Equilibrium static

The first step in any dynamical calculation is to establish static equilibrium one has to verify, before imposing any displacements, that the ab initio method chosen correctly describes the starting point, and that the undistorted structure is in static equilibrium. This is usually done by calculating the total energy at various lattice constants i.e. for a crystal under (positive °fppegative) compression (Fig. 3.1). The minimum of the function E (a) determines the equilibrium lattice constant a, the second 

We notice in Fig. 3.1 that the function E ° (a) is rather flat at its minimum, so that a precise determination of the equilibrium 

It is explained in Ref. 12 how the self-consiste LDF calculation yields the total energy, forces on atoms and average macroscopic stress a. Knowledge of forces is of little use for establishing static equilibrium because, in crystals as symmetric as or T, the site-symmetry makes the net force on each atom 

If the calculations shown schematically in Fig. 3.1 were to extend over a large range of lattice constants, the simple linear relation for p(a) would have to replaced by some p(V) equation of state the Murnaghan equation 

All Fourier series have to be made finite when performed numerically the choice of the number of waves used in any calculation is a compromise between the computational effort and the errors caused by the truncation they are difficult to estimate and one usually resorts to numerical testing. An example of a convergence test for a, B - calculated from p(a) as described above (Fig. 3.1) - is s iown in Tab. 3.1 the behavior of truncation errors is typical for many similar situations. Whereas the absolute values of both pressure and energy vary considerably with increasing number of waves, the a, B calculated from them evolve only slowly. Apparently a large part of the truncation error is systematic. Detailed convergence tests for the different potentials used can be found e.g. in Ref. 24 so far the most detailed study 

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Anzellotti G., Giaquinta M. (1982) On the existence of the fields of stresses and displacements for an elasto-perfectly plastic body in static equilibrium. J. Math. Pure Appl. 61, 219-244. 

Adsorption for gas purification comes under the category of dynamic adsorption. Where a high separation efficiency is required, the adsorption would be stopped when the breakthrough point is reached. The relationship between adsorbate concentration in the gas stream and the solid may be determined experimentally and plotted in the form of isotherms. These are usually determined under static equilibrium conditions but dynamic adsorption conditions operating in gas purification bear little relationship to these results. Isotherms indicate the affinity of the adsorbent for the adsorbate but do not relate the contact time or the amount of adsorbent required to reduce the adsorbate from one concentration to another. Factors which influence the service time of an adsorbent bed include the grain size of the adsorbent depth of adsorbent bed gas velocity temperature of gas and adsorbent pressure of the gas stream concentration of the adsorbates concentration of other gas constituents which may be adsorbed at the same time moisture content of the gas and adsorbent concentration of substances which may polymerize or react with the adsorbent adsorptive capacity of the adsorbent for the adsorbate over the concentration range applicable over the filter or carbon bed efficiency of adsorbate removal required. 

If there are no unbalanced forces acting on a particle, tbe particle is said to be in static equilibrium, and Newton s second law reduces to... 

A beam subjected to a simple transverse load (Figure 2-29a) will bend. Furthermore, if the beam is cut (Figure 2-29b) and free body diagrams of the remaining sections are constructed, then a shear force V and a moment M must be applied to the cut ends to maintain static equilibrium. 

The magnitude of the shearing force and the moment can be determined from the conditions of static equilibrium of the beam section. Thus, for the cut shown in Figure 2-29b, the shear force and the moment on the left-hand section are... 

The stress in the axial direction in a cylindrical vessel is found by taking a cross-section perpendicular to the longitudinal axis and imposing the conditions of static equilibrium. This yields... 

Barrett and Thomas (10)proposed that these effects of differential monomer adsorption could be modeled by correcting homogeneous solution copolymerization reactivity ratios with the monomer s partition coefficient between the particles and the diluent. The partition coefficient is measured by static equilibrium experiments. Barrett s suggested equations are ... 

In the stress analysis of pressure vessels and pressure vessel components stresses are classified as primary or secondary. Primary stresses can be defined as those stresses that are necessary to satisfy the conditions of static equilibrium. The membrane stresses induced by the applied pressure and the bending stresses due to wind loads are examples of primary stresses. Primary stresses are not self-limiting if they exceed the yield point of the material, gross distortion, and in the extreme situation, failure of the vessel will occur. 

Consider the forces acting on the elemental section of the wall of the cylinder shown in Figure 13.43. The cylinder is under an internal pressure P, and an external pressure Pe. The conditions for static equilibrium, with the forces resolved radially, give ... 

Chhabra RP, PHT Uhlherr. Static equilibrium and motion of spheres in viscoplastic liquids. In NP Cheremisinoff, ed. Encyclopedia of Fluid Mechanics, Vol. 7. Houston, TX Gulf Pub Co, 1988, Chap 21. 

