Simple extension

If an isotropic cubical element is elongated in one direction and the dimensional changes in the two mutually perpendicular directions are equal (Fig. 1-14), the experiment is termed simple extension if 7 u is positive, 722 and 733 are equal to each other and negative (or possibly zero). The stresses are t ] = ar 22 -o 33 == Pa. where ar is the tensile stress resulting from the applied force and Pa is the ambient pressure. This type of deformation results when a rod, strip, or fiber is subjected to a tensile force. Equation 42 for this case becomes  

Elimination of K t) from equations 47 and 48 gives the analog of equation 7 for this deformation  

Although 7i 1 is constant by definition during the stress relaxation experiment, 722 will in general change. The relation between axial extension and lateral contraction for a perfectly elastic solid is specified by Poisson s ratio /it  

The quantity in brackets is 2 E t)/2, where E t) is the tensile relaxation modulus or time-dependent Young s modulus. Hence, 

For a perfectly elastic solid, or a viscoelastic solid at equilibrium, the equilibrium Young s modulus is related to the shear and bulk moduli more simply as follows  

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The next step towards increasing the accuracy in estimating molecular properties is to use different contributions for atoms in different hybridi2ation states. This simple extension is sufficient to reproduce mean molecular polarizabilities to within 1-3 % of the experimental value. The estimation of mean molecular polarizabilities from atomic refractions has a long history, dating back to around 1911 [7], Miller and Sav-chik were the first to propose a method that considered atom hybridization in which each atom is characterized by its state of atomic hybridization [8]. They derived a formula for calculating these contributions on the basis of a theoretical interpretation of variational perturbation results and on the basis of molecular orbital theory. 

Phofoelectron spectroscopy is a simple extension of the photoelectric effect involving the use of higher-energy incident photons and applied to the study not only of solid surfaces but also of samples in the gas phase. Equations (8.1) and (8.2) still apply buf, for gas-phase measuremenfs in particular, fhe work function is usually replaced by fhe ionization energy l so fhaf Equation (8.2) becomes... 

As a simple extension we discuss a two-layer adsorbate with ontop stacking in one dimension with a Hamiltonian... 

The formation of bicyclic imines (263,264) from piperidine enamines and y-bromopropyl amines may appear at first sight to be a simple extension of the reactions of enamines with alkyl halides. However, evidence has been found that the products are formed by an initial enamine exchange, followed by an intramolecular enamine alkylation. Thus y-bromodiethylamino-propane does not react with piperidinocyclohexene under conditions suitable for the corresponding primary amine. Furthermore, the enamine of cyclopentanone, but not that of cyclohexanone, requires a secondary rather than primary y-bromopropylamine, presumably because of the less favorable imine to enamine conversion in this instance. 

The use of Polya s Theorem in a specialized context such as the above, has led to the extension of the theorem along certain useful lines. One such derivation pertains to the situation where the boxes are not all filled from the same store of figures. More specifically, the boxes are partitioned into a number of subsets, and there is a store of figures peculiar to each subset. To make sense of this we must assume that no two boxes in different subsets are in the same orbit of the group in question. A simple extension of Polya s Theorem enables us to tackle problems of this type. Instead of the cycle index being a function of a single family of variables, the 5j, we have other families of variables, one for each subset. An example from chemical enumeration will make this clear. 

MW + 1 ion), and this reaction is 35-40 kcal./mole exothermic. We might expect that the rate for this reaction would be appreciably greater than the rate for primary H abstraction, and we consequently postulate that primary H abstraction to yield MW — 1 ions will not occur to any significant extent. The other process we must consider is a simple extension of Reaction 12—namely, beta fission at a branch point. However, we now wish to consider the case where the branch point is a quaternary carbon. As a typical example ... 

Bi-layer adsorption is not uncommon and the development of the bi-layer adsorption isotherm equation is a simple extension of that used for the mono-layer equation. The Langmuir equation for bi-layer adsorption is as follows ... 

Finally, there are groups of liquid crystals where, at the current time, force fields are not particularly useful. These include most metal-containing liquid crystals. Some attempts have been made to generalise traditional force fields to allow them to cover more of the periodic table [40, 43]. However, many of these attempts are simple extensions of the force fields used for simple organic systems, and do not attempt to take into account the additional strong polarisation effects that occur in many metal-containing liquid crystals, and which strongly influence both molecular structure and intermolecular interactions. 

This is Mooney s equation for the stored elastic energy per unit volume. The constant Ci corresponds to the kTvel V of the statistical theory i.e., the first term in Eq. (49) is of the same form as the theoretical elastic free energy per unit volume AF =—TAiS/F where AaS is given by Eq. (41) with axayaz l. The second term in Eq. (49) contains the parameter whose significance from the point of view of the structure of the elastic body remains unknown at present. For simple extension, ax = a, ay — az—X/a, and the retractive force r per unit initial cross section, given by dW/da, is... 

