Skip to main content
SpringerLink
Log in

A mixed finite element procedure of gradient Cosserat continuum for second-order computational homogenisation of granular materials

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

A mixed finite element (FE) procedure of the gradient Cosserat continuum for the second-order computational homogenisation of granular materials is presented. The proposed mixed FE is developed based on the Hu–Washizu variational principle. Translational displacements, microrotations, and displacement gradients with Lagrange multipliers are taken as the independent nodal variables. The tangent stiffness matrix of the mixed FE is formulated. The advantage of the gradient Cosserat continuum model in capturing the meso-structural size effect is numerically demonstrated. Patch tests are specially designed and performed to validate the mixed FE formulations. A numerical example is presented to demonstrate the performance of the mixed FE procedure in the simulation of strain softening and localisation phenomena, while without the need to specify the macroscopic phenomenological constitutive relationship and material failure model. The meso-structural mechanisms of the macroscopic failure of granular materials are detected, i.e. significant development of dissipative sliding and rolling frictions among particles in contacts, resulting in the loss of contacts.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

References

  1. Pasternak E, Muhlhaus HB (2005) Generalised homogenisation procedures for granular materials. J Eng Math 52:199–229

    Article  MathSciNet  MATH  Google Scholar 

  2. Chang CS, Kuhn MR (2005) On virtual work and stress in granular media. Int J Solids Struct 42:3773–3793

    Article  MATH  Google Scholar 

  3. Ehlers W, Ramm E, Diebels S, d’Addetta G (2003) From particle ensembles to Cosserat continua: homogenization of contact forces towards stresses and couple stresses. Int J Solids Struct 40:6681–6702

    Article  MathSciNet  MATH  Google Scholar 

  4. d’Addetta G, Ramm E, Diebels S, Ehlers W (2004) A particle center based homogenization strategy for granular assemblies. Eng Comput 21:360–383

    Article  MATH  Google Scholar 

  5. d’Addetta G, Ramm E, Diebels S, Ehlers W (2004) Static equations of the Cosserat continuum derived from intra-granular stresses. Granular Matter 13:189–196

    Google Scholar 

  6. Mühlhaus H-B, Oka F (1996) Dispersion and wave propagation in discrete and continuous models for granular materials. Int J Solids Struct 33:2841–2858

    Article  MATH  Google Scholar 

  7. Hicher PY, Chang CS (2005) Evaluation of two homogenization techniques for modeling the elastic behaviors of granular matrials. J Eng Mech 131:1184–1194

    Article  Google Scholar 

  8. Nemat-Nasser S, Hori M (1999) Micromechanics: overall properties of heterogeneous materials. Elsevier, Amsterdam

    Google Scholar 

  9. Chang CS, Ma L (1992) Elastic material constants for isotropic granular solids with particle rotation. Int J Solids Struct 29:1001–1018

  10. Suiker ASJ, de Borst R (2005) Enhanced continua and discrete lattices for modelling granular assemblies. Proc R Soc A 363:2543–2580

    MATH  Google Scholar 

  11. Bigoni D, Drugan WJ (2007) Analytical derivation of Cosserat moduli via homogenization of heterogeneous elastic materials. J Appl Mech 74:741–753

    Article  MathSciNet  Google Scholar 

  12. Sab K, Pradel F (2009) Homogenisation of periodic Cosserat media. Int J Comput Appl Technol 34:60–71

    Article  Google Scholar 

  13. Jenkins JT, Koenders MA (2004) The incremental response of random aggregates of identical round particles. Eur Phys J E 13:113–123

    Article  Google Scholar 

  14. La Ragione L, Jenkins JT (2007) The initial response of an idealized granular material. Proc R Soc A 463:735–758

    Article  MATH  Google Scholar 

  15. Gardiner BS, Tordesillas A (2005) Micromechanical constitutive modeling of granular media: evolution and loss of contact in particle clusters. J Eng Mech 52:93–106

    MathSciNet  MATH  Google Scholar 

  16. Papanicolopulos SA, Veveakis E (2011) Sliding and rolling dissipation in Cosserat plasticity. Granular Matter 13:197–204

    Article  Google Scholar 

  17. Luscher DJ, McDowell DL, Bronkhorst CA (2010) A second gradient theoretical framework for hierarchical multiscale modeling of materials. Int J Plasticity 26:1248–1275

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Kouznetsova V, Brekelmans W, Baaijens F (2001) An approach to micro-macro modeling of heterogeneous materials. Comput Mech 27:37–48

    Article  MATH  Google Scholar 

  20. Wagner GJ, Liu WK (2003) Coupling of atomistic and continuum simulations using a bridging scale decomposition. J Comput Phys 190:249–274

