2-D constitutive equations for orthotropic Cosserat type laminated shells in finite element analysis - Publication - MOST Wiedzy


2-D constitutive equations for orthotropic Cosserat type laminated shells in finite element analysis


We propose 2-D Cosserat type orthotropic constitutive equations for laminated shells for the purpose of initial failure estimation in a laminate layer. We use nonlinear 6-parameter shell theory with asymmetric membrane strain measures and Cosserat kinematics as the framework. This theory is specially dedicated to the analysis of irregular shells, inter alia, with orthogonal intersections, since it takes into account the drilling rotation degree of freedom. Therefore, the shell is endowed naturally with 6 degrees of freedom: 3 translations and 3 rotations. The proposed equations are formulated from the statement of the generalized Cosserat plane stress with additional transverse shear components and integrated over the shell's thickness using the equivalent single layer approach (ESL). The resulting formulae are implemented into the own Fortran code enabling nonlinear shell analysis. Some numerical results are presented to show the performance of the proposed approach.


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artykuł w czasopiśmie wyróżnionym w JCR
Published in:
COMPOSITES PART B-ENGINEERING no. 165, pages 335 - 353,
ISSN: 1359-8368
Publication year:
Bibliographic description:
Chróścielewski J., Sabik A., Sobczyk B., Witkowski W.: 2-D constitutive equations for orthotropic Cosserat type laminated shells in finite element analysis// COMPOSITES PART B-ENGINEERING. -Vol. 165, (2019), s.335-353
Digital Object Identifier (open in new tab) 10.1016/j.compositesb.2018.11.101
Bibliography: test
  1. Singh DB, Singh BN. New higher order shear deformation theories for free vibration and buckling analysis of laminated and braided composite plates. Int J Mech Sci 2017;131- 132:265-77. doi:10.1016/j.ijmecsci.2017.06.053. open in new tab
  2. Pagani A, Carrera E. Large-deflection and post-buckling analyses of laminated composite beams by Carrera Unified Formulation. Compos Struct 2017;170:40-52. doi:10.1016/j.compstruct.2017.03.008. open in new tab
  3. Teter A, Mania RJ, Kolakowski Z. Non-linear stability and load-carrying capacity of thin- walled laminated columns in aspects of coupled buckling and coupled stiffness submatrix. open in new tab
  4. Compos Struct 2018;192:72-81. doi:10.1016/j.compstruct.2018.02.070. open in new tab
  5. Hasim KA. Isogeometric static analysis of laminated composite plane beams by using refined zigzag theory. Compos Struct 2018;186:365-74. doi:10.1016/j.compstruct.2017.12.033. open in new tab
  6. Ou X, Zhang X, Zhang R, Yao X, Han Q. Weak form quadrature element analysis on nonlinear bifurcation and post-buckling of cylindrical composite laminates. Compos Struct 2018;188:266-77. doi:10.1016/j.compstruct.2018.01.007. open in new tab
  7. Wan L, Yang D, Ismail Y, Sheng Y. 3D particle models for composite laminates with anisotropic elasticity. Compos Part B Eng 2018;149:110-21. doi:10.1016/j.compositesb.2018.05.022. open in new tab
  8. Liao BB, Jia LY. Finite element analysis of dynamic responses of composite pressure vessels under low velocity impact by using a three-dimensional laminated media model. Thin-Walled Struct 2018;129:488-501. doi:10.1016/j.tws.2018.04.023. open in new tab
  9. Arruda MRT, Garrido M, Castro LM., Ferreira AJM, Correia JR. Numerical modelling of the creep behaviour of GFRP sandwich panels using the Carrera Unified Formulation and Composite Creep Modelling. Compos Struct 2018;183:103-13. doi:10.1016/j.compstruct.2017.01.074. open in new tab
  10. Tornabene F, Fantuzzi N, Bacciocchi M. Free vibrations of free-form doubly-curved shells made of functionally graded materials using higher-order equivalent single layer theories. open in new tab
  11. Compos Part B Eng 2014;67:490-509. doi:10.1016/j.compositesb.2014.08.012. open in new tab
