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Exact corotational shell for finite strains and fracture

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Abstract

The corotational method for frame-invariant elements is generalized to obtain a consistent large-strain shell element incorporating thickness extensibility. The resulting element allows arbitrary in-plane deformations and is distinct from the traditional corotational methods (either quadrature-based or element-based) in the sense that the corotational frame is exact. The polar decomposition operation is performed in two parts, greatly simplifying the linearization calculations. Expressions for the strain-degrees-of-freedom matrices are given for the first time. The symbolic calculations are performed with a well-known algebraic system with a code generation package. Classical linear benchmarks are shown with excellent results. Applications with hyperelasticity and finite strain plasticity are presented, with asymptotically quadratic convergence and very good benchmark results. An example of finite strain plasticity with fracture is solved successfully, showing remarkable robustness without the need of enrichment techniques.

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References

  1. Allman DJ (1988) Evaluation of the constant strain triangle with drilling rotations. Int J Numer Methods Eng 26: 2645–2655

    Article  MATH  Google Scholar 

  2. Alvin K, de la Fuente HM, Haugen B, Felippa CA (1992) Membrane triangles with corner drilling freedoms: I. The EFF element. Finite Elem Anal Des 12: 165–187

    Article  Google Scholar 

  3. Antman SS (2005) Nonlinear problems of elasticity, 2nd edn. Springer, New York

    MATH  Google Scholar 

  4. Areias P Simplas. http://home.uevora.pt/~pmaa/SimplasWebsite/Simplas.html

  5. Areias P, Dias-da-Costa D, Alfaiate J, Júlio E (2009) Arbitrary bi-dimensional finite strain cohesive crack propagation. Comput Mech 45(1):61–75

    Article  MathSciNet  MATH  Google Scholar 

  6. Areias P, Rabczuk T (2010) Smooth finite strain plasticity with non-local pressure support. Int J Numer Methods Eng 81: 106–134

    MATH  Google Scholar 

  7. Areias P, Ritto-Corrêa M, Martins JAC (2010) Finite strain plasticity, the stress condition and a complete shell model. Comput Mech 45: 189–209

    Article  MathSciNet  MATH  Google Scholar 

  8. Areias P, Van Goethem N, Pires EB (2010) A damage model for ductile crack initiation and propagation. Comput Mech (in Press)

  9. Basar Y, Ding Y (1992) Finite rotation shell elements for the analysis of finite rotation shell problems. Int J Numer Methods Eng 34: 165–169

    Article  Google Scholar 

  10. Batoz J-L (1980) A study of three-node triangular plate bending elements. Int J Numer Methods Eng 15: 1771–1812

    Article  MATH  Google Scholar 

  11. Battini J-M (2004) On the choice of local element frame for corotational triangular shell elements. Commun Numer Methods Eng 20: 819–825

    Article  MATH  Google Scholar 

  12. Battini J-M, Pacoste C (2006) On the choice of the linear element for corotational triangular shells. Comput Method Appl Mech Eng 195: 6362–6377

    Article  MATH  Google Scholar 

  13. Belytschko T, Liu Wk, Moran B (2000). Nonlinear finite elements for continua and structures. Wiley, New York

    MATH  Google Scholar 

  14. Bergan PG, Felippa CA (1985) A triangular membrane element with rotational degrees of freedom. Comput Methods Appl Mech Eng 50: 25–69

    Article  MATH  Google Scholar 

  15. Bergan PG, Nygaard MK (1984) Finite elements with increased freedom in choosing shape functions. Int J Numer Methods Eng 20: 643–663

    Article  MATH  Google Scholar 

  16. Brank B, Ibrahimbegovic A (2001) On the relation between different parametrizations of finite rotations for shells. Eng Comput 18: 950–973

    Article  MATH  Google Scholar 

  17. Crisfield MA, Moita GF (1996) A unified co-rotational framework for solids, shells and beams. Int J Solids Struct 33(20–22): 2969–2992

    Article  MATH  Google Scholar 

  18. Dvorkin E, Pantuso D, Repetto E (1995) A formulation of the MITC4 shell element for finite strain elasto-plastic analysis. Comput Methods Appl Mech Eng 125: 17–40

    Article  Google Scholar 

  19. Eberlein R, Wriggers P (1999) Finite element concepts for finite elastoplastic strains and isotropic stress response in shells: theoretical and computational analysis. Comput Methods Appl Mech Eng 171: 243–279

    Article  MATH  Google Scholar 

  20. Felippa CA (2003) A study of optimal membrane triangles with drilling freedoms. Technical report CU-CAS-03-02, University of Colorado, College of Engineering, Campus Box 429, Boulder, Colorado 80309, February 2003

