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Journal of Elasticity
Published in: Journal of Elasticity 1/2020

23-09-2020

Stable Spatially Localized Configurations in a Simple Structure—A Global Symmetry-Breaking Approach

Authors: Shrinidhi S. Pandurangi, Ryan S. Elliott, Timothy J. Healey, Nicolas Triantafyllidis

Published in: Journal of Elasticity | Issue 1/2020

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Abstract

We revisit the classic stability problem of the buckling of an inextensible, axially compressed beam on a nonlinear elastic foundation with a semi-analytical approach to understand how spatially localized deformation solutions emerge in many applications in mechanics. Instead of a numerical search for such solutions using arbitrary imperfections, we propose a systematic search using branch-following and bifurcation techniques along with group-theoretic methods to find all the bifurcated solution orbits (primary, secondary, etc.) of the system and to examine their stability and hence their observability. Unlike previously proposed methods that use multi-scale perturbation techniques near the critical load, we show that to obtain a spatially localized deformation equilibrium path for the perfect structure, one has to consider the secondary bifurcating path with the longest wavelength and follow it far away from the critical load. The novel use of group-theoretic methods here illustrates a general methodology for the systematic analysis of structures with a high degree of symmetry.
previous article Higher-Order Peridynamic Material Correspondence Models for Elasticity

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Appendix
Available only for authorised users
Footnotes
1
The reader should note that we employ a linear beam model for its simplicity, both analytically and numerically, in this work. Thus, undue weight should not be given to the physical interpretation of fine-scale details associated with the highly-deformed solutions obtained here. Instead, the primary focus here is an analysis of the simplest model leading to the existence of such spatially localized solutions. As we show, all of the important phenomena (primary bifurcation, cascades of secondary bifurcations, etc.) occur well within the small-displacement regime. Thus, the linear beam model poses no limitation on our ability to determine the correct mechanics of the nonlinear beam–foundation system of interest.
 
2
Such as those illustrated in Fig. 8(a).
 
3
Assuming adequate continuity, the fourth-order Euler–Lagrange equation in \([-L/2, L/2]\) requires four boundary conditions to be satisfied.
 
4
In fact, for systems with Lie symmetry groups (continuous infinite groups), such as the one considered here, the continuous orbits of solutions are known as relative equilibria. In general, relative equilibria of the system’s equations of motion can correspond to an orbit of equilibria, traveling or rotating waves, dynamic trajectories that appear time-periodic in a suitable moving frame, or other more complicated motions. For a discussion of the theory of relative equilibria and their stability see [13]. In this work, we explore only orbits of equilibria, and so the distinction between a relative equilibrium of a system with continuous symmetry and an orbit of equilibria of a system with discrete symmetry is immaterial.
 
5
In Sect. 4.1.1 it is shown that \(L_{c} = 2\pi \) for the perfect beam–foundation system.
 
6
Indeed, with (as below) \(q=20\), the primary bifurcation branch will have period \(L_{c}\). Then each secondary bifurcating branch may be associated with one of the periods \(L_{m} = m L_{c}\) with \(m \in \{1,2,\dots ,q\}\).
 
7
Actually, its symmetry is a subgroup of \(D_{\infty h}\) that is conjugate to \(D_{4d}\). Here, we will not be concerned with this distinction.
 
8
As mentioned above, in the problem at hand, we find continuous orbits of equilibria, all having the same energy. This implies the existence of a zero eigenvalue of the stability operator \(\mathcal{E}_{,ww}\) in Eq. (2.9). Thus, all equilibria are, at best, neutrally stable. Accordingly, in this work we ignore the zero eigenvalue associated with the solution orbit and require that all other eigenvalues be positive for stability.
 
9
This is adequate for correct integration of the higher-order gradient term in the stiffness matrix.
 
10
As described in Sect. 2.3, the primary bifurcation orbit consists of an infinite set of configurations generated by the symmetries of \(D_{\infty h}\); here we select one specific representative of the orbit.
 
