Witten's lectures on crumpling

Physica A 313 (2002) 83 – 109

www.elsevier.com/locate/physa

Witten’s lectures on crumpling A.J. Wood (Editor) Oxford University, Department of Theoretical Physics, 1 Keble Road, Oxford, UK

Abstract Crumpling is a distortion brought about by a strong compression of a surface in which energy condenses in extremely small regions rather than being stored uniformly. Scaling laws describe how much energy goes into how little space. As crumpling of a sheet proceeds, a network of ridges and vertices develops. The precise nature of these geometrical singularities depends on the stretching and bending characteristics. It is shown that for two-dimensional surfaces in three-dimensional space energy mainly condenses in stretching ridges. The scaling arguments are put on a /rm basis by using the mathematical description of an elastic membrane through the von K1arm1an equations, and the universal properties of ridges are discussed. From this theory one can understand that crumpling confers strength. Finally, it is sketched how energy condensation and its scaling laws depend on the dimensions of the con/ned sheet and embedding space. c 2002 Elsevier Science B.V. All rights reserved.
Keywords: Crumpling; Energy condensation; Stretching and bending elasticity; Membranes; von K1arm1an equations

1. Introduction A common topic of research in condensed matter physics is the properties and manner in which external energy is stored or compressed in a particular substance. If we think of the additional external energy taking the form of a simple perturbation to an existing system then how will the system react? If this perturbation is weak or uniform in some sense then we can reasonably expect that the energy will be stored uniformly throughout the substance. If the perturbation is strong however, it often happens that a new phenomenon called ‘condensation’ occurs. This condensation phenomenon is a central feature of many outstanding problems in the /eld of physics, encompassing 

This paper was written by Jamie Wood in consultation with Tom Witten, based on Witten’s transparencies. Joseph Indekeu and Henk van Beijeren did extensive editing on this draft, while Witten provided references and /gures. E-mail address: [email protected] (A.J. Wood). c 2002 Elsevier Science B.V. All rights reserved. 0378-4371/02/$ - see front matter
PII: S 0 3 7 8 - 4 3 7 1 ( 0 2 ) 0 1 2 6 0 - 8

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turbulence, Buid interfaces and gravitational collapse of galactic matter. What all these instances have in common is that the energetic perturbation is condensed into an indefinitely small fraction of the space, sometimes into a lower dimensional subset of the original system. In the most well understood case above, that of the Buid interface, this is clearly just the two-dimensional surface that forms in the three-dimensional space. In the other examples things are less clear, and the subset could even have a fractal dimension. The question is, can we /t the often observed phenomena of crumpling into the list of examples above? Crumpling is something we are all familiar with; we must have crumpled at least an object a day for most of our lives, but very few of those times have many of us looked at what eCect our sudden input of energy has had on the sheet of paper or other unfortunate inanimate object. What structures have been produced in order to absorb the energy we have put in and how much work did we actually have to do to eCect this change? Throughout this document the paradigm object to be crumpled will be an elastic membrane that is con/ned within a sphere whose radius is reduced—imagine a rubber sheet in a deBating balloon. In the crumpling process it is clear that some form of hierarchical structure is produced which potentially has some scale invariant properties associated with it. Under a weak compressive force these features will primarily be sharp edges or ridges and as the compression increases sharp corners or vertices will appear. This does not happen if we crush a one-dimensional /bre in a balloon, so why is there a diCerence? The article is divided into three more sections that correspond to (a large part of) the lectures given at the summer school. In the next section, we shall look at the properties of the stretching ridge and see how it dominates the energy density of the crushed sheet. In particular we shall derive a power law for the fraction of the membrane the energy is condensed into and show how we can regard a crumpled object as a network of ridges. In Section 3 we look in more detail at the mathematics of a thin sheet, in particular the von K1arm1an equations and a set of scaling relations for properties of the ridges. In the /nal section we take a brief look at the crumpling properties of sheets and hyper-sheets in d-dimensional space. 2. Stretching ridges and their properties In the past many forms of distorted surface have been examined in the literature. These studies diCered in the type of surface studied and in the way that surface is perturbed. One important class of surface is a liquid-like membrane. The thermal distortions of these liquid sheets, most commonly composed of surfactants, is a subject of considerable recent interest. This is due to their many biophysical implications for lipids and consequently cell walls and other biological membranes. A second fascinating class is the surface of an elastic solid such as a gel. The buckling instabilities at the surface of a three-dimensional elastic solid is another related topic, and this has been studied by Tanaka in the form of a drying gel [1]. In this article we will deal instead with elastic membranes such as a piece of paper or rubber. One important

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type of perturbation to consider is thermal Buctuations. In Ref. [2] the distortion due to thermal Buctuations of two-dimensional elastic sheets, a ‘crumpling transition’, was examined and a second type of distortion that yields rich response is to put topological defects—dislocations and disclinations—into the membrane. Such internal defects of elastic sheets, in particular distortion saddles, have been considered by Seung and Nelson [3] and analogously, vortex ‘blimps’ in liquid sheets have been investigated by Mackintosh and Lubensky [4]. Here we wish to consider another type of perturbation, in which the membrane is forced to distort by a con/ning external force, such as the sphere mentioned above. More directly related to the work summarised here are the observations of crumpled sheets by Kantor et al. [5]. There has also been considerable work by other authors on crushing as well. Scheidl and Troger [6] have looked at the scaling properties of singularities in the sheets and other work by Witten and co-workers will be reviewed here. 2.1. Energy of an elastic sheet Consider a large, thin elastic disk of radius R0 and thickness h. The energetic properties of the sheet are determined by the bending modulus which we denote  and a stretching modulus, G. The total energy, ET , of the sheet, in a given con/guration, is given schematically by the sum of the bending and stretching energies, respectively,  
1 2 2 (1) ET = d s  2 + G L ; RL where R−1 is the local curvature and 2L is the local deformation factor. This simL pli/ed form of the energy is suOcient for our present scaling arguments. We will be more precise later. Importantly, the bending modulus has dimensions of energy and the stretching modulus has dimensions of energy per unit area. This leads to two important conclusions.  Firstly, there is a characteristic length scale implicit in this energy given by =G. This length arises from the /nite thickness of the sheet h. Indeed, a sheet of uniform, isotropic material has the following relation between the thickness and the moduli (once we have speci/ed these precisely): h2 =

