B. Yasara DharmadasaAnn and H.J. Smead Department of Aerospace

Engineering Sciences,University of Colorado Boulder,

Boulder, CO 80309e-mail: buwaneth.dharmadasa@colorado.edu

Matthew W. McCallumAnn and H.J. Smead Department of Aerospace

Engineering Sciences,University of Colorado Boulder,

Boulder, CO 80309e-mail: matthew.mccallum@colorado.edu

Seyon MierunalanDepartment of Civil Engineering,

University of Moratuwa,Katubedda 10400, Sri Lanka

e-mail: 178038c@uom.lk

Sahangi P. DassanayakeDepartment of Civil Engineering,

University of Moratuwa,Katubedda 10400, Sri Lankae-mail: 188037G@uom.lk

Chinthaka H. M. Y.Mallikarachchi

Department of Civil Engineering,University of Moratuwa,

Katubedda 10400, Sri Lankae-mail: yasithcm@uom.lk

Francisco Lopez Jimenez1

Ann and H.J. Smead Department of AerospaceEngineering Sciences,

University of Colorado Boulder,Boulder, CO 80303

e-mail: Francisco.LopezJimenez@colorado.edu

Formation of Plastic Creasesin Thin Polyimide FilmsWe present a combined experimental and analytical approach to study the formation ofcreases in tightly folded Kapton polyimide films. In the experiments, we have developed arobust procedure to create creases with repeatable residual fold angle by compressing ini-tially bent coupons. We then use it to explore the influence of different control parameters,such as the force applied, and the time the film is being pressed. The experimental resultsare compared with a simplified one-dimensional elastica model, as well as a high fidelityfinite element model; both models take into account the elasto-plastic behavior of thefilm. The models are able to predict the force required to create the crease, as well asthe trend in the residual angle of the fold once the force is removed. We non-dimensionalizeour results to rationalize the effect of plasticity, and we find robust scalings that extend ourfindings to other geometries and material properties. [DOI: 10.1115/1.4046002]

Keywords: constitutive modeling, material properties, thin-films, plastic creases

1 IntroductionGossamer space structures often make use of thin films and mem-

branes that are tightly compacted and stowed before launch, andthen deployed in space. As an early example, in 1960, NASA suc-cessfully launched ECHO, a reflective satellite balloon that inflatedto a diameter of 100 feet [1]. Inspired by the success of ECHO, theInflatable Antenna Experiment (IAE) satellite was launched whereinflated tubes formed a rigidized space antenna structure [2]. Thinfilms are also used for solar sails, a novel propulsion concept inwhich the thrust is generated from the impulse of solar photons.A milestone in solar sailing is the IKAROS solar sail demonstratorproject, where JAXA successfully carried out an orbital deploymentof a 196 square meters sail [3]. This technology is ideal for low-cost,lightweight CubeSats. It is currently implemented in CubeSail andLightSail [4,5] and has been proposed for future missions such asthe near-Earth asteroid scout [6]. Other uses of thin films in spaceinclude the HabEx starshade [7] and the deployable sun-shield forthe James Webb Space Telescope [8]. Thermally stable metallic

polymer films such as Kapton and Mylar are commonly used forthese space applications.Two different surface disturbances are observed in membranes

and thin films: wrinkles and creases. Figure 1 shows images ofthe IKAROS solar sail demonstrator during stowage and afterbeing deployed, where both surface irregularities are clearlyvisible [9]. Wrinkles are temporary distortions that occur due tocompressive buckling in thin films, which remain in the elasticregime [10,11]. As such, they are a consequence of the loading con-ditions and geometry and disappear once those are corrected and themembrane is under tension [12]. Creases, on the other hand, are per-manent features caused by the highly localized plastic deformationthat takes place when the film is folded to a very tight radius of cur-vature, which is often the case during the packaging of deployablestructures. Figure 2 summarizes the process in which a crease iscreated, and how it alters the mechanical response of a film.Under folding force F, the deformation is highly localized in thecrease region, which results in permanent deformation of the film.Once the force is removed, the permanent curvature in the creaseresults in an equilibrium fold angle φ0. Under tensile loadingFtensile, the fold angle varies, but since φ < 180 deg, there isalways a shortening of the in-plane film length compared to the pris-tine condition. The pretension due to this shortening, as well as theincreased bending stiffness due to the out-of-plane deformation, hasa significant effect on the natural frequencies of a membrane

1Corresponding author.Contributed by the Applied Mechanics Division of ASME for publication in

the JOURNAL OF APPLIED MECHANICS. Manuscript received November 9, 2019; finalmanuscript received January 12, 2020; published online January 17, 2020. Assoc.Editor: Yihui Zhang.

