Towards a quantitative understanding of period-doubling wrinkling patterns occurring in film/substrate bilayer systems

Zhao, Yan; Cao, Yanping; Hong, Wei; Wadee, M. Ahmer; Feng, Xi‐Qiao

doi:10.1098/rspa.2014.0695

Cited by 33 publications

(36 citation statements)

References 33 publications

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“…For the special case when both the layer and substrate are neo-Hookean, fully numerical simulations have been carried out by Cao and Hutchinson [8], and more recently by Zhao et al [17]. Our asymptotic results show excellent agreement with their numerical results in a number of ways.…”

Section: Numerical Resultssupporting

confidence: 71%

“…Although successful at providing important first insight into the period-doubling phenomenon, justification of the various assumptions behind these approximate theories can be very subtle [7,1], and their self-consistency in the asymptotic sense remains unclear. Thus, it is not entirely unexpected that the analysis of Brau et al [4] predicts nonexistence of period-doubling if the substrate is incompressible, which contradicts the results from fully numerical simulations given by Cao and Hutchinson [8] and Zhao et al [17]. In this paper we examine the period-doubling phenomenon based on the exact theory of nonlinear elasticity without making any approximating assumptions about the interfacial conditions or the nonlinear behavior of the bilayer structure.…”

mentioning

confidence: 81%

“…Of particular interest is the period-doubling secondary bifurcation that has been observed experimentally by Pocivavsek et al [15], Brau et al [4], and Sun et al [16], and numerically simulated by Sun et al [16], Cao and Hutchinson [8], and Budday, Kuhl, and Hutchinson [5]. This phenomenon has also been analyzed theoretically based on various approximate theories by Brau et al [4], Zhao et al [17], and Zhuo and Zhang [18]. Although successful at providing important first insight into the period-doubling phenomenon, justification of the various assumptions behind these approximate theories can be very subtle [7,1], and their self-consistency in the asymptotic sense remains unclear.…”

mentioning

confidence: 87%

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An Asymptotic Analysis of the Period-Doubling Secondary Bifurcation in a Film/Substrate Bilayer

Fu¹,

Cai²

2015

SIAM J. Appl. Math.

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Abstract. It has previously been observed experimentally and simulated numerically that when a thin film bonded to a much softer substrate is subjected to a uni-axial compression parallel to the interface, the initial buckled pattern will suffer a secondary bifurcation that doubles the period of the original pattern when the compressive strain reaches a critical value. This perioddoubling phenomenon is analyzed in this paper using an asymptotically self-consistent approach based on the exact theory of nonlinear elasticity. The predicted critical strain based on a fourterm expansion shows good agreement with that obtained using fully numerical simulations, and it is demonstrated that four is the minimum number of terms that should be included in order to give realistic predications. Although our illustrative calculations are conducted for neo-Hookean materials, the proposed approach can deal with any material models and can be extended to higher orders.

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Section: Numerical Resultssupporting

confidence: 71%

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confidence: 81%

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confidence: 87%

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An Asymptotic Analysis of the Period-Doubling Secondary Bifurcation in a Film/Substrate Bilayer

Fu¹,

Cai²

2015

SIAM J. Appl. Math.

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“…While early analytical investigations were restricted to linear elastic materials and primary wrinkling modes [1, 3], more recent studies consider nonlinear elastic material behavior [13, 29] and explain secondary bifurcations in weakly nonlinear confined systems [5, 36]. However, those studies do not consider growing materials and fail to provide profound insight into the wrinkle-to-fold transition and multiple bifurcations in highly nonlinear confined layers [30].…”

Section: Motivationmentioning

confidence: 99%

Period-doubling and period-tripling in growing bilayered systems

Budday

Kuhl

Hutchinson

2015

Philosophical Magazine

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Growing layers on elastic substrates are capable of creating a wide variety of surface morphologies. Moderate growth generates a regular pattern of sinusoidal wrinkles with a homogeneous energy distribution. While the critical conditions for periodic wrinkling have been extensively studied, the rich pattern formation beyond this first instability point remains poorly understood. Here we show that upon continuing growth, the energy progressively localizes and new complex morphologies emerge. Previous studies have often overlooked these secondary bifurcations; they have focused on large stiffness ratios between layer and substrate, where primary instabilities occur early, long before secondary instabilities emerge. We demonstrate that secondary bifurcations are particularly critical in the low stiffness ratio regime, where the critical conditions for primary and secondary instabilities move closer together. Amongst all possible secondary bifurcations, the mode of period-doubling plays a central role - it is energetically favorable over all other modes. Yet, we can numerically suppress period-doubling, by choosing boundary conditions, which favor alternative higher order modes. Our results suggest that in the low stiffness regime, pattern formation is highly sensitive to small imperfections: surface morphologies emerge rapidly, change spontaneously, and quickly become immensely complex. This is a common paradigm in developmental biology. Our results have significantly applications in the morphogenesis of living systems where growth is progressive and stiffness ratios are low.

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“…This intriguing property of the band structure inspired us to utilize the load-insensitive band gap to fabricate the tunable defect mode. However, the periodic-doubling mode may occur when the overall compressive strain reaches a value of approximately 19%[27], which may lead to a significant variation in the band gap[25]. The model used in the present study.…”

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confidence: 99%

Tunable defect mode in a soft wrinkled bilayer system

Zheng

Cao

2016

Extreme Mechanics Letters

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Here we study the defect mode in a wrinkled soft bilayer induced by a local geometrical defect introduced into the system. We show that the band gap of the studied composite system depends on the wrinkling-induced stress pattern and is not sensitive to the compression strain ε in the given loading range. However, the frequency of the defect mode apparently varies with ε . This interesting phenomenon enables us to widely tune the spatial extension of the defect mode from highly localized to completely extended by simply controlling the imposed external load.Our strategy to create a tunable defect mode in a soft layered composite is simple and straightforward and may find such broad applications as the development of advanced soft metamaterials and corresponding devices.

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Towards a quantitative understanding of period-doubling wrinkling patterns occurring in film/substrate bilayer systems

Cited by 33 publications

References 33 publications

An Asymptotic Analysis of the Period-Doubling Secondary Bifurcation in a Film/Substrate Bilayer

An Asymptotic Analysis of the Period-Doubling Secondary Bifurcation in a Film/Substrate Bilayer

Period-doubling and period-tripling in growing bilayered systems

Tunable defect mode in a soft wrinkled bilayer system

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