Hybrid-Mixed Solid-Shell Element for Stress Analysis of Laminated Piezoelectric Shells through Higher-Order Theories

  • Gennady M. KulikovEmail author
  • Svetlana V. Plotnikova
  • Erasmo Carrera
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 81)


A geometrically exact hybrid-mixed four-node piezoelectric solid-shell element by using the sampling surfaces (SaS) method is developed. The SaS formulation is based on choosing inside the layers the arbitrary number of SaS parallel to the middle surface and located at Chebyshev polynomial nodes in order to introduce the displacements and electric potentials of these surfaces as basic shell unknowns. The external surfaces and interfaces are also included into a set of SaS because of the variational formulation. Such a choice of unknowns with the consequent use of Lagrange polynomials in the through-thickness approximations of displacements, strains, electric potential and electric field leads to a very compact piezoelectric shell element formulation. To implement the efficient analytical integration throughout the element, the enhanced assumed natural strain (ANS) method is employed. The proposed hybrid-mixed four-node piezoelectric shell element is based on the Hu-Washizu variational equation and exhibits a superior performance in the case of coarse meshes. It could be useful for the three-dimensional (3D) stress analysis of thick and thin doubly-curved laminated piezoelectric shells since the SaS formulation gives the possibility to obtain the numerical solutions with a prescribed accuracy, which asymptotically approach the exact solutions of piezoelectricity as the number of SaS tends to infinity.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



This work was supported by the Russian Ministry of Education and Science (Grants No. 9.1148.2017/4.6 and 9.4914.2017/6.7).