Journal of Applied Mechanics MAY 2020, Vol. 87 / 051009-1Copyright © 2020 by ASME

structure. This is the proposed explanation for the differencebetween simulations and experimental results for the deploymentdynamics of IKAROS [13], which can result in problems with atti-tude control and tearing or entangling of the film. Out-of-plane dis-tortions caused by the creases can also affect the thrust vector at lowmembrane tension [14], and tight packaging can destroy the protec-tive coatings on the film [15]. Hence, understanding and modelingthe crease behavior is crucial for the successful design of thin filmstructures, from space deployable structures to origami-basedmetamaterials.Several models have been presented to describe the effect of

creases in the mechanical properties of films. Murphey [16] calcu-lated the homogenized stress–strain relationship of a randomlycreased sheet when the crease amplitudes and wavelengths are mea-sured, without explicitly resolving the deformed film profile.Hossain et al. [17] characterized the non-linear stress–strain rela-tionship of a crease and modeled it in a finite element frameworkby defining a softer non-linear material strip. To capture theout-of-plane stiffness, Nishizawa et al. [13] proposed to add beamelements with a second moment of area equivalent to that of thecreased region; this model was able to capture the natural frequen-cies of the membrane without explicitly accounting for thedeformed geometry. The effect of creases on pressurized cylindershas been modeled numerically, taking into account the mechanicsof a single crease [18,19]. However, the most common approachto model creases in the solar sail deployment simulations is to ide-alize them as hinges with torque springs, see Fig. 3. This techniqueaccounts for the stiffness of the hinge, as well as for its effect on thefilm profile [20,21]. A similar approach is used to model the stiff-ness of fold lines in origami [22–24], although in this case, panels

are often assumed to remain flat. In both cases, it is necessary tocarefully characterize the mechanical behavior of the crease; inthe case in which it can be assumed to be linear [25], it can bedescribed by the equivalent rotational stiffness (k) and the equilib-rium angle under no applied loading (φ0).The mechanical properties of creases have been explored exper-

imentally in single creases [21,26] and Z-folds [27,28], as well asusing a linkage mechanism able to provide pure bending to arbi-trarily high curvature [29]. The method to create the crease can sig-nificantly affect the results, and even for a given method, it is oftendifficult to achieve repeatable folds to use in the experiments[17,26,30,31]. Furthermore, films used in space structures oftenhave a thickness in the micrometer range [32], so measuring

(a) (b)

Fig. 3 Torque spring idealization of a crease: (a) unstressedstate and (b) under tensile loading, showing opening of thecrease as well as bending in the film

(a)

(b)

(c)

(d)

(e)

Fig. 2 States of the film geometry during the creasing process:(a) initially flat film, (b) film bent under the folding force F,(c) equilibrium fold angle when the force is released, (d ) foldangle expanding (φ1 >φ0) and film bending when subjected totensile load Ftensile, and (e) shortening of the projected filmlength due to the crease

(a)

(b)

Fig. 1 (a) Stowed and (b) deployed state of IKAROS solar sail [9](Reprinted with permission of Elsevier Ltd. © 2011)

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creasing and unfolding forces becomes challenging. Numericalsimulations have been used to analyze the creasing process forthicknesses and loading conditions that present difficulties forexperimental exploration. These include one-dimensional simpli-fied models as well as solid-based numerical simulations accountingfor the elasto-plastic nature of the film [33–36]. However, there is alack of experimental validation of the predictions for the equilib-rium angle. Furthermore, the effect of the viscoelasticity of thefilm is usually neglected, and so, its influence on the process isnot well understood.In this study, we combine experiments and analysis to rationalize

the different factors that contribute in the formation of a crease, andhow they affect the equilibrium fold angle. In particular, we willfocus on establishing a repeatable experimental procedure tocrease formation that can be used in future experimental studies.We then use the experimental results, as well as analysis, toexplore the relative effect of the folding parameters on the resultingcrease. The paper is organized as follows. In Sec. 2, we describe theexperimental setup, discussing key parameters that lead to a robustprotocol. Section 3 describes the two modeling approaches fol-lowed: a simplified one-dimensional elastica model and a fullyresolved finite element model. Both models account for the elasto-plastic behavior in the film. Section 4 presents and compares ourexperimental and analytical results. Section 5 summarizes our find-ings and discusses planned future work.

2 Experimental SetupThe present study focuses on the creation of a single crease in a

rectangular piece of polyimide film, by applying compressive forceusing compression platens in a universal testing machine. The mate-rial used is Kapton (DuPont) film, whose high durability and stableproperties over a large temperature range make it ideal for spaceapplications [37]. Kapton properties are shown in Table 1, as pro-vided by the manufacturer.

