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Archive of Applied Mechanics

, Volume 89, Issue 12, pp 2577–2590 | Cite as

Mesh bias and shear band inclination in standard and non-standard continua

  • Sepideh Alizadeh Sabet
  • René de BorstEmail author
Original
  • 116 Downloads

Abstract

A severe, spurious dependence of numerical simulations on the mesh size and orientation can be observed in elasto-plastic models with a non-associated flow rule. This is due to the loss of ellipticity and may also cause a divergence in the incremental-iterative solution procedure. This paper first analyses the dependence of the shear band inclination in a biaxial test on the mesh size as well as on the mesh orientation. Next, a Cosserat continuum model, which has been employed successfully for strain-softening plasticity, is proposed to prevent loss of ellipticity. Now, numerical solutions result for shear band formation which are independent of the size and the orientation of the discretisation.

Keywords

Non-associated plasticity Cosserat continuum Mesh bias Strain localisation Shear band Ellipticity 

Notes

Acknowledgements

Financial support of the European Research Council under Advanced Grant 664734 ‘PoroFrac’ is gratefully acknowledged.

References

  1. 1.
    Altenbach, H., Eremeyev, V.A. (ed.): Cosserat media. In: Generalized Continua - From the Theory to Engineering Applications, pp. 65–130. Springer, Heidelberg (2013) Google Scholar
  2. 2.
    Arthur, J.R.F., Dunstan, T.: Rupture layers in granular media. In: P.A. Vermeer, H.J. Luger (eds.) Proceedings IUTAM Symposium on Deformation and Failure in Granular Media, pp. 453–460. A. A. Balkema, Rotterdam (1982)Google Scholar
  3. 3.
    Arthur, J.R.F., Dunstan, T., Al-Ani, Q.A.J.L., Assadi, A.: Plastic deformation and failure in granular media. Géotechnique 27, 53–74 (1977)Google Scholar
  4. 4.
    Bardet, J.P.: A comprehensive review of strain localization in elastoplastic soils. Comput. Geotech. 10, 163–188 (1990)Google Scholar
  5. 5.
    Bažant, Z.P., Belytschko, T.B., Chang, T.P.: Continuum theory for strain-softening. ASCE J. Eng. Mech. 110, 1666–1692 (1984)Google Scholar
  6. 6.
    Bažant, Z.P., Pijaudier-Cabot, G.: Nonlocal continuum damage, localization instability and convergence. ASME J. Appl. Mech. 55, 287–293 (1988)zbMATHGoogle Scholar
  7. 7.
    Cosserat, E., Cosserat, F.: Théorie des Corps Déformables. A. Hermann et fils, Paris (1909)zbMATHGoogle Scholar
  8. 8.
    Coulomb, C.A.: Essai sur une application des règles des maximis et minimis à quelques problemès de statique. Mem. Acad. R. des. Sci. 7, 343–382 (1776)Google Scholar
  9. 9.
    de Borst, R.: Bifurcations in finite element models with a non-associated flow law. Int. J. Numer. Anal. Methods Geomech. 12, 99–116 (1988)zbMATHGoogle Scholar
  10. 10.
    de Borst, R.: Simulation of strain localization: a reappraisal of the Cosserat continuum. Eng. Comput. 8, 317–332 (1991)Google Scholar
  11. 11.
    de Borst, R.: A generalisation of J2-flow theory for polar continua. Comput. Methods Appl. Mech. Eng. 103, 347–362 (1993)zbMATHGoogle Scholar
  12. 12.
    de Borst, R., Crisfield, M.A., Remmers, J.J.C., Verhoosel, C.V.: Nonlinear Finite Element Analysis of Solids and Structures. Wiley, Chichester (2012)zbMATHGoogle Scholar
  13. 13.
    de Borst, R., Mühlhaus, H.B.: Gradient-dependent plasticity: formulation and algorithmic aspects. Int. J. Numer. Methods Eng. 35, 521–539 (1992)zbMATHGoogle Scholar
  14. 14.
