scholarly journals Wrinkling instabilities in soft bilayered systems

2017 ◽  
Vol 375 (2093) ◽  
pp. 20160163 ◽  
Author(s):  
Silvia Budday ◽  
Sebastian Andres ◽  
Bastian Walter ◽  
Paul Steinmann ◽  
Ellen Kuhl
Keyword(s):  
Period Doubling ◽  
Media Theory ◽  
Analytical Models ◽  
Control Parameters ◽  
Computational Simulations ◽  
Smart Surfaces ◽  
Post Buckling ◽  
Surface Patterns ◽  
Secondary Instabilities ◽  
Secondary Bifurcations

Wrinkling phenomena control the surface morphology of many technical and biological systems. While primary wrinkling has been extensively studied, experimentally, analytically and computationally, higher-order instabilities remain insufficiently understood, especially in systems with stiffness contrasts well below 100. Here, we use the model system of an elastomeric bilayer to experimentally characterize primary and secondary wrinkling at moderate stiffness contrasts. We systematically vary the film thickness and substrate prestretch to explore which parameters modulate the emergence of secondary instabilities, including period-doubling, period-tripling and wrinkle-to-fold transitions. Our experiments suggest that period-doubling is the favourable secondary instability mode and that period-tripling can emerge under disturbed boundary conditions. High substrate prestretch can suppress period-doubling and primary wrinkles immediately transform into folds. We combine analytical models with computational simulations to predict the onset of primary wrinkling, the post-buckling behaviour, secondary bifurcations and the wrinkle-to-fold transition. Understanding the mechanisms of pattern selection and identifying the critical control parameters of wrinkling will allow us to fabricate smart surfaces with tunable properties and to control undesired surface patterns like in the asthmatic airway. This article is part of the themed issue ‘Patterning through instabilities in complex media: theory and applications.’

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

2015 ◽  
Vol 471 (2173) ◽  
pp. 20140695 ◽  
Author(s):  
Yan Zhao ◽  
Yanping Cao ◽  
Wei Hong ◽  
M. Khurram Wadee ◽  
Xi-Qiao Feng
Keyword(s):  
Potential Energy ◽  
Critical Strain ◽  
Deformation Behaviour ◽  
Compressive Strain ◽  
Period Doubling ◽  
Careful Examination ◽  
Deformation State ◽  
Quantitative Understanding ◽  
Surface Patterns ◽  
Film Substrate

Compression of a stiff film on a soft substrate may lead to surface wrinkling when the compressive strain reaches a critical value. Further compression may cause a wrinkling–folding transition, and the sinusoidal wrinkling mode can then give way to a period-doubling bifurcation. The onset of the primary bifurcation has been well understood, but a quantitative understanding of the secondary bifurcation remains elusive. Our theoretical analysis of the branching of surface patterns reveals that the wrinkling–folding transition depends on the wrinkling strain and the prestrain in the substrate. A characteristic strain in the substrate is adopted to determine the correlation among the critical strain of the period-doubling mode, the wrinkling strain and the prestrain in an explicit form. A careful examination of the total potential energy of the system reveals that beyond the critical strain of period-doubling, the sinusoidal wrinkling mode has a higher potential energy in comparison with the period-doubling mode. The critical strain of the period-doubling mode strongly depends on the deformation state of the hyperelastic solid, indicating that the nonlinear deformation behaviour of the substrate plays a key role here. The results reported here on the one hand provide a quantitative understanding of the wrinkling–folding transition observed in natural and synthetic material systems and on the other hand pave the way to control the wrinkling mode transition by regulating the strain state in the substrate.

scholarly journals The origin of compression influences geometric instabilities in bilayers

