Coupled systems mechanics

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Notable highlights on locking-free techniques of Reissner-Mindlin plate finite elements in elastostatics

  • Nguyen, Cong-Uy (Centre de Recherche Royallieu, Universite de Technologie de Compiegne/ Alliance Sorbonne Universite) ;
  • Batoz, Jean-Louis (Centre de Recherche Royallieu, Universite de Technologie de Compiegne/ Alliance Sorbonne Universite) ;
  • Ibrahimbegovic, Adnan (Centre de Recherche Royallieu, Universite de Technologie de Compiegne/ Alliance Sorbonne Universite)
  • Received : 2021.04.22
  • Accepted : 2021.05.25
  • Published : 2021.06.25
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Abstract

In this paper, we discuss about the notable locking-free techniques of several simple plate bending finite elements for the Reissner-Mindlin plate bending theory. The brief background for Reissner-Mindlin plate theory is presented, in which stress and strain derivation are given along with one-field and two-field variational approaches. Afterwards, we classify several efficient robust techniques in a cluster of main categories to present sequentially, which are all able to overcome the locking phenomenon in thick plate bending problems. Only selective algorithms are programmed to conduct numerical simulations. The corresponding results are compared between these elements to show their performances.

Keywords

Acknowledgement

This work is financially supported under the research project SELT-TUM by Agence Nationale de la Recherche (ANR-16-CE92- 0036). Moreover, Prof. Adnan Ibrahimbegovic is also supported by Institut Universitaire de France (IUF). These sources of funding are gratefully acknowledged.