2.1 Specimen Preparation and Test Procedure. All speci-mens are obtained from initially flat films, in order to avoid theresidual curvature observed in rolled films. We consider two differ-ent thicknesses: h= 50.8 μm and h= 127 μm (0.002 in and 0.005 in,respectively). We cut rectangular coupons of width W= 25.4 mmand total length 101.6 mm and check for visible defects such ascuts and stretch marks (Fig. 4(a)). The coupons are lightly bentand held in that shape using scotch tape, such that they can beplaced between the compression platens of a universal testingmachine (Instron 5969, 1 kN load cell). The length of the samplesis such that this initial curvature results in strains at least oneorder of magnitude smaller than the yield strain of Kapton.Once the specimens are placed between the compression platens,

a compressive force is applied by moving the top platen at a rate of10 mm/min, until the desired applied force F or platen distance d isreached. The specimens are then held under compression over agiven pressing time tpress in which either the applied force or the dis-tance are held constant, see Fig. 4(b). The reason to distinguishbetween the two control parameters is that, due to the viscoelasticnature of Kapton, there is stress relaxation in the samples, and

possible rearrangement of their deformed geometry. As such, thecreases created following both approaches, loading and displace-ment control, are different. This is addressed in detail in Sec. 2.2.The creasing process is imaged with a USB video microscope(Mighty Scope 5M).After the hold period tpress, the top platen is raised at a rate of

100 mm/min, and the scotch tape is cut, allowing the crease tounfold. The coupons are then suspended from one end such thatthe crease line is parallel to gravity. This helps to minimize thegravity effects and friction with a possible support. As soon asthe coupons are suspended (which takes approximately 30 s afterthe end of the test), we capture images of the equilibrium foldangle every 10 s for a period of 5 min using a Nikon D610camera (24.3 megapixels, AF-S Micro Nikkor 60 mm lens;Fig. 4(c)). The reason to track the evolution of the angle overtime is that previous studies have shown that the viscoelasticnature of thin films results in the opening of the crease angle overtime [31,38]. The equilibrium fold angle in each image is measuredwith the help of a MATLAB script that fits straight lines to the flat por-tions right next to the crease, see Fig. 4(d ).

2.2 Influence of Test Control Parameter. Figure 5 comparesthe results of experiments under force control and platen distancecontrol, in order to illustrate the difference between both proce-dures. We use F and d to indicate the values at which both forceand distance are held constant, respectively. First, a coupon ofthickness h= 127 μm is compressed to a platen distance d =380 μm and held at that configuration for tpress= 120 s. As the topplaten is lowered at a uniform rate, the force increases exponen-tially, and once the platen halts at the final pressing configuration,the force starts reducing with time, at a decaying rate, seeFigs. 5(a) and 5(b). We observe a peak force of 51 N when theplaten halts and a plateau value of 32 N at the end of the holdingperiod. Next, two samples of the same thickness are pressed up toF = 51N and F = 32N and held for the same amount of time,tpress= 120 s. When the force is held constant, there is a slightdecrease in platen distance in order to account for the relaxationin the film (Fig. 5(a)). We believe material viscoelasticity is respon-sible for the force and curvature rearrangements during the holdingphase. Although this is similar to the effect observed in creep and

Table 1 Material properties of Kapton HN polyimide

Property Value

Density (kg/m3) 1420Ultimate tensile strength (MPa) 231Young’s modulus (GPa) 2.5Poisson’s ratio 0.34Yield stress (MPa) 69Yield strain (%) 3

Fig. 4 Experimental setup: (a) test coupon bent using a tape,(b) universal testing machine applying compression to the speci-men, (c) set-up used to capture images of suspended couponusing a magnet, and (d ) fold-angle measurement using MATLAB

script

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relaxation tests, in the present experiments neither stress nor strainis constant along the length of the film. This important differencemeans that changes in the global response are due to both stressrelaxation, as well as small readjustments in geometry due to differ-ential softening between regions of the film. Figure 5(c) shows theevolution angle for the three specimens, with large differencesobserved. The angles increase over time, which is consistent withprevious studies [31,38].An important difference between the two creasing approaches is

the repeatability of the results. In Fig. 6, we fold three coupons each(127 μm thick) to either d = 635 μm or F = 30N. The compressionplatens were detached and reassembled between each experiment,to increase the possible differences between tests. We observethat controlling the applied force produces a repeatable fold angle(deviations of ±2 deg between samples), while the fold anglesobtained when controlling the platen distance have higher devia-tions (±15 deg), see Fig. 6(c). This can be explained by consideringthe force curve in Fig. 6(b): the three samples with the same d reachvery different maximum forces, due to the high slope of the force–displacement curve at that point. The images from the USB micro-scope during testing showed that the compression platens are notperfectly parallel and can have a small misalignment (up to 0.05deg), which varies from test to test. This creates small variationsin d, on the order of 10 μm, which nevertheless result in large var-iations in F, which explains the deviations between samples. Byspecifying F, the fold angles are more consistent, and it is themethod used to obtain the results in Sec. 4.

