Abstract

Crumpling a thin sheet of material into a small volume requires energy for creating a network of deformations such as vertices and ridges^{1, 2}. Scaling properties of a single elastic vertex^{3, 4, 5} or ridge have been analysed theoretically^{6, 7, 8}, and crumpling of a sheet by numerical simulations^{1, 9, 10}. Real materials are however elasto-plastic^{11, 12, 13, 14, 15} and large local strains induce irreversible plastic deformations. Hence, a numerical model that can be purely elastic or elasto-plastic is introduced. In crumpled elastic sheets, the ridge patterns are found to be similar, independent of the width to thickness (L/h) ratio of the sheet, and the fractal dimension of crumpled sheets is given by scaling properties of the energy and average length of ridges. In crumpled elasto-plastic sheets, such a similarity does not appear as the L/h ratio affects the deformations, and the fractal dimension (D_pl) is thereby reduced. Evidence is also found of D_pl not being universal but dependent on the plastic yield point of the material.

Introduction

Crumpling a sheet of paper is a typical everyday example of crumpling processes. Such processes seem to have generic features, and they have become a fascinating research topic. Interesting applications of crumpling are also found at microscopic scales, in, for example, polymerized membranes¹⁶ and graphene^{17, 18}. It would thus be important to know the universal properties of crumpling. So far there has been some degree of incompatibility, however, between crumpling experiments and numerical simulations. For one thing, the simulated sheets have been fully elastic^{1, 9, 10}, whereas the ones used in experiments have been made of elasto-plastic materials such as mylar^{7, 12}, paper^{11, 14, 15, 19}, aluminium foil¹³ and even layers of cream²⁰. Self-avoidance and nonlinear deformations make the full process very complex²¹, but the theory describing individual structures such as sharp vertices (or developable cones) and ridges is rather well established^{4, 5, 6, 8, 22}. The cones and ridges analysed have been elastic, but the results obtained are expected to hold for elasto-plastic sheets of high L/h ratios^{2, 5}. A very recent suggestion¹³ is that plasticity affects crumpling only slightly. It is not clear however, to what extent the previous analyses could make a distinction between elastic and elasto-plastic behaviour.

The fractal dimension D relates the size R of a crumpled sheet at a given compression force to its linear size L such that RL^2/D. A value of D close to its lower physical limit, D=2, means a loose 'packing' of the crumpled sheet and that much force is required for crumpling large sheets. On the other hand, a value of D close to its upper physical limit, D=3, means a compact structure resulting from application of much less force. A natural assumption is that D is determined by the density of vertices, facets and ridges, and by scaling of the energies of these structures with their size. To test this assumption for both elastic and elasto-plastic sheets, we construct a numerical model described below.

We model a sheet of material as a triangular lattice with spacing a and a size up to 1,0001,000 lattice sites connected by massless beams with bending, shear and tensile stiffness²³. The beams are (linearly) elastic or elasto-plastic, and their motion is opposed by small viscous damping. We use a large-rotation formulation²⁴ so as to correctly describe large displacements of beams of width a, Young's modulus Y_b and Poisson ratio =1/3. Elasto-plasticity of beams is implemented by elastic/perfectly plastic stress–strain relations for tensile, rotational and torsional strains. Self-avoidance of the sheet is introduced by having an elastic frictionless sphere of radius a/2, mass m and Young's modulus Y_s=Y_b at each lattice site. Spheres do not interact with their nearest neighbours so as not to affect the in-plane compressibility of the sheet. In the simulations, a rectangular sheet is confined inside a spherical shell of shrinking radius, as in Fig. 1 (see Supplementary Information, Videos), and crumpling is initiated by gently and randomly pushing the corners of the sheet. A more detailed description of the model is provided in Supplementary Information, Section S1.

Figure 1: Crumpling thin sheets inside a spherical shell.

a–d, An elastic sheet takes the form of a cone at radius R=0.44R₀ (a) and has only few vertices at R=0.25R₀ (b), whereas an elasto-plastic sheet (c,d) does not undergo elastic relaxation under crumpling and thus appears stiffer. In c R=0.44R₀ and in d R=0.25R₀.

