The effect of plasticity in crumplingof thin sheets: Supplementary

Information

T. Tallinen, J.A. Astrom and J. Timonen

Video S1. The video shows crumpling of an elastic sheet with a width tothickness ratio of L/h = 500 and a Young’s modulus of Y = 1 GPa.

Video S2. Crumpling of an elasto-plastic sheet. The sheet has the sameparameters as the one in Video S1, and in addition a yield stress of 10 MPa.

1. Simulation model

A model for thin sheets of elastic or elasto-plastic material was constructedas a triangular lattice with spacing a and size up to 1,000 x 1,000 latticepoints. Each lattice point had mass m and moment of inertia I, and theywere connected by beam elements. The beams had a 12 x 12 stiffness matrixcorresponding to three translational and three rotational degrees of freedomat both ends of the beam. The large thickness of the beams was accountedfor by including shear effects in the formulation of the stiffness matrix. Mo-tion of local degrees of freedom in the beams was opposed by small viscousdamping. The magnitude of damping was such that the motion of any sin-gle beam was under-damped. Large displacements of beams were taken intoaccount by separating the rigid body rotation of the beam from its localdeformation. This kind of formulation for handling large displacements hasbeen used in FEM (see e.g. Crisfield, M. A. A. Comput. Methods Appl. Mech.Engrg. 81, 131 (1990)) and in the dynamics of deformable bodies (Erleben,K. et al. Physics-based animation, Charles River Media (2005)). The beamshad width a, Young’s modulus Yb and Poisson ratio υ = 1/3. To account

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Figure 1: Illustration of the simulation model. Stripped visualizationshows spheres and skeletons of beams. The sheet thickness h is the sameas the sphere diameter d, lattice constant a and the length and width of abeam. The confining shell around the sheet is shown (semi)transparent.

for elasto-plasticity, all deformations exceeding a plastic yield point of beamsin tensile strain, bending or torsion were irreversible and resulted in con-stant stress beyond the yield point. For example, if the tensile strain of abeam exceeded its maximum elastic strain, its tension remained at the valueof the yield tension. When the tension was released, the beam recoveredits original length added by the amount of plastic strain beyond the yieldpoint. Derivation of the elements of the stiffness matrix as well as the elasticlimits was made in accordance with standard methods of structural analy-sis (e.g. Timoshenko, S. Strength of materials, 3rd ed., Krieger Publishing(1976)). Self-avoidance of the sheet was introduced by having an elastic fric-tionless sphere of radius a/2 and Young’s modulus Ys = Yb at each latticepoint. Spheres did not interact with their nearest neighbours so that theydid not affect the in-plane compressibility of the sheet. Otherwise, overlap-ping spheres had a repulsive (compression) force proportional to their depthof overlap and the Young’s modulus Ys. Elastic or elasto-plastic energy ofthe sheet was calculated as a sum of deformation energies of the individ-ual beams (energies of stretching/compression, bending and torsion of thebeams were summed up) and compression energies of the spheres. Crum-pling was induced by a spherical shell enclosing the sheet, as in Fig. 1 and inthe Supplementary videos. If a sphere was in contact with the shell, it was

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given a force towards the center of the shell. The magnitude of this forcewas also proportional to the depth of overlap and the Young’s modulus Y .The shell radius was let to shrink slowly such that the kinetic energy of thesheet remained very small compared to its deformation energy (except forpossible spontaneous bucklings of the sheet). The total confining force wasdetermined as the sum of radial compression forces of the spheres in contactwith the enclosing shell. Newton’s equations of motion were explicitly solvedat each time step to propagate the simulation in time. The time step was setas dt =

√m

4aYb. Crumpling of sheets with 106 lattice points required about

107 time steps.

2. Scaling of ridge energy

Theoretical results indicate that in the limit of high aspect ratio L/h, theenergy of a single ridge in a fully elastic sheet is proportional to (L/h)1/3

(Lobkovsky, A. et al. Science 270, 1482 (1995)). As a test for our numericalmodel we simulated the energy of a single ridge as a function of sheet sizeL (fixed h), for both elastic and elasto-plastic sheets. In elastic sheets theridge energy indeed became proportional to (L/h)1/3 very soon the sheet sizeexceeded L/h ≈ 100 (see Fig. 2).

