Crumpled paper - École Normale Supérieure › ~benamar › paper › crumpling.pdf · Crumpled paper By M. Ben Amar1 and Y. Pomeau2 1Laboratoire de Physique Statistique, associ

Crumpled paper

B y M. Ben Amar1 and Y. Pomeau2

1Laboratoire de Physique Statistique, associe au CNRS, a l’Ecole NormaleSuperieure, et aux Universites Paris VI et VII 24 Rue Lhomond,

75231 Paris Cedex 05, France2Department of Mathematics, Universite of Arizona, Tucson, AZ 85721, USA

The crumpling of a piece of paper leaves permanent scars, showing a focusing ofthe stress. This is explained by looking at the geometry of the developable surfaces.According to Gauss, such a surface should have everywhere an infinite principalradius of curvature. The same condition holds when one minimizes the elastic energyof a bended plate; up to a small flexural part, this energy is minimum when theplate follows a developable surface. By considering the developable surfaces that arebounded by given closed curves in R3, we show that such a curve does not alwaysbound a piece of developable surface. But one can find a special class of conicalsurfaces, the d-cones, that are still developable in the sense that they can be mappedon a plane by conserving the distances. This d-cone gives the outer solution of theelasticity equations, although the vicinity of the tip is described by the full equations,including the flexural term.

1. Introduction

The crumpling of a piece of paper, or better of a sheet used for making transparen-cies, leaves permanent marks, very localized and with a typical crescent shape. Thisshows that, by some mechanism, the crumpling has focused the stress at some points,exceeding there the limit yield of the material and leading thus to irreversible plas-tic deformations. This kind of remark has more than an anecdotical character, asthis is also relevant for crashed cars, where the body (sheet-metal) is often stronglydeformed at well definite points, showing most likely the same phenomenon of stress-focusing. One of us (Pomeau 1995) has recently drawn the attention to the relation-ship between this and the geometry of developable surfaces. The present paper isdevoted to a discussion of this point, with its relation to elasticity theory, some-thing that was not considered in Pomeau (1995). Independently, in a review paper,E. Sanchez-Palencia (1995) has underlined the role of the geometry of surfaces forthe understanding of the behaviour of constrained plates and shells. The problemwe are going to consider is the minimization of the energy of deformation of a plateconstrained to pass through a given (non-planar) curve Γ in R3.

As explained in §2, it will appear that, after a convenient rescaling, this energy canbe made formally minimum when the plate is close to a smooth developable surface.The problem then turns into the one of knowing whether or not the curve Γ bounds

Proc. R. Soc. Lond. A (1997) 453, 729–755 c© 1997 The Royal SocietyPrinted in Great Britain 729 TEX Paper

730 M. Ben Amar and Y. Pomeau

a piece of smooth developable surface with the topology of a disc. This depends onan equation (3.5) with no simple general solutions (although some progress can bemade with numerical means). As suggested by Pomeau (1995) and Amar & Pomeau(1995), this equation may have no physically convenient solutions for certain curvesΓ (although a mathematical solution still exists). By this, we mean that the surfacebuilt out of the solution of the equation is not smooth and intersects itself in R3.As was predicted, and as the numerical studies presented here show, this happenswhen—generically—the mathematical surface gets a swallow-tail inside Γ ; we giveexamples of that in §3 c.

It will appear that the analysis of this situation yields quite naturally to a boundarylayer technique: in the limit of large deformations, the configuration minimizing theenergy of deformation is for a surface that is almost everywhere developable, butnear a special point (the tip of a ‘d-cone’) where the Gaussian curvature, that shouldbe zero for a developable surface stricto sensu, takes ‘finite’ values, in a sense to bemade clearer. Near these points, the stress field is much larger than elsewhere and asheet of crumpled paper becomes permanently scarred there because the limit yieldof the paper is reached.

The present work leads to considerations on the geometry of surfaces that wehave not found in the literature. The problem we had is that the Gauss criterium(the ‘theorema egregium’ , Spivak 1979), stating that the Gaussian curvature of adevelopable surface is zero, assumes that this curvature can be computed, namelythat this developable surface is C2-smooth till the second derivatives-included. Thed-cones that we have found are such that they can still be applied one-to-one ona plane continuously and isometrically, as an ‘ordinary’ developable surface, butthey do not satisfy the Gauss condition everywhere, being not C2-smooth at thetip. Hence the solution of the minimization problem splits into an outer solution ofzero Gaussian curvature matched with an inner solution (near the tip of the d-cone)where the flexural energy-neglected in the outer solution-has to be taken into account.Indeed, the developability in the original sense (mapping on a plane by preserving thedistances) does not require the C2-smoothness; think, for instance, to a wedge wheretwo half planes merge—a counter-example given seemingly by Lebesgue attending asa student a lecture by Hadamard around 1900—a story that one of us (Y. Pomeau)learned from Alain Chenciner and known too to Jacques des Cloizeaux, having thusa good chance of being true according to the criteria of truth in historical research.At the end, we present some speculations concerning the general shape (avoiding atoo precise word here) of a developable surface that is not necessarily C2-smooth.We restrict ourselves to a unique and localized singularity which is a d-cone, soavoiding stronger deformations as ridges (Witten & Li 1993; Lobkovsky 1996). Inthis case, given a contour Γ , the family of solutions is a 3 parameter manifold in R3.Our condition of developability, plus the minimization of the bending energy, selectscompletely the sought surface which mathematically represents our plate in the limitof vanishing thickness.

This paper is organized as follows. Section 2 is devoted to the static equations ofelasticity, both in the limit of small and strong deformations. Section 3 is concernedwith the construction of a developable surface bounded by a given skewed curve.When it is possible, these developable surfaces are regular (C2-smooth); when it isnot, they have the shape of a developable d-cone. Section 4 gives the way to selectthe d-cone solution among a three-parameter manifold.

