The effect of plasticity in crumpling of thin sheets: Supplementary Information

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1 The effect of plasticity in crumpling of thin sheets: Supplementary Information T. Tallinen, J. A. Åström and J. Timonen Video S1. The video shows crumpling of an elastic sheet with a width to thickness 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 same parameters 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 constructed as a triangular lattice with spacing a and size up to 1,000 x 1,000 lattice points. Each lattice point had mass m and moment of inertia I, and they were connected by beam elements. The beams had a 12 x 12 stiffness matrix corresponding to three translational and three rotational degrees of freedom at both ends of the beam. The large thickness of the beams was accounted for by including shear effects in the formulation of the stiffness matrix. Motion of local degrees of freedom in the beams was opposed by small viscous damping. The magnitude of damping was such that the motion of any single beam was under-damped. Large displacements of beams were taken into account by separating the rigid body rotation of the beam from its local deformation. This kind of formulation for handling large displacements has been 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 beams had width a, Young s modulus Y b and Poisson ratio υ = 1/3. To account 1

2 Figure 1: Illustration of the simulation model. Stripped visualization shows spheres and skeletons of beams. The sheet thickness h is the same as the sphere diameter d, lattice constant a and the length and width of a beam. The confining shell around the sheet is shown (semi)transparent. for elasto-plasticity, all deformations exceeding a plastic yield point of beams in tensile strain, bending or torsion were irreversible and resulted in constant stress beyond the yield point. For example, if the tensile strain of a beam exceeded its maximum elastic strain, its tension remained at the value of the yield tension. When the tension was released, the beam recovered its original length added by the amount of plastic strain beyond the yield point. Derivation of the elements of the stiffness matrix as well as the elastic limits was made in accordance with standard methods of structural analysis (e.g. Timoshenko, S. Strength of materials, 3rd ed., Krieger Publishing (1976)). Self-avoidance of the sheet was introduced by having an elastic frictionless sphere of radius a/2 and Young s modulus Y s = Y b at each lattice point. Spheres did not interact with their nearest neighbours so that they did not affect the in-plane compressibility of the sheet. Otherwise, overlapping spheres had a repulsive (compression) force proportional to their depth of overlap and the Young s modulus Y s. Elastic or elasto-plastic energy of the sheet was calculated as a sum of deformation energies of the individual beams (energies of stretching/compression, bending and torsion of the beams were summed up) and compression energies of the spheres. Crumpling was induced by a spherical shell enclosing the sheet, as in Fig. 1 and in the Supplementary videos. If a sphere was in contact with the shell, it was 2

3 given a force towards the center of the shell. The magnitude of this force was 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 the sheet remained very small compared to its deformation energy (except for possible spontaneous bucklings of the sheet). The total confining force was determined as the sum of radial compression forces of the spheres in contact with the enclosing shell. Newton s equations of motion were explicitly solved at each time step to propagate the simulation in time. The time step was set as dt = m 4aY b. Crumpling of sheets with 10 6 lattice points required about 10 7 time steps. 2. Scaling of ridge energy Theoretical results indicate that in the limit of high aspect ratio L/h, the energy 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 numerical model we simulated the energy of a single ridge as a function of sheet size L (fixed h), for both elastic and elasto-plastic sheets. In elastic sheets the ridge energy indeed became proportional to (L/h) 1/3 very soon the sheet size exceeded L/h 100 (see Fig. 2). In elasto-plastic sheets the ridge energy was initially (small L/h) clearly smaller than in the corresponding elastic sheets, but approached the latter for increasing sheet size, and became at the same time approximately proportional to (L/h) 1/3. This scaling result is also shown in Fig. 2 together with an example of a plastic ridge. For short ridges plastic yielding appeared along the whole ridge, but as L/h increased, a threshold was reached beyond which the middle part of the ridge remained elastic. This threshold strongly depended on the bending angle and the yield point (for an angle of π/2 and a yield point of σ y /Y = 0.01 the threshold value of L/h was few hundred). It is evident that in long enough ridges the plastic deformations are concentrated in relatively small areas in the vicinities of vertices (as suggested based on the behaviour of fully elastic sheets in Witten, T. A. Rev. Mod. Phys. 79, 643 (2007)), and that the elastic deformation energy dominates the total energy in this limit. Validity of elastic theory for elasto-plastic vertices has also been shown experimentally in Mora, T. & Boudaoud, A. Europhys. Lett. 59, 643 (2007). 3

