This work presents an approximate analytical study of the problem of dynamic wrinkling of a thin metal sheet under a specified time varying tension. The problem is investigated in the framework of the dynamic stability of a nonlinear plate model on elastic foundation which namely takes into account the nonlinear mechanics of mid-plane stretching and the dependence of the membrane force on this mechanics. The plate is assumed to be a wide rectangular slab, hinged at two opposite ends and free at the long ends, which can be deformed in a cylindrical shape so that the governing in-plane bending equation of motion takes the same form as that of a beam (e.g. lateral strip) element. An approximate analytical analysis of the beam wrinkling behavior under sinusoidal parametric excitation is carried out by using the assumed single mode wrinkling motion to reduce the beam field nonlinear partial differential equation to that of a single degree of freedom non-linear oscillator. A first order stability analysis of an approximate analytical solution obtained using the Multi-Time-Scales (MMS) method is used to derive a criterion defining critical driving frequency in terms of system parameters for the initiation of wrinkling motion in the thin metal sheet. Results obtained using this criterion is presented for selected values of system parameters.
P. Tugcu, K. W. Naele, P. D. Wu and S. R. Mac Ewen, “Effect of Planar Anisotropy on Wrinkling Formation Tendencies in Curved Sheets,” International Journal of Mechanical Sciences, Vol. 43, No. 12, 2001, pp. 2883-2897. doi:10.1016/S0020-7403(01)00065-0
H. Lu, H. S. Cheng, J. Cao and W. K. Liu, “Adaptive Enrichment Meshfree Simulation and Experiment on Buckling and Post Buckling Analysis in Sheet Metal Forming,” Computer Methods in Applied Mechanics and Engineering, Vol. 194, No. 21-24, 2005, pp. 2569-2590. doi:10.1016/j.cma.2004.07.046
M. A. Wadee and G. W. Hunt, “Interactively Induced Localized Buckling in Sandwich Structures with Core Orthotropy,” Transactions of ASME Journal of Applied Mechanics, Vol. 65, 1998, pp. 523-528.
V. Birman, “Dynamic Wrinkling in Sandwich Beams,” Composites Part B: Engineering, Vol. 35, No. 6-8, 2004, pp. 665-672. doi:10.1016/j.compositesb.2003.08.012
J. Hohe, L. Librescu and S. Y. Oh, “Dynamic Buckling of Flat and Curved Sandwich Panels with Transversely Compressible Core,” Composite Structures, Vol. 74, No. 1, 2006, pp. 10-24. doi:10.1016/j.compstruct.2005.03.003
H. Hassani and K. Neale, “On the Analysis of Sheet Metal Wrinkling,” International Journal of Mechanical Science, Vol. 33, No. 1, 1991, pp. 13-30. doi:10.1016/0020-7403(91)90024-W
B. Davidovitch, R. Schroll, D. Vella, M. Adda-Bedia and E. Creda, “Prototypical Model for Tensional Wrinkling in Thin Sheets,” Applied Physical Sciences, Vol. 108, 2011, pp. 18227-18232.
Y. Mei, S. Kiravittaya, S. Harazim and O. G. Schmidt, “Principles and Applications of Micro and Nanoscale Wrinkles,” Materials Science and Engineering: Reports, Vol. 70, No. 3-6, 2010, pp. 209-224. doi:10.1016/j.mser.2010.06.009
J. W. Hutchinson, “Plastic Buckling,” Advances in Applied Mechanics, Vol. 14, 1974, pp. 67-144. doi:10.1016/S0065-2156(08)70031-0
J. W. Hutchinson, “Post-Bifurcation Behavior in the Plastic Range,” Journal of the Mechanics and Physics of Solids, Vol. 21, No. 3, 1973, pp. 163-190. doi:10.1016/0022-5096(73)90017-3
K. W. Neale and P. Tugcu, “A Numerical Analysis of Wrinkling Formation Tendencies in Sheet Metals,” International Journal of Numerical Methods in Engineering, Vol. 30, No. 8, 1990, pp. 1595-1608. doi:10.1002/nme.1620300816
J. Cao, “Prediction of Plastic Wrinkling Using the Energy Method,” Transactions of ASME Journal of Applied Mechanics, Vol. 66, 1999, pp. 646-652.
J. Cao and M. Boyce, “Wrinkling Behavior of Rectangular Plates under Lateral Constraint,” International Journal of Solids and Structures, Vol. 34, No. 20, 1997, pp. 153-176. doi:10.1016/S0020-7683(96)00008-X
Y. Kim and Y. Son, “Study on Wrinkling Limit Diagrams of Anistropic Sheet Metals,” Journal of Materials Processing Technology, Vol. 97, No. 1-3, 2000, pp. 88-94. doi:10.1016/S0924-0136(99)00346-5
J. P. De Magalhaes Correia and G. Ferron, “Wrinkling of Anisotropic Metal Sheets under Deep-Drawing: Analytical and Numerical Study,” Journal of Materials Processing Technology, Vol. 155-156, 2004, pp. 1604-1610. doi:10.1016/j.jmatprotec.2004.04.270
A. S. Benson and J. Mayers, “General Instability and Face Wrinkling of Sandwich Plates—A Unified Theory and Applications,” American Institute of Aeronautics and Astronautics Journal, Vol. 5, No. 4, 1967, pp. 729-739.