  12. Fazzolari FA. Reissner's Mixed Variational Theorem and variable kinematics in the modelling of laminated composite and FGM doubly-curved shells. Compos Part B Eng 2016;89:408-23. doi:10.1016/j.compositesb.2015.11.031. open in new tab
  13. Chróścielewski J, Miśkiewicz M, Pyrzowski Ł, Sobczyk B, Wilde K. A novel sandwich footbridge -Practical application of laminated composites in bridge design and in situ measurements of static response. Compos Part B Eng 2017;126:153-61. doi:10.1016/j.compositesb.2017.06.009. open in new tab
  14. Chróścielewski J, Miśkiewicz M, Pyrzowski Ł, Rucka M, Sobczyk B, Wilde K. Modal properties identification of a novel sandwich footbridge -Comparison of measured and FEA. Compos Part B Eng 2018. doi:10.1016/j.compositesb.2018.06.016. open in new tab
  15. Yademellat H, Nikbakht A, Saghafi H, Sadighi M. Experimental and numerical investigation of low velocity impact on electrospun nanofiber modified composite laminates. Compos Struct 2018;200:507-14. doi:10.1016/j.compstruct.2018.05.146. open in new tab
  16. Li Y, Pimenta S, Singgih J, Nothdurfter S, Schuffenhauer K. Experimental investigation of randomly-oriented tow-based discontinuous composites and their equivalent laminates. open in new tab
  17. Compos Part A Appl Sci Manuf 2017;102:64-75. doi:10.1016/j.compositesa.2017.06.031. open in new tab
  18. Kharghani N, Guedes Soares C. Experimental, numerical and analytical study of bending of rectangular composite laminates. Eur J Mech -A/Solids 2018;72:155-74. doi:10.1016/j.euromechsol.2018.05.007. open in new tab
  19. Tan W, Naya F, Yang L, Chang T, Falzon BG, Zhan L, et al. The role of interfacial properties on the intralaminar and interlaminar damage behaviour of unidirectional composite laminates: Experimental characterization and multiscale modelling. Compos Part B Eng 2018;138:206-21. doi:10.1016/j.compositesb.2017.11.043. open in new tab
  20. Kaddour AS, Hinton MJ, Soden PD. A comparison of the predictive capabilities of current failure theories for composite laminates: additional contributions. Compos Sci Technol 2004;64:449-76. doi:10.1016/S0266-3538(03)00226-4. open in new tab
  21. Hinton M., Kaddour A., Soden P. A further assessment of the predictive capabilities of current failure theories for composite laminates: comparison with experimental evidence. open in new tab
  22. Compos Sci Technol 2004;64:549-88. doi:10.1016/S0266-3538(03)00227-6. open in new tab
  23. Soden P., Kaddour A., Hinton M. Recommendations for designers and researchers resulting from the world-wide failure exercise. Compos Sci Technol 2004;64:589-604. doi:10.1016/S0266-3538(03)00228-8. open in new tab
  24. Sabik A. Progressive failure analysis of laminates in the framework of 6-field non-linear shell theory. Compos Struct 2018; 200: 195-203. open in new tab
  25. Koh R, Madsen B. Strength failure criteria analysis for a flax fibre reinforced composite. Mech Mater 2018;124:26-32. doi:10.1016/j.mechmat.2018.05.005. open in new tab
  26. Gu J, Chen P. Some modifications of Hashin's failure criteria for unidirectional composite materials. Compos Struct 2017;182:143-52. doi:10.1016/j.compstruct.2017.09.011. open in new tab
  27. Thomson DM, Cui H, Erice B, Hoffmann J, Wiegand J, Petrinic N. Experimental and numerical study of strain-rate effects on the IFF fracture angle using a new efficient implementation of Puck's criterion. Compos Struct 2017;181:325-35. doi:10.1016/j.compstruct.2017.08.084. open in new tab
  28. Gu J, Chen P. Extension of Puck's inter fibre fracture (IFF) criteria for UD composites. open in new tab
  29. Compos Sci Technol 2018;162:79-85. doi:10.1016/j.compscitech.2018.04.019. open in new tab
  30. Obert E, Daghia F, Ladevèze P, Ballere L. Micro and meso modeling of woven composites: Transverse cracking kinetics and homogenization. Compos Struct 2014;117:212-21. doi:10.1016/j.compstruct.2014.06.035. open in new tab
  31. Wang B, Fang G, Liu S, Fu M, Liang J. Progressive damage analysis of 3D braided composites using FFT-based method. Compos Struct 2018;192:255-63. doi:10.1016/j.compstruct.2018.02.040. open in new tab
