Fractal Geometry and Mechanics of Randomly Folded Thin Sheets

Fractal Geometry and Mechanics of RandomlyFolded Thin Sheets

Alexander S. Balankin and Orlando Susarrey Huerta

Abstract This work is devoted to the statistical geometry of crumpling network andits effect on the geometry and mechanical properties of randomly folded materials.We found that crumpling networks in randomly folded sheets of different kindsof paper exhibit statistical self-similarity characterized by the universal fractal di-mension DN = 1.83 ± 0.03. The balance of bending and stretching energy storedin the folded creases determines the fractal geometry of folded sheets displayingintrinsically anomalous self-similarity with the universal local fractal dimensionDl = 2.67 ± 0.05 and the material dependent global fractal dimension D ≤ Dl .Moreover, we found that the entropic rigidity of crumpling network governs themechanical behavior of randomly crumpled sheets under uniaxial compression.

Keywords Folded matter · fractal · scaling · mechanical properties

1 Introduction and Background

Folded configurations of thin matter are very common in nature and technology [1].Examples range from virus capsids and polymerized membranes to folded engi-neering materials and the earth’s crust buckling. Quite recently, randomly foldedmaterials became a subject of great interest because of their fascinating topologicaland mechanical properties [2–12]. It was found that despite the complicated ap-pearance of folded configurations, the folding phenomenon is in itself very robust,because almost any thin material crumple in such a way that folding energy is con-centrated in the network of narrow ridges that meet in the point-like vertices [13].This leads to anomalously low compressibility of folded materials [2–4] and to avery slow stress-strain relaxation in randomly crumpled sheets [4, 10, 14]. Further-more, the balance of bending and stretching energy stored in the folded creasesdetermines the scaling properties of folded state as a function of the sheet size L ,

A.S. Balankin (B)Instituto Politecnico Nacional, Ed. 5, 3piso, ESIME, Av. Politecnico Nacional,Mexico D.F., 07738, Mexicoe-mail: [email protected]

F.M. Borodich (ed.), IUTAM Symposium on Scaling in Solid Mechanics,IUTAM Bookseries 10 DOI 10.1007/978-1-4020-9033-2 22,C© Springer Science+Business Media B.V. 2009

233

234 A.S. Balankin and O.S. Huerta

thickness h, material density ρm , two dimensional Young modulus �, confinementratio K = L I R, where R is the diameter of folded sheet, and the confinement forceF [8, 10, 11]. Specifically, it was shown that

R

h∝(

L

h

)2/D ( F

�h

)−δ(1)

where D is the mass fractal dimension and δ is the force scaling exponent. Accord-ingly, the set of randomly folded sheets of different sizes folded under the sameconfinement force, obeys a fractal law for the ball mass

M ∝ L2 ∝ RD. (2)

For randomly folded phantom sheets (self-intersections are allowed) with bend-ing rigidity D = 8/3 and δ = 3/8 [8], while for the self-avoiding sheets (cannot self intersect) numerical simulations suggest that D = 7/3 and δ = 1/4 [8],whereas the self-avoiding membranes without bending rigidity are characterized byD = 2.5 [15]. In the case of predominantly plastic sheets, such as aluminum foils,the shape and sizes of which do not change after the folding force is withdrawn, thescaling properties are determined by the effect of self-avoidance and characterizedby the universal scaling exponents D = 2.30 ± 0.01 and δ = 0.21 ± 0.02 [11]. Incontrast to this, the diameter of randomly folded elasto-plastic sheet increases whenthe folding force is withdrawn [10], such that a set of randomly folded sheets ischaracterized by the material dependent mass fractal dimension [10]. Furthermore,it was found that the surface of randomly folded balls and unfolded sheets bothexhibit an intrinsically anomalous roughness characterized by the universal localroughness exponent ζ = 0.72 ± 0.04 and the material dependent global roughnessexponent α = 2/D ≥ ζ [10]. However, despite a large amount of studies devotedto crumpling phenomena, the physics of folding remains poorly understood.

In this paper we report the results of experimental and theoretical studies offolding geometry and its effect on the mechanical properties of folded matter. Forexperimental studies we used different kinds of paper which offer a convenient andeconomical means for studying the crumpling phenomena in the laboratory.

2 Statistical Geometry of Folding

To study the statistical geometry of folding configurations we used three kinds ofpaper of different thickness and bending rigidity (see Table 1). It should be notedthat these paper were previously used for experiments reported in [10]. We used thesquare sheets with edge length L which was varied from L0 = 2 to 100 cm with therelation L = kL0 for scaling factor k = 1, 2, 4, 8, 16, 25, and 50. At least 30 sheetsof the same size of each paper were folded in hands into approximately sphericalballs (see Fig. 1).

