Longitudinal response

For systems with parallel axes (e.g., single crystals of magnetic molecular clusters or a ferrofluid frozen in a strong field), the coefficients for the longitudinal response read... 

The complex susceptibility components %Y(co) can be evaluated from Eq. (147) by calculation of the eigenvalues Xyk for normal rotational diffusion (see Section III.C). However, /.,( or) may be much more effectively calculated by using the continued fraction method (see Ref. 103 for detail). Let us first evaluate the longitudinal response. By expanding the distribution function W(i9, t) in a Fourier series (here W is independent of 9)... 

Just as the longitudinal response, Eq. (A2.7) can be solved exactly for the Fourier-Laplace transform gi(i ) in terms of ordinary continued fractions to yield... 

Strengthening method is to use external posttensioning as shown in Fig. 13a. Steel jackets bonded to the sides of the joint and anchored to the bent cap are also effective (Thewalt and Stojadinovic 1995). Finally, FRP jackets in the form of an ankle wrap have been tested (Gergely et al. 2000) and used to strengthen column to cap beam-colunm joints as shown in Fig. 13b (Pantelides et al. 2004). The nominal principal tensile stress developed in the beam-column joint is used to design the carbon FRP composite layers. When strengthening of these joints is required to improve the longitudinal response, transversely prestressed bolsters may be used. 

SCC has been defined as failure by cracking under the combined action of corrosion and stress (Fig. 9.1). The stress and corrosion components interact S3mergistically to produce cracks, which initiate on the surface exposed to the corrodent and propagate in response to the stress state. They may run in any direction but are always perpendicular to the principal stress. Longitudinal or transverse crack orientations in tubes are common (Figs. 9.2 and 9.3). Occasionally, both longitudinal and transverse cracks are present on the same tube (Fig. 9.4). Less frequently, SCC is a secondary result of another primary corrosion mode. In such cases, the cracking, rather than the primary corrosion, may be the actual cause of failure (Fig. 9.5). 

Figure 7.2. Response of elastic-plastic solid to planar impact at X = 0 u = longitudinal particle velocity. Measurements are made as a function of time at fixed Lagrangian position X.

The shock-induced micromechanical response of <100>-loaded single crystal copper is investigated [18] for values of (WohL) from 0 to 10. The latter value results in W 10 Wg at y = 0.01. No distinction is made between total and mobile dislocation densities. These calculations show that rapid dislocation multiplication behind the elastic shock front results in a decrease in longitudinal stress, which is communicated to the shock front by nonlinear elastic effects [pc,/po > V, (7.20)]. While this is an important result, later recovery experiments by Vorthman and Duvall [19] show that shock compression does not result in a significant increase in residual dislocation density in LiF. Hence, the micromechanical interpretation of precursor decay provided by Herrmann et al. [18] remains unresolved with existing recovery experiments. 

There is no MOKE response for normally incident p- or j-light in either the longitudinal or transverse geometries. 

There is a larger MOKE response as the angle of incidence becomes more oblique for the longitudinal geometry up to a maximum at an angle of 60° to 80°, depending on the specific material. 

The dispersion of a solute band in a packed column was originally treated comprehensively by Van Deemter et al. [4] who postulated that there were four first-order effect, spreading processes that were responsible for peak dispersion. These the authors designated as multi-path dispersion, longitudinal diffusion, resistance to mass transfer in the mobile phase and resistance to mass transfer in the stationary phase. Van Deemter derived an expression for the variance contribution of each dispersion process to the overall variance per unit length of the column. Consequently, as the individual dispersion processes can be assumed to be random and non-interacting, the total variance per unit length of the column was obtained from a sum of the individual variance contributions. 

Axial flow, in which the liquid enters the impeller and discharges along a parallel path to the axis, is shown in Figure 9. The radial and longitudinal components are primarily responsible for the derived mixing action. The tangential component is important when the shaft has a vertical orientation and is positioned near the center of the tank. 

In crystals, the response of the crystal to a longitudinal loading may produce deformation controlled by the crystal symmetry that is not uniaxial... 

Fig. 4.3. Typical normalized piezoelectric current-versus-time responses are compared for x-cut quartz, z-cut lithium niobate, and y-cut lithium niobate. The y-cut response is distorted in time due to propagation of both longitudinal and shear components. In the other crystals, the increases of current in time can be described with finite strain, dielectric constant change, and electromechanical coupling as predicted by theory (after Davison and Graham [79D01]).

Secondly, these quotations emphasize the fact that the same river input that fuels longitudinal heterogeneity in reservoirs also forms a strong link between the reservoir and its watershed (e.g., [6]). This link has been conceptualized mostly in the form of load-response empirical models [7, 8], or mass-balance approaches [9]. Curiously, empirical modelers usually consider reservoirs as stirred reactors, ignoring the longitudinal spatial heterogeneity present in most situations and processes. 

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