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The parameters of the model are chosen to simulate crumpling of real materials. Sheets of width L up to 10 cm, thickness h=0.1 mm, density 1,000 kg m^-3 and Young's modulus 1 GPa (bending modulus 10^-4 N m is typical of polymeric materials and paper) are crumpled in one second to a volume fraction of =0.5. This corresponds to R0.1R₀ for a sheet with L/h=1,000, where R₀L is the initial radius of the shell. Crumpling is slow enough to make inertial effects small, and the sheet is close to an elastic (or elasto-plastic) equilibrium at any time. The same parameters are used for elasto-plastic sheets except for a plastic yield point, which is varied in a wide interval to find its influence on crumpling. Yield points of polymeric materials and paper²⁵ are included in this interval.

When the shell enclosing an elastic sheet begins to shrink, the corners of the sheet bend, and the first vertex appears at R0.75R₀, whereby the sheet is transformed into a cone as in Fig. 1a. This cone persists until R0.4R₀ when its tip splits and the (relatively) flat areas begin to buckle. This 'cone regime' is similar to that observed in previous simulations on circular sheets⁹. After the cone breaks, ridges and facets (areas of relatively small curvature of uniform sign surrounded by ridges and vertices) appear in the sheet. As is evident from Fig. 2, elastic sheets crumple first by forming layers of facets. Systematic compactification of a sheet by repeated folding would result in a pattern of rectangular facets, the size of which would scale as XL(R/L) with =1 (see Supplementary Information, Section S6). Repeated folding can, to some extent, be traced in the ridge patterns of crumpled elastic sheets, but the facet size decreases faster than the radius of the shell, resulting in an average facet size () scaling with an >1.

Figure 2: Intersections of crumpled sheets.

Snapshots are taken at volume fractions of =0.25 (left, inside the scaling regime) and =0.5 (right, beyond the scaling regime) for sheets with L/h=1,000. a,b, A crumpled elastic sheet (a) is more effectively folded, indicated by its more layered structure, so that it has a larger characteristic facet size and a lower energy than a crumpled elasto-plastic sheet (b).

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It has been suggested that the energy of a crumpled configuration (E_t) is given by the energy stored in the ridges between vertices^{1, 2}: if we express the number of ridges as and multiply it by the well-known scaling form for the energy of a single ridge⁶, E_X(X/h)(L/h)(R/L), we find that the total elastic energy scales as (no bending angle dependence^{2, 6} is included, as simulation results indicate it is averaged out)

An equivalent scaling form (see Supplementary Information, Section S3) has also been derived by dimensional analysis⁹, and found to hold for elastic sheets⁹.

In Fig. 3a, we show the average facet diameter as a function of the radius of the confining shell, R/R₀, for elastic and elasto-plastic sheets of varying size. The average ridge length should scale as the facet diameter, although its distribution may have a larger variance (see Supplementary Information, Section S5).

Figure 3: Development of facet size and deformation energy during crumpling.

a, Average linear size of areas with uniform sign of mean curvature for elastic and elasto-plastic sheets of size L/h=250, 500 and 1,000. In a crumpled state (R<0.4R₀), this size describes the characteristic size of facets and ridges in the sheet. b, The total energy of elastic and elasto-plastic sheets of size L/h=250, 500 and 1,000, scaled by 1/(L/h)^1/3. Transitions in the energy of the elastic sheets at R0.75R₀ and R0.4R₀ indicate the formation of a cone and the end of a single-cone regime, respectively. The plots shown are averages of three simulations, and the yield point of the elasto-plastic sheets is _y/Y =0.01.

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For elastic sheets, we find _el1.65 when R<0.4R₀. For ridges with large X/h, it has been shown that _el=1/3 (ref. 6) (we confirmed this result by simulating a single ridge, see Supplementary Information, Section S2). For these values of _el and _el, equation (1) gives E_t^elR^-2.76 for elastic sheets, which is within error bars the same as our numerical result E_t^elR^-2.830.11 (Fig. 3b). Such scaling of energy seems to hold to volume fractions beyond 0.3, where after the energy begins to be increasingly dominated by self-contacts instead of the ridges, and increases faster with further compression. The distribution of elastic energy in this contact-energy-dominated regime has previously been shown to be similar however to that in the scaling regime considered here¹⁰. Our scaling results for elastic sheets are comparable to the simulation results of ref. 9, where the total energy was found to scale as E_tR^-3 and the number of ridges as NR^-4, corresponding to _el2. Elastic sheets without self-avoidance have also been found to obey the same scaling form, but with =1 (refs 1,9).

