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Handbook of Mechanics of Materials pp 1–40Cite as

Coupling of Discrete and Continuum Approaches in Modeling the Behavior of Materials

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Abstract

For computer simulation of the mechanical behavior of materials and various media, methods of continuum mechanics are mainly used. Continuum approach uses highly developed mathematical apparatus of continuous functions, and capabilities of this approach are extremely wide and well known. However, for a number of very important processes, such as severe plastic deformation, mass mixing, damage initiation and development, material fragmentation, and so on, continuum methods of solid mechanics face certain hard difficulties. As a result, a great interest for the approach based on a discrete description of materials and media has been growing up in recent years.

It is obvious that both continuum and discrete approaches have their own advantages and disadvantages. A great number of commercial software has been created for solving numerous scientific and engineering problems based on continuum mechanics. Hence, the main line of discrete approach development seems to be not a substitute but a supplement to continuum methods in solving complex specific problems based on a coupling of the continuum and discrete approaches.

This chapter shows how to solve this problem on the example of the finite-difference method for numerical solution of dynamic problems on elastic-plastic deformation of continua, which is based on the continuum approach, and the movable cellular automaton method based on the discrete approach. Two particular applications of the coupled method are considered. The first example concerns simulation of target penetration by long rod at the macroscale. The second one deals with simulation of sliding friction at the scale of contact patch (mesoscale).

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References

  1. Darrigol O. Between hydrodynamics and elasticity theory: the first five births of the Navier-stokes equation. Arch Hist Exact Sci. 2002;56:95–150.

    Article  MathSciNet  MATH  Google Scholar 

  2. Cundall PA, Strack ODL. A discrete numerical model for granular assemblies. Geotechnique. 1979;29(1):47–65.

    Article  Google Scholar 

  3. Herrmann HJ. Simulating granular media on the computer. In: Garrido PL, Marro J, editors. 3rd Granada lectures in computational physics. Heidelberg: Springer; 1995.

    Google Scholar 

  4. Luding S. Granular materials under vibration: simulation of rotating spheres. Phys Rev E. 1995;52(4):4442–57.

    Article  Google Scholar 

  5. Poschel T. Granular material flowing down an inclined chute: a molecular dynamics simulation. J Phys II. 1993;3(1):27–40.

    Google Scholar 

  6. Herrmann HJ, Luding S. Modeling granular media on the computer. Contin Mech Thermodyn. 1998;10(4):189–231.

    Article  MathSciNet  MATH  Google Scholar 

  7. Psakhie SG, Horie Y, Korostelev SY, Smolin AY, Dmitriev AI, Shilko EV, Alekseev SV. Method of movable cellular automata as a tool for simulation within the framework of physical mesomechanics. Russ Phys J. 1995;38(11):1157–68.

    Article  Google Scholar 

  8. Psakhie SG, Ostermeyer GP, Dmitriev AI, Shilko EV, Smolin AY, Korostelev SY. Movable cellular automata method as a new trend in computational mechanics. I. Theoretical description. Phys Mesomech. 2000;3(2):5–12.

    Google Scholar 

  9. Psakhie SG, Chertov MA, Shilko EV. Interpretation of the parameters of the method of movable cellular automata on the basis of continuum description. Phys Mesomech. 2000;3(3):89–92.

    Google Scholar 

  10. Psakhie SG, Horie Y, Ostermeyer GP, Korostelev SY, Smolin AY, Shilko EV, Dmitriev AI, Blatnik S, Spegel M, Zavsek S. Movable cellular automata method for simulating materials with mesostructure. Theor Appl Fract Mech. 2001;37(1–3):311–34.

    Article  Google Scholar 

  11. Psakhie SG, Ruzhich VV, Smekalin OP, Shilko EV. Response of the geological media to dynamic loading. Phys Mesomech. 2001;4(1):63–6.

    Google Scholar 

  12. Goldin SV, Psakhie SG, Dmitriev AI, Yushin VI. Structure rearrangement and “lifting” force phenomenon in granular soil under dynamic loading. Phys Mesomech. 2001;4(3):91–7.

    Google Scholar 

  13. Smolin AY, Shilko EV, Astafurov SV, Konovalenko IS, Buyakova SP, Psakhie SG. Modeling mechanical behaviors of composites with various ratios of matrix–inclusion properties using movable cellular automaton method. Def Technol. 2015;11(1):18–34.

    Article  Google Scholar 

  14. Psakhie SG, Smolin AY, YuP S, Makarov PV, Shilko EV, Chertov MA, Evtushenko EP. Simulation of behavior of complex media on the basis of a discrete-continuous approach. Phys Mesomech. 2003;6(5–6):47–56.

    Google Scholar 

  15. Psakhie SG, Smolin AY, Stefanov YP, Makarov PV, Chertov MA. Modeling the behavior of complex media by jointly using discrete and continuum approaches. Tech Phys Lett. 2004;30(9):712–4.

