PSI - Issue 6

Yurii Meshcheryakov / Procedia Structural Integrity 6 (2017) 109–114 Author name / Structural Integrity Procedia 00 (2017) 000 – 000

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Themesostructurehasthefollowing importantfeatures: 1. Mesostructure isa totality ofpolarizedinsigndislocationcharges. 2 Interaction of mesoparticles is of long-range character.

In this situation, there is a certain analogy of mesostructure and plasma of charged particles – electrons and ions. We use this formal analogy for description the coupling between macroscopic response of medium on shock loading and kinetics of mesostructure.The behavior of assembles with random character of interaction is known to be described on the basis Fokker-Plank kinetic equation: f t   + v f r   + u  f u   = - u   ( F 1 f ) + 2 2 u   ( F 2 f ). (5) Thelefthandsideofthisequationdescribesaconvectivetransportationofprobabilitydensitywhereas the right part indicates for what mechanisms this transportation occurs.The first item in the right part of the equation, the so-called ―drift‖ item, characterizes an averaged change of probability density in the velocity space whereas the second item describes an influence of random velocity fluctuations. Coefficients F 1 and F 2 are the diffusion coefficients of Fokker-Plank equation: F 1 = 2 u t   isthecoefficientofdynamicfrictionand F 2 is the diffusion coefficient in the velocity space which characterizes the rate ofchange of the particle velocity dispersion: F 2 = 1 1 u u t    . (6) Thedifferencebetween 1 u  and 2 u  followsfromtheassumptionintroducedbyHubbard (1960) for assemble of charged particles.By analogy with assemble of charged particles, themotionofassembleofchargedmesoparticlesintherandomstressfields (assequenceoflong rangecharacterofinteraction) canbepresentedinformofsuperpositionoftwomodesofmotion: (i) ―Mode 1‖ is an averaged motion of approximately plane shock front on the time interval between two successive interactions of mesoparticles and (ii) ―Mode 2‖ is quickly fluctuating motions of mesoparticles due to action of random stress fields. Inthisapproachthevalue 1 u  isthemeandeviationofvelocitywhichissmallascomparedtomacroscopicchangeofvelocity,forexample, attheplasticfrontofshockwave. However, it istoolargeascomparedtovelocitychangecausedbythe mutualinteractionsof mesoparticleswitheachother. Δ t isthetimeintervalforwhichthemeanvelocitychanges by 1 u  . Since the Mode 1 is the motion of mesoparticles between two successive correlations,thismotionflowsatmeanvelocity u , which is determined by action of stress σ ( u τ , τ ). In this case, the change of velocity is determined by integral: 2 1 Here σ isthestress which is determined by mean displacement of particle between two successive correlations, s isthesquareofmesoparticleand m isthemassofmesoparticle. Dynamic friction coefficient in the Fokker-Plank equation, 1 1 u F t    ,characterizes the mean acceleration (deceleration) of mesoparticles. In dynamically deformed medium, the first and second diffusion coefficients prove to be not independent. To show that we can use the formalism of correlation functions which is used to characterize the fluctuative features of media. Similar formalism has been applied by Hubburd (1960) and Kihara and Aono (1963) for determination the diffusion coefficients of Fokker-Plank equation in plasma. The basis for application of this approach to mesostructure is the long-range character of interaction of mesoparticles, similar to charged particles of plasma. In determination oftheseconddiffusioncoefficient, the Mode M2 of dynamic straining should be taken into account: 0 ( , )     u d t     u s m (7)