32  : 3 G

(2)

Secondly, as the stretching is area dependent we see that the relative energetic cost of stretching the sheet grows with size whereas the bending energy is independent of any scale transformations. To see this we can examine the behaviour of (1) under a scale transformation by a factor of . That is, we make a second sheet of factor larger than the /rst, made of the same material (and thus having the same ; G, and h). We then distort the two sheets in a similar way, so that the vector displacement between each pair of points in the new sheet is just times larger than the vector displacement between the corresponding points in the original sheet. We can readily relate the

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energy ET of the original sheet to the energy ET( ) of the expanded one:  
1 ( ) 2 2 2 S( ) : ET
→ ET = d s  2 2 + G L

RL  
1 = d 2 s  2 + 2 G 2L ; RL

(3)

where this behaviour is evident. We can conclude from this the important relationship that the large sheet limit (R0 =h1) is the same as the unstretchable limit (GR20 ). This gives the familiar property that ‘large’ sheets do not stretch! The energy cost for stretching one of these sheets becomes arbitrarily larger than the bending cost in this limit. The original sheet and the expanded sheet look exactly the same, yet the proportion of stretching energy in the expanded sheet is arbitrarily larger. With this motivation, we shall try to analyse the crumpling process by supposing that our sheet is simply unstretchable. 2.2. Con2ning an unstretchable sheet If we consider a sheet in the above, unstretchable limit, then an immediate constraint occurs if we compress such a sheet. If we pick an arbitrary point on the surface and attach to it two orthogonal (one and two) directions, then the radii of curvature in these directions are RL; 1 and RL; 2 . If both of these curvatures are /nite numbers then at the chosen point the surface will bulge outwards. This implies that the surface is being stretched at this point, which is forbidden. Therefore, the only allowable con/gurations in the unstretchable limit are those that have at least one of RL; 1 and RL; 2 in/nite for every point on the surface (of course this is only a local requirement—diCerent points can and will have diCerent choices of orthogonal coordinates). This constraint implies that the deformation induced in a sheet by a compression will take a cone-like structure. The energy absorbed by such a structure can be estimated by considering such a cone of edge length X . The local radius of curvature is proportional to the distance r of the point from the apex along the side. We will introduce the thickness of the sheet h as a lower cutoC. This gives an approximate calculation of the energy for such a simple deformation, assuming approximate circular symmetry, as
X 1 Energy ≈  r dr 2 R (r) L h X : h This simple calculation leads us to the following conclusions: =  log

(4)

• The deformation energy for a given compression factor is logarithmically related to the size of the sheet. • The cone-like structure hinted at here implies that there exist singular ‘disclinations’. • If there are two disclinations, then there must be a perfectly sharp crease connecting them, otherwise we are forced to conclude that stretching cannot be neglected.

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Optimal curvature radius R if stretching is allowed: "kite" model must bend down to

If R > 0, then If cross-line

must pull away

is of fixed length,

a distance y

from

R

x y

R

by about y2 / X....stretching

This lengthens Stretching factor γ

y2 / X2. ... Energy

Gγ 2 (area)

G R5 X3

RX Bending energy Optimal R: R5 X3

κ R-2 (area)

κ R-1X

R-1X or R ~ X2/3 << X

Energy ~ X1/3 Fig. 1. A depiction of the kite model used in scaling arguments in the text.

• If the sheet undergoes further compression then more such disclinations are required to permit the sheet to /t in the allowed space. However, we could instead think of a diCerent model. Suppose we consider a kite-like shape (see Fig. 1) of our elastic sheet, but this time allow the sheet to stretch by a small amount. We consider the kite to be of length X and to be clamped securely in a

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diamond-shaped frame. This frame is constructed in such a way that it is hinged along one of its main diagonals. In this way when the hinge is used the kite is simply bent along its ridge line. This non-compressed model allows to see what happens when we force two sharp bends to exist. The local deformation factor induced by bending the kite in this way is  y 2 r−X = : (5) ≈ X X Because the diagonal of the kite is pulled away from its original position, its length has increased to a value r. As with any stretched string that is pulled laterally a distance y, the length r ≈ X (1 + (y=X )2 ). The radius of curvature can now take a /nite value at the ridge mid-point and we denote this RL . The distance y above will then be of order RL . We now make the critical assumption that the deformation, and the stretching in particular, is con/ned to a strip also of order RL . We motivate this assumption by claiming that the stretching ridge is an apparent minimal con/guration for the elastic sheet and so any deformation outside the ridge will not be likely on energetic grounds. This is plausible since any deviation from the ridge con/guration will involve strain over a much wider area than that in the ridge. Using (1) we can now estimate the bending and stretching energy contribution to the total energy of the sheet
1 Bending energy; B =  d 2 s 2 RL ∼ XRL
Stretching energy; S = G

1 = XR−1 L ; R2L d 2 s 2

R4L = GX −3 R5L : (6) X4 The minimum of the energy will be when these two terms are approximately equal and we can use (2) to reintroduce the thickness and hence the large sheet parameter X=h. A scaling relation for the energy can be thus derived: ∼ GXRL

ET ≡ S + B ∼ (X=h)1=3 :

(7)

We can draw several important conclusions from this scaling relation. • RL =X ∼ (X=h)−1=3 ⇒ energy is con/ned to an arbitrarily small fraction of the membrane. The anticipated energy condensation is indeed occurring. • The inverse curvature RL ∼ h(X=h)2=3 h and the strain, ≈ (RL =X )2 , are both weak. This means that our weak deformation assumption claimed earlier is indeed justi/ed. • (X=h)1=3 log(X=h) meaning that the ridge energy is considerably greater than the vertex energy. This means that the ridges dominate the energetic properties of the crumpled material and also that the ridges create a con2ning potential for the vertices.

