Robust four-node elements based on Hu–Washizu principle for nonlinear analysis of Cosserat shells - Publication - MOST Wiedzy

Search

Robust four-node elements based on Hu–Washizu principle for nonlinear analysis of Cosserat shells

Abstract

Mixed 4-node shell elements with the drilling rotation and Cosserat-type strain measures based onthe three-field Hu–Washizu principle are proposed. In the formulation, apart from displacement and rotationfields, both strain and stress resultant fields are treated as independent. The elements are derived in the frame-work of a general nonlinear 6-parameter shell theory dedicated to the analysis of multifold irregular shells.The novelty of the developed elements stems from the fact that the measures of assumed strains and stressresultants are asymmetric. The original interpolation of drilling and bending components of strains and stressresultants is proposed. In the formulation of new mixed elements, two variants of the interpolation of membranecomponents are used and interpolation of the independent fields is defined in the natural or skew coordinates.Accuracy and efficiency of the developed elements are tested in several linear and nonlinear numerical exam-ples. It is shown that smaller number of independent parameters in the interpolation of membrane componentsgives more accurate results. The proposed mixed 4-node elements enable the use of large load steps in non-linear computations. Moreover, they require significantly less equilibrium iterations than other shell elementsformulated in the 6-parameter shell theory.

Citations

  • 1

    CrossRef

  • 1

    Web of Science

  • 1

    Scopus

Details

Category:
Articles
Type:
artykuł w czasopiśmie wyróżnionym w JCR
Published in:
CONTINUUM MECHANICS AND THERMODYNAMICS no. 31, pages 1757 - 1784,
ISSN: 0935-1175
Language:
English
Publication year:
2019
Bibliographic description:
Daszkiewicz K., Witkowski W., Burzyński S., Chróścielewski J.: Robust four-node elements based on Hu–Washizu principle for nonlinear analysis of Cosserat shells// CONTINUUM MECHANICS AND THERMODYNAMICS. -Vol. 31, (2019), s.1757-1784
DOI:
Digital Object Identifier (open in new tab) 10.1007/s00161-019-00767-1
Bibliography: test
  1. Belytschko, T., Tsay, C.-S.: A stabilization procedure for the quadrilateral plate element with one-point quadrature. Int. J. Numer. Methods Eng. 19, 405-419 (1983). https://doi.org/10.1002/nme.1620190308 open in new tab
  2. Belytschko, T., Leviathan, I.: Physical stabilization of the 4-node shell element with one point quadrature. Comput. Methods Appl. Mech. Eng. 113, 321-350 (1994). https://doi.org/10.1016/0045-7825(94)90052-3 open in new tab
  3. Reese, S.: A large deformation solid-shell concept based on reduced integration with hourglass stabilization. Int. J. Numer. Methods Eng. 69, 1671-1716 (2007). https://doi.org/10.1002/nme.1827 open in new tab
  4. Pian, T.H.H.: State-of-the-art development of hybrid/mixed finite element method. Finite Elem. Anal. Des. 21, 5-20 (1995). https://doi.org/10.1016/0168-874X(95)00024-2 open in new tab
  5. Pian, T.H.H.: Derivation of element stiffness matrices by assumed stress distributions. AIAA J. 2, 1333-1336 (1964). https:// doi.org/10.2514/3.2546 open in new tab
  6. Spilker, R.L.: Hybrid-stress eight-node elements for thin and thick multilayer laminated plates. Int. J. Numer. Methods Eng. 18, 801-828 (1982). https://doi.org/10.1002/nme.1620180602 open in new tab
