PROPOSAL ABSTRACT:

Name of Principal Investigator: ENRIQUE CERDA

Proposal Title:FOLDING & CREASING: LOCALIZATION PHENOMENA IN LOW

DIMENSIONAL SYSTEMS

Describe the main issues to be addressed: goals, methodology and expected results. The maximum length for this sectionis 1 page (Verdana font size 10, letter size is suggested).

A filament or membrane embedded or attached to a fluid or elastic substrate show a rich new set of behaviors undercompression. Although buckling of these systems have been studied since a long time, its important role in cell mechanics,morphogenesis, morphology, flexible electronics, or nanomechanics, between others, has stimulated a resurge of interest inthe problem. Wrinkling and localization phenomena as creasing and folding (far from the threshold for buckling) have beenreported in di↵erent systems and conditions showing that few elements are neccesary to have these structures to emerge.We hypothesize that these patterns are more a geometrical artifact than anything related to material properties and willimplement an experimental, numerical and theoretical approach to study them.

The proposal aims to find a unified description based on the concept of mismatch, Em/Es, the ratio of the modulus ofthe filament or membrane to the modulus of the substrate. Large mismatch is related to the wrinkling to fold transitionwhile a mismatch close to one, Em/Es ⇠ 1, leads to creasing. We will experimentally study the wrinkling to fold transitionin floated membranes induced by indentation and systematically study the interaction between defects or indentation pointsby using a novel setup. To study elastic substrates, we will carry out numerical simulations in ABAQUS to test di↵erentconstitutive material models and perform systematic numerical analysis to understand the importance of di↵erent materialparameters as mismatch. Numerical work will also guide us to find the main elements that are necessary to include in asimplified theoretical model that captures the di↵erent instabilities.

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PROPOSED RESEARCH

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PROPOSAL DESCRIPTION

INTRODUCTION

The interest in buckling behavior of thin filaments and membranes under compression has been driven by a wide rangeof natural systems: the compression of intracellular microtubules [1–3], the compression of lung surfactants [4–9], drying-induced buckling of pollen grains [10], crumpling of paper [11, 12], wrinkles on skin [13, 14], and the internal morphologyof a host of biological tissues including arteries [15, 16], intestines [17, 18], and brain [19–22]. In most of these cases, thefilament or membrane undergoing compression is embedded or attached to an elastic substrate. Over the last two decades,crumpling [11,12], wrinkling [23–52]/folding [6–9,53–59], creasing [19,60–63], and other surface instabilities such as herringbonepatterns in biaxially compressed supported membranes [14, 39, 47], edge instabilities [41, 45], capillary-tensile wrinkling [28,31, 41, 43, 44], etc. have received immense attention in the literature, often within the context of highly idealized models,unified under the umbrella of stress focusing instabilities [11, 12]. This proposal aims to find a unified description based onthe concept of “mismatch”, E

m

/Es

, the ratio of the modulus of the filament or membrane to the modulus of the substrate.Creasing, wrinkling and folding are competing instabilities that emerge depending on mismatch and compressive strainapplied (see Fig. 1). Physical models have explored these instabilities in the range of E

m

/Es

⇠ 1 � 102, however, thevariation of mismatch in nature is even larger. The outer and inner layer of gray and white matter in the brain have amismatch E

m

/Es

⇠ 0.7 [22]; the endothelial and epithelial lining of arteries and veins have a mismatch Em

/Es

⇠ 102 [9];our skin resting on top of a soft subcutaneous layer have a mismatch of order E

m

/Es

⇠ 103 � 104 depending on bodyregion [64, 65]; and mismatch has the astonishing large value of E

m

/Es

⇠ 108 for cytoskeletal biopolymers in living cells [1].

B.

A.

Figure 1: A and B show histology slides of human arterial cross-sections. A. shows collapse via a creasing instability while B.shows nearly uniform folding of the internal boundary.

The choice of mismatch as the important parameter totrack is not arbitrary. A large mismatch implies that thedescription for the filament or membrane can be made as aseparate entity from the substrate, the interaction with thesubstrate being modelled as an e↵ective external force ap-plied over them. Classical rod and plate theory [66, 67] arethen commonly used to describe the sti↵er structures withimportant consequences: the e↵ective parameter taking intoaccount the flexibility of the filament or membrane is a com-bination of thickness and Young modulus. A cylindrical rodof radius R has a bending sti↵ness E

m

I, where I = R4/4is the moment of inertia. Similarly, the bending sti↵ness fora membrane of thickness h is defined by the combinationB = E

m

t3/12(1 � ⌫2m

), where t is the thickness and ⌫m

isthe Poisson ratio of the membrane. Dimensional analysis im-plies that the only possible length scale is � ⇠ (E

m

I/Es

)1/4

for a filament and � ⇠ (B/Es

)1/3 for a membrane. Theselength scales give the wavelength of wrinkling that has beenexperimentally measured [1, 25] and theoretically computedwith precise methods. Two formulaes are extensively citedin the literature for the wavelength �

c

and critical strain ✏c

for wrinkling of a membrane under plane-strain conditions

�c

=2⇡

31/3t(E

m

/Es

)1/3

✏c

=32/3

4(E

s

/Em

)2/3 (1)

where Em

= Em

/(1�⌫2m

) and Es

= Es

/(1�⌫2s

). Here ⌫m

and ⌫s

are the Poisson ratio of membrane and substrate respectively.These relations are found throughout the literature since Bowden et al’s work [25] and, certainly, can be considered as almost“classical”. However, their determination is based in the wrong assumption of zero shear stress at the interface betweenmembrane and substrate. The same boundary condition has been applied in more sophisticated calculations for the buckling

1

analysis in recent years [26–30] and intensively cited in di↵erent applications [34, 35]. Although the di↵erence between usingthe wrong or right boundary condition is just a numerical prefactor in front of relations (1) it shows the weakness of theapproach: the buckling calculation is already an approximation that is based on assuming the membrane on top to followthe equations of a plate. Obviously this can only be right if the mismatch is large and the thickness of the membrane smallerthan any other length scale in the problem (for example the depth of the substrate or the wavelength of the wrinkles). Thus,the assumption of plate theory is not a systematic approximation and can be only considered as giving the leading term whenE

m

/Es

! 1. This condition is not clearly noted in the literature and shows why there is certain freedom for not applyingits physical requirements, and one of them is shear stress must not be neglected at the interface. It also opens the question offor which values of the mismatch plate theory gives a good approximation for the buckling analysis. Certainly this is not truefor the case E

m

= Es

when the problem is transformed into a di↵erent one since a physical membrane cannot be recognized.

