Formation of Plastic Creases in Thin Polyimide Films

Home , CubeSail, Polyimide

B. Yasara Dharmadasa Ann and H.J. Smead Department of Aerospace Engineering Sciences, University of Colorado Boulder, Boulder, CO 80309 e-mail: [email protected] Matthew W. McCallum Ann and H.J. Smead Department of Aerospace Engineering Sciences, University of Colorado Boulder, Boulder, CO 80309 e-mail: [email protected] Formation of Plastic Creases Seyon Mierunalan in Thin Polyimide Films Department of Civil Engineering, University of Moratuwa, We present a combined experimental and analytical approach to study the formation of Katubedda 10400, Sri Lanka creases in tightly folded Kapton polyimide films. In the experiments, we have developed a e-mail: [email protected] robust procedure to create creases with repeatable residual fold angle by compressing initially bent coupons. We then use it to explore the influence of different control parameters, Sahangi P. Dassanayake such as the force applied, and the time the film is being pressed. The experimental results Department of Civil Engineering, are compared with a simplified one-dimensional elastica model, as well as a high fidelity University of Moratuwa, finite element model; both models take into account the elasto-plastic behavior of the Katubedda 10400, Sri Lanka film. The models are able to predict the force required to create the crease, as well as e-mail: [email protected] the trend in the residual angle of the fold once the force is removed. We non-dimensionalize our results to rationalize the effect of plasticity, and we find robust scalings that extend our Chinthaka H. M. Y. findings to other geometries and material properties. [DOI: 10.1115/1.4046002] Mallikarachchi Keywords: constitutive modeling, material properties, thin-films, plastic creases Department of Civil Engineering, University of Moratuwa, Katubedda 10400, Sri Lanka e-mail: [email protected] Francisco Lopeź Jimeneź 1 Ann and H.J. Smead Department of Aerospace Engineering Sciences, University of Colorado Boulder, Boulder, CO 80303 e-mail: [email protected]

1 Introduction polymer films such as Kapton and Mylar are commonly used for these space applications. Gossamer space structures often make use of thin films and mem- Two different surface disturbances are observed in membranes branes that are tightly compacted and stowed before launch, and and thin films: wrinkles and creases. Figure 1 shows images of then deployed in space. As an early example, in 1960, NASA suc- the IKAROS solar sail demonstrator during stowage and after cessfully launched ECHO, a reflective satellite balloon that inflated being deployed, where both surface irregularities are clearly to a diameter of 100 feet [1]. Inspired by the success of ECHO, the visible [9]. Wrinkles are temporary distortions that occur due to Inflatable Antenna Experiment (IAE) satellite was launched where compressive buckling in thin films, which remain in the elastic inflated tubes formed a rigidized space antenna structure [2]. Thin regime [10,11]. As such, they are a consequence of the loading con- films are also used for solar sails, a novel propulsion concept in ditions and geometry and disappear once those are corrected and the which the thrust is generated from the impulse of solar photons. membrane is under tension [12]. Creases, on the other hand, are per- A milestone in solar sailing is the IKAROS solar sail demonstrator manent features caused by the highly localized plastic deformation project, where JAXA successfully carried out an orbital deployment that takes place when the film is folded to a very tight radius of cur- of a 196 square meters sail [3]. This technology is ideal for low-cost, vature, which is often the case during the packaging of deployable lightweight CubeSats. It is currently implemented in CubeSail and structures. Figure 2 summarizes the process in which a crease is LightSail [4,5] and has been proposed for future missions such as created, and how it alters the mechanical response of a film. the near-Earth asteroid scout [6]. Other uses of thin films in space Under folding force F, the deformation is highly localized in the include the HabEx starshade [7] and the deployable sun-shield for crease region, which results in permanent deformation of the film. the James Webb Space Telescope [8]. Thermally stable metallic Once the force is removed, the permanent curvature in the crease φ results in an equilibrium fold angle 0. Under tensile loading Ftensile, the fold angle varies, but since φ < 180 deg, there is 1Corresponding author. always a shortening of the in-plane film length compared to the pris- Contributed by the Applied Mechanics Division of ASME for publication in tine condition. The pretension due to this shortening, as well as the the JOURNAL OF APPLIED MECHANICS. Manuscript received November 9, 2019; final manuscript received January 12, 2020; published online January 17, 2020. Assoc. increased bending stiffness due to the out-of-plane deformation, has Editor: Yihui Zhang. a significant effect on the natural frequencies of a membrane

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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) fold angle expanding (φ1 > φ0) and film bending when subjected to tensile load Ftensile, and (e) shortening of the projected film length due to the crease