  7. Pian, T.H.H., Sumihara, K.: Rational approach for assumed stress finite elements. Int. J. Numer. Methods Eng. 20, 1685-1695 (1984). https://doi.org/10.1002/nme.1620200911 open in new tab
  8. Dvorkin, E.N., Bathe, K.-J.: A continuum mechanics based four-node shell element for general non-linear analysis. Eng. Comput. 1, 77-88 (1984). https://doi.org/10.1108/eb023562 open in new tab
  9. Simo, J.C., Fox, D.D.: On a stress resultant geometrically exact shell model. Part I: formulation and optimal parametrization. Comput. Methods Appl. Mech. Eng. 72, 267-304 (1989). https://doi.org/10.1016/0045-7825(89)90002-9 open in new tab
  10. Simo, J.C., Fox, D.D., Rifai, M.S.: On a stress resultant geometrically exact shell model. Part II: the linear theory; com- putational aspects. Comput. Methods Appl. Mech. Eng. 73, 53-92 (1989). https://doi.org/10.1016/0045-7825(89)90098- 4 open in new tab
  11. Lee, S.W., Pian, T.H.H.: Improvement of plate and shell finite elements by mixed formulations. AIAA J. 16, 29-34 (1978). https://doi.org/10.2514/3.60853 open in new tab
  12. Lee, S.W., Rhiu, J.J.: A new efficient approach to the formulation of mixed finite element models for structural analysis. Int. J. Numer. Methods Eng. 23, 1629-1641 (1986). https://doi.org/10.1002/nme.1620230905 open in new tab
  13. Simo, J.C., Rifai, M.S.: A class of mixed assumed strain methods and the method of incompatible modes. Int. J. Numer. Methods Eng. 29, 1595-1638 (1990). https://doi.org/10.1002/nme.1620290802 open in new tab
  14. Simo, J.C., Armero, F.: Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes. Int. J. Numer. Methods Eng. 33, 1413-1449 (1992). https://doi.org/10.1002/nme.1620330705 open in new tab
  15. Brank, B.: Assessment of 4-node EAS-ANS shell elements for large deformation analysis. Comput. Mech. 42, 39-51 (2008). https://doi.org/10.1007/s00466-007-0233-3 open in new tab
  16. Wagner, W., Gruttmann, F.: A robust non-linear mixed hybrid quadrilateral shell element. Int. J. Numer. Methods Eng. 64, 635-666 (2005). https://doi.org/10.1002/nme.1387 open in new tab
  17. Gruttmann, F., Wagner, W.: Structural analysis of composite laminates using a mixed hybrid shell element. Comput. Mech. 37, 479-497 (2006). https://doi.org/10.1007/s00466-005-0730-1 open in new tab
  18. Wisniewski, K., Turska, E.: Four-node mixed Hu-Washizu shell element with drilling rotation. Int. J. Numer. Methods Eng. 90, 506-536 (2012). https://doi.org/10.1002/nme.3335 open in new tab
  19. Shang, Y., Cen, S., Li, C.-F.: A 4-node quadrilateral flat shell element formulated by the shape-free HDF plate and HSF membrane elements. Eng. Comput. 33, 713-741 (2016). https://doi.org/10.1108/EC-04-2015-0102 open in new tab
  20. Winkler, R., Plakomytis, D.: A new shell finite element with drilling degrees of freedom and its relation to existing formula- tions. In: Papadrakakis, M., Papadopoulos, V., Stefanou, G., and Plevris, V. (eds.) ECCOMAS Congress 2016-Proceedings of the 7th European Congress on Computational Methods in Applied Sciences and Engineering. pp. 1-40., Crete Island, Greece (2016) open in new tab
  21. Li, Z.X., Zhuo, X., Vu-Quoc, L., Izzuddin, B.A., Wei, H.Y.: A four-node corotational quadrilateral elastoplastic shell element using vectorial rotational variables. Int. J. Numer. Methods Eng. 95, 181-211 (2013). https://doi.org/10.1002/nme.4471 open in new tab
