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A coupled refined high-order global-local theory and finite element model for static electromechanical response of smart multilayered/sandwich beams

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Abstract

In the present study, a coupled refined high-order global-local theory is developed for predicting fully coupled behavior of smart multilayered/sandwich beams under electromechanical conditions. The proposed theory considers effects of transverse normal stress and transverse flexibility which is important for beams including soft cores or beams with drastic material properties changes through depth. Effects of induced transverse normal strains through the piezoelectric layers are also included in this study. In the presence of non-zero in-plane electric field component, all the kinematic and stress continuity conditions are satisfied at layer interfaces. In addition, for the first time, conditions of non-zero shear and normal tractions are satisfied even while the bottom or the top layer of the beam is piezoelectric. A combination of polynomial and exponential expressions with a layerwise term containing first order differentiation of electrical unknowns is used to introduce the in-plane displacement field. Also, the transverse displacement field is formulated utilizing a combination of continuous piecewise fourth-order polynomial with a layerwise representation of electrical unknowns. Finally, a quadratic electric potential is used across the thickness of each piezoelectric layer. It is worthy to note that in the proposed shear locking-free finite element formulation, the number of mechanical unknowns is independent of the number of layers. Excellent correlation has been found between the results obtained from the proposed formulation for thin and thick piezoelectric beams with those resulted from the three-dimensional theory of piezoelasticity. Moreover, the proposed finite element model is computationally economic.

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References

  1. Chopra I.: Review of state of art of smart structures and integrated systems. AIAA J. 40(11), 2145–2187 (2002)

    Article  Google Scholar 

  2. Gaudenzi P.: Smart Structures: Physical Behavior, Mathematical Modeling and Applications. Wiley, London (2009)

    Google Scholar 

  3. Crawley E.F., de Luis J.: Use of piezoelectric actuators as element of intelligent structures. AIAA J. 25, 1373–1385 (1987)

    Article  Google Scholar 

  4. Tzou H.S., Gadre M.: Theoretical analysis of a multi-layered thin shell coupled with piezoelectric shell actuators for distributed vibration controls. J. Sound Vib. 132, 433–450 (1989)

    Article  Google Scholar 

  5. Wang B.T., Rogers C.A.: Laminate plate theory for spatially distributed induced strain actuators. J. Compos. Mater. 25(4), 433–452 (1991)

    Google Scholar 

  6. Sung C.K., Chen T.F., Chen S.G.: Piezoelectric modal sensor/actuator design for monitoring /generating flexural and torsional vibrations of cylindrical shells. J. Sound Vib. 118, 48–55 (1996)

    Google Scholar 

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

    Article  Google Scholar 

  8. Saravanos D.A., Heyliger P.R.: Mechanics and computational models for laminated piezoelectric beams, plates, and shells. Appl. Mech. Rev. 52(10), 305–320 (1999)

    Article  Google Scholar 

  9. Brooks S., Heyliger P.: Static behavior of piezoelectric laminates with distributed and patched actuators. J. Intell. Mater. Syst. Struct. 5, 635–646 (1994)

    Article  Google Scholar 

  10. Ray M.C., Rao K.M., Samanta B.: Exact analysis of coupled electroelastic behavior of a piezoelectric plate under cylindrical bending. Comput. Struct. 45(4), 667–677 (1992)

    Article  MATH  Google Scholar 

  11. Ray M.C.H., Rao K.M., Samanta B.: Exact solution for static analysis of an intelligent structure under cylindrical bending. Comput. Struct. 47(6), 1031–1042 (1993)

    Article  MATH  Google Scholar 

  12. Allik H., Hughes T.J.R.: Finite element method for piezoelectric vibration. Int. J. Numer. Methods Eng. 2, 151–157 (1970)

    Article  Google Scholar 

  13. Tzou H.S., Tseng C.I.: Distributed piezoelectric sensor/actuator design for dynamic measurement /control of distributed parameter systems: a piezoelectric finite element approach. J. Sound Vib. 138(1), 17–34 (1990)