A.

B.

C.

I.

II.A.

B.

C.

III.A.

B.

C.

Figure 2: Wrinkle-to-fold instability I. on waterin a 10µm polyester film (wavelength � ⇠ O(cm),II. in a 100µm latex sheet bonded to a soft rub-ber (� ⇠ O(mm)), and III. in a 15nm gold nano-particle tri-layer at an air/water interface (� ⇠O(µm). These transitions appear very similar de-spite the the wide variation in film thickness andsubstrate sti↵ness

Of central interest to the proposal is to explore the infinite mismatchlimit E

m

/Es

⇠ 1. We argue that in that limit the substrate can beconsidered as a fluid with no resistance to shear although there is a re-maining vertical e↵ective sti↵ness because of gravity. This model wasfirst proposed in [8] and has motivated further work with elastic sub-strates [53, 54]. When the membrane rest on top of a fluid the wrinkle-to-fold (W-t-F) transition is observed when small amplitude, uniformlydistributed, wrinkles transform into large amplitude, highly localizedfolds [8, 9, 53–56, 58]. Two canonical cases, both in uniaxial compres-sion, are illustrated in Fig. 2. Figure 2 I. shows an elastic membranefloating on a liquid surface [8, 9, 55, 56]. Small uniform wrinkles developat low levels of compression. With increasing compression, most of thewrinkles relax whereas a single one grows into a fold with high local cur-vature. The nature of this transition resembles a second order transition(the relevant order parameter, a ratio of amplitudes of the fold to thewrinkles, diverges continuously) [8]. Figure 2 II. shows similar transitionin a membrane attached to a soft elastic solid [8, 53, 54]. Although bothtwo phenomena are examples of strain focusing and appear visually sim-ilar, the underlying physics di↵ers in crucial ways to be discussed later.The W-t-F instability of liquid-supported films in two dimensional sys-tems is now well-understood, in particular, Diamant and Witten haveprovided an exact analytical solution that describes the transition inquantitative detail [54–56]. However, there are still many unansweredquestions for cylindrical geometry where the W-t-F transition has beenreported [59] and there is not a clear theoretical model as the one pro-posed in Refs. [8, 55, 56].

Similarly, the W-t-F transition of membranes on soft solid substrates(finite mismatch) is still poorly-understood [8, 53, 54, 58]. Some exper-iments have reported strain focusing occurs via period-doubling transi-tions [53, 54, 58]. Our own experiments (Fig. 2 II) do not show suchregularity. The e↵ect of the mismatch remains uncertain. We predictedpreviously in [8, 9] that localization should be observed in the limit oflarge mismatch to connect with what we observe when the substrate isa fluid. However, this prediction has been elusive because most physi-cal models are in the range E

m

/Es

⇠ 1 � 102. Recently, Refs. [62, 63]presented a numerical work for mismatch close to E

m

/Es

⇠ 103 show-ing signs of localization in the sense of the W-t-F transition. The samebehavior has been observed in our own numerical simulations so that we

aim to develop a theoretical model to understand the conditions for localization for elastic substrates in the large mismatchregime .

The analysis in the opposite limit of low mismatch Em

/Es

! 1 o↵ers another important theoretical motivation. A di↵erentsurface instability, first studied by M. Biot in his book “Mechanics of Incremental Deformations” [60] and named “creasing”or “Biot instability”, happens when the sti↵ness of the membrane equals the sti↵ness of the substrate. He showed that asurface of an elastic slab under compression is unstable for all wavelengths for strains larger than 10%. A breakthrough forthis problem was given by Hohlfeld and Mahadevan [19] where they carefully studied numerically the instability in the limitof vanishing bending sti↵ness of an upper “thin membrane” that added bending energy to the system. Their analysis showedthat the instability is extremely sensitive to imperfections at the free surface and the system is regularized by the presenceof a membrane. However, the highly nonlinear nature of the instability makes harder its study. The work of Holfeld andMahadevan and others that have followed [62] show that material and geometric nonlinearities seem important. They use aneo-hookean or hyperelastic material model and assume incompressibility. Hence, the numerical and theoretical analysis have

2

elements related to the specific nature of the model used.There are now numerous recent works that relate the instability to the development of form and function during morpho-

genesis and have implemented heavy numerical simulations to study the process under di↵erent geometrical configurations.The large strain threshold for the creasing instability requires the use of all the machinery of finite elasticity to implementhighly complex mathematical tools and use energy methods (Arruda-Boyce, Mooney-Rivlin, Neo Hooke, Ogden, etc.) validfor large strains. This has turned the current e↵orts to understand the creasing instability in a highly technical subject withlittle room for physical intuition. Fortunately, the same system but coated with a sti↵er membrane will have a lower criticalstrain and the standard methods of linear elasticity will be plainly justified for some lower bound of E

m

/Es

. It is importantto define what we understand for linear elasticity: stress-strain relations are linear but the equations of elasticity could behighly nonlinear because of rotation e↵ects. In other words, we hypothesize that material nonlinearities are not necessaryto capture the instability. A simple examination of Biot’s calculation in Ref. [60] shows that the instability is driven byrotation nonlinearities but the exact number of the critical strain is corrected by material nonlinearities and area changesdue to Poisson e↵ects. We believe that studying the problem in the linear elasticity framework and approaching the Biotinstability from the limit E

m

/Es

! 1 not only regularized the problem, but also helps to understand the creasing instabilitywithout using material nonlinearities. Thus, the instability is more a geometrical artifact than anything related to materialproperties. An important insight in this direction is what we have obtained in our preliminary work: linear elasticity of theequations for zero Poisson ratio gives a critical threshold curve that corresponds to our numerical simulations for low andlarge mismatch (see Fig. 8). Thus incompressibility and material nonlinearities are not a key factor and the system can havea creasing instability even when it is constrained to deform in the plane x� y. Having a system with a simple linear stabilityanalysis increase our chances to include nonlinearities in a simplified model.