  22. Li, Z.X., Li, T.Z., Vu-Quoc, L., Izzuddin, B.A., Zhuo, X., Fang, Q.: A 9-node co-rotational curved quadrilateral shell element for smooth, folded and multi-shell structures. Int. J. Numer. Methods Eng. (2018). https://doi.org/10.1002/nme.5936 open in new tab
  23. Tang, Y.Q., Zhou, Z.H., Chan, S.L.: A simplified co-rotational method for quadrilateral shell elements in geometrically nonlinear analysis. Int. J. Numer. Methods Eng. 112, 1519-1538 (2017). https://doi.org/10.1002/nme.5567 open in new tab
  24. Ko, Y., Lee, P.S., Bathe, K.J.: A new MITC4+ shell element. Comput. Struct. 182, 404-418 (2017). https://doi.org/10.1016/ j.compstruc.2016.11.004 open in new tab
  25. Kulikov, G.M., Carrera, E., Plotnikova, S.V.: Hybrid-mixed quadrilateral element for laminated plates composed of func- tionally graded materials. Adv. Mater. Technol. 44-55, (2017). https://doi.org/10.17277/amt.2017.01.pp.044-055 open in new tab
  26. Boutagouga, D.: A new enhanced assumed strain quadrilateral membrane element with drilling degree of freedom and modified shape functions. Int. J. Numer. Methods Eng. 110, 573-600 (2017). https://doi.org/10.1002/nme.5430 open in new tab
  27. Kulikov, G.M., Plotnikova, S.V., Carrera, E.: A robust, four-node, quadrilateral element for stress analysis of functionally graded plates through higher-order theories. Mech. Adv. Mater. Struct. 1-20, (2017). https://doi.org/10.1080/15376494. 2017.1288994 open in new tab
  28. Wisniewski, K., Turska, E.: Improved nine-node shell element MITC9i with reduced distortion sensitivity. Comput. Mech. 62, 499-523 (2018). https://doi.org/10.1007/s00466-017-1510-4 open in new tab
  29. Cazzani, A., Serra, M., Stochino, F., Turco, E.: A refined assumed strain finite element model for statics and dynamics of laminated plates. Contin. Mech. Thermodyn. (2018). https://doi.org/10.1007/s00161-018-0707-x open in new tab
  30. Chróścielewski, J., Makowski, J., Stumpf, H.: Genuinely resultant shell finite elements accounting for geometric and material non-linearity. Int. J. Numer. Methods Eng. 35, 63-94 (1992). https://doi.org/10.1002/nme.1620350105 open in new tab
  31. Chróścielewski, J., Makowski, J., Stumpf, H.: Finite element analysis of smooth, folded and multi-shell structures. Comput. Methods Appl. Mech. Eng. 141, 1-46 (1997). https://doi.org/10.1016/S0045-7825(96)01046-8 open in new tab
  32. Altenbach, J., Altenbach, H., Eremeyev, V.A.: On generalized Cosserat-type theories of plates and shells: a short review and bibliography. Arch. Appl. Mech. 80, 73-92 (2010). https://doi.org/10.1007/s00419-009-0365-3 open in new tab
  33. Neff, P.: A geometrically exact Cosserat shell-model including size effects, avoiding degeneracy in the thin shell limit. Part I: formal dimensional reduction for elastic plates and existence of minimizers for positive Cosserat couple modulus. Contin. Mech. Thermodyn. 16, 577-628 (2004). https://doi.org/10.1007/s00161-004-0182-4 open in new tab
  34. Chróścielewski, J., Sabik, A., Sobczyk, B., Witkowski, W.: 2-D constitutive equations for orthotropic Cosserat type laminated shells in finite element analysis. Compos. Part B Eng. 165, 335-353 (2019). https://doi.org/10.1016/j.compositesb.2018.11. 101 open in new tab
