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Computational Mechanics

, Volume 39, Issue 4, pp 381–399 | Cite as

On the Analysis of an EFG Method Under Large Deformations and Volumetric Locking

  • Rodrigo RossiEmail author
  • Marcelo Krajnc Alves
Original Paper

Abstract

This work investigates a modified element-free Galerkin (MEFG) method when applied to large deformation processes. The proposed EFG method enables the direct imposition of the essential boundary conditions, as a result of the kronecker delta property of the special shape functions, constructed in the neighborhood of the essential boundary. The plasticity model assumes a multiplicative decomposition of the deformation gradient into an elastic and a plastic part and considers a J 2 elasto-plastic constitutive relation that accounts for a nonlinear isotropic hardening. The constitutive model is written in terms of the rotated Kirchhoff stress and of the conjugate logarithmic strain measure. A total Lagrangian formulation is considered in order to improve the computational performance of the proposed algorithm. Here, aspects related to the volumetric locking are numerically investigated and an F-bar approach is considered. Some numerical results are presented, under axisymmetric and plane strain assumption, in order to attest the performance of the proposed method.

Keywords

Finite strains Mesh-free EFG Volumetric locking 

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References

  1. 1.
    Akkaram S, Zabaras N (2001) “An updated Lagrangian finite element sensitivity analysis of large deformations using quadrilateral elements”. Int J Numer Methods Eng 52:1131–1163CrossRefzbMATHGoogle Scholar
  2. 2.
    Alves MK, Rossi R (2003) “A modified element-free Galerkin method with essential boundary conditions, enforced by an extended partition of unity finite element weight function”. Int J Numer Methods Eng 57:1523–1552CrossRefzbMATHGoogle Scholar
  3. 3.
    Alves MK, Rossi R (2005) “An extension of the partition of unity finite element method. J Barz Soc Mech Sci Eng XXVII(3):209–216Google Scholar
  4. 4.
    Askes H, De Borst R, Heeres O (1999) “Conditions for locking-free elasto-plastic analyses in the Element-Free Galerkin method”. Comput Methods Appl Mechan Eng 173:99–109CrossRefzbMATHGoogle Scholar
  5. 5.
    Belytschko T, Tabbara M (1996) “Dynamic fracture using element-free Galerkin methods”. Int J Numer Methods Eng 39:923–938CrossRefzbMATHGoogle Scholar
  6. 6.
    Belytschko T, Lu YY, Gu L (1994) “Element-free Galerkin methods”. Int J Numer Methods Eng 37:229–256CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Belytschko T, Organ D, Krongauz Y (1995) “A coupled finite element-element-free Galerkin method”. Comput Mechanics 17(3):186–195zbMATHMathSciNetGoogle Scholar
  8. 8.
    Dolbow J, Belytschko T (1999) “Volumetric locking in the element free Galerkin method”. Int J Numer Methods Eng 46:925–942CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Duarte AC, Oden J,T (1996) ”An h-p adaptive method using clouds”. Comput Methods Appl Mechanics Eng 139(1–4):237–262CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Eterovic AL, Bathe KJ (1990) “A hyperelastic-based large strain elasto-plastic constitutive formulation with combined isotropic-kinematic hardening using the logarithmic stress and strain measures”. Int J Numer Methods Eng 30:1099–1114CrossRefzbMATHGoogle Scholar
  11. 11.
    Gavete L, Benito JJ, Falcón S, Ruiz A (2000) “Penalty functions in constrained variational principles for element free Galerkin method”. Eur J Mech A/Solids 19(4):699–720CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Günther FC, Liu WK (1998) “Implementation of boundary conditions for meshless methods”. Comput Methods Appl Mechanics Eng 163:205–230CrossRefzbMATHGoogle Scholar
  13. 13.
    Hegen D (1996) “Element-free Galerkin methods in combination with finite element approaches”. Comput Methods Appl Mechanics Eng 135(1–2):143–166CrossRefzbMATHGoogle Scholar
  14. 14.
    Hill R (1978) “Aspects of invariance in solid mechanics”. Adv Appl Mech 18:1–75zbMATHCrossRefGoogle Scholar
  15. 15.
    Huerta A, Méndez SF (2000) “Enrichment and coupling of the finite element and meshless methods”. Int J Numer Methods Eng 48:1615–1636CrossRefzbMATHGoogle Scholar