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Fig. 2. The Seungium sheet. The G and  spring are shown.

2.3. Numerical simulations to test the scaling predictions In order to test the scaling predictions above, a computational model of an elastic sheet is required that can be compressed in a controlled way or indeed made to form speci/c shapes that should exhibit scaling in a prescribed way. Such a model was constructed by Seung and Nelson [3] in the late eighties. Work then and afterwards [7] has shown that such a sheet, a ‘Seungium’ lattice model, does indeed behave like an isotropic rubber material, exactly what is required to test the scalings above (See Fig. 2). The sheet is composed of a series of lattice sites on an icosahedral lattice that are interconnected by springs to the nearest neighbours. In addition, there are bending springs that exert forces to keep adjacent triangles from bending relative to each other. One can readily show that these springs give the lattice an energetic cost for stretching or bending. From the spring constants, one can derive the stretching modulus G and the bending modulus . Consequently, we call the nearest-neighbour springs G-springs and the next-to-nearest-neighbour springs -springs in an obvious notation. This is important as via the relationship for the thickness in terms of the two moduli (2) we can determine and control an eCective thickness for such a sheet that will enable the scaling predictions above to be eCectively scrutinised. 2.4. A virial theorem Another intriguing result that can be derived from the analysis presented so far is a result concerning the ratio of the bending energy to the stretching energy. This can be

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interpreted as a virial theorem for crumpling. In the thin-sheet limit, we have explored in the analysis above the two separate terms that make up the energy can be computed separately in terms of the ridge curvature, R−1 L . We can determine Bending energy; B = k1 RL ; Stretching energy; S = k2 R−5 L :

(8)

The total energy is thus the sum of these two terms. We can then minimise the energy in terms of the ridge curvature by taking a simple derivative w. r. t. the curvature. This yields dET dS dB + = dRL dRL dRL = k1 − 5k2 R−6 L = (B − 5S)=RL :

(9)

So at the energetic minimum we have the result that the ratio of the bending energy to the stretching energy is a universal number which takes the value 5. This virial theorem has been con/rmed by numerical results from the Seungium tetrahedron. These results also show the strong dependence of the analytical results on the large sheet limit. To approach the limiting ratio, the size must be several hundred times the thickness. 2.5. Interactions From the analysis of the Seungium sheets it is also possible to gain some numerical understanding of the way in which the component parts of the crumpled sheet interact with each other. We have already demonstrated through the scaling predictions that adjacent vertices feel a mutual attraction owing to the distorted ridge joining them but now we need to primarily understand how the ridges, and secondarily the vertices, interact with each other in order to compose a more complete picture of the crumpled sheet. Fig. 4 shows a numerically determined plot, with its chi-squared /tting, for the energy of the sheet for variable spacings between adjacent ridges. It is immediately clear that the interaction energy falls oC exponentially quickly but importantly it does so with a range that is determined by the width of the ridge, not the length. As the width of the ridges is small this means that the ridges are eCectively independent of each other and only interact weakly. As we shall see this is very diCerent from vertices that we shall treat shortly. We also must take care to look at the diCerence in interactions between ridges of a diCerent orientation. We regard ridges of the same type ones that curve in the same direction forming a U shape and opposite ones, as ones that bend in the opposite sense forming a Z shape or a kink-like defect. As can be seen from Fig. 5 there is a qualitative diCerence in the form of the potentials. The interaction potential for two ridges of the same type is the familiar Lennard–Jones or wetting binding

A.J. Wood / Physica A 313 (2002) 83 – 109

91

Fig. 3. Seungium sheet models of a kite, a boat, a tetrahedron and a “poked” cone. The darker shading shows the areas of greatest energy.

potential with an attractive tail and a small repulsive core. For interactions between diCering types we see precisely the opposite eCect, with a repulsive tail but what seems to be an attractive core. The net eCect is to create a steep activation boundary that must be crossed in order for the ridges to combine and the kink to be ‘ironed out’. For vertices, the interactions have a considerably diCering character. Weak vertices interact by a strongly attractive mechanism which leads them to coalescence. The con/guration of two vertices, of solid angles 1 and 2 , connected by a ridge of length X have a combined energy which is dominated by the energy of the ridge ∼ X 1=3 . The two vertices in this situation will attempt to combine into a single vertex of solid angle 1 +2 which encompasses the energy of the previous ridge and the new con/guration possesses no ridge. However, this process has a strong limiting factor, the resulting larger vertex must be an allowable con/guration which will not be possible if the sum of the too solid angles is too great, or more speci/cally we must have 1 + 2 6 2. We have then the conclusion that weak vertices attract strongly, but strong vertices do not.

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A.J. Wood / Physica A 313 (2002) 83 – 109 1.62

Total energy ( κ = 0.05)

1.6

1.58

1.56

1.54

1.52

1.5 20

30

40

50

60

70

80

90

Distance between ridges (in units of lattice const) Fig. 4. A plot showing the exponential decay of the interaction strength between ridges as the distance is increased.

Fig. 5. Total elastic energy of a two-ridge con/guration in the units of  found from a 100b × 130b strip ◦ bent in two places by 90 versus the distance between the two ridges, D. The thickness is = 0:0002. Square symbols correspond to the ridges that have the same orientation (con/guration U), whereas the circles correspond to two ridges of the opposite orientation (con/guration Z).

