Mathematical Physics - Volume II - Numerical Methods

6.3 Particle Dynamics

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Table 6.2: Examples of representative temporal and spatial scales necessary to observe some typical deformation mechanisms under step-pressure loading of amorphous brittle materials (adopted from [59]).

Mechanism

Representative Length Scale Representative Time Scale

100 µ m

Crack nucleation Crack coalescence

10 ns

1 mm 1 mm 1 mm

100 ns

1 µ s 1 µ s

Comminution Fragment flow

10 µ s 10 µ s

Interfragment friction Interfragment rotation

10 mm 10 mm

(iv) Output oversaturation (“data glut”). The above-mentioned detailed insight into the mechanisms of the studied phenomena on the atomic scale with extremely fine time resolution can lead to the oversaturation with the raw MD output data ( r , v , a ).

6.3 Particle Dynamics Particle dynamics (PD) is one of many computational methods developed to bridge the gap between the microscopic and macroscopic spatial scales (Figure 6.2). As presented herein, it is an engineering offshoot of MD on an arbitrarily selected spatial scale (Figure 6.11). Since PD has MD techniques at the root, it is sometimes called quasi-MD. Thus, the basic distinguishing features of PD in relation to MD lie in the coarser spatial scale and, in that regard, the phenomenological constitutive model that defines the interparticle interaction. It will be shown in this chapter that this constitutive model may or may not have a functional form of the empirical interatomic potentials. Other than that, the computational simulation techniques used in PD modeling are largely the same as those well known from the traditional MD literature [5], [15], [17]. Greenspan [60] contributed the most to the early development of particle methods as presented in this introduction. 6.3.1 Basic Idea of PD Basically, the PD system consists of material points (referred to as particles in this chapter) of known masses m i , and positions r i ( i = 1 , . . . , N ). Depending on the material being modeled (fluid, amorphous or crystalline solids), these material points can be arranged randomly or regularly according to the topology of an underlying network. As discussed in Chapter 6.2, the known initial configuration defines the reference state. The calculation methodology requires an approximate solution of the system of differential equations with given initial conditions. At an arbitrary time ( t > 0), the position and momentum of each particle are completely determined by Newton’s laws of motion (6.1) 1 . Therefore, the movement of each particle is deterministic. As already mentioned, the well-established MD techniques have been adapted to simulate such coarser-scale material systems with the role of atoms being taken over by a different kind of material points—the large chunks of material—often called a continuum particle or quasi-particle. For simplicity, these material points often interact with each