2018 ◽  
Vol 474 (2217) ◽  
pp. 20180267 ◽  
Author(s):  
Sebastian Andres ◽  
Paul Steinmann ◽  
Silvia Budday
Keyword(s):  
Film Growth ◽  
Numerical Study ◽  
Real Life ◽  
Critical Conditions ◽  
Pattern Selection ◽  
Smart Surfaces ◽  
Life Problems ◽  
Wide Range ◽  
Post Buckling ◽  
The Impact

Geometric instabilities in bilayered structures control the surface morphology in a wide range of biological and technical systems. Depending on the application, different mechanisms induce compressive stresses in the bilayer. However, the impact of the chosen origin of compression on the critical conditions, post-buckling evolution and higher-order pattern selection remains insufficiently understood. Here, we conduct a numerical study on a finite-element set-up and systematically vary well-known factors contributing to pattern selection under the four main origins of compression: film growth, substrate shrinkage and whole-domain compression with and without pre-stretch. We find that the origin of compression determines the substrate stretch state at the primary instability point and thus significantly affects the critical buckling conditions. Similarly, it leads to different post-buckling evolutions and secondary instability patterns when the load further increases. Our results emphasize that future phase diagrams of geometric instabilities should incorporate not only the film thickness but also the origin of compression. Thoroughly understanding the influence of the origin of compression on geometric instabilities is crucial to solving real-life problems such as the engineering of smart surfaces or the diagnosis of neuronal disorders, which typically involve temporally or spatially combined origins of compression.

NONLINEAR DYNAMICS OF A BOUNCING BALL DRIVEN BY ELECTRIC FORCES

2003 ◽  
Vol 13 (10) ◽  
pp. 2959-2975 ◽  
Author(s):  
A. KHAYARI ◽  
A. T. PÉREZ
Keyword(s):  
Time Series ◽  
Fractal Structure ◽  
Period Doubling ◽  
Control Parameters ◽  
Conducting Liquid ◽  
Lower Electrode ◽  
Bouncing Ball ◽  
Onset Of Chaos ◽  
Experimental Time Series ◽  
Route To Chaos

This paper is devoted to a theoretical and experimental study of the dynamics of a bouncing ball driven by an electric force. The experimental model consists of a metallic ball immersed in a poorly conducting liquid between two horizontal electrodes. The ball bounces upon the lower electrode as a high voltage is applied between the two plates. The measurement of the time between successive impacts produces a time series, which depends on two control parameters, the amplitude and the frequency of the applied voltage. A theoretical model is proposed, which provides a discrete nonlinear map, and discussed in comparison with the experimental results. It is shown that the system exhibits a period doubling route to chaos and a non-Feigenbaum universal scaling at the onset of chaos. Chaotic motion is investigated using the usual tools: Lyapunov exponents, correlation dimensions and entropies. Fractal structure of the chaotic attractor is also brought to evidence in experimental time series as well as in numerical simulations.

Global Stability Behaviour og Single-Layer Reticulated Domes Created Using a New Nomenclature

2021 ◽  
Author(s):  
Ranjith Kolakkattil ◽  
Arul Jayachandran
Keyword(s):  
Physical Meaning ◽  
Load Capacity ◽  
Single Layer ◽  
Primary Objective ◽  
Surface Pattern ◽  
Uniform Loading ◽  
Post Buckling ◽  
Surface Patterns ◽  
Reticulated Domes ◽  
Effect Of Surface

The primary objective of the paper is to investigate the post buckled behaviour of the single-layered Kite geometry dome developed using a novel crystallographic parameterisation principle. Both triangulated and non-triangulated domes are evolved based on the crystallographic parameterisation principles. It brings in a unique nomenclature for identifying different tessellations in reticulated single-layer dome configurations. This nomenclature brings in a physical meaning to dome tessellations instead of being called by the inventors such as Schwedler dome etc. In this paper, the effect of surface pattern on the load capacity of dome configuration is demonstrated with the comparison of domes having different surface patterns. The comparison of post-buckling behaviour of two different single-layer dome configurations - Kiewitt dome and Kite dome is presented. Despite having rigid nodal joints, the load capacity of the dome is significantly reduced when subjected to unsymmetrical and collateral loads due to the localised effect of these loads and the increased chance of snap-through compared to symmetrical uniform loading acting all over the structure. The Kite geometry have higher performance under uniform gravity loading with a low rise to span ratio.