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93

2.6. Networks of ridges All our above analysis leads us to conclude that a useful description of a crumpled sheet is as a network of stretching ridges. These ridges, rather than either the vertices or the facets, dominate the energy of a given crumpled con/guration. As they only weakly interact we can regard each ridge as an independent contribution to the energy and we can write down the total energy absorbed in the sheet as   Xi 1=3 Energy ≈  : (10) h i We now look at some properties of this network of ridges and the intervening facets. Some information about the distribution of the ridges and consequently the facets in the enclosing container is an important /rst step. Are the networks of ridges fairly uniform across the surface or do the edges crush /rst, leaving a giant facet in the centre of the con/ning volume? We can get some idea of this by a simple argument concerning the mean size and spacing of the randomly oriented facets. Consider a given con/guration of a crumpled sheet of relaxed edge size length, L, which has been crumpled inside a spherical container of radius, R. In this con/guration we assume that the average size of a facet has edge length  and that the mean spacing between the centres of adjacent facets is of order b. The number of facets N is evidently  2 L N ≈ (11)  which is related to the mean spacing between the facets via  2  R3 3 3 b ≈ ≈R N L

(12)

if the facets are well de/ned, and hence do not intersect or align with each other; then we expect b¿ ⇒ 3 6 b3  2  L  2 R ⇒6R

R L ⇒ 3 6 R3

(13)

which tells us that in a crumpled con/guration the average size is much less than the size of the container. This argument assumed that the ridge network is fairly uniform across the entire crumpled sheet. Experiments suggest that in real crumpled sheets, the ridge network is not uniform; instead there is a broad distribution of ridge lengths. Experiments which involved

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Fig. 6. The results from an experiment measuring the distribution of noises when crumpling an elastic sheet. The results show the broad distribution of ridge energies and thus the ridge lengths themselves, see text and Ref. [8].

measuring the crackling sounds and pictures of crumpled sheets support this idea, see Fig. 6. This seemingly broad distribution of sizes across the sheet is good evidence of a hierarchical structure, and hence a hierarchical mechanism to produce it, as we alluded to earlier in the article. A simple, and plausible, mechanism for such buckling eCects would be to consider the repeated uneven breaking of an initial ‘ridge’ of length L. For simplicity let us assume that we break the ridge into one-third and two-third fractions, as depicted in Fig. 6 (lower left), k times. This means that any one of the random 2k

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95

fragments will have a length, ‘, given by k factors

       1 1 1 ‘=L

3

3

3

2 3

2 3

2 3

 1  ···

   pick one pick one pick one

3 2 3

(14)

 pick one

which is a product of random factors. Thus the log of ‘ is the sum of random terms: a random walk. More precisely we can infer that u ≡ log ‘ is the length of a k-step random walk distribution. This is de/ned in the usual way via the probability density function, P 2 2 dP 1 (15) = √ e−(u−u) =k du k which in terms of the variable ‘ is 2 2 dP dP du 1 e−(log ‘−log ‘) =k : (16) = =√ d‘ du d‘ k‘ For a broad range of ‘, such ‘log-normal’ distributions are dominated by the 1=‘ prefactor. One way to obtain information about P(‘) is to analyse the sound that a sheet makes when the facets are stressed and forced to buckle. This is the familiar sound of crumpling paper, consisting of a series of discrete pops. By recording this sound, one can measure the sound energy E in each pop and thus determine the distribution of energies P(E). It is natural to suppose that the sound emitted by a large facet should be louder than that emitted by a small one, so that E grows with ‘. Assuming that this growth takes the form of a power law, we can readily determine P(E) from the P(‘) found above. Indeed, P(E) itself must have a log-normal form. Kramer and Lobkovsky [8] did this experiment and found a broad distribution of pop energies, consistent with this prediction (see Fig. 6). 3. The von Karman equations In order to con/rm the heuristic scaling arguments of the previous section we shall derive and use the more mathematical approach, that of the von K1arm1an equations. The language we shall adopt is as follows. We shall describe the position of the membrane in a given con/guration by its vector position r(x). The deformation of such a sheet is measured by two quantities, the strain, ij , de/ned by   1 @r @r ij ≡ (17) · − "ij 2 @xi @xj and the curvature, Cij , given by  @ @ Cij ≡ r(x) · nˆ ; @xi @xj

(18)

forms that should be familiar. We can determine the shape of a membrane, r(x), then, by minimising the deformation energy. A more precise de/nition of the deformation

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energy than (1) is thus required in terms of (17) and (18) above. The energy should be a local, uniform and expandable function of and C everywhere. It also must be a positive function, A natural choice for ET is then

unless and C are zero everywhere. ET = 

ˆ +G ˆ 2 + # det C] d 2 x[(Tr C)

d 2 x[(Trˆ )2 + #G detˆ ]

+ higher-order terms ; (19) ˆ ˆ where A is a matrix with entries [A]ij ≡ Aij . The trace and the determinant of the curvature ‘matrix’—under the above de/nition—are equivalent to the (algebraic) mean and the gaussian (geometric mean) curvatures, respectively. Both the strain and the curvature also have some geometric constraints in order that the resulting set of equations do indeed describe a well-de/ned surface. These relations can be determined in a precise way through the use of ChristoCel symbols, but here we wish to simply motivate their origin. The /rst relation is simply @k Cij = @j Cik ;

(20)

which is the direct tensor analogy of the condition satis/ed by an irrotational vector /eld. We can immediately verify this property by using the de/nition Cij =@j ri . In terms of the ChristoCel symbols this is found by examining the change in the curvature tensor when it is parallelly transporting around some arbitrary closed curve. This condition gives two constraints on the curvature and means that we have only one remaining free degree of freedom. We can thus write the tensor down in terms of a curvature potential function, f(x), which is related to Cij by the simple form Cij = @i @j f(x)

(21)

which one can readily check satis/es (20). The second constraint concerns the eCect of localised gaussian curvature on the global strain of the rest of the sheet. Consider the eCect of introducing a small cap of a sphere smoothly onto an otherwise undeformed circular elastic sheet. Such a cap has uniform gaussian curvature. In order to compensate for the ‘cap’, the entire circle most shrink into a cone-like structure which has non-zero strain everywhere. We assert @i @j ij = det Cˆ : (22) The proof of this relationship is a standard subject of diCerential geometry [9]. This is Gauss’ Theorema Egregium applied to a membrane that is Bat in its unconstrained state. This relationship tells us that localised gaussian curvature acts as a source of strain, in the Buid dynamical sense. Equivalently, the gaussian curvature can be thought of as mimicking a charge and the strain as the corresponding electrostatic potential. In addition to these geometric constraints the membrane must also be in local mechanical equilibrium at all points. The important material quantities for the mechanical constraints are the stress, ij and the in-plane moment, Mij which, by the theory of elasticity, are linearly related to the strain and the curvature, respectively. We shall return to this relationship shortly. The /rst mechanical constraint is that the in-plane forces need to be balanced. But since only a non-uniform stress can make forces, this condition is simply met by requiring that the stress be a divergence-free tensor /eld @i ij = 0;

j = 1; 2 :