    Article  MATH  Google Scholar 

  21. Li XK, Wan K (2001) A bridging scale method for granular materials with discrete particle assembly-Cosserat continuum modeling. Comput Geotech 38:1052–1068

    Article  Google Scholar 

  22. Wellmann C, Wriggers P (2012) A two scale model of granular materials. Comput Methods Appl Mech Eng 205–208:46–58

    Article  MathSciNet  Google Scholar 

  23. Vernerey F, Liu WK, Moran B (2007) Multi-scale micromorphic theory for hierarchical materials. J Mech Phys Solids 55:2603–2651

    Article  MathSciNet  MATH  Google Scholar 

  24. Terada K, Kikuchi N (2001) A class of general algorithms for multi-scale analyses of heterogeneous media. Comput Methods Appl Mech Eng 190:5427–5464

    Article  MathSciNet  MATH  Google Scholar 

  25. Kaneko K, Terada K, Kyoya T, Kishino Y (2003) Global-local analysis of granular media in quasi-static equilibrium. Int J Solids Struct 40:4043–4069

    Article  MATH  Google Scholar 

  26. Li XK, Liu QP, Zhang JB (2010) A micro-macro homogenization approach for discrete particle assembly-Cosserat continuum modeling of granular materials. Int J Solids Struct 47:291–303

    Article  MATH  Google Scholar 

  27. Geers MGD, Kouznetsova VG, Brekelmans WAM (2010) Multi-scale computational homogenization: trends and challenges. J Comput Appl Math 234:2175–2182

    Article  MATH  Google Scholar 

  28. Kouznetsova V, Geers MGD, Brekelmans WAM (2002) Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. Int J Numer Methods Eng 54:1235–1260

    Article  MATH  Google Scholar 

  29. Kouznetsova V, Geers MGD, Brekelmans WAM (2004) Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy. Comput Methods Appl Mech Eng 193:5525–5550

    Article  MATH  Google Scholar 

  30. Zhang HW, Galvanetto U, Schrefler BA (1999) Local analysis and global nonlinear behaviour of periodic assemblies of bodies in elastic contact. Comput Mech 24:217–229

    Article  MATH  Google Scholar 

  31. Zhang HW, Schrefler BA (2000) Global constitutive behaviour of periodic assemblies of inelastic bodies in contact. Mech Compos Mater Struct 7:355–382

    Article  Google Scholar 

  32. Li XK, Zhang JB, Zhang X (2011) Micro-macro homogenization of gradient-enhanced Cosserat media. Eur J Mech A 30:362–372

    Article  MATH  Google Scholar 

  33. Larsson R, Diebels S (2007) A second-order homogenization procedure for multi-scale analysis based on micropolar kinematics. Int J Numer Methods Eng 69:2485–2512

    Article  MathSciNet  MATH  Google Scholar 

  34. Burst H, Forest S, Sab K (1998) Cosserat overall modeling of heterogeneous materials. Mech Res Commun 25:449–454

    Article  Google Scholar 

  35. Kaczmarczyk Ł, Pearce CJ, Bićanić N (2008) Scale transition and enforcement of RVE boundary conditions in second-order computational homogenization. Int J Numer Methods Eng 74:506–522

    Article  MATH  Google Scholar 

  36. Kaczmarczyk Ł, Pearce CJ, Bićanić N (2010) Studies of microstructural size effect and higher-order deformation in second-order computational homogenization. Comput Struct 88:1383–1390

    Article  Google Scholar 

  37. Li XK, Du YY, Duan QL (2013) Micromechanically informed constitutive model and anisotropic damage characterization of Cosserat continuum for granular materials. Int J Damage Mech 22:643–682

    Article  Google Scholar 

  38. Kruyt NP (2003) Statics and kinematics of discrete Cosserat-type granular materials. Int J Solids Struct 40:511–534

    Article  MATH  Google Scholar 

  39. Zhang HW, Wang H, Wriggers P, Schrefler BA (2005) Some investigations on finite element method for contact analysis of multiple Cosserat bodies. Comput Mech 36:444–458

    Article  MATH  Google Scholar 

  40. Hill R (1963) Elastic properties of reinforced solids: some theoretical principles. J Mech Phys Solids 11:357–372

    Article  MATH  Google Scholar 

  41. Li XK, Zhang X, Zhang JB (2010) A generalized Hill’s lemma and micromechanically based macroscopic constitutive model for heterogeneous granular materials. Comput Methods Appl Mech Eng 199:3137–3152

    Article  MATH  Google Scholar 

  42. Zhang HW, Schrefler BA (2000) Gradient dependent plasticity model and dynamic strain localisation analysis of saturated and partially saturated porous media: one dimensional model. Eur J Mech A 19:503–524