    Article  Google Scholar 

  14. Xu K.M., Noor A.K., Tang Y.: Three-dimensional solutions for coupled thermo-electro-elastic response of multi-layered plates. Comput. Methods Appl. Mech. Eng. 126, 355–371 (1995)

    Article  Google Scholar 

  15. Reddy J.N.: An evaluation of equivalent-single-layer and layerwise theories of composite laminates. Compos. Struct. 25(1–4), 21–35 (1993)

    Article  Google Scholar 

  16. Hwang W.S., Park H.C.: Finite element modeling of piezoelectric sensors and actuators. AIAA J. 31, 930–937 (1993)

    Article  Google Scholar 

  17. Suleman A., Venkaya V.B.: A simple finite element formulation for a laminated composite plate with piezoelectric layers. J. Intell. Mater. Syst. Struct. 6, 776–782 (1995)

    Article  Google Scholar 

  18. Sheikh A.H., Topdar P., Halder S.: An appropriate FE model for through thickness variation of displacement and potential in thin/moderately thick smart laminates. Compos. Struct. 51, 401–409 (2001)

    Article  Google Scholar 

  19. Yang J.S.: Equations for the flexural motion of elastic plates with partially electroded piezoelectric actuators. Smart Mater. Struct. 6, 485–490 (1997)

    Article  Google Scholar 

  20. Yang J.S.: Equations for thick elastic plates with partially electroded piezoelectric actuators and higher order electric fields. Smart Mater. Struct. 8, 73–82 (1999)

    Article  Google Scholar 

  21. Chee C.Y.K., Tong L., Steven P.G.: A mixed model for composite beams with piezoelectric actuators and sensors. Smart Mater. Struct. 8, 417–432 (1999)

    Article  Google Scholar 

  22. Jiang J.P., Li D.X.: A new finite element model for piezothermoelastic composite beam. J. Sound Vib. 306, 849–864 (2007)

    Article  Google Scholar 

  23. Shu X.: Free-vibration of laminated piezoelectric composite plates based on an accurate theory. Compos. Struct. 67, 375–382 (2005)

    Article  Google Scholar 

  24. Thornburgh R.P., Chattopadhyay A.: Simultaneous modeling of mechanical and electrical response of smart composite structures. AIAA J. 40(8), 1603–1610 (2002)

    Article  Google Scholar 

  25. Fukunaga H., Hu N., Ren G.X.: Finite element modeling of adaptive composite structures using a reduced higher-order plate theory via penalty functions. Int. J. Solids Struct. 38, 8735–8752 (2001)

    Article  MATH  Google Scholar 

  26. Mitchell J.A., Reddy J.N.: A refined plate theory for composite laminates with piezoelectric laminate. Int. J. Solids Struct. 32(16), 2345–2367 (1995)

    Article  MATH  Google Scholar 

  27. Heyliger P.R., Saravanos D.A.: Coupled discrete-layer finite elements for laminated piezoelectric plates. Commun. Numer. Methods Eng. 10(12), 971–981 (1994)

    Article  MATH  Google Scholar 

  28. Saravanos D.A., Heyliger P.R.: Coupled layer-wise analysis of composite beams with embedded piezoelectric sensors and actuators. J. Intell Mater. Syst. Struct. 6, 350–363 (1995)

    Article  Google Scholar 

  29. Saravanos D.A., Heyliger P.R., Hopkins D.A.: Layer-wise mechanics and finite element model for the dynamic analysis of piezoelectric composite plates. Int. J. Solids Struct. 34(3), 359–378 (1997)

    Article  MATH  Google Scholar 

  30. Kusculuoglu Z.K., Fallahi B., Royston T.Y.: Finite element model of a beam with a piezoelectric patch actuator. J. Sound Vib. 276, 27–44 (2004)