3 Modeling and AnalysisWe model the creasing process using two different approaches.

First, we idealize the film as a one-dimensional element usingEuler’s theory of elastica. The approach is similar to the analysispresented by Secheli et al. [34], but we use experimental data forthe plastic behavior of Kapton instead of an idealized behavior.

This prevents us from producing a closed-form solution, and wesolve the equations numerically. Second, we carry out a moredetailed finite element analysis of the film using the commercialfinite element package ABAQUS. Both models consider large deflec-tions and the elasto-plastic behavior of Kapton. The effect of gravityduring the pressing of sample was neglected in modeling, since theself-weight is at least three orders of magnitude lower than theapplied pressing force. Electrostatic forces are also neglected,since they have been shown to not have a significant influence onfilms with the range of thickness used in our experiments [39].

3.1 One-Dimensional Elastica Model. The theory of elasticais a geometrically non-linear model that describes the behavior ofslender structures undergoing large deflections [40]. Figure 7(a)shows that the crease region can be idealized as a one-dimensionalelement with symmetric shape (indicated by green dashes). Further-more, we idealize the load exerted by the compression platens ontothe sample as a vertical point load F, at the point in which the speci-men is tangent to the platen. According to these assumptions, weonly need to analyze a cantilever of length Lc, which is fixed per-pendicular to the symmetry plane (Fig. 7(b)). The force F isapplied on a single point at the opposite end, and we neglect therest of the specimen beyond the contact point. We assume the speci-men to be inextensible, and the plane sections to remain plane whiledeformed. Besides considering the non-linear geometry, we need toconsider the material non-linearity and the varying length of Lc asthe test progresses, which represents a significant variation fromthe traditional elastica analysis. The internal bending moment at apoint of arc-length s∈ (0, Lc), measured from the point of symmetrycan be expressed as

M(s) = F Lx − x s( )( ) (1)

where Lx is the distance in the X-direction between the point of sym-metry and the applied loading. The shape of the specimens is

Fig. 5 Comparison of creasing test on 127 μm thick coupons under force and platen distance control. Variation of(a) platen distance, (b) applied force during the test, and (c) equilibrium fold angle after the test.

Fig. 6 Comparison of repeatability in experiments under force and platen distance control. Variation of (a) platendistance, (b) applied force during the test, and (c) equilibrium fold angle after the test. Line and point styles indicatedifferent control parameters, colors indicate different nominally identical tests.

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obtained by solving the following system of equations:

θ(s) =∫s0κ(ξ) dξ

x(s) =∫s0cos(θ(ξ)) dξ

y(s) =∫s0sin(θ(ξ)) dξ

(2)

where κ(s) is the curvature at the point of arc-length s, x(s) and y(s)are the position in the X− Y coordinate system, and θ(s) is the anglebetween the tangent vector and the X-axis, as defined in Fig. 7(b).The boundary conditions to solve the equations are θ(0) = π/2and θ(Lc) = 0. In addition, we need to enforce that Lx= x(Lc). Thevertical position of the end point is y(Lc)= d′/2, which is relatedto the platen distance by d′ = d− h to account for the thickness ofthe film, which becomes important for small values of d.In order to solve the system of Eqs. (1) and (2), we need a mate-

rial relationship between the moment M(s) and the curvature κ(s) atevery point of the specimen. In the case of a linear elastic material,the stress–strain curve is injective, and this relationship can beuniquely defined by M = DEκ, where DE is the linear elasticbending stiffness. However, this is not the case when consideringthe material plasticity of Kapton, where the loading history needsto be considered. This is illustrated in Fig. 8(a), which shows thestress–strain response of Kapton [37]. If the load remains in theelastic region (path OA), loading and unloading follow the samepath in the stress–strain plane, and the stress can be uniquelydefined by the strain at that point. Once the loading at any pointin history results in strains higher than the yield strain, the stressat a given strain depends also on the loading history. The loadingpaths OBC and OB′C′ end with the same strains εC = εC′ , but dif-ferent stresses σC≠ σC′. Assuming that the unloading follows thesame slope as the initial linear regime, with stiffness E, each ofthe stresses can be calculated using:

σ(εmax, ε) = σmax − E(εmax − ε) (3)

where σmax and εmax are the maximum stress and strain experiencedby that material point in its loading history, respectively.In our problem, the strain at every point is given by the curvature

and the through the thickness distance to the neutral axis ζ − ζn:

ε = κ ζ − ζn( )

(4)

with no dependance on the loading history. The position of theneutral axis with respect to the geometric centroid ζn is obtained

by enforcing that the axial force is zero:

Faxial(κmax, κ) =∫h/2−ζn−h/2−ζn

σ(εmax, ε) dζ = 0 (5)

where κ is the current curvature at a specific point and κmax is themaximum curvature experienced by the same material point atany point in its time history. Equation (4) is used to calculateεmax and ε as a function of κmax and κ, respectively. In our case,we assume that the stress–strain relationship is the same intension and compression, and the calculation is performed on theundeformed geometry, which yields ζn = 0.The moment–curvature relationship is then calculated as:

M(κmax, κ) =∫h/2−h/2

σ(εmax, ε) ζ dζ (6)

Equation (6) shows that the moment at a material point is definedby a combination κ and κmax. To speed up the analysis, we calculateand save the moment values, Msaved =M(κmax, κ), by numericallyintegrating Eq. (6), taking 100 through thickness points and usingthe material properties of Kapton. The κmax and κ were discretizedat increments of Δκmax = 0.02/h and Δκ = 0.02/h, correspondingto increments of 1% strain at the surface of the film. Figure 8(b)

(a) (b)

Fig. 7 (a) Snapshots of the film from the microscope. Dashedlines emphasize the region between the points of contact, thatare idealized as (b) a cantilever with a point load at the oppositeend (Color version online.)

(a)

(b)

Fig. 8 (a) In-plane tensile stress–strain relationship for Kapton[37], showing different loading histories. The curve has beenextended beyond the last reported value in the data, at ɛ≈0.7,with a straight line with the same slope. (b) Moment-curvaturerelationship obtained integrating the stress relationship. Theplot shows loading (red) as well as the unloading (blue dashed)paths when the surface strain at the point of maximum loadingis 0.03, 0.3, 0.6 and 0.99, respectively. (Color version online.)

Journal of Applied Mechanics MAY 2020, Vol. 87 / 051009-5

plots the calculated moment–curvature relationship, normalized toeliminate the thickness dependence, and uniquely defines thecurrent curvature κ for a given momentM and the maximum curva-ture during the loading history κmax.As the test progress, the applied force F increases and the beam

length Lc needs to shorten to satisfy the boundary conditions. Thechange in Lc is the main reason for the need to account for κmaxin the moment–curvature relationship. For a given arc-length,s (0 < s < Lc), M(s = 0) >M(s = s) >M(s = Lc) = 0. Initially,M(s) increases as the applied force is increased, but decreases asLc approaches s, since M (s = Lc) = 0.The final algorithm needs to take into account the non-linear

geometry, the material non-linearity, and the fact that the lengthof the specimen under loading decreases as the point of contactwith the platens approaches the symmetry line. The following iter-ative algorithm is utilized to find a solution that satisfies all therequirements.

(1) First, we choose a value of the initial specimen length Lc=Lc0, which is sufficiently large to ensure that all points arein the elastic regime.

(2) We initialize all maximum curvatures to be equal to zero,κmax(s) = 0 for all values of s. The arc-length is discretizedusing n points (in our analysis, n= 3000 and Lc0= 30 mm).

(3) We solve the elastica problem:(a) Guess the values of F and Lx.(b) Integrate Eqs. (1) and (2) to obtain the applied moment

and coordinates at every point. The value of κ(s) atevery point is the one that verifies Msaved(κ(s),κmax(s)) =M(s).

(c) Iterate the values of F and Lx until the boundary condi-tions are satisfied. This is accomplished by minimizingthe function G = Lx − x Lc( )( )2 + θ(Lc)( )2.

(4) The values of the maximum curvature are updated if neces-sary, so that κnewmax(s) =max κmax(s), κ(s)( ).

(5) We then decrease the value of the specimen length, Lnewc , andgo back to Step 3. The value of Lnewc is a multiple of Lc0/n, sothat the same discretization of the arc-length dimension inStep 2 can be used throughout the simulation.

The output of the algorithm is the values of F and d′ for everyvalue of Lc considered, and the value of the maximum curvatureat every point κmax(s), for the original range of the arc-length 0 <Lc <Lc0. The residual curvature at every point κres is obtained byimposing zero moment:

M κmax, κres( ) = 0 (7)

which corresponds to complete unloading in the moment–curvaturerelationship. The equilibrium fold angle is obtained by integratingthe residual curvature along the arc-length.