In summary, we propose a theoretical, numerical, and experimental study focus on the analysis of localization problemsfor di↵erent values of mismatch. They are detailed as follows:

Problem 1: Wrinkling to Fold Transition in Floated Membranes

Over several decades, theoretical approaches have been developed to treat particular cases of wrinkling [23–36,38–41,43–52]. InRef. [9] we integrated wrinkling and folding as part of a more general surface instability: the W-t-F transition, unifying diversesystems based purely on geometry and linear elasticity. Similar transition has been observed in other systems [53–56, 58, 59]Figure 3 illustrates the geometry of the specific problem in the simplest configuration, rectilinear compression of a membranesupported on a liquid. An end-displacement � is applied to impose a nominal strain ✏ = �/L, where L is membrane length.

A0

A1

Liquid

Membrane

Figure 3: Geometry of wrinkling and folding: membrane isshown in blue, substrate (liquid) occupies lower half-plane.Here � is wrinkle wavelength, L is the length of the membrane,v is the vertical displacement of the membrane, and � the hor-izontal wall displacement. In the folded membrane we definethe relevant amplitude parameters, where by convention we la-bel the fold amplitude as A0 and decaying wrinkling amplitudeas A1.

Assuming the floating membrane as an inextensible thinsheet (i.e. contour length remains L) and considering only itsbending energy (U

b

) while defining a general potential for thesubstrate (U

s

) to include gravity e↵ects, we defined the totalenergy to be U = U

b

+ Us

. Specifically, Ub

= (B/2)RdL2

and Us

= (K/2)RdL v2 where is the curvature of the

surface, v is the vertical displacement, and K = ⇢g. Weimplicitly found in our modelization the perfect “Winklermodel” [23] in the sense of having an e↵ective springinessthat acts only against vertical displacement of the membrane.

For liquid substrates, the above formalism predicted thatuniform wrinkles of wavelength appeared at small strain.At larger displacements, a localized fold appeared. Themodel correctly captured the entire wrinkle to fold insta-bility, i.e. the entire sequence of initial growth of A1, andsubsequent growth of the order parameter A0/A1. Its nu-merical solution correctly predicted the amplitude scaling inboth regimes. Importantly, it identified the wrinkle wave-length � as the key length scale for the W-t-F transition:specifically, amplitude data from a wide range of experi-ments superposed when plotted as a function of d = ✏⇥L/�.Furthermore, subsequent theoretical work by Witten andDiamant elegantly showed that the Euler-Lagrange equa-tion derived from energy minimization was a higher ordermember of the stationary-sine-Gordon-modified-Korteweg-de Vries hierarchy (sG-mKdV) [54–56]. This is why theIntroduction above stated that “W-t-F instability of liquid-

supported films is now well-understood, at least for rectilinear compression”.When the geometry is circular a similar scenario is observed when a force is applied to the center of a floated circular

membrane. The out of plane displacement of the indentor generates a compressive hoop stress leading to wrinkling of themembrane. In our current Fondecyt proyect we are studying, theoretically and experimentally, the wrinkling instability

3

for the circular configuration. Specifically, the transition between wrinkling patterns in the near threshold (NT) and farfrom threshold (FT) regimes [40, 41]. However, winkled patterns collapse into folds for large indentation depth. The W-t-Ftransition in circular geometry was first observed by Holmes and Crosby [59] but there is not a well establish model thatcaptures the transition as in the case of rectilinear compression. Additionally, more detailed experiments are needed to trackthe W-t-F transition in circular geometry. We plan to use our experimental setup to study the transition in circular geometry.

Figure 4: Figure extracted fromRef. [37] showing the induction of theW-t-F transition when two defects in-teract.

We also aim to explore other configurations with our experimental system. In-dentation of a thin floating membrane can be applied in di↵erent directions and withtwo or more indentors to know more about winkling in the NT and FT regimes, andW-t-F transitions. In [37], the authors show that two water drops resting on top ofa membrane and approaching makes the wrinkles around them to collapse into folds(See Fig. 4). Our own experiments show that a defect in a membrane induces theW-t-F transition. We plan to study how two point forces interact in a membrane bycontrolling the distance, indentation depth, and membrane parameters.

Problem 2: Wrinkling to Fold Transition for Large Mismatch

We now turn from liquid substrates to soft solid substrates. While the qualitativesimilarity between the W-t-F instability on liquid and elastic substrates is clear, thedetails are far more complex. Wrinkling on soft elastic substrates can be studied inthe limit of small deformations and linear elasticity. This is almost a classical problemin elasticity and has been studied during di↵erent periods of time and for di↵erentreasons. The first works on wrinkling were focus in engineering applications of sand-wich panels mainly focus on the aeronautical industry. The book of Howard Allen(1969) is frequently cited to account for these early e↵orts to understand wrinkling.While the engineering approach was more interested in the e↵ective mechanical prop-erties of composite materials, new interest came from the development of nanofilmsand thin coatings to understand their natural tendence to wrinkle. The seminal workof Bowden et al. (1998) showed how wrinkles can naturally appear in small scalestructures.