(23)

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As with the curvature, this implies that there exists only one remaining free parameter. As with the curvature the stress can be expressed in terms of a scalar potential that we denote as '. In this case the potential is related to ij via ij = ‘curl’ij ' = *ik *jl @k @l ' ; where the antisymmetric two-tensor *ij has the obvious de/nition   0 1 : *ij ≡ *ij3 ≡ −1 0

(24)

(25)

One can again readily check that any ij of the form (24) satis/es (23). In this form the potential function ' is commonly called the Airy force function. A second constraint is that the torque on an element of the membrane induced by the normal shear stress, Qi , should balance. The internal torque can only emerge through non-uniformities in the moments and thus that these two eCects should balance gives us Qi = @j Mij :

(26)

The /nal constraint is that the normal forces to the sheet should also balance. We can also take into account some additional external pressure term P in the calculation. The vertical force from the sheet comes from both the stress and the curvature and is expressed as the tensor convolution of the two. A contribution from the shear stress can only come from any non-uniformity and so the equation we derive for balance is @i Qi = ij Cij + P

(27)

which becomes @i @j Mij = ij Cij + P

(28)

in terms of the moments from (26), thus combining the latter two, (26) and (27), into a single constraint. The stress in Eq. (27) arises from deforming the material; thus, it can be expressed in terms of the elastic energy. By considering the work dET needed to increase the strain, one readily veri/es that ij =@ET =@ ij . Since the energy is quadratic in the strain, the stress  is evidently proportional to the strain: ˆ ˙ ˆ . The proportionality constants are known as the elasticity tensor. These are evidently proportional to the G and #G coeOcients from Eq. (19). By parallel reasoning, the moments are Mij = @ET =@Cij , and ˆ with proportionality constants given by the bending stiCness  and the thus Mˆ ˙ C, associated # . We stress here that this is a much simpli/ed de/nition and that it is possible to derive all of the results above systematically with far less assumptions than have been made here [10]. The potentials de/ned for and for C enable us to simplify the two constraints expressed in Eqs. (22) and (28). In this form they are denoted the (/rst and second)

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von K1arm1an equations 1 4 1 ∇ ' = − [f; f] ; G 2

(29)

∇4 f = ['; f] + P(x) ;

(30)

where we de/ne ∇4 = (∇2 )2 = @i @i @j @j

(31)

[a; b] = *ik *jl (@i @j a)(@k @l b) :

(32)

and Eq. (29) is known as the geometric von K1arm1an equation; Eq. (30) is called the force equation. 3.1. Scaling relations from the equations Now that we have a set of equations that describe the behaviour of the elastic membrane we can utilise them in order to extract and con/rm some scaling relations. We will once more consider the simple model of the minimal ridge that was presented in the previous section, see Fig. 1. Firstly, we need to de/ne some units to work in; a convenient scheme is to take the ridge length as the dimensionless length scale in the following way 1 X ≡ 1;  ≡ 1; G ≡ 2 (33) h under which we can write the von K1arm1an Eqs. (29) and (30) as 1

2 ∇4 ' = − [f; f] ; 2 ∇4 f = ['; f] ;

(34)

where is a new parameter which is evidently the length (=G)1=2 . Furthermore we de/ne x = x1 and y = x2 . The angle ., denoting the deviation of the ridge from the horizontal, determines the distant behaviour of f, which we can write as a boundary condition
(35) 2. = dyCyy = @y f|∞ −∞ ; which means that for y → ±∞; f → .|y|. If we can solve, or extract the scaling form of, the von K1arm1an equations with this boundary condition then the limit as → 0 should reveal the minimal scaling ridge. The variables in this limit need to be /nite so we can try scaling of the form ˜ f = −/f;

' = −" '˜

⇒ y = −/ y; ˜

x = x; ˜

@y = / @y˜

(36)

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under which the two von K1arm1an equations become   4 4 @4 '˜ 2−" @ '˜ 2/ 4/ @ '˜ ˜ f] ˜ ; = 2/−2/ [f;

+ + 2 @y˜ 4 @x˜ 4 @x˜2 @y˜ 2

99

(37)

where we include the / exponents explicitly so as to make their origin clear, and   4˜ 4˜ 4˜ −/ @ f 2/ @ f 4/ @ f ˜ :

= −/−"+2/ ['; + 2 + ˜ f] (38) @x˜ 4 @x˜2 @y˜ 2 @y˜ 4 Since in the limit → 0 we expect to form narrow ridges, corresponding to the exponent / ¡ 0, we can anticipate that the y derivatives in (37) and (38) will be dominant which leads to the following diCerential equations which must be /nite and well de/ned in the limit as → 0

2−"+4/

3/

@4 '˜ ˜ f] ˜ ; = 0 [f; @y˜ 4

@4f˜ ˜ : = /−" ['; ˜ f] @y˜ 4

(39)

From the /rst equation we infer 2 − " + 4/ = 0; from the second we infer 3/ = / − ". The solution of these two equations for the scaling is simple and we /nd 1 2 /=− ; "= (40) 3 3 which implies, by substituting y˜ for the ridge length in (36) which is normalised to 1  1=3 h X (41) y≈ X that is ridge width ∼ length



thickness length

1=3 (42)

which elegantly con/rms the scaling picture we established in the previous section. A similar analysis shows the eCect of changing the ridge angle . (See Fig. 7). We may increase this angle by a factor of A by merely multiplying the f potential by A. The eCect of this change of scale on the von K1arm1an equations can readily be deduced. We suppose that f, ' are solutions of Eq. (34). Then we substitute f=Af˜ and '=Ap ', ˜ without altering the scale of the co-ordinates x or y. Evidently, the new f˜ has its ridge angle . reduced by a factor A relative to f. As we change f, some corresponding rescaling for ' may be necessary. We have postulated a generic scaling law with an as yet unknown exponent p. We now consider the von K1arm1an equations satis/ed by f˜ and '. ˜ The force von K1arm1an equation is linear in f, so changing from f to f˜ has no eCect on it. The only change is a factor of Ap in front of ['; f]. Evidently we can avoid any eCect of A by taking p = 0. Now the geometric von K1arm1an equation reads ˜ f], ˜ or ( =A)2 ∇4 '˜ = − 1 [f; ˜ f]. ˜ This equation shows that the eCect of