Proc. R. Soc. Lond. A (1997)

Crumpled paper 731

2. The equations of elasticity of deformed plates

(a ) The Foppl–von Karman equationsA plate is a thin sheet of elastic material of constant thickness which is a plane in

its rest state. The equations of elasticity of plates were published in 1907 by Foppl(1907), although they are widely known as the von-Karman equations (Karman1910, below the FvK equations). They allow to compute the deformation energyof a plate of material-elastic in the Hookean sense (which forbids one to considerirreversible deformations), this being the energy needed to pull the plate out of itsplanar ‘ground’ state. This piece of 3D material has a constant thickness h, muchless than any characteristic length of the sample in any other dimension. In thislimit of a large aspect ratio, and by assuming that the plate remains close to anhorizontal plane, the FvK equation are derived by assuming that the bending of theplate is small enough to keep it close to its undisturbed state (see below for a moredetailed derivation of the FvK equations in a covariant form), taken as the z = 0plane. This last point is not clearly emphasized in many textbooks. For instance, inthe classical treatise by Love (1927), the FvK equations are derived under a vaguelydefined assumption of ‘neglecting the quadratic terms compared to the linear terms’.

The standard form of the FvK equations for an equilibrium deformation reads:

h3E

12(1− σ2)∆2ξ − h[ξ, χ] = 0, (2.1 a)

where χ is the so-called Airy potential, ξ the (small) deviation of the plate out of thez = 0 plane, E (σ) its Young modulus (Poisson ratio) and ∆2 is for the bilaplacian.The functions χ and ξ depend on (x, y), the rectangular coordinates in the z = 0plane. The equation (2.1 a) is to be completed by the second FvK equation:

∆2χ+ E[ξ, ξ] = 0. (2.1 b)

The symbol [ξa, ξb] in (2.1 a), (2.1 b) means

[ξa, ξb] =12∂2ξa∂x2

∂2ξb∂y2 +

12∂2ξa∂y2

∂2ξb∂x2 −

∂2ξa∂x∂y

∂2ξb∂x∂y

.

Note that [ξa, ξb] can be written as the divergence of a vector, a very useful remarkfor the following. Let us define a vector P (bold-italic characters denote vectors) suchthat

[ξ, ξ] = − 12 div(P ),

with

P =[∂ξ

∂y

∂2ξ

∂x∂y− ∂ξ

∂x

∂2ξ

∂y2

]ex +

[∂ξ

∂x

∂2ξ

∂x∂y− ∂ξ

∂y

∂2ξ

∂x2

]ey. (2.2)

In (2.2), ex and ey are perpendicular unit vectors in the (x, y) plane. Beforewe embark on calculations, let us notice that [ξ, ξ] is proportional to the Gaussiancurvature, so that the equation for a C2-smooth developable surface is [ξ, ξ] = 0, theCartesian equation of the surface being z = ξ(x, y). This remark will play a basicrole in the following. Although we shall not use this remark, note that the generalformal solution of the (Monge–Ampere) equation [ξ, ξ] = 0 is

ξ(x, y|s) = a(s) + b(s)x+ c(s)y,

where a(s), b(s) and c(s) are smooth arbitrary functions of a parameter s that is



related itself to x and y by the condition

∂ξ

∂s= 0 for s = s(x, y).

The general solution of the Monge–Ampere equation is then ξ = ξ(x, y|s(x, y)).In the following, we shall deal with three basic points related to the FvK equation.(i) These equations are the Euler–Lagrange equations for an energy functional

(derived through a consistent schema from the general equations of elasticity, theEuler–Lagrange structure carries over). This functional is given below. Note that theequations (2.1 a), (2.1 b) assume external stresses on the boundary only.

(ii) The Euler–Lagrange functional can be generalized to a plate that is not nec-essarily close to a plane on a large scale, but remains smoothly deformed with itsradius of curvature everywhere much larger than its thickness.

(iii) Although this looks a simple question, the order of magnitude consistent withthe FvK equations is not so trivial. The main point we are going to show is that,for large deformations, the so-called flexural term (that is, h3E∆2ξ/(12(1 − σ2)) inequation (2.1 a)) is negligible compared to the others, because it involves the largestpower of the small parameter h. Without this term, and for convenient boundaryconditions, the energy minimum is a purely geometrical problem that reduces itselfto the one of developable surface. But this last one may have no smooth solution, sothat a more complicated ‘basic’ solution has to be found, where the flexural term maybecome locally relevant (this corresponds then to a typical boundary layer situationwhere the order of magnitude for the radius of curvature of the surface is not theone of the solution elsewhere).

(b ) The energy functional for the FvK equationsFrom the general principles of elasticity theory, the solutions of the FvK equations

make stationnary the functional defining the elastic energy, that reads in general:

Fξ = h

∫Ω

dxdy dz uijσij ,

where uij is the deformation tensor and σij the stress. When the plate is very thin(h → 0) and the vertical (or the bending displacement uz) very small (uz = ξ oforder or less than h), this energy reduces to a bending contribution:

Fξ = h

∫Ω

dxdy

Eh2

24(1− σ2)(∇2ξ)2 − 2(1− σ)[ξ, ξ]

.