C. C. Lin and C. D. Mote, “Eigenvalue Solutions Predicting the Wrinkling of Rectangular Webs under Non-Linearly Distributed Edge Loading,” Journal of Sound and Vibration, Vol. 197, No. 2, 1996, pp. 179-189. doi:10.1006/jsvi.1996.0524
B. K. Hadi and F. L. Matthew, “Development of Benson-Mayers Theory on the Wrinkling of Anisotropic Sandwich Panels,” Composite Structures, Vol. 49, No. 4, 2001, pp. 425-434. doi:10.1016/S0263-8223(00)00077-5
B. K. Hadi, “Wrinkling of Sandwich Column: Comparison between Finite Element Analysis and Analytical Solutions,” Composite Structures, Vol. 53, No. 4, 2001, pp. 477-482. doi:10.1016/S0263-8223(01)00060-5
H. Li, H. Yang, M. Zhan and R. J. Gu, “A New Method to Accurately Obtain Wrinkling Limit Diagram in NC Bending Process of Thin-Walled Tube with Large Diameter under Different Loading Paths,” Journal of Materials Processing Technology, Vol. 17, No. 1-3, 2006, pp. 192-196. doi:10.1016/j.jmatprotec.2006.03.191
X. Wang and L. H. N. Lee, “Postbifurcation Behavior of Wrinkles in Square Metal Sheet under Yoshida Test,” International Journal of Plasticity, Vol. 9, 1993, pp. 1-19.
H. S. Lee, D. W. Hung, J. H. Jeong and S. Im, “Finite Element Analysis of Lateral Buckling for Beam Structures,” Computers and Structures, Vol. 53, No. 6, 1994, pp. 1357-1371. doi:10.1016/0045-7949(94)90400-6
J. B. Kim, D. Y. Yang, J. W. Yoon and F. Barlat, “The Effect of Plastic Anisotropy on Compressive Instability in Sheet Metal Forming,” International Journal of Plasticity, Vol. 16, No. 6, 2000, pp. 649-676. doi:10.1016/S0749-6419(99)00064-9
C. Loganathan and R. Naryaanasamy, “Effect of Mechanical Properties on the Wrinkling Behavior of Three Different Commercially Pure Aluminum Grades When Drawn through Conical Tractrix Dies,” Materials Science and Engineering: A, Vol. 406, No. 1-2, 2005, pp. 229-253. doi:10.1016/j.msea.2005.06.037
J. L. Grenestedt and J. Reany, “Wrinkling of Corrugated Skin Sandwich Panels,” Composites Part A, Vol. 38, No. 2, 2007, pp. 576-589. doi:10.1016/j.compositesa.2006.02.007
E. S. Lee and S. K. Youn, “Finite Element Analysis of Wrinkling Membrane Structures with Large Deformations,” Finite Elements in Analysis and Design, Vol. 42, No. 8-9, 2006, pp. 780-791. doi:10.1016/j.finel.2006.01.004
G. P. Dube, P. C. Dumir and A. Mallik, “Dynamic Buckling of Laminated Thick Shallow Sphere Cap Based on a Static Analysis,” Mechanical Research Communications, Vol. 27, No. 5, 2000, pp. 561-566. doi:10.1016/S0093-6413(00)00130-0
D. S. Lee, “Nonlinear Dynamic Buckling of Orthotropic Cylindrical Shell Subjected to Rapid Loading,” Journal of Engineering Mathematics, Vol. 38, No. 2, 2000, pp. 141-154. doi:10.1023/A:1004729816574
S. Timoshenko and S. Woinowsky-Krieger, “Theory of Plates and Shells,” McGraw-Hill, New York, 1987.
R. H. Plaut and E. R. Johnson, “The Effect of Initial Thrust and Elastic Foundation on the Vibration Frequencies of a Shallow Arc,” Journal of Sound and Vibration, Vol. 78, No. 4, 1981, pp. 565-571. doi:10.1016/S0022-460X(81)80125-3
W. Lacarbonara, H. N. Arafat and A. H. Nayfeh, “Large Non-Linear Interactions in Imperfect Beams at Veering,” International Journal of Non-linear Mechanics, Vol. 40, No. 7, 2005, pp. 987-1003. doi:10.1016/j.ijnonlinmec.2004.10.006
A. A. Al-Qaisia and M. N. Hamdan, “Non-Linear Frequency Veering in a Beam Resting on Elastic Foundation,” Journal of Vibration and Control, Vol. 15, No. 11, 2009, pp. 1627-1647. doi:10.1177/1077546309103262
C. H. Kim, N. C. Perkins and C. W. Lee, “Parametric Resonance of Plates in a Sheet Metal Coating Process,” Journal of Sound and Vibration, Vol. 268, 2003, pp. 679-697. doi:10.1016/S0022-460X(02)01538-9
A. H. Nayfeh and T. D. Mook, “Nonlinear Oscillations,” Wiley-Interscience, New York, 1979.