  35. Sabik, A.: Progressive failure analysis of laminates in the framework of 6-field non-linear shell theory. Compos. Struct. 200, 195-203 (2018). https://doi.org/10.1016/j.compstruct.2018.05.069 open in new tab
  36. Burzyński, S., Chróścielewski, J., Witkowski, W.: Elastoplastic law of Cosserat type in shell theory with drilling rotation. Math. Mech. Solids. 20, 790-805 (2015). https://doi.org/10.1177/1081286514554351 open in new tab
  37. Burzyński, S., Chróścielewski, J., Daszkiewicz, K., Witkowski, W.: Geometrically nonlinear FEM analysis of FGM shells based on neutral physical surface approach in 6-parameter shell theory. Compos. Part B Eng. 107, 203-213 (2016). https:// doi.org/10.1016/j.compositesb.2016.09.015 open in new tab
  38. Burzyński, S., Chróścielewski, J., Daszkiewicz, K., Witkowski, W.: Elastoplastic nonlinear FEM analysis of FGM shells of Cosserat type. Compos. Part B Eng. 154, 478-491 (2018). https://doi.org/10.1016/j.compositesb.2018.07.055 open in new tab
  39. Atluri, S.N., Murakawa, H.: On hybrid finite element models in nonlinear solid mechanics. In: Bergan, P.G. (ed.) Finite Elements in Nonlinear Mechanics, pp. 25-69. Tapir Press, Norway (1977) open in new tab
  40. Murakawa, H., Atluri, S.N.: Finite elasticity solutions using hybrid finite elements based on a complementary energy principle. J. Appl. Mech. 45, 539-547 (1978) open in new tab
  41. Cazzani, A., Atluri, S.N.: Four-noded mixed finite elements, using unsymmetric stresses, for linear analysis of membranes. Comput. Mech. 11, 229-251 (1993). https://doi.org/10.1007/BF00371864 open in new tab
  42. Seki, W., Atluri, S.N.: Analysis of strain localization in strain-softening hyperelastic materials, using assumed stress hybrid elements. Comput. Mech. 14, 549-585 (1994). https://doi.org/10.1007/BF00350837 open in new tab
  43. Seki, W., Atluri, S.N.: On newly developed assumed stress finite element formulations for geometrically and materially nonlinear problems. Finite Elem. Anal. Des. 21, 75-110 (1995). https://doi.org/10.1016/0168-874X(95)00028-X open in new tab
  44. Sansour, C., Bednarczyk, H.: The Cosserat surface as a shell model, theory and finite-element formulation. Comput. Methods Appl. Mech. Eng. 120, 1-32 (1995). https://doi.org/10.1016/0045-7825(94)00054-Q open in new tab
  45. Sansour, C., Bocko, J.: On hybrid stress, hybrid strain and enhanced strain finite element formulations for a geometrically exact shell theory with drilling degrees of freedom. Int. J. Numer. Methods Eng. 43, 175-192 (1998). https://doi.org/10. 1002/(SICI)1097-0207(19980915)43:1<175::AID-NME448>3.0.CO;2-9 open in new tab
  46. Chróścielewski, J.: The family of C0 finite elements in the nonlinear six parameter shell theory (in Polish), Zeszyty Naukowe Politechniki Gdańskiej, 540, Budownictwo Lądowe, Nr 53., Gdańsk (1996) open in new tab
  47. Chróścielewski, J., Witkowski, W.: Four-node semi-EAS element in six-field nonlinear theory of shells. Int. J. Numer. Methods Eng. 68, 1137-1179 (2006). https://doi.org/10.1002/nme.1740 open in new tab
  48. Witkowski, W.: 4-node combined shell element with semi-EAS-ANS strain interpolations in 6-parameter shell theories with drilling degrees of freedom. Comput. Mech. 43, 307-319 (2009). https://doi.org/10.1007/s00466-008-0307-x open in new tab