The basic configuration for our numerical and theoretical work is similar to Fig. 3,but here we impose a nominal strain ✏ = �/L to both membrane and substrate (seeFig. 5). As we showed in the introduction, dimensional analysis explains wrinklingfor the deep limit case as � ⇠ t(E

m

/Es

)1/3, for the case of a membrane of thicknesst, and � ⇠ R(E

m

/Es

)1/4, for the case of a filament of radius R. The prefactor canthen be computed by using the right matching conditions at the interface. It can be found in Allen’s book the followingequilibrium condition is obtained at the interface

�s

xy

= Em

trx

"mxx

(2)

where �s

xy

is the shear of the substrate at the interphase and "mxx

is the strain of the membrane. Neglecting the couplingbetween shear forces at the interphase and compressive axial forces over the membrane would mean that �s

xy

= 0. However, wecan estimate �s

xy

⇠ Em

ry

u ⇠ Em

ku where k is the wavenumber of the wrinkle and u the horizontal displacement. Similarly,the strain in the membrane is "m

xx

⇠ ku. We conclude that the ratio between the two terms is Em

trx

"mxx

/�s

xy

⇠ (Em

/Es

)tk ⇠(E

m

/Es

)2/3 � 1 for large mismatch since k�1 ⇠ �. Thus, the right hand side term of (2) is more relevant and the rightboundary condition for large mismatch is r

x

"mxx

= 0 at the interfase. Applying the correct boundary condition gives

�c

= 2⇡

✓3� 4⌫

s

12(1� ⌫s

)2

◆1/3

t(Em

/Es

)1/3

✏c

=

✓3(1� ⌫

s

)2

2(3� 4⌫s

)

◆2/3

(Es

/Em

)2/3 (3)

The classical relation in (1) reproduces the correct prefactor only for incompressible materials (⌫s

= 1/2). It could be arguethat the di↵erence in the prefactor is negligible but it is easily detectable in nowadays numerical simulations. One importantaim of the grant will be to incorporate finite element analysis (FEA) as a tool in our laboratory. The commercial packageABAQUS will be used (current license belonging to one of our collaborators, Luka Pocivavsek). Preliminary simulationspresented throughout this grant show the capability of FEA to accurately capture period doubling structure and creasinginstabilities at the correct strains under multiple boundary conditions (see Fig. 5). In this regard, the correct theoreticalrelations shown in (3) has been important to validate our simulations. Figure 8 shows the critical strain predicted by ournumerical simulations compared with relations (1) and (3) for a compressible material of zero Poisson ratio.

With the help of numerical simulations our plan is to explore the parameter space to search for the W-t-F transition atlarge mismatch. At the same time we plan to develop a theoretical modelization relaxing four key assumptions from most

4

previous analysis: (i) we will no longer assume that geometric nonlinearity associated with substrate rotation is negligible(large rotation of the film was already included in previous analyses), (ii) we will not assume the system is incompressible,instead our analysis will be focus on the zero Poisson ratio case, (iii) we will not longer assume rubber elasticity to take intoaccount material nonlinearities, and (iv) we will no longer assume plate theory to describe the deflection of the membrane.Relaxation of these assumptions leads to a rich new set of behaviors.

Em

/Es =

10

0

!=0.4

Em

/Es =

1!=0.41

!=0.20

!=0.26

!=0.33

!=0.40

!local=0.4-0.6

!local > 0.6

!=0.41

Figure 5: Finite element simulations of an elastic sub-strate/membrane system with E

m

/Es

= 1 and Em

/Es

= 100,both materials are treated as linearly elastic solids with zeroPoisson ratio. The systems are uniformly compressed in thehorizontal direction, boundary conditions being applied to theend planes and moreover the bottom plane being constrainedto simple horizontal motion. The deformation geometry isshown with the strain field imposed in color; only compres-sive (negative in this notation) strains are colorized, any ten-sile strains are uniformly gray. The membrane has been re-moved to allow visualization of developed surface strains at thesubstrate/membrane interface. In the zero-mismatch case, theblock undergoes uniform compression as expected up to ✏ = 0.4,at this point multiple creasing instabilities develop along theunconstrained surface. The instability shows strong signs ofcriticality as the instabilities appear randomly and grow un-controllably. In the higher mismatch system, wrinkling appearsuniformly. The near-surface strain field in the substrate (✏

local

)is not uniform showing an alternating pattern of compressionand tension following the wrinkle imposed local displacements.

Relaxing (i) is important to understand localization. Asin the W-t-F transition for liquid substrates, we expectthat substrate nonlinearities are required to explain local-ized structures. Important work by Brau et al. [53] coupledplate theory to describe the membrane with a linear elasticdescription of the substrate. Their model includes rotatione↵ects for the membrane while ignoring finite rotations forthe substrate. They made a weakly nonlinear analysis of themembrane while neglecting geometric nonlinearities for thesubstrate. Their analysis shows that nonlinearities for themembrane leads to period-doubling which is consistent withtheir own experiments, but it does not predict localizationin the sense of the W-t-F transition for liquid substrates.However their experiments, and current experimental datafrom other references, are largely confined to a modulus mis-match regime of E

m

/Es

⇠ 50� 250. Numerical simulationsshow that localization in the sense of the W-t-F transitionobserved for liquid substrate is possible for E

m

/Es

� 103.We will explore the large mismach limit with numerical sim-ulations and try to produce a model to capture localizationin that limit.

Additionally, assumptions (ii) and (iii) simplifies enor-mously the analysis. The study of compressible materialswith zero Poisson ratio makes irrelevant the di↵erence be-tween Cauchy and Piola-Kircho↵ stresses for a system underuniaxial compression. Similarly, relaxing (iii) aims to sim-plify the problem by reducing the e↵ect of material nonlin-earities while keeping geometric nonlinearities. We will alsorelax in (iv) which is an important assumption of previousworks. Including the deformation accross the thickness ofthe membrane by studying the full equations of elasticity formembrane and substrate will help to understand how platetheory emerge from the large mismatch limit. It will also helpto connect the results in the large mismatch limit with ourresults for lower mismatch (see next problem below) whereplate theory is not longer used. Most importantly, assump-tions (i-iv) still preserve the buckling instabilities observedfor the bilayer system as Fig. 5 shows, stressing our observa-tion that the main elements for the di↵erent instabilities arehidden in geometrical nonlinearities.