In elastic sheets, the scaling of as a function of compression, R/R₀ (or R/L), does not depend on sheet thickness h. This indicates that if ridge lengths in sheets of different L are scaled by L, all ridge patterns will become indistinguishable. To test such similarity of ridge patterns for varying L at fixed R/R₀ and h, the probability distributions of linear facet sizes were determined (Fig. 4c). As expected from the scaling behaviour of , these distributions are independent of L/h, and the ridge patterns are indeed statistically similar. Distributions of facet size could be described by lognormal distributions with fairly similar parameters as those found for ridge length distributions in previous simulations⁹, although better fits were provided by gamma distributions (see Supplementary Information, Section S5). A gamma distribution has been shown to arise from layered structures in simulated crumpling of a one-dimensional model²⁶, reminiscent of the layering observed here in elastic sheets.

Figure 4: Mean curvature fields and facet size distributions of crumpled sheets.

a,b, The mean curvatures of an elastic (a) and elasto-plastic (b) sheet, for L/h=1,000 and R=0.18R₀. Dark and bright shading represent different signs of mean curvature. c,d, Distributions of the relative linear size of facets (X) in elastic (c) and elasto-plastic (d) sheets, for L/h=250 and 1,000, and R=0.18R₀. The distributions shown are averages over six crumpled sheets.

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In elasto-plastic sheets, the situation is more complicated. It is evident from Fig. 3 that their average linear facet size and energy scale similarly as a function of compression to the elastic sheets. There are however differences in the two crumpling processes, which arise at early phases of crumpling. In vertices in particular, plastic deformations appear already for R/R₀ close to unity. An elasto-plastic sheet is not able to transform into a cone necessary for a folding type of initial deformations, and large numbers of vertices and ridges appear soon after crumpling begins. This becomes increasingly pronounced for increasing L/h so that the relative facet diameter then decreases as shown in Fig. 3a. It is evident from this figure that the average ridge length scales now as , where function g(z) has a power-law form in a fairly large range of the argument. It is difficult to determine by simulations a functional form for f(z). The above scaling form means however that elasto-plastic sheets of different L/h can only have the same average (relative) ridge length for different degrees of compression. Consequently, the similarity of ridge patterns found for crumpled elastic sheets does not appear in elasto-plastic sheets. The lack of such similarity is also evident in the L/h-dependent distributions of the linear facet size in elasto-plastic sheets (Fig. 4d): sheet thickness must be scaled together with the other spatial dimensions to preserve the form of the distribution. The distributions of linear facet size are now well fitted by lognormal distributions, found previously for experimental facet size¹⁵ and ridge length¹⁴ distributions in crumpled paper.

The L/h dependence in the crumpling of elasto-plastic sheets arises from the L/h dependence of deformations that involve plastic yielding. We determined the fraction of the energy related to plastic deformations and the fraction of plastically deformed area for elasto-plastic sheets at R/R₀=0.4, where self-contacts of the sheet begin to appear. The plastic fraction of energy (deformed area) was about 80% (10%) for L/h=250, and about 50% (1%) for L/h=1,000. For increasing L/h, the plastic fraction of energy became increasingly concentrated near the tips of vertices (see Supplementary Information, Sections S2,S4). For small L/h, the more extended plasticity of vertices and ridges makes them relatively weaker. Such sheets can, to some extent, accommodate further confinement by extending the size of existing plastic deformations instead of forming new initially elastic deformations. An existing plastic vertex can then move so as to form a plastically deformed crease. The length of a plastic crease cannot scale proportional to sheet size (fixed h) because of its rapidly increasing energy with increasing length. Formation of new ridges and vertices becomes preferred as L/h is increased. Elasto-plastic sheets of small L/h may thus deform rather like elastic sheets during their early phase of compression, and thereby sustain a relatively high facet size and low energy during further crumpling in comparison with sheets of high L/h.

The focusing of plastic deformations described above is in agreement with increased stress focusing for increasing L/h found for elastic sheets². We can also confirm a previous theoretical prediction²⁷ that in elastic sheets the area fraction in which the energy density exceeds a given value scales as ^-5/4 (see Supplementary Information, Section S4).

The fractal dimension of crumpled sheets is a measure for how sheet size affects the compactification, and it has been measured for sheets of different materials^{11, 13, 19, 28}. We can derive a scaling expression for the fractal dimension of crumpled elastic sheets by considering the dependence of their energy on their size R, and determining the point at which a predefined total force FdE/dR=const is reached. Inserting a constant force in equation (1), and expressing it in the form R(L), the fractal dimension defined by RL^2/D_el can be found such that

Similarity of elastic ridge patterns thus leads to the result that the fractal dimension of crumpled elastic sheets depends only on the scaling properties under crumpling of the average ridge length and ridge energy. Using the numerically obtained values for these scaling properties, _el1.65 and _el0.33, equation (2) gives D_el2.43.