  49. Daszkiewicz, K.: A family of hybrid mixed elements in 6-parameter shell theory, geometrically nonlinear analysis of func- tionally graded shells. Doctoral Thesis (in Polish) (2017)
  50. Chróścielewski, J., Burzyński, S., Daszkiewicz, K., Witkowski, W.: Mixed 4-node shell element with assumed strain and stress in 6-parameter theory. In: Pietraszkiewicz, W., Witkowski, W. (eds.) Shell Structures: Theory and Applications, vol. 4, pp. 359-362. Taylor & Francis Group, London (2018) open in new tab
  51. Libai, A., Simmonds, J.G.: The Nonlinear Theory of Elastic Shells. Cambridge University Press, Cambridge (1998) open in new tab
  52. Reissner, E.: Linear and nonlinear theory of shells. In: Fung, Y.C., Sechler, E.E. (eds.) Thin Shell Structures, pp. 29-44. Prentice-Hall, Englewood Cliffs (1974)
  53. Miśkiewicz, M.: Structural response of existing spatial truss roof construction based on Cosserat rod theory. Contin. Mech. Thermodyn. 31(1), 79-99 (2019). https://doi.org/10.1007/s00161-018-0660-8 open in new tab
  54. Eremeyev, V.A., Zubov, L.M.: On constitutive inequalities in nonlinear theory of elastic shells. ZAMM Zeitschrift fur. Angew. Math. und Mech. 87, 94-101 (2007). https://doi.org/10.1002/zamm.200610304 open in new tab
  55. Chróścielewski, J., Witkowski, W.: Discrepancies of energy values in dynamics of three intersecting plates. Int. J. Numer. Method. Biomed. Eng. 26, 1188-1202 (2010). https://doi.org/10.1002/cnm.1208 open in new tab
  56. Eremeyev, V.A., Lebedev, L.P.: Existence theorems in the linear theory of micropolar shells. ZAMM Zeitschrift fur Angew. Math. und Mech. 91, 468-476 (2011). https://doi.org/10.1002/zamm.201000204 open in new tab
  57. Eremeyev, V.A., Lebedev, L.P., Cloud, M.J.: The Rayleigh and Courant variational principles in the six-parameter shell theory. Math. Mech. Solids 20, 806-822 (2015). https://doi.org/10.1177/1081286514553369 open in new tab
  58. Pietraszkiewicz, W., Eremeyev, V.A.: Natural lagrangian strain measures of the non-linear cosserat continuum. In: Maugin, G.A., Metrikine, A.V. (eds.) Mechanics of Generalized Continua, pp. 79-86. Springer, Berlin (2010) open in new tab
  59. Pietraszkiewicz, W., Eremeyev, V.A.: On vectorially parameterized natural strain measures of the non-linear Cosserat con- tinuum. Int. J. Solids Struct. 46, 2477-2480 (2009). https://doi.org/10.1016/j.ijsolstr.2009.01.030 open in new tab
  60. Pietraszkiewicz, W., Konopińska, V.: On unique kinematics for the branching shells. Int. J. Solids Struct. 48, 2238-2244 (2011). https://doi.org/10.1016/j.ijsolstr.2011.03.029 open in new tab
  61. Nowacki, W.: Theory of Asymmetric Elasticity. Pergamon Press, Oxford (1986) open in new tab
  62. Washizu, K.: On the variational principles of elasticity and plasticity. Aeroelastic and Structures Research Laboratory Technical Report No. 25-18., Cambridge (1955)
  63. Wisniewski, K., Turska, E.: Improved four-node Hellinger-Reissner elements based on skew coordinates. Int. J. Numer. Methods Eng. 76, 798-836 (2008). https://doi.org/10.1002/nme.2343 open in new tab
  64. Wisniewski, K., Turska, E.: Improved 4-node Hu-Washizu elements based on skew coordinates. Comput. Struct. 87, 407-424 (2009). https://doi.org/10.1016/j.compstruc.2009.01.011 open in new tab
  65. Yuan, K.-Y., Huang, Y.-S., Pian, T.H.H.: New strategy for assumed stresses for 4-node hybrid stress membrane element. Int. J. Numer. Methods Eng. 36, 1747-1763 (1993). https://doi.org/10.1002/nme.1620361009 open in new tab