11
The higher-order derivatives that must vanish by symmetry are thus only approximately satisfied.
 
12
Note, as written it is necessary to take \(\delta w_{s0} := 0\).
 
13
In fact, even the employed parameterization is problematic, since it is restricted to \(\xi \geq 0\), and is therefore unable to distinguish between the two “halves” of the bifurcated path.
 
14
There are also simple bifurcations of symmetry \(D_{1}\) which occur at loads \(\lambda _{cia}\), but \(\lambda _{cia} > \lambda _{cis}\) (see Appendix A.4). Here we are interested in the first (i.e. lowest load) bifurcation point, so we do not present the imperfect bifurcations with \(D_{1}\) symmetry.
 
15
From Appendix A.4, it is clear that \(2(1+\zeta )^{1/2} < \lambda _{cis} < 2\), for \(\zeta < 0\).
 
16
See Footnote 8.
 
17
Compare Eq. (A.6) to Eq. (4.6). In Eq. (4.6), no assumptions about symmetry are made, and one must ensure that the entire operator is non-singular. In Eq. (A.6), equivariance is assumed and this suffices to ensure that \(\mathcal{E}^{b}_{,w\lambda } {\stackrel{i}{w}} = 0\;,\ i=1,2,\dots ,n_{ \mu }\) and that the operator \(\mathcal{E}^{0}_{,ww}\) is a scalar multiple of the identity. Thus, the two criteria are equivalent when the assumptions of equivariant bifurcation theory are satisfied.
 