Fig. 1 (a) Stowed and (b) deployed state of IKAROS solar sail [9] (Reprinted with permission of Elsevier Ltd. © 2011) are often assumed to remain flat. In both cases, it is necessary to carefully characterize the mechanical behavior of the crease; in the case in which it can be assumed to be linear [25], it can be structure. This is the proposed explanation for the difference described by the equivalent rotational stiffness (k) and the equilib- φ between simulations and experimental results for the deployment rium angle under no applied loading ( 0). dynamics of IKAROS [13], which can result in problems with atti- The mechanical properties of creases have been explored exper- tude control and tearing or entangling of the film. Out-of-plane dis- imentally in single creases [21,26] and Z-folds [27,28], as well as tortions caused by the creases can also affect the thrust vector at low using a linkage mechanism able to provide pure bending to arbi- membrane tension [14], and tight packaging can destroy the protec- trarily high curvature [29]. The method to create the crease can sig- tive coatings on the film [15]. Hence, understanding and modeling nificantly affect the results, and even for a given method, it is often the crease behavior is crucial for the successful design of thin film difficult to achieve repeatable folds to use in the experiments structures, from space deployable structures to origami-based [17,26,30,31]. Furthermore, films used in space structures often metamaterials. have a thickness in the micrometer range [32], so measuring Several models have been presented to describe the effect of creases in the mechanical properties of films. Murphey [16] calculated the homogenized stress–strain relationship of a randomly creased sheet when the crease amplitudes and wavelengths are mea- (a)(b) sured, without explicitly resolving the deformed film profile. Hossain et al. [17] characterized the non-linear stress–strain relationship of a crease and modeled it in a finite element framework by defining a softer non-linear material strip. To capture the out-of-plane stiffness, Nishizawa et al. [13] proposed to add beam elements with a second moment of area equivalent to that of the creased region; this model was able to capture the natural frequencies of the membrane without explicitly accounting for the deformed geometry. The effect of creases on pressurized cylinders has been modeled numerically, taking into account the mechanics of a single crease [18,19]. However, the most common approach to model creases in the solar sail deployment simulations is to idealize them as hinges with torque springs, see Fig. 3. This technique accounts for the stiffness of the hinge, as well as for its effect on the Fig. 3 Torque spring idealization of a crease: (a) unstressed film profile [20,21]. A similar approach is used to model the stiff- state and (b) under tensile loading, showing opening of the ness of fold lines in origami [22–24], although in this case, panels crease as well as bending in the film

051009-2 / Vol. 87, MAY 2020 Transactions of the ASME creasing and unfolding forces becomes challenging. Numerical possible rearrangement of their deformed geometry. As such, the simulations have been used to analyze the creasing process for creases created following both approaches, loading and displace- thicknesses and loading conditions that present difficulties for ment control, are different. This is addressed in detail in Sec. 2.2. experimental exploration. These include one-dimensional simpli- The creasing process is imaged with a USB video microscope fied models as well as solid-based numerical simulations accounting (Mighty Scope 5M). for the elasto-plastic nature of the film [33–36]. However, there is a After the hold period tpress, the top platen is raised at a rate of lack of experimental validation of the predictions for the equilib- 100 mm/min, and the scotch tape is cut, allowing the crease to rium angle. Furthermore, the effect of the viscoelasticity of the unfold. The coupons are then suspended from one end such that film is usually neglected, and so, its influence on the process is the crease line is parallel to gravity. This helps to minimize the not well understood. gravity effects and friction with a possible support. As soon as In this study, we combine experiments and analysis to rationalize the coupons are suspended (which takes approximately 30 s after the different factors that contribute in the formation of a crease, and the end of the test), we capture images of the equilibrium fold how they affect the equilibrium fold angle. In particular, we will angle every 10 s for a period of 5 min using a Nikon D610 focus on establishing a repeatable experimental procedure to camera (24.3 megapixels, AF-S Micro Nikkor 60 mm lens; crease formation that can be used in future experimental studies. Fig. 4(c)). The reason to track the evolution of the angle over We then use the experimental results, as well as analysis, to time is that previous studies have shown that the viscoelastic explore the relative effect of the folding parameters on the resulting nature of thin films results in the opening of the crease angle over crease. The paper is organized as follows. In Sec. 2, we describe the time [31,38]. The equilibrium fold angle in each image is measured experimental setup, discussing key parameters that lead to a robust with the help of a MATLAB script that fits straight lines to the flat por- protocol. Section 3 describes the two modeling approaches fol- tions right next to the crease, see Fig. 4(d). lowed: a simplified one-dimensional elastica model and a fully fi resolved nite element model. Both models account for the elasto- 2.2 Influence of Test Control Parameter. Figure 5 compares plastic behavior in the film. Section 4 presents and compares our fi the results of experiments under force control and platen distance experimental and analytical results. Section 5 summarizes our nd- control, in order to illustrate the difference between both proce- ings and discusses planned future work. dures. We use F and d to indicate the values at which both force and distance are held constant, respectively. First, a coupon of 2 Experimental Setup thickness h = 127 μm is compressed to a platen distance d = 380 μm and held at that configuration for t = 120 s. As the top The present study focuses on the creation of a single crease in a press fi platen is lowered at a uniform rate, the force increases exponen- rectangular piece of polyimide lm, by applying compressive force tially, and once the platen halts at the final pressing configuration, using compression platens in a universal testing machine. The mate- fi the force starts reducing with time, at a decaying rate, see rial used is Kapton (DuPont) lm, whose high durability and stable Figs. 5(a) and 5(b). We observe a peak force of 51 N when the properties over a large temperature range make it ideal for space platen halts and a plateau value of 32 N at the end of the holding applications [37]. Kapton properties are shown in Table 1, as pro- period. Next, two samples of the same thickness are pressed up to vided by the manufacturer. F = 51 N and F = 32 N and held for the same amount of time, tpress = 120 s. When the force is held constant, there is a slight 2.1 Specimen Preparation and Test Procedure. All speci- decrease in platen distance in order to account for the relaxation mens are obtained from initially flat films, in order to avoid the in the film (Fig. 5(a)). We believe material viscoelasticity is respon- residual curvature observed in rolled films. We consider two differ- sible for the force and curvature rearrangements during the holding ent thicknesses: h = 50.8 μm and h = 127 μm (0.002 in and 0.005 in, phase. Although this is similar to the effect observed in creep and respectively). We cut rectangular coupons of width W = 25.4 mm and total length 101.6 mm and check for visible defects such as cuts and stretch marks (Fig. 4(a)). The coupons are lightly bent and held in that shape using scotch tape, such that they can be placed between the compression platens of a universal testing machine (Instron 5969, 1 kN load cell). The length of the samples is such that this initial curvature results in strains at least one order 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 of 10 mm/min, until the desired applied force F or platen distance d is reached. The specimens are then held under compression over a given pressing time tpress in which either the applied force or the distance are held constant, see Fig. 4(b). The reason to distinguish between the two control parameters is that, due to the viscoelastic nature of Kapton, there is stress relaxation in the samples, and