In elasto-plastic sheets, the lack of similarity of ridge patterns means that there is an extra L/h dependence in the average ridge length, which would appear in an equation for the total elastic energy of the ridges similar otherwise to equation (1) (the exponent in the L/h term would increase), and thereby in a subsequent expression for the fractal dimension. Even without deriving such an expression, we can draw some conclusions about the elasto-plastic fractal dimension. As the ridge density will in this case increase with increasing sheet size, the energy and 'strength' of crumpled elasto-plastic sheets will increase faster with increasing sheet size than in the elastic case (the energy to create a single ridge also approaches that of an elastic ridge, see Supplementary Information, Section S2). The fractal dimension of elasto-plastic sheets, D_pl, must thus be smaller than D_el (a plastic sheet becomes less densely packed for a given force). D_pl is also expected to depend on the plastic yield point (dependence on L/h becomes weaker for increasing yield point) such that D_plD_el when .

To find numerical values for the fractal dimensions, we crumpled elastic and elasto-plastic sheets of size varying from 1 to 100 cm². The thickness of the sheets was 0.1 mm and their Young's modulus was 1 GPa. A predefined force of 50 N was chosen, and the plastic yield stress was _y/Y =0.002,0.01 or 0.05. For elastic sheets we found D_el2.50 in excellent agreement with the scaling result above, and in elasto-plastic sheets D_pl increased from about 2.11 to about 2.37 for increasing values of _y (see Fig. 5). When the compression force was increased, we observed a slight increase in the dimensions (see Supplementary Information, Section S7).

Figure 5: Relation of sheet width to the final radius of the crumpled configuration.

Sheets of varying size (fixed h) were crumpled until the confining force reached 50 N. Scaling fits to simulated final radii as a function of L indicate a fractal dimension of D_el=2.500.03 for elastic sheets and D_pl=2.200.03 for elasto-plastic sheets with the yield point _y/Y =0.01. The plots shown are averages over three simulations. Inset: Fractal dimensions for different yield points.

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We can conclude that in elastic materials, crumpling is independent of L/h, and the energy of crumpled sheets satisfies a scaling expression⁹ in the regime where it is dominated by that of ridges, and where it is also given by the scaling properties of an individual ridge⁶ and the average facet size. The resulting similarity of the elastic ridge patterns leads to a scaling expression for the fractal dimension of crumpled sheets, in excellent agreement with our numerical result for D_el. We expect the similarity of elastic ridge patterns to result from the ability of elastic structures to relax towards minimal energy configurations if the compression rate and friction are not too high.

In elasto-plastic materials, however, relaxation towards minimal energy configurations is restricted by the presence of plastic deformations. Ridges appear early under compression, and the average facet size is affected by the L/h ratio of the sheet. Later during crumpling, the linear facet size and energy seem to scale very much as in the elastic case. We found that the related scaling exponents are similar, _plel and _plel. Because of restricted relaxation, compactification of the sheet under crumpling is less effective than in the elastic case, and thus the fractal dimension D_pl is smaller than D_el. This difference arises from the lack of similarity of the elasto-plastic ridge patterns. By varying the plastic yield point in a wide interval, we also found evidence of a yield point dependence of D_pl such that it approached the elastic value for high yield points and decreased towards two in the opposite limit. For materials with low yield points, the difference of an elasto-plastic fractal dimension from the elastic value should be detectable.

Acknowledgements

We are grateful to A. Resnick, M. Kardar, M. Karttunen, R. Metzler and M. den Nijs for very useful discussions. We acknowledge CSC—the Finnish IT Centre for Science for the computing time grant jyy2517.

Received 8 February 2008; Accepted 13 November 2008; Published online 7 December 2008.