In this case, one assumes that there exists a middle plane (which is the plane z =0, when it is unperturbed) which sustains a bending strain only. The extensionaldisplacement in this middle plane is negligible. When x increases and becomes muchlarger than the plate thickness, but is still small enough to keep the plate closeto a plane: h < ξ < L (L being the characteristic size of the sample), one cannotdisregard anymore the extension in this middle plane. This is the situation where theFvK equations can be derived from the general equations of elasticity. One assumesthen the radius of curvature of the plate to be much bigger than the thickness.This assumption is fundamental for the validity of the FvK equations, although theassumption that the plate is everywhere close to a plane is not that crucial; as shownin the next subsection below, the FvK equation may be written in a fully covariantform, in a way allowing to take into account that the normal to the plate varies by


Crumpled paper 733

a finite amount over the whole extent of the plate, although this one has a radius ofcurvature that is everywhere much less than the thickness. Let

X = x+ ux(x, y), Y = y + uy(x, y) and Z = ξ(x, y)

be the material coordinates of the strained plate, (X,Y, Z) being the coordinates ofthe material point that has coordinates (x, y, z = 0) in the undisturbed state. Thestrain tensor is introduced by computing the transformed element of length:

dL2 = dX2 + dY 2 + dξ2,

where dX = dx(1 + ∂ux/∂x) + dy (∂ux/∂y), and similar formulae for dY and dξ, sothat dL2 can be written as

dL2 = dx2 + dy2 + Uαβ dxα dxβ,

where the greek indices take the values x and y. They are defined in such a way thatdxx = dx and dxy = dy, and the Einstein convention of summation on like indicesis used. Using the expression of dX in terms of the derivatives of ux(x, y) and ofξ(x, y), one obtains at once the explicit form of uαβ that involves terms linear andquadratic in the derivatives of ux(x, y) and uy(x, y), and quadratic in the derivativesof ξ(x, y). One can neglect the square of the derivatives ux(x, y) and uy(x, y) whenthe plate is close to the z = 0 plane, since these derivatives are basically the localangle of the normal to the plane with respect to the vertical. This assumption is notcrucial as shown in §2 c. For the moment, we shall deal with this limit, wherein thedeformation tensor reads

uαβ =12

[∂uα∂xβ

+∂uβ∂xα

+∂ξ

∂xα

∂ξ

∂xβ

].

The FvK approximation consists in keeping nonlinearities of the vertical displace-ment only. In this case, the elastic energy can be split into two parts: the bendingand the extensional energy, a fact which is not so obvious. So, it reads

Fξ, uαβ = h

∫Ω

dxdy[

Eh2

24(1− σ2)(∇2ξ)2 − 2(1− σ)[ξ, ξ]+ 1

2uαβσαβ

], (2.3 a)

with uαβ related to σαβ by the bidimensional Hooke law

uαβ =1 + σ

E

[σαβ − σ

1 + σσααδαβ

].

Since the last integral in (2.3 a) contains only in-plane contributions, equation(2.3 a) can be written in term of ξ and χ, the 2D stress tensor σαβ being simplyrelated to the Airy potential, so that the in-plane conditions of elastic equilibriumare automatically satisfied:

σxx =∂2χ

∂y2 , σyy =∂2χ

∂x2 and σxy = − ∂2χ

∂x∂y,

so that

Fξ, χ = h

∫Ω

dxdy

Eh2

24(1− σ2)(∇2ξ)2−2(1−σ)[ξ, ξ]+ 1

2E(∇2χ)2−1 + σ

E[χ, χ]

.

(2.3 b)This integral extends over the whole area of the plate (

∫Ω), assumed to be close



to the plane z = 0. The variationnal scheme which allows us to recover the FvKequations from (2.3 b) is not obvious since the physical quantities to vary indepen-dently are the three displacements: ux(x, y), uy(x, y) and ξ(x, y). These quantitiesappear explicitly in (2.3 a) but not in (2.3 b). Nevertheless, the FvK equations areusually given with ξ and χ. Finally, one needs to mention that some doubts havebeen expressed about the validity of the FvK equations (Schaeffer & Golubitsky 1979;Pomeau 1981) but we will keep in mind that these equations include the first non-linear effects when the displacements become important. Moreover, these equationsappear as the simplest ones for the study of certain bifurcation problems (Schaeffer& Golubitsky 1979; Pomeau 1981).

The boundary conditions have been explicited in part by Landau & Lifchitz (1990),who restrict themselves to the case without stresses on the boundary. In the case offree boundary (the edge of the plate is nowhere fixed so δξ and its normal derivativeare arbitrary) but with exterior forces, one gets the following set of three equations:

Eh2

24(1− σ2)

− ∂∆ξ

∂n

−(1− σ)∂

∂l

[cos(θ) sin(θ)

∂2ξ

∂x2 −∂2ξ

∂y2

+ (sin2(θ)− cos2(θ))

∂2ξ

∂x∂y

]+[

cos(θ)∂2χ

∂y2 − sin(θ)∂2χ

∂x∂y

]∂ξ

∂x+[

sin(θ)∂2χ

∂x2 − cos(θ)∂2χ

∂x∂y

]∂ξ

∂y= 0, (2.4 a)

∆ξ + (1− σ)[2 cos(θ) sin(θ)

∂2ξ

∂x∂y− sin2(θ)

∂2ξ

∂x2 − cos2(θ)∂2ξ

∂y2

]= 0, (2.4 b)

withσijnj = Pi; (2.4 c)

θ means the angle between the x axes and the normal to the boundary, ∂/∂l (resp.∂/∂n) is the derivative with respect to the arclength (resp. the normal coordinate)along the boundary. These boundary conditions have to be verified for the freelyhanging part of the plate where ξ and its derivatives can be arbitrary. For clampedsides of the plates, only (2.4 c) remains valid, while for simply supported part, onecan forget (2.4 a). Let us also notice that one can get rid of the Young modulus Ein (2.3 b) and (2.4 a), (2.4 b) by redefining a scaled Airy potential χ′ = χ/E, so thatthe functional to be minimized becomes

F ′ξ, χ′ = h

∫Ω

dxdy

h2

24(1− σ2)(∇2ξ)2 − 2(1− σ)[ξ, ξ]

+ 12(∇2χ′)2 − (1 + σ)[χ′, χ′]

. (2.5 a)

From its definition, F ′ξ, χ′ is automatically positive so it is bounded from be-low, the state of zero energy being the unperturbed plate. For simply supported orclamped plates, another Euler–Lagrange functional can be defined, where ξ and χcan be varied independently:

E′ξ, χ′ = h

∫Ω

dxdy

h2

24(1− σ2)(∇2ξ)2 − 2(1− σ)[ξ, ξ]+

12

(∇2χ′)2 − χ[ξ, ξ]

(2.5 b)


Crumpled paper 735

It is easy to show that E′ is also bounded from below, since the two last terms ofequation (2.5 b) give a positive contribution for solutions of the second FvK equations(2.1 b).

