MOVABLE CELLULAR AUTOMATON METHOD AS A TREND IN DISCRETE COMPUTATIONAL MECHANICS

The paper presents the basics of movable cellular automaton method
aimed for simulating deformation and fracture of materials and media at different scales. Initially, the particle method has been  employed in mechanics of materials only at microscale as molecular  dynamics. Its further development has been led to a group of  methods which are usually called as discrete element method and  used for simulation of loose and granular materials at the  macroscale. The presented method of movable cellular automata  was developed for simulating deformation and fracture of materials  at different scales: at mesoscale with an explicit account for material structure, and at macroscale within the framework of a media with effective properties. The main advantages and differences of  the approach compared with the other methods of discrete  computational mechanics are considered. These advantages, first of all, are determined by the fact that the considered approach is based on two basic methods of discrete simulation: particle method and  cellular automaton method. Employing the formalism of cellular  automata allows explicit description of both processes of damage  generation and evolution as well as of crack healing and  microwelding. More of that, it is possible to describe heat transfer, chemical reactions and phase transitions as well. The second  important advantage of the movable cellular automaton method is the many-body type of interaction among its elements. The use of  many-body interaction allows us to avoid artificial effect of the  particle packing and locality of their interaction on the resulting  behavior of the modeled material that is extremely important for  modeling elastic-plastic matereials. As a further development of the  considered approach, two techniques are discussed which enable to describe contact interaction of solid bodies surfaces at the microand mesoscopic scales within the framework of the particle method.

1. Darrigol, O. 2002, “Between hydrodynamics and elasticity theory: the first five births of the Navier-Stokes equation”, Arch. Hist. Exact Sci., vol. 56. pp. 95-150.

2. Levin, V. A. 2015, Nonlinear computational mechanics of strength. V. 1. Models and methods. Nucleation and development of defects. Eds. V. A. Levin , Physmatlit, Moscow. (inRussian)

3. Levin, V. A. & Vershinin, A. V. 2015, Nonlinear computational mechanics of strength. V. 3. Numerical methods. Implementation for high-performance computers. Eds. V. A. Levin, Physmatlit, Moscow. (inRussian)

4. Morozov, E. M., Levin, V. A. & Vershinin, A. V. 2015, Strength analysis. Fidesis for engineers, URRS, Moscow. (inRussian)

5. Cundall, P. A. & Strack, O. D. L. 1979, “A discrete numerical model for granular assemblies”, Geotechnique, vol. 29. No. 1. pp. 47-65.

6. Cundall, P. A. 1988, “A computer simulations of dense sphere assembles”, Micromechanics of granular materialsEds. Eds. by M. Satake and J.T. Jenkins. Amsterdam: Elsever Sci. Publ., pp. 113-123.

7. Herrmann, H. J. 1995, “Simulating granular media on the computer”, 3rd Granada lectures in computational physics. Eds. by P.L. Garrido and J. Marro. Heidelberg: Springer, pp. 67-114.

8. Hemmingsson, J., Herrmann, H. J. & Roux, S. 1997, “On stress networks in granular media”, J. Phys. I, vol. 7. pp. 291-302.

9. Walton, O. R. 1993, “Numerical simulation of inclined chute flows of monodisperse, inelastic, frictional spheres”, Mechanics of Materials, vol. 16. pp. 239-247.

10. Luding, S. 1995, “Granular materials under vibration: Simulations of rotating spheres”, Phys. Rev. E, vol. 52. No. 4. pp. 4442.

11. P¨oschel, T. 1993, “Granular material flowing down an inclined chute: A molecular dynamic simulation”, J. Phys. II, vol. 3. pp. 27.

12. Greenspan, D. 1988, “Particle modeling in science and technology”, Coll. Math. Societatis Janos Bolyai., №. 50. pp. 51.

13. Ostermayer, G.P. 2000, “Friction models with discrete layers”, Z. Angew. Math. Mech., vol. 80. pp. 61-64.

14. Mustoe, G. G. W. 1992, “A generalized formulation of the discrete element method”, Engineering computations, vol. 9. pp. 181-190.