3.2 Finite Element Model. The accuracy of the elastica modeldecreases when the film is pressed to very small platen distances d≈2h, since at that point the specimen can no longer be idealized as aslender beam. Furthermore, Poisson’s effect causes an expansion ofthe region under compression and a reduction of the region undertension, an effect neglected in our one-dimensional model thatresults in a shift of the neutral axis. To account for such effects,we used the commercial package ABAQUS/STANDARD to create ahigh fidelity elasto-plastic finite element model. We use the samecoupon thickness as in the experiments, 50.8 μm and 127 μm, andan isotropic hardening plasticity model. The nominal stress (σnom)and strain (εnom) data obtained from the manufacturer [37] are con-verted to true stress (σtr) and true plastic strain (εpl) using

σtr = σnom (1 + ε)εtr = ln (1 + ε)

εpl = εtr −1Eσtr

(8)

We use symmetry along the fold line to model half of the film andincrease the computational efficiency. We assume plane strain beha-vior to further simplify the problem. Four-node plane strain quadri-lateral elements (CPE4) are used in the analysis, and the meshdensity is varied along the film length, so that the region near thesymmetry end has around 50 elements through the thickness. Atypical finite element model consisted of 9898 nodes and 8631 ele-ments. The compression platen is modeled as a rigid beam usingtwo node beam elements (RB2D2), and the compression force istransferred by defining frictionless contact between the film andthe beam.The creasing of the film occurs over several steps. First the nodes

along the symmetry line are fixed (Ux=Uy= 0) and the free edge isdisplaced in order to bend the film, as shown in Fig. 9(b). The rigidplate moves closer to the symmetry line and the free edge is released,allowing the film and the plate to make contact (Fig. 9(c)). Next, thesymmetric edge boundary is set to rollers in the X-direction (Uy= 0),while fixing a single node to prevent rigid body movements. Thefilm is pressed by moving the rigid plate up to a specified platen dis-tance d/2 (Fig. 9(d )), and then the film is released by turning off thecontact definition. Figure 9(e) illustrates the resulting equilibriumfold angle, which can be measured by calculating the angle of thestraight portion with the X-axis.

Fig. 9 Finite element model for folding of thin films using rigidplates: (a) The film is flat in the undeformed configuration, (b) itis initially bent by applying displacement boundary conditions,(c) subsequently by contact with a rigid plate, (d ) with the curva-ture increasing as the plate moves closer to the symmetry line,and (e) the final geometry shows permanent deformation oncethe loading is removed

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4 Results4.1 Folding Force. We first focus on the folding process, and

in particular, the evolution of the applied force F as a function of theplaten distance d. Figure 10(a) plots the results from different exper-imental runs on nominally identical samples, as well as the predic-tions from the elastica analysis and the finite element model. Bothmodels are able to predict the force–displacement variation accu-rately over the whole range of forces used in the experiments.The two variables are inversely related, with the force increasingexponentially as the distance between the plates decreases. Theexperimental results correspond to 15 different tests with h=127 μm and 10 tests with h= 50.8 μm.In order to further rationalize our results, Fig. 10(b) shows the

same data plotted in a logarithmic scale, and non-dimensionalizedso that the results from both values of h collapse onto a singlecurve. To identify the effect of plasticity, we also plot the resultsof an elastica analysis with a linearly elastic constitutive materialmodel. The elastic regime corresponds to the range d/(2h) > 20,where the slope is −2, indicating a power-law relationshipbetween normalized force and normalized platen distance. Atd/(2h)≈ 20, we observe deviations between the elastic and plasticbehavior. Remarkably, the behavior in the plastic regime is alsoclose to a power law of slope approximately equal to −1.6 withinthe whole range considered. The smallest value of the normalizedplaten distance d/(2h)= 1 corresponds to the case in which theseparation between the plates is equal to the combined thickness

of the two flat regions of the specimen. In practice, values ofd/(2h) < 2 correspond to situations in which the crease region isunder significant compressive stress, which explains the higherdeviations between experiments and analysis.The choice of non-dimensionalization in Fig. 10(b), as well as the

power law revealed in the results, can be understood by consideringthe sketch of the sample geometry in Fig. 11(a). Considering firstthe elastic regime, the maximum bending moment, which islocated at the center of the sample, is equal to

Mmax = FLx = DEκmax = DE1R

(9)

where DE∝EWh3 is the elastic bending stiffness and R is the radiusof curvature at the center of the crease. In the case of an elastica withelastic material properties and negligible width, the geometry isself-similar, meaning that we can assume that all geometric param-eters scale together:

R

h∝d

h

Lxh∝d

h

(10)

Combining Eqs. (9) and (11) yields the expression:

F

EWh∝

d

h

( )−2

(11)

which explains the power law observed in the elastic regime inFig. 10(b). In the plastic regime, there are two possible differences.First, the softening on the material due to the yielding will affectEq. (9). Second, the geometry will deviate from the self-similar

Fig. 10 (a) Applied force versus platen distance, showing datafrom experiments, elastica analysis and the finite elementmodel. (b) Non-dimensionalized results, where E is the elasticmodulus, W is the sample width, and h is the film thickness.The transition between the elastic and the plastic regime is atd/(2h)≈20.

