SpringerLink
Log in

A wrinkling model for pneumatic membranes and the complementarity computational framework

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

The paper proposes a complementarity computational framework for the wrinkling analysis of pneumatic membranes under follower loads. Geometric and material nonlinearities are separated by using the co-rotational finite element method. A reasonable wrinkling model is proposed based on the constitutive law of bi-modulus materials in the principal stress space. To improve the convergence, a linear complementarity framework is constructed in the local frame and embedded into the global Newton–Raphson iteration. The proposed method requires no extra solution techniques to ensure convergence, compared with other solution strategies, such as a pseudo-dynamic method and a penalty stabilization method. Three benchmark tests are employed to verify the proposed model and method. The numerical results have a good agreement with the existing numerical and experimental data. Importantly, the proposed computational method can even make a more accurate prediction on the displacement response and wrinkling regions than the post-buckling analysis of thin shells in some situation.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Germany)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Similar content being viewed by others

References

  1. Geppert U, Biering B, Lura F, Block J, Straubel M, Reinhard R (2011) The 3-step DLR-ESA Gossamer road to solar sailing. Adv Space Res 48:1695–1701

    Google Scholar 

  2. Rogers JA, Someya T, Huang Y (2010) Materials and mechanics for stretchable electronics. Science 327:1603–1607

    Google Scholar 

  3. Li B, Cao YP, Feng XQ, Gao H (2011) Surface wrinkling of mucosa induced by volumetric growth: theory, simulation and experiment. J Mech Phys Soilds 59:758–774

    MathSciNet  Google Scholar 

  4. Li B, Cao YP, Feng XQ, Gao H (2012) Mechanics of morphological instabilities and surface wrinkling in soft materials: a review. Soft Matter 8:5728–5745

    Google Scholar 

  5. Miyamura T (2000) Wrinkling on stretched circular membrane under in-plane torsion: bifurcation analyses and experiments. Eng Struct 23:1407–1425

    Google Scholar 

  6. Wong YW, Pellegrino S (2006) Wrinkled membranes, part I: experiments. J Mech Mater Struct 1:1–23

    Google Scholar 

  7. Wong YW, Pellegrino S (2006) Wrinkled membranes, part II: analytical models. J Mech Mater Struct 1:25–59

    Google Scholar 

  8. Wong YW, Pellegrino S (2006) Wrinkled membranes, part III: numerical simulations. J Mech Mater Struct 1:63–95

    Google Scholar 

  9. Cerda E, Mahadevan L (2003) Geometry and physics of wrinkling. Phys Rev Lett 90(7):1–4

    Google Scholar 

  10. van der Heijden A (2009) W.T. Koiter’s elastic stability of solids and structures. Cambridge University Press, Cambridge

    Google Scholar 

  11. Steigmann DJ (2012) A well-posed finite-strain model for thin elastic sheets with bending stiffness. Math Mech Solids 18(1):103–112

    MathSciNet  Google Scholar 

  12. Taylor M, Bertoldi K, Steigmann DJ (2014) Spatial resolution of wrinkle patterns in thin elastic sheets at finite strain. J Mech Phys Soilds 62:163–180

    MathSciNet  MATH  Google Scholar 

  13. Patil A, Nordmark A, Eriksson A (2015) Wrinkling of cylindrical membranes with non-uniform thickness. Eur J Mech A Solids 54:1–10

    MathSciNet  MATH  Google Scholar 

  14. Patil A, Nordmark A, Eriksson A (2016) Instabilities of wrinkled membranes with pressure loadings. J Mech Phys Soilds 94:298–315

    MathSciNet  Google Scholar 

  15. Patil A, Nordmark A, Eriksson A (2015) Instability investigation on fluid-loaded pre-stretched cylindrical membranes. Proc R Soc A 471:20150016

    Google Scholar 

  16. Wang CG, Du XW, Tan HF, He XD (2009) A new computational method for wrinkling analysis of gossamer space structures. Int J Solids Struct 46:1516–1526

    MATH  Google Scholar 

  17. Wang CG, Tan HF (2010) Experimental and numerical studies on wrinkling control of an inflated beam using SMA wires. Smart Mater Struct 19(10):105019

    Google Scholar 

  18. Ji QX, Wang CG, Tan HF (2017) Multi-scale wrinkling analysis of the inflated beam under bending. Int J Mech Sci 126:1–11

    Google Scholar 

  19. Taylor M, Davidovitch B, Qiu Z, Bertoldi K (2015) A comparative analysis of numerical approaches to the mechanics of elastic sheets. J Mech Phys Soilds 79:92–107

    MathSciNet  Google Scholar 

  20. Miller RK, Hedgepeth JM (1982) An algorithm for finite element analysis for partly wrinkled membranes. AIAA J 20:1761–1763

    MATH  Google Scholar 

  21. Ding HL, Yang BE (2003) The modeling and numerical analysis of wrinkled membranes. Int J Numer Methods Eng 58:1785–1801

    MATH  Google Scholar 

  22. Zhang HW, Zhang L, Gao Q (2011) An efficient computational method for mechanical analysis of bimodular structures based on parametric variational principle. Comput Struct 89:2352–2360

    Google Scholar 

  23. Jarasjarungkiat A, Wuchner R, Bletzinger KU (2008) A wrinkling model based on material modification for isotropic and orthotropic membranes. Comput Methods Appl Mech Eng 197:773–788

