PSI - Issue 6

Yurii Meshcheryakov / Procedia Structural Integrity 6 (2017) 109–114 Author nam / Structural Integrity Procedia 00 (2017) 000 – 0 0

114 6

    - - 2 u

2 D D x u u     2

    - 2

     .

D D t u x    

u D D t u x 



+ 2 u

(26)



If   1 D D u u   

(27)

   

.The expression   D D u x  

2

u

u D D





 

can be transformed as

the Eq. (26) transforms into equation

2

t

t u x





  D D u x  

1 D D u u u u x t C       

1

  2 u 

, and Eq.(26) transform into u  

0  .

=

(28)

C

0

0

Solution to this equation is: u = u 0 + u 1 ln t.

(29)

Eq. (27) can be transformed into:     1 D D D D u u u u       . (30)

Thus, the condition (30) can be considered as criterion for change of uniform regime of shock-wave propagation to ‖catastrophic‖ regime. The condition for change of the regime of dynamic flow depends both on the ratio of the velocity variance and mean particle velocity and on the ratio of their time derivatives. Note, that analogous criterion for change of the regime of energy exchange has independently been obtain by Khantuleva (2016) by using the non-local approach.Thus, itmaybeconcludedthatstochasticnature of dynamic deformation plays a key role in dynamic heterogenization of materials. Acknowledgments This work is conducted in the frame of Russian Scientific Fund, Project No 17-11-01053. References Fradkov A.L. Cybernetic Physics. 2003. Saint-Petersburg. Science.208 p. Holian B.L., Lomdahl. P.S. 1998. Plasticity induced by shock waves in nonequilibrium molecular-dynamics simulations. Science,. 280. 2085 2088. Hubburd J. 1960. The friction and diffusion coefficients of the Fokker-Plank equation. Proc, Roy. Soc. A 260, 114-126. Khantuleva T.A. 2000.Microstructure formation in the framework of the non-local theory of interfaces. Material Phys. Mech.. 2, 51-62. Khantuleva T.A. 2002.The Shock Wave as a Nonequilibrium Transport Process. In: High-Pressure Shock Compression of Solids VI. Editors: Yu Ya. Horie, L. Davison, N.N. Thadhani. N.Y. Springer.215-254. Khantuleva T.A., Meshcheryakov Yu.I. 2016. On the instability of plastic flow at the mesoscale under high-velocity deformation of solid. Physical mesomechanics. 19, 5-13. Kihara T.,and Aono O. 1963. Unified theory of relaxation in plasma.Basic theorem. J. PhysSoc. Japan.18, 837-851. Krivtzov A.M. 1999.Relation between spall strength and mesoparticle velocity dispersion. Int. J. Impact Engineering..23 (1).477-487. Mescheryakov Yu.I. Meso-Macro energy exchange in shock deformed and fractured s olids.In.: ―High - Pressure Shock Compression of Solids VI‖. Springer, 2002, p p.169-213. Meshcheryakov Yu.I., Divakov A.K., Zhigacheva N.I., Makarevich I.P., Barakhtin B.K. 2008. Shock-induced structures in copper.Physical Review .B. 78. 64301 - 64316. Yano K. and Horie Y . 1999. , Discrete-element modeling of shock compression of polycrystalline copper. Physical Review B. 59. 13672-13680. Zubarev D N. 1961. Statistical operator for non-equilibrium systems.Reports of Russian Academy of Sciences USSA., 140, 92-95.