For the experimental investigation of the problem we willrely on our collaborations. We don’t have the expertise toprepare polymer materials with the approriate material prop-erties that emulate our physical models but we have initiateda collaboration with Professor Sachin Velankar, a faculty inChemical Engineering of University of Pittsburgh, to helpus with the experimental part. He has an important back-ground in polymer science, and strong interest in bucklingphenomena [33, 57]. His recent research includes swelling-induced folding of films of cross-linked polymers, reversibly-buckling surfaces, and (in progress) buckling of membranessupported on viscous films. Most relevant to this proposal is his experimental expertise and practical knowledge of polymermaterials. This is critical to design clean experiments that isolate the physical phenomena of interest.

Similarly, we have established a collaboration work with the “Group of Intracellular Dynamics and Transport” from Uni-versidad de Buenos Aires. They have developed nice optical techniques for studying the buckling of cytoskeleton microtubules

5

in-vivo [3]. Microtubules are biofilaments with a radius of approximately 10 nm that may have importance for the structureand mechanics of the cell. They have also importance for transport inside the cell since molecular motors use them as higwaysystem to move around the cell. Previous works have shown microtubules to buckle into sinusoidal shapes with a well definedwavelength when compressive forces are applied, for example by the motion of molecular motors along them. These eventsare frequently observed in the cell and have been explained by the presence of a soft lateral reinforcement. However, it is alsoobserved localized deformations similar to the W-t-F for liquid substrates. It could be possible that these localized eventsare related to the position where the compressive forces are applied or the structure of the sourrounding cytosqueleton thatavoids a sinusoidal shape to develop. But another explanation is given by the W-t-F transition because the same wavelengthobserved implies an extremely large mismatch, E

m

/Es

⇠ 108. To know the answer to the mistery, we will use numericalsimulations in collaboration with the experimental work provided by the argentinian group.

Problem 3: Folding & Creasing for Low Mismatch

One of the key assumptions in the previous works already cited is the large mismatch in sti↵ness. But the whole richness ofthe system is not captured in this limit. A di↵erent surface instability, first studied by M. Biot in his book “Mechanics ofIncremental Deformations” [60] and named “creasing” or “Biot instability”, happens in the opposite limit when the sti↵nessof the membrane equals the sti↵ness of the substrate. The limit is singular since the only length scale of the problem, itsthickness, is meaningless. A breakthrough for this problem was given by Holfeld and Mahadevan [19] where they carefullystudied numerically the instability in the limit of vanishing bending sti↵ness of the upper membrane. Their analysis showedthat the instability is extremely sensitive to imperfections at the free surface and one example is the presence of a “skin”or membrane on top. However, they implicitly assume large mismatch since plate theory is used to describe the membrane.Although this help to understand the behavior of the system in the Biot’s limit, it is not helpful to connect the creasinginstability with other buckling instabilities as wrinkling. Additionally, the work of Holfeld and Mahadevan and others thathave followed [62] show that material and geometric nonlinearities seem important. They use a neo-hookean or hyperelasticmaterial model and assume incompressibility. Hence, the numerical and theoretical analysis have elements related to thespecific nature of the model used.

We plan to approach the Biot instability from the limit Em

/Es

! 1. This limit is more physical in the sense that anyslab of soft material will easily develop a skin or crust because of aging or other reasons. In fact to study a material withouta gradient of sti↵ness near a surface requires very special preparation and manipulation conditions. We expect that studyingthe problem in the low mismatch limit not only regularized the problem, but also helps to understand the creasing instabilityin relation with wrinkling and folding. We will also work under the four assumptions (i-iv) described above. Our preliminarywork shows that these assumptions lead to a similar marginal stability curve than the one obtained for M. Biot. It also showsthat geometrical nonlinearities are the culprit to explain creasing.

An important tool to understand the low mismatch limit will be again numerical simulations. We have already testedthat FEA in ABAQUS have the su�cient precision to capture the instability. Figure 5 shows FEA simulations where surfaceinstabilities (creasing) are captured within the limit of low mismatch and linear elasticity. Importantly these instabilitiesbegin to occur as critical strain approaches 0.4, as is captured by preliminary calculations plotted in Fig. 8 and discussedbelow. It is noteworthy that in this simulation no characteristic length scale appears, the crease is infinitely sharp, and thespacing between creases is random. In contrast, in the lower half of Fig. 5, where a bilayer system is modeled, at small strain, acharacteristic wrinkle wavelength appears which agrees with previous linear stability analysis. Upon further compression, thesurface relaxes into folds in a general pattern strongly resembling experiments (see Fig. 2 II.). These preliminary simulationsgive further insight into local strain fields, not captured in previous experiments. Specifically, critical creasing strains arelocally approached in the substrate within a near surface layer even when nominal substrate strains are much lower. Thoseregions of higher substrate strain, to first approximation, have access to the creasing instability; moreover, the simulationsshow that this is where membrane folds nucleate. This strongly points to the hypothesis of a wrinkle-crease coupling. Thegoal of our simulations will be to study the role of modulus mismatch, with particular attention paid to the evolution of nearsurface strains and their deviation from nominal strain in the system.

6

METHODOLOGY & WORK IN PROGRESS

Problem 1

Experiments

Force Sensor

Rear Projection Glass

Camera

Liquid

Membrane

Collimated Light

Needle (Point Force)

Linear Nanopositioner

Figure 6: Setup description. We use the shadowscast by the defomation of the liquid to observe thewrinkles and folds. The physical principle is thesame that allows to observe striders in a shallowstream (see inset).