(a)

(b)

Fig. 11 (a) Schematic of all dimensions associated with the filmbending during creasing and (b) variation of different ratiosbetween the dimensions, as a function of the normalized platendistance d/(2h)

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solution behind the relationships in Eq. (10), with curvature local-izing at the crease due to material softening. To explore the relativeinfluence of both factors, we plot the two geometric parameters (R/hand Lx/h) and the normalized instantaneous bending stiffness (D/DE) as a function of the normalized platen distance d/(2h). In all

cases, the values are obtained using the one-dimensional model.The values for d/(2h) > 20 correspond to the elastic response dis-cussed previously. There is a slight decrease in the slope of R/hand an increase in the slope of Lx/h for small values of d/(2h),which agrees with the expected localization of curvature at thecrease. The effect is however small, and the slope of both curvesdoes not change significantly, meaning that the shape of theplastic elastica is still close to the self-similar elastic solution. Thevariation in bending stiffness, obtained through numerical differen-tiation of the results in Fig. 8(b), is much more significant, due to thehigh plastic strains at the crease. Our results indicate, therefore, thatthe reduced slope in the plastic power law in Fig. 10(b) (approxi-mately −1.6) is largely driven by the material softening and notby the subsequent change in geometry.

4.2 Predicting the Fold Angle. We now focus on the equilib-rium fold angle resulting from the folding procedure, i.e., the angleof the crease under zero applied moment. Previous studies [31,38],as well as the experimental results presented in Fig. 5(c), showedthat the equilibrium angle expands over time, due to the viscoelasticnature of Kapton. The results presented in this section correspond tothe initial equilibrium angle, which is the first-angle recorded afterthe creation of the crease (30 s after the pressing force is removed).We first explore the influence of the other timescale relevant to

the problem: the press time tpress, i.e., the amount of time that thefolding force F is applied for a given specimen. Figure 12 showsthe initial equilibrium angle for five coupons with h= 127 μm,which have been pressed with a force F = 30N for different

Fig. 12 Initial equilibrium fold angles for coupons with h=127 μm, pressed with a force F = 30N, as a function of thepress time tpress. Inset plots the same data in logarithmic scale.The line is not a fit, and simply illustrates exponential decay.

Fig. 13 (a) Initial equilibrium fold angle when the tpress and F parameters are varied in the experiments, along withpredictions from elastica and finite element model. (b) Same data, with non-dimensionalized force in a logarithmicscale. (c) Initial equilibrium fold angle as a function of the normalized platen distance d/(2h) for elastica and finiteelement predictions.

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holding times. When the force is applied for 1 s, the equilibriumfold angle is approximately 130 deg, and it decreases as the presstime is increased. The equilibrium fold angle reaches a plateauvalue around 95 deg for samples pressed for over 2 h, which repre-sents a 35-deg reduction. This behavior is consistent with the Arrhe-nius activation mechanism suggested by Thiria and Adda-Bedia [38].Next, we investigate the dependence of the equilibrium fold angle

on the pressing force for 50.8 μm and 127 μm thick samples, withdata from 36 different experiments shown in Fig. 13(a). Three dif-ferent values of the press time tpress have been applied: 1 s, 2 min,and 24 h. The results show that φ decreases as either the appliedforce F or the press time tpress increases. The equilibrium foldangles resulting from the elastica and the finite element model arealso plotted for comparison. Since viscoelasticity has not beenmodeled, the results from the model do not depend on the presstime. They capture the same trend as the experimental results, butoverpredict the effect on the resulting crease (i.e., predict smallervalues of φ).Figure 13(b) shows the same data, with the force normalized

using the same scaling as in Fig. 10(b) and presented in a logarith-mic scale. This eliminates the thickness dependence, and revealstwo regimes. For values of F/(E W h) < 50 × 10−4, the equilibriumangle transitions from φ = 180 deg (flat, i.e., no plasticity) tovalues in the range 160 deg ≤ φ ≤ 180 deg. This corresponds tocases in which the plastic deformation has not yet concentrated ina sharp crease. For F/(EWh) > 50 × 10−4, there is a clear relationshipof the form F/(EWh) ∝ 10−φ/φ0 , with φ ≈ 100 deg, which applies toboth experiments and analysis. The results now show a clear dis-tinction between the different values of tpress in the experimentalresults and that they approach the predictions as the press timeincreases.The elastica and finite element predictions in Fig. 13(b) show a

small disagreement in the equilibrium angle. To further explorethis difference, Fig. 13(c) shows the equilibrium angle as a function

of the normalized platen distance d/(2h) for both models. In thiscase, the agreement is very close, which indicates that the equilib-rium angle (and therefore the plastic deformation) is dictated bygeometry, while the deviations in applied force are attributed tothe different assumptions in the elastica constitutive model andour finite element analysis. However, as shown in Fig. 6, force isthe preferred control parameter for the experiments, due to theextreme sensitivity to small deviations in the platen distance.