Table 1 Material properties of Kapton HN polyimide

Property Value

Density (kg/m3) 1420 Ultimate tensile strength (MPa) 231 Young’s modulus (GPa) 2.5 Fig. 4 Experimental setup: (a) test coupon bent using a tape, Poisson’s ratio 0.34 (b) universal testing machine applying compression to the speci- Yield stress (MPa) 69 men, (c) set-up used to capture images of suspended coupon Yield strain (%) 3 using a magnet, and (d) fold-angle measurement using MATLAB script

Journal of Applied Mechanics MAY 2020, Vol. 87 / 051009-3 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.

relaxation tests, in the present experiments neither stress nor strain This prevents us from producing a closed-form solution, and we is constant along the length of the film. This important difference solve the equations numerically. Second, we carry out a more means that changes in the global response are due to both stress detailed finite element analysis of the film using the commercial relaxation, as well as small readjustments in geometry due to differ- finite element package ABAQUS. Both models consider large deflec- ential softening between regions of the film. Figure 5(c) shows the tions and the elasto-plastic behavior of Kapton. The effect of gravity evolution angle for the three specimens, with large differences during the pressing of sample was neglected in modeling, since the observed. The angles increase over time, which is consistent with self-weight is at least three orders of magnitude lower than the previous studies [31,38]. applied pressing force. Electrostatic forces are also neglected, An important difference between the two creasing approaches is since they have been shown to not have a significant influence on the repeatability of the results. In Fig. 6, we fold three coupons each films with the range of thickness used in our experiments [39]. (127 μm thick) to either d = 635 μmorF = 30 N. The compression platens were detached and reassembled between each experiment, to increase the possible differences between tests. We observe 3.1 One-Dimensional Elastica Model. The theory of elastica that controlling the applied force produces a repeatable fold angle is a geometrically non-linear model that describes the behavior of (deviations of ±2 deg between samples), while the fold angles slender structures undergoing large deflections [40]. Figure 7(a) obtained when controlling the platen distance have higher devia- shows that the crease region can be idealized as a one-dimensional tions (±15 deg), see Fig. 6(c). This can be explained by considering element with symmetric shape (indicated by green dashes). Further- the force curve in Fig. 6(b): the three samples with the same d reach more, we idealize the load exerted by the compression platens onto very different maximum forces, due to the high slope of the force– the sample as a vertical point load F, at the point in which the speci- displacement curve at that point. The images from the USB micro- men is tangent to the platen. According to these assumptions, we scope during testing showed that the compression platens are not only need to analyze a cantilever of length Lc, which is fixed per- perfectly parallel and can have a small misalignment (up to 0.05 pendicular to the symmetry plane (Fig. 7(b)). The force F is deg), which varies from test to test. This creates small variations applied on a single point at the opposite end, and we neglect the in d, on the order of 10 μm, which nevertheless result in large var- rest of the specimen beyond the contact point. We assume the speci- iations in F, which explains the deviations between samples. By men to be inextensible, and the plane sections to remain plane while specifying F, the fold angles are more consistent, and it is the deformed. Besides considering the non-linear geometry, we need to method used to obtain the results in Sec. 4. consider the material non-linearity and the varying length of Lc as the test progresses, which represents a significant variation from the traditional elastica analysis. The internal bending moment at a 3 Modeling and Analysis point of arc-length s ∈ (0, Lc), measured from the point of symmetry We model the creasing process using two different approaches. can be expressed as fi First, we idealize the lm as a one-dimensional element using M(s) = FL()− xs() (1) Euler’s theory of elastica. The approach is similar to the analysis x presented by Secheli et al. [34], but we use experimental data for where Lx is the distance in the X-direction between the point of sym- the plastic behavior of Kapton instead of an idealized behavior. metry and the applied loading. The shape of the specimens is

Fig. 6 Comparison of repeatability in experiments 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. Line and point styles indicate different control parameters, colors indicate different nominally identical tests.

