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A variational discrete element method for the computation of Cosserat elasticity

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Abstract

The variational discrete element method developed in Marazzato et al. (Int J Numer Methods Eng, 121(23):5295–5319, 2020) for dynamic elasto-plastic computations is adapted to compute the deformation of elastic Cosserat materials. In addition to cellwise displacement degrees of freedom (dofs), cellwise rotational dofs are added. A reconstruction is devised to obtain \(P^1\) non-conforming polynomials in each cell and thus constant strains and stresses in each cell. The method requires only the usual macroscopic parameters of a Cosserat material and no microscopic parameter. Numerical examples show the robustness of the method for both static and dynamic computations in two and three dimensions.

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References

  1. Fundamental Algorithms for Scientific Computing in Python (2020) SciPy 1.0. Nat Methods 17:261–272

    Article  Google Scholar 

  2. André D, Girardot J, Hubert C (2019) A novel DEM approach for modeling brittle elastic media based on distinct lattice spring model. Comput Methods Appl Mech Eng 350:100–122

    Article  MathSciNet  Google Scholar 

  3. André D, Iordanoff I, Charles J-L, Néauport J (2012) Discrete element method to simulate continuous material by using the cohesive beam model. Comput Methods Appl Mech Eng 213:113–125

    Article  Google Scholar 

  4. André D, Jebahi M, Iordanoff I, Charles J-L, Néauport J (2013) Using the discrete element method to simulate brittle fracture in the indentation of a silica glass with a blunt indenter. Comput Methods Appl Mech Eng 265:136–147

    Article  Google Scholar 

  5. Arnold D (1982) An interior penalty finite element method with discontinuous elements. SIAM J Numer Anal 19(4):742–760

    Article  MathSciNet  Google Scholar 

  6. Avci B, Wriggers P (2012) A DEM-FEM coupling approach for the direct numerical simulation of 3D particulate flows. J Appl Mech 79(1):2698

    Article  Google Scholar 

  7. Balay S, Abhyankar S, Adams M, Brown J, Brune P, Buschelman K, Dalcin L, Dener A, Eijkhout V, Gropp W (2019) Petsc users manual

  8. Belytschko T, Hughes TJR (1983) Computational methods for transient analysis, vol 1. North-Holland(Computational Methods in Mechanics, Amsterdam

    MATH  Google Scholar 

  9. Bleyer J (2018) Numerical tours of computational mechanics with fenics

  10. Boiveau T, Burman E (2016) A penalty-free Nitsche method for the weak imposition of boundary conditions in compressible and incompressible elasticity. IMA J Numer Anal 36(2):770–795

    Article  MathSciNet  Google Scholar 

  11. Brocato M, Capriz G (2001) Gyrocontinua. Int J Solids Struct 38(6–7):1089–1103

    Article  Google Scholar 

  12. Burman E (2012) A penalty-free nonsymmetric Nitsche-type method for the weak imposition of boundary conditions. SIAM J Numer Anal 50(4):1959–1981

    Article  MathSciNet  Google Scholar 

  13. Cosserat E, Cosserat F (1909) Théorie des corps déformables. A. Hermann et fils

  14. Cundall P, Strack O (1979) A discrete numerical model for granular assemblies. Geotechnique 29(1):47–65

    Article  Google Scholar 

  15. Dalcin L, Paz R, Kler P, Cosimo A (2011) Parallel distributed computing using python. Adv Water Resour 34(9):1124–1139

    Article  Google Scholar 

  16. Di Pietro DA (2012) Cell centered Galerkin methods for diffusive problems. ESAIM Math Modell Numer Anal 46(1):111–144

    Article  MathSciNet  Google Scholar 

  17. Ericksen JL (1974) Liquid crystals and Cosserat surfaces. Q J Mech Appl Math 27(2):213–219

    Article  Google Scholar 

  18. Eymard R, Gallouët T, Herbin R (2009) Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces. IMA J Numer Anal 30(4):1009–1043

    Article  MathSciNet  Google Scholar 

  19. David C-LF, Saunders M (2011) Lsmr: an iterative algorithm for sparse least-squares problems. SIAM J Sci Comput 33(5):2950–2971

    Article  MathSciNet  Google Scholar 

  20. Forest S, Pradel F, Sab K (2001) Asymptotic analysis of heterogeneous Cosserat media. Int J Solids Struct 38(26–27):4585–4608

    Article  MathSciNet  Google Scholar 

  21. Godio M, Stefanou I, Sab K, Sulem J, Sakji S (2017) A limit analysis approach based on Cosserat continuum for the evaluation of the in-plane strength of discrete media: application to masonry. Eur J Mech A/Solids 66:168–192

    Article  MathSciNet  Google Scholar 

  22. Hoover WG, Ashurst WT, Olness RJ (1974) Two-dimensional computer studies of crystal stability and fluid viscosity. J Chem Phys 60(10):4043–4047

    Article  Google Scholar 

  23. Jebahi M, André D, Terreros I, Iordanoff I (2015) Discrete element method to model 3D continuous materials. Wiley, New York