Fractal Geometry and Mechanics of Randomly Folded Thin Sheets 235

Table 1 Sample table

Commercial name Carbon Biblia Albanene

Thickness h (mm) 0.024±0.004 0.039±0.002 0.068±0.005Areal density (g/m2) 22±0.8 35.6±0.5 63±1D from (2) 2.13±0.05 2.30±0.05 2.54±0.06Dl from (4) 2.68±0.05 2.67±0.04 2.66±0.05Dl from (5) 2.67±0.07 2.66±0.07 2.68±0.06DS 2.06±0.04 2.13±0.05 2.22±0.05Φ 1.38±0.09 1.40±0.07 1.4±0.1DN 1.82±0.04 1.83±0.03 1.83±0.03φ 1.26±0.07 1.23±0.07 1.29±0.07

Fig. 1 (a) Balls folded from sheets of the same size 40 × 40 cm of different papers (from left toright: Carbon, Biblia and Albanene); (b) cut through a crumpled ball of Biblia paper; and (c) foldedball crossed over by three silk strings along the chords

After the folding force is withdrawn, the ball diameter logarithmically increasesduring approximately ten days (see [10]). Accordingly, to reduce the uncertaintiescaused by variations in the squeezing force and strain relaxation, all measurementsreported below were performed two weeks after the sheets were folded, when nochanges in the ball dimensions were observed. The global fractal dimension of eachset of folded sheets was determined from the scaling relation (2), where R is theensemble averaged diameter (see Fig. 3a). Notice that the value of D measured inthis work (see Table 1) coincide with those reported in [10].

2.1 Intrinsically Anomalous Self-Similarity of RandomlyFolded Matter

It should be pointed out that the scaling relation (2) characterizes the self-similarityof the set of randomly folded sheets of different sizes. For self-similar fractals likeMenger sponge or porous media the self-similarity of internal structure is charac-terized by the same fractal dimension D. However, the local fractal dimension Dl ,which characterizes the self-similarity of folded configurations can differs from D,such that, generally, Dl ≥ D [12]. Accordingly, we performed the study of scal-ing properties of internal structure of randomly folded sheets. For this purpose, thefolded balls were crossed over with the silk strings in such a way that the Euclideandistance r (chord length) between the entry and exit points in the folded state wasvaried from 0.1R to R (see Figs. 1c and 2).


Fig. 2 The unfolded state ofsheet crossed over by silkstring. The crossing pointnumbering corresponds to thestring path

We found that the ensemble averaged number of intersections between the foldedsheet and straight line (one-dimensional string) scales as

N = aK 2( r

R

)Dl−2, (4)

where the local fractal dimension Dl = 2.67 ± 0.05 is found to be the same forall folded paper sheets (see Fig. 3 b). Hence, when the ball is performed alongits diameter (r = R) the numbed of intersections (see Fig. 1a, b and c) scale asND = aK 2 ∝ RD−2.

Scaling behavior (4) implies that the local mass density of the folded sheet scalesas ρ ∝ r−(3−Dl ) R−(Dl−D). So, in contrast to the case of (statistically) self-similarfractals, the local mass density within a randomly folded ball depends not only onthe size of local volume, but also on the ball diameter. Accordingly, in analogy withthe concept of intrinsically anomalous kinetic roughening, (see [16]), the scalingbehavior of randomly folded sheets we termed as the intrinsically anomalous self-similarity [12].

Furthermore, we found that the mean distance between the entry and exit pointsin the unfolded sheet satisfies the following scaling relation

ξ

L∝( r

R

)ψ(5)

with ψ = 0.30± 0.06 for all kinds of paper used in this work (see Fig. 3c). We alsofound that the total length of the string path along the intersection in the unfoldedstate scales as Γ = K 3r (see Fig. 3d), while the mean distance between intersections


Fig. 3 Log-log plots of: (a) R (mm) versus L (mm): the slopes of straight lines are equal to 0.94(dashed line), 0.86 (solid line), and 0.79 (dash-dotted line); (b) N ∗ = N/ND versus r/R the slopeof straight line is 0.68; (c) ξ ∗ = ξ/ξ (R) versus r/R in arbitrary units, where ξ (R) ∝ K R, and (d)Γ/r versus K 3 in arbitrary units for balls folded from three different papers: Carbon (◦), Biblia (•),and Albanene (♦).Straight lines are the best fittings (after [12])

is found to be proportional to ξ , and so, ψ = 3 − Dl . Further, the surfaces of ballswere painted with the Chinese ink. After unfolding, the painted area S is found toscale with ball diameter as S ∝ RDS , where the surface fractal dimension is relatedto the global mass dimension as DS = 3−2/D (see Table 1). Besides, we found thatthe ratio of areas painted in two sides of sheet increases with the ball diameter as

Ω = (R/Rc)Φ (6)

up to Rc, while for balls of diameter R ≥ Rc the painted areas in both sides of sheetare statistically equal. The scaling exponent Φ = 1.4 ± 0.1 is found to be the samefor three papers (see Table 1) and seems to be universal.