  1. Bakhvalov NS (1977) Numerical Methods: Analysis, Algebra, Ordinary Differential Equations. MIR, MoscowGoogle Scholar
  2. Bathe KJ, Dvorkin EN (1986) A formulation of general shell elements-the use of mixed interpolation of tensorial components. Int J Numer Meth Engng 22(3):697–722Google Scholar
  3. Bathe KJ, Lee PS, Hiller JF (2003) Towards improving the MITC9 shell element. Comput Struct 81(8):477–489Google Scholar
  4. Betsch P, Stein E (1995) An assumed strain approach avoiding artificial thickness straining for a non-linear 4-node shell element. Comm Num Meth Engng 11(11):899–909Google Scholar
  5. Carrera E (1999) Multilayered shell theories accounting for layerwise mixed description. Part 1: Governing equations. AIAA J 37(9):1107–1116Google Scholar
  6. Carrera E (2003) Theories and finite elements for multilayered plates and shells: A unified compact formulation with numerical assessment and benchmarking. Arch Comput Meth Engng 10(3):215–296Google Scholar
  7. Carrera E, Valvano S (2017) Analysis of laminated composite structures with embedded piezoelectric sheets by variable kinematic shell elements. J Intel Mater Systems Struct online:1–29,
  8. Carrera E, Brischetto S, Nali P (2011) Plates and Shells for Smart Structures: Classical and Advanced Theories for Modeling and Analysis. John Wiley & Sons Ltd, ChichesterGoogle Scholar
  9. Carrera E, Cinefra M, Petrolo M, Zappino E (2014) Finite Element Analysis of Structures Through Unified Formulation. John Wiley & Sons Ltd, ChichesterGoogle Scholar
  10. Chen WQ, Ding HJ, Xu RQ (2001) Three-dimensional static analysis of multi-layered piezoelectric hollow spheres via the state space method. Int J Solids Struct 38(28):4921–4936Google Scholar
  11. Cinefra M, Carrera E, Valvano S (2015) Variable kinematic shell elements for the analysis of electro-mechanical problems. Mech Adv Mater Struct 22:77–106Google Scholar
  12. Dunn ML, Taya M (1994) Electroelastic field concentrations in and around inhomogeneities in piezoelectric solids. Trans ASME J Appl Mech 61(2):474–475Google Scholar
  13. Heyliger P (1997) A note on the static behavior of simply supported laminated piezoelectric cylinders. Int J Solids Struct 34(29):3781–3794Google Scholar
  14. Hughes TJR, Tezduyar TE (1981) Finite elements based upon Mindlin plate theory with particular reference to the four-node bilinear isoparametric element. Trans ASME J Appl Mech 48(3):587–596Google Scholar
  15. Klinkel S, Wagner W (2006) A geometrically non-linear piezoelectric solid shell element based on a mixed multi-field variational formulation. Int J Numer Meth Eng 65(3):349–382Google Scholar
  16. Klinkel S, Wagner W (2008) A piezoelectric solid shell element based on a mixed variational formulation for geometrically linear and nonlinear applications. Comp Struct 86(1):38–46Google Scholar
  17. Ko Y, Lee PS, Bathe KJ (2017) A new MITC4+ shell element. Comput Struct 182:404–418Google Scholar
  18. Kulikov GM (2001) Refined global approximation theory of multilayered plates and shells. J Eng Mech 127(2):119–125Google Scholar
  19. Kulikov GM, Carrera E (2008) Finite deformation higher-order shell models and rigid-body motions. Int J Solids Struct 45(11):3153–3172Google Scholar
  20. Kulikov GM, Plotnikova SV (2008) Geometrically exact four-node piezoelectric solid-shell element. Mech Adv Mater Struct 15(3-4):199–207Google Scholar
  21. Kulikov GM, Plotnikova SV (2010) Solution of a coupled problem of thermopiezoelectricity based on a geometrically exact shell element. Mech Comp Mater 46(4):349–364Google Scholar
  22. Kulikov GM, Plotnikova SV (2011a) Exact geometry piezoelectric solid-shell element based on the 7-parameter model. Mech Adv Mater Struct 18(2):133–146Google Scholar
  23. Kulikov GM, Plotnikova SV (2011b) Finite rotation piezoelectric exact geometry solid-shell element with nine degrees of freedom per node. Comput Mater Continua 23:233–264Google Scholar
  24. Kulikov GM, Plotnikova SV (2011c) Non-linear exact geometry 12-node solid-shell element with three translational degrees of freedom per node. Int J Numer Meth Engng 88(13):1363–1389Google Scholar
  25. Kulikov GM, Plotnikova SV (2011d) On the use of a new concept of sampling surfaces in shell theory. In: Altenbach H, Eremeyev VA (eds) Shell-like Structures: Non-classical Theories and Applications, Springer, Berlin, Heidelberg, pp 715–726Google Scholar
  26. Kulikov GM, Plotnikova SV (2013) A sampling surfaces method and its application to threedimensional exact solutions for piezoelectric laminated shells. Int J Solids Struct 50(11):1930–1943Google Scholar
  27. Kulikov GM, Plotnikova SV (2014) Exact electroelastic analysis of functionally graded piezoelectric shells. Int J Solids Struct 51(1):13–25Google Scholar
  28. Kulikov GM, Plotnikova SV (2015) The use of 9-parameter shell theory for development of exact geometry 12-node quadrilateral piezoelectric laminated solid-shell elements. Mech Adv Mater Struct 22(6):490–502Google Scholar
  29. Kulikov GM, Plotnikova SV (2017) Assessment of the sampling surfaces formulation for thermoelectroelastic analysis of layered and functionally graded piezoelectric shells. Mech Adv Mater Struct 24(5):392–409Google Scholar
  30. Kulikov GM, Mamontov AA, Plotnikova SV (2015) Coupled thermoelectroelastic stress analysis of piezoelectric shells. Compos Struct 124:65–76Google Scholar
  31. Lee S, Goo NS, Park HC, Yoon KJ, Cho C (2003) A nine-node assumed strain shell element for analysis of a coupled electro-mechanical system. Smart Mater Struct 12:355–362Google Scholar
  32. Lentzen S (2009) Nonlinearly coupled thermopiezoelectric modelling and FE-simulation of smart structures. No. 419 in Fortschritt-Berichte VDI, Reihe 20, VDI-Verlag GmbH, DüsseldorfGoogle Scholar
  33. Macneal RH (1982) Derivation of element stiffness matrices by assumed strain distributions. Nuclear Engineering and Design 70(1):3–12Google Scholar
  34. Park KC, Stanley GM (1986) A curved C0 shell element based on assumed natural-coordinate strains. Trans ASME J Appl Mech 53(2):278–290Google Scholar
  35. Sze KY, Yao LQ (2000) A hybrid stress ANS solid-shell element and its generalization for smart structure modelling. Part I: Solid-shell element formulation. Int J Numer Meth Eng 48:545–564Google Scholar
  36. Sze KY, Yao LQ, Yi S (2000) A hybrid stress ANS solid-shell element and its generalization for smart structure modelling. Part II: Smart structure modelling. Int J Numer Meth Eng 48:565–582Google Scholar
  37. Zheng S, Wang X, Chen W (2004) The formulation of a refined hybrid enhanced assumed strain solid shell element and its application to model smart structures containing distributed piezoelectric sensors/actuators. Smart Mater Struct 13:N43–N50Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • Gennady M. Kulikov
    • 1
    Email author
  • Svetlana V. Plotnikova
    • 1
  • Erasmo Carrera
    • 2
  1. 1.Laboratory of Intelligent Materials and StructuresTambov State Technical UniversityTambovRussia
  2. 2.Department of Mechanical and Aerospace EngineeringPolitecnico di TorinoTurinItaly

Personalised recommendations