In elasto-plastic sheets the ridge energy was initially (small L/h) clearlysmaller than in the corresponding elastic sheets, but approached the latterfor increasing sheet size, and became at the same time approximately pro-portional to (L/h)1/3. This scaling result is also shown in Fig. 2 togetherwith an example of a plastic ridge. For short ridges plastic yielding appearedalong the whole ridge, but as L/h increased, a threshold was reached beyondwhich the middle part of the ridge remained elastic. This threshold stronglydepended on the bending angle and the yield point (for an angle of π/2 and ayield point of σy/Y = 0.01 the threshold value of L/h was few hundred). It isevident that in long enough ridges the plastic deformations are concentratedin relatively small areas in the vicinities of vertices (as suggested based onthe behaviour of fully elastic sheets in Witten, T. A. Rev. Mod. Phys. 79,643 (2007)), and that the elastic deformation energy dominates the total en-ergy in this limit. Validity of elastic theory for elasto-plastic vertices has alsobeen shown experimentally in Mora, T. & Boudaoud, A. Europhys. Lett. 59,643 (2007).

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101

102

103

100

101

L/hE

/κ

ElasticElasto−plastic

∼ (L/h)1/3

Figure 2: Deformation energy of a single ridge as a function of itslength. To form a ridge, two opposing sides of a sheet were bent to an angleπ/2. An example of a ridge in an elasto-plastic sheet is shown on the left.The areas which contain plastic yielding are marked red. On the right theenergy of the ridge is shown as a function of L/h for both elastic and elasto-plastic sheets. The expected (L/h)1/3 scaling is marked with a dashed line.This kind of configuration was also called the minimal ridge by Lobkovsky(Phys. Rev. E 53, 3750 (1996)).

3. Scaling of total energy

A scaling form for the total energy has been derived by dimensional analy-sis in the form Et ∼ κ(K0R

20/κ)βV G/αV G(Rf/R0)

1−1/αV G , where K0 is a 2-dYoung’s modulus, R0 the initial radius of a spherical shell enclosing a flat cir-cular sheet and Rf its final radius (Vliegenthart, G. A. & Gompper, G. NatureMaterials 5, 216 (2006)). The exponents ’alpha’ and ’beta’ used in the ex-pression above are denoted here with a subscript V G. Noting that κ ∼ Y h3,K0 ∼ Y h and R0 ∼ L, we find that Et ∼ κ(L/h)2βV G/αV G(Rf/L)1−1/αV G .This expression is similar to equation (1) in the main article, and pro-vides a mapping between the ’alphas’ and ’betas’: 2βV G/αV G = β and1− 1/αV G = α(β − 2).

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0.2 0.4 0.6 0.8 10.7

0.75

0.8

0.85

0.9

0.95

1

Fraction of energy

φ A

L/h = 1000L/h = 500L/h = 250

100

102

10−4

10−2

ε

φ A

Crumpled sheet

∼ε −5/4

a b

Figure 3: Focusing of energy. In a cumulative distributions of deformationenergy in crumpled elastic sheets (R/R0 = 0.18) are shown. In b the fractionφA of the sheet area in which the energy density exceeds ε is shown. Acorresponding energy map with logarithmic colour coding is shown in theinset.

4. Focusing of deformation energy

Previous studies on elastic sheets indicate also that deformation energy isfocused on an increasingly smaller fraction of the area of the crumpledsheet when the sheet size is increased (Kramer, E. M. & Witten, T. A.Phys. Rev. Lett. 78, 1303 (1997)). We tested this conclusion with positiveresults by simulating distributions of local deformation energy with our nu-merical model, and show results for elastic sheets with different aspect ratiosin Fig. 3a. In addition, focusing of energy in a loosely crumpled elastic sheet(Fig. 3b) was in good agreement with the prediction that the area fractionof the sheet in which the energy density exceeds a given value ε should scaleas ε−5/4 (Didonna, B. A. et al. Phys. Rev. E 65, 016603 (2001)).

We can thus conclude that our numerical model correctly describes the knownindividual ridge energy and energy focusing behaviours of fully elastic sheets,and seems also to extend such behaviours into elasto-plastic sheets in a reli-able manner.

