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

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Analysis and Modelling of Advanced Structures and Smart Systems

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 81))

Abstract

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.

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References

  • Bakhvalov NS (1977) Numerical Methods: Analysis, Algebra, Ordinary Differential Equations. MIR, Moscow

    Google Scholar 

  • 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):697–722

    Google Scholar 

  • Bathe KJ, Lee PS, Hiller JF (2003) Towards improving the MITC9 shell element. Comput Struct 81(8):477–489

    Google Scholar 

  • 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):899–909

    Google Scholar 

  • Carrera E (1999) Multilayered shell theories accounting for layerwise mixed description. Part 1: Governing equations. AIAA J 37(9):1107–1116

    Google Scholar 

  • 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):215–296

    Google Scholar 

  • 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, http://dx.doi.org/10.1177/1045389X17704913

  • 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, Chichester

    Google Scholar 

  • Carrera E, Cinefra M, Petrolo M, Zappino E (2014) Finite Element Analysis of Structures Through Unified Formulation. John Wiley & Sons Ltd, Chichester

    Google Scholar 

  • 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):4921–4936

    Google Scholar 

  • Cinefra M, Carrera E, Valvano S (2015) Variable kinematic shell elements for the analysis of electro-mechanical problems. Mech Adv Mater Struct 22:77–106

    Google Scholar 

  • Dunn ML, Taya M (1994) Electroelastic field concentrations in and around inhomogeneities in piezoelectric solids. Trans ASME J Appl Mech 61(2):474–475

    Google Scholar 

  • Heyliger P (1997) A note on the static behavior of simply supported laminated piezoelectric cylinders. Int J Solids Struct 34(29):3781–3794

    Google Scholar 

  • 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):587–596

    Google Scholar 

  • 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):349–382

    Google Scholar 

  • 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):38–46

    Google Scholar 

  • Ko Y, Lee PS, Bathe KJ (2017) A new MITC4+ shell element. Comput Struct 182:404–418

    Google Scholar 

  • Kulikov GM (2001) Refined global approximation theory of multilayered plates and shells. J Eng Mech 127(2):119–125

    Google Scholar 

  • Kulikov GM, Carrera E (2008) Finite deformation higher-order shell models and rigid-body motions. Int J Solids Struct 45(11):3153–3172

    Google Scholar 

  • Kulikov GM, Plotnikova SV (2008) Geometrically exact four-node piezoelectric solid-shell element. Mech Adv Mater Struct 15(3-4):199–207

    Google Scholar 

  • Kulikov GM, Plotnikova SV (2010) Solution of a coupled problem of thermopiezoelectricity based on a geometrically exact shell element. Mech Comp Mater 46(4):349–364

    Google Scholar 

  • Kulikov GM, Plotnikova SV (2011a) Exact geometry piezoelectric solid-shell element based on the 7-parameter model. Mech Adv Mater Struct 18(2):133–146

    Google Scholar 

  • Kulikov GM, Plotnikova SV (2011b) Finite rotation piezoelectric exact geometry solid-shell element with nine degrees of freedom per node. Comput Mater Continua 23:233–264

    Google Scholar 

  • 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):1363–1389

    Google Scholar 

  • 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 715–726

    Google Scholar 

  • 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):1930–1943

    Google Scholar 

  • Kulikov GM, Plotnikova SV (2014) Exact electroelastic analysis of functionally graded piezoelectric shells. Int J Solids Struct 51(1):13–25

    Google Scholar 

  • 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):490–502

    Google Scholar 

  • 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):392–409

    Google Scholar 

  • Kulikov GM, Mamontov AA, Plotnikova SV (2015) Coupled thermoelectroelastic stress analysis of piezoelectric shells. Compos Struct 124:65–76

    Google Scholar 

  • 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 12:355–362

    Google Scholar 

  • Lentzen S (2009) Nonlinearly coupled thermopiezoelectric modelling and FE-simulation of smart structures. No. 419 in Fortschritt-Berichte VDI, Reihe 20, VDI-Verlag GmbH, Düsseldorf

    Google Scholar 

  • Macneal RH (1982) Derivation of element stiffness matrices by assumed strain distributions. Nuclear Engineering and Design 70(1):3–12

    Google Scholar 

  • Park KC, Stanley GM (1986) A curved C0 shell element based on assumed natural-coordinate strains. Trans ASME J Appl Mech 53(2):278–290

    Google Scholar 

  • 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 48:545–564

    Google Scholar 

  • 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 48:565–582

    Google Scholar 

  • 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–N50

    Google Scholar 

Download references

Acknowledgements

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).

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Authors and Affiliations

  1. Laboratory of Intelligent Materials and Structures, Tambov State Technical University, Sovetskaya Street, 106, Tambov, 392000, Russia

    Gennady M. Kulikov & Svetlana V. Plotnikova

  2. Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Corso Duca degli Abbruzzi 24, 10129, Turin, Italy

    Erasmo Carrera

Authors
  1. Gennady M. Kulikov
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  2. Svetlana V. Plotnikova
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  3. Erasmo Carrera
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Corresponding author

Correspondence to Gennady M. Kulikov .

Editor information

Editors and Affiliations

  1. Faculty of Mechanical Engineering, Otto-von-Guericke-University Magdeburg , Magdeburg, Germany

    Holm Altenbach

  2. Mechanical and Aerospace Engineering, Politecnico di Torino, Torino, Italy

    Erasmo Carrera

  3. Laboratory of Intelligent Mat. & Struct., Tambov State Technical University , Tambov, Russia

    Gennady Kulikov

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Kulikov, G.M., Plotnikova, S.V., Carrera, E. (2018). Hybrid-Mixed Solid-Shell Element for Stress Analysis of Laminated Piezoelectric Shells through Higher-Order Theories. In: Altenbach, H., Carrera, E., Kulikov, G. (eds) Analysis and Modelling of Advanced Structures and Smart Systems. Advanced Structured Materials, vol 81. Springer, Singapore. https://doi.org/10.1007/978-981-10-6895-9_4

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  • DOI: https://doi.org/10.1007/978-981-10-6895-9_4

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