  66. Wiśniewski, K.: Finite Rotation Shells. Springer, Barcelona (2010) open in new tab
  67. Klinkel, S., Gruttmann, F., Wagner, W.: A mixed shelf formulation accounting for thickness strains and finite strain 3d material models. Int. J. Numer. Methods Eng. 74, 945-970 (2008). https://doi.org/10.1002/nme.2199 open in new tab
  68. Wisniewski, K., Wagner, W., Turska, E., Gruttmann, F.: Four-node Hu-Washizu elements based on skew coordinates and contravariant assumed strain. Comput. Struct. 88, 1278-1284 (2010). https://doi.org/10.1016/j.compstruc.2010.07.008 open in new tab
  69. Pietraszkiewicz, W.: The resultant linear six-field theory of elastic shells: what it brings to the classical linear shell models? ZAMM J. Appl. Math. Mech./Zeitschrift für Angew. Math. und Mech. 96, 899-915 (2016). https://doi.org/10.1002/zamm. 201500184 open in new tab
  70. Kasper, E.P., Taylor, R.L.: Mixed-enhanced strain method. Part II: geometrically nonlinear problems. Comput. Struct. 75, 251-260 (2000). https://doi.org/10.1016/S0045-7949(99)00135-2 open in new tab
  71. Piltner, R., Taylor, R.L.: A systematic construction of B-bar functions for linear and non-linear mixed-enhanced finite elements for plane elasticity problems. Int. J. Numer. Methods Eng. 44, 615-639 (1999). https://doi.org/10.1002/(SICI)1097- 0207(19990220)44:5<615::AID-NME518>3.0.CO;2-U open in new tab
  72. Burzyński, S., Chróścielewski, J., Witkowski, W.: Geometrically nonlinear FEM analysis of 6-parameter resultant shell theory based on 2-D Cosserat constitutive model. ZAMM J. Appl. Math. Mech./Zeitschrift für.Angew. Math. und Mech. 96, 191-204 (2016). https://doi.org/10.1002/zamm.201400092 open in new tab
  73. Chróścielewski, J., Pietraszkiewicz, W., Witkowski, W.: On shear correction factors in the non-linear theory of elastic shells. Int. J. Solids Struct. 47, 3537-3545 (2010). https://doi.org/10.1016/j.ijsolstr.2010.09.002 open in new tab
  74. Macneal, R.H., Harder, R.L.: A proposed standard set of problems to test finite element accuracy. Finite Elem. Anal. Des. 1, 3-20 (1985). https://doi.org/10.1016/0168-874X(85)90003-4 open in new tab
  75. Cook, R.D.: Improved two-dimensional finite element. J. Struct. Div. 100, 1851-1863 (1974)
  76. Piltner, R., Taylor, R.L.: A quadrilateral mixed finite element with two enhanced strain modes. Int. J. Numer. Methods Eng. 38, 1783-1808 (1995). https://doi.org/10.1002/nme.1620381102 open in new tab
  77. Argyris, J.H., Balmer, H., Doltsinis, J.S., Dunne, P.C., Haase, M., Kleiber, M., Malejannakis, G.A., Mlejnek, H.-P., Müller, M., Scharpf, D.W.: Finite element method: the natural approach. Comput. Methods Appl. Mech. Eng. 17(18), 1-106 (1979) open in new tab
  78. Simo, J.C., Fox, D.D., Rifai, M.S.: On a stress resultant geometrically exact shell model. Part III: computational aspects of the nonlinear theory. Comput. Methods Appl. Mech. Eng. 79, 21-70 (1990). https://doi.org/10.1016/0045-7825(90)90094-3 open in new tab
  79. Stander, N., Matzenmiller, A., Ramm, E.: An assessment of assumed strain methods in finite rotation shell analysis. Eng. Comput. 6, 58-66 (1989). https://doi.org/10.1108/eb023760 open in new tab
Sources of funding:
Verified by:
Gdańsk University of Technology

seen 107 times

Recommended for you

Meta Tags