The sum of these expressions represents the entropy of mixing. The result obtained by substituting for the free volumes from Eqs. (13), (14), and (15), is identical with Eq. (10). According to this simple derivation, the terms appearing in Eq. (10) represent contributions to the entropy which originate in the greater spatial freedom of the molecules in the solution. By a simple extension of the derivation to a mixture of polymer species, Eq. (12) may be obtained. 

Thus, adds and bases do not react directly but as solvent cations and anions. Since emphasis is placed upon ionization interactions, inherent addity and basidty is neglected, as are interactions in the non-ionic state. The theory is a simple extension of the Arrhenius theory and suffers from... 

In fact, the Valley theorem is a simple extension of the Cusp theorem. However, the Cusp theorem provides only a local information (for r=0), while the Valley theorem... 

The operational variants in tongue movement so far detected, are a simple extension/retraction and oscillations — occurring singly or as multiple bouts Fig. 7.3(a) the latter type can be divided into those with slow motion and those of normal duration (Toubeau, 1994). 

The energy systems of the 21st century probably will not be a simple extension of an improved copy of what they are now. Even today one can see new trends developing however, it is impossible to predict which ones will mature technologically and take an... 

A CD process for the production of DAA from acetone was developed previously in our laboratory and the kinetics of the aldol condensation of acetone were characterized (13,14). CD is a green reactor technology that provides enhanced yield and selectivity in addition to significant energy savings (15). The one-step synthesis of MIBK by CD appears to be a simple extension of this process. However, the introduction of hydrogen to this system opens... 

In fact, we have already used a modeling strategy when Po(AU) was approximated as a Gaussian. This led to the second-order perturbation theory, which is only of limited accuracy. A simple extension of this approach is to represent Pq(AU) as a linear combination of n Gaussian functions, p, (AU), with different mean values and variances [40]... 

Figure 2. The principle of the two-network method for cross-linking in a state of simple extension. First network with modulus Gy is entirely due to chain entangling. Second network with modulus Gx is formed by cross-linking in the strained state. Both Gy and Gx can be calculated from the two-network theory.

Figure 3. Modulus contributions from chemical cross-links (Cx, filled triangles) and from chain entangling (Gx, unfilled symbols) plotted against the extension ratio during cross-linking, A0, for 1,2-polybutadiene. Key O, GN, equibiaxial extension , G.v, pure shear A, Gx, simple extension Gx°, pseudo-equilibrium rubber plateau modulus for a polybutadiene with a similar microstructure. See Ref. 10.

Figure 4. Experimental setup for stress-relaxation and cross-linking at constant simple extension. Key A, electron accelerator B, beam aperture C, force transducer D, thermostated box E, sample F, stretching device G, connecting rods.

The two-network method has been carefully examined. All the previous two-network results were obtained in simple extension for which the Gaussian composite network theory was found to be inadequate. Results obtained on composite networks of 1,2-polybutadiene for three different types of strain, namely equibiaxial extension, pure shear, and simple extension, are discussed in the present paper. The Gaussian composite network elastic free energy relation is found to be adequate in equibiaxial extension and possibly pure shear. Extrapolation to zero strain gives the same result for all three types of strain The contribution from chain entangling at elastic equilibrium is found to be approximately equal to the pseudo-equilibrium rubber plateau modulus and about three times larger than the contribution from chemical cross-links. 

The Fourier method is not a requirement, and direct sinusoidal fitting procedures are also used to fit the data from a set of images. A number of specialized procedures have been described over the years and it is worth noting that extracting the amplitude and phase may be done as a simple extension to conventional linear regression. 

Nebenvalenz) to describe the chemical forces underlying formation of inner complexes, and G. N. Lewis s general concept of the Lewis-acid-base adduct allowed many types of coordinate bonding to be recognized as simple extensions of Lewis-like covalent concepts. 

However, at low Reynolds number (e.g., the near-wall region), the dissipation tensor can be highly anisotropic and other models are required (Pope 2000). For example, a simple extension of (4.53) yields (Rotta 1951)... 

A simple extension of the competition technique is to the comparison of scavenger efficiencies. Thus pairs of spin traps have been allowed to compete for a variety of radicals, including t-butoxyl, phenyl, and primary alkyl. Much more revealing, however, is the type of experiment in which the bimolecular trapping of a radical is allowed to compete with some other reaction of that radical whose absolute rate constant is known. In this way, the rate constant for the trapping reaction itself is accessible. 

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