Structural Engineering and Mechanics

  • 제69권6호
  • /
  • Pages.693-698
  • /
  • 2019
  • /
  • 1225-4568(pISSN)
  • /
  • 1598-6217(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Closed-form solution of axisymmetric deformation of prestressed Föppl-Hencky membrane under constrained deflecting

  • Lian, Yong-Sheng (School of Civil Engineering, Chongqing University) ;
  • Sun, Jun-Yi (School of Civil Engineering, Chongqing University) ;
  • Dong, Jiao (School of Civil Engineering, Chongqing University) ;
  • Zheng, Zhou-Lian (School of Civil Engineering, Chongqing University) ;
  • Yang, Zhi-Xin (School of Civil Engineering, Chongqing University)
  • 투고 : 2018.05.03
  • 심사 : 2019.02.05
  • 발행 : 2019.03.25
⟨ 이전 논문 다음 논문 ⟩

초록

In this study, the problem of axisymmetric deformation of prestressed $F{\ddot{o}}ppl-Hencky$ membrane under constrained deflecting was analytically solved and its closed-form solution was presented. The small-rotation-angle assumption usually adopted in membrane problems was given up, and the initial stress in membrane was taken into account. Consequently, this closed-form solution has higher calculation accuracy and can be applied for a wider range in comparison with the existing approximate solution. The presented numerical examples demonstrate the validity of the closed-form solution, and show the errors of the contact radius, profile and radial stress of membrane in the existing approximate solution brought by the small-rotation-angle assumption. Moreover, the influence of the initial stress on the contact radius is also discussed based on the numerical examples.

