Mathematical Physics - Volume II - Numerical Methods

Chapter 6. Introduction to Computational Mechanics of Discontinua

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other according to a simple central-force rule that completely determines the strain energy density function based on their mutual position. A system of Newton differential equations of motion for a system of particles with defined momenta p i = m i ˙ r i is then approximated by an appropriate system of finite difference equations and then solved using one of the many available integration algorithms as outlined in Chapter 6.2.3.

Figure 6.11: Schematic illustration of mapping between a molecular (or atomic) structure and the PD model (represented by a coarse-grained MD model).

It is interesting to note that the time-reversible equation for calculating the next particle position of the system (6.9) can be modified into the following form r i ( t + δ t ) = r i ( t )+ η [ r i ( t ) − r i ( t − δ t )] + δ t 2 a i ( t ) (6.23) where η marks the dumping coefficient. The reversible scheme (6.23) is used (often with η = 0 . 95) in PD simulations of quasi-static problems for the purpose of dissipating kinetic energy in order to obtain an equilibrium configuration in a time-efficient manner. Assuming that the interparticle forces are conservative, the intensities of the central force with which the particle j acts on the neighboring particle i and the resulting force with which all the first neighbors act on the particle i can be calculated using expression (6.2). The standard form of time-reversible Equation (6.9) is recovered from (6.23) for η = 1 (no dissiption). 6.3.2 Interparticle Potentials The interparticle potential (which, as already mentioned, in the PD models plays the role of constitutive law) must be adopted as an initial modeling step. Along with the spatio temporal scale of simulations, this potential represents the basic difference between MD and PD. One of the most commonly used pair potentials for interparticle interactions φ ( r i j ) = − 1 p − 1 P r p − 1 i j + 1 q − 1 Q r q − 1 i j , q > p > 1 , f i j = − d φ ( r i j ) d r i j = − P r p i j + Q r q i j (6.24) represents a generalization of the well-known Lennard-Jones 6-12 potential (6.7). The limit case of the exponent p = 1 was analyzed in detail by Wang and Ostoja-Starzewski [61] with an alternative form of potential that is necessary due to the singularity of expression