    MathSciNet  MATH  Google Scholar 

  24. Raible T, Tegeler K, Lohnert S, Wriggers P (2005) Development of a wrinkling algorithm for orthotropic membrane materials. Comput Methods Appl Mech Eng 194:2550–2568

    MATH  Google Scholar 

  25. Contri P, Schrefler BA (1988) A geometrically nonlinear finite element analysis of wrinkling membrane surfaces by a no-compression material model. Commun Appl Numer Methods 4:5–15

    Google Scholar 

  26. Lee ES, Youn SK (2006) Finite element analysis of wrinkling membrane structures with large deformations. Finite Elem Anal Des 42:780–791

    Google Scholar 

  27. Ziegler R, Wagner W, Bletzinger KU (2003) A finite element model for the analysis of wrinkled membrane structures. Int J Space Struct 18(1):1–14

    Google Scholar 

  28. Gal E, Zelkha M, Levy R (2011) A simple co-rotational geometrically non-linear membrane finite element wrinkling analysis. Int J Struct Stab Dyn 11(1):181–195

    MathSciNet  MATH  Google Scholar 

  29. Steigmann DJ (1990) Tension-field theory. Proc R Soc A Math Phys 429:141–173

    MathSciNet  MATH  Google Scholar 

  30. He XT, Zheng ZL, Sun JY, Li YM, Chen SL (2009) Convergence analysis of a finite element method based on different moduli in tension and compression. Int J Solids Struct 46:3734–3740

    MATH  Google Scholar 

  31. Belytschko T, Hsieh BJ (1973) Nonlinear transient finite element analysis with convected coordinates. Int J Numer Methods Eng 7:255–271

    MATH  Google Scholar 

  32. Crisfield MA (1997) Nonlinear finite element analysis of solids and structures. Essentials, vol 1. Wiley, Chichester

    MATH  Google Scholar 

  33. Felippa CA, Haugen B (2005) A unified formulation of small-strain corotational finite elements: I. Theory. Comput Methods Appl Mech Eng 194:2285–2335

    MATH  Google Scholar 

  34. Battini JM (2008) A non-linear corotational 4-node plane element. Mech Res Commun 35(6):408–413

    MATH  Google Scholar 

  35. Zhang L, Lu MK, Zhang HW, Yan B (2015) Geometrically nonlinear elasto-plastic analysis of clustered tensegrity based on the co-rotational approach. Int J Mech Sci 93:154–165

    Google Scholar 

  36. Eriksson A, Faroughi S (2013) Quasi-static inflation simulations based on co-rotational triangular space membrane elements. Int J Struct Stab Dyn 13(3):1250067

    MathSciNet  MATH  Google Scholar 

  37. Faroughi S, Eriksson A (2017) Co-rotational formulation for dynamic analysis of space membranes based on triangular elements. Int J Mech Mater Des 12(2):229–241

    Google Scholar 

  38. Du ZL, Guo X (2014) Variational principles and the related bounding theorems for bi-modulus materials. J Mech Phys Soilds 73:183–211

    MathSciNet  MATH  Google Scholar 

  39. Du ZL, Zhang YP, Zhang WS, Guo X (2016) A new computational framework for materials with different mechanical responses in tension and compression and its applications. Int J Solids Struct 100–101:54–73

    Google Scholar 

  40. Cottle RW, Pang JS, Stone RE (1993) The linear complementarity problem. Academic Press, New York

    MATH  Google Scholar 

  41. Horning J, Schoop H, Herbrich U (2006) Wrinkling analysis of thermo-elastic membranes. Tech Mech 26(1):33–43

    Google Scholar 

  42. Yoo EJ, Roh JH, Han JH (2007) Wrinkling control of inflatable booms using shape memory alloy wires. Smart Mater Struct 16(2):340–348

    Google Scholar 

  43. Deng X, Pellegrino S (2012) Wrinkling of orthotropic viscoelastic membranes. AIAA J 50(3):668–681

    Google Scholar 

Download references

Acknowledgements

Supports from National Natural Science Foundation of China (No. 11872133), SAST Funding (No. SAST2017-022), Chongqing Research Program of Basic Research and Frontier Technology (No. cstc2016jcyjA0058), Fundamental Research Funds for the Central Universities (No. 2019CDQYHK039) at Chongqing University and Research Foundation (No. GZ18110) of State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology are gratefully acknowledged.

Author information

Authors and Affiliations

  1. College of Aerospace Engineering, Chongqing University, Chongqing, 400044, People’s Republic of China

    Liang Zhang

  2. State Key Laboratory for Strength and Vibration of Mechanical Structures, **’an Jiaotong University, **’an, 710049, People’s Republic of China

    Liang Zhang

  3. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, 116024, People’s Republic of China

    Kaijun Dong, Mengkai Lu & Hongwu Zhang

  4. Department of Mechanical Engineering and Mechanics, Ningbo University, Ningbo, 315211, People’s Republic of China

    Mengkai Lu

Authors
  1. Liang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kaijun Dong
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mengkai Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hongwu Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Liang Zhang or Mengkai Lu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, L., Dong, K., Lu, M. et al. A wrinkling model for pneumatic membranes and the complementarity computational framework. Comput Mech 65, 119–134 (2020). https://doi.org/10.1007/s00466-019-01755-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-019-01755-7

Keywords

Search

Navigation