As mentionned, among several possible solutions of equations (2.1) and (2.4), wewill choose the one of less energy. Note that the second and fourth terms in (2.5 a)depend only on the boundary conditions. When the Gaussian curvature of the surfaceis continuous, the Gauss–Bonnet theorem yields the integral curvature,

∫Ω dxdy[ξ, ξ],

in terms of the geodesic curvature kg (that is the curvature in the tangent plane) ofthe contour Γ : ∫

Γdxdy [ξ, ξ] +

∫Γ

ds kg = 2π. (2.6)

(Here, we suppose that Γ is a smooth curve without slope discontinuity.)

(c ) Elastic energy for large deformationsBy arbitrary deformations, we mean deformations that may be large over large

distances, but such that the radius of curvature of the surface is everywhere muchgreater than the plate thickness h, the condition of validity of the FvK equations. Theidea we shall use below is that locally the FvK equations remain true, because theradius of curvature of the surface is large enough to make valid the FvK approxima-tion of the elasticity equations, the FvK equations being written in the coordinatesof the local tangent plane. Indeed, this is not possible everywhere if this tangentplane rotates by a finite angle along the surface. In other terms, we need to writethe FvK equations in a covariant form, not attached to a frame of reference relateditself to the local orientation of the surface. We shall use a simple method for doingso: we shall try to transform the FvK equation into a form that is independent ofany system of coordinates. The difficulty in doing so is that these equations involvespace derivatives of high order, so that the transformation laws for these derivativesare very cumbersome. When first derivatives are involved, the change of variables israther elementary, as it depends only on the quantities defining the local orientationof the tangent plane. We shall use this remark when computing the covariant formof the square of a gradient along the surface. The knowledge of this gradient is suf-ficient to compute higher order derivatives, as the covariant Laplacian. We shall nottry to make systematically the connection between our calculations and the equa-tions of elasticity. We shall use a simple approach and write the various quantitiesthat appear in the FvK equations in their manifestly covariant form. This is withoutproblem for those quantities with an obvious geometrical meaning as the mean andGaussian curvature, but things are less obvious for the high order derivatives of theAiry potential.

It is far more easy to write E′ (2.5 b) then to write the FvK equation themselvesin a covariant form, although the transition from the former to the latter is withoutmystery, involving only a variation of an explicitly given functional. We shall contentourselves with the covariant writing of E′. The integration element dxdy is the areaelement on the surface, denoted as dS later. Two terms in the integrand of equations(2.5 a) or (2.5 b) can be written at once in a covariant form, they are the first and lastone, since ∇2ξ (Spivak 1979) is the mean curvature, or 1

2(1/R1 + 1/R2), R1 and R2being the principal radii of curvature of the surface, although [ξ, ξ] is the Gaussiancurvature, 1/(R1R2).

The expression of the mean and Gaussian curvature in rectangular coordinates iswell known, and the corresponding formulae were derived first by Laplace and Gauss



respectively. Let Z(X,Y ) be the Cartesian equation of the surface, then its meancurvature is

12

(1R1

+1R2

)=

∂

∂X

∂Z

∂X√1 +

(∂Z

∂X

)2

+(∂Z

∂Y

)2

+∂

∂Y

∂Z

∂Y√1 +

(∂Z

∂X

)2

+(∂Z

∂Y

)2

, (2.7 a)

and its Gaussian curvature is

1R1R2

=

∂2Z

∂X2

∂2Z

∂Y 2 −(

∂2Z

∂X∂Y

)2

1 +(∂Z

∂X

)2

+(∂Z

∂Y

)2 . (2.7 b)

The covariant form of the term proportional to 12(∇2χ′)2 in the FvK energy requires

two things: first, one has to find the covariant form of the equation (2.1 b) relatingχ′ to the Cartesian equation of the plate, and then to find the expression of thecorresponding contribution to the energy. Actually, the two things are related throughthe following observation. The equation (2.1 b) as it is written follows formally fromthe minimization of the functional

E′′χ′ =∫

dxdy−12(∇2χ′)2 − χ′[ξ, ξ], (2.8)

where [ξ, ξ] is considered as given and where the variation is on χ′. In order to write(2.1 b) in a covariant form, it is sufficient to write (2.8) covariantly as well, so thatthe minimization with respect to χ′, supposing the shape of the plate given (i.e.supposing [ξ, ξ], given in the usual approximation for FvK, or here for Z(X,Y ),and the Gaussian curvature given) will yield the covariant equivalent of (2.1 b). Thecovariant equivalent of dxdy is the area element on the surface, i.e.

dX dY√1 +

(∂Z

∂X

)2

+(∂Z

∂Y

)2,

while the covariant Gaussian curvature has been given in (2.7 b).It now remains to find the covariant form of∇2χ′. In this expression, the derivatives

are with respect to Lagrangian coordinates in the undisturbed state of the plate thatis a plane. This plane can be taken as locally tangent to the plate, where the Laplacianis computed. This covariant Laplacian is found by using the following trick. On aflat plane, the Laplacian operator is related to the gradient operator by a classicalformula of the calculus of variations. Let GT (x, y) be a functional of a smootharbitrary function T (x, y); x, y being the rectangular coordinates in a plane:

GT = −∫

dxdy12

(∇2T (x, y)′)2,


Crumpled paper 737

then, the variation of G. can be written as

δGT =∫

dxdy δT (x, y)∇2T (x, y).