2.2 Geometry of Crumpling Network

Paper is an elasto-plastic material and so crumpling creases formed in randomlyfolded paper produce permanent marks associated with plastic deformations, leav-ing imprinted the crumpling network as it is shown in Fig 4 (left). The scannedimages were used to reconstruct crumpling networks formed by straight ridgeswhich meet in the point-like vertices (see Fig. 4 (right)). In this work we studiedthe statistical geometry of crumpling networks formed in sheets of different sizesof different kinds of paper. We found that the vertex connectivity degree (numberof ridges connected with a vertex) distribution has an exponential tail. The ridgelength distributions are found to conform to the Gamma distribution (see Fig 5a)with probability density

p(l) = mm

Γ (m)

(l

lm

)m−1

exp

(−m

l

lm

)(7)

where the mean length lm ∝ R (see also [10]), Γ (.) is the Gamma function, and mis the shape parameter which is found to be proportional to the number of layers nin the randomly folded ball (see Fig. 1b) .

Furthermore, we found that the crumpling networks are statistically auto-similarwithin the wide range of length scales (see Fig. 5b). The fractal dimension of crum-pling network is found to be independent on the paper thickness, sheet size, andconfinement ratio. Specifically, we found that all crumpling networks studied in thiswork are characterized by the fractal (self-similarity) dimension DN = 1.83± 0.03(see Table 1); and so, one can expect that this value is universal for random crum-pling phenomena.

Fig. 4 The unfolded state of sheet of Biblia paper of size 10 × 10 cm (left) and the correspondingnetwork of crumpling creases (right)


Fig. 5 (a) Statistical distributions of ridge length in an unfolded sheet of Biblia paper of size10 × 10 cm (bins – experimental data, solid line – fitting with the Gamma distribution) and (b)fractal graph of number of boxes NB versus box size Δ for crumpling network from Fig. 4 (right)

3 Mechanical Properties of Folded Sheets

Mechanical properties of randomly folded mater are governed by rigid network ofcrumpling creases, which accumulates more than 95% of folding energy [1]. Themechanical response of crumpling network is determined by the shape and volumedependence of its free energy. In the three-dimensional stress state the mechanicalresponse of crumpling network is controlled by the volume dependence of networkenthalpy, which leads to the power law behavior (1). In this work, randomly foldedthin sheets were tested under uniaxial compression using a universal test machine atdifferent loading rates in the range from 0.01 to 10 mm/sec. Figure 6a shows typical

Fig. 6 (a) Typical force-displacement curve of randomly folded sheet of paper under uniaxialcompression and (b) the loading part of force-compression curve (a) in coordinates −F versusΛ = (1 − c)/(1 − λ) (solid line – fitting with (8))


Fig. 7 (a) Force-displacement curve (maximal um deformation and u p deformation after unload-ing) and (b) strain relaxation after unloading of randomly folded ball

force – compression behavior F(u = R − H ) of randomly folded paper ball underuniaxial compression.

Despite to the irreversible nature of ball deformation (see Fig. 7a), we foundthat in all cases, the loading part of the experimental force-compression curveF (λ = H/R) does not depend on the loading rate in the range from 0.01 to10 mm/sec. and may be precisely fitted by the following simple relationship

F = Y

(1 − c

λ− c− 1

)(8)

as it is shown in Fig. 6b in which the force is plotted versus Λ = (1 − c)/(λ − c),where the fitting parameter c is found to be equal to c = nh/R ∝ R, n ∝ R2 is thenumber of layers of thickness h in randomly folded ball and Y is the ball stiffness.We noted that from (7) follows that the entropy of crumpling network depends onthe compression ratio λ as S ∝ (λ − c) − ln(λ − c), and so the entropic rigidity ofcrumpling network obeys the force-compression behavior (8). Hence, in contrast tothe three-dimensional stress state, the response of randomly folded sheet to uniaxialloading (8) is predominantly of the entropic nature. The elastic modulus E ∝ Y/R2

is found to scale as E = E0 R−φ with the universal scaling exponent φ = 1.26±0.07(see Table 1).

After unloading, the strains slowly relaxes during approximately ten days,such that

λ(t) = H (t)

R= λP

[1 + κλm ln

(t

τ

)](9)

where κ is the material dependent constant, τ ∝ (L/h)γ , where γ ∼= 3 ± 1,λP = (R − u p)/R, and λm = (R − um)/R. The logarithmic strain relaxation can beattributed to the fractal nature of crumpling network with the gamma distribution ofridge lengths.


Acknowledgement This work has been supported by CONACYT of the Mexican Governmentunder Project No. 55736. The technical help from Antonio Horta Rangel, Didier Samayoa Ochoa,Ernesto Pineda Leon, Rolando Cortes Montes de Oca, and Maribel Mendoza is acknowledged

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Fractal Geometry and Mechanics of Randomly Folded Thin Sheets

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