Boundary displacement

The following restriction imposed upon the boundary displacements u =... 

Here is a normal component of the boundary displacements vector u defined by the decomposition... 

Ni3C decomposition is included in this class on the basis of Doremieux s conclusion [669] that the slow step is the combination of carbon atoms on reactant surfaces. The reaction (543—613 K) obeyed first-order [eqn. (15)] kinetics. The rate was not significantly different in nitrogen and, unlike the hydrides and nitrides, the mobile lattice constituent was not volatilized but deposited as amorphous carbon. The mechanism suggested is that carbon diffuses from within the structure to a surface where combination occurs. When carbon concentration within the crystal has been decreased sufficiently, nuclei of nickel metal are formed and thereafter reaction proceeds through boundary displacement. 

During the experiment the boundary displacement is recorded. Its derivative with regard to time is the velocity v of grain boundary motion, which is related to the driving force p by the boundary mobility m = v p. For convenience we use the reduced boundary mobility... 

As a result of applying a point force on the boundary, displacements in the semiinfinite solid are produced. These displacements can be calculated from Hooke s law and the displacement-strain relationships. The displacement in the r-direction, /r, is given by Eq. (2.20) and Eq. (2.23) as... 

Here t is the unit tangent along C and t X k is drawn outward. Two kinds of displacement must be considered the virtual displacement, Sx, required for the element of virtual work, which is of a mechanical nature, and the actual boundary displacement, 8 x, which corresponds to the actual increase in area and applies to the free energy relationships. Since the identical parameter y appears in both types of work, it is demonstrated that y is simultaneously surface tension and possesses excess surface free energy properties. 

Plot of grain boundary displacement versus anneal time for growth of a-plane sapphire seeds into 500 ppm Ti-doped and undoped aluminas at 1600°C. In comparison to results obtained previously with a lower purity powder, mobilities in undoped material are increased. However, the most striking feature is the increased migration rate for the Ti-doped material. 

For another quantity of electricity, f, the boundary displacement, v, will be v a Ff/F so that... 

One class of optimization problems encountered in nuclear engineering is the search for the material composition distribution or the geometry of a nuclear system that will extremize (minimize or maximize) one property of the system while keeping other properties within given bounds. Numerous methods have been developed for attacking this type of optimization problem. Reviews and references for these methods are provided (56, 114, 115). One approach to optimization is the perturbation approach. It makes use of material density or boundary displacement sensitivity functions. 

Perturbation-based optimization methods were introduced, both for inhomogeneous and homogeneous systems, by the Russian school (52) and by Lewins (5J). Methods based on material density perturbations have been developed and applied (53-55, 57, 61, 69, 70) to several problems described by the inhomogeneous Boltzmann equation and related to radiation shields (5i, 55), fission reactors (54, 57), and fusion reactors (61, 69, 116). Similar methods based on boundary displacement perturbations have also been developed and applied (91, 117) for the optimization of radiation shield problems. Perturbation-based optimization methods associated with the homogeneous Boltzmann equations have been developed and applied to several fission reactor problems (56, 92, 96 99). 

If the material has not been strained beyond the limit of plasticity, the stress field of the material recovers to the initial state when the external force and boundary displacement are removed. In contrast, if the material has been strained permanently beyond the limit of plasticity, the stress field of the material carmot recover to the initial state but transits to a third state, as illustrated in Fig. 2c, owing to the constraints on deformation during... 

The mechanics of stability and localization of deforma tion for plastic rigid materials has been considered by Hill (1957), who stated that, in general, the conditions for loss of load-carrying capacity under controlled boundary displacements or large distortions under constant boundary loads are different from the conditions for localization by inhomogeneous deformation. The conditions for the latter are more akin to those of uniqueness. For the plastic rigid bar the two conditions coincide. As Hart (1967) has shown, and as was developed in more detail by Hutchinson and Neale (1977), the two conditions are always separate for strain-rate-sensitive materials, where much stable extension can follow a load maximum. 

Babcock, S. E. and Ballufifi, R. W., Grain boundary kinetics—I. In situ observations of coupled grain boundary dislocation motion, crystal translation and boundary displacement, Acta MetalL, 37, 2357 5, 1989. 

Despite the stress, the void in strain does not vanished it is defined in term of the boundary displacements of the voids. So, because of the void fraction is usually negligible. Therefore, last term in Eq. (26) could be neglected and the equation corrected to ... 

Linear peristaltic pumps transport fluid through a flexible duct using traveling contraction waves. In a typical linear peristaltic pump, discrete translational elements rhythmically compress a straight section of flexible tube, moving fluid volumes. In contrast to rotary peristaltic pumps, linear peristaltic pumps usually do not use rollers or sliding contact elements. Because a moving boundary displaces fluid and induces the flow, linear peristaltic pumps are an example of positive-displacement pumps. 

Wang and Sun (2001) developed another numerical method to simulate textile processes and to determine the micro-geometry of textile fabrics. They called it a digital-element model. It models yams by pin-connected digital-rod-element chains. As the element length approaches zero, the chain becomes fully flexible, imitating the physical behavior of the yams. The interactions of adjacent yarns are modeled by contact elements. If the distance between two nodes on different yarns approaches the yam diameter, contact occurs between them. The yarn microstructure inside the fabric is determined by process mechanics, such as yarn tension and interyam friction and compression. The textile process is modeled as a nonlinear solid mechanics problem with boundary displacement (or motion) conditions. This numerical approach was identified as digital-element simulation rather than as finite element simulation because of a special yam discretization process. With the conventional finite element method, the element preserves... 

Several numerical methods are available to solve Che Stokes flow field equations. In the elastohydrodynamic problem considered here Che elastic boundary displacements and tractions are most important. We are less interested in Che displacement and stress field within Che elastic... 

This is the boundary displacement from the stress-free configuration which must be imposed to each arm of the double cantilever specimen to initiate crack growth from an initial crack length of a. 

Fig. 4.10. The variation of total potential energy and crack surface fracture energy with crack length is illustrated qualitatively for the configuration depicted in Figure 4.9 for the cases of (a) imposed boundary displacement 6 and (b) imposed boundary force P. The results illustrate the stability of crack advance under increasing displacement in case (a) and instability under increasing force in case (b). Adapted from Freund (1990).

Therefore, the displacement is given by the elastic form, but with q r, t), a known quantity, substituted for the specified stress (up to a multiplying constant). If the specified stress is always zero, as happens in certain contact problems for example, g(r, t) will also be zero, and the normal boundary displacement will be given precisely by the elastic form. In contrast to the special case of expanding or stationary 5 (0, we can make no useful statement about the stress, in the general case. 

The turbulent flame velocities in the container with fans have been measured by two methods. In the first method, the front boundary displacement of the turbulent flame in the Schlieren photos was observed, similar to the case of the laminar flame the equivalent spherical front radius was found by a mathematical treatment (planimetry, for example). The up-dated Schlieren photography devices have allowed registration in two projections for calculating the flame source volume with better accuracy. 

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