References

  1. Allaire, G. and Delgado, G. (2016), "Stacking sequence and shape optimization of laminated composite plates via a level-set method", J. Mech. Phys. Solid., 97, 168-196. https://doi.org/10.1016/j.jmps.2016.06.014.
  2. Auricchio, F. and Taylor, R.L. (1994), "A shear deformable plate element with an exact thin limit", Comput. Meth. Appl. Mech. Eng., 118(3-4), 393-412. https://doi.org/10.1016/0045-7825(94)90009-4.
  3. Auricchio, F. and Taylor, R.L. (1994), "A triangular thick plate finite element with an exact thin limit", Finite Elem. Anal. Des., 19(1-2), 57-68. https://doi.org/10.1016/0168-874X(94)00057-M.
  4. Babuska, I. and Scapolla, T. (1989), "Benchmark computation and performance evaluation for a rhombic plate bending problem", Int. J. Numer. Meth. Eng., 28, 155-179. https://doi.org/10.1002/nme.1620280112.
  5. Bathe, K.J. (2016), Finite Element Procedures, Prentice Hall, Pearson Education Inc, Massachusetts, USA.
  6. Batoz, J.L. (1982), "An explicit formulation for an efficient triangular plate-bending element", Int. J. Numer. Meth. Eng., 18(7), 1077-1089. https://doi.org/10.1002/nme.1620180711.
  7. Batoz, J.L. and Dhatt, G. (1990), Modelisation des Structures par Elements Finis Vol. 2: Poutres et Plaques, Herms, Paris, France.
  8. Batoz, J.L. and Dhatt, G. (1992), Modelisation des Structures par Elements Finis Vol. 3: Coques, Herms, Paris, France.
  9. Batoz, J.L. and Katili, I. (1992), "On a simple triangular reissner/mindlin plate element based on incompatible modes and discrete constraints", Int. J. Numer. Meth. Eng., 35(8), 1603-1632. https://doi.org/10.1002/nme.1620350805.
  10. Batoz, J.L., Bathe, K.J. and Ho, L.W. (1980), "A study of three-node triangular plate bending elements", Int. J. Numer. Meth. Eng., 15, 1771-1812. https://doi.org/10.1002/nme.1620151205.
  11. Bazeley, G.P., Cheung, Y.K., Irons, B.M. and Zienkiewicz, O.C. (1966), "Triangular elements in plate bending-Conforming and non-conforming solutions", Air Force Flight Dynamics Laboratory, 547-576.
  12. Belblidia, F., Bae, J., Rechak, S. and Hinton, E. (2001), "Topology optimization of plate structures using a single- or three- layered artificial material model", Adv. Eng. Softw., 32(2), 159-168. https://doi.org/10.1016/S0045-7949(00)00141-3.
  13. Brugnoli, A., Alazard, D., Valerie, P.B. and Matignon, D. (2019), "Port-Hamiltonian formulation and symplectic discretization of plate models Part I: Mindlin model for thick plates", Appl. Math. Model., 75, 940-60. https://doi.org/10.1016/j.apm.2019.04.035.
  14. Brugnoli, A., Alazard, D., Valerie, P.B. and Matignon, D. (2019), "Port-Hamiltonian formulation and symplectic discretization of plate models Part II: Kirchhoff model for thin plates", Appl. Math. Model., 75, 961-981. https://doi.org/10.1016/j.apm.2019.04.036.
  15. Cirak, F. and Ortiz, M. (2001), "Fully C1-conforming subdivision elements for finite deformation thin-shell analysis", Int. J. Numer. Meth. Eng., 51, 813-833. https://doi.org/10.1002/nme.182.
  16. Clough, R.W. and Felippa, C.A. (1968), "A refined quadrilateral element for analysis of plate bending", Air Force Flight Dynamics Laboratory, USA.
  17. Dolbow, J., Moes, N. and Belytschko, T. (2000), "Modeling fracture in MindlinReissner plates with the extended finite element method", Int. J. Solid. Struct., 37, 48-50. https://doi.org/10.1016/S0020-7683(00)00194-3.
  18. Goo, S., Wang, S., Hyun, J. and Jung, J. (2016), "Topology optimization of thin plate structures with bending stress constraints", Comput. Struct., 175, 134-143. https://doi.org/10.1016/j.compstruc.2016.07.006.
  19. Gruttmann, F. and Wagner, W. (2004), "A stabilized one-point integrated quadrilateral Reissner-Mindlin plate element", Int. J. Numer. Meth. Eng., 61, 2273-2295. https://doi.org/10.1002/nme.1148.
  20. Guo, Y.Q., Batoz, J.L., Naceur, H., Bouabdallah, S., Mercier, F. and Barlet, O. (2000), "Recent developments on the analysis and optimum design of sheet metal forming parts using a simplified inverse approach", Comput. Struct., 78(1-3), 133-148. https://doi.org/10.1016/S0045-7949(00)00095-X.
  21. Hrabok, M.M. and Hrudey, T.M. (1984), "A review and catalogue of plate bending finite elements", Comput. Struct., 19(3), 479-495. https://doi.org/10.1016/0045-7949(84)90055-5.
  22. Huang, H.C. and Hinton. E. (1984), "A nine node Lagrangian Mindlin plate element with enhanced shear interpolation", Eng. Comput., 1(4), 369-379. https://doi.org/10.1108/eb023593.
  23. Hughes, T.J.R. and Brezzi, F. (1989), "On drilling degrees of freedom", Comput. Meth. Appl. Mech. Eng., 72(1), 105-121. https://doi.org/10.1016/0045-7825(89)90124-2.
  24. Hughes, T.J.R. and Tezduyar, T. (1981), "Finite elements based upon mindlin plate theory with particular reference to the four-node bilinear isoparametric element", J. Appl. Mech., 48(3), 587-596. https://doi.org/10.1115/1.3157679.
  25. Hughes, T.J.R., Taylor, R.L. and Kanoknukulchai, W. (1977), "A simple and efficient finite element for plate bending", Int. J. Numer. Meth. Eng., 11(10), 1529-1543. https://doi.org/10.1002/nme.1620111005.
