Width effects

It is often found that, in practice, the measured value of Gic or K c varies with the width of the specimen over a certain range of widths, and this usually arises because the state of stress near the crack tip varies from plane stress in a very thin specimen to plane strain near the centre of a wide plate. The general form of the relationship between Kic (or Gic) and specimen width, b, is shown schematically in Fig. 7.6. The value for fracture in plane strain conditions is usually less than under plane stress. This arises because the tensile stress at which a material yields is greater in a triaxial stress field (plane strain) than in a biaxial one (plane stress) and thus, in the former, a more limited degree of plasticity develops at the crack tip cf. Equations 7.26 and 7.27. The lower, conservative, plane strain value is usually the one required for engineering design and life prediction studies. The width, fe, necessary to achieve this condition is usually taken to be  

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As we will see in Chapter 4, g-matrices are often difficult to interpret reliably. The interpretation of isotropic g-values is even less useful and subject to misinterpretation. Thus isotropic ESR spectra should be used to characterize a radical by means of the hyperfine coupling pattern, to study its dynamical properties through line width effects, or to measure its concentration by integration of the spectrum and comparison with an appropriate standard but considerable caution should be exercised in interpreting the g-value or nuclear hyperfine coupling constants. 

Since these terms are proportional to tr, they increase with decreasing temperature.1 There are several line-width contributions, included in oc0, which do not depend on m,-. These include magnetic field inhomogeneity and the spin rotation interaction, the latter increasing with 1/tr and thus with increasing temperature. These and other line-width effects have been studied in some detail and are discussed elsewhere.13... 

Casagrande et al.5 have used line-width effects to study the rate of fluxiona-lity in (Ph2C2)Co(CO)[P(OMe)3]2. The experimental spectrum (Figure 5.9a),... 

Another example of alternating line width effect was found in the spectra of durosemiquinone (6),9,10 where the effect is due to alkali metal ions hopping back and forth from one oxygen atom to the other. The rates depend on the alkali metal as shown in Table 5.3. 

P. D. Sullivan and J. R. Bolton, Adv. Magn. Reson. 4 39 (1970). The Alternating Line-Width Effect. ... 

From the point of view of the solvent influenee, there are three features of an electron spin resonance (ESR) speetrum of interest for an organic radical measured in solution the gf-factor of the radical, the isotropie hyperfine splitting (HFS) constant a of any nucleus with nonzero spin in the moleeule, and the widths of the various lines in the spectrum [2, 183-186, 390]. The g -faetor determines the magnetic field at which the unpaired electron of the free radieal will resonate at the fixed frequency of the ESR spectrometer (usually 9.5 GHz). The isotropie HFS constants are related to the distribution of the Ti-electron spin density (also ealled spin population) of r-radicals. Line-width effects are correlated with temperature-dependent dynamic processes such as internal rotations and electron-transfer reaetions. Some reviews on organic radicals in solution are given in reference [390]. 

In the asperity filtering regime, the Greenwood and Williamson theory no longer properly models the local contact pressure since the model contains no notion of asperity width. We describe a simple statistical method for incorporating width effects in two-dimensional polishing. A more sophisticated but more complex approach based on elasticity theory that takes into account asperity shape and the interaction of the asperity with trench structures of similar size can be found in Reference 27. The statistical approach assumes that... 

An ingenious approach to the treatment of finite gradient pulse width effects has been provided by Wang et al. (1995). They demonstrate that it is possible to approximate the temporal behavior of any gradient pulse by a sum of impulses, each being in the narrow gradient pulse limit. By this means one can derive an analytical solution to the echo attenuation. 

Symmetry requirements on the x s that result from the permutation of the order of application of the fields, from causality requirements and from the requirement that real fields must give rise to real polarizations (in the absence of line-width effects) ensure that the x s are real, that, [Pg.251]

The spectrum of the dvuenesemiquinone radical-cation exhibits a marked alternating line-width effect which is temperature-dependent it is ascribed to cis-trans isomerism of (68) and (69), the life-time of each isomer being comparable with the inverse frequency separation between the methyl proton-splittings (Bolton and Carrington, 1962). Carrington (1962) has discussed the theoretical line-shapes to be expected for different rates of isomerization and has estimated that the life-times of the isomers at room temperature are about 10 sec. 

Alternating hue-width effects were found in the ESR spectra of some protonated phenoxyl radicals and were interpreted in terms of jumps between geometrical isomers involving the direction of the 0" -H group in relation to the ring . ... 

The presented technique, finite pulse r.f. driven recoupling (fpRFDR), restores homonuclear dipolar interactions based on constructive usage of finite pulse-width effects in a phase- and symmetry-cycled 7c-pulse train in which a rotor-synchronous n pulse is applied every rotation period. The restored effective dipolar interaction has the form of a ZQ dipolar Hamiltonian for static solids, whose symmetry in spin space is different from that obtained by conventional r.f. driven recoupling (RFDR) techniques. It has been demonstrated that the efficiency of recoupling by fpRFDR is not strongly dependent on chemical shift differences or resonance offsets in contrast to previous recoupling... 

Fig. 8.21. The vanishing particle width effect in a doubly-excited series of the Ca spectrum. Note the narrowing of the n — 6 member, although the remainder of the series members are broad. The perturber (hardly visible on account of its breadth and weakness) is indicated by an X in the figure (after U. Griesmann et al. [390]).

The mechanical slit is placed before the detector to limit further the number of frequencies that impinge simultaneously upon it. Transfer optics may differ from one type of monochromator to another and yield different spatial spreads of the frequencies that arrive at the mechanical slit. An adjustment of the slit width effects... 

Finally, of great importance are finite width effects. The strength of a composite laminate in the presence of a hole, a crack, or impact damage is drastically reduced as the size of the notch covers a significant percentage of one of the dimensions of the structure. Finite width correction factors have been obtained by Tan for holes [19]. 

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