There, the Laplacian appears as the coefficient of dxdy δT (x, y) in the integral forδGT. This gives way to a direct calculation of the covariant Laplacian: one writesthe functional GT by an integral

∫dΣ (∇LT )2, where ∇LT is the gradient of a

function T defined on the surface. In the rectangular coordinates along the tangentplane to the surface, the components of ∇LT are (∂T/∂x, ∂T/∂y), which allows towrite (∇LT )2 by standard change of coordinates in terms of the ‘absolute coordinates’X and Y , as well as in function of the derivatives of Z(X,Y ) defining the orientationof the tangent plane (note that this is possible because the gradient involves firstorder derivatives only). The result is

(∇LT )2 =

∂2T

∂X2

(1 +

(∂Z

∂Y

)2)+∂2T

∂Y 2

(1 +

(∂Z

∂X

)2)− 2

∂2T

∂X∂Y

∂Z

∂X

∂Z

∂Y

1 +

(∂Z

∂X

)2

+(∂Z

∂Y

)2 .

Consider now the integral along the surface of Cartesian equation Z = Z(X,Y ):

Gχ′ =∫

dΣ (∇Lχ′)2,

where (∇χ′)2 is just the above expression, but with χ′ instead of T . The Laplacian∇2

Lχ′ is now defined through the first variation of Gχ′ that is written as

δGχ′ =∫

dX dY(

1 +(∂Z

∂X

)2

+(∂Z

∂Y

)2)1/2

δχ′∇2Lχ′.

The final result is a rather long expression that can be written in a number of ways,for instance as

∇2Lχ′=

∂2χ′

∂X2

(1+(∂Z

∂Y

)2)+∂2χ′

∂Y 2

(1+(∂Z

∂X

)2)−2

∂2χ′

∂X∂Y

∂Z

∂X

∂Z

∂Y

1+(∂Z

∂X

)2

+(∂Z

∂Y

)2 +A∂χ′

∂X+B

∂χ′

∂Y,

where

A =[1 +

(∂Z

∂X

)2

+(∂Z

∂Y

)2]−1/2∂

∂X

1 +(∂Z

∂Y

)2

[1 +

(∂Z

∂X

)2

+(∂Z

∂Y

)2]1/2

−2∂

∂Y

∂Z

∂X

∂Z

∂Y[1 +

(∂Z

∂X

)2

+(∂Z

∂Y

)2]1/2 ,Proc. R. Soc. Lond. A (1997)


B =[1 +

(∂Z

∂X

)2

+(∂Z

∂Y

)2]−1/2∂

∂Y

1 +(∂Z

∂X

)2

[1 +

(∂Z

∂X

)2

+(∂Z

∂Y

)2]1/2

−2∂

∂X

∂Z

∂X

∂Z

∂Y[1 +

(∂Z

∂X

)2

+(∂Z

∂Y

)2]1/2 .This completes our writing of the intrinsic (Beltrami) Laplacian. Putting it into

the integral (2.8) that relates the Gaussian curvature to the Airy potential, one can,in principle, find the covariant form of the second FvK equation. This would yielda rather complicated expression, of no much help as far as explicit calculation areconcerned. However, all this calculation shows that the theory we present is notrestricted to surfaces uniformly close to a plane; the flexural energy is still mimimumand equal to zero when the Gaussian curvature of the surface vanishes. This Gaussiancurvature is still the ‘source term’ for the Airy potential and the extensional energyis quadratic positive in this Airy potential. This energy is thus minimum, and zerofor any developable surface, being it close or not to a plane.

(d ) Scaling of the FvK equationsIn the present work, we are considering plates that are deformed in such a way

that they have to follow a given non-planar curve. This implies in particular thatthe strength of an external force cannot appear as a scaling parameter, contraryto the case considered by Landau & Lifchitz (1990). Let us first examine the usualFvK equations. Accordingly, and as it follows from the form of the Euler–Lagrangefunctional given by equation (2.5 a), there appear quantities with the dimensions ofa length (1 − σ2 is a pure number of order 1) only. The three natural length scalesare: the plate thickness h, the size of the sample L and some typical length scalecoming from the boundary conditions, that will be denoted as R, and could be seen,for instance, as the typical value of the inverse torsion of the (non-planar) curve thatbounds the plate. The deviation ξ is such that R ≈ L2/ξ (here we assume that thelength of variation is L itself, so ∇ ≈ 1/L). With this assumption, the radius ofcurvature of the surface should also be of order R, which is at once verified since[ξ, ξ] ∼= (∇2ξ)2 ∼= 1/R2. Balancing both terms in equation (2.1 b) (remember thatχ′ = χ/E), one gets χ′ ∼= ξ2, so that the order of magnitude of the first and secondterms in the integrand of equation (2.5 a) is h2ξ2/L4, although the order of magnitudeof the third and last one (extensional term) is of order ξ4/L4. In the limit where theFvK equations apply (h ξ L), the flexural term (that is h3E/(12(1− σ2))∆2ξin equation (2.1 a)) is negligible. In the case of strong deformation, where ξ ≈ L,R also becomes of order L but our conclusions are not changed; the flexural termremains smaller than the extensional term. Nevertheless in this case, one needs toconsider the most complicated energy expression in the previous subsection.