4 10 1 E/κ Elastic Elasto plastic (L/h) 1/ L/h Figure 2: Deformation energy of a single ridge as a function of its length. To form a ridge, two opposing sides of a sheet were bent to an angle π/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 the energy of the ridge is shown as a function of L/h for both elastic and elastoplastic 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 analysis in the form E t κ(k 0 R 2 0/κ) β V G/α V G (R f /R 0 ) 1 1/α V G, where K 0 is a 2-d Young s modulus, R 0 the initial radius of a spherical shell enclosing a flat circular sheet and R f its final radius (Vliegenthart, G. A. & Gompper, G. Nature Materials 5, 216 (2006)). The exponents alpha and beta used in the expression above are denoted here with a subscript V G. Noting that κ Y h 3, K 0 Y h and R 0 L, we find that E t κ(l/h) 2β V G/α V G (R f /L) 1 1/α V G. This expression is similar to equation (1) in the main article, and provides a mapping between the alphas and betas : 2β V G /α V G = β and 1 1/α V G = α(β 2). 4

5 1 a b 10 2 Crumpled sheet ε 5/4 φ A L/h = 1000 L/h = 500 L/h = 250 φ A Fraction of energy ε Figure 3: Focusing of energy. In a cumulative distributions of deformation energy in crumpled elastic sheets (R/R 0 = 0.18) are shown. In b the fraction φ A of the sheet area in which the energy density exceeds ε is shown. A corresponding energy map with logarithmic colour coding is shown in the inset. 4. Focusing of deformation energy Previous studies on elastic sheets indicate also that deformation energy is focused on an increasingly smaller fraction of the area of the crumpled sheet when the sheet size is increased (Kramer, E. M. & Witten, T. A. Phys. Rev. Lett. 78, 1303 (1997)). We tested this conclusion with positive results by simulating distributions of local deformation energy with our numerical model, and show results for elastic sheets with different aspect ratios in 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 fraction of the sheet in which the energy density exceeds a given value ε should scale as ε 5/4 (Didonna, B. A. et al. Phys. Rev. E 65, (2001)). We can thus conclude that our numerical model correctly describes the known individual ridge energy and energy focusing behaviours of fully elastic sheets, and seems also to extend such behaviours into elasto-plastic sheets in a reliable manner. 5

6 5. Facet extraction To determine the facet size distributions of crumpled sheets, 2-d mean curvature maps were thresholded resulting in binary images where areas of positive and negative curvature were marked respectively as black and white (Fig. 4). The local mean curvature of the sheet was extracted from the mesh of lattice sites (Desburn, M. SIGGRAPH 99, ). The black and white areas were then split into separate roughly convex regions by applying the watershed algorithm (see e.g. Meyer, F. Signal processing 38, (1994)). These regions describe relatively flat parts of the sheet surrounded by features of clearly higher local curvature, called ridges and vertices when the curvature becomes high enough. We call these regions facets. Facet areas where determined in pixels and their relative linear sizes were determined as square roots of the areas divided by the linear size L of the sheet. Facets with a size smaller than L/100 were omitted from the analysis. This procedure does not rely on any assumption regarding the detailed shape or energy content of the ridges. It is thus straightforward to apply at any degree of crumpling and in sheets of varying width to thickness ratio. Facet size distributions in crumpled sheets were reasonably well described by a lognormal distribution N(x) exp[ (ln(x) µ) 2 /(2σ 2 )]/(xσ) (Fig. 5). The found standard deviations σ 0.5 for the logarithms of linear facet sizes correspond to σ 1.0 for the facet areas in excellent agreement with the σ 1.17 found for crumpled paper in Andresen, C. A., Hansen, A. & Schmittbuhl, J. Phys. Rev. E 76, (2007). For ridge lengths l in simulated crumpled elastic sheets a lognormal distribution given in the form N(l) exp[ (log(l/l 0 )) 2 /b)]/( bl) has earlier been found with b = 0.95 (Vliegenthart, G. A. & Gompper, G. Nature Materials 5, 216 (2006)). This corresponds to σ 0.7, and is also in good agreement with the present results. The lognormal distribution found for the ridge lengths in crumpled paper (Blair, D. L. & Kudrolli, A. Phys. Rev. Lett. 94, (2005)) is as well in agreement with the present result although is a bit wider (σ ). A wider ridge length distribution may arise from the fact that a single facet is surrounded by multiple ridges of varying length. In the case of elastic sheets, a slightly better fit in comparison with a lognormal fit was provided by a gamma distribution N(x) x a 1 /[b a Γ(a)]exp( x/b) with the shape parameter a = 4.0 (Fig. 5). A gamma distribution has previously been found for the segment lengths of a 1-d model of crumpling, owing to interaction at high confinement of segment layers (Sultan, E. & Boudaoud, 6