4.3 Through Thickness Stress Profile. The one-dimensionalelastica model and the finite element analysis are in good agreementwith respect to the two macroscopic predictions considered, forceduring creasing and equilibrium angle. We now explore their agree-ment with respect to the local stress and strain fields, which areimportant to evaluate possible failure of the film or damage in thecoating. In this case, we will compare analysis with the sameplaten distance d, since as shown in Fig. 13(c), this is a goodcontrol parameter in simulations that do not suffer from the smallimperfections in alignment observed in the experiments.Figure 14(a) compares the axial stress predicted by both models

at the crease line, for a sample with h= 50.8 μm pressed to platendistances d= {240, 400, 800} μm. The horizontal axis representsthe through-the-thickness dimension in the undeformed configura-tion. Two main differences are observed. First, the stress distribu-tion produced by the finite element is not symmetric with respectto the geometric centroid. The reason is that, due to the Poisson’seffect, the region in tension contracts, and the region in compressionexpands, which results in asymmetries between the tension andcompression sides of the specimen. This also results in an overalldecrease in the film thickness (up to 2% variation for d=240 μm). The second difference is that the finite element model pre-dicts higher maximum stress than the elastica, with deviations of17%, 29%, and 50% for decreasing values of d. This is attributed

Fig. 14 (a) Axial stress profile when a 50.8 μm thick sample is pressed to d=240 μm, 400 μm, and 800 μm,(b) the strain profile when the force is removed, (c) the stress arrangement when the force is removed, and(d ) stress contours obtained from the finite element analysis, for the case d=400 μm

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to the idealizations behind the elastica model, which include the factthat stresses are calculated in the undeformed geometry.Figure 14(b) shows the axial plastic strain in the crease. The finite

element predictions show different slopes in tension and compres-sion, again due to the relative compression and expansion of bothregions, respectively. The maximum strains are significant, with|εp| ≥ 0.3 for d= 240 μm. These plastic strains result in significantresidual axial stresses in the unloaded configuration, see Fig. 14(c),with three inversion points along the film thickness. The predictionsare again very close for moderate platen distance (d= 800 μm), withsignificant deviations for highly compressed creases (d = 240 μm).The maximum residual stresses are on the order of 25% of themaximum stresses observed during loading.Finally, Fig. 14(d ) plots the axial stress contours during creasing,

and the residual axial stress, for the case with d= 400 μm. Thevalues at the symmetry line are therefore identical to those inFigs. 14(a) and 14(c), and in both cases, the stress decreasesrapidly with the arc-length. The crease is concentrated on aregion whose size is on the same order of magnitude as the filmthickness, which further validates the usual modeling approach oftreating the crease as a hinge of negligible width.

5 Conclusion and Future WorkWe have investigated the mechanics controlling the creation of a

plastic fold on thin Kapton polyimide films. In our experiment, wecompressed a previously bent coupon between two parallel com-pression platens. By controlling the pressing force, the platen dis-tance, and the total pressed time, we have been able to identifythe relative influence of each parameter, as well as the importanceof the viscoelastic properties of the film. The resulting procedureis able to create creases with consistent equilibrium angle. Theexperiments were then compared with the results from a one-dimensional elastica model and a high fidelity finite element simu-lation. By considering the non-linear geometry of the experimentand the elasto-plastic properties of Kapton, the two models wereable to predict the force-platen distance relationship accurately,while the equilibrium fold angles were under-predicted. Webelieve the discrepancy is due to the material viscoelasticity,which was not included in either of the models. Future work isfocused on developing a viscoelastic-plastic constitutive modelfor our numerical simulations, using the results from creep andrelaxation tests on Kapton coupons. We anticipate that capturingtime dependence will not only be important to predict the residualangle of the crease, but also its rotational stiffness, which is a keyelement to understand multistability in origami structures [41].The results from the elastica model have been non-

dimensionalized using scaling analysis. This has identified thekey parameters in the problem, as well as the different powerlaws governing the elastic and elasto-plastic regimes during thecompression process. These relations can be useful to extend ourpredictions to other geometries and materials, such as ultra-thinfilms where experiments are challenging. We also hope to extendour work to different geometries and loading conditions, in orderto find a connection with the scaling laws observed in the crumplingof thin sheets [42–44], where the interaction of non-parallel foldsand self-contact play an important role.

AcknowledgmentThis work has been supported by National Research Council,

Sri Lanka, under grant No. 14-023 and Ministry of Science Tech-nology and Research, Sri Lanka, under Indo-Sri Lanka ResearchGrant MSTR/TRD/AGR/3/02/09.

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