scholarly journals OSCILLATION AND PERIOD DOUBLING IN TCP/RED SYSTEM: ANALYSIS AND VERIFICATION

2008 ◽  
Vol 18 (05) ◽  
pp. 1459-1475 ◽  
Author(s):  
XI CHEN ◽  
SIU-CHUNG WONG ◽  
CHI K. TSE ◽  
FRANCIS C. M. LAU
Keyword(s):  
Characteristic Frequency ◽  
System Analysis ◽  
Transmission Control Protocol ◽  
Period Doubling ◽  
Oscillatory Behavior ◽  
Analytical Models ◽  
System Characteristic ◽  
Random Events ◽  
Average System ◽  
The Difference

It has been known that a bottleneck RED (Random Early Detection) gateway can become oscillatory when regulating multiple identical TCP (Transmission Control Protocol) flows. However, a systematic explanation of such oscillatory behavior is not available. In this paper, we first use the fluid-flow model to derive the system characteristic frequency, and then compare with the frequencies of the RED queue length waveforms observed from "ns-2" simulations. The "ns-2" simulator is the only viable simulation tool accepted by industry for verification purposes. Analysis of the TCP source frequency distribution reveals the occurrence of period doubling when the system enters the instability region as the filter resolution varies. Since random events and a large number of TCP flows are involved in the process of generating the average system dynamics, a statistical viewpoint is taken in the analysis. Our results reflect the true system behavior as they are based on data from "ns-2" simulations rather than numerical simulations of analytical models. The physical mechanism of oscillation is explained in terms of the difference in the TCP source frequency and the TCP-RED system characteristic frequency.

Nonlinear Analysis of Jacket-Type Offshore Platforms Using Fiber Elements

2006 ◽  
Vol 128 (3) ◽  
pp. 224-232 ◽  
Author(s):  
B. Asgarian ◽  
A. A. Aghakouchak ◽  
R. G. Bea
Keyword(s):  
Lateral Displacement ◽  
Nonlinear Behavior ◽  
Offshore Structures ◽  
Buckling Load ◽  
Hysteretic Behavior ◽  
Analytical Models ◽  
Offshore Platforms ◽  
Nonlinear Program ◽  
Post Buckling ◽  
Strength And Stiffness

A nonlinear fiber element for analysis of jacket type offshore structures is formulated and implemented in the nonlinear program DRAIN-3DX. This element can be used for modeling the nonlinear behavior of both strut and portal members. The element predicts buckling load and post buckling behavior of strut members accurately. It also produces fairly accurate results for yield load and post yield behavior of portal members. This element is verified using the experimental data for individual strut and portal members subjected to cyclic displacements. The element is then used to predict nonlinear behavior of two tested X-braced jackets made of tubular members under cyclic lateral displacement. The results are in good agreement with experiments and the results of other analytical models in terms of frame hysteretic behavior, energy dissipation, buckling load, load-deformation curve, strength and stiffness degradation.

On Bifurcations Inducing Glacial Cycle Lengthening During Pliocene/Pleistocene Epoch

2014 ◽  
Vol 24 (08) ◽  
pp. 1440018 ◽  
Author(s):  
N. N. Ivashchenko ◽  
V. M. Kotlyakov ◽  
D. M. Sonechkin ◽  
N. V. Vakulenko
Keyword(s):  
Dynamical System ◽  
Global Climate ◽  
Period Doubling ◽  
Climate Forcing ◽  
Dynamical System Theory ◽  
Glacial Cycle ◽  
Climatic Response ◽  
Climatic Fluctuations ◽  
Paleoclimatic Records ◽  
Secondary Bifurcations

