Abstract
The prebuckling deformation of structures is neglected in most of the conventional buckling theory (CBT) and numerical method (CNM), because it is usually very small in conventional concepts. In the preceding paper (Su et al., 2019), we found a class of structures from the emerging field of stretchable electronics, of which the prebuckling deformation became large and essential for determining the critical buckling load, and developed a systematic buckling theory for 3D beams considering the effects of finite prebuckling deformation (FPD). For bulk structures that appear vastly in the advanced structures, a few buckling theories consider the effects of the prebuckling deformation in constitutive equations by energy method, which are significantly important but not straightforward and universal enough. In this paper, a systematic and straightforward theory for the FPD buckling of bulk structures is developed with the use of two constitutive models. The variables for the prebuckling deformation serve as the coefficients of the incremental displacements, deformation components, and stress in the buckling analysis. Four methods, including the CBT, CNM, DLU (disturbing-loading-unloading method) method and FPD buckling theory, are applied to the classic problems, including buckling of an elastic semi-plane solid and buckling of an elastic rectangular solid, respectively. Compared with the accurate buckling load from the DLU method, the FPD buckling theory is able to give a good prediction, while the CBT and CNM may yield unacceptable results (with 70% error for the buckling of an elastic semi-plane solid).
Issue Section:
Research Papers
References
1.
Euler
, L.
, 1744
, Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes
, Apud Marcum-Michaelem Bousquet
.2.
Lagrange
, J. L.
, Serret
, J. A.
, and Darboux
, G.
, 1867
, Oeuvres de Lagrange
, Gauthier-Villars
, Paris
.3.
Kim
, D. H.
, Song
, J.
, Choi
, W. M.
, Kim
, H. S.
, Kim
, R. H.
, Liu
, Z.
, Huang
, Y. Y.
, Hwang
, K. C.
, Zhang
, Y. W.
, and Rogers
, J. A.
, 2008
, “Materials and Noncoplanar Mesh Designs for Integrated Circuits With Linear Elastic Responses to Extreme Mechanical Deformations
,” Proc. Natl. Acad. Sci. U. S. A.
, 105
(48
), pp. 18675
–18680
. 4.
Su
, Y.
, Wu
, J.
, Fan
, Z.
, Hwang
, K.
, Song
, J.
, Huang
, Y.
, and Rogers
, J. A.
, 2012
, “Postbuckling Analysis and Its Application to Stretchable Electronics
,” J. Mech. Phys. Solids
, 60
(3
), pp. 487
–508
. 5.
Jiang
, H.
, Khang
, D. Y.
, Song
, J.
, Sun
, Y.
, Huang
, Y.
, and Rogers
, J. A.
, 2007
, “Finite Deformation Mechanics in Buckled Thin Films on Compliant Supports
,” Proc. Natl. Acad. Sci. U. S. A.
, 104
(40
), pp. 15607
–15612
. 6.
Fan
, Z.
, Hwang
, K. C.
, Rogers
, J. A.
, Huang
, Y.
, and Zhang
, Y.
, 2018
, “A Double Perturbation Method of Postbuckling Analysis in 2D Curved Beams for Assembly of 3D Ribbon-Shaped Structures
,” J. Mech. Phys. Solids
, 111
, pp. 215
–238
. 7.
Timoshenko
, S. P.
, and Gere
, J. M.
, 1961
, Theory of Elastic Stability
, 2nd ed., McGraw-Hill
, New York
.8.
Khang
, D.
, Jiang
, H.
, Huang
, Y.
, and Rogers
, J. A.
, 2006
, “A Stretchable Form of Single-Crystal Silicon for High-Performance Electronics on Rubber Substrates
,” Science
, 311
(5758
), pp. 208
–212
. 9.
Fu
, Y. B.
, and Ogden
, R. W.
, 1999
, “Nonlinear Stability Analysis of Pre-Stressed Elastic Bodies
,” Continuum Mech. Thermodyn.
, 11
(3
), pp. 141
–172
. 10.
Fu
, Y.
, and Rogerson
, G. A.
, 1994
, “A Nonlinear Analysis of Instability of a Pre-Stressed Incompressible Elastic Plate
,” Proc. R. Soc. London, A
, 446
(1927
), pp. 233
–254
. 11.
Chadwick
, P.
, and Ogden
, R.
, 1971
, “On the Definition of Elastic Moduli
,” Arch. Ration. Mech. Anal.
, 44
(1
), pp. 41
–53
. 12.
Su
, Y.
, Ping
, X.
, Yu
, K. J.