    de Borst, R., Sluys, L.J.: Localisation in a Cosserat continuum under static and dynamic loading conditions. Comput. Methods Appl. Mech. Eng. 90, 805–827 (1991)Google Scholar
  15. 15.
    de Borst, R., Sluys, L.J., Mühlhaus, H.B., Pamin, J.: Fundamental issues in finite element analysis of localisation of deformation. Eng. Comput. 10, 99–122 (1993)Google Scholar
  16. 16.
    Desrues, J.: La localisation de la déformation dans les matériaux granulaires. Ph.D. thesis, Institut National Polytechnique de Grenoble, Grenoble (1984)Google Scholar
  17. 17.
    Drucker, D.C., Prager, W.: Soil mechanics and plastic analysis or limit design. Q. Appl. Math. 10, 157–165 (1952)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Duthilleul, B.: Rupture progressive: Simulation physique et numérique. Ph.D. thesis, Institut National Polytechnique de Grenoble, Grenoble (1983)Google Scholar
  19. 19.
    Ehlers, W., Scholz, B.: An inverse algorithm for the identification and the sensitivity analysis of the parameters governing micropolar elasto-plastic granular material. Arch. Appl. Mech. 77, 911–931 (2007)zbMATHGoogle Scholar
  20. 20.
    Hadamard, J.: Leçons sur la Propagation des Ondes et les Équations de l’Hydrodynamique. A. Hermann, Paris (1903)zbMATHGoogle Scholar
  21. 21.
    Hill, R.: A general theory of uniqueness and stability in elastic-plastic solids. J. Mech. Phys. Solids 6, 236–249 (1958)zbMATHGoogle Scholar
  22. 22.
    Hill, R.: Acceleration waves in solid. J. Mech. Phys. Solids 10, 1–16 (1962)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Le Pourhiet, L.: Strain localization due to structural softening during pressure sensitive rate independent yielding. Bull. de la Société géologique de Fr. 184, 357–371 (2013)Google Scholar
  24. 24.
    Loret, B., Prévost, J.H.: Dynamic strain localization in elasto-(visco-) plastic solids, Part 1. General formulation and one-dimensional examples. Comput. Methods Appl. Mech. Eng. 83, 247–273 (1990)zbMATHGoogle Scholar
  25. 25.
    Mandel, J.: Conditions de stabilité et postulat de Drucker. In: Kravtchenko, C.J., Sirieys, P.M. (eds.) Proceedings IUTAM Symposium on Rheology and Soil Mechanics, pp. 58–68. Springer, Berlin (1966)Google Scholar
  26. 26.
    Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Prentice Hall, Englewood Cliffs (1983)zbMATHGoogle Scholar
  27. 27.
    Mazière, M., Forest, S.: Strain gradient plasticity modeling and finite element simulation of Lüders band formation and propagation. Contin. Mech. Thermodyn. 27, 83–104 (2015)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Mühlhaus, H.B.: Application of Cosserat theory in numerical solutions of limit load problems. Ingenieur-Archiv. 59, 124–137 (1989)Google Scholar
  29. 29.
    Mühlhaus, H.B., Aifantis, E.C.: A variational principle for gradient plasticity. Int. J. Solids Struct. 28, 845–857 (1991)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Mühlhaus, H.B., Vardoulakis, I.: The thickness of shear bands in granular materials. Géotechnique 37, 271–283 (1987)Google Scholar
  31. 31.
    Needleman, A.: Non-normality and bifurcation in plane strain tension and compression. J. Mech. Phys. Solids 27, 231–254 (1979)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Needleman, A.: Material rate dependence and mesh sensitivity in localization problems. Comput. Methods Appl. Mech. Eng. 67, 69–85 (1988)zbMATHGoogle Scholar
  33. 33.
    Pamin, J., Askes, H., de Borst, R.: Two gradient plasticity theories discretized with the element-free Galerkin method. Comput. Methods Appl. Mech. Eng. 192, 2377–2407 (2003)zbMATHGoogle Scholar
  34. 34.