Literature
1.
Allgower, E.L., Georg, K.: Introduction to Numerical Continuation Methods. Classics in Applied Mathematics, vol. 45. SIAM, Philadelphia (2003) MATH
2.
Amazigo, J.C., Budiansky, B., Carrier, G.F.: Asymptotic analyses of the buckling of imperfect columns on nonlinear elastic foundations. Int. J. Solids Struct. 6(10), 1341–1356 (1970) MATH
3.
Audoly, B.: Localized buckling of a floating elastica. Phys. Rev. E 84, 011605 (2011) ADS
4.
Auguste, A., Jin, L., Hayward, R.C., Suo, Z.: Post-wrinkle bifurcations in elastic bilayers with modest contrast in modulus. Extreme Mech. Lett. 11, 30–36 (2017)
5.
Beardmore, R., Peletier, M., Budd, C., Wadee, M.: Bifurcations of periodic solutions satisfying the zero-Hamiltonian constraint in reversible differential equations. SIAM J. Math. Anal. 36(5), 1461–1488 (2005) MathSciNetMATH
6.
Biot, M.A.: Surface instability of rubber in compression. Appl. Sci. Res., Sect. A 12, 168–182 (1963) MATH
7.
Budd, C., Hunt, G., Kuske, R.: Asymptotics of cellular buckling close to Maxwell load. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 457(2016), 2935–2964 (2001) ADSMathSciNetMATH
8.
Cao, Y., Hutchinson, J.W.: From wrinkles to creases in elastomers: the instability and imperfection-sensitivity of wrinkling. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 468(2137), 94–115 (2012) ADSMathSciNetMATH
9.
Champneys, A.R., Toland, J.: Bifurcation of a plethora of multi-modal homoclinic orbits for autonomous Hamiltonian systems. Nonlinearity 6(5), 665–721 (1993) ADSMathSciNetMATH
10.
Chen, D., Cai, S., Suo, Z., Hayward, R.C.: Surface energy as a barrier to creasing of elastomer films: an elastic analogy to classical nucleation. Phys. Rev. Lett. 109, 038001 (2012) ADS
11.
Chen, D., Jin, L., Suo, Z., Hayward, R.C.: Controlled formation and disappearance of creases. Mater. Horiz. 1(2), 207–213 (2014)
12.
Chen, Y.C., Yang, S., Wheeler, L.: Surface instability of elastic half-spaces by using the energy method. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 474(2213), 20170854 (2018). ADSMathSciNetMATH
13.
Chossat, P., Lauterbach, R.: Methods in Equivariant Bifurcations and Dynamical Systems. Advanced Series in Nonlinear Dynamics, vol. 15. World Scientific, Singapore (2000) MATH
14.
Ciarletta, P.: Matched asymptotic solution for crease nucleation in soft solids. Nat. Commun. 9(496), 1–7 (2018)
15.
Coman, C.D.: Inhomogeneities and localised buckling patterns. IMA J. Appl. Math. 71(1), 133–152 (2006) MathSciNetMATH
16.
Coman, C.D.: Localized elastic buckling: non-linearities versus inhomogeneities. IMA J. Appl. Math. 75(3), 461–474 (2010) MathSciNetMATH
17.
Diab, M., Kim, K.S.: Ruga-formation instabilities of a graded stiffness boundary layer in a neo-Hookean solid. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 470(2168), 20140218 (2014) ADS
18.
Diamant, H., Witten, T.A.: Compression induced folding of a sheet: an integrable system. Phys. Rev. Lett. 107, 164302 (2011) ADS
19.
Everall, P.R., Hunt, G.: Mode jumping in the buckling of struts and plates: a comparative study. Int. J. Non-Linear Mech. 35(6), 1067–1079 (2000) MATH
20.
21.
Gent, A.N., Cho, I.S.: Surface instabilities in compressed or bent rubber blocks. Rubber Chem. Technol. 72(2), 253–262 (1999)
22.
Golublitsky, M., Stewart, I., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory. Applied Mathematical Sciences, vol. 2. Springer, Berlin (1988)
23.
Healey, T.J.: A group-theoretic approach to computational bifurcation problems with symmetry. Comput. Methods Appl. Mech. Eng. 67(3), 257–295 (1988) ADSMathSciNetMATH
24.
Hohlfeld, E., Mahadevan, L.: Unfolding the sulcus. Phys. Rev. Lett. 106, 105702 (2011) ADS
25.
Hohlfeld, E., Mahadevan, L.: Scale and nature of sulcification patterns. Phys. Rev. Lett. 109, 025701 (2012) ADS
26.
Hong, W., Zhao, X., Suo, Z.: Formation of creases on the surfaces of elastomers and gels. Appl. Phys. Lett. 95, 111901 (2009) ADS
27.
Hunt, G., Everall, P.R.: Arnold tongues and mode-jumping in the supercritical post-buckling of an archetypal elastic structure. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 455(1981), 125–140 (1999) ADSMathSciNetMATH
28.
Hunt, G.W., Wadee, M.K.: Comparative Lagrangian formulations for localized buckling. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 434(1892), 485–502 (1991) ADSMathSciNetMATH
29.
