Stress trajectory density

The stress distribution in a component can be visualised using so-caUed stress trajectories. These trajectories always run in the direction of the maximum principal stress. Their distance is inversely proportional to the stress so that the stress trajectory density is a measure of the locally acting stress. Each abrupt change in cross section deflects the stress trajectories which then move closer together. Thus, a local stress concentration arises. 

Similar method was used in [3] with heat flow and in [4, 5] with stress trajectories. The purpose of this work is to develop efficient method for identification of current densities within a domain where some data on electric potential are provided. 

Fig- 4.1. Stress trajectories in notched components. The stress trajectories are aligned with the maximum principal stress, their density is a measure of the stress level. At the notch root, there is a stress concentration in both geometries... 

There exists a second reason why Latora and Baranger have been forced to depart from the adoption of a density equation, thereby rather adopting the supposedly equivalent time evolution of a bunch of trajectories. This is due to the fact that the Lyapunov coefficients are local and might change with moving from one point of the phase space to another. It is important to stress this second reason because it is closely related to the directions which need to be followed to reveal by means of experiments the breakdown of the density perspective, and with it of quantum mechanics, in spite of the fact that so far the predictions of quantum mechanics have been found to fit very satisfactorily the experimental observation. 

In a sheared suspension, the effects are two-fold. First, the expression for bulk stress itself must be modified. Second, the probability density is affected since the continuity equation for the latter must be replaced by a convection-diffusion equation. As a consequence, the distinction between open and closed trajectories loses some of its meaning. Batchelor (1977) gives the equivalent viscosity of a sheared suspension subject to strong Brownian motion as... 

Nonequilibrium conditions may occur with respect to disturbances in the interior of a system, or between a system and its surroundings. As a result, the local stress, strain, temperature, concentration, and energy density may vary at each instance in time. This may lead to instability in space and time. Constantly changing properties cannot be described properly by referring to the system as a whole. Some averaging of the properties in space and time is necessary. Such averages need to be clearly stated in the utilization and correlation of experimental data, especially when their interpretations are associated with theories that are valid at equilibrium. Components of the generalized flows and the thermodynamic forces can be used to define the trajectories of the behavior of systems in time. A trajectory specifies the curve represented by the flow and force components as functions of time in the flow-force space. 

The bead dynamics is realized by the integration of the equations of motion for the beads. A trajectory is generated through the system s phase space. All thermodynamic observables (e.g. density fields, order parameters, correlation functions, stress tensor, etc.) can be constructed from suitable averages. An immense advantage over conventional molecular dynamics and Brownian dynamics is that all forces are soft , thus allowing... 

FIG. 8. Profiles of the density of dissociated atoms, p/Po ( ) of Fig-7, vs lattice plane number at different times. Data have been smoothed by averaging over 2 neighboring planes ahead of the reaction front, and 6 behind the reaction front, and over 15 time steps. Solid lines are the steady-state trajectories of the head and foot of the reaction front. Dashed lines are the corresponding trajectories of the shock front obtained from a similar plot of the stress profiles vs time and distance. Dotted line marks the trajectories of some local reaction sites initiated by shock compression. Velocities refer to the stationary mirror plane. From reference [38b]. 

The fact that GeTe is getting more stressed-rigid upon doping is not the only parameter that influences the stability of the compound. If we compute the vibrational density of states by performing a Fourier transform of the velocity autocorrelation function computed for the AIMD trajectory, we obtain curves plotted in Fig. 18.16. As already found experimentally [62], amorphous GeTe is at the same time elastically... 

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