051009-4 / Vol. 87, MAY 2020 Transactions of the ASME (a)(b) (a)

Fig. 7 (a) Snapshots of the film from the microscope. Dashed lines emphasize the region between the points of contact, that are idealized as (b) a cantilever with a point load at the opposite end (Color version online.) (b) obtained by solving the following system of equations: s θ(s) = κ(ξ) dξ 0 s x(s) = cos(θ(ξ)) dξ (2) 0 s y(s) = sin(θ(ξ)) dξ 0 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 angle between the tangent vector and the X-axis, as defined in Fig. 7(b). The boundary conditions to solve the equations are θ(0) = π/2 and θ(Lc) = 0. In addition, we need to enforce that Lx = x(Lc). The ′ vertical position of the end point is y(Lc) = d /2, which is related to the platen distance by d′ = d − h to account for the thickness of Fig. 8 (a) In-plane tensile stress–strain relationship for Kapton the film, which becomes important for small values of d. [37], showing different loading histories. The curve has been In order to solve the system of Eqs. (1) and (2), we need a mate- extended beyond the last reported value in the data, at ɛ ≈ 0.7, b rial relationship between the moment M(s) and the curvature κ(s)at with a straight line with the same slope. ( ) Moment-curvature relationship obtained integrating the stress relationship. The every point of the specimen. In the case of a linear elastic material, – plot shows loading (red) as well as the unloading (blue dashed) the stress strain curve is injective, and this relationship can be paths when the surface strain at the point of maximum loading fi = κ uniquely de ned by M DE , where DE is the linear elastic is 0.03, 0.3, 0.6 and 0.99, respectively. (Color version online.) bending stiffness. However, this is not the case when considering the material plasticity of Kapton, where the loading history needs to be considered. This is illustrated in Fig. 8(a), which shows the by enforcing that the axial force is zero: stress–strain response of Kapton [37]. If the load remains in the / −ζ elastic region (path OA), loading and unloading follow the same h 2 n path in the stress–strain plane, and the stress can be uniquely Faxial(κmax, κ) = σ(εmax, ε) dζ = 0 (5) − / −ζ defined by the strain at that point. Once the loading at any point h 2 n in history results in strains higher than the yield strain, the stress where κ is the current curvature at a specific point and κ is the at a given strain depends also on the loading history. The loading max ′ ′ maximum curvature experienced by the same material point at ε = ε ′ paths OBC and OB C end with the same strains C C , but dif- any point in its time history. Equation (4) is used to calculate ferent stresses σC ≠ σC′. Assuming that the unloading follows the εmax and ε as a function of κmax and κ, respectively. In our case, same slope as the initial linear regime, with stiffness E, each of we assume that the stress–strain relationship is the same in the stresses can be calculated using: tension and compression, and the calculation is performed on the ζ = undeformed geometry, which yields n 0. σ(εmax, ε) = σmax − E(εmax − ε) (3) The moment–curvature relationship is then calculated as: h/2 where σmax and εmax are the maximum stress and strain experienced M(κmax, κ) = σ(εmax, ε) ζ dζ (6) by that material point in its loading history, respectively. −h/2 In our problem, the strain at every point is given by the curvature ζ − ζ fi and the through the thickness distance to the neutral axis n: Equation (6) shows that the moment at a material point is de ned by a combination κ and κmax. To speed up the analysis, we calculate = κ , κ ε = κζ− ζ and save the moment values, Msaved M( max ), by numerically n (4) integrating Eq. (6), taking 100 through thickness points and using the material properties of Kapton. The κmax and κ were discretized with no dependance on the loading history. The position of the at increments of Δκmax = 0.02/h and Δκ = 0.02/h, corresponding ζ fi neutral axis with respect to the geometric centroid n is obtained to increments of 1% strain at the surface of the lm. Figure 8(b)