  32. Rozylo P, Debski H, Kubiak T. A model of low-velocity impact damage of composite plates subjected to Compression-After-Impact (CAI) testing. Compos Struct 2017;181:158-70. doi:10.1016/j.compstruct.2017.08.097. open in new tab
  33. Chróścielewski J, Makowski J, Stumpf H. Genuinely resultant shell finite elements accounting for geometric and material non-linearity. Int J Numer Meth Eng 1992; 35: 63- 94. open in new tab
  34. Chróścielewski J, Makowski J, Stumpf H. Finite element analysis of smooth, folded and multi-shell structures. Comp. Meth. Appl. Mech. Engng 1997; 141: 1-46. open in new tab
  35. Altenbach J, Altenbach H, Eremeyev VA. On generalized Cosserat-type theories of plates and shells: a short review and bibliography. Archive of Applied Mechanics, 2010;80:73-92. open in new tab
  36. Miśkiewicz M. Structural response of existing spatial truss roof construction based on Cosserat rod theory. Contin Mech Thermodyn 2018. doi:10.1007/s00161-018-0660-8. open in new tab
  37. Altenbach H, Eremeyev VA. Actual developments in the nonlinear shell theory-state of the art and new applications of the six-parameter shell theory. Shell structures: Theory and applications vol. 3, 2014:3-12. open in new tab
  38. Eremeyev VA, Lebedev LP, Cloud MJ. The Rayleigh and Courant variational principles in the six-parameter shell theory. Mathematics and Mechanics of Solids, 2015; 20(7): 806- 822. open in new tab
  39. Bîrsan M, Neff P. Existence of minimizers in the geometrically non-linear 6-parameter resultant shell theory with drilling rotations, Mathematics and Mechanics of Solids 2014;19(4):376-397. open in new tab
  40. Chróścielewski J, Sabik A, Sobczyk B, Witkowski W. Nonlinear FEM 2D failure onset prediction of composite shells based on 6-parameter shell theory. Thin-Walled Struct 2016;105:207-19. open in new tab
  41. 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 2016;96:191-204. open in new tab
  42. 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 2016;107:203-13. open in new tab
  43. Burzyński S, Chróścielewski J, Daszkiewicz K, Witkowski W. Elastoplastic nonlinear FEM analysis of FGM shells of Cosserat type. Compos Part B Eng 2018. doi:10.1016/j.compositesb.2018.07.055. open in new tab
  44. Eremeyev VA, Pietraszkiewicz W. Local symmetry group in the general theory of elastic shells. Journal of Elasticity 2006;85(2):125-152. open in new tab
  45. Libai A, Simmonds JG. The Nonlinear Theory of Elastic Shells, 2nd ed. 1998, Cambridge University Press, Cambridge open in new tab
  46. Toupin RA. Theories of Elasticity with Couple-stress, Archive for Rational Mechanics and Analysis 1964;17(2):85-112. open in new tab
  47. Eremeyev VA, Lebedev LP, Altenbach H. Foundations of Micropolar Mechanics, 2013, Springer. open in new tab
  48. Pietraszkiewicz W. Eremeyev VA, On vectorially parameterized natural strain measures of the non-linear Cosserat continuum. International Journal of Solids and Structures, 2009;46(11):2477-2480 open in new tab
  49. Stuelpnagel J. On the parameterization of the three-dimensional rotation group. SIAM Review 1964;6:422-430. open in new tab
  50. Chróścielewski J, Kreja I, Sabik A, Witkowski W. Modeling of composite shells in 6- parameter nonlinear theory with drilling degree of freedom, Mech. Adv. Mater. Struct. 2011;18:403-419. open in new tab
  51. Nakamura S, Benedict R, Lakes R. Finite element method for orthotropic micropolar elasticity. Int. J. Engng. Sci. Eng. 1984;22:319-330.
  52. Fantuzzi N, Leonetti L, Trovalusci P, Tornabene F. Some Novel Numerical Applications of Cosserat Continua. International Journal of Computational Methods 2018; Vol. 15, No. 3, 1850054 (38 pages). open in new tab
  53. Tornabene F, Fantuzzi N, Bacciocchi M. Mechanical behaviour of composite Cosserat solids in elastic problems with holes and discontinuities. Composite Structures 2017; 179:468-481. open in new tab
  54. Kaw AK. Mechanics of Composite Materials, Second edition. Boca Raton, London, New York: Taylor & Francis Group; 2006.