    Peerlings, R.H.J., de Borst, R., Brekelmans, W.A.M., De Vree, J.H.P.: Gradient enhanced damage for quasi-brittle materials. Int. J. Numer. Methods Eng. 39, 3391–3403 (1996)zbMATHGoogle Scholar
  35. 35.
    Pijaudier-Cabot, G., Bažant, Z.P.: Nonlocal damage theory. ASCE J. Eng. Mech. 113, 1512–1533 (1987)zbMATHGoogle Scholar
  36. 36.
    Read, H.E., Hegemier, G.A.: Strain softening of rock, soil and concrete—a review article. Mech. Mater. 3, 271–294 (1984)Google Scholar
  37. 37.
    Rice, J.R.: The localization of plastic deformation. In: W.T. Koiter (ed.) Proceedings of the 14th International Congress on Theoretical and Applied Mechanics, pp. 207–220. North-Holland, Amsterdam (1976)Google Scholar
  38. 38.
    Rice, J.R., Rudnicki, J.W.: A note on some features of the theory of localization of deformation. Int. J. Solids Struct. 16, 597–605 (1980)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Roscoe, K.H.: The influence of strains in soil mechanics. Géotechnique 20, 129–170 (1970)Google Scholar
  40. 40.
    Rudnicki, J.W., Rice, J.R.: Conditions for the localization of deformation in pressure sensitive dilatant materials. J. Mech. Phys. Solids 23, 371–394 (1975)Google Scholar
  41. 41.
    Sabet, S.A., de Borst, R.: Structural softening, mesh dependence, and regularisation in non-associated plastic flow. Int. J. Numer. Anal. Methods Geomech. 43, 2170–2183 (2019)Google Scholar
  42. 42.
    Scarpelli, G., Wood, D.M.: Experimental observations of shear patterns in direct shear tests. In: P.A. Vermeer, H.J. Luger (eds.) Proceedings IUTAM Symposium on Deformation and Failure in Granular Media, pp. 473–483. A. A. Balkema, Rotterdam (1982)Google Scholar
  43. 43.
    Schaefer, H.: Das Cosserat-Kontinuum. Zeitschrift für Angew. Math. Mech. 47, 485–498 (1967)zbMATHGoogle Scholar
  44. 44.
    Sluys, L.J.: Wave Propagation, Localisation and Dispersion in Softening Solids. Ph.D. thesis, TU Delft, Delft (1992)Google Scholar
  45. 45.
    de Souza Neto, E.A., Peric, D., Owen, D.R.J.: Computational Methods for Plasticity: Theory and Applications. Wiley, Chichester (2011)Google Scholar
  46. 46.
    Thomas, T.Y.: Plastic Flow and Fracture in Solids. Academic Press, Cambridge (1961)zbMATHGoogle Scholar
  47. 47.
    Triantafyllidis, N., Aifantis, E.C.: A gradient approach to localization of deformation. I. Hyperelastic materials. J. Elast. 16, 225–237 (1986)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Vardoulakis, I.: Shear band inclination and shear modulus of sand in biaxial tests. Int. J. Numer. Anal. Methods Geomech. 4, 103–119 (1980)zbMATHGoogle Scholar
  49. 49.
    Vermeer, P.A.: The orientation of shear bands in biaxial tests. Géotechnique 40, 223–234 (1990)Google Scholar
  50. 50.
    Vermeer, P.A., de Borst, R.: Non-associated plasticity for soils, concrete and rock. Heron 29(3), 1–64 (1984)Google Scholar
  51. 51.
    Wang, W.M., Sluys, L.J., de Borst, R.: Viscoplasticity for instabilities due to strain softening and strain-rate softening. Int. J. Numer. Methods Eng. 40, 3839–3864 (1997)zbMATHGoogle Scholar
  52. 52.
    Wawersik, W.R., Brace, W.F.: Post-failure behavior of a granite and diabase. Rock Mech. 3, 61–85 (1971)Google Scholar
  53. 53.
    Wawersik, W.R., Fairhurst, C.H.: A study of brittle rock fracture in laboratory compression experiments. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 7, 561–575 (1970)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Civil and Structural EngineeringUniversity of SheffieldSheffieldUK

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