Abstract
We consider the wrinkling of highly stretched, thin rectangular sheets—a problem that has attracted the attention of several investigators in recent years, nearly all of which employ the classical Föppl–von Kármán (F–K) theory of plates. We first propose a rational model that correctly accounts for large mid-plane strain. We then carefully perform a numerical bifurcation/continuation analysis, identifying stable solutions (local energy minimizers). Our results in comparison to those from the F–K theory (also obtained herewith) show: (i) For a given fine thickness, only a certain range of aspect ratios admit stable wrinkling; for a fixed length (in the highly stretched direction), wrinkling does not occur if the width is too large or too small. In contrast, the F–K model erroneously predicts wrinkling in those very same regimes for sufficiently large applied macroscopic strain. (ii) When stable wrinkling emerges as the applied macroscopic strain is steadily increased, the amplitude first increases, reaches a maximum, decreases, and then returns to zero again. In contrast, the F–K model predicts an ever-increasing wrinkling amplitude as the macroscopic strain is increased. We identify (i) and (ii) as global isola-center bifurcations—in terms of both the macroscopic-strain parameter and an aspect-ratio parameter. (iii) When stable wrinkling occurs, for fixed parameters, the transverse pattern admits an entire orbit of neutrally stable (equally likely) possibilities: These include reflection symmetric solutions about the mid-plane, anti-symmetric solutions about the mid-line (a rotation by π radians about the mid-line leaves the wrinkled shape unchanged) and a continuously evolving family of shapes “in-between”, say, parametrized by an arbitrary phase angle, each profile of which is neither reflection symmetric nor anti-symmetric.
Similar content being viewed by others
References
Antman, S.S., Pierce, J.F.: The intricate global structure of buckled states of compressible columns. SIAM J. Appl. Math. 50, 95–419 (1990)
Berger, M.S.: Nonlinearity and Functional Analysis. Academic Press, New York (1977)
Cerda, E., Mahadevan, L.: Geometry and physics of wrinkling. Phys. Rev. Lett. 90, 1–4 (2003)
Cerda, E., Ravi-Chandar, K., Mahadevan, L.: Wrinkling of an elastic sheet under tension. Nature 419, 579–580 (2002)
Chen, Y.-C., Healey, T.J.: Bifurcation to pear-shaped equilibria of pressurized spherical membranes. Int. J. Non-Linear Mech. 26, 279–291 (1991)
Ciarlet, P.G.: Mathematical Elasticity, vol. I. North-Holland, Amsterdam (1988)
Ciarlet, P.G.: An introduction to differential geometry with applications to elasticity. J. Elast. 78–79, 3–201 (2005)
Dacorogna, B.: Direct Methods in the Calculus of Variations, 2nd edn. Springer, New York (2008)
Dym, C.L., Shames, I.H.: Solid Mechanics: A Variational Approach. McGraw-Hill, New York (1973)
Evans, L.C.: Quasiconvexity and partial regularity in the calculus of variations. Arch. Ration. Mech. Anal. 95, 227–252 (1986)
Friedl, N., Rammerstorfer, F.G., Fischer, F.D.: Buckling of stretched strips. Comput. Struct. 78, 185–190 (2000)
Golubitsky, M., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory. Springer, New York (1985)
Healey, T.J., Miller, U.: Two-phase equilibria in the anti-plane shear of an elastic solid with interfacial effects via global bifurcation. Proc. R. Soc. A, Math. Phys. Eng. Sci. 463, 1117–1134 (2007)
Healey, T.J., Simpson, H.S.: Global continuation in nonlinear elasticity. Arch. Ration. Mech. Anal. 143, 1–28 (1998)
Hetényi, M.: Beams on Elastic Foundation. University Michigan Press, Ann Arbor (1946)
Jacques, N., Potier-Ferry, M.: On mode localization in tensile plate buckling. C. R., Méc. 333, 804–809 (2005)
Keller, H.B.: Numerical Methods in Bifurcation Problems. Tata Institute of Fundamental Research/Springer, Bombay/New York (1987)
Le Dret, H., Raoult, A.: Quasiconvex envelopes of stored energy densities that are convex with respect to the strain tensor. In: Calculus of Variations, Applications and Computations. Pitman Research Notes in Mathematics, vol. 326, pp. 138–146 (1995)
Meirovitch, L.: Analytical Methods in Vibrations. Macmillan, New York (1967)
Nayyar, V., Ravi-Chandar, K., Huang, R.: Stretch-induced stress patterns and wrinkles in hyperelastic thin sheets. Int. J. Solids Struct. 48, 3471–3483 (2011)
Puntel, E., Deseri, L., Fried, E.: Wrinkling of a stretched thin sheet. J. Elast. 105, 137–170 (2011)
Raoult, A.: Non-polyconvexity of the stored energy function of a Saint Venant–Kirchhoff material. Apl. Mat. 31, 417–419 (1986)
Reddy, J.N.: Nonlinear Finite Element Analysis. Oxford University Press, New York (2004)
Steigmann, D.J.: Koiter’s shell theory from the perspective of three-dimensional elasticity. J. Elast. 111, 91–107 (2013)
Valent, T.: Boundary Value Problems of Finite Elasticity. Springer, New York (1988)
Von Kármán, T., Edson, L.: The Wind and Beyond. Little, Brown and Company, Boston (1967)
Acknowledgements
The work of R.-B.C. and T.J.H. was supported in part by the National Science Foundation through grant DMS-0707715 and that of T.J.H. also through NSF grant DMS-1007830. The work of Q.L. was supported in part by China Scholarship Council 2010850685 and by the National Science Foundation of China 61104150. Each of these is gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Healey, T.J., Li, Q. & Cheng, RB. Wrinkling Behavior of Highly Stretched Rectangular Elastic Films via Parametric Global Bifurcation. J Nonlinear Sci 23, 777–805 (2013). https://doi.org/10.1007/s00332-013-9168-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00332-013-9168-3