2 ∇4 '˜ = − 12 A2 [f; 2 changing the opening angle . is equivalent to the eCect of changing the thickness . Thus, for example, the width w of the ridge scales as 1=3 . The A scaling shows that

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A.J. Wood / Physica A 313 (2002) 83 – 109

Ridge Width Ridge Sag

Ridge Radius of Curvature

2α

C

-1

Fig. 7. Angle-scaling.

π − 2α

h

X

Fig. 8. A bent strip.

w scales as the ridge angle . to the −1=3 power. From this we can infer the scaling of the energy. We are free to focus on the bending energy Eb , since the stretching energy is a /xed multiple of the bending energy. This bending energy is of order 2 Cyy wX . Here Cyy is the transverse curvature in the middle of the ridge. Integrating the curvature gives the turning angle .. Thus Cyy w ∼ = .. Inserting this in the bending energy, we /nd Eb ∼ .2 =w. Now using w ∼ .−1=3 we infer Eb ∼ .7=3 . The scaling of the stretching ridge width as we move from the vertex to mid-ridge is also of interest. We consider a stretching ridge on an elastic membrane of thickness h as before. The ridge itself has a total length X and a position-dependent width W (x) measured from the left (say) vertex. We can immediately infer, as a geometrical constraint, that W (x) x → 0 as →0 (43) X X W( 2 ) in the limit h W x X (See Fig. 8). We now ask how the left side goes to zero in this limit. We assume that W (x) ∼ xp X q in this limit with both p; q ¿ 0 so that (43)

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101

is satis/ed. From earlier calculation we know that 1

2

W (x)|x≈X ∼ h 3 X 3 ∼ X p X q

(44)

and hence that p + q = 2=3. If we take W (x) to have the strongest x behaviour possible then the inequalities imposed on q imply that we can take q = 0, so that W (x) ∼ x2=3 . Under this assumption of strong x dependence we can look at the behaviour this implies for various energetic quantities associated with the ridge. We see that at a distance x from a vertex, the ridge energy is contained in an area xW (x) ∼ x5=3 . This region has an energy density given by * ∼ W (x)−2 ∼ x−4=3 , and hence Energy density ∼ Area−4=5 :

(45)

This relation implies that a ridge contains a wide range of energy densities from low near the middle to high near the vertices. Given some threshold energy density *, there exists some area of the surface where the energy density exceeds that. Eq. (45) implies that the threshold energy falls oC as the area increases according to a −4=5 power law. This prediction of how uniformly energy density is distributed over the area of the sheet can be directly tested numerically. Further, this way of describing the energy distribution is useful, since it characterises the condensation of energy without regard to the shape of the region where the energy is concentrated. We should note that our −4=5 power law was not derived rigorously, but rather was based on the hypothesis that W (x) becomes independent of X for x X . 3.2. Boundaries One question we can ask is how does any enforced behaviour at the boundaries aCect the properties of the ridge. In our minimal scaling ridge picture, then, only normal forces are relevant, but if we were to clamp the edges of the ridge then tangential forces are also present. One could anticipate just such a system arising in an experimental situation wanting to look at the scaling behaviour of the ridges in isolation. We need to investigate how such boundaries aCect both the ridge-line stress, xx and the Bank stress yy . Such investigations have been done numerically, e.g. in Ref. [11]. The minimal ridge is analytically more tractible. But the clamped ridge is probably a better representation of the ridges in a crumpled sheet or the ridges in the tetrahedron of Fig. 3. To explain this point, let us imagine taking this tetrahedron and cutting out a diamond-shaped piece extending to the midpoints of the adjacent faces, thus separating one ridge from the /ve others. We can create a ridge with the same ridge angle that the tetrahedron has by exerting normal forces on the boundary of this diamond, thereby forming a minimal ridge. But this ridge diCers from the ridge in the intact tetrahedron. The middle of the tetrahedon face feels a tensile stress arising from a pull from the three ridges bounding it. The diamond-shape minimal ridge feels no such stress. Thus, when we cut the diamond out of the tetrahedron, we expect the two edges at the opposite sides of the cut to pull apart, thus reducing