7 Figure 4: Illustration of facet segmentation. a, Mean curvature field of a crumpled sheet. b, Thresholded areas of positive (white) and negative (black) curvature. c, The thresholded image segmented into regions which approximate facets N(x) 10 0 Elastic Elasto plastic Lognormal Lognormal Gamma x/l Figure 5: Facet size distributions. Distributions of linear facet size are shown for elasto-plastic (yield stress σ y = 0.01Y ) and elastic sheets of size L/h = Both distributions are averages over those for six sheets crumpled to R/R 0 = The parameters of the lognormal distribution fit (see text) for elastic sheets are µ = 2.90 and σ = 0.52 and for elasto-plastic sheets they are µ = 3.15 and σ = The parameters of the gamma distribution fit for elastic sheets are a = 4.0 and b = A. Phys. Rev. Lett. 96, (2006)). Crumpled elastic sheets display a much more stronger layering than elasto-plastic sheets, and this may explain their somewhat different facet size distributions. 7

8 6. Packing by repeated folding Folding a sheet repeatedly by the simplest possible (symmetric) way results in a tiled pattern of facets in the sheet. A pattern after four such foldings is illustrated in Fig. 6a. The pattern is always similar after a fixed number of repeated folds, no matter what is the size of the sheet. In about a half of the simulations of elastic sheets (Tallinen, Åström and Timonen, to be published) an almost symmetrically folded pattern appeared in the beginning of crumpling. Such patterns for two different sizes of the elastic sheet are shown in the top row of Fig. 6b. It is evident that these patterns are statistically similar (ridge patterns for more randomly crumpled elastic sheets are 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/R 0 (the bottom row of Fig. 6b) are not similar since the amounts of ridges and vertices clearly increase with increasing size of the sheet. In repeated folding the size (X) of the facets scales linearly with the width (R) of the folded configuration, X R. Under crumpling we thus expect a similar power law scaling of facet size, but as crumpling is a random process, the mean size of Figure 6: Repeated folding. a, A ridge pattern resulting from repeated folding along the central line. Black and white indicate the two possible directions of folds. b, Ridge patterns at R/R for elastic and elastoplastic sheets. The sheets with L/h = 250 are scaled to the same size as the sheets with L/h =

9 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 A 227, (1996)) to generalize ideal folding into a model with random folds. 7. Fractal dimensions To find the fractal dimensions of crumpled elastic and elasto-plastic sheets we crumpled sheets with width to thickness ratios in the interval [100, 1000]. Crumpling was done slowly and the confining force was monitored. The force at which the final radius of the crumpled configuration was measured was chosen such that the volume fraction of the configuration was reasonable. For example, 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 the biggest sheets simulated (L/h = 1000) were around 10%. The final radii of the crumpled sheets were plotted as a function of L to determine if there was a relationship R L 2/D (this is equivalent to M R D, since M L 2 ). From this relationship the mass fractal dimensions D were extracted. The fractal dimension smoothly decreased from its elastic value when the plasticity of material was increased (that is, the yield stress σ y was decreased). For a confining force of 50 Newtons, we found D el 2.50 for elastic sheets, and D pl 2.37, D pl 2.20 and D pl 2.11 for elasto-plastic sheets with the yield points σ y /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 found D pl 2.21 and D pl 2.24 (σ y /Y = 0.01), and D el 2.45 and D el 2.56, respectively (see Fig. 7b). A slight increase in D for increasing force may arise from the high volume fractions of the final configurations. For a very high force the sheet would fill the entire compressing shell and the result would be close to D = 3. Examples of ridge patterns at the final radius are shown in Fig. 8. 9

10 a R[mm] Elastic 25N Elasto pl. 25N Elastic 100N Elasto pl. 100N L 2/2.45 L 2/2.22 L 2/2.56 L 2/2.24 b R [mm] σ y /Y = σ y /Y = 0.05 L 2/2.11 L 2/ L [mm] L [mm] Figure 7: Relation between the sheet width and the radius at a fixed force of the crumpled configuration. a, The final radius (R) as a function of the sheet width (L) for fully elastic sheets (blue) and for elastoplastic sheets (red) at total confining forces of 25 N (open symbols) and 100 N (solid symbols). For elastic sheets at 25 N, R L 2/D el25 with a fractal dimension of D el For elastic sheets at 100 N, and for elasto-plastic sheets at 25 N and 100 N, the fractal dimensions D el , D pl and D pl were found. The yield point of the elasto-plastic sheets in a is 1% of the Young s modulus (σ y /Y = 0.01). In b R(L) for elasto-plastic sheets of materials with a high and low yield point is shown. For weakly plastic sheets (σ y /Y = 0.05) D pl 2.37 was found and for very plastic sheets (σ y /Y = 0.002) D pl In a and b the physical thickness of the sheets was h = 0.1 mm and the Young s modulus was Y = 1 GPa. The plots are averages of three simulations. 10

11 Figure 8: Ridge patterns of elastic and elasto-plastic sheets crumpled by the same force. a, b, c and d show the mean curvature field of a crumpled fully elastic sheet, and crumpled elasto-plastic sheets with yield points at 5%, 1% and 0.2% of the Young s modulus, respectively. All sheets have the thickness 0.1 mm, width 80 mm and Young s modulus 1 GPa. The confining force was 100 N in all cases. 11