Coupling elastic

Fluid-Elastic Coupling Fluid flowing over tubes causes them to vibrate with a whirling motion. The mechanism of fluid-elastic coupling occurs when a critical velocity is exceeded and the vibration then becomes self-excited and grows in amplitude. This mechanism frequently occurs in process heat exchangers which suffer vibration damage. 

The transition temperatures which we find for the austenite - martensite transition in simulations without vacancies, are much to low. The introduction of vacancies lead to much more physical results, while the martensite - austenite transition is not affected by vacancies. From this we conclude that vacancies lower the energy barrier which the system has to overcome during the transition. The reason for this might be a weakening of long-range elastic couplings in the lattice. 

FIG. 2 Energy of elastically coupled charges with an additional short-range repulsion term. Copyright 2001 Marcel Dekker, Inc. 

Here pg and p f are the mass densities of the gel and the solvent, respectively, K is a bulk modulus, c0 is the speed of sound, and i s is the solvent shear viscosity. The solvent bulk viscosity has been neglected. The terms proportional to / arise from an elastic coupling in the free energy between the density deviation of gel and that of solvent The p in Eq. (6.1) coincides with the shear modulus of gels treated so far. We neglect the frequency-dependence of the elastic moduli. It can be important in dynamic light scattering, however, as will be discussed in the next section. 

The tensor of stress, Oy, has the meaning of the force in direction j on an infinitesimal area with normal in the direction i and is again a symmetric tensor with 6 independent components. In classical elasticity only the force resultant at any point is considered, the couple that must also exist is assumed to be negligible by comparison. However, in polar field theories of elasticity, couple stresses are considered and additional equations of equilibirum required. Classically however only the equation of stress equilibrium... 

Mitome, N., Suzuki, T., Hayashi, S., and Yoshida, M. (2004). Thermophilic ATP synthase has a decamer c-ring Indication of noninteger 10 3 H+/ATP ratio and permissive elastic coupling. Proc. Natl. Acad. Sci. USA 101, 12159-12164. 

Rief M, Fernandez JM, Gaub HE (1998a) Elastically coupled two-level-systems as a model for biopolymer extensibility. Phys. Rev. Lett. 81 4764-4767... 

The filler network break-down with increasing deformation amplitude and the decrease of moduli level with increasing temperature at constant deformation amplitude are sometimes referred to as a thixotropic change of the material. In order to represent the thixotropic effects in a continuum mechanical formulation of the material behavior the viscosities are assumed to depend on temperature and the deformation history [31]. The history-dependence is implied by an internal variable which is a measure for the deformation amplitude and has a relaxation property as realized in the constitutive theory of Lion [31]. More qualitatively, this relaxation property is sometimes termed viscous coupling1 [26] which means that the filler structure is viscously coupled to the elastomeric matrix, instead of being elastically coupled. This phenomenological picture has... 

The key feature distinguishing the OOA from the cooperative JT effect is the way the chemical bonding effects are included. This will be an important part of the present study. In Sect. 2, we present a simple qualitative description of the cooperative JT effect. First, in Sect. 2.1, we demonstrate the chemical nature of the JT instability. In Sect. 2.2, the cooperative JT effect is presented as interplay of short-range chemical bonding effects with the long-range intercell elastic coupling. To reveal the most important differences of the two approaches from one another, in Sect. 3 we present a simplified version of the OOA. In Sect. 4, we discuss some additional effects that are closely related to JT instability in crystals. 

Developed by Englman and Halperin [4], the above steps represent the traditional way in the cooperative JT problem. It s conceptual advantage and important part is solving the respective one-cell JT problem (9) including the low-symmetry mean field (10) of all other distorted cells. In this way, both effects, the dynamic strengthening chemical bonds with low-symmetry lattice distortions and the intercell elastic coupling, are included. 

Here is inverse of the respective dynamical matrix. In (17) it describes elastic coupling of tetragonal distortions, Qe(n) and Qs n), the ones active in the JT case E IS) e. Obviously, the only important contribution comes from the orbital exchange of close neighbors. In cubic symmetry, (17) simplifies to... 

Kupferman, R. Kawaguchi, M.N. Denn, M.M. Emergence of structure in a model of liquid crystalline polymers with elastic coupling. J. Non-Newton. Fluid Mech. 2000, 91, 255-271. 

A paper by Prandtl [18] on the kinetic theory of solid bodies, which was published in 1928, one year prior to Tomlinson s paper [17], never achieved the recognition in the tribology community that it deserves. PrandtI s model is similar to the Tomlinson model and likewise focused on elastic hysteresis effects within the bulk. Nevertheless, Prandtl did emphasize the relevance of his work to dry friction between solid bodies. In particular, he formulated the condition that can be considered the Holy Grail of dry, elastic friction If the elastic coupling of the mass points is chosen such that at every instance of time a fraction of the mass points possesses several stable equilibrium positions, then the system shows hysteresis. In the context of friction, hysteresis translates to finite static friction or to a finite kinetic friction that does not vanish in the limit of small sliding velocities. Note that the dissipative term that is introduced ad hoc in Eq. (19) does vanish linearly with small velocities. 

PrandtI s condition is satisfied if the elastic coupling to a lattice site is sufficiently small, namely k [Pg.209]

Temperatiue can be incorporated into the PT model by including random forces in the equation of motion, Eq. (22), similar to those introduced in the rigid-wall model, Eqs. (12) and (13). The additional complication in the PT model is the elastic coupling of the central particle to a (moving) equilibrium site, which makes it difficult to apply the rigorous concepts developed for the Brownian motion of a particle in a periodic potential to the PT model. 

Consider a 3-dimensional, corrugated solid placed on a smooth substrate as a simplified model for a mechanical contact, which is a subtle case (Section III. C.5). The macroscopic contact will then consist of individual junctions where asperities from the corrugated solid touch the substrate. A microscopic point of contact p then carries a normal load Ip, and a shear force fp will be exerted from the substrate to the asperity and vice versa. These random forces fp will try to deform both solids. For the sake of simplicity, let us only consider elastic deformations in the top solid. Asperities in intimate contact with the substrate will be subject to a competition between the (elastic) coupling to the top solid and the interaction with the substrate. If the elastic stress exceeds the local critical shear stress c,p of junction p, the contact will break and asperity p will find a new mechanical equilibrium position. In order for (5 to exceed Uep, the area A = tiL- over which the random forces accumulate must be sufficiently laige. The value of L where this condition is satisfied is called the elastic... 

To operate a stirrer, knowledge is in the first instance required concerning the required power input Pm of the electrical motor. This has to take into account both the stirrer power and all power losses Py in the shaft bearings, in the gear box and in the shaft seals. In addition the start up power Pa has to be included. This considerably depends upon the stirrer type and upon the type of drive (stirrer shaft rigidly or elastically coupled with the motor shaft belt drive) ... 

In this section the interaction of sliding speed and friction, in particular the interpretation of the instrumental displays in the measurement of motion, will be examined more closely than it was in Sections 8.3 and 8.4. We are interested specifically in the reliable determination of with an apparatus that employs elastic coupling of the... 

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