  26. Ibrahimbegovic, A. (1992), "Plate quadrilateral finite element with incompatible modes", Commun. Appl. Numer. Meth., 8, 497-504. https://doi.org/10.1002/cnm.1630080803.
  27. Ibrahimbegovic, A. (1993), "Quadrilateral finite elements for analysis of thick and thin plates", Comput. Meth. Appl. Mech. Eng., 110, 195-209. https://doi.org/10.1016/0045-7825(93)90160-Y.
  28. Ibrahimbegovic, A. (2009), Nonlinear Solid Mechanics: Theoretical Formulations and Finite Element Solution Methods, Springer, Berlin, Germany.
  29. Katili, I. (1993), "A new discrete Kirchhoff-Mindlin element based on Mindlin-Reissner plate theory and assumed shear strain fields - Part I: An extended DKT element for thick plate bending analysis", Int. J. Numer. Meth. Eng., 36, 1859-1883. https://doi.org/10.1002/nme.1620361106.
  30. Katili, I. (1993), "A new discrete Kirchhoff-Mindlin element based on Mindlin-Reissner plate theory and assumed shear strain fields - Part II: An extended DKQ element for thick plate bending analysis", Int. J. Numer. Meth. Eng., 36, 1885-1908. https://doi.org/10.1002/nme.1620361107.
  31. Katili, I., Batoz, J.L., Maknun, I.J., Hamdouni, A. and Millet, O. (2014), "The development of DKMQ plate bending element for thick to thin shell analysis based on the Naghdi/Reissner/Mindlin shell theory", Finite Elem. Anal. Des., 100, 12-27. https://doi.org/10.1016/j.finel.2015.02.005.
  32. Kiendl, J., Ambati, M., Lorenzis, L.D, Gomez, H. and Reali, A. (2016), "Phase-field description of brittle fracture in plates and shells", Comput. Meth. Appl. Mech. Eng., 312, 374-394. https://doi.org/10.1016/j.cma.2016.09.011.
  33. Konno, J. and Stenberg, R. (2010), "Finite element analysis of composite plates with an application to the paper cockling problem", Finite Elem. Anal. Des., 46(3), 265-272. https://doi.org/10.1016/j.finel.2009.10.001.
  34. Lavrencic, M. and Brank, B. (2021), "Hybrid-mixed low-order finite elements for geometrically exact shell models: Overview and comparison", Arch. Comput. Meth. Eng., 1-35. https://doi.org/10.1007/s11831-021-09537-2.
  35. Macchelli, A., Melchiorri, C. and Bassi, L. (2005), "Port-based modelling and control of the Mindlin plate", Proceedings of the 44th IEEE Conference on Decision and Control, Seville, Spain.
  36. Mindlin, R. (1951), "Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates", J. Appl. Mech., 38, 18-31. https://doi.org/10.1115/1.4010217.
  37. Moleiro, F., Mota Soares, C.M., Mota Soares, C.A. and Reddy, J.N. (2009), "Mixed least-squares finite element models for static and free vibration analysis of laminated composite plates", Comput. Meth. Appl. Mech. Eng., 198(21-26), 1848-1856. https://doi.org/10.1016/j.cma.2008.12.023.
  38. Morley, L.S.D. (1971), "The constant-moment plate-bending element", J. Strain Anal. Eng. Des., 6(1), 6-20. https://doi.org/10.1243/03093247V061020.
  39. Nguyen, C.U. and Ibrahimbegovic, A. (2020), "Hybrid-stress triangular finite element with enhanced performance for statics and dynamics", Comput. Meth. Appl. Mech. Eng., 372, 113381. https://doi.org/10.1016/j.cma.2020.113381.
  40. Nguyen, C.U. and Ibrahimbegovic, A. (2020), "Visco-plasticity stress-based solid dynamics formulation and time-stepping algorithms for stiff case", Int. J. Solid. Struct., 196-197, 154-170. https://doi.org/10.1016/j.ijsolstr.2020.04.018.
  41. Ortiz, M. and Morris, G.R. (1988), "C0 finite element discretization of Kirchhoff's equations of thin plate bending", Int. J. Numer. Meth. Eng., 26, 1551-1566. https://doi.org/10.1002/nme.1620260707.
  42. Reissner, E. (1945), "The effect of transverse shear deformation on the bending of elastic plates", J. Appl. Mech., 12, A69-A77. https://doi.org/10.1115/1.4009435.
  43. Reissner, E. (1976), "On the theory of transverse bending of elastic plates", Int. J. Solid. Struct., 12(8), 545-554. https://doi.org/10.1016/0020-7683(76)90001-9.
  44. Robinson, J. and Haggenmacher, G.W. (1979), "LORA-An accurate four node stress plate bending element", Int. J. Numer. Meth. Eng., 14(2), 296-306. https://doi.org/10.1002/nme.1620140214.
  45. Schollhammer, D. and Fries. T.P. (2019), "Kirchhoff-Love shell theory based on tangential differential calculus", Comput. Mech., 64, 113-131. https://doi.org/10.1007/s00466-018-1659-5.
  46. Taylor, R.L. (2014), "FEAP-A finite element analysis program", University of California, Berkeley.
  47. Tessler, A. and Hughes, T.J.R. (1983), "An improved treatment of transverse shear in the Mindlin-Type four-node quadrilateral element", Comput. Meth. Appl. Mech. Eng., 39, 311-335. https://doi.org/10.1016/0045-7825(83)90096-8.
  48. Ulmer, H., Hofacker, M. and Miehe, C. (2012), "Phase field modeling of fracture in plates and shells", Proc. Appl. Math. Mech., 12, 171-172. https://doi.org/10.1002/pamm.201210076.
  49. Viebahn, N., Pimenta, P.M. and Schroder, J. (2017), "A simple triangular finite element for nonlinear thin shells: statics, dynamics and anisotropy", Comput. Mech., 59, 281-297. https://doi.org/10.1007/s00466-016-1343-6.
  50. Zienkiewicz, O.C., Taylor, R.L. and Zhu, J.Z. (2005), The Finite Element Nethod: Its Basis and Fundamentals, Prentice Hall, Pearson Education Inc, USA.