Mathematical Physics - Volume II - Numerical Methods

6.3 Particle Dynamics

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Figure 6.13: Nonlinear hybrid potential (6.26). Schematic representation of the interaction between particles that (a) are or were first-nearest neighbors, and (b) were not first-nearest neighbors initially. The yellow arrow indicates the effect of increasing the parameter B , which defines the slope of the repulsive wall [62]. The latter can be identified in dynamic simulations, for example, by matching the ballistic equation of state. Thus, the average link stiffness of the interparticle potential is determined uniquely by the value of modulus of elasticity of the pristine material The expression (6.27) is equivalent to (6.41) of the corresponding triangular lattice model with central interactions. This equivalence also implies the fixed value of Poisson’s ratio, v ( 3 D ) = 1 / 4 (6.39) 1 . The hybrid potential (6.26) was introduced to reproduce some underlying features of the deformation process typical for the considered materials, such as: brittle behavior in tension, increase of shock wave velocity and decrease of compressibility with increasing pressure. It should be noted that linear interparticle interactions, characteristic of elastic-brittle behavior of materials (and the traditional spring-network models of Chapter 6.4.2), can be considered a modification of the above hybrid potential, with the attractive part of the potential (6.26) 2 being used in the repulsion domain as well [63]. Finally, Watson and Steinhauser [64] used a conceptual solution of interparticle inter action very similar to the one of Mastilovic´ and Krajcˇinovic´ [62] to model the phenomena of hypervelocity impact. The main difference is that their interparticle repulsion instead of the Born-Meyer (6.26) 1 used the ubiquitous Lennard-Jones 6-12 potential (6.7), while the Hooke potential (6.26) 2 was used in unaltered form in the attractive branch. It is interesting to note that for the purposes of their 3D simulations, Watson and Steinhauser [64] adjusted the remaining two parameters of the model—corresponding to the depth of the potential well in (6.7) and the bond stiffness in (6.26) 2 — by a fitting procedure based on a series of hypervelocity impact experiments (a sphere colliding with a thin plate). This nicely illustrates the connection between physical (laboratory) and virtual experiments that is becoming an integral part of contemporary computational modeling. k i j = 8 5 √ 3 E ( 3 D ) 0 . (6.27)