Temperature fictitious

Using the equilibrium equations of the elasticity theory enables one to determine the stress tensor component (Tjj normal to the plane of translumination. The other stress components can be determined using additional measurements or additional information. We assume that there exists a temperature field T, the so-called fictitious temperature, which causes a stress field, equal to the residual stress pattern. In this paper we formulate the boundary-value problem for determining all components of the residual stresses from the results of the translumination of the specimen in a system of parallel planes. Theory of the fictitious temperature has been successfully used in the case of plane strain [2]. The aim of this paper is to show how this method can be applied in the general case. 

Many authors have shown that residual stresses in glass articles can be formally considered as the thermal stresses due to a certain fictitious temperature field. In the general case... 

Theory of the fictitious temperature field allows us to analyze the problems of residual stresses in glass using the mathematical apparatus of thermoelasticity. In this part we formulate the boundary-value problem for determining the internal stresses. We will Lheretore start from the Duhamel-Neuinan relations... 

In integrated photoelasticity it is impossible to achieve a complete reconstruction of stresses in samples by only illuminating a system of parallel planes and using equilibrium equations of the elasticity theory. Theory of the fictitious temperature field allows one to formulate a boundary-value problem which permits to determine all components of the stress tensor field in some cases. If the stress gradient in the axial direction is smooth enough, then perturbation method can be used for the solution of the inverse problem. As an example, distribution of stresses in a bow tie type fiber preforms is shown in Fig. 2 [2]. 

The practical implication is the fact that in the CP MD simulation the molecular system does not evolve right on the Born-Oppenheimer PES, but stays close to it. A measure of deviations from the BO PES is the fictitious kinetic energy (wave-function temperature). Figure 4-2 demonstrates that this deviation is minor, as the electronic (fictitious) temperature is relatively low. The wave function stays cold (compared to the hot nuclei) in the MD terminology the term cold electrons is often used in this context. 

The gas molecules and the suspended partic e.s in the atmosphere emit radiation as well as absorbing it. The atmospheric emission is primarily due to the COj and H2O molecules and is concentrated in the regions front 5 to 8 p.m and above 13 p.m. Although this emission is far from resembling the distribution of radiation from a blackbody, it is found convenient in radiation calculations to treat the atmosphere as a blackbody at some lower fictitious temperature that emits an equivalent amount of radiation energy. This fictitious temperature is called the effective sky teniperatur Then the radiation emission from the atmosphere to the earth s surface is expressed as... 

Phonon velocity is constant and is the speed of sound for acoustic phonons. The only temperature dependence comes from the heat capacity. Since at low temperature, photons and phonons behave very similarly, the energy density of phonons follows the Stefan-Boltzmann relation oT lvs, where o is the Stefan-Boltzmann constant for phonons. Hence, the heat capacity follows as C T3 since it is the temperature derivative of the energy density. However, this T3 behavior prevails only below the Debye temperature which is defined as 0B = h( DlkB. The Debye temperature is a fictitious temperature which is characteristic of the material since it involves the upper cutoff frequency ooD which is related to the chemical bond strength and the mass of the atoms. The temperature range below the Debye temperature can be thought as the quantum requirement for phonons, whereas above the Debye temperature the heat capacity follows the classical Dulong-Petit law, C = 3t)/cb [2,4] where T is the number density of atoms. The thermal conductivity well below the Debye temperature shows the T3 behavior and is often called the Casimir limit. 

In this approach, the nuclei are simulated at some finite temperature, T, which ultimately dictates the kinetic energy of the nuclei. The electronic structure, however, is kept close to the Born-Oppenheimer surface. The fictitious temperature of the electrons must therefore be close to zero. In simulating the dynamics for a specific system, the electrons must remain cold while the atoms must remain hot and thus maintain a nearly adiabatic system. The fictitious mass of the electron and the time step>s for the dynamics must be carefully structured so as to prevent energy transfer from the hot nuclei into the cold electrons. The Verlet algorithm is typically used to integrate these equations. 

However, the activation energy which was deduced by using the activation-relaxation technique, with a Metropolis accept-reject criterion and a fictitious temperature of 0.5eV, ranged from 0.22 to l.OeV. It also exhibited a steep increase at low temperature. The very large pre-exponential factor suggested that the interatomic forces which resulted from the Tersoff potential were very strong. These predictions were consistent, to some extent, with recent experimental results for liquid Si. 

The CPMD approach exploits in another way the separation of fast electronic and slow nuclear motions, as compared to BOMD. The KS orbitals are imbued with a fictitious time dependence, that is, y/i t)—> y/i t,t), and a dynamics for the orbitals is introduced that propagates an initially fully minimized set of orbitals to subsequent minima corresponding to each new nuclear configuration. This task is accomplished by designing the orbital dynamics in such a way that the orbitals are maintained at a fictitious temperature that is much smaller than the real nuclear temperature T but are still allowed to relax qitiddy in response to the nuclear motion. 

One way in which this shift in the neutron density may be taken into account is by introducing an effective neutron temperature Tn. Thus we define some fictitious temperature Tn which, when used in the function m of (4.196), gives a new Gaussian that yields a better fit to the distorted density distribution arising from the presence of the absorber. The determination of this parameter is deferred until a later section. For the present, all we require is that such a number can be specified for a given system. 

Fig. 54. - Example of thermodilatometric data 1396J obtained by using tlic NETZSCH instrument TMA 402, which shows the actually measured curve (thin solid line) and simultaneously calculated data of the relative defoimation (solid points) by means of the Narayanaswam s model [400J. Right, there is the graph of the time dependence of thermodynamic temperature (thin solid line) and calculated fictitious temperature (solid points). Courtesy of Marie Cfiromcikovd and Marek Liska, Trenilin, Slovakia. ...

One of the most convenient tools for practical determination of fictitious temperatures is thermomechanometry [396] see Fig. 54, where the time dependence of fictitious temperature can be obtained on the basis of the Tool-Narayanaswami relation [391,396,400] by the optimization of viscosity measurements (logr] T,Tf versus temperatures) using the Vogel-Fulcher equation again. 

Where 6[H[X, X2,..., Xn is the function that maps the point in multi-dimensional configuration space Ai, A2,..., A to a point in single dimensional property space. Note that this equation corresponds to a minimization of the desired property function 0[H X, X2,...,Xn. T is a fictitious temperature parameter, kept constant when performing Monte Carlo optimizations. However, it can be ramped up and slowly decreased to trap the system in the optimal Xj, A, ..., A . This is known as simulated annealing." ... 

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