    Article  Google Scholar 

  16. Wilkins ML. Calculation of elastic-plastic flow. In: Alder B, Fernbach S, Rotenberg M, editors. Methods of computational physics, Fundamental methods in hydrodynamics, vol. 3. New York: Academic Press; 1964.

    Google Scholar 

  17. Wilkins ML. Computer simulation of dynamic phenomena. Berlin: Springer; 1999.

    Book  MATH  Google Scholar 

  18. Daw MS, Foiles SM, Baskes MI. The embedded-atom method: a review of theory and applications. Materials Science Reports. 1993;9:251–310.

    Article  Google Scholar 

  19. Psakhie SG, Shilko EV, Smolin AY, Dimaki AV, Dmitriev AI, Konovalenko IS, Astafurov SV, Zavshek S. Approach to simulation of deformation and fracture of hierarchically organized heterogeneous media, including contrast media. Phys Mesomech. 2011;14(5–6):224–48.

    Article  Google Scholar 

  20. Psakhie SG, Shilko EV, Grigoriev AS, Astafurov SV, Dimaki AV, Smolin AY. A mathematical model of particle-particle interaction for discrete element based modeling of deformation and fracture of heterogeneous elastic-plastic materials. Eng Fract Mech. 2014;130:96–115.

    Article  Google Scholar 

  21. Shilko EV, Psakhie SG, Schmauder S, Popov VL, Astafurov SV, Smolin AY. Overcoming the limitations of distinct element method for multiscale modeling of materials with multimodal internal structure. Comput Mater Sci. 2015;102:267–85.

    Article  Google Scholar 

  22. Potyondy DO, Cundall PA. A bonded-particle model for rock. Int J Rock Mech Min Sci. 2004;41(8):1329–64.

    Article  Google Scholar 

  23. Smolin AY, Roman NV, Dobrynin SA, Psakhie SG. On rotation in the movable cellular automaton method. Phys Mesomech. 2009;12(3–4):124–9.

    Article  Google Scholar 

  24. Noh VF. CEL: a time-dependent two-space-dimensional coupled Eulerian-Lagrangian code. In: Alder B, Fernbach S, Rotenberg M, editors. Methods of computational physics, Fundamental methods in hydrodynamics, vol. 3. New York: Academic Press; 1964.

    Google Scholar 

  25. Johnson GR. Dynamic response of axisymmetric solids subjected to impact and spin. AIAA J. 1979;17(9):975–9.

    Article  Google Scholar 

  26. Gulidov AI, Fomin VM, Shabalin II. Mathematical simulation of fracture in impact problems with formation of fragments. International Journal of Fracture. 1999;100(2):183–96.

    Article  Google Scholar 

  27. Stefanov YP. Wave dynamics of cracks and multiple contact surface interaction. Theor Appl Fract Mech. 2000;34(2):101–8.

    Article  Google Scholar 

  28. Stefanov YP, Bakeev RA, Rebetsky YL, Kontorovich VA. Structure and formation stages of a fault zone in a geomedium layer in strike-slip displacement of the basement. Phys Mesomech. 2014;17(3):204–15.

    Article  Google Scholar 

  29. Stefanov Y, Bakeev RA. Deformation and fracture structures in strike-slip faulting. Eng Fract Mech. 2014;129:102–11.

    Article  Google Scholar 

  30. Johnson GR Stryk RA, Holmquist TR, Souka OA. Recent EPIC code developments for high velocity impact: 3D element arrangements and 2D fragment distributions. International Journal of Impact Engineering. 1990;10:281–94.

    Article  Google Scholar 

  31. Beissel SR, Johnson GR. Large-deformation triangular and tetrahedral element formulations for unstructured meshes. Comput Methods Appl Mech Eng. 2000;187:469–82.

    Article  MATH  Google Scholar 

  32. Johnson GR, Stryk RA, Beissel SR. SPH for high velocity impact computations. Comput Methods Appl Mech Eng. 1996;139:347–73.

    Article  MATH  Google Scholar 

  33. Johnson GR, Beissel SR, Stryk RA. An improved generalized particle algorithm that includes boundaries and interfaces. Int J Numer Methods Eng. 2002;53:875–904.

    Article  Google Scholar 

  34. Chertov MA, Smolin AY, Shilko EV, Psakhie SG. Simulation of complex plane targets penetration by long rod using MCA method. In: Sih GC, Kermanidis TB, Pantelakis SG, editors. Multiscaling in applied science and emerging technology, fundamentals and applications in mesomechanics: proceedings of the sixth international conference for mesomechanics. Patras; 2004.

