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An efficient finite element with layerwise mechanics for smart piezoelectric composite and sandwich shallow shells

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Abstract

In this work, we present a new efficient four-node finite element for shallow multilayered piezoelectric shells, considering layerwise mechanics and electromechanical coupling. The laminate mechanics is based on the zigzag theory that has only seven kinematic degrees of freedom per node. The normal deformation of the piezoelectric layers under the electric field is accounted for without introducing any additional deflection variables. A consistent quadratic variation of the electric potential across the piezoelectric layers with the provision of satisfying the equipotential condition of electroded surfaces is adopted. The performance of the new element is demonstrated for the static response under mechanical and electric potential loads, and for free vibration response of smart shells under different boundary conditions. The predictions are found to be very close to the three dimensional piezoelasticity solutions for hybrid shells made of not only single-material composite substrates, but also sandwich substrates with a soft core for which the equivalent single layer (ESL) theories perform very badly.

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References

  1. Achary GGS (2005) Coupled exact 3D solutions and 2D zigzag theory for thermo-electro-mechanical analysis of hybrid piezoelectric plates. PhD Thesis, Indian Institute of Technology Delhi, India

  2. Allik H, Hughes TJR (1970) Finite element method for piezoelectric vibration. Int J Num Methods Eng 2:151–157

    Article  Google Scholar 

  3. Balamurugan V, Narayanan S (2001a) Active vibration control of smart shells using distributed piezoelectric sensors and actuators. Smart Mater Struct 10:173–180

    Article  Google Scholar 

  4. Balamurugan V, Narayanan S (2001b) Shell finite element for smart piezoelectric composite plate/shell structures and its applications to the study of active vibration control. Finite Elem Anal Des 37:713–738

    Article  MATH  Google Scholar 

  5. Balamurugan V, Narayanan S (2008) A piezolaminated composite degenerated shell finite element for active control of structures with distributed piezosensors and actuators. Smart Mater Struct 17: article no. 035031

    Google Scholar 

  6. Balamurugan V, Narayanan S (2009) Multilayer higher order piezolaminated smart composite shell finite element and its application to active vibration control. J Intell Mater Syst Struct 20:425–441

    Article  Google Scholar 

  7. Benjeddou A (2000) Advances in piezoelectric finite element modeling of adaptive structural elements: a survey. Comput Struct 76:347–363

    Article  Google Scholar 

  8. Bhattacharya P, Suhail H, Sinha PK (2002) Finite element analysis and distributed control of laminated composite shells using LQR/IMSC approach. Aerosp Sci Technol 6:273–281

    Article  MATH  Google Scholar 

  9. Correia IFP, Soares CMM, Soares CAM, Herskovits J (2002) Active control of axisymmetric shells with piezoelectric layers: a mixed laminated theory with a high order displacement field. Comput Struct 80:2256–2275

    Google Scholar 

  10. Daneshmehr AR, Shakeri M (2007) Three-dimensional elasticity solution of cross-ply shallow and non-shallow panels with piezoelectric sensors under dynamic load. Compos Struct 80:429–439

    Article  Google Scholar 

  11. Gabbert U, Köoppe H, Seeger F, Berger H (2002) Modeling of smart composite shell structures. J Theor Appl Mech 40: 575–593

    Google Scholar 

  12. Heyliger P, Pei KC (1996) Layerwise mechanics and finite element model for laminated piezoelectric shells. AIAA J 34(11):2353–2360

    Article  MATH  Google Scholar 

  13. Jeyachandrabose C, Kirkhope J, Meekisho L (1987) An improved discrete Kirchhoff quadrilateral thin-plate bending element. Int J Num Methods Eng 24:635–654

    Article  MATH  Google Scholar 

  14. Kapuria S, Kulkarni SD (2007) An improved discrete Kirchhoff quadrilateral element based on third-order zigzag theory for static analysis of composite and sandwich plates. Int J Num Methods Eng 69:1948–1981

    Article  MATH  Google Scholar 

  15. Kapuria S, Kulkarni SD (2008) An efficient quadrilateral element based on improved zigzag theory for dynamic analysis of hybrid plates with electroded piezoelectric actuators and sensors. J Sound Vib 315:118–145

    Article  Google Scholar 

  16. Kapuria S, Kumari P (2010) Three-dimensional piezoelasticity solution for dynamics of cross-ply cylindrical shells integrated with piezoelectric fiber reinforced composite actuators and sensors. Compos Structures 92:2431–2444

    Article  Google Scholar 

  17. Kapuria S, Dumir PC, Sengupta S (1997) Nonaxisymmetric exact piezothermoelastic solutions for laminated cylindrical shell. AIAA J 35:1792–1795

    Article  MATH  Google Scholar 

  18. Khdeir AA, Aldraihem OJ (2011) Exact analysis for static response of cross ply laminated smart shells. Compos Struct 94:92–101

    Article  Google Scholar 

  19. Kioua H, Mirza S (2000) Piezoelectric induced bending and twisting of laminated composite shallow shells. Smart Mater Struct 9:476–484

    Article  Google Scholar 

  20. Kioua H, Mirza S (2001) Free vibration of laminated composite shallow shells containing piezoelectric layers. Mech Compos Mater Struct 8:181–197

    Article  Google Scholar 

  21. Klinkel S, Wagner W (2008) A piezoelectric solid shell element based on a mixed variational formulation for geometrically linear and nonlinear applications. Comput Struct 86:38–46

    Article  Google Scholar 

  22. Kogl M, Bucalem ML (2005) Analysis of smart laminates using piezoelectric MITC plate and shell elements. Comput Struct 83:1153–1163