Journal of Applied Mechanics MAY 2020, Vol. 87 / 051009-5 plots the calculated moment–curvature relationship, normalized to We use symmetry along the fold line to model half of the film and eliminate the thickness dependence, and uniquely defines the increase the computational efficiency. We assume plane strain beha- current curvature κ for a given moment M and the maximum curva- vior to further simplify the problem. Four-node plane strain quadri- ture during the loading history κmax. lateral elements (CPE4) are used in the analysis, and the mesh As the test progress, the applied force F increases and the beam density is varied along the film length, so that the region near the length Lc needs to shorten to satisfy the boundary conditions. The symmetry end has around 50 elements through the thickness. A change in Lc is the main reason for the need to account for κmax typical finite element model consisted of 9898 nodes and 8631 ele- in the moment–curvature relationship. For a given arc-length, ments. The compression platen is modeled as a rigid beam using s (0 M(s =s) > M(s = Lc) = 0. Initially, two node beam elements (RB2D2), and the compression force is M(s) increases as the applied force is increased, but decreases as transferred by defining frictionless contact between the film and Lc approachess, since M(s = Lc) = 0. the beam. The final algorithm needs to take into account the non-linear The creasing of the film occurs over several steps. First the nodes geometry, the material non-linearity, and the fact that the length along the symmetry line are fixed (Ux = Uy = 0) and the free edge is of the specimen under loading decreases as the point of contact displaced in order to bend the film, as shown in Fig. 9(b). The rigid with the platens approaches the symmetry line. The following iter- plate moves closer to the symmetry line and the free edge is released, ative algorithm is utilized to find a solution that satisfies all the allowing the film and the plate to make contact (Fig. 9(c)). Next, the requirements. symmetric edge boundary is set to rollers in the X-direction (Uy = 0), while fixing a single node to prevent rigid body movements. The (1) First, we choose a value of the initial specimen length L = c film is pressed by moving the rigid plate up to a specified platen dis- L , which is sufficiently large to ensure that all points are c0 tance d/2 (Fig. 9(d)), and then the film is released by turning off the in the elastic regime. contact definition. Figure 9(e) illustrates the resulting equilibrium (2) We initialize all maximum curvatures to be equal to zero, fold angle, which can be measured by calculating the angle of the κ (s) = 0 for all values of s. The arc-length is discretized max straight portion with the X-axis. using 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)at every point is the one that verifies Msaved(κ(s), κmax(s)) = M(s). (c) Iterate the values of F and Lx until the boundary conditions are satisfied. This is accomplished by minimizing 2 2 the function G = ()Lx − xL()c + ()θ(Lc) . (4) The values of the maximum curvature are updated if neces- κnew = ()κ , κ sary, so that max(s) max max(s) (s) . new (5) We then decrease the value of the specimen length, Lc , and new go back to Step 3. The value of Lc is a multiple of Lc0/n,so that the same discretization of the arc-length dimension in Step 2 can be used throughout the simulation. The output of the algorithm is the values of F and d′ for every value of Lc considered, and the value of the maximum curvature at every point κmax(s), for the original range of the arc-length 0 < Lc < Lc0. The residual curvature at every point κres is obtained by imposing zero moment:

M()κmax, κres = 0 (7) which corresponds to complete unloading in the moment–curvature relationship. The equilibrium fold angle is obtained by integrating the residual curvature along the arc-length.

3.2 Finite Element Model. The accuracy of the elastica model decreases when the film is pressed to very small platen distances d ≈ 2h, since at that point the specimen can no longer be idealized as a slender beam. Furthermore, Poisson’s effect causes an expansion of the region under compression and a reduction of the region under tension, an effect neglected in our one-dimensional model that results in a shift of the neutral axis. To account for such effects, we used the commercial package ABAQUS/STANDARD to create a high fidelity elasto-plastic finite element model. We use the same coupon thickness as in the experiments, 50.8 μm and 127 μm, and an isotropic hardening plasticity model. The nominal stress (σnom) and strain (εnom) data obtained from the manufacturer [37] are con- verted to true stress (σ ) and true plastic strain (ε ) using tr pl Fig. 9 Finite element model for folding of thin films using rigid a fi fl fi b σ = σ (1 + ε) plates: ( ) The lm is at in the undeformed con guration, ( )it tr nom is initially bent by applying displacement boundary conditions, εtr = ln (1 + ε) (c) subsequently by contact with a rigid plate, (d) with the curva- (8) ture increasing as the plate moves closer to the symmetry line, ε = ε − 1 σ and (e) the final geometry shows permanent deformation once pl tr E tr the loading is removed