    Google Scholar 

  35. Gust WH. High impact deformation of metal cylinders at elevated temperatures. J Appl Phys. 1982;53(5):3566–75.

    Article  Google Scholar 

  36. Fomin VM, Gulidov AI, Sapozhnikov GA, et al. High-velocity interaction of solids. Novosibirsk: Izdatelstvo SO RAN; 1999. (in Russian)

    Google Scholar 

  37. Smolin AY, Konovalenko IS, Psakhie SG. Identification of elastic waves generated in the contact zone of a friction couple. Tech Phys Lett. 2007;33(7):600–3.

    Article  Google Scholar 

  38. Psakhie SG, Popov VL, Shilko EV, Smolin AY, Dmitriev AI. Spectral analysis of the behavior and properties of solid surface layers. Nanotribospectroscopy. Phys Mesomech. 2009;12(5–6): 221–34.

    Article  Google Scholar 

  39. Dmitriev AI, Smolin AY, Popov VL, Psakhie SG. A multilevel computer simulation of friction and wear by numerical methods of discrete mechanics and a phenomenological theory. Phys Mesomech. 2009;12(1–2):11–9.

    Article  Google Scholar 

  40. Blackmore D, Zhou J. A general fractal distribution function for rough surface profiles. SIAM J Appl Math. 1996;56(6):1694–719.

    Article  MathSciNet  MATH  Google Scholar 

  41. Venkatachalam R. Mechanical vibrations. Delhi: PHI Learning Private Limited; 2014.

    Google Scholar 

  42. Dmitriev AI, Popov VL, Psakhie SG. Simulation of surface topography with the method of movable cellular automata. Tribol Int. 2006;39(5):444–9.

    Article  Google Scholar 

  43. Mallat S. A wavelet tour of signal processing. San Diego: Academic Press; 1999.

    MATH  Google Scholar 

  44. Bacry E (2002) LastWave home page [Internet]. Updated 20 Jun 2002; Accessed 19 Sept 2016. Available from: http://www.cmap.polytechnique.fr/~lastwave

  45. Oñate E, Rojek J. Combination of discrete element and finite element methods for dynamic analysis of geomechanics problems. Comput Methods Appl Mech Eng. 2004;193(27–29): 3087–128.

    Article  MATH  Google Scholar 

  46. Johnson GR. Numerical algorithms and material models for high-velocity impact computations. International Journal of Impact Engineering. 2011;38(6):456–72.

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Strength Physics and Materials Science of Siberian Branch of Russian Academy of Sciences, Tomsk, Russia

    Alexey Yu. Smolin, Igor Yu. Smolin, Evgeny V. Shilko, Yuri P. Stefanov & Sergey G. Psakhie

  2. Tomsk State University, Tomsk, Russia

    Alexey Yu. Smolin, Igor Yu. Smolin & Evgeny V. Shilko

  3. Trofimuk Institute of Petroleum Geology and Geophysics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia

    Yuri P. Stefanov

  4. Tomsk Polytechnic University, Tomsk, Russia

    Sergey G. Psakhie

  5. Skolkovo Institute of Science and Technology, Skolkovo, Russia

    Sergey G. Psakhie

Authors
  1. Alexey Yu. Smolin
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  2. Igor Yu. Smolin
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  3. Evgeny V. Shilko
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  4. Yuri P. Stefanov
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  5. Sergey G. Psakhie
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Corresponding author

Correspondence to Alexey Yu. Smolin .

Editor information

Editors and Affiliations

  1. Materials Science and Strength of Materials, University of Stuttgart Institute for Materials Testing, Stuttgart, Germany

    Siegfried Schmauder

  2. Department of Civil Engineering, National Taiwan University Department of Civil Engineering, Taipei, T´ai-pei, Taiwan

    Chuin-Shan Chen

  3. BEC 254, University of Alabama at Birmingham, Birmingham, Alabama, USA

    Krishan K. Chawla

  4. Fulton School of Engineering, Arizona State University, Tempe, Arizona, USA

    Nikhilesh Chawla

  5. Engineering Mechanics, Zhejiang University Engineering Mechanics, Hangzhou, China

    Weiqiu Chen

  6. Katayanagi Advanced Research Laboratorie, Tokyo University of Technology Katayanagi Advanced Research Laboratorie, Tokyo, Japan

    Yutaka Kagawa

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© 2018 Springer Nature Singapore Pte Ltd.

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Smolin, A.Y., Smolin, I.Y., Shilko, E.V., Stefanov, Y.P., Psakhie, S.G. (2018). Coupling of Discrete and Continuum Approaches in Modeling the Behavior of Materials. In: Schmauder, S., Chen, CS., Chawla, K., Chawla, N., Chen, W., Kagawa, Y. (eds) Handbook of Mechanics of Materials. Springer, Singapore. https://doi.org/10.1007/978-981-10-6855-3_35-1

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  • DOI: https://doi.org/10.1007/978-981-10-6855-3_35-1

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  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-10-6855-3

  • Online ISBN: 978-981-10-6855-3

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