One of the key experimental ingredients to understand the wrinklingof thin films was developed in the last decade by the collaboration betweenpolymerists and physicists in University of Amherst. They managed toprepare polystyrene (PS) films with very well defined thickness rangingfrom 50 to 500 nanometers [28,37]. Thickness is an important parameterin the scaling relations obtained from theory and its variation gives a wayto discard or validate models. We have learned from them how to preparethese films by spin coating with the specified thickness. A central part ofour approach will be to make experiments with varying thickness to testtheoretical scaling laws.

A di�culty when studying the deformation of these films is that theyare transparent. Previous works used di↵usive light provided by the mi-croscope or other light source to unveils the deformation of the membrane.The final result is a pattern of shadows and light projected by the lightsource over the same surface. Accordingly, the wrinkled structure showslonger wrinkles in some angular directions because light has always aprefered direction. This is a problem when studying structures with aradial geometry because light enhances some structures but diminishesothers. We have built a di↵erent setup where the visualization takesadvantage of the film being translucent. Figure 6 explains the physicalprinciple. The out of plane displacement of the film deforms also thewater surface and the distorsion of the surface deflects the incident light.It produces a pattern that is projected in the bottom of the container byusing a rear projection glass. The projected pattern shows no preferedangular directions and large angular sectors can be used to obtain data.

Figure 7: Two di↵erent methods to observe the deformationof the surface. Left: Image extracted from Ref. [37]. Thedeformation is detected by using a stereo microscope in normaltransmitted mode. Light goes from the right bottom corner tothe top left corner, so that wrinkles are not well defined in thatdirection. Right: The method we propose is not sensitive tothe direction of light since it only follows the deformation ofthe water surface.

We have already used our setup to study the transitionbetween wrinkling under NT and FT conditions. The samesetup with some improvements will be used to study the W-t-F transition in circular geometry and the interaction betweenpoint forces or defects applied at di↵erent locations. Figure7 shows some images obtained with our current setup to giveevidence of the good optical control of our system.

Precise control of position of the indentors can be madewith the XYZ nanopositioners (Thorlabs) available in ourlab. In addition we have succesfully worked with commer-cially available force sensors (Futek) that have the sensitivityto track the forces required to deform our membranes. Weplan to decrease the noise in our force measurments by propercushion of the setup. In addition, a water evacuation systemwill be added to decrease contamination of the liquid bathby dust in the air.

One of the remaining experimental challenges to solveis to decrease the presence of microparticles contaminatingthe membrane and water surface. These are responsible ofinhomogeneities in the final membranes that could lead touncontrolled W-t-F transition (see Fig. 4) or add consider-able noise to the process of wrinkle formation. We plan totake some measures to reduce these e↵ects by perfecting theprocedure to fabricate our membranes, working in a cleaneratmosphere and installing our experiment in a room speciallydesigned for this purpose.

Theory

For the W-t-F transition in floated membranes there is a well defined model that accounts for the phenomena observedwhen the geometry is rectangular. The energy for this case can be written as [8] U = B

2

RdL2 + K

2

RdL v2 (see Problem 1

above and Fig. 3) that must be minimized under the constrain of inextensibility. The result of the minimization process is

7

a form of the Elastica equation that was analitically solved in [55, 56]. Since the system deforms in the plane x-y, stretchingenergy is not necessary and the deformation can be considered as an isommetry. The analysis is more di�cult in circulargeometry because stretching energy plays an important role. Previous theoretical works have studied wrinkling of an annularmembrane due to in-plane forces applied to its inner boundary and the wrinkling due to a point force applied to the center ofa circular membrane [41,42]. In both cases there is an important role of stretching. These works show that the system movesfrom a state described by buckling analysis to a far from threshold regime (FT) where the azhimutal stress is negligible. Theanalysis is made by using the Foppl von Karman equations that accounts for large deflections, small slope limit. However, itremains to be understood how the system moves from a FT regime highly wrinkled to a folded regime.

The Foppl von Karman equations cannot account for the large slope limit needed to describe localization. We hypothesizethat an equivalent description to the Elastica equation exists for a system with circular geometry describing the transitionfrom the FT regime to an isometric regime where localization is a solution. One of our goal will be to develop a model showingthe transition.

Problem 2

Numerics

1 10 100 1000 1040.001

0.010

0.100

1.000

Figure 8: Threshold function of mismatch for ⌫m

= ⌫s

= 0compared with numerical simulations in ABAQUS. The thicksolid line in both figures shows the exact calculation obtainedfrom minimizing the marginal stability curves in Fig. 9, thedashed line gives the “classical relation” defined in (1), andthe thin solid line gives the scaling (3).

Our preliminary simulations in ABAQUS have shownthat it has the neccesary precision to compute the largeand small scale structures for wrinkling, folding and creas-ing. Given the highly non-linear buckling and post-bucklingstructures of interest and complex contact conditions at theinterface, the dynamic-explicit solver will be used primar-ily. In particular, the simulations will provide a method todefine parts of parameter space where desired instabilitiesare observed; allowing experiments to be more e�cientlytailored. We also have the capability to directly import-ing experimentally derived membrane geometries (profiles)or substrate strain fields into the FEA solver. These canbe used create predefined strain fields as initial conditionsfor the numerical analysis, and serve as further validationof the numerical procedures. Also importantly, many of thedesired instabilities occur at high strains, 0.2 and greater.While we have chosen materials that remain linearly elasticover a broad range, eventually non-linear stress-strain e↵ectswill become important. FEA simulations easily allow for avariety of constitutive models including ones where experi-mental data from material characterization (as is describedabove) are directly used in the analysis. By doing simulationswith both idealized linearly elastic and real material proper-ties allows us to gauge the e↵ect material non-linearity hason the instabilities. Moreover, since theoretical work will befocused on geometric non-linearities within the confines oflinear elasticity, these simulations will provide an importantbridge and validation between theory and experiment. Figure 5 contain examples of preliminary simulations and highlightour ability to carry out the research plan. ABAQUS will be our main tool to check the assumptions for the theoreticalmodelization.