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the elastic ridge energy. Similar tension acts at the boundaries of a single-clamped ridge. Thus we expect a clamped ridge to have higher elastic energy than its unclamped counterpart. We can estimate the energy required to add the clamping constraint to the ridge. For de/niteness we now consider our original roof-shaped minimal ridge. When we deform a Bat sheet into the roof shape, there is no constraint holding the eves of the roof to a particular distance from the peak. Thus the material is free to withdraw towards the peak. In the clamped ridge, we do not allow this withdrawal. By knowing the displacement of the eves in the minimal ridge, we can /nd the extra strain required to impose the clamping constraint. From this we can infer the corresponding stress and the added elastic energy. Arguments of this type suggest that the energy increase does not outweigh the energy of the original minimal ridge. Thus the clamping constraint is not expected to ruin the ridge scaling. Numerical studies of Ref. [11] con/rm this view. They suggest that clamped ridges have a clear diCerence in the ridge shape and forces. But these diCerences are not so great as to alter the scaling properties discussed above for the minimal ridge. 3.3. Response to buckling forces Arguments presented above have outlined a complete picture for a crumpled elastic sheet at rest. We have a sound understanding of the way the energy is distributed throughout the sheet and the scaling laws associated with this. The one-third power law for the energy scaling on the ridges is the dominant term and hence much of our analysis has been devoted to this. One question we can now ask is; how will the energy scaling alter if additional force is applied to a ridge? A way of answering exactly this question numerically is to construct a cubic elastic sheet that is constructed by attaching edges together in the familiar way described in Section 2.3. This action costs a certain amount of energy, E0 , and creates the stretching ridges. A buckling force can now be applied to such an object by ‘pinching’ the corners, equivalent to shrinking a hard sphere around the cube as shown in Fig. 9. In practice this is accomplished by introducing repulsive potentials which are located just outside the vertices of the cube. For computational speed, the symmetry of the cube is exploited so that the numerical grid covers only a single edge of the cube. The cores of the pinching potentials must thus lie on lines which pass through diagonally opposite vertices to preserve this symmetry. When the force is applied we /nd that the additional buckling energy, Eb , is added to the sheet energy E0 in a simplistic way, thus preserving all the energy scaling we have previously found. Fig. 9 shows that the elastic energy measured at the buckling threshold exhibits exactly the same scaling as was found on the minimal ridges when at rest. We can also conclude that as the buckling energy is proportional to the rest energy, E0 , that the crumpled sheet, with its constituent stretching ridges, is inde/nitely stronger than a smoothly bent sheet such as an axially compressed cylinder the same size as the cube and made of the same material [13].

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Fig. 9. The eCects of deformation of a cube by pushing inwards at the corners. The plot shows the total elastic energy (Eb + E0 ) in the sheet, at rest and at the buckling threshold, after minimisation. ranges from 1:25 × 10−3 to 1:77 × 10−5 . Straight lines are least-squares /ts to a scaling form y = axb . The scaling exponent /t for the resting ridge values (lower line) was −0:32, the /t at the buckling threshold was −0:31.

4. How fundamental is crumpling? In the preceding sections of this article we have presented and reviewed results pertaining exclusively to a thin, two-dimensional sheet which has been constrained in a three-dimensional sheet. In this /nal section we want to ask how fundamental are the rules we have established for this special case when we begin to generalise to diCerent dimensions. To generalise our perspective then we need to consider more general objects, namely a d-dimensional manifold with an associated metric, or distance function, de/ned on the manifold. In addition to these standard mathematical structures we also need to have a well-de/ned energy cost for changing this metric, giving us an elasticity of the manifold. This would be equivalent to the time-dependent changes in a Buid system or the temporal or spatial variation of the metric within a spacetime. We must

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Fig. 10. Ridge structures and energy distributions for sheets crumpled in three and higher dimensions.

also consider subsets of a small spatial extent inside the d dimensions and hence the compacti/cation of these subsets. More speci/cally, we are now in a position to ask a number of more detailed questions regarding the outcome of a given crumpling procedure. First of all—will it crumple at all? We have already asserted that a two-dimensional manifold (sheet) in a three-dimensional compacting space (the balloon) will crumple but a one-dimensional /bre will not; it can just coil up on itself. Can we establish a general law relating to this? If the con/ned manifold does crumple then what will any condensed regions look like? Will con/ning ridges, or their equivalents, form? 4.1. Crumpling a three-sheet We can see from Fig. 10 an excellent illustration of the eCect of crumpling a two-dimensional sheet into a three-dimensional subspace. The plot shows the successive emergence of high-energy regions in the sheet as it is crumpled and the ridges form.

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In three dimensions we can anticipate that the counterpart of these ridges are surfaces which will appear when we crumple the three-sheet in a four-dimensional space. This also occurs in the /ve-dimensional space but not in a six-dimensional one. In this section we attempt to establish why this happens and to understand the diCerences in the diCerent dimensions. Let us /rst look at the diCerence between an embedding dimension of four or /ve. Consider a three-sheet with co-ordinates x = (x1 ; x2 ; x3 ) that is embedded in a space with co-ordinates r=(r1 ; r2 ; : : : ; rd ) where d is four or /ve. The material surface vector, d s˜, is determined by the dxi through  ds˜2 = (46) dxi2 ; material ; and similarly the space vector is given by ds2 = dr · dr; space ;

(47)

which we can rewrite in the following way:    dr dr ds2 = d xj d xi · d xi d xj ij 

= d s˜2 +

 ij



 dr dr    d xi  · − "ij  d xj ;  d xi d xj 



(48)

Strain Tensor; ij

which naturally introduces a de/nition of the strain tensor, ij . The material is unstretched if and only if the strain tensor is zero, so (48) gives us a way of understanding how stretching emerges from the curvature of the material. If we look /rst at the case d = 4 and consider just two components of the material then the curvature of the material can only be in the remaining dimension, the 4ˆ direction. Thus, an expression for r for an area of local curvature would be of the form: r = x11ˆ + x22ˆ + (x12 + x22 ) 4ˆ ;

(49)

which by insertion into (48) shows that the strain tensor, ij = 0, and hence that ds2 ¿ d s˜2

(50)

and hence that the material occupies more space than it did originally and therefore must be stretched. In /ve dimensions however, there is an extra dimension to curve into and hence we can imagine an expression like: r = x11ˆ + x22ˆ + x12 4ˆ + x225ˆ ;

(51)

which does indeed have curvature, but in two di;erent directions. Again by insertion into (48) we /nd the ij = 0 and hence that ds2 = d s˜2 and the material is not stretched by this type of bending. For the sheet to, therefore, be con/ned without stretching, at least one of the directions must be uncurved. Having established that in both four and /ve dimensions we expect a three-sheet to crumple we would also like to understand how the crumpling is altered by this

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Fig. 11. A three-dimensional sheet crushed by a factor of 2 in four dimensions.

Fig. 12. A three-dimensional sheet crushed by a factor of 2 in /ve dimensions.

change of embedding dimension. In Figs. 11 and 12 we can see the eCect of crushing a three-sheet by of factor of two in four and /ve dimensions, respectively, as shown by preliminary computer investigations. The topological diCerence between the two is clearly distinct. The computer investigations also show that the three-sheet is completely uniform when crushed in a six-dimensional space [14]. 4.2. The upper limit, d = 2M With the results found previously we are in a position to make a conjecture about when and when not we expect an M -sheet to be crumpled in a d-dimensional space [12]. For each in turn we present an argument as to why this is the case.