051009-6 / Vol. 87, MAY 2020 Transactions of the ASME 4 Results of the two flat regions of the specimen. In practice, values of d/(2h) < 2 correspond to situations in which the crease region is 4.1 Folding Force. We first focus on the folding process, and under significant compressive stress, which explains the higher in particular, the evolution of the applied force F as a function of the deviations between experiments and analysis. platen distance d. Figure 10(a) plots the results from different exper- The choice of non-dimensionalization in Fig. 10(b), as well as the imental runs on nominally identical samples, as well as the predic- power law revealed in the results, can be understood by considering tions from the elastica analysis and the finite element model. Both the sketch of the sample geometry in Fig. 11(a). Considering first models are able to predict the force–displacement variation accu- the elastic regime, the maximum bending moment, which is rately over the whole range of forces used in the experiments. located at the center of the sample, is equal to The two variables are inversely related, with the force increasing exponentially as the distance between the plates decreases. The 1 experimental results correspond to 15 different tests with h = Mmax = FLx = DEκmax = DE (9) R 127 μm and 10 tests with h = 50.8 μm. 3 In order to further rationalize our results, Fig. 10(b) shows the where DE ∝ EWh is the elastic bending stiffness and R is the radius same data plotted in a logarithmic scale, and non-dimensionalized of curvature at the center of the crease. In the case of an elastica with so that the results from both values of h collapse onto a single elastic material properties and negligible width, the geometry is curve. To identify the effect of plasticity, we also plot the results self-similar, meaning that we can assume that all geometric param- of an elastica analysis with a linearly elastic constitutive material eters scale together: model. The elastic regime corresponds to the range d/(2h) > 20, where the slope is −2, indicating a power-law relationship R ∝ d h h between normalized force and normalized platen distance. At (10) d/(2h) ≈ 20, we observe deviations between the elastic and plastic Lx ∝ d behavior. Remarkably, the behavior in the plastic regime is also h h close to a power law of slope approximately equal to −1.6 within the whole range considered. The smallest value of the normalized platen distance d/(2h) = 1 corresponds to the case in which the Combining Eqs. (9) and (11) yields the expression: separation between the plates is equal to the combined thickness − F d 2 ∝ (11) EWh h which explains the power law observed in the elastic regime in Fig. 10(b). In the plastic regime, there are two possible differences. First, the softening on the material due to the yielding will affect Eq. (9). Second, the geometry will deviate from the self-similar

~~(a)~~

~~(b)~~

Fig. 10 (a) Applied force versus platen distance, showing data from experiments, elastica analysis and the finite element model. (b) Non-dimensionalized results, where E is the elastic Fig. 11 (a) Schematic of all dimensions associated with the film modulus, W is the sample width, and h is the film thickness. bending during creasing and (b) variation of different ratios The transition between the elastic and the plastic regime is at between the dimensions, as a function of the normalized platen d/(2h) ≈ 20. distance d/(2h)

Journal of Applied Mechanics MAY 2020, Vol. 87 / 051009-7 solution behind the relationships in Eq. (10), with curvature local- cases, the values are obtained using the one-dimensional model. izing at the crease due to material softening. To explore the relative The values for d/(2h) > 20 correspond to the elastic response dis- influence of both factors, we plot the two geometric parameters (R/h cussed previously. There is a slight decrease in the slope of R/h and Lx/h) and the normalized instantaneous bending stiffness (D/ and an increase in the slope of Lx/h for small values of d/(2h), DE) as a function of the normalized platen distance d/(2h). In all which agrees with the expected localization of curvature at the crease. The effect is however small, and the slope of both curves does not change significantly, meaning that the shape of the plastic elastica is still close to the self-similar elastic solution. The variation in bending stiffness, obtained through numerical differen- tiation of the results in Fig. 8(b), is much more significant, due to the high plastic strains at the crease. Our results indicate, therefore, that the reduced slope in the plastic power law in Fig. 10(b) (approximately −1.6) is largely driven by the material softening and not by the subsequent change in geometry.

4.2 Predicting the Fold Angle. We now focus on the equilibrium fold angle resulting from the folding procedure, i.e., the angle of the crease under zero applied moment. Previous studies [31,38], as well as the experimental results presented in Fig. 5(c), showed that the equilibrium angle expands over time, due to the viscoelastic nature of Kapton. The results presented in this section correspond to the initial equilibrium angle, which is the first-angle recorded after the creation of the crease (30 s after the pressing force is removed). We first explore the influence of the other timescale relevant to Fig. 12 Initial equilibrium fold angles for coupons with h = the problem: the press time tpress, i.e., the amount of time that the 127 μm, pressed with a force F = 30 N, as a function of the folding force F is applied for a given specimen. Figure 12 shows fi = μ press time tpress. Inset plots the same data in logarithmic scale. the initial equilibrium angle for ve coupons with h 127 m, The line is not a fit, and simply illustrates exponential decay. which have been pressed with a force F = 30 N for different