Theory

The analysis will be constrained to plain stress or plain strain conditions. This assumption also cancels out anticlasticcurvatures and simplifies greatly the analysis. Our aim is to reduce the problem to the x-y plane and focus on the mainelements explaining the buckling instabilities. In the same spirit, assumptions (i-iv) will help to reduce the problem evenfurther. An important theoretical insight obtained from ABAQUS is that folding and creasing are instabilities related togeometrical nonlinearities. We will use that input to obtain a simplified model in the limit of large mismatch that accountsfor large rotations of the membrane and substrate. We expect that the large mismatch limit has a nontrivial model thataccount for geometrical nonlinearities.

We have already computed the linear stability analysis of a model under the framework defined by assumptions (i-iv). Itshows that the correct critical threshold for wrinkling and large mismatch is given by relations (3) confirming the appropriateboundary conditions at the interface. The same calculation has been important to test our numerical simulations.

8

Problem 3

Numerics

We will do a systematic analysis in ABAQUS to understand how the creasing and wrinkling instability compete in the lowmismatch limit. Although the linear stability analysis capture correctly the threshold line observed in ABAQUS (see Fig. 5),the postbuckling result is completly di↵erent for E

m

/Es

! 1 compared what it is observed for larger values (for exampleE

m

/Es

⇠ 5). Creasing is the pattern observed for Em

/Es

! 1 at the threshold line, but wrinkling is the preferred mode ofdeformation for larger values at the threshold. It predicts a crossover between these two modes at some value of mismatch.We aim to characterize that crossover numerically and understand the mechanism for the transition.

ABAQUS will also help us to track local quantities as strain, amplitude, and extensibility of di↵erent layers to look forimportant simplifications of the problem. We also plan to track the amplitude of the creases to study its behavior with strainas we did in [8].

Theory

0 2 4 6 8 10 12 140.0

0.1

0.2

0.3

0.4

0.5

Figure 9: Marginal stability curve ✏ = ✏(kt) forzero Poisson ratio and three di↵erent values of mis-match (Y = E

m

/Es

). The minimum value definesthe critical strain ✏

c

. Strains lower than ✏c

arestables. On the contrary, if ✏ > ✏

c

the system isunstable with a most unstable mode k

c

.

We believe that studying the problem in the linear elasticity frame-work and approaching the Biot instability from the limit E

m

/Es

! 1not only regularizes the problem, but also helps to understand the creas-ing instability without using material nonlinearities. We plan to do aglobal analysis of the buckling instability without assuming plate theoryto describe the membrane. In particular, this involves including the roleof finite rotations both in the membrane and substrate. Such rotationale↵ects are routinely taken into account for the membrane, and indeedthis is the hallmark of plate theory. However, given the preliminary sim-ulations, it is apparent that substrate near-surface shear stress becomescomparable to nominal compressive stress (�

xy

⇠ �xx

), therefore finiterotations in the substrate should not be ignored. To avoid unnecessarycomplexity, we will first restrain the analysis for zero Poisson ratio ma-terials that avoids coupling in the z direction. This assumption alsocancels out anticlastic curvatures and simplifies greatly the analysis. Weexpect that most of the linear and nonlinear behavior can be capture bythis assumption. Specifically, we will obtain the instability for low mis-match and include the thickness of the layer H. The few examples inthe literature for finite elasticity where the general problem of bucklingis studied in this sense [19, 62] mix the e↵ect of rotation (or geometricalnonlinearities) with the nonlinear response of the material (or materialnonlinearities). Buckling is a linear problem that couples compressivestresses to the transversal direction through a Laplace type of e↵ect so

that material nonlinearities are not necessary to understand the mechanism (although they are important to understand spe-cific details). Hence, we expect that a linear analysis of the full elastic equations will capture the instability for low mismatch.We have done preliminary calculations showing that this analysis works and can give the marginal stability curve ✏ = ✏(kt)for a given value of membrane thickness and modulus mismatch (see Fig. 9).

An important insight in this direction is that the linear analysis of the equations for zero poisson ratio gives a similarmarginal stability curve to the one obtained by M. Biot in the limit of E

m

/Es

! 1 [60]. The marginal curve becomes flatfor small mismatch (see Fig. 9) and all modes are unstables in that limit which is a sign of creasing. Thus incompressibilityand material non-linearity are not key factors, and the system can have a creasing instability even when it is constrained todeform in the plane x � y. This preliminary work coupled with the simulations, also done using linear elasticity, shows thecreasing instability has more a geometrical origin than anything related to material properties.

Our preliminary work shows that we can continuously approach the limit Em

/Es

! 1: we replace an ideal isotropicmaterial with no membrane, where all the modes are unstable, by a bilayer where the marginal stability curve has a minimum.Moreover, we can approach creasing from a direction where material nonlinearities are not relevant opening a way to explainthis instability by using geometric nonlinearities only. We plan to apply amplitude analysis to the equations of elasticity tostudy the postbuckling behavior in a similar way was made by Brau et al. [53]. However, we will include finite rotations forthe substrate and membrane.

9

HYPOTHESIS

The main hypothesis for the three problems proposed is that elasticity is su�cient to describe the physical behavior of oursystems. We will work in the framework of linear and finite elasticity. Thus, irreversible e↵ects as plasticity will be neglectedin our analysis. We detail below the main elements in our analysis.

Problem 1

The experimental part will be straightforward since our work in the framework of our Fondecyt project No. 1130579 havegiven as the necessary expertise to fabricate our membranes with the specified thickness. In addition, we have already aworking setup that only requires some improvements to carry out the specific goals of the proposal.