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5. Facet extraction

To determine the ’facet’ size distributions of crumpled sheets, 2-d mean cur-vature maps were thresholded resulting in binary images where areas of pos-itive and negative curvature were marked respectively as black and white(Fig. 4). The local mean curvature of the sheet was extracted from themesh of lattice sites (Desburn, M. SIGGRAPH’99, 317-324). The black andwhite areas were then split into separate roughly convex regions by applyingthe watershed algorithm (see e.g. Meyer, F. Signal processing 38, 113-125(1994)). These regions describe relatively flat parts of the sheet surroundedby features of clearly higher local curvature, called ridges and vertices whenthe curvature becomes high enough. We call these regions facets. Facet areaswhere determined in pixels and their relative linear sizes were determined assquare roots of the areas divided by the linear size L of the sheet. Facetswith a size smaller than L/100 were omitted from the analysis. This proce-dure does not rely on any assumption regarding the detailed shape or energycontent of the ridges. It is thus straightforward to apply at any degree ofcrumpling and in sheets of varying width to thickness ratio.

Facet size distributions in crumpled sheets were reasonably well describedby a lognormal distribution N(x) ∼ exp[−(ln(x)−µ)2/(2σ2)]/(xσ) (Fig. 5).The found standard deviations σ ≈ 0.5 for the logarithms of linear facetsizes correspond to σ ≈ 1.0 for the facet areas in excellent agreement withthe σ ≈ 1.17 found for crumpled paper in Andresen, C. A., Hansen, A.& Schmittbuhl, J. Phys. Rev. E 76, 026108 (2007). For ridge lengths l insimulated crumpled elastic sheets a lognormal distribution given in the formN(l) ∼ exp[−(log(l/l0))

2/b)]/(√

bl) has earlier been found with b = 0.95(Vliegenthart, G. A. & Gompper, G. Nature Materials 5, 216 (2006)). Thiscorresponds to σ ≈ 0.7, and is also in good agreement with the presentresults. The lognormal distribution found for the ridge lengths in crumpledpaper (Blair, D. L. & Kudrolli, A. Phys. Rev. Lett. 94, 166107 (2005)) isas well in agreement with the present result although is a bit wider (σ ≈1.2− 1.4). A wider ridge length distribution may arise from the fact that asingle facet is surrounded by multiple ridges of varying length.

In the case of elastic sheets, a slightly better fit in comparison with a lognor-mal fit was provided by a gamma distribution N(x) ∼ xa−1/[baΓ(a)]exp(−x/b)with the shape parameter a = 4.0 (Fig. 5). A gamma distribution has previ-ously been found for the segment lengths of a 1-d model of crumpling, owingto interaction at high confinement of segment layers (Sultan, E. & Boudaoud,

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Figure 4: Illustration of facet segmentation. a, Mean curvature fieldof a crumpled sheet. b, Thresholded areas of positive (white) and negative(black) curvature. c, The thresholded image segmented into regions whichapproximate facets.

0.025 0.05 0.1 0.2

100

101

x/L

N(x

)

ElasticElasto−plasticLognormalLognormalGamma

Figure 5: Facet size distributions. Distributions of linear facet size areshown for elasto-plastic (yield stress σy = 0.01Y ) and elastic sheets of sizeL/h = 1000. Both distributions are averages over those for six sheets crum-pled to R/R0 = 0.18. The parameters of the lognormal distribution fit (seetext) for elastic sheets are µ = −2.90 and σ = 0.52 and for elasto-plasticsheets they are µ = −3.15 and σ = 0.47. The parameters of the gammadistribution fit for elastic sheets are a = 4.0 and b = 0.015.

A. Phys. Rev. Lett. 96, 136103 (2006)). Crumpled elastic sheets display amuch more stronger layering than elasto-plastic sheets, and this may explaintheir somewhat different facet size distributions.

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6. Packing by repeated folding

Folding a sheet repeatedly by the simplest possible (symmetric) way resultsin a tiled pattern of facets in the sheet. A pattern after four such foldingsis illustrated in Fig. 6a. The pattern is always similar after a fixed numberof repeated folds, no matter what is the size of the sheet. In about a halfof the simulations of elastic sheets (Tallinen, Astrom and Timonen, to bepublished) an almost symmetrically folded pattern appeared in the beginningof crumpling. Such patterns for two different sizes of the elastic sheet areshown in the top row of Fig. 6b. It is evident that these patterns arestatistically similar (ridge patterns for more randomly crumpled elastic sheetsare also statistically similar) and resemble that of a repeatedly folded sheet.In contrast with this, the ridge patterns in elasto-plastic sheets at fixed R/R0

(the bottom row of Fig. 6b) are not similar since the amounts of ridges andvertices clearly increase with increasing size of the sheet. In repeated foldingthe size (X) of the facets scales linearly with the width (R) of the foldedconfiguration, X ∼ R. Under crumpling we thus expect a similar power lawscaling of facet size, but as crumpling is a random process, the mean size of

Figure 6: Repeated folding. a, A ridge pattern resulting from repeatedfolding along the central line. Black and white indicate the two possibledirections of folds. b, Ridge patterns at R/R0 ≈ 0.25 for elastic and elasto-plastic sheets. The sheets with L/h = 250 are scaled to the same size as thesheets with L/h = 1000.