References

1. Kramer, E. M. & Witten, T. A. Stress condensation in crushed elastic manifolds. Phys. Rev. Lett. 78, 1303–1306 (1997). | Article | ISI | ChemPort |
2. Witten, T. A. Stress focusing in elastic sheets. Rev. Mod. Phys. 79, 643–675 (2007). | Article |
3. Cerda, E., Chaieb, S., Melo, F. & Mahadevan, L. Conical dislocations in crumpling. Nature 401, 46–49 (1999). | Article | ISI | ChemPort |
4. Liang, T. & Witten, T. A. Crescent singularities in crumpled sheets. Phys. Rev. E 71, 016612 (2005). | Article | ChemPort |
5. Mora, T. & Boudaoud, A. Thin elastic plates: on the core of developable cones. Europhys. Lett. 59, 41–47 (2002). | Article | ChemPort |
6. Lobkovsky, A. et al. Scaling properties of stretching ridges in a crumpled elastic sheet. Science 270, 1482–1485 (1995). | Article | ISI | ChemPort |
7. Boudaoud, A., Patricio, P., Couder, Y. & BenAmar, M. Dynamics of singularities in a constrained elastic plate. Nature 407, 718–720 (2000). | Article | PubMed | ChemPort |
8. DiDonna, B. A. & Witten, T. A. Anomalous strength of membranes with elastic ridges. Phys. Rev. Lett. 87, 206105 (2001). | Article | PubMed | ChemPort |
9. Vliegenthart, G. A. & Gompper, G. Forced crumpling of self-avoiding elastic sheets. Nature Mater. 5, 216–221 (2006). | Article | ISI | ChemPort |
10. Åström, J. A., Timonen, J. & Karttunen, M. Crumpling of a stiff tethered membrane. Phys. Rev. Lett. 93, 244301 (2004). | Article | PubMed | ChemPort |
11. Gomes, M. A. F. Paper crushes fractally. J. Phys. A 20, L283 (1987). | Article |
12. Matan, K., Williams, R. B., Witten, T. A. & Nagel, S. R. Crumpling a thin sheet. Phys. Rev. Lett. 88, 076101 (2002). | Article | PubMed | ChemPort |
13. Balankin, A. S., Silva, I. C., Martinez, O. A. & Huerta, O. S. Scaling properties of randomly folded plastic sheets. Phys. Rev. E 75, 051117 (2007). | Article | ChemPort |
14. Blair, D. L. & Kudrolli, A. Geometry of crumpled paper. Phys. Rev. Lett. 94, 166107 (2005). | Article | PubMed | ChemPort |
15. Andresen, C. A., Hansen, A. & Schmittbuhl, J. Ridge network in crumpled paper. Phys. Rev. E 76, 026108 (2007). | Article | ChemPort |
16. Chaieb, S., Natrajan, V. K. & El-rahman, A. A. Glassy conformations in wrinkled membranes. Phys. Rev. Lett. 96, 078101 (2006). | Article | PubMed | ChemPort |
17. Fasolino, A., Los, J. H. & Katsnelson, M. I. Intrinsic ripples of graphene. Nature Mater. 6, 858–861 (2007). | Article | ChemPort |
18. Stankovich, S. et al. Graphene-based composite materials. Nature 442, 282–286 (2006). | Article | PubMed | ISI | ChemPort |
19. Balankin, A. S. et al. Intrinsically anomalous roughness of randomly crumpled thin sheets. Phys. Rev. E 74, 061602 (2006). | Article | ChemPort |
20. Gomes, M. A. F., Donato, C. C., Campello, S. L., de Souza, R. E. & Cassia-Moura, R. Structural properties of crumpled cream layers. J. Phys. D 40, 3665–3668 (2007). | Article | ChemPort |
21. Nelson, D., Piran, T. & Weinberg, S. Statistical Mechanics of Membranes and Surfaces 2nd edn (World Scientific, 2004).
22. Das, M., Vaziri, A., Kudrolli, A. & Mahadevan, L. Curvature condensation and bifurcation in an elastic shell. Phys. Rev. Lett. 98, 014301 (2007). | Article | PubMed | ChemPort |
23. Timoshenko, S. Strength of Materials (D. van Nostrand Company, 1966).
24. Crisfield, M. A. A consistent co-rotational formulation for non-linear, three dimensional, beam elements. Comput. Methods Appl. Mech. Eng. 81, 131–150 (1990). | Article |
25. Alava, M. & Niskanen, K. The physics of paper. Rep. Prog. Phys. 69, 669–723 (2006). | Article | ISI |
26. Sultan, E. & Boudaoud, A. Statistics of crumpled paper. Phys. Rev. Lett. 96, 136103 (2006). | Article | PubMed | ChemPort |
27. Didonna, B. A. et al. Singularities, structures, and scaling in deformed m-dimensional elastic manifolds. Phys. Rev. E 65, 016603 (2001). | Article | ChemPort |
28. Kantor, Y., Kardar, M. & Nelson, D. Tethered surfaces: Statics and dynamics. Phys. Rev. A 35, 3056–3071 (1987). | Article | PubMed | ChemPort |

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