0
Research Papers

Symplectic Analysis of Wrinkles in Elastic Layers With Graded Stiffnesses

[+] Author and Article Information
Jianjun Sui

School of Aerospace Engineering and
Applied Mechanics,
Tongji University,
Shanghai 200092, China;
Department of Mechanical and
Aerospace Engineering,
Syracuse University,
Syracuse, NY 13244

Junbo Chen, Xiaoxiao Zhang

Department of Mechanical and
Aerospace Engineering,
Syracuse University,
Syracuse, NY 13244

Guohua Nie

School of Aerospace Engineering and
Applied Mechanics,
Tongji University,
Shanghai 200092, China
e-mail: ghnie@tongji.edu.cn

Teng Zhang

Department of Mechanical
and Aerospace Engineering,
Syracuse University,
Syracuse, NY 13244
e-mail: tzhang48@syr.edu

1Corresponding authors.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received August 8, 2018; final manuscript received September 25, 2018; published online October 18, 2018. Assoc. Editor: Pedro Reis.

J. Appl. Mech 86(1), 011008 (Oct 18, 2018) (8 pages) Paper No: JAM-18-1467; doi: 10.1115/1.4041620 History: Received August 08, 2018; Revised September 25, 2018

Wrinkles in layered neo-Hookean structures were recently formulated as a Hamiltonian system by taking the thickness direction as a pseudo-time variable. This enabled an efficient and accurate numerical method to solve the eigenvalue problem for onset wrinkles. Here, we show that wrinkles in graded elastic layers can also be described as a time-varying Hamiltonian system. The connection between wrinkles and the Hamiltonian system is established through an energy method. Within the Hamiltonian framework, the eigenvalue problem of predicting wrinkles is defined by a series of ordinary differential equations with varying coefficients. By modifying the boundary conditions at the top surface, the eigenvalue problem can be efficiently and accurately solved with numerical solvers of boundary value problems. We demonstrated the accuracy of the symplectic analysis by comparing the theoretically predicted displacement eigenfunctions, critical strains, and wavelengths of wrinkles in two typical graded structures with finite element simulations.

FIGURES IN THIS ARTICLE
<>
Copyright © 2019 by ASME
Your Session has timed out. Please sign back in to continue.