When the beam of a triple balance stops swinging, the balance has reached a static equilibrium. A body is said to be in static equilibrium if a body at rest will stay at rest. That statement is definitely true in this case. 

As mentioned in Section 2.1, Earnshaw s theorem establishes that there can be no stable static equilibrium arrangement of classical ions and dipoles. Nevertheless, quantum mechanics allows numerous stable arrangements of ions, such as those... 

Van der Waals, whose theory has been further developed by Hulshoff and by Bakker, went one step further than Gibbs by assuming that there exists a perfectly continuous transition from one medium to the other at the boundary. This assumption limits him to the consideration of one particular case that of a liquid in contact with its own saturated vapour, and mathematical treatment becomes possible by the further assumption that the Van der Waals equation (see Chapter II.) holds good throughout the system. The conditions of equilibrium thus become dynamical, as opposed to the statical equilibrium of Laplace s theory. Van der Waals arrives at the following principal results (i) that a surface tension exists at the boundary liquid-saturated vapour and that it is of the same order of magnitude as that found by Laplace s theory (2) that the surface tension... 

The existence of the limit (3) guarantees that, after a large enough number of steps, the different configurations are generated following a probability density II. Then it is said that a distribution of stationary probability or situation of static equilibrium has been reached. If II has been previously chosen, the method consists of selecting pij so that the conditions (2) and (4) are fulfilled. We must stress the fact that the condition of microscopic reversibility ... 

Stress Theorem in the Determination of Static Equilibrium by the Density Functional... 

Figure 11. The gravitational mass (in units of the solar mass M ) versus the normalized central energy density (eo = 156 MeV fm-3) (left panel) and versus the equatorial radius (right panel). The thin lines represent static equilibrium configurations, whereas the thick fines display configurations rotating at their respective Kepler frequencies. Several different stellar matter compositions are considered (see text for details).

By virtue of its yield stress, an unsheared viscoelastic material is capable of supporting the immersed weight of a particle for an indefinite period of time, provided that the immersed weight of the particle does not exceed the maximum upward force which can be exerted by virtue of the yield stress of the fluid. The conditions for the static equilibrium of a sphere are now discussed. 

Many investigators 35 have reported experimental results on the necessary conditions for the static equilibrium of a sphere. The results of all such studies may be represented by a factor Z which is proportional to the ratio of the forces due to the yield stress xY and those due to gravity. 

The reported initial anisotropy(87) was r(0) = 0.39 0.01, which is within experimental uncertainty of the theoretical value 0.40. Evidently, there is no significant amplitude of faster motions. However, internal motions of the dye on a time scale much longer than 120 ns would not be detected in FPA experiments, as they would provide only an effectively static equilibrium distribution of dye orientations over the observation period. 

In Fig. 3.16 dynamic structure factor data from a A =36 kg/mol PE melt are displayed as a function of the Rouse variable VWt (Eq. 3.25) [4]. In Fig. 3.6 the scaled data followed a common master curve but here they spht into different branches which come close together only at small values of the scahng variable. This splitting is a consequence of the existing dynamic length scale, which invalidates the Rouse scaling properties. We note that this length is of purely dynamic character and cannot be observed in static equilibrium experiments. 

Guo, G. and Long, Y. (2001) Static equilibrium studies on separation of dichlorobenzene isomers on binder-free hydrophobic adsorlaent of MFI type zeolite. Sep. Purif. Technol,... 

Guo, G.-Q., Chen, H., and Ying-Cai, L. (2000) Separation of p-xylene from C8 aromatics on binder-free hydrophobic adsorbent of MFI zeolite. I. Studies on static equilibrium. Micropor. Mesopor. Mater., 39, 149-161. 

Note Subscript 2 indicates the conditions at C-J plane, while subscript 3 indicates the expl state corresponding to products of. deton in static equilibrium within the volume of original expl (Ref 52, p 376)... 

If life ceased, the planet would gradually settle towards a static equilibrium that would be very different from today s environment. 

Static equilibrium value of Q , Q/% can be obtained by setting dWdQ,i=0. This gives... 

Thus, within the context of the Newtonian force atom and the caloric theory of heat, solids, liqitids, and gases were all viewed as organized arrays of particles produced by a static equilibrium between the attractive interparticle forces, on the one hand, and the repulsive intercaloric forces, on the other. The sole difference was that the position of eqitilibriitm became greater as one passed from the solid to the liqitid to the gas, due to the increasing size of the caloric envelopes siuToittrding the component atoms (Figures 5 and 6). 

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