Skip to main content
SpringerLink
Log in

Variation principle of piezothermoelastic bodies, canonical equation and homogeneous equation

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

Combining the symplectic variations theory, the homogeneous control equation and isoparametric element homogeneous formulations for piezothermoelastic hybrid laminates problems were deduced. Firstly, based on the generalized Hamilton variation principle, the non-homogeneous Hamilton canonical equation for piezothermoelastic bodies was derived. Then the symplectic relationship of variations in the thermal equilibrium formulations and gradient equations was considered, and the non-homogeneous canonical equation was transformed to homogeneous control equation for solving independently the coupling problem of piezothermoelastic bodies by the incensement of dimensions of the canonical equation. For the convenience of deriving Hamilton isoparametric element formulations with four nodes, one can consider the temperature gradient equation as constitutive relation and reconstruct new variation principle. The homogeneous equation simplifies greatly the solution programs which are often performed to solve nonhomogeneous equation and second order differential equation on the thermal equilibrium and gradient relationship.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Pagano N J. Exact solutions for rectangular bidirectional composites and sandwich plates[J]. Journal of Composite Materials, 1970, 4(1): 20–34.

    Google Scholar 

  2. Ray M C, Rao K M, Samanta B. Exact solution for static analysis of an intelligent structure under cylindrical bending[J]. Computers and Structures, 1993, 47(6): 1031–1042.

    Article  MATH  Google Scholar 

  3. Batra R C, Liang X Q. Vibration of a rectangular laminated elastic plate with embedded piezoelectric sensors and actuators[J]. Computer and Structures, 1997, 63(2): 203–216.

    Article  MATH  Google Scholar 

  4. Ray M C, Bhattacharya R, Samanta B. Exact solutions for dynamic analysis of composite plate with distributed piezoelectric layers[J]. Compute and Structures, 1998, 66(6): 737–743.

    Article  MATH  Google Scholar 

  5. Vel S S, Batra R C. Generalized plane strain thermopiezoelectric analysis of multilayered plates[J]. Journal of Thermal Stresses, 2003, 26(4): 353–377.

    Google Scholar 

  6. Heyliger P R, Brooks P. Exact solution for laminated piezoelectric plates in cylindrical bending[J]. Journal of Applied Mechanics, 1996, 63(6): 903–910.

    MATH  Google Scholar 

  7. Xu K, Noor A K, Tang Y Y. Three-dimensional solutions for coupled thermoelectroelastic response of multilayered plates[J]. Computer Methods in Applied Mechanics and Engineering, 1995, 126(3/4): 355–371.

    Article  Google Scholar 

  8. Ootao Y, Tanigawa Y. Control of transient thermoelastic displacement of a two-layered composite plate constructed of isotropic elastic and piezoelectric layers due to nonuniform heating[J]. Archive of Applied Mechanics, 2001, 61(4/5): 207–230.

    Google Scholar 

  9. Tarn Jiann-Quo. A state space formalism for piezothermoelasticity[J]. International Journal of Solids and Structures, 2002, 39(20): 5173–5184.

    Article  MATH  MathSciNet  Google Scholar 

  10. Kapuria S, Dumir P C. Sengupta S. Three-dimensional solution for shape control of a simply supported rectangular hybrid plate[J]. Journal of Thermal Stresses, 2003, 22(2): 159–176.

    Article  Google Scholar 

  11. Zhang C, Cheung Y K, Di S, Zhang N. The exact solution of coupled thermoelectroelastic behavior of piezoelectric laminates[J]. Computers and Structures, 2002, 80(13): 1201–1212.

    Article  Google Scholar 

  12. Ding H J, Wang H M, Ling D S. Analytical solution of a pyroelectric hollow cylinder for piezothermoelastic axisymmetric dynamic problems[J]. Journal of Thermal Stresses, 2003, 26(3): 261–276.

    Google Scholar 

  13. Zhong Wanxie. A new systematic for theory of elasticity[M]. Dalian: Dalian University of Technology Press, 1995 (in Chinese).

    Google Scholar 

  14. Zhong Wanxie. Symplectic system of applied mechanicas[M]. Beijing: Science Press, 2003 (in Chinese).

    Google Scholar 

  15. Qing Guanghui, Qiu Jiajun, Liu Yanhong. Modified H-R variational principle for magnetoelectroelastic bodies and state-vector equation[J]. Applied Mathematics and Mechanics (English Edition), 2005, 26(6): 722–728.

    Google Scholar 

  16. Gu Yuanxian, Chen Biaosong, Zhang Hongwu. Precise time-integration with dimension expanding method[J]. Acta Mechanica Sinca, 2000, 24(4): 447–456 (in Chinese).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. State Key Laboratory of Engines, Tianjin University, Tianjin, 300072, P. R. China

    Liu Yan-hong Doctor  (刘继红) (Associate Professor) & Zhang Hui-ming  (张惠明)

  2. Aeronautical Mechanics and Avionics Engineering College, Civil Aviation University of China, Tianjin, 300300, P. R. China

    Liu Yan-hong Doctor  (刘继红) (Associate Professor)

Authors
  1. Liu Yan-hong Doctor  (刘继红)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhang Hui-ming  (张惠明)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Liu Yan-hong Doctor  (刘继红).

Additional information

Communicated by ZHONG Wan-xie

Project supported by the National Natural Science Foundation of China (No. 50276041)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, Yh., Zhang, Hm. Variation principle of piezothermoelastic bodies, canonical equation and homogeneous equation. Appl Math Mech 28, 193–200 (2007). https://doi.org/10.1007/s10483-007-0207-y

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-007-0207-y

Key words

Chinese Library Classification

2000 Mathematics Subject Classification

Search

Navigation