References

Yang, S. , Khare, K. , and Lin, P. C. , 2010, “ Harnessing Surface Wrinkle Patterns in Soft Matter,” Adv. Funct. Mater., 20(16), pp. 2550–2564. [CrossRef]
Li, B. , Cao, Y.-P. , 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. [CrossRef]
Wang, Q. , and Zhao, X. , 2016, “ Beyond Wrinkles: Multimodal Surface Instabilities for Multifunctional Patterning,” MRS Bull, 41(2), pp. 115–122. [CrossRef]
Genzer, J. , and Groenewold, J. , 2006, “ Soft Matter With Hard Skin: From Skin Wrinkles to Templating and Material Characterization,” Soft Matter, 2(4), pp. 310–323. [CrossRef]
Budday, S. , Steinmann, P. , and Kuhl, E. , 2014, “ The Role of Mechanics During Brain Development,” J. Mech. Phys. Solids, 72, pp. 75–92. [CrossRef] [PubMed]
Tallinen, T. , Chung, J. Y. , Biggins, J. S. , and Mahadevan, L. , 2014, “ Gyrification From Constrained Cortical Expansion,” Proc. Natl. Acad. Sci. U.S.A, 111(35), pp. 12667–12672. [CrossRef] [PubMed]
Tallinen, T. , Chung, J. Y. , Rousseau, F. , Girard, N. , Lefèvre, J. , and Mahadevan, L. , 2016, “ On the Growth and Form of Cortical Convolutions,” Nat. Phys., 12(6), p. 588. [CrossRef]
Yin, J. , Cao, Z. , Li, C. , Sheinman, I. , and Chen, X. , 2008, “ Stress-Driven Buckling Patterns in Spheroidal Core/Shell Structures,” Proc. Natl. Acad. Sci. U.S.A, 105(49), pp. 19132–19135. [CrossRef] [PubMed]
Yin, J. , Chen, X. , and Sheinman, I. , 2009, “ Anisotropic Buckling Patterns in Spheroidal Film/Substrate Systems and Their Implications in Some Natural and Biological Systems,” J. Mech. Phys. Solids, 57(9), pp. 1470–1484. [CrossRef]
Breid, D. , and Crosby, A. J. , 2011, “ Effect of Stress State on Wrinkle Morphology,” Soft Matter, 7(9), pp. 4490–4496. [CrossRef]
Kim, D.-H. , Lu, N. , Ma, R. , Kim, Y.-S. , Kim, R.-H. , Wang, S. , Wu, J. , Won, S. M. , Tao, H. , and Islam, A. , 2011, “ Epidermal Electronics,” Science, 333(6044), pp. 838–843. [CrossRef] [PubMed]
Rogers, J. A. , Someya, T. , and Huang, Y. , 2010, “ Materials and Mechanics for Stretchable Electronics,” Science, 327(5973), p. 1603. [CrossRef] [PubMed]
Guvendiren, M. , Yang, S. , and Burdick, J. A. , 2009, “ Swelling‐Induced Surface Patterns in Hydrogels With Gradient Crosslinking Density,” Adv. Funct. Mater., 19(19), pp. 3038–3045. [CrossRef]
Kang, M. K. , and Huang, R. , 2010, “ Swell-Induced Surface Instability of Confined Hydrogel Layers on Substrates,” J. Mech. Phys. Solids, 58(10), pp. 1582–1598. [CrossRef]
Toh, W. , Ding, Z. , Yong Ng, T. , and Liu, Z. , 2015, “ Wrinkling of a Polymeric Gel During Transient Swelling,” ASME J. Appl. Mech., 82(6), p. 061004. [CrossRef]
Amar, M. B. , and Ciarletta, P. , 2010, “ Swelling Instability of Surface-Attached Gels as a Model of Soft Tissue Growth Under Geometric Constraints,” J. Mech. Phys. Solids, 58(7), pp. 935–954. [CrossRef]
Glatz, B. A. , Tebbe, M. , Kaoui, B. , Aichele, R. , Kuttner, C. , Schedl, A. E. , Schmidt, H.-W. , Zimmermann, W. , and Fery, A. , 2015, “ Hierarchical Line-Defect Patterns in Wrinkled Surfaces,” Soft Matter, 11(17), pp. 3332–3339. [CrossRef] [PubMed]
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. [CrossRef] [PubMed]
Diab, M. , Zhang, T. , Zhao, R. , Gao, H. , and Kim, K.-S. , 2013, “ Ruga Mechanics of Creasing: From Instantaneous to Setback Creases,” Proc. R. Soc. A., 469(2157), p. 20120753.
Diab, M. , and Kim, K.-S. , 2014, “ Ruga-Formation Instabilities of a Graded Stiffness Boundary Layer in a neo-Hookean Solid,” Proc. R. Soc. A, 470(2168), p. 20140218. [CrossRef]
Wu, Z. , Meng, J. , Liu, Y. , Li, H. , and Huang, R. , 2014, “ A State Space Method for Surface Instability of Elastic Layers With Material Properties Varying in Thickness Direction,” ASME J. Appl. Mech., 81(8), p. 081003. [CrossRef]
Zhang, T. , 2017, “ Symplectic Analysis for Wrinkles: A Case Study of Layered Neo-Hookean Structures,” ASME J. Appl. Mech., 84(7), p. 071002. [CrossRef]
Zhong, W. , and Williams, F. W. , 1993, “ Physical Interpretation of the Symplectic Orthogonality of the Eigensolutions of a Hamiltonian or Symplectic Matrix,” Comput. Struct., 49(4), pp. 749–750.
Zhong, W. , and Wei-an, Y. , 1999, “ New Solution System for Plate Bending and Its Application,” Acta Mech. Sini., 31(2), pp. 173–184.
Zhong, W. , 2001, “ Symplectic Energy Band Analysis for Periodical Electromagnetic Waveguide,” J. Comput. Mech., 18(4), pp. 379–387.
Zhang, H. , and Zhong, W. , 2003, “ Hamiltonian Principle Based Stress Singularity Analysis Near Crack Corners of Multi-Material Junctions,” Int. J. Solids Struct., 40(2), pp. 493–510. [CrossRef]
Yao, W. , Zhong, W. , and Lim, C. W. , 2009, Symplectic Elasticity, World Scientific, Singapore.
Peng, H. , Gao, Q. , Wu, Z. , and Zhong, W. , 2012, “ Symplectic Approaches for Solving Two-Point Boundary-Value Problems,” J. Guid. Control. Dyn, 35(2), pp. 653–659. [CrossRef]
Hu, X. , and Yao, W. , 2011, “ A Novel Singular Finite Element on Mixed-Mode Bimaterial Interfacial Cracks With Arbitrary Crack Surface Tractions,” Int. J. Fract., 172(1), pp. 41–52. [CrossRef]