키워드

과제정보

연구 과제 주관 기관 : National Natural Science Foundation of China

참고문헌

  1. Alekseev, S.A. (1953), "Elastic circular membranes under the uniformly distributed loads", Eng. Corp., 14, 196-198.
  2. Arthurs, A.M. and Clegg, J. (1994), "On the solution of a boundary value problem for the nonlinear Foppl-Hencky equation", Z. Angew. Math. Mech., 74(7), 281-284. https://doi.org/10.1002/zamm.19940740713
  3. Beck, A. and Grabmuller, H. (1992), "Wrinkle-free solutions of circular membrane problems", Z. Angew. Math. Phys., 43(3), 481-504. https://doi.org/10.1007/BF00946242
  4. Chien, W.Z. (1948), "Asymptotic behavior of a thin clamped circular plate under uniform normal pressure at very large deflection", Sci. Rep. Nat. Tsinghua Univ., 5(1), 193-208.
  5. Chucheepsakul, S., Kaewunruen, S. and Suwanarat, A. (2009), "Large deflection analysis of orthotropic, elliptic membranes", Struct. Eng. Mech., 31(6), 625-638. https://doi.org/10.12989/sem.2009.31.6.625
  6. Ersoy, H., Ozpolat, L. and Civalek, O. (2009), "Free vibration of circular and annular membranes with varying density by the method of discrete singular convolution", Struct. Eng. Mech., 32(5), 621-634. https://doi.org/10.12989/sem.2009.32.5.621
  7. Feng, L.Y., Li, X.W. and Shi, T.L. (2015), "Nonlinear large deflection of thin film overhung on compliant substrate using shaft-loaded blister test", J. Appl. Mech., 82(9), 091001. https://doi.org/10.1115/1.4030739
  8. Hasheminejad, S.M. and Ghaheri, A. (2015), "Exact solution for vibration analysis of an eccentric elliptical membrane", Z. Angew. Math. Mech., 95(7), 730-741. https://doi.org/10.1002/zamm.201300118
  9. Hencky, H. (1915), "Uber den spannungszustand in kreisrunden platten mit verschwindender biegungssteifigkeit", Zeitschrift Fur Mathematik und Physik, 63, 311-317.
  10. Lian, Y.S., He, X.T., Liu, G.H., Sun, J.Y. and Zheng, Z.L. (2017a), "Application of perturbation idea to well-known Hencky problem: A perturbation solution without smallrotation-angle assumption", Mech. Res. Commun., 83(7), 32-46. https://doi.org/10.1016/j.mechrescom.2017.05.001
  11. Lian, Y.S., Sun, J.Y., Ge, X.M., Yang, Z.X., He, X.T. and Zheng, Z.L. (2017b), "A theoretical study of an improved capacitive pressure sensor: Closed-form solution of uniformly loaded annular membranes", Measurement, 111, 84-92. https://doi.org/10.1016/j.measurement.2017.07.025
  12. Lian, Y.S., Sun, J.Y., Yang, Z.X., He, X.T. and Zheng, Z.L. (2016), "Closed-form solution of well-known Hencky problem without small-rotation-angle assumption", Z. Angew. Math. Mech., 96(12), 1434-1441. https://doi.org/10.1002/zamm.201600059
  13. Molla-Alipour, M. and Ganji, B.A. (2015), "Analytical analysis of mems capacitive pressure sensor with circular diaphragm under dynamic load using differential transformation method (DTM)", Acta Mech. Sol. Sin., 28(4), 400-408. https://doi.org/10.1016/S0894-9166(15)30025-2
  14. Sun, J.Y., Hu, J.L., He, X.T. and Zheng, Z.L. (2010), "A theoretical study of a clamped punch-loaded blister configuration: The quantitative relation of load and deflection", Int. J. Mech. Sci., 52(7), 928-936. https://doi.org/10.1016/j.ijmecsci.2010.03.009
  15. Sun, J.Y., Hu, J.L., Zheng, Z.L., He, X.T. and Geng, H.H. (2011), "A practical method for simultaneous determination of Poisson's ratio and Young's modulus of elasticity of thin films", J. Mech. Sci. Technol., 25(12), 3165-3171. https://doi.org/10.1007/s12206-011-1002-y
  16. Sun, J.Y., Lian, Y.S., Li, Y.M., He, X.T. and Zheng, Z.L. (2015), "Closed-form solution of elastic circular membrane with initial stress under uniformly-distributed loads: Extended Hencky solution", Z. Angew. Math. Mech., 95(11), 1335-1341. https://doi.org/10.1002/zamm.201400032
  17. Sun, J.Y., Lian, Y.S., Li, Z.L., He, X.T. and Zheng, Z.L. (2014), "Theoretical study on shaft-loaded blister test technique: Synchronous characterization of surface and interfacial mechanical properties", Int. J. Adhes. Adhes., 51, 128-139. https://doi.org/10.1016/j.ijadhadh.2014.03.004
  18. Sun, J.Y., Rong, Y., He, X.T., Gao, X.W. and Zheng, Z.L. (2013), "Power series solution of circular membrane under uniformly distributed loads: Investigation into Hencky transformation", Struct. Eng. Mech., 45(5), 631-641. https://doi.org/10.12989/sem.2013.45.5.631
  19. Todorovic, D.M., Rabasovic, M.D., Markushev, D.D. and Sarajlic, M. (2014), "Photoacoustic elastic bending in thin film-substrate system: Experimental determination of the thin film parameters", J. Appl. Phys., 116(5), 273-318. https://doi.org/10.1007/s00339-013-8118-0
  20. Wang, T.F., He, X.T. and Li, Y.H. (2018), "Closed-form solution of a peripherally fixed circular membrane under uniformlydistributed transverse loads and deflection restrictions", Math. Probl. Eng., 5989010.
  21. Weinitschke, H.J. (1973), "Endliche deformationen elastischer membranen", Z. Angew. Math. Mech., 53(12), T89-T90.
  22. Yang, Z.X., Sun, J.Y., Li, K., Lian, Y.S., He, X.T. and Zheng, Z.L. (2018), "Theoretical study on synchronous characterization of surface and interfacial mechanical properties of thin-filmsubstrate systems with residual stress based on pressure blister test technique", Polymers, 10(1), 49. https://doi.org/10.3390/polym10010049
  23. Yang, Z.X., Sun, J.Y., Ran, G.M. and He, X.T. (2017), "A new solution to Foppl-Hencky membrane equation", J. Mech., 33(4), N7-N11. https://doi.org/10.1017/jmech.2016.119