    Article  Google Scholar 

  43. Zhang HW, Schrefler BA (2001) Uniqueness and localisation analysis of elastic-plastic saturated porous media. Int J Numer Anal Methods Geomech 25:29–48

    Article  MATH  Google Scholar 

  44. Zienkiewicz OC, Taylor RL (2000) The finite element method: solid mechanics. Butterworth-Heinemann, Oxford

    Google Scholar 

  45. Zervos A, Papanastasiou P, Vardoulakis I (2001) A finite element displacement formulation for gradient elastoplasticity. Int J Numer Methods Eng 50:1369–1388

    Article  MATH  Google Scholar 

  46. Xia ZC, Hutchinson JW (1996) Crack tip fields in strain gradient plasticity. J Mech Phys Solids 44:1621–1648

    Article  Google Scholar 

  47. Shu JY, Fleck NA (1998) The prediction of a size effect in microindentation. Int J Solids Struct 35:1363–1383

    Article  MATH  Google Scholar 

  48. Shu JY, King WE, Fleck NA (1999) Finite elements for materials with strain gradient effects. Int J Numer Methods Eng 44:373–391

    Article  MATH  Google Scholar 

  49. Matsushima Chambon TR, Caillerie D (2002) Large strain finite element analysis of a local second gradient model: application to localization. Int J Numer Methods Eng 54:499–521

    Article  MATH  Google Scholar 

  50. Zervos A (2008) Finite elements for elasticity with microstructure and gradient elasticity. Int J Numer Methods Eng 73:564–595

    Article  MathSciNet  MATH  Google Scholar 

  51. Askes H, Aifantis EC (2011) Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results. Int J Solids Struct 48:1962–1990

    Article  Google Scholar 

  52. Onck PR (2002) Cosserat modeling of cellular solids. Comptes Rendus Mecanique 330:717–722

    Article  MATH  Google Scholar 

  53. Fleck NA, Hutchinson J (1997) Strain gradient plasticity. Adv Appl Mech 33:295–361

    Article  Google Scholar 

  54. Jetteur PH, Cescotto S (1991) Mixed finite element for the analysis of large inelastic strains. Int J Numer Methods Eng 31:229–239

    Article  MATH  Google Scholar 

  55. Li XK, Cescotto S (1996) Finite element method for gradient plasticity at large strains. Int J Numer Methods Eng 39:619–633

    Article  MATH  Google Scholar 

  56. Crisfield MA, Moita GF (1996) A co-rotational formulation for 2-D continua including incompatible modes. Int J Numer Methods Eng 39:2619–2633

    Article  MathSciNet  MATH  Google Scholar 

  57. Li XK, Chu XH, Feng YT (2005) A discrete particle model and numerical modeling of the failure modes of granular materials. Eng Comput 22:894–920

    Article  MATH  Google Scholar 

  58. Providas E, Kattis M (2002) Finite element method in plane Cosserat elasticity. Comput Struct 80:2059–2069

    Article  Google Scholar 

  59. De Borst R, Sluys LJ (1991) Localisation in a Cosserat continuum under static and dynamic loading conditions. Comput Methods Appl Mech Eng 90:805–827

    Article  Google Scholar 

  60. Li XK, Tang HX (2005) A consistent return mapping algorithm for pressure-dependent elastoplastic Cosserat continua and modelling of strain localisation. Comput Struct 83:1–10

  61. De Borst R, Sluys LJ, Muhlhaus HB, Pamin J (1993) Fundamental issues in finite element analyses of localisation of deformation. Eng Comput 10:99–121

    Article  Google Scholar 

Download references

Acknowledgments

The authors are pleased to acknowledge the support for this work by the National Natural Science Foundation of China through Contract/Grant numbers 11372066, 11072046, 11102036 and the National Key Basic Research and Development Program (973 Program) through Contract number 2010CB731502.

Author information

Authors and Affiliations

  1. The State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian , 116024, People’s Republic of China

    Xikui Li, Yuanbo Liang, Qinglin Duan & Youyao Du

  2. Department of Civil, Environmental and Architectural Engineering, University of Padua, via Marzolo 9, 35131 , Padua, Italy

    B.A. Schrefler

Authors
  1. Xikui Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuanbo Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Qinglin Duan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. B.A. Schrefler
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Youyao Du
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xikui Li.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, X., Liang, Y., Duan, Q. et al. A mixed finite element procedure of gradient Cosserat continuum for second-order computational homogenisation of granular materials. Comput Mech 54, 1331–1356 (2014). https://doi.org/10.1007/s00466-014-1062-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-014-1062-9

Keywords

Search

Navigation