Zhou, Z. , Xu, X. , Leung, A. Y. , and Huang, Y. , 2013, “ Stress Intensity Factors and T-Stress for an Edge Interface Crack by Symplectic Expansion,” Eng. Fract. Mech., 102, pp. 334–347. [CrossRef]
Xu, X. , Ma, Y. , Lim, C. , and Chu, H. , 2006, “ Dynamic Buckling of Cylindrical Shells Subject to an Axial Impact in a Symplectic System,” Int. J. Solids Struct., 43(13), pp. 3905–3919. [CrossRef]
Li, R. , Tian, Y. , Wang, P. , Shi, Y. , and Wang, B. , 2016, “ New Analytic Free Vibration Solutions of Rectangular Thin Plates Resting on Multiple Point Supports,” Int. J. Mech. Sci., 110, pp. 53–61. [CrossRef]
Li, R. , Zheng, X. , Wang, H. , Xiong, S. , Yan, K. , and Li, P. , 2018, “ New Analytic Buckling Solutions of Rectangular Thin Plates With All Edges Free,” Int. J. Mech. Sci., 144, pp. 67–73. [CrossRef]
Wang, B. , Li, P. , and Li, R. , 2016, “ Symplectic Superposition Method for New Analytic Buckling Solutions of Rectangular Thin Plates,” Int. J. Mech. Sci., 119, pp. 432–441. [CrossRef]
Liu, L. , and Bhattacharya, K. , 2009, “ Wave Propagation in a Sandwich Structure,” Int. J. Solids Struct., 46(17), pp. 3290–3300. [CrossRef]
Qiang, G. , Zhong, W. , and Howson, W. , 2004, “ A Precise Method for Solving Wave Propagation Problems in Layered Anisotropic Media,” Wave Motion, 40(3), pp. 191–207. [CrossRef]
Gao, Q. , Lin, J. , Zhong, W. , Howson, W. P. , and Williams, F. W. , 2006, “ A Precise Numerical Method for Rayleigh Waves in a Stratified Half Space,” Int. J. Numer. Methods Eng., 67(6), pp. 771–786. [CrossRef]
Biot, M. A. , 1965, Mechanics of Incremental Deformations, Wiley, New York.
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. [CrossRef]
Cao, Y. , and Hutchinson, J. W. , 2012, “ Wrinkling Phenomena in Neo-Hookean Film/Substrate Bilayers,” ASME J. Appl. Mech., 79(3), p. 031019. [CrossRef]
Zhao, R. , Zhang, T. , Diab, M. , Gao, H. , and Kim, K.-S. , 2015, “ The Primary Bilayer Ruga-Phase Diagram I: Localizations in Ruga Evolution,” Extreme Mech. Lett., 4, pp. 76–82. [CrossRef]
Cai, S. , Breid, D. , Crosby, A. J. , Suo, Z. , and Hutchinson, J. W. , 2011, “ Periodic Patterns and Energy States of Buckled Films on Compliant Substrates,” J. Mech. Phys. Solids, 59(5), pp. 1094–1114. [CrossRef]
Brau, F. , Vandeparre, H. , Sabbah, A. , Poulard, C. , Boudaoud, A. , and Damman, P. , 2011, “ Multiple-Length-Scale Elastic Instability Mimics Parametric Resonance of Nonlinear Oscillators,” Nat. Phys., 7(1), pp. 56–60. [CrossRef]
Sun, J.-Y. , Xia, S. , Moon, M.-W. , Oh, K. H. , and Kim, K.-S. , 2012, “ Folding Wrinkles of a Thin Stiff Layer on a Soft Substrate,” Proc. R. Soc. A, 468(2140), pp. 932–953. [CrossRef]
Zang, J. , Zhao, X. , Cao, Y. , and Hutchinson, J. W. , 2012, “ Localized Ridge Wrinkling of Stiff Films on Compliant Substrates,” J. Mech. Phys. Solids, 60(7), pp. 1265–1279. [CrossRef]
Jin, L. , Takei, A. , and Hutchinson, J. W. , 2015, “ Mechanics of Wrinkle/Ridge Transitions in Thin Film/Substrate Systems,” J. Mech. Phys. Solids, 81, pp. 22–40. [CrossRef]
Jin, L. , Auguste, A. , Hayward, R. C. , and Suo, Z. , 2015, “ Bifurcation Diagrams for the Formation of Wrinkles or Creases in Soft Bilayers,” ASME J. Appl. Mech., 82(6), p. 061008. [CrossRef]
Budday, S. , Kuhl, E. , and Hutchinson, J. W. , 2015, “ Period-Doubling and Period-Tripling in Growing Bilayered Systems,” Philos. Mag., 95(28–30), pp. 3208–3224. [CrossRef]
Fu, Y. , and Cai, Z. , 2015, “ An Asymptotic Analysis of the Period-Doubling Secondary Bifurcation in a Film/Substrate Bilayer,” SIAM J. Appl. Math., 75(6), pp. 2381–2395. [CrossRef]
Zhao, R. , and Zhao, X. , 2017, “ Multimodal Surface Instabilities in Curved Film–Substrate Structures,” ASME J. Appl. Mech., 84(8), p. 081001. [CrossRef]
Li, B. , Jia, F. , Cao, Y. P. , Feng, X. Q. , and Gao, H. , 2011, “ Surface Wrinkling Patterns on a Core-Shell Soft Sphere,” Phys. Rev. Lett., 106(23), p. 234301. [CrossRef] [PubMed]
Cao, Y. P. , Li, B. , and Feng, X. Q. , 2012, “ Surface Wrinkling and Folding of Core–Shell Soft Cylinders,” Soft Matter, 8(2), pp. 556–562. [CrossRef]
Ciarletta, P. , Balbi, V. , and Kuhl, E. , 2014, “ Pattern Selection in Growing Tubular Tissues,” Phys. Rev. Lett., 113(24), p. 248101. [CrossRef] [PubMed]
Stoop, N. , Lagrange, R. , Terwagne, D. , Reis, P. M. , and Dunkel, J. , 2015, “ Curvature-Induced Symmetry Breaking Determines Elastic Surface Patterns,” Nat. Mater., 14(3), pp. 337–342. [CrossRef] [PubMed]
Xu, F. , and Potier-Ferry, M. , 2016, “ On Axisymmetric/Diamond-like Mode Transitions in Axially Compressed Core–Shell Cylinders,” J. Mech. Phys. Solids, 94, pp. 68–87. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Schematics of wrinkles in an elastic neo-Hookean material with exponentially decaying stiffness underneath the top surface (μS/μ=1,μ0/μ=100,h=10l): (a) the reference flat configuration, and the longitudinal and transverse directions are denoted as X1 and X2 respectively, and (b) the deformed wrinkling configuration. Note that the schematics of the wrinkle wavelength are on the deformed configuration, while our theoretical analysis is on the undeformed configuration.