  16. 16.
    Huerta A, Méndez SF (2001) “Locking in the incompressible limit for the element-free Galerkin method”. Int J Numer Methods Eng 51:1361–1383CrossRefzbMATHGoogle Scholar
  17. 17.
    Hughes TJR (1980) “Generalization of seletive integration procedures to anisotropic and nonlinear media”. Int J Numer Methods Eng 15:1413–1418CrossRefzbMATHGoogle Scholar
  18. 18.
    Kaljević I, Saigal S (1997) “An improved element free Galerkin formulation”. Int J Numer Methods Eng 40(16):2953–2974CrossRefzbMATHGoogle Scholar
  19. 19.
    Krongauz Y, Belytschko T (1996) “Enforcement of essential boundary conditions in meshless approximations using finite elements”. Comput Methods Appl Mechanics Eng 131(1–2):133–145CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Lancaster P, Salkuskas K (1981) “Surfaces generated by moving least square methods”. Math Comput 37:141–158zbMATHCrossRefGoogle Scholar
  21. 21.
    Liu WK, Li S, Belytschko T (1997) “Moving least-square reproducing kernel methods (I) Methodology and convergence”. Comput Methods Appl Mechanics Eng 147:113–154CrossRefMathSciNetGoogle Scholar
  22. 22.
    Ortiz M, Radovitzky RA, Repeto EA (2001) “The computation of exponential and logarithmic mappings and their fisrt and second linearizations”. Int J Numer Methods Eng 52:1431–1441CrossRefzbMATHGoogle Scholar
  23. 23.
    Pannachet T, Askes H (2000) “Some observations on the enforcement of constraint equations in the EFG method”. Commun Numer Methods Eng 16(12):819–930CrossRefzbMATHGoogle Scholar
  24. 24.
    Rossi R, Alves MK (2004) “Recovery Based Error Estimation and Adaptivity Applied to a Modified Element-Free Galerkin Method”. Comput Mech 33(3):194–205CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    Rossi R, Alves MK (2005) “An h-adaptive modified element-free Galerkin method”. Eur J Mech A/Solids 24:782–799CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    Simo JC, Armero F (1992) “Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes”. Int J Numer Methods Eng 33:1413–1449CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Simo JC, Rifai S (1990) “A class of mixed assumed strain methods and the method of incompatible modes”. Int J Numer Methods Eng 29:1595–1638CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Souza Neto EA, Peric D, Dutko M, Owen DRJ (1996) “Design of simple low order finite elements for large strain analysis of nearly incompressible solids”. Int J Solids Struct 33:3277–3296CrossRefzbMATHGoogle Scholar
  29. 29.
    Souza Neto EA, Peric D, Owen DRJ (2002) Computational plasticity: small and large strain finite element analysis of elastic and inelastic solids. Classroom notes University College of Swansea, WalesGoogle Scholar
  30. 30.
    Ventura G (2002) “An augmented Lagrangian approach to essential boundary conditions in meshless methods”. Int J Numer Methods Eng 53:825–842CrossRefGoogle Scholar
  31. 31.
    Vidal Y, Villon P Huerta A (2003) “Locking in the incompressible limit:pseudo-divergence-free element free Galerkin”. Commun Numer Methods Eng 19:725–735CrossRefzbMATHGoogle Scholar
  32. 32.
    Weber G, Anand L (1990) “Finite deformation constitutive equations and a time integration procedure for isotropic” hyperelastic-viscoplastic solids. Comput Methods Appl Mechanics Eng 79:173–202CrossRefzbMATHGoogle Scholar
  33. 33.
    Wells GN, Sluys LJ, De Borst R (2002) “A p-adaptive scheme for overcoming volumetric locking during plastic flow”. Comput Methods Appl Mechanics Eng 191:3153–3164CrossRefzbMATHGoogle Scholar
  34. 34.
    Zhang X, Liu X, Song KZ, Lu MW (2001) “Imposition of essential boundary conditions by displacement constraint equations in meshless methods”. Commun Numer Methods Eng 17(3): 165–178CrossRefzbMATHGoogle Scholar
  35. 35.
    Zhu T, Atluri N (1998) “Modified collocation method and a penalty function for enforcing the essential boundary conditions in the element free Galerkin method”. Comput Mech 21: 211–222CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Departamento de Engenharia MecânicaUniversidade de Caxias do Sul Cidade UniversitáriaCaxias do SulBrazil
  2. 2.Departamento de Engenharia MecânicaUniversidade Federal de Santa CatarinaFlorianópolisBrazil

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