Since we will work with very thin membranes (50-500 nm), it is plainly justified to use linear elasticity to describe thedeformation of these films. The analysis of the transition from wrinkles to folds in circular geometry will be made by using theFoppl von Karman equations for small slope deflections [66,67]. However, we will need to implement a theoretical frameworkto study the large slope limit to describe localization. In that case, stretching energy will be neglected and the transitionfrom wrinkles to fold will be studied as an isommetry

Problems 2 and 3

The main hypothesis for our numerical work is that ABAQUS have the necessary accuracy to capture the highly singularsolutions observed for creasing and folding. Our preliminary work shows that this is the case. Similarly, recent works showthat ABAQUS is a reliable software to study bubkling instabilities.

The numerical and theoretical work will be made in the framework of finite elasticity. This is a vast subject plenty ofdi↵erent models to describe material nonlinearities (Arruda-Boyce, Mooney-Rivlin, Neo Hooke, Ogden, etc.). The large strainthreshold for the instability for low mismatch has given a reason for the lovers of finite elasticity to implement highly complexmathematical tools. This has turned the current e↵orts to understand the creasing instability in a highly technical subjectwithout appeal for physical intuition. The main hypohesis to understand the structures observed for low and large mismatchis that geometrical nonlinearities are su�cient to explain them.

10

GENERAL AND SPECIFIC GOALS

Problem 1

1. We will prepare our current experimental setup to study the W-t-F transition in circular geometry.

2. A positive air pressure cleanroom will be established where we can work free of contamination.

3. We will obtain experimental data to characterize the wrinkling to fold transition in circular geometry.

4. We expect to develop a theoretical model describing the W-t-F transition in circular geometry. A paper will be published.

5. We will study interacting defects inducing the W-t-F transition by using two or more indentors.

6. We expect to obtain a model to understand the W-t-F induced by interacting defects. A paper will be published.

Problem 2

1. The ABAQUS software will be installed in the certified hardware.

2. We plan to numerically study a membrane on top of an elastic substrate in the limit of large mismatch and connectwith our results for liquid substrates.

3. We expect to develop a model describing the buckling instability for large mismatch including geometrical nonlinearitiesof the substrate.

4. We will confront our numerical and theoretical results with the experimental work made with polymer membranes andsubstrates carry out by Prof. Sachin Velankar from University of Pittsburgh. A paper will be published.

5. We plan to numerically study a filament embeded in an elastic substrate in the limit of large mismatch.

6. We will confront our numerical and theoretical results with the in-situ studies of microtubules buckling in cells made bythe “Group of Intracellular Dynamics and Transport” from Universidad de Buenos Aires. A paper will be published.

7. We will use our expertise in ABAQUS to explore other configurations where wrinkling, folding and creasing are competinginstabilities.

Problem 3

1. We plan to numerically study a membrane on top of an elastic substrate in the limit of low mismatch and understandthe connection between wrinkling, folding and creasing.

2. We expect to develop a model describing the buckling instability for low mismatch including geometrical nonlinearitiesof the substrate.

3. We will confront our numerical and theoretical results with the experimental work made with polymer membranes andsubstrates carry out by Prof. Sachin Velankar.

AVAILABLE RESOURCES

Our laboratory counts with the necessary equipment to fabricate thin polymer films. We have a spin coater (Laurell) thatwe have used to obtain films of thickness ranging 50-500 nm. We also count with another device to fabricate films byelectrospinning (Spraybase) and a system to measure the thickness of our films (Filmetrics).

Our laboratory also counts with two optical tables (Thorlabs), a stereoscopic microscope (Olympus) and accompanyingcamera (Jenoptik), basic optical equipment to carry out measurements (lenses, cameras, etc), and systems to control forcesand displacements (Omega, Futek, etc.). The laboratory consists in a workspace with a surface of more than 50 m2 where 4students and/or researchers can work comfortably. We have also a new space of 35m2 where we plan to establish a positiveair pressure cleanroom.

In addition, we count with the normal resources of a well established institution: internet access, a specialized library withmore than 4000 titles in the Department of Physics and subscription to the CINCEL network containing the most importantjournals in physics and engineering.

11

WORK PLAN

Here P1=Problem 1, P2=Problem 2, P3=Problem 3.

Problem Goals Activities YearsYearsYearsYears

1 2 3 4

P1

1 •We will prepare our current setup to study the W-t-F transition in circular geometry. X

P1

2 •A positive air pressure cleanroom will be established where we can work free of contamination. X X

P13 •We will obtain experimental data to characterize the wrinkling to fold transition in circular geometry. X X

P1 4 •We expect to develop a theoretical model describing the W-t-F transition in circular geometry. A paper will be published. X XP15 •We will study interacting defects inducing the W-t-F transition by using two or more indentors. X X

P1

6 •We will study interacting defects inducing the W-t-F transition by using two or more indentors. A paper will be published. X X

P1

P2

1 •The ABAQUS software will be installed in the certified hardware. X

P2

2 •We plan to numerically study a membrane on top of an elastic substrate in the limit of large mismatch. X X

P23 •We expect to develop a model describing the buckling instability for large mismatch including geometrical nonlinearities of the

substrate.X

P2 4 •We will confront our numerical and theoretical results with the experimental work made with polymer membranes and substrates carry out by Prof. Sachin Velankar from University of Pittsburgh. A paper will be published.

XP25 •We plan to numerically study a filament embeded in an elastic substrate in the limit of large mismatch. X X

P2

6 •We will confront our numerical and theoretical results with the in-situ studies of microtubules buckling in cells made by the “Group of Intracellular Dynamics and Transport” from Universidad de Buenos Aires. A paper will be published.

X X

P2

7 •We will use our expertise in ABAQUS to explore other configurations where wrinkling, folding and creasing are competing instabilities.

X X

P3

1 •We plan to numerically study a membrane on top of an elastic substrate in the limit of low mismatch and understand the connection between wrinkling, folding and creasing.

X X

P3

2 •We expect to develop a model describing the buckling instability for low mismatch including geometrical nonlinearities of the substrate.

X X

P33 •We will confront our numerical and theoretical results with the experimental work made with polymer membranes and substrates

carry out by Prof. Sachin Velankar. A paper will be published.X X

P3P3P3

12

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