Grahic Jump Location
Fig. 2

Demonstration of solving the critical strain and wavenumber of onset wrinkle through symplectic analysis of an elastic layer with exponential decaying moduli (i.e., λ=100,μ0=100,μs=1,h=50 in the dimensionless form of Eq. (2)): (a) search the critical strain εc for a given wavenumber k to let p2=0 at the top surface, and (b) the minimum strain for all the possible wavenumbers is the critical wrinkle strain εw and the associated dimensionless wavenumber is the corresponding wrinkle wavenumber kw.

Grahic Jump Location
Fig. 3

Schematics of exponential decaying function case: (a) modulus variation along the transverse direction, (b) comparison of longitudinal displacement eigenfunction U1 between symplectic and FE results, (c) comparison of transverse displacement eigenfunction U2 between symplectic and FE results, (d) a FE simulated configuration in abaqus standard solver (scale bar is 25), (e) wrinkle profiles of the perturbed displacement u1 varying with X1 at different depths, and (f) wrinkle profiles of the perturbed displacement u2 at different depths. The color in (d) indicates the maximum principal logarithmic strain.

Grahic Jump Location
Fig. 4

(a) The critical wrinkle strain and (b) the corresponding dimensionless wavenumber for exponentially decaying graded elastic layers with μ0/μs ranging from 10 to 104

Grahic Jump Location
Fig. 5

Schematics of Fermi–Dirac distribution case: (a) modulus varying along the transverse direction, (b) comparison of longitudinal displacement eigenfunction U1 between symplectic and FE results, (c) comparison of transverse displacement eigenfunction U2 between symplectic and FE results, (d) a FE-simulated configuration in abaqus standard solver (scale bar is 20), (e) wrinkle profiles of the perturbed displacement u1 varying with X1 at different depths, and (f) wrinkle profiles of the perturbed displacement u2 at different depths. The color in (d) indicates the maximum principal logarithmic strain.

Grahic Jump Location
Fig. 6

(a) The critical wrinkle strain and (b) the corresponding dimensionless wavenumber for Ferm–Dirac distribution graded elastic layers with μ0/μs ranging from 2 to 104

Tables

Errata

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In