformed in Section 2.6 to coordinate systems that are not aligned with the principal material... 

First, we need a common, unambiguous manner of writing how a laminate is specified to be laid-up, i.e., stacking-sequence notation. Then, quasi-isotropic, balanced, and hybrid laminates are defined. 

For a specially orthotropic square boron-epoxy plate with stiffness ratios 0 /022= 10 and (Di2-t-2D66) = 1, the four lowest frequencies are displayed in Table 5-3 along with the four lowest frequencies of an isotropic plate. There, the factor k is defined as... 

Tsai and Pagano further defined the isotropic stiffness and shear rigidity [7-16] to be... 

EXAMPLE Consider the isotropic peripheral PCA defined in equation 7.69 ... 

EXAMPLES (1) Isotropic Peripheral PGA - consider the isotropic version of the peripheral PCA defined by equations 7.61 and 7.63 i.e. take a-2 — os = 02.3- In this case, the detailed balance condition is satisfied when the 3-spin coupling constant hi23 = 0. From equation 7.96, we see that this condition translates to... 

Phase composition changes as a function of temperature are similar to these that were observed for the lithium-containing system (see Fig. 21), except that individual stable ammonium-cobalt-oxyfluoroniobate occurs prior to the formation of CoNbOF5 (Fig.21, curve 4). It is assumed that the composition of this intermediate phase, formed at 330-350°C, is (NH CoNbOFy [129]. Complete removal of ammonium occurs at about 400°C (Fig. 21, curve 5) and leads to the formation of CoNbOFs. The compound is defined as practically isotropic rose-colored ciystals with a refractive index of N = 1.500. The... 

The wide choice available in plastics makes it necessary to select not only between TPs, TSs, reinforced plastics (RPs), and elastomers, but also between individual materials within each family of plastic types (Chapters 6 and 7). This selection requires having data suitable for making comparisons which, apart from the availability of data, depends on defining and recognizing the relevant plastics behavior characteristics. There can be, for instance, isotropic (homogeneous) plastics and plastics that can have different directional properties that run from the isotropic to anisotropic. Here, as an example, certain... 

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 present review shows how the microhardness technique can be used to elucidate the dependence of a variety of local deformational processes upon polymer texture and morphology. Microhardness is a rather elusive quantity, that is really a combination of other mechanical properties. It is most suitably defined in terms of the pyramid indentation test. Hardness is primarily taken as a measure of the irreversible deformation mechanisms which characterize a polymeric material, though it also involves elastic and time dependent effects which depend on microstructural details. In isotropic lamellar polymers a hardness depression from ideal values, due to the finite crystal thickness, occurs. The interlamellar non-crystalline layer introduces an additional weak component which contributes further to a lowering of the hardness value. Annealing effects and chemical etching are shown to produce, on the contrary, a significant hardening of the material. The prevalent mechanisms for plastic deformation are proposed. Anisotropy behaviour for several oriented materials is critically discussed. 

An alternative compact expansion with coefficients which are independent of the optical process can be derived for the isotropic parallel average of the second hyperpolarizability 7 defined as [13]... 

Numerical results for the shielding field of the benzene molecule are collected in Table 1 for the center of the molecule (labelled COM), and for points along a quarter circle of radius 2.47 A from the -ajcis to the x-axis, see Figure 3 for specification of axes. The radius of the circle corresponds to the distance from the ring center to a proton but, as defined, the points lie in the entirely nucleus-free xz-plane. Except for COM, the entries in the table are labelled by the angle between the z-axes and the direction to the field point. The table includes the isotropic part of the shielding, and the principal... 

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