Mathematical Physics - Volume II - Numerical Methods

6.2 Molecular Dynamics

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material properties and the limited number of available experiments. Comprehensive DEM reviews were published in [9-11]. Regardless of the type of the CMD model, the evolution of a system of particles (the discrete structural elements in the most general sense) of known masses m i , moments of inertia I i , and positions r i ( i = 1 , 2 , . . . , N ) is obtained by solving a system of equations of motion for each particle. Within classical mechanics, the equations that define the translational and rotational motion are those that correspond to Newton’s second law

d p i d t d L i d t

d d t d d t

( m i ˙ r i ) = F i ,

=

(6.1)

( I i ω i ) = M i

=

where p i and L i mark, respectively, the linear and angular momenta of i -th particle and F i and M i corresponding forces and moments. Obviously, the motion of material points (all CMD systems except DEM) is completely defined by Equation (6.1) 1 . (Hereinafter, symbols in boldface designate vectors and tensors.) Lastly, the following brief introduction to CMD stems out of the two-part essay published in coauthorship with Antonio Rinaldi in “Handbook of Damage Mechanics: Nano to Macro Scale for Materials and Structures” [1]. Despite renewed efforts to make it as representative and comprehensive as possible, its content suffers unavoidably from the author’s bias due to research backround and interests.

6.2 Molecular Dynamics

This short introduction aims to outline the basics of the traditional molecular dynamics (MD) method based on the classical mechanics. This computational simulation technique allows prediction of the temporal evolution of a system of material points (Figure 6.4) interacting via empirical interatomic potentials or molecular mechanics force fields by numerically solving Newton’s equations of motion. Simulation methods based on quantum mechanics are beyond the scope of this overview, as well as many other advanced topics. The first research article [12] in which Alder and Wainwright used MD to simulate perfectly elastic collisions of hard spheres was published in 1957. In 1960, Gibson and co-authors used a Born-Mayer potential to simulate a radiation damage of solid copper. Rahman (1964) simulated liquid argon by using 864 atoms interacting with a Lennard Jones potential [13]. The first computer simulation of a simplified protein folding was produced in 1975 [14]. These pioneering articles were published more than half a century ago. Therefore, it is not surprising that many outstanding monographs devoted to computer simulations in condensed matter physics are available to the interested researcher (e.g., [5,15-17]).