By neglecting this flexural term, and as shown in §2 c, one may find Euler–Lagrange minima represented by non-C2-smooth functions ξ (which verify [ξ, ξ] = 0,almost but not everywhere), making then h3E/(12(1−σ2))∆2ξ infinite at some welldefined points and/or along some curves. In these situations, the flexural term can-not be neglected there and other scalings have to be used. The solution of the FvK


Crumpled paper 739

equations has to be split into an outer and an inner part. This scaling of the innersolution does not depend upon the outer scale R, and is such that all terms in theFvK equations are of the same order. According to the general principles of boundarylayer theory, the outer solution is defined by the balance of the second and last termin the FvK equation. Near the singularities, the neglected flexural energy diverges,so that the scaling of the inner solution is defined by the constraint that the firstterm of FvK—the flexural term—becomes of the same order of magnitude as theothers. This is realized by the condition that every length scale is of order h andthat χ′ ∼= h2 (or χ of order Eh2) for the inner solution, seemingly in contradictionwith the basic assumption of the FvK theory, namely that every radius of curvaturehas to be much larger than h. The FvK theory may remain applicable in the innerdomain, when the angle at the tip is small. The order of magnitude of this angle iscontrolled by the dimensionless number g0 introduced later.

To show that the stresses are focused near the singularities would require morerefined estimates; actually, the stress scales like a second derivative of χ′ and so itdoes not seem to make a difference to have χ′ of order h2 in an area of size h or χ′of order R2 in an area of size R, because in both cases the order of magnitude ofthe stress remains of order 1. Actually, the situation is more complicated than that,since the outer solution is a piece of developable surface for which χ′ is zero at theformally dominant order R2, which does not change the scaling just obtained. Wereconsider this question of the stress-focusing later (§4 c).

3. Developable surface bounded by a given curve

In this section we study the following problem: given a skewed (non-planar curve)Γ , what is the shape of the surface bounded by this curve that minimizes the FvKenergy? Indeed, this problem has no general formal solution, due to the stronglynonlinear character of the FvK equations. Nevertheless, we shall use now two crucialproperties:

(i) The flexural term is negligible if R h, where R is a typical length scale of Γand the solution obtained by neglecting the flexural energy is smooth (has nowhereinfinite curvature).

(ii) The FvK functional without the flexural contribution reads:

F ′′ξ, χ′ = h

∫Ω

dxdy 12uαβσαβ = h

∫Ω

dxdy(12(∇2χ′)2 − (1 + σ)[χ′, χ′]). (3.1)

Since F ′′ represents a pure extensional energy as shown by the first integral, bydefinition, it is positive and bounded from below. An obvious minimum is got when∆χ′ vanishes, which implies that γ(r) = [ξ, ξ] = 0 from equation (2.1 b). Followingthe definition of χ′, this also means that there is no (or a negligible) strain in the planeof the plate. The minimization of F ′′ becomes the problem of finding a developablesurface (recall that γ(r) = 0 is the equation for a C2-developable surface) boundedby Γ . Since this question does not seem to have been looked at before, contrary tothe problem of minimal surfaces, let us first recall some elementary useful results indifferential geometry.

(a ) Ruled surfaces and developable surfacesIn textbooks of differential geometry, the set of developable surfaces is often intro-

duced as a subset of the ruled surfaces. The classical examples of developable surfaces



are the cone, the cylinder and the tangent developable (Spivak 1979; Gray 1993).They can be described with the following parametric representation for a surface:

r(s, t) = a(s) + tb(s). (3.2)

In equation (3.2), a(s) and b(s) are parametric representations of two differentskewed curves in R3. Equation (3.2) is the general two-parameter representation of aruled surface. The next step for defining a developable surface is to impose that eachgeneratrix of the ruled surface is a line of principal curvature (or equivalently that itsGaussian curvature is zero). Following the calculation by Gauss for a ruled surface,the Gaussian curvature which is proportional to γ(r) vanishes when the determinant∂r(s, t)/∂t, ∂r(s, t)/∂s, ∂2r(s, t)/∂t∂s also vanishes. In the case of a ruled surface,this gives the condition

∂a(s)∂s

, b(s),∂b(s)∂s

= 0. (3.3)

The cone would correspond to a(s) = a0 (the curve a(s) is reduced to a single point),a cylinder to b(s) = b0 so γ(r) = 0. As often quoted in textbooks, an example ofdevelopable surface is the so-called tangent developable for which b(s) is the tangentto the curve a(s). Note that the normal and the binormal developable surfaces (withobvious definition: b(s) is the normal or the binormal to a(s)) are not developablesurfaces since equation (3.3) is not verified. Let us make two remarks:

(i) When wrapping a sheet of paper on a cylinder, very little energy is needed andit is possible to recover the initial state of the sheet of paper even though we haveperformed a rather strong distortion, when measured in terms of the variation of theorientation of the normal to the sheet. Since the cylinder is a developable surface,this supports our idea that distortions such that the sheet remains a developablesurface cost very little energy and remain a reversible process.

(ii) When a(s) is arbitrary (which is the case of our boundary which follows the ex-terior forces), the only developable surface referred to in the literature is the tangentdevelopable discussed above. Nice representations can be obtained by Mathematica(Wolfram 1992) as explained and shown by Gray (1993). Nevertheless, this surfacecannot help us since it is not topologically a disc bounded by Γ , Γ being topolog-ically equivalent to a circle. Since it seems to us that we are faced with a problemyet unsolved, let us explain our strategy for the construction of surfaces satisfyingγ(r) = 0 at least almost everywhere.

(b ) Making a developable surfaceWe transform the search and making of a developable surface into the one of

finding a diffeomorphism f(s) of Γ parametrized by s, that has to be the solution ofan equation ((3.5) below). Instead of drawing the surface with generatrices joiningpair of points on two arbitrary curves, we will impose to the generatrix to go fromone point to another point of the same skewed curve which is our boundary Γ . Wewill define a quasi-ruled surface by the two parameters (s, t) representation:

r(s, t) = tr(s) + (1− t)r(f(s)). (3.4)

One needs to define on Γ a one-to-one map f(s) linking two points at the two endsof a generatrix, this generatrix being defined as the segment of the straight line that ison the developable surface. Our plate would correspond to values of t between 0 and1, although one can extend t to ±∞. Let s be the curvilinear coordinate along Γ , thusa quasi-ruled surface is given by a mapping s → s′ = f(s). From its definition, the


Crumpled paper 741

mapping f(.) is idempotent: s = f(f(s)). Furthermore, we shall consider cases (theyare other possibilities that we discuss briefly at the end) where this diffeomorphismof the circle (Γ is topologically a circle) has topological degree (−1). Hence the maps → f(s) has two fixed points corresponding each to generatrices of zero length(points of vanishing torsion, as we are going to show).

