Mathematical Physics - Volume II - Numerical Methods

6.3 Particle Dynamics

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(6.24) for this value of the exponent. With regards to relation (6.24), P , Q , p and q are positive constants, and q > p in order to obtain a repulsive force necessarily much stronger than the attractive one. The Greenspan trailblazing method [60], based on interparticle interaction (6.24), exerted a profound influence on development of PD. Note that the condition of zero-stress state in the equilibrium configuration (6.24) 2 results in the expression for the equilibrium interparticle distance r i j 0 = Q P 1 q − p (6.25) defined in tems of the four parameters of the interparticle potential (6.24) 1 . In principle, the equilibrium interparticle distance can be shrinked to the level of a few angstoms in which case the PD model is reduced to the MD model. An ubiquitous step, common to all CMD methods, is to ensure that the assignment of model parameters results in appropriate physical and mechanical properties of the material. With regards to that crucial step, Wang and Ostoja-Starzewski [61] introduced a novel principle of equivalence between the meso-scale particle model and the micro-scale MD model, while using the Greenspan modeling approach based on the interparticle potential (6.24). This was achieved by establishment of four equality conditions (releted to mass, elastic strain energy, modulus of elasticity and tensile strength) for the purpose of determination of four unknown model parameters in (6.24). In this way, the equations for parameters P , Q , p and q are derived and an appropriate parametric study conducted in order to determine the model parameters under given conditions. For example, Figure 6.12 shows the dependence of the modulus of elasticity and tensile strength on the parameters ( p , q ) of the interparticle potential (6.24) 1 , arrived at by simulations at constant values of the sample volume and the equilibrium interparticle distance [61].

Figure 6.12: Modulus of elasticity and tensile strength dependance upon a pair of parameters ( p , q ) at fixed values of the interparticle distance (0.2 [cm]) and sample volume (8 . 0 × 11 . 4 × 3 . 1 [ cm 3 ] ). (Reproduced from reference data [61].)