    Article  Google Scholar 

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

    Article  Google Scholar 

  24. Kulkarni SA, Bajoria KM (2003) Finite element modeling of smart plates/shells using higher order shear deformation theory. Compos Struct 62:41–50

    Article  Google Scholar 

  25. Kumar R, Mishra BK, Jain SC (2008) Static and dynamic analysis of smart cylinderical shell. Finite Elem Anal Des 45:13–24

    Article  Google Scholar 

  26. Lammering R, Mesecke-Rischmann S (2003) Multi-field variational formulations and related finite elements for piezoelectric shells. Smart Mater Struct 12:904–913

    Article  Google Scholar 

  27. Macneal RH, Harder RL (1985) A proposed standard set of problems to test finite element accuracy. Finite Elem Anal Des 1:3–20

    Article  Google Scholar 

  28. Marinkocic D, Koppe H, Gabbert U (2006) Numerically efficient finite element formulation for modeling active composite laminates. Mech Adv Mater Struct 13:379–392

    Article  Google Scholar 

  29. Narayanan S, Balamurugan V (2003) Finite element modelling of piezolaminated smart structures for active vibration control with distributed sensors and actuators. J Sound Vib 262:529–562

    Article  Google Scholar 

  30. Nath JK, Kapuria S (2009) Improved smeared and zigzag third order theories for piezoelectric angle-ply laminated cylindrical shells under electrothermomechanical loads. J Mech Mater Struct 4:1157–1184

    Google Scholar 

  31. Nath JK, Kapuria S (2012) Coupled efficient layerwise and smeared third order theories for vibration of smart piezolaminated cylindrical shells. Compos Struct 94:1886–1899

    Article  Google Scholar 

  32. Neto MA, Leal RP, Yu W (2012) A triangular finite element with drilling degrees of freedom for static and dynamic analysis of smart laminated structures. Comput Struct 108–109:61–74

    Article  Google Scholar 

  33. Niemi AH (2010) A bilinear shell element based on a refined shallow shell model. Int J Numer Methods Eng 81:485–512

    Google Scholar 

  34. Oh J, Cho M (2007) Higher order zigzag theory for smart composite shells under mechanical-thermo-electric loading. Int J Solids Struct 44:100–127

    Google Scholar 

  35. Petyt M (2010) Introduction to finite element vibration analysis. Cambridge University Press, New York

  36. Qatu MS (2004) Vibration of laminated shells and plates. Elsevier, New York

    Google Scholar 

  37. Raja S, Sinha PK, Prathap G, Dwarakanathan D (2004) Thermally induced vibration control of composite plates and shells with piezoelectric active damping. Smart Mater Struct 13:939–950

    Article  Google Scholar 

  38. Reddy JN (1984) Exact solutions of moderately thick laminated shells. J Eng Mech 110:794–809

    Article  Google Scholar 

  39. Roy T, Chakraborty D (2009) Optimal vibration control of smart fiber reinforced composite shell structures using improved genetic algorithm. J Sound Vib 319:15–40

    Article  Google Scholar 

  40. Saravanos DA (1997) Mixed laminate theory and finite element for smart piezoelectric composite shell structures. AIAA J 9:10–18

    Google Scholar 

  41. Schulz K, Klinkel S, Wagner W (2011) A finite element formulation for piezoelectric shell structures considering geometrical and material non-linearities. Int J Numer Methods Eng 87:491–520

    Article  MATH  MathSciNet  Google Scholar 

  42. 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 Methods Eng 48:545–564

    Article  MATH  Google Scholar 

  43. 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 Methods Eng 48:565–582

    Article  MATH  Google Scholar 

  44. Sze KY, Yang XM, Fan H (2004) Electric assumptions for piezoelectric laminate analysis. Int J Solids Struct 41:2363–2382

    Article  MATH  Google Scholar 

  45. Tan XG, Vu-Quoc L (2005) Optimal solid shell element for large deformable composite structure with piezoelectric layers and active vibration control. Int J Numer Methods Eng 64:1981–2013

    Google Scholar 

  46. Tiersten HF (1969) Linear piezoelectric plate vibrations. Plenum Publishing Corporation, New York

    Book  Google Scholar 

  47. Varelis D, Saravanos DA (2006) Coupled mechanics and finite element for non-linear laminated piezoelectric shallow shells undergoing large displacements and rotations. Int J Numer Methods Eng 66(8):1211–1233

    Article  MATH  Google Scholar 

  48. Vidal P, D’Ottavio M, Thaier MB, Polit O (2011) An efficient finite shell element for the static response of piezoelectric laminates. J Intell Mater Syst Struct 22:671–690

    Article  Google Scholar 

  49. Wu CP, Lo JY (2006) An asymptotic theory for dynamic response of laminated piezoelectric shells. Acta Mech 183:177–208

    Article  MATH  Google Scholar 

  50. Yasin MY, Kapuria S (2013) An efficient layerwise finite element for shallow composite and sandwich shells. Compos Struct 98:202–214

    Google Scholar 

  51. Zemčík R, Rolfes R, Rose M, Tessmer J (2006) High-performance 4-node shell element with piezoelectric coupling. Mech Adv Mater Struct 13:393–401

    Article  Google Scholar 

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Acknowledgments

S Kapuria is thankful to the Department of Science and Technology (DST), Government of India, for supporting this research work through a project grant.

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  1. Department of Applied Mechanics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110016, India

    M. Yaqoob Yasin & S. Kapuria

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  1. M. Yaqoob Yasin
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  2. S. Kapuria
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Correspondence to S. Kapuria.

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Yasin, M.Y., Kapuria, S. An efficient finite element with layerwise mechanics for smart piezoelectric composite and sandwich shallow shells. Comput Mech 53, 101–124 (2014). https://doi.org/10.1007/s00466-013-0896-x

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