    Article  Google Scholar 

  31. Garcia Lage R., Mota Soares C.M., Mota Soares C.A., Reddy J.N.: Analysis of adaptive plate structures by mixed layerwise finite elements. Compos. Struct. 66, 269–276 (2004)

    Article  Google Scholar 

  32. Garcia Lage R., Mota Soares C.M., Mota Soares C.A., Reddy J.N.: Modeling of piezolaminated plates using layer-wise mixed finite element models. Comput. Struct. 82, 1849–1863 (2004)

    Article  Google Scholar 

  33. Robaldo A., Carrera E., Benjeddou A.: A unified formulation for finite element analysis of piezoelectric adaptive plates. Comput. Struct. 84, 1494–1505 (2006)

    Article  Google Scholar 

  34. Tzou H.S., Ye R.: Analysis of piezoelastic structures with laminated piezoelectric triangle shell elements. AIAA J. 34, 110–115 (1996)

    Article  MATH  Google Scholar 

  35. Ambartsumyan, S.A.: Theory of anisotropic plates. In: Ashton, J.E. (Ed.), Technomic, Stamford. Translated from Russian by Cheron T (1969)

  36. Whitney J.M.: The effects of transverse shear deformation on the bending of laminated plates. J. Compos. Mater. 3, 534–547 (1969)

    Article  Google Scholar 

  37. Icardi U.: Higher-order zig-zag model for analysis of thick composite beams with inclusion of transverse normal stress and sublaminates approximations. Compos. Part B 32, 343–354 (2001)

    Article  Google Scholar 

  38. Icardi U.: A three-dimensional zig-zag theory for analysis of thick laminated beams. Compos. Struct. 52, 123–135 (2001)

    Article  Google Scholar 

  39. Reissner E.: On a mixed variational theorem and on a shear deformable plate theory. Int. J. Numer. Method Eng. 23, 193–198 (1986)

    Article  MATH  Google Scholar 

  40. Murakami H.: A laminated beam theory with interlayer slip. J. Appl. Mech. 51, 551–559 (1984)

    Article  MATH  Google Scholar 

  41. Murakami H.: Laminated composite plate theory with improved in-plane responses. J. Appl. Mech. 53, 661–666 (1986)

    Article  MATH  Google Scholar 

  42. Carrera E.: A study of transverse normal stress effects on vibration of multilayered plates and shells. J. Sound Vib. 225, 803–829 (1999)

    Article  Google Scholar 

  43. Carrera E.: Single-layer vs multi-layers plate modeling on the basis of Reissner’s mixed theorem. AIAA J. 38, 342–343 (2000)

    Article  Google Scholar 

  44. Carrera E.: Historical review of zig-zag theories for multilayered plates and shells. Appl. Mech. Rev. 56, 287–308 (2003)

    Article  Google Scholar 

  45. Oh J., Cho M.: A finite element based on cubic zig-zag plate theory for the prediction of thermo-electric-mechanical behaviors. Int. J. Solids Struct. 41(5–6), 1357–1375 (2004)

    Article  MATH  Google Scholar 

  46. Kapuria S.: An efficient coupled theory for multilayered beams with embedded piezoelectric sensory and active layers. Int. J. Solids Struct. 38, 9179–9199 (2001)

    Article  MATH  Google Scholar 

  47. Kapuria S., Dumir P.C., Ahmed A.: An efficient coupled layerwise theory for dynamic analysis of piezoelectric composite beams. J. Sound Vib. 261, 927–944 (2003)

    Article  Google Scholar 

  48. Kapuria S., Alam N.: Efficient layerwise finite element model for dynamic analysis of laminated piezoelectric beams. Comput. Method Appl. Mech. Eng. 195, 2742–2760 (2006)