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the facets necessarily decreases faster than the size of the whole configuration(repeated folding has the maximal ridge length at each stage).

An attempt has already been made (Plouraboue, F. & Roux, S. Physica A227, 173-182 (1996)) to generalize ideal folding into a model with randomfolds.

7. Fractal dimensions

To find the fractal dimensions of crumpled elastic and elasto-plastic sheetswe crumpled sheets with width to thickness ratios in the interval [100, 1000].Crumpling was done slowly and the confining force was monitored. The forceat which the final radius of the crumpled configuration was measured waschosen such that the volume fraction of the configuration was reasonable. Forexample, the final volume fractions of the smallest elastic sheets (L/h = 100)with a confining force of 50 N were around one third, while those of thebiggest sheets simulated (L/h = 1000) were around 10%. The final radii ofthe crumpled sheets were plotted as a function of L to determine if therewas a relationship R ∼ L2/D (this is equivalent to M ∼ RD, since M ∼ L2).From this relationship the mass fractal dimensions D were extracted.

The fractal dimension smoothly decreased from its elastic value when theplasticity of material was increased (that is, the yield stress σy was decreased).For a confining force of 50 Newtons, we found Del ≈ 2.50 for elastic sheets,and Dpl ≈ 2.37, Dpl ≈ 2.20 and Dpl ≈ 2.11 for elasto-plastic sheets with theyield points σy/Y = 0.05, 0.01 and 0.002, respectively (see Fig. 7a and Fig.5 in the main article). For compressing forces of 25 N and 100 N we foundDpl ≈ 2.21 and Dpl ≈ 2.24 (σy/Y = 0.01), and Del ≈ 2.45 and Del ≈ 2.56,respectively (see Fig. 7b). A slight increase in D for increasing force mayarise from the high volume fractions of the final configurations. For a veryhigh force the sheet would fill the entire compressing shell and the resultwould be close to D = 3. Examples of ridge patterns at the final radius areshown in Fig. 8.

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10 20 30 40 60 80100

2

4

6

8

1012141618

L [mm]

R[m

m]

10 20 30 40 60 80100

2

4

6

8

1012141618

L [mm]

R [m

m]

Elastic 25NElasto−pl. 25NElastic 100NElasto−pl. 100N

∼ L2/2.45

∼ L2/2.22

∼ L2/2.56

∼ L2/2.24

σy /Y = 0.002

σy /Y = 0.05

∼ L2/2.11

∼ L2/2.37

a b

Figure 7: Relation between the sheet width and the radius at afixed force of the crumpled configuration. a, The final radius (R) as afunction of the sheet width (L) for fully elastic sheets (blue) and for elasto-plastic sheets (red) at total confining forces of 25 N (open symbols) and 100N (solid symbols). For elastic sheets at 25 N, R ∼ L2/Del25 with a fractaldimension of Del25 ≈ 2.45. For elastic sheets at 100 N, and for elasto-plasticsheets at 25 N and 100 N, the fractal dimensions Del100 ≈ 2.56, Dpl25 ≈ 2.22and Dpl100 ≈ 2.24 were found. The yield point of the elasto-plastic sheets ina is 1% of the Young’s modulus (σy/Y = 0.01). In b R(L) for elasto-plasticsheets of materials with a high and low yield point is shown. For weaklyplastic sheets (σy/Y = 0.05) Dpl ≈ 2.37 was found and for very plasticsheets (σy/Y = 0.002) Dpl ≈ 2.11. In a and b the physical thickness of thesheets was h = 0.1 mm and the Young’s modulus was Y = 1 GPa. The plotsare averages of three simulations.

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Figure 8: Ridge patterns of elastic and elasto-plastic sheets crum-pled by the same force. a, b, c and d show the mean curvature field ofa crumpled fully elastic sheet, and crumpled elasto-plastic sheets with yieldpoints at 5%, 1% and 0.2% of the Young’s modulus, respectively. All sheetshave the thickness 0.1 mm, width 80 mm and Young’s modulus 1 GPa. Theconfining force was 100 N in all cases.

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