  55. Reddy JN. Mechanics of laminated composite plates and shells. Theory and analysis. 2 th edition. 2004, CRC Press. open in new tab
  56. Jeong J, Ramezani H, Münch I, Neff P. A numerical study for linear isotropic Cosserat elasticity with conformally invariant curvature, Z. Angew. Math. Mech. 2009;89(7):552 - 569 (2009) open in new tab
  57. Li X, Yu K, Han J, Song H, Zhao R. Buckling and vibro-acoustic response of the clamped composite laminated plate in thermal environment. Int J Mech Sci 2016;119:370-82. doi:10.1016/j.ijmecsci.2016.10.021 open in new tab
  58. Sander O, Neff P, Bîrsan M. Numerical treatment of a geometrically nonlinear planar Cosserat shell model, arXiv:1412.3668v1 [math.NA] 11 Dec 2014. open in new tab
  59. Hashin Z. Failure Criteria for Unidirectional Fiber Composites. J Appl Mech 1980;47:329- 34. doi:10.1115/1.3153664. open in new tab
  60. Yuan-Sheng C. Physical interpretation of Hashin's criterion of fatigue failure under multiaxial stress. Eng Fract Mech 1986;24:165-7. doi:10.1016/0013-7944(86)90048-2. open in new tab
  61. Wang FS, Yu XS, Jia SQ, Li P. Experimental and numerical study on residual strength of aircraft carbon/epoxy composite after lightning strike. Aerosp Sci Technol 2018;75:304- 14. doi:10.1016/j.ast.2018.01.029. open in new tab
  62. Bsisu KA-D, Hussein HH, Sargand SM. The Use of Hashin Damage Criteria, CFRP- Concrete Interface and Concrete Damage Plasticity Models in 3D Finite Element Modeling of Retrofitted Reinforced Concrete Beams with CFRP Sheets. Arab J Sci Eng 2017;42:1171-84. doi:10.1007/s13369-016-2356-3. open in new tab
  63. Yu G, Ren Y, Zhang T, Xiao W, Jiang H. Hashin Failure Theory Based Damage Assessment Methodology of Composite Tidal Turbine Blades and Implications for the Blade Design. China Ocean Eng 2018;32:216-25. doi:10.1007/s13344-018-0023-z. open in new tab
  64. Gu J, Chen P. Some modifications of Hashin's failure criteria for unidirectional composite materials. Compos Struct 2017;182:143-52. doi:10.1016/j.compstruct.2017.09.011. open in new tab
  65. Duarte APC, Díaz Sáez A, Silvestre N. Comparative study between XFEM and Hashin damage criterion applied to failure of composites. Thin-Walled Struct 2017;115:277-88. doi:10.1016/j.tws.2017.02.020. open in new tab
  66. Pietraszkiewicz W, Konopińska V. Drilling couples and refined constitutive equations in the resultant geometrically non-linear theory of elastic shells. Int J Solids Struct 2014;51:2133-43. doi:10.1016/j.ijsolstr.2014.02.022. open in new tab
  67. Stander N, Matzenmiller A, Ramm E. An assessment of assumed strain methods in finite rotation shell analysis, Engineering Computations 1989;6:58-66. open in new tab
  68. Klinkel S, Gruttmann F, Wagner W. A continuum based three-dimensional shell element for laminated structures, Computers and Structures 1999;71:43-62. open in new tab
  69. Sze KY, Liu XH, Lo SH. Popular benchmark problems for geometric nonlinear analysis of shells, Fin. Elem. Anal. Des. 2004;40:1151-1569. open in new tab
  70. Arciniega RA, Reddy JN.,Tensor-based finite element formulation for geometrically nonlinear analysis of shell structures, Comput. Methods Appl. Mech. Engrg. 2007;196: 1048-1073. open in new tab
  71. Wiśniewski K. Finite Rotation Shells. Barcelona: Springer, 2010. open in new tab
  72. Padhi GS, Shenoi RA, Moy SSJ, Hawkins GL. Progressive failure and ultimate collapse of laminated composite plates in bending, Compos Struct 1998; 40: 277-291. open in new tab
  73. Moy SSJ, Shenoi RA, Allen HG. Strength and stiffness of fibre-reinforced plastic plates, Proc. Instn Cio Engrs Structs & Bldgs 1996; 116: 204-220. open in new tab
  74. Davila CG, Camanho PP, Rose CA. Failure Criteria for FRP Laminates. J. Compos. Mater 2005; 39: 323-345. open in new tab
  75. Knight NF, Rankin CC, Brogan FA. STAGS computational procedure for progressive failure analysis of laminated composite structures. International Journal of Non-linear Mechanics 2002;37: 833-849. open in new tab
  76. Sleight DW. Progressive failure analysis methodology for laminated composite structures, NASA/TP-1999-209107. open in new tab
  77. Sobczyk B. FEM analysis of composite materials failure in nonlinear six field shell theory, PhD thesis, Gdańsk University of Technology, 2017.
  78. Gliszczyński A, Kubiak T. Progressive failure analysis of thin-walled composite columns subjected to uniaxial compression. Composite Structures 2017;169:52-61. open in new tab
  79. Chróścielewski J, Sabik A, Sobczyk B, Witkowski W. Analysis of laminates with the use of 2-D Cosserat constitutive model, 41st Solid Mechanics Conference, August 27-31, 2018 open in new tab
  80. Warsaw, Poland, http://solmech2018.ippt.pan.pl/abstracts/0289.pdf
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