In the Pliocene (about two–five million years ago) global climate fluctuated with a period corresponding well to the 41-thousand-year cycle of changes in the Earth's axis obliquity. Then, this period disappeared, despite the fact that the obliquity cycle even slightly increased its swing and, therefore, the climatic response to this external climate forcing would have only strengthened. By analyzing paleoclimatic records covering the whole Pliocene and Pleistocene epoch, we show that the climatic response to the obliquity cycle simply became unstable, and therefore unobservable. At the same time, through the period-doubling bifurcation, which is well-known in dynamical system theory, new stable, and so observable, climatic fluctuations have been excited with an approximately doubled period. Further, these fluctuations experienced several secondary bifurcations, and, as a result, their periods increased even more.

VARIETY OF TYPES OF CRITICAL BEHAVIOR AND MULTISTABILITY IN PERIOD-DOUBLING SYSTEMS WITH UNIDIRECTIONAL COUPLING NEAR THE ONSET OF CHAOS

1993 ◽  
Vol 03 (01) ◽  
pp. 139-152 ◽  
Author(s):  
A.P. KUZNETSOV ◽  
S.P. KUZNETSOV ◽  
I.R. SATAEV
Keyword(s):  
Parameter Space ◽  
Critical Behavior ◽  
Period Doubling ◽  
Control Parameters ◽  
Multicritical Point ◽  
Scaling Properties ◽  
Unidirectional Coupling ◽  
Critical Indices ◽  
Onset Of Chaos ◽  
Behavior Types

The dynamics of two unidirectionally coupled period-doubling systems is investigated depending on three relevant parameters (control parameters of subsystems and coupling). There is a hierarchy of critical behavior types. Feigenbaum’s critical surfaces existing in the parameter space are bounded by tricritical lines and intersect along the bicritical line. These lines, in turn, intersect at a new multicritical point BT. Universality and scaling properties for all the critical situations are discussed, and the table of critical indices is given.

The cascade structure of linear instability in collapsible channel flows

2008 ◽  
Vol 600 ◽  
pp. 45-76 ◽  
Author(s):  
X. Y. LUO ◽  
Z. X. CAI ◽  
W. G. LI ◽  
T. J. PEDLEY
Keyword(s):  
Reynolds Number ◽  
Solid Mechanics ◽  
Period Doubling ◽  
Linear Instability ◽  
Analytical Models ◽  
Beam Model ◽  
Neutral Curves ◽  
Wall Stiffness ◽  
Cascade Structure ◽  
Steady Solutions

This paper studies the unsteady behaviour and linear stability of the flow in a collapsible channel using a fluid–beam model. The solid mechanics is analysed in a plane strain configuration, in which the principal stretch is defined with a zero initial strain. Two approaches are employed: unsteady numerical simulations solving the nonlinear fully coupled fluid–structure interaction problem; and the corresponding linearized eigenvalue approach solving the Orr–Sommerfeld equations modified by the beam. The two approaches give good agreement with each other in predicting the frequencies and growth rates of the perturbation modes, close to the neutral curves. For a given Reynolds number in the range of 200–600, a cascade of instabilities is discovered as the wall stiffness (or effective tension) is reduced. Under small perturbation to steady solutions for the same Reynolds number, the system loses stability by passing through a succession of unstable zones, with mode number increasing as the wall stiffness is decreased. It is found that this cascade structure can, in principle, be extended to many modes, depending on the parameters. A puzzling ‘tongue’ shaped stable zone in the wall stiffness–Re space turns out to be the zone sandwiched by the mode-2 and mode-3 instabilities. Self-excited oscillations dominated by modes 2–4 are found near their corresponding neutral curves. These modes can also interact and form period-doubling oscillations. Extensive comparisons of the results with existing analytical models are made, and a physical explanation for the cascade structure is proposed.

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