Skip to main content
SpringerLink
Log in

Higher-order asymptotics for edge-buckling of pre-stressed thin plates under in-plane bending

  • Published:
Journal of Engineering Mathematics Aims and scope Submit manuscript

Abstract

Singular perturbation techniques are used to investigate the linear eigenvalue problem that describes the partial wrinkling of a pre-stressed rectangular thin elastic plate under in-plane bending. The dependence of the critical load and the wavelength of the localised oscillatory pattern on the non-dimensional bending rigidity of the plate is captured with a higher-order boundary-layer analysis. Comparisons with direct numerical simulations of the original eigenproblem confirm the accuracy and versatility of the asymptotic technique explored in this work.

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

  • Alfutov NA (2000). Stability of elastic structures. Springer-Verlag, Berlin

    MATH  Google Scholar 

  • Coman CD, Bassom AP (2007) Wrinkling of pre-stressed annular thin films under azimuthal shearing. Math Mech Solids (in press)

  • Stein M, Hedgepeth JM (1961) Analysis of partly wrinkled membranes. Technical Note, NASA-TN D-813

  • Mikulas MM (1964) Behaviour of a flat stretched membrane wrinkled by the rotation of an attached hub. Technical report, NASA-TN D-2456

  • Miller RK, Hedgepeth JM, Weingarten VI, Das P and Kahyai S (1985). Finite element analysis of partly wrinkled membranes. Comput Struct 20: 631–639

    Article  Google Scholar 

  • Pipkin AC (1986). The relaxed energy density for isotropic elastic membranes. IMA J Appl Math 36: 85–89

    Article  MATH  MathSciNet  Google Scholar 

  • Steigmann D (1990). Tension field theory. Proc Roy Soc Lond A 429: 141–173

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Coman CD (2007). Edge-buckling in stretched thin films under in-plane bending. Z angew Math Phys 58: 510–525

    Article  MATH  MathSciNet  Google Scholar 

  • Coman CD and Bassom AP (2007). On the wrinkling of a pre-stressed annular film in tension. J Mech Phys Solids 55: 1601–1617

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Reissner E (1959). The edge effect in symmetric bending of shallow shells of revolutions. Commun Pure Appl Math 7: 385–398

    Article  MathSciNet  Google Scholar 

  • Ranjan GV and Steele CR (1980). Nonlinear correction for edge bending of shells. ASME J Appl Mech 47: 861–865

    MATH  MathSciNet  Google Scholar 

  • Taber LA (1985). Nonlinear asymptotic solution of Reissner plate equations. ASME J Appl Mech 52: 907–912

    Article  MATH  Google Scholar 

  • Movchan AB and Movchan NV (1995). Mathematical modelling of solids with non-regular boundaries. CRC Press, Boca Raton

    Google Scholar 

  • Fu YB (1998). Some asymptotic results concerning the buckling of a spherical shell of arbitrary thickness. Int J Non-Lin Mech 33: 1111–1122

    Article  MATH  Google Scholar 

  • Friedl N, Rammerstorfer FG and Fischer FD (2000). Buckling of stretched strips. Comput Struct 78: 185–190

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Jacques N and Potier-Ferry M (2005). On mode localisation in tensile plate buckling. C R Mecanique 333: 804–809

    MATH  ADS  Google Scholar 

  • Géminard JC, Bernal R and Melo F (2004). Wrinkle formations in axisymmetrically stretched membranes. Eur Phys J E 15: 117–126

    Article  Google Scholar 

  • Mora T and Boudaoud A (2006). Buckling of swelling gels. Eur Phys J E 20: 119–124

    Article  Google Scholar 

  • Coman CD and Haughton DM (2006). Localised wrinkling instabilities in radially stretched annular thin films. Acta Mech 185: 179–200

    Article  MATH  Google Scholar 

  • Harris AK, Wild P and Stopak D (1980). Silicone rubber substrata: a new wrinkle in cell locomotion. Science 208: 177–179

    Article  ADS  Google Scholar 

  • Burton K and Taylor DL (1997). Traction forces of cytokinesis measured using optically modified elastic substrata. Nature 385: 450–454

    Article  ADS  Google Scholar 

  • Månsson J, Söderqvist J (2003) Finite element analysis of thin membrane wrinkling. M.Sc. Thesis, Royal Institute of Technology (Stockholm)

  • Timoshenko SP (1961). Theory of elastic stability. McGraw-Hill, New York

    Google Scholar 

  • Doedel EJ, Champneys AR, Fairgrieve TF, Kuznetsov YA, Sandstede B, Wang Y (1997) AUTO97: continuation and bifurcation software for ordinary differential equations. Technical report, Dept. of Computer Science, Concordia University, Montreal, Canada. Available by ftp from ftp://cs.concordia.caint/pub/doedel/auto

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Glasgow, University Gardens, Glasgow, G12 8QW, Scotland, UK

    Ciprian D. Coman

  2. School of Mathematics and Statistics, University of Western Australia, Crawley, 6009, Australia

    Andrew P. Bassom

Authors
  1. Ciprian D. Coman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrew P. Bassom
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ciprian D. Coman.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Coman, C.D., Bassom, A.P. Higher-order asymptotics for edge-buckling of pre-stressed thin plates under in-plane bending. J Eng Math 63, 327–338 (2009). https://doi.org/10.1007/s10665-007-9196-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10665-007-9196-9

Keywords

Search

Navigation