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

051009-8 / Vol. 87, MAY 2020 Transactions of the ASME holding times. When the force is applied for 1 s, the equilibrium of the normalized platen distance d/(2h) for both models. In this fold angle is approximately 130 deg, and it decreases as the press case, the agreement is very close, which indicates that the equilib- time is increased. The equilibrium fold angle reaches a plateau rium angle (and therefore the plastic deformation) is dictated by value around 95 deg for samples pressed for over 2 h, which repre- geometry, while the deviations in applied force are attributed to sents a 35-deg reduction. This behavior is consistent with the Arrhe- the different assumptions in the elastica constitutive model and nius activation mechanism suggested by Thiria and Adda-Bedia [38]. our finite element analysis. However, as shown in Fig. 6, force is Next, we investigate the dependence of the equilibrium fold angle the preferred control parameter for the experiments, due to the on the pressing force for 50.8 μm and 127 μm thick samples, with extreme sensitivity to small deviations in the platen distance. data from 36 different experiments shown in Fig. 13(a). Three different 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 applied 4.3 Through Thickness Stress Profile. The one-dimensional force F or the press time tpress increases. The equilibrium fold elastica model and the finite element analysis are in good agreement angles resulting from the elastica and the finite element model are with respect to the two macroscopic predictions considered, force also plotted for comparison. Since viscoelasticity has not been during creasing and equilibrium angle. We now explore their agree- modeled, the results from the model do not depend on the press ment with respect to the local stress and strain fields, which are time. They capture the same trend as the experimental results, but important to evaluate possible failure of the film or damage in the overpredict the effect on the resulting crease (i.e., predict smaller coating. In this case, we will compare analysis with the same values of φ). platen distance d, since as shown in Fig. 13(c), this is a good Figure 13(b) shows the same data, with the force normalized control parameter in simulations that do not suffer from the small using the same scaling as in Fig. 10(b) and presented in a logarith- imperfections in alignment observed in the experiments. mic scale. This eliminates the thickness dependence, and reveals Figure 14(a) compares the axial stress predicted by both models − two regimes. For values of F/(EWh) < 50 × 10 4, the equilibrium at the crease line, for a sample with h = 50.8 μm pressed to platen angle transitions from φ = 180 deg (flat, i.e., no plasticity) to distances d = {240, 400, 800} μm. The horizontal axis represents values in the range 160 deg ≤ φ ≤ 180 deg. This corresponds to the through-the-thickness dimension in the undeformed configura- cases in which the plastic deformation has not yet concentrated in tion. Two main differences are observed. First, the stress distribu- − a sharp crease. For F/(EWh) > 50 × 10 4, there is a clear relationship tion produced by the finite element is not symmetric with respect −φ/φ of the form F/(EWh) ∝ 10 0 , with φ ≈ 100 deg, which applies to to the geometric centroid. The reason is that, due to the Poisson’s both experiments and analysis. The results now show a clear dis- effect, the region in tension contracts, and the region in compression tinction between the different values of tpress in the experimental expands, which results in asymmetries between the tension and results and that they approach the predictions as the press time compression sides of the specimen. This also results in an overall increases. decrease in the film thickness (up to 2% variation for d = The elastica and finite element predictions in Fig. 13(b) show a 240 μm). The second difference is that the finite element model pre- small disagreement in the equilibrium angle. To further explore dicts higher maximum stress than the elastica, with deviations of this difference, Fig. 13(c) shows the equilibrium angle as a function 17%, 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