15. Bi´cani´c, N. 2004, “Discrete element methods”, In: E. Stein, R. Borst, T.J.R. Hughes (Eds.), Encyclopedia of computational mechanics. Vol. 1: Fundamentals, Wiley, Chichester, pp. 311-337.

16. Hockney, R. W. & Eastwood, J. W. 1981, Numerical Simulation by the Particle Method, McGraw-Hill International Book Company.

17. Potter, D. 1973, Computational Physics, John Wiley & Sons Ltd., London.

18. Psakhie, S. G., Horie, Y., Korostelev, S.Yu., Smolin, A.Yu., Dmitriev, A. I., Shilko, E. V. & Alekseev, S. V., 1995, “Method of movable cellular automata as a tool for simulation within the framework of mesomechanics”, Russian Physics Journal, vol. 38. № 11, pp. 1157-1168.

19. Psakhie, S. G., Korostelev, S.Yu., Smolin, A.Yu., Dmitriev, A. I., Shilko, E. V., Moiseyenko, D,D., Tatarintsev, E. M. & Alexeev, S. V. 1998, “Movable cellular automata method as a tool for physical mesomechanics of materials”, Physical Mesomechanics, vol. 1, No. 1 pp. 89-102.

20. Dmitriev, A. I., Korostelev, S.Yu., Ostermeyer, G.P., Psakhie, S. G., Smolin, A. Yu. & Shilko, E.V. 1999, “Movable cellular automata method as a tool for simulation at the mesolevel”, Proc. of RAS. Mechanics of solids, vol. 6. pp. 87-94.

21. Psakhie, S., Shilko, E., Smolin, A., Astafurov, S. & Ovcharenko, V. 2013, “Development of a formalism of movable cellular automaton method for numerical modeling of fracture of heterogeneous elastic-plastic materials”, Frattura ed Integrit`a Strutturale, vol. 24. pp. 26-59.

22. Shilko, E. V., Psakhie, S. G., Schmauder, S., Popov, V. L., Astafurov, S. V. & Smolin, A.Yu. 2015, “Overcoming the limitations of distinct element method for multiscale modeling of materials with multimodal internal structure”, Computational Materials Science, vol. 102. pp. 267-285.

23. Mikhailov, A. S. 1994, Foundations of Synergetics I. Distributed Active Systems. Springer, Berlin.

24. Smolin, A. Y., Roman, N. V., Dobrynin, S. A. & Psakhie, S. G. 2009, “On rotation in the movable cellular automaton method”, Physical Mesomechanics vol. 12. № 2. pp. 17-22.

25. Foiles, S. M., Baskes, M. I. & Daw, M. S. 1986, Embeded-atom-method functions for the f.c.c. metals Cu, Ag, Au, Ni, Pd, Pt and their alloys”, Physical Review B, vol. 33. No. 12. pp. 7983-7991.

26. Psakhie, S. G., Zolnikov, K.P. & Kryzhevich, D. S. 2008, “Protodefect as a basis of multilevel nanoscale plasticity of crystal materials”, American Institute of Physics, vol. 999. pp. 20-31.

27. Korchuganov, A. V., Zolnikov, K.P., Kryzhevich, D. S. Chernov, V. M. & Psakhie, S. G. 2016, “MD simulation of plastic deformation nucleation in stressed crystallites under irradiation”, Physics of Atomic Nuclei, vol. 79. iss. 7. pp. 1193-1198.

28. Kragelskii, I. V. 1962, Friction and Wear, Mashgiz, Moscow.

29. Tworzydlo, W. W., Cecot, W., Oden, J. T. & Yew, C. H. 1999, “Computational micro- and macroscopic models of contact and friction: formulation, approach and applications”, Wear, vol. 220. P. 113-140.

30. Rozman, M. G., Urbakh, M. & Klafter, J. 1998, “Stick-slip dynamics of interfacial friction”, Physica A, vol. 249. pp. 184-189.

31. Raharijaona, F., Roizard, X. & Stebut, J. 1999. “Usage of 3D roughness parameters adapted to the experimental simulation of sheet-tool contact during a drawing operation”, Tribology International, vol. 32, pp. 59-67.