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4.2.1. Strain-free con2nement when d ¿ 2M For example, let us consider a three-sheet in a six-dimensional space. Again we let x be the co-ordinates of the three-sheet and r be the co-ordinates of the space. Consider a curve de/ned by   sin( xR1 )    sin( x2 )  R      sin( x3 )  R   r(x) = R  (52)  :  cos( x1 )  R      cos( x2 )  R   cos( xR3 )

This then de/nes a curve that /ts entirely within a (hyper-)sphere of radius R with each of the xi curving in an independent direction. Therefore, it does not have to stretch in order to be con/ned by the arguments we have presented previously. 4.2.2. No con2nement without stretching if d ¡ 2M For example let us consider a three-sheet in a /ve-dimensional space and use a similar co-ordinate system as before. Let us argue for a contradiction as follows. First, suppose that the sheet does not stretch (and therefore could be con/ned within a small sphere). This implies that there is no curvature in at least one direction at each point, without loss of generality let us call this the two-direction. Along this uncurved direction, perpendicular directions must remain curved, let us suppose not. A schematic of this appears in Fig. 13 where the sheet is next to the boundary. We can see from this diagram that the section against the boundary cannot be Bat if stretching is to be prevented. Therefore we have a contradiction, since the uncurved direction must stay uncurved to the boundary. The three-sheet cannot then /t inside a small sphere as we originally supposed. 5. Conclusions In Tom Witten’s lectures an understanding was provided of how energy can be condensed in an arbitrarily small fraction of matter, in response to a strong perturbation. While weak perturbations usually lead to energy being stored uniformly, a violent or persistent strong compression of a material will cause the response to be concentrated in small regions. Crushing an elastic membrane in a sphere, or crumpling a piece of paper, are examples of this kind of energy condensation. The deformation energy associated with the bending and stretching of elastic sheets of /nite thickness, con/ned to a shrinking embedding sphere, has been studied with the aid of a scaling analysis. The resulting scaling laws teach us that the energy of a crumpled two-dimensional sheet embedded in a three-dimensional sphere is condensed mainly in the ridges, rather than in the vertices, and that the scaling properties as a function of size are fairly universal.

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No confinement without stretching if d < 2M Eric Kramer eg. 3-sheet in 5 space 4 x1

1

x2

2

no stretch

no curvature in at least one direction at each point.

call it the 2 direction along uncurved direction, perpendicular directions must remain curved ....suppose not ∴ can't be flat must stretch

∴ uncurved direction must stay uncurved to the boundary can't fit in small sphere Fig. 13. No con/nement without stretching if d ¡ 2M .

The sheet, which can take a great variety of forms, is essentially a network of stretching ridges, which result from a competition between bending and stretching energy. The energy in a region of linear size X in a sheet of thickness h is found to condense into a fraction of order (h=X )(1=3) . This is con/rmed by numerical studies. Neighbouring ridges interact and the new characteristic length scale of this interaction is the width of the ridge, intermediate between the sheet thickness and the size of the object. The deformation of a membrane is measured by the strain and curvature tensors, which constitute the energy functional. Minimisation of this energy results in the von K1arm1an equations, which give a mathematical description of an elastic membrane. The scaling properties of these equations con/rm the anticipated scaling characteristics of the ridge and put them on a more rigorous basis. An energy distribution or energy density pro/le quanti/es the condensation in the vicinity of vertices. From this analysis one can understand that ridge condensation confers strength to the crumpled sheet, which is therefore stronger than a smoothly bent one. Finally, these phenomena are studied in general dimensionalities. For membrane dimension M and embedding space dimension d, the energy condensation is found to

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depend in a non-trivial way on d and the number of bending directions d − M . For d − M = 1, the energy is concentrated on ridges while for d − M ¿ 1 the vertices dominate. In all of this M ¿ d − M is assumed, otherwise the membrane is smooth. References [1] T. Tanaka, Nature 358 (1992) 6386. [2] D. Nelson, T. Piran, S. Weinberg (Eds.), Statistical Mechanics of Membranes and Surfaces, World Scienti/c, Singapore, 1989; D.R. Nelson, L. Peliti, J. Phys. (Paris) 48 (1987) 1085. [3] H.S. Seung, D.R. Nelson, Phys. Rev. A 38 (1988) 1005. [4] F.C. Mackintosh, T.C. Lubensky, Phys. Rev. Lett. 67 (1991) 1169. [5] Y. Kantor, M. Kardar, D.R. Nelson, Phys. Rev. Lett. 57 (1986) 791; J.B.C. Garcia, M.A.F. Gomes, T.I. Jyh, T.I. Ren, J. Phys. A 25 (1992) L353. [6] R. Scheidl, H. Troger, Comput. Struct. 27 (1987) 157. [7] A. Lobkovsky, S. Gentes, H. Li, D. Morse, T.A. Witten, Science 270 (1995) 1482. [8] E.M. Kramer, A.E. Lobkovsky, Universal power law in the noise from a crumpled elastic sheet, Phys. Rev. E 53 (1995) 1465. [9] R.M. Wald, General Relativity, The University of Chicago Press, Chicago, 1984. [10] A.E. Lobkovsky, Ph.D. Thesis, University of Chicago, 1996. [11] A.E. Lobkovsky, T.A. Witten, Phys. Rev. E 38 (1997) 1577. [12] E.M. Kramer, Ph.D. Thesis, University of Chicago, 1993. [13] B.A. DiDonna, T.A. Witten, Phys. Rev. Lett. 87 (2001) 206,105. [14] B.A. DiDonna, S.C. Venkataramani, T.A. Witten, E.M. Kramer, Singularities, structures, and scaling in deformed m-dimensional elastic manifolds, Phys. Rev. E 65 (2002) 016,603 math-ph=0101002.