    Book  Google Scholar 

  24. Jeong J, Neff P (2010) Existence, uniqueness and stability in linear Cosserat elasticity for weakest curvature conditions. Math Mech Solids 15(1):78–95

    Article  MathSciNet  Google Scholar 

  25. Lamb H (1904) On the propagation of tremors over the surface of an elastic solid. Philos Trans R Soc London Ser A 203(359–371):1–42

    MATH  Google Scholar 

  26. Logg A, Mardal K-A, Wells G (2012) Automated solution of differential equations by the finite element method: the FEniCS book, vol 84. Springer, Berlin

    Book  Google Scholar 

  27. Logg A, Wells GN (2010) Dolfin: automated finite element computing. ACM Trans Math Softw 37(2):10298

    Article  MathSciNet  Google Scholar 

  28. Marazzato F, Ern A, Monasse L (2020) A variational discrete element method for quasistatic and dynamic elastoplasticity. Int J Numer Methods Eng 121(23):5295–5319

    Article  MathSciNet  Google Scholar 

  29. Marazzato F, Ern A, Monasse L (2021) Quasi-static crack propagation with a Griffith criterion using a discrete element method

  30. Mariotti C (2007) Lamb’s problem with the lattice model Mka3D. Geophys J Int 171(2):857–864

    Article  Google Scholar 

  31. Michael M, Vogel F, Peters B (2015) DEM-FEM coupling simulations of the interactions between a tire tread and granular terrain. Comput Methods Appl Mech Eng 289:227–248

    Article  MathSciNet  Google Scholar 

  32. Monasse L, Mariotti C (2012) An energy-preserving Discrete Element Method for elastodynamics. ESAIM Math Modell Numer Anal 46:1527–1553

    Article  MathSciNet  Google Scholar 

  33. Neff P (2006) A finite-strain elastic-plastic Cosserat theory for polycrystals with grain rotations. Int J Eng Sci 44(8–9):574–594

    Article  MathSciNet  Google Scholar 

  34. Neff P, Chelminski K (2007) Well-posedness of dynamic Cosserat plasticity. Appl Math Optim 56(1):19–35

    Article  MathSciNet  Google Scholar 

  35. Notsu H, Kimura M (2014) Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. Netw Heterog Media 9(4):10265

    MathSciNet  MATH  Google Scholar 

  36. Potyondy D, Cundall P (2004) A bonded-particle model for rock. Int J Rock Mech Min Sci 41(8):1329–1364

    Article  Google Scholar 

  37. Providas E, Kattis MA (2002) Finite element method in plane Cosserat elasticity. Comput Struct 80(27–30):2059–2069

    Article  Google Scholar 

  38. Puscas MA, Monasse L, Ern A, Tenaud C, Mariotti C (2015) A conservative Embedded Boundary method for an inviscid compressible flow coupled with a fragmenting structure. Int J Numer Methods Eng 103(13):970–995

    Article  MathSciNet  Google Scholar 

  39. Rattez H, Stefanou I, Sulem J (2018) The importance of Thermo-Hydro-Mechanical couplings and microstructure to strain localization in 3D continua with application to seismic faults. Part i: Theory and linear stability analysis. J Mech Phys Solids 115:54–76

    Article  MathSciNet  Google Scholar 

  40. Rattez H, Stefanou I, Sulem J, Veveakis M, Poulet T (2018) Numerical analysis of strain localization in rocks with thermo-hydro-mechanical couplings using Cosserat continuum. Rock Mech Rock Eng 51(10):3295–3311

    Article  Google Scholar 

  41. Ries A, Wolf D, Unger T (2007) Shear zones in granular media: three-dimensional contact dynamics simulation. Phys Rev E 76(5):051301

    Article  Google Scholar 

  42. Sautot C, Bordas S, Hale J (2014) Extension of 2D FEniCS implementation of Cosserat non-local elasticity to the 3D case. Technical report, Université du Luxembourg

  43. Shelukhin VV, Ružička M (2013) On Cosserat-Bingham fluids. ZAMM-J Appl Math Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 93(1):57–72

    Article  MathSciNet  Google Scholar 

  44. Stefanou I, Sulem J, Vardoulakis I (2008) Three-dimensional Cosserat homogenization of masonry structures: elasticity. Acta Geotech 3(1):71–83

    Article  Google Scholar 

  45. Sulem J, Vardoulakis IG (1995) Bifurcation analysis in geomechanics. CRC Press, London

    Book  Google Scholar 

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Acknowledgements

The author would like to thank A. Ern from Inria and Ecole Nationale des Ponts et Chaussées for stimulating discussions and L. Monasse from Inria for carefully proof-reading this manuscript. The author would also like to thank the anonymous reviewers for their contributions which helped substantially improve this paper.

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  1. Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803, USA

    Frédéric Marazzato

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  1. Frédéric Marazzato
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Correspondence to Frédéric Marazzato.

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https://github.com/marazzaf/DEM_cosserat.git

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Marazzato, F. A variational discrete element method for the computation of Cosserat elasticity. Comput Mech 68, 1097–1109 (2021). https://doi.org/10.1007/s00466-021-02060-y

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