, Lee
, J. W.
, Fan
, J. A.
, Wang
, B.
, Li
, M.
, et al, 2017
, “In-Plane Deformation Mechanics for Highly Stretchable Electronics
,” Adv. Mater.
, 29
(8
), p. 1604989
. 13.
Su
, Y.
, Zhao
, H.
, Liu
, S.
, Li
, R.
, Wang
, Y.
, Wang
, Y.
, Bian
, J.
, and Huang
, Y.
, 2019
, “Buckling of Beams With Finite Prebuckling Deformation
,” Int. J. Solids Struct.
, 165
(15
), pp. 148
–159
. 14.
Biot
, M. A.
, 1938
, “Theory of Elasticity With Large Displacements and Rotations
,” Proceedings of the Fifth International Congress of Applied Mechanics
, pp. 117
–122
.15.
Biot
, M. A.
, 1959
, “Folding of a Layered Viscoelastic Medium Derived From an Exact Stability Theory of a Continuum Under Initial Stress
,” Q. Appl. Math.
, 17
(2
), pp. 185
–204
. 16.
Biot
, M. A.
, 1963
, “Surface Instability of Rubber in Compression
,” Appl. Sci. Res.
, 12
(2
), pp. 168
–182
. 17.
Biot
, M. A.
, and Romain
, J. E.
, 1965
, “Mechanics of Incremental Deformations
,” J. Appl. Mech.
, 32
(4
), p. 957
.18.
Novozhilov
, V. V.
, 1953
, Foundations of the Nonlinear Theory of Elasticity
, Graylock Press
, New York
.19.
Kerr
, A. D.
, and Tang
, S.
, 1967
, “The Instability of a Rectangular Elastic Solid
,” Acta Mechanica
, 4
(1
), pp. 43
–63
. 20.
Brunelle
, E. J.
, 1973
, “Surface Instability Due to Initial Compressive Stress
,” Bull. Seismol. Soc. Am.
, 63
(6-1
), pp. 1885
–1893
. 21.
Triantafyllidis
, N.
, 1983
, “On the Bifurcation and Postbifurcation Analysis of Elastic-Plastic Solids Under General Prebifurcation Conditions
,” J. Mech. Phys. Solids
, 31
(6
), pp. 499
–510
. 22.
Jiménez
, F. L.
, and Triantafyllidis
, N.
, 2013
, “Buckling of Rectangular and Hexagonal Honeycomb Under Combined Axial Compression and Transverse Shear
,” Int. J. Solids Struct.
, 50
(24
), pp. 3934
–3946
. 23.
Triantafyllidis
, N.
, 1980
, “Bifurcation Phenomena in Pure Bending
,” J. Mech. Phys. Solids
, 28
(3
), pp. 221
–245
. 24.
Lee
, D.
, Triantafyllidis
, N.
, Barber
, J. R.
, and Thouless
, M. D.
, 2008
, “Surface Instability of an Elastic Half Space With Material Properties Varying With Depth
,” J. Mech. Phys. Solids
, 56
(3
), pp. 858
–868
. 25.
Ogden
, R.
, and Fu
, Y.
, 1996
, “Nonlinear Stability Analysis of a Pre-Stressed Elastic Half-Space
,” Contemp. Res. Mech. Math. Mater.
, pp. 164
–175
.26.
Ogden
, R. W.
, 1984
, Non-Linear Elastic Deformations
, Dover Publications, Inc.
, New York
.27.
Bažant
, Z.
, 1971
, “A Correlation Study of Formulations of Incremental Deformation and Stability of Continuous Bodies
,” ASME J. Appl. Mech.
, 38
(4
), pp. 919
–928
. 28.
Goodier
, J. N.
, and Plass
, H. J.
, 1952
, “Energy Theorems and Critical Load Approximations in the General Theory of Elastic Stability
,” Q. Appl. Math.
, 9
(4
), pp. 286
–287
. 29.
Eringen
, A.
, and Paslay
, P. R.
, 1964
, “Non-Linear Theory of Continuous Media
,” ASME J. Appl. Mech.
, 31
(4
), p. 368
. 30.
Huang
, Z. Y.
, Hong
, W.
, and Suo
, Z.
, 2005
, “Nonlinear Analyses of Wrinkles in a Film Bonded to a Compliant Substrate
,” J. Mech. Phys. Solids
, 53
(9
), pp. 2101
–2118
. 31.
Huang
, R.
, 2005
, “Kinetic Wrinkling of an Elastic Film on a Viscoelastic Substrate
,” J. Mech. Phys. Solids
, 53
(1
), pp. 63
–89
. 32.