Journal of Applied Mechanics MAY 2020, Vol. 87 / 051009-9 to the idealizations behind the elastica model, which include the fact [2] Freeland, R., and Veal, G., 1998, “Significance of the Inflatable Antenna that stresses are calculated in the undeformed geometry. Experiment Technology,” 39th AIAA/ASME/ASCE/AHS/ASC Structures, fi Structural Dynamics, and Materials Conference and Exhibit, Long Beach, CA, Figure 14(b) shows the axial plastic strain in the crease. The nite Apr. 20–23, AIAA, p. 2104. element predictions show different slopes in tension and compres- [3] Tsuda, Y., Mori, O., Funase, R., Sawada, H., Yamamoto, T., Saiki, T., Endo, sion, again due to the relative compression and expansion of both T., Yonekura, K., Hoshino, H., and Kawaguchi, J., 2013, “Achievement of regions, respectively. The maximum strains are significant, with IKAROS—Japanese Deep Space Solar Sail Demonstration Mission,” Acta |ε |≥ . = μ fi Astronaut., 82(2), pp. 183–188. p 0 3 for d 240 m. These plastic strains result in signi cant [4] Biddy, C., and Svitek, T., 2012, “Lightsail-1 Solar Sail Design and Qualification,” residual axial stresses in the unloaded configuration, see Fig. 14(c), Proceedings of the 41st Aerospace Mechanisms Symposium, Jet Propulsion Lab., with three inversion points along the film thickness. The predictions National Aeronautics and Space Administration, Pasadena, CA, pp. 451–463. are again very close for moderate platen distance (d = 800 μm), with [5] Leipold, M., 2000, “Solar Sail Mission Design,” Ph.D. thesis, Deutsches Zentrum fi = μ für Luft-und Raumfahrt, Munich, Germany. signi cant deviations for highly compressed creases (d 240 m). [6] McNutt, L., Johnson, L., Kahn, P., Castillo-Rogez, J., and Frick, A., 2014, The maximum residual stresses are on the order of 25% of the “Near-Earth Asteroid (NEA) Scout,” AIAA Space 2014 Conference and maximum stresses observed during loading. Exposition, San Diego, CA, Aug. 4–7, AIAA, p. 4435. Finally, Fig. 14(d) plots the axial stress contours during creasing, [7] Mennesson, B., Gaudi, S., Seager, S., Cahoy, K., Domagal-Goldman, S., = μ Feinberg, L., Guyon, O., Kasdin, J., Marois, C., Mawet, D., Tamura, M., and the residual axial stress, for the case with d 400 m. The Mouillet, D., Prusti, T., Quirrenbach, A., Robinson, T., Rogers, L., Scowen, P., values at the symmetry line are therefore identical to those in Somerville, R., Stapelfeldt, K., Stern, D., Still, M., Turnbull, M., Booth, J., Figs. 14(a) and 14(c), and in both cases, the stress decreases Kiessling, A., Kuan, G., and Warfield, K., 2016, “The Habitable Exoplanet rapidly with the arc-length. The crease is concentrated on a (HabEx) Imaging Mission: Preliminary Science Drivers and Technical fi Requirements,” Space Telescopes and Instrumentation 2016: Optical, Infrared, region whose size is on the same order of magnitude as the lm and Millimeter Wave, Edinburgh, UK, June 26–July 1, Vol. 9904, International thickness, which further validates the usual modeling approach of Society for Optics and Photonics, p. 99040L. treating the crease as a hinge of negligible width. [8] Lightsey, P. A., Atkinson, C. B., Clampin, M. C., and Feinberg, L. D., 2012, “James Webb Space Telescope: Large Deployable Cryogenic Telescope in Space,” Opt. Eng., 51(1), p. 011003. [9] Tsuda, Y., Mori, O., Funase, R., Sawada, H., Yamamoto, T., Saiki, T., Endo, T., 5 Conclusion and Future Work and Kawaguchi, J., 2011, “Flight Status of IKAROS Deep Space Solar Sail Demonstrator,” Acta Astronaut., 69(9–10), pp. 833–840. We have investigated the mechanics controlling the creation of a [10] Cerda, E., Ravi-Chandar, K., and Mahadevan, L., 2002, “Thin Films: Wrinkling plastic fold on thin Kapton polyimide films. In our experiment, we of An Elastic Sheet Under Tension,” Nature, 419(6907), p. 579. compressed a previously bent coupon between two parallel com- [11] Wong, W., and Pellegrino, S., 2006, “Wrinkled Membranes I: Experiments,” pression platens. By controlling the pressing force, the platen dis- J. Mech. Mater. Struct., 1(1), pp. 3–25. [12] Sakamoto, H., Park, K., and Miyazaki, Y., 2005, “Dynamic Wrinkle Reduction tance, and the total pressed time, we have been able to identify Strategies for Cable-Suspended Membrane Structures,” J. Spacecraft Rockets, the relative influence of each parameter, as well as the importance 42(5), pp. 850–858. of the viscoelastic properties of the film. The resulting procedure [13] Nishizawa, T., Sakamoto, H., Okuma, M., Furuya, H., Sato, Y., Okuizumi, N., is able to create creases with consistent equilibrium angle. The Shirasawa, Y., and Mori, O., 2014, “Evaluation of Crease Effects on Out-of-Plane Stiffness of Solar Sails,” Trans. Jpn. Soc. Aeronaut. Space Sci., experiments were then compared with the results from a one- Aerosp. Technol. Jpn., 12(ists29), pp. Pc_107–Pc_113. dimensional elastica model and a high fidelity finite element simu- [14] Campbell, B. A., and Thomas, S. J., 2014, “Realistic Solar Sail Thrust,” Advances lation. By considering the non-linear geometry of the experiment in Solar Sailing, M. 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[19] Secheli, G., Viquerat, A., and Lappas, V., 2015, “An Examination of Crease The results from the elastica model have been non- Removal in Rigidizable Inflatable Metal-Polymer Laminate Cylinders,” 2nd dimensionalized using scaling analysis. This has identified the AIAA Spacecraft Structures Conference, Kissimmee, FL, Jan. 5–9, AIAA, p. 0681. key parameters in the problem, as well as the different power [20] Okuizumi, N., Muta, A., Matsunaga, S., Sakamoto, H., Shirasawa, Y., and Mori, laws governing the elastic and elasto-plastic regimes during the O., 2011, “Small-Scale Experiments and Simulations of Centrifugal Membrane compression process. These relations can be useful to extend our Deployment of Solar Sail Craft ‘IKAROS’,” 52nd AIAA/ASME/ASCE/AHS/ ASC Structures, Structural Dynamics and Materials Conference 19th AIAA/ predictions to other geometries and materials, such as ultra-thin – fi ASME/AHS Adaptive Structures Conference, Denver, CO, Apr. 4 7, AIAA, lms where experiments are challenging. We also hope to extend p. 1888. our work to different geometries and loading conditions, in order [21] Miyazaki, Y., Shirasawa, Y., Mori, O., Sawada, H., Okuizumi, N., Sakamoto, H., to find a connection with the scaling laws observed in the crumpling Matunaga, S., Furuya, H., and Natori, M., 2011, “Conserving Finite Element of thin sheets [42–44], where the interaction of non-parallel folds Dynamics of Gossamer Structure and Its Application to Spinning Solar Sail ‘IKAROS’,” 52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural and self-contact play an important role. 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