  21. Felippa CA, Alexander S (1992) Membrane triangles with corner drilling freedoms: III. Implementation and performance evaluation. Finite Elem Anal Des 12: 203–239

    Article  MATH  Google Scholar 

  22. Felippa CA, Haugen B (2005) Unified formulation of small-strain corotational finite elements: I. Theory. Comput Methods Appl Mech Eng 194: 2285–2335

    Article  MATH  Google Scholar 

  23. Felippa CA, Militello C (1992) Membrane triangles with corner drilling freedoms: II. The ANDES element. Finite Elem Anal Des 12: 189–201

    Article  MATH  Google Scholar 

  24. Goldstein H, Poole CP, Safko JL (2001) Classical mechanics, 3rd edn. Addison-Wesley, New York

    Google Scholar 

  25. Hughes TJR (2000) The finite element method. Dover, 2000. Reprint of Prentice-Hall edition, 1987

  26. Hughes TJR, Carnoy E (1983) Nonlinear finite element formulation accounting for large membrane stress. Comput Methods Appl Mech Eng 39: 69–82

    Article  MATH  Google Scholar 

  27. Jetteur P (1987) Improvement of the quadrilateral jet shell element for a particular class of shell problems. Technical report IREM Internal Report 87/1, Ecole Polytechnique Fédérale de Lausanne

  28. Klinkel S, Gruttmann F, Wagner W (2008) A mixed shell formulation accounting for thickness strains and finite strain 3D material models. Int J Numer Methods Eng 74: 945–970

    Article  MATH  Google Scholar 

  29. Korelc J (2002) Multi-language and multi-environment generation of nonlinear finite element codes. Eng Comput 18(4): 312–327

    Article  Google Scholar 

  30. Liu WK, Guo Y, Belytschko T (1998) A multiple-quadrature eight-node hexahedral finite element for large deformation elastoplastic analysis. Comput Methods Appl Mech Eng 154: 69–132

    Article  MathSciNet  MATH  Google Scholar 

  31. MacNeal RH, Harder RL (1985) A proposed standard set of problems to test finite element accuracy. Finite Elem Anal Des 1: 1–20

    Article  Google Scholar 

  32. Wolfram Research Inc. Mathematica (2007). http://www.wolfram.com/mathematica/

  33. Sansour C, Kollmann FG (2000) Families of 4-node and 9-node finite elements for a finite deformation shell theory. an assessment of hybrid stress, hybrid strain and enhanced strain elements. Comput Mech 24: 435–447

    Article  MATH  Google Scholar 

  34. Simo JC, Kennedy JG (1992) On the stress resultant geometrically exact shell model. Part V: nonlinear plasticity: formulation and integration algorithms. Comput Methods Appl Mech Eng 96: 133–171

    Article  MathSciNet  MATH  Google Scholar 

  35. Wagner W, Gruttmann F (2005) A robust non-linear mixed hybrid quadrilateral shell element. Int J Numer Methods Eng 64: 635–666

    Article  MATH  Google Scholar 

  36. Wagner W, Klinkel S, Gruttmann F (2002) Elastic and plastic analysis of thin-walled structures using improved hexahedral elements. Comput Struct 80: 857–869

    Article  Google Scholar 

  37. Wisniewski K (1998) A shell theory with independent rotations for relaxed Biot stress and right stretch strain. Comput Mech 21(2): 101–122

    Article  MATH  Google Scholar 

  38. Wisniewski K, Turska E (2009) Improved 4-node Hu-Washizu elements based on skew coordinates. Comput Struct 87: 407–424

    Article  Google Scholar 

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Authors and Affiliations

  1. Departamento de Física, Colégio Luís António Verney, Universidade de Évora, Rua Romão Ramalho 59, 7002-554, Évora, Portugal

    P. Areias, J. Garção & J. Infante Barbosa

  2. ICIST, Lisbon, Portugal

    P. Areias & E. B. Pires

  3. IDMEC, Lisbon, Portugal

    J. Garção & J. Infante Barbosa

  4. Departamento de Engenharia Civil e Arquitectura, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001, Lisbon, Portugal

    E. B. Pires

Authors
  1. P. Areias
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  2. J. Garção
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  3. E. B. Pires
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  4. J. Infante Barbosa
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Correspondence to P. Areias.

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Areias, P., Garção, J., Pires, E.B. et al. Exact corotational shell for finite strains and fracture. Comput Mech 48, 385–406 (2011). https://doi.org/10.1007/s00466-011-0588-3

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  • DOI: https://doi.org/10.1007/s00466-011-0588-3

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