Chen
, X.
, and Hutchinson
, J. W.
, 2004
, “Herringbone Buckling Patterns of Compressed Thin Films on Compliant Substrates
,” ASME J. Appl. Mech.
, 71
(5
), pp. 597
–603
. 33.
Bowden
, N.
, Brittain
, S.
, Evans
, A.
, Hutchinson
, J.
, and Whitesides
, G.
, 1998
, “Spontaneous Formation of Ordered Structures in Thin Films of Metals Supported on an Elastomeric Polymer
,” Nature
, 393
(6681
), pp. 146
–149
. 34.
Song
, J.
, Jiang
, H.
, Liu
, Z. J.
, Khang
, D. Y.
, Huang
, Y.
, Rogers
, J. A.
, Lu
, C.
, and Koh
, C.
, 2008
, “Buckling of a Stiff Thin Film on a Compliant Substrate in Large Deformation
,” Int. J. Solids Struct.
, 45
(10
), pp. 3107
–3121
. 35.
Jiang
, H.
, Khang
, D.
, Fei
, H.
, Kim
, H.
, Huang
, Y.
, Xiao
, J.
, and Rogers
, J. A.
, 2008
, “Finite Width Effect of Thin-Films Buckling on Compliant Substrate: Experimental and Theoretical Studies
,” J. Mech. Phys. Solids
, 56
(8
), pp. 2585
–2598
. 36.
Bo
, L.
, Huang
, S. Q.
, and Feng
, X. Q.
, 2010
, “Buckling and Postbuckling of a Compressed Thin Film Bonded on a Soft Elastic Layer: A Three-Dimensional Analysis
,” Arch. Appl. Mech.
, 80
(2
), p. 175
. 37.
Li
, B.
, Zhao
, H.
, and Feng
, X.
, 2011
, “Spontaneous Instability of Soft Thin Films on Curved Substrates Due to van der Waals Interaction
,” J. Mech. Phys. Solids
, 59
(3
), pp. 610
–624
. 38.
Li
, B.
, Cao
, Y.
, Feng
, X.-Q.
, and Gao
, H.
, 2012
, “Mechanics of Morphological Instabilities and Surface Wrinkling in Soft Materials: A Review
,” Soft Matter
, 8
(21
), pp. 5728
–5745
. 39.
Holland
, M. A.
, Li
, B.
, Feng
, X. Q.
, and Kuhl
, E.
, 2017
, “Instabilities of Soft Films on Compliant Substrates
,” J. Mech. Phys. Solids
, 98
, pp. 350
–365
. 40.
Huang
, S.
, and Li
, B.
, 2008
, “Three-Dimensional Analysis of Spontaneous Surface Instability and Pattern Formation of Thin Soft Films
,” J. Appl. Phys.
, 103
(8
), p. 545
.41.
Wang
, P.
, Casadei
, F.
, Shan
, S.
, Weaver
, J. C.
, and Bertoldi
, K.
, 2014
, “Harnessing Buckling to Design Tunable Locally Resonant Acoustic Metamaterials
,” Phys. Rev. Lett.
, 113
(1
), p. 014301
. 42.
Bertoldi
, K.
, Reis
, P. M.
, Willshaw
, S.
, and Mullin
, T. J. A. M.
, 2010
, “Negative Poisson’s Ratio Behavior Induced by an Elastic Instability
,” Adv. Mater.
, 22
(3
), pp. 361
–366
. 43.
Bertoldi
, K.
, Boyce
, M. C.
, Deschanel
, S.
, Prange
, S. M.
, and Mullin
, T.
, 2008
, “Mechanics of Deformation-Triggered Pattern Transformations and Superelastic Behavior in Periodic Elastomeric Structures
,” J. Mech. Phys. Solids
, 56
(8
), pp. 2642
–2668
. 44.
Biot
, M. A.
, 1965
, Mechanics of Incremental Deformations
, John Wiley and Sons Inc.
, New York
.45.
Cao
, Y.
, and Hutchinson
, J. W.
, 2012
, “From Wrinkles to Creases in Elastomers: The Instability and Imperfection-Sensitivity of Wrinkling
,” Proc. R. Soc. A
, 468
(2137
), pp. 94
–115
. 46.
Dassault-Systèmes
, 2010
, Abaqus Analysis User's Manual v.6.10
, Dassault Systèmes Simulia Corp.
, Johnston, RI
.Copyright © 2022 by ASME
You do not currently have access to this content.
