Mathematical Physics - Volume II - Numerical Methods

Chapter 6. Introduction to Computational Mechanics of Discontinua

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volume (surface in 2D) plays a role of a virtual sensor (thermometer in this case). This approach is reminiscent of the division of the motion of molecules, in the kinetic theory of gases, into a "random" and a "systematic" part [44].

Figure 6.6: (a) A typical time history of atomic speed recorded in the course of 0.408 [km/s] ballistic Taylor test [45]. The full black line represents the systematic velocity of (correlative) motion while the dashed red line represents the corresponding total velocity of the individual atom. The difference between the total velocity and the associated systematic velocity results in the velocity of vibratory motion which defines the thermal energy of random vibrations related to the instantaneous kinetic temperature (6.16) (Adopted from [43]). (b) Averaging area (in general, volume) superimposed on an irregular lattice for evaluation of the macroscopic state parameters in MD simulations ( virtual sensor ). The instantaneous kinetic temperature (6.16) is averaged in both time and space. In 2D simulations, the averaging area is usually taken to be the same circular region (shaded in Figure 6.6b) that was previously used to calculate the correlative motion velocity. Therefore, each node in the network also represents the center of the averaging area of radius R av , characterized by the corresponding velocity of correlative motion and the temperature obtained by averaging within the specified area. The size of the averaging region is the result of a compromise between contradictory requirements for the largest possible size of statistical sample and the finest possible calculation (measurement) resolution. An example of the arrangement of the averaging regions is shown in Figure 6.7 for the case of a slender flat-head projectile hitting a rigid wall (the Taylor ballistic test [45]). The circular averaging areas (6.6b), which play the role of measuring gages, follow the movement of the atoms on which they are centered during the sample deformation. Accordingly, if during deformation the atoms, on which the averaging surfaces are centered, approach the edge of the deformed object, incomplete averaging may occur, which has different consequences for different macroscopic parameters (depending on their definitions; Equations (6.13)-(6.15)) which should be carefully examined in each specific case.