Hunt, G.W., Bolt, H.M., Thompson, J.M.T.: Structural localization phenomena and dynamical phase-space analogy. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 425(1869), 245–267 (1989) ADSMathSciNetMATH
30.
Hunt, G.W., Wadee, M.K., Shiacolas, N.: Localized elasticae for the strut on the linear foundation. J. Appl. Mech. 60(4), 1033–1038 (1993) ADSMATH
31.
Hunt, G., Peletier, M.A., Champneys, A.R., Woods, P.D., Wadee, M.A., Budd, C.J., Lord, G.J.: Cellular buckling in long structures. Nonlinear Dyn. 21(1), 3–29 (2000) MathSciNetMATH
32.
Ikeda, K., Murota, K.: Imperfect Bifurcation in Structures and Materials: Engineering Use of Group-Theoretic Bifurcation Theory, 2nd edn. Applied Mathematical Sciences, vol. 149. Springer, Berlin (2010) MATH
33.
Iooss, G., Joseph, D.D.: Elementary Stability and Bifurcation Theory, 2nd edn. Springer, Berlin (1990) MATH
34.
Jin, L., Chen, D., Hayward, R.C., Suo, Z.: Creases on the interface between two soft materials. Soft Matter 10(2), 303–311 (2014) ADS
35.
Jin, L., Auguste, A., Hayward, R.C., Suo, Z.: Bifurcation diagrams for the formation of wrinkles or creases in soft bilayers. J. Appl. Mech. 82(6), 061008 (2015) ADS
36.
Keller, H.B.: Numerical Methods in Bifurcation Problems. Tata Institute of Fundamental Research/Springer, Bombay/New York (1987)
37.
Li, B., Cao, Y.P., Feng, X.Q., Gao, H.: Mechanics of morphological instabilities and surface wrinkling in soft materials: a review. Soft Matter 8(21), 5728–5745 (2012) ADS
38.
Luongo, A.: On the amplitude modulation and localization phenomena in interactive buckling problems. Int. J. Solids Struct. 27(15), 1943–1954 (1991) MATH
39.
40.
McWeeny, R.: Symmetry: An Introduction to Group Theory and Its Applications. Dover, New York (2002) MATH
41.
Peletier, M.: Sequential buckling: a variational analysis. SIAM J. Math. Anal. 32(5), 1142–1168 (2001) MathSciNetMATH
42.
Pocivavsek, L., Dellsy, R., Kern, A., Johnson, S., Lin, B., Lee, K.Y.C., Cerda, E.: Stress and fold localization in thin elastic membranes. Science 320(5878), 912–916 (2008) ADS
43.
Potier-Ferry, M.: Amplitude Modulation, Phase Modulation and Localization of Buckling Patterns. Cambridge University Press, Cambridge (1983)
44.
Potier-Ferry, M.: Foundations of Elastic Postbuckling Theory, vol. 288. Springer, Berlin (1987) MATH
45.
Rivetti, M.: Non-symmetric localized fold of a floating sheet. C. R., Méc. 341(3), 333–338 (2013) ADS
46.
Silling, S.A.: Creasing singularities in compressible elastic materials. J. Appl. Mech. 58(1), 70–74 (1991) ADS
47.
48.
Wadee, M.A.: Effects of periodic and localized imperfections on struts on nonlinear foundations and compression sandwich panels. Int. J. Solids Struct. 37(8), 1191–1209 (2000) MATH
49.
Wadee, M.K., Bassom, A.P.: Effects of exponentially small terms in the perturbation approach to localized buckling. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 455(1986), 2351–2370 (1999) ADSMathSciNetMATH
50.
Wadee, M.K., Bassom, A.P.: Restabilization in structures susceptible to localized buckling: an approximate method for the extended post-buckling regime. J. Eng. Math. 38(1), 77–90 (2000) MATH
51.
Wadee, M.K., Hunt, G.W., Whiting, A.I.M.: Asymptotic and Rayleigh-Ritz routes to localized buckling solutions in an elastic instability problem. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 453(1965), 2085–2107 (1997) ADSMathSciNetMATH
52.
Wadee, M.K., Coman, C.D., Bassom, A.P.: Solitary wave interaction phenomena in a strut buckling model incorporating restabilisation. Physica D 163(1–2), 26–48 (2002) ADSMathSciNetMATH
53.
Wang, Q., Zhao, X.: A three-dimensional phase diagram of growth-induced surface instabilities. Sci. Rep. 5(8887), 1–10 (2015)
54.
Woods, P., Champneys, A.: Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian–Hopf bifurcation. Phys. D: Nonlinear Phenom. 129(3), 147–170 (1999) ADSMathSciNetMATH
55.
Zhao, R., Zhang, T., Diab, M., Gao, H., Kim, K.-S.: The primary bilayer ruga-phase diagram I: localizations in ruga evolution. Extreme Mech. Lett. 4, 76–82 (2015)
Metadata
Title
Stable Spatially Localized Configurations in a Simple Structure—A Global Symmetry-Breaking Approach
Authors
Shrinidhi S. Pandurangi
Ryan S. Elliott
Timothy J. Healey
Nicolas Triantafyllidis
Publication date
23-09-2020
Publisher
Springer Netherlands
Published in
Journal of Elasticity / Issue 1/2020
Print ISSN: 0374-3535
Electronic ISSN: 1573-2681
DOI
https://doi.org/10.1007/s10659-020-09794-5

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