    Article  MATH  Google Scholar 

  49. Polit O., Touratier M.: High-order triangular sandwich plate finite element for linear and non-linear analyses. Comput. Method Appl. Mech. Eng. 185(2–4), 305–324 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  50. Dau F., Polit O., Touratier M.: An efficient C1 finite element with continuity requirements for multilayered/sandwich shell structures. Comput. Struct. 82(23–26), 1889–1899 (2004)

    Article  Google Scholar 

  51. Ossadzow-David C., Touratier M.: A multilayered piezoelectric shell theory. Compos. Sci. Technol. 64, 2121–2137 (2004)

    Article  Google Scholar 

  52. Fernandes A., Pouget J.: Analytical and numerical approaches to piezoelectric bimorph. Int. J. Solids Struct. 40, 4331–4352 (2003)

    Article  MATH  Google Scholar 

  53. Vidal P., Polit O.: A family of sinus finite elements for the analysis of rectangular laminated beams. Compos. Struct. 84, 56–72 (2008)

    Article  Google Scholar 

  54. Beheshti-Aval S.B., Lezgy-Nazargah M.: A finite element model for composite beams with piezoelectric layers using a sinus model. J. Mech. 26(2), 249–258 (2010)

    Article  Google Scholar 

  55. Beheshti-Aval S.B., Lezgy-Nazargah M.: Assessment of velocity-acceleration feedback in optimal control of smart piezoelectric beams. Smart Struct. Syst. 6(8), 921–938 (2010)

    Google Scholar 

  56. Beheshti-Aval S.B., Lezgy-Nazargah M., Vidal P., Polit O.: A refined sinus finite element model for the analysis of piezoelectric laminated beams. J. Intell. Mater. Syst. Struct. 22, 203–219 (2011)

    Article  Google Scholar 

  57. D’Ottavio M., Kröplin B.: An extension of Reissner mixed variational theorem to piezoelectric laminates. Mech. Adv. Mater. Struct. 13(2), 139–150 (2006)

    Article  Google Scholar 

  58. Carrera E., Nali P.: Mixed piezoelectric plate elements with direct evaluation of transverse electric displacement. Int. J. Numer. Methods Eng. 80, 403–424 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  59. Li X., Liu D.: Generalized laminate theories based on double superposition hypothesis. Int. J. Numer. Methods Eng. 40(7), 1197–1212 (1997)

    Article  MATH  Google Scholar 

  60. Shariyat M.: A generalized global-local high-order theory for bending and vibration analyses of sandwich plates subjected to thermo-mechanical loads. Int. J. Mech. Sci. 52, 495–514 (2010)

    Article  Google Scholar 

  61. Lezgy-Nazargah M., Beheshti-Aval S.B., Shariyat M.: A refined mixed global-local finite element model for bending analysis of multi-layered rectangular composite beams with small widths. Thin Walled Struct. 49, 351–362 (2011)

    Article  Google Scholar 

  62. Lezgy-Nazargah M., Shariyat M., Beheshti-Aval S.B.: A refined high-order global-local theory for finite element bending and vibration analyses of the laminated composite beams. Acta Mech. 217(3–4), 219–242 (2011)

    Article  MATH  Google Scholar 

  63. Zhen W., Wanji C.: Refined triangular element for laminated elastic-piezoelectric plates. Compos. Struct. 78, 129–139 (2007)

    Article  Google Scholar 

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  1. Civil Engineering Faculty, K.N. Toosi University of Technology, Tehran, Iran

    S. B. Beheshti-Aval & M. Lezgy-Nazargah

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  1. S. B. Beheshti-Aval
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  2. M. Lezgy-Nazargah
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Correspondence to S. B. Beheshti-Aval.

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Beheshti-Aval, S.B., Lezgy-Nazargah, M. A coupled refined high-order global-local theory and finite element model for static electromechanical response of smart multilayered/sandwich beams. Arch Appl Mech 82, 1709–1752 (2012). https://doi.org/10.1007/s00419-012-0621-9

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