Adapting to this situation the calculation by Gauss (Spivak 1979), one gets thatthe developability condition requires

r(s)− r(f(s)), t(s), t(f(s)) = 0, (3.5)

where t(s) = dr(s)/ds is the tangent unit vector. This states that the generatricesbetween the two end points and the tangent vectors at these points are coplanar.Equation (3.5) is an equation for the map f(.), given Γ .

The equation for this diffeomorphism of Γ does not seem to have a general solutionfor Γ arbitrary, at least a solution representing a bent plate. Some constraints uponΓ do follow from it. In particular, a necessary condition for Γ to bound a smoothpiece of developable surface is that it should have at least two points of zero torsion.This follows from the remark that a map of the circle with topological degree (−1),as f , has at least two fixed points, i.e. the equation s = f(s) has at least two roots.Let us expand the equation (3.5) near these roots. For that purpose, let us considerthe function of two variables s and s′,

D(s, s′) = r(s)− r(s′), t(s), t(s′),and expand it for s close to s′, by using the standard expansion, valid up to the thirdorder in the difference (s− s′):

r(s′) = r(s) + (s′ − s)t(s) +(s− s′)2

2Rn(s)

+(s′ − s)3

6R

[− 1R

dRdsn(s) +

(t(s)R

+ τb(s))]

.

In this expansion, n is the normal to Γ , R its radius of curvature, b the binormaland τ the torsion. From this, one deduces at once that

D(s, s′) ∼= τ(s′ − s)4

4R2 n(s), t(s), b(s),for s close to s′. This shows that at any generic point, the equation D(s, s′) = 0has no other solution but s = s′, since the next term in the Taylor expansion ofD(s, s′) will be a fifth power of (s′ − s), and no other root of D(s, s′) = 0 will existnear s = s′ but s = s′ if the coefficient of the dominant fourth power is not zero.Consider now what happens if this coefficient, τ/4R2, is close to zero. Let us expandthis as τ/4R2 ∼= C(s−s0), where s0 is the value of s such that τ/4R2 is zero and C aconstant, generically not zero. Thus the function D(s, s′) expands near s = s′ = s0 as

D(s, s′) ∼= C(s− s0)(s′ − s)4 +D(s′ − s)5,

D being a constant, also assumed to be non-zero. Thus the equation D(s, s′) = 0has, near s = s0, two solutions s = s′ and s′ = s− C/D(s− s0. This solution is theone representing the generatrix of vanishing length at s = s0, s0 being defined bythe condition that τ/4R2 is zero there. Since we assume Γ is smooth, its radius ofcurvature is never zero, and the only remaining possibility is that the torsion vanishesat s0.



It happens that it is quite easy to build curves Γ with a non-vanishing torsion(Pomeau 1995) and these curves cannot bound a piece of smooth developable surface.On the other end, many curves do bound a piece of developable surface, as all planarcurves for instance. Thus it makes sense to study the problem of the transition, undercontinuous deformation, from a curve Γ that bounds a piece of smooth developablesurface to a Γ that does not. This surely implies a condition on the geodesic curvatureintegral, following the Gauss–Bonnet theorem (equation (2.6)) but we do not findthis condition very explicit. This transition can be understood by looking at thechange of shape with Γ of the mapping solving equation (3.5). Missing a generalanalytical method, we had to rely upon numerical methods. Although we did notexamine the problem systematically, we met three different cases.

(i) The mapping is regular, has topological degree (−1) and defines a one-to-onecorrespondance between points of Γ .

(ii) The mapping does not exist on some interval [s1, s2] and, of course, for theimage interval [f(s1), f(s2)]. In this case, the surface is not compact.

(iii) The mapping is many to one for some interval [s1, s2]. Indeed, there is nothingin equation (3.5) itself imposing that f(.) is a diffeomorphism of topological degree(−1). Actually, it happens quite often that the numerical solution of (3.5) has twoextrema, that is that there is an arc [s1, s2] such that df(s)/ds is positive insideand negative outside. By continuity, df(s)/ds is zero at s = s1 and s2, although, bysymmetry, df(s)/ds is infinite at f(s1) and f(s2). The surface that can be built outof such a non-invertible mapping gets a fold, via a swallowtail transition, that canbe characterized quite accurately (see figure 7a, b).

Note that, given Γ , several mappings can be obtained as solutions of equation(3.5). So these three cases may appear simultaneously in the sense that, for the sameΓ , three different solutions of equation (3.5) fit into each of the 3 cases just men-tionned above. We may stress on the fact that equation (3.5) is solved numericallywithout difficulty. The main trouble consists in the separation of the different so-lutions, which is absolutely necessary for drawing the corresponding surfaces. Thenumerical examples given below are in broad agreement with the ideas developedhere, that is that changing the shape of Γ changes the shape of f(.) in a rathergeneric way.

Finally, the mapping f(s), which depends on the whole shape of Γ , depending onthe non-local equation (3.5), gives a very simple and clear diagnostic of the possibilityto find a developable surface.

(c ) Examples of developable surfaces bounded by a given skewed curve in R3

In this subsection, we do not solve mathematically the question of existence andmultiplicity of developable surfaces resting on a skewed curve in space. Our objectiveis more modest and consists simply of presenting a few typical examples. Our taskis to find f(s), given a parametrization r(s) of Γ . Remember that a ‘good mapping’is symmetric across the first bissectrix in the Cartesian plane (s, f(s)) and crosses itat the point of zero torsion. We choose a three-parameter (α, β, δ) manifold of Γ ’sgiven by

x = α cos(s) + β cos3(s) + δ sin2(s),y = α sin(s) + β sin3(s) + δ cos2(s),z = β sin(s) + βα cos3(s) + αδ cos2(s).

(3.6)


Crumpled paper 743

Figure 1. The mapping s′ = f(s), solution of equation (3.5) for a boundary Γ which satisfiesequation (3.6) with α = β = 1, δ = 0.

Figure 2. The developable surface which rests on Γ with α = β = 1, δ = 0. It turns out that inthis case, the surface is a piece of cylinder so the generatrices are parallel.



If δ = 0, Γ is symmetric with respect to a central point, which is O(x = y = z = 0):i.e. r(s + π) = −r(s). So δ is a symmetry-breaking parameter. We obtained themapping f(s) by solving numerically equation (3.5) with the Newton method. Thegraph for f is presented in figure 1, when α = β = 1 and δ = 0. One can noticeseveral solutions, some not corresponding to physical solutions. One of these solutionscorresponds to a diffeomorphism of topological degree 1 such that f(s) = s + π. Inthis case, the generatrices join symmetric points and all generatrices cross at thecentre of symmetry. This corresponds to a cone solution. On the other hand, a goodexample is given by the mapping (s → f(s) = s − 1

2π), which gives a piece ofcylinder with parallel generatrices, as shown in figure 2. Of course, our special choiceof parameters yields symmetries for Γ which allow the construction of standarddevelopable surfaces. Finally, figure 1 also shows two closed loops for f(s) whichmeans a non-compact mapping, i.e. a surface with holes. This surface has not thetopology of a disk and so cannot be a slightly distorted plate. What we plan now isto break the symmetry which appears for this particular Γ . Let us fix the parametersas follows: α = 1, β = 0.5 and δ = 0. Figure 3 displays the different mapping f(s):since O is always the centre of symmetry, we again get the conic solution. But themost important result is that, here, there exist three ‘good’ mappings which definethree different developable surfaces. In figure 4, we have drawn one of them sincethese three surfaces are very close to each other. However, a plane cross section showsus that they are different. So, for suitable boundaries Γ , it is possible to find several(in this case three) developable surfaces which could represent an elastic plate. Wealso have considered non-centre-symmetric boundaries by varying the parameter δfrom 0 to ±1, with α = β = 1. Two typical graphs for f(s) have been calculateddepending on the sign of δ. Nevertheless, both of them exhibit some kind of bifurcatedeigenvalue spectrum; the cone and the cylinder solutions disappear, making way fora mapping which sometimes follows the cone one, sometimes the cylinder one, butbeing attracted by the loops as shown in figure 5. This kind of situation reminds oneof the (no)crossing of eigenvalues in quantum mechanics under a weak perturbation(Landau & Lifchitz 1990). In order to show that such a complicated map can begiven by rather smooth and regular boundaries, we have drawn these boundaries forδ = 0,±0.3 in figure 6.

When δ is positive (see figure 5), we have constructed the surface correspondingto the continuous underlined map (figure 7b). More exactly, we have drawn out thespecific interval which looks like some Maxwell curve (see figure 7a). Note that someparts of the map show a one-to-one relationship between two points of Γ while, forsome specific interval [s1, s2], three generatrices merge for every point, a situationstrongly reminescent of geometrical optics (inside caustics three rays cross at everypoint). As for geometrical optics, two of these generatrices are tangent to a smoothcurve (that would be the caustics in 2D geometrical optics). Those two ‘caustics’merge at a cusp point, that is a ‘swallowtail’ (see figure 8, which is a cut of thesurface by a transverse plane) on the developable surface and, at the other end, theymeet Γ at f(s1) and f(s2). Each caustic is a line of cusp points on the surface, as itis well known for developable surfaces made of the bundle of tangents to a skewedcurve (Spivak 1979) (note that the curve under consideration here is the caustics,not Γ ).

Indeed, all this makes obvious that, when f(s) is not a diffeomorphism of topo-logical degree (−1), no smooth developable surface is bounded by Γ , so that theminimization of the energy functional (2.5 a) is not realized by such a smooth C2-developable surface. This leads one to come back to our original formulation of the


Crumpled paper 745

Figure 3. The mapping s′ = f(s), solution of equation (3.5) for a boundary Γ which satisfiesequation (3.6) with α = 1, β = 0.5, δ = 0.

Figure 4. One of the three developable surfaces which rests on Γ with α = 1, β = 0.5, δ = 0.This surface corresponds to the dotted curve in figure 3. In this case, the surface is not a pieceof cylinder; the reader can see that the generatrices are note parallel.

problem; that is with the flexural energy included. The argument for this is quiteclear: neglecting this energy, one finds (for instance, in the case of figure 7a) a for-mal solution to the minimization with an infinite flexural energy (because the mean



Figure 5. The mapping s′ = f(s), solution of equation (3.5) for a boundary Γ which satisfiesequation (3.6) with α = 1, β = 1, δ = ±0.3. The continuous and the dashed curves correspondto δ = 0.3, while the dots and the dot-dashed line correspond to δ = −0.3.

Figure 6. A comparison of the boundaries Γ , satisfying equation (3.6) with α = 1, β = 1,δ = 0, ±0.3.

curvature diverges along the ‘caustics’), so that one has to take into account thisflexural energy in one way or another to get the solution of the full minimizationproblem. This is what is studied in the next section.


Crumpled paper 747

Figure 7. (a) Extension of the continuous mapping of figure 5 (corresponding to α = 1, β = 1,δ = 0.3). This extension focuses on the specific interval which looks like a Maxwell curve inphase transition. (b) Singular surface corresponding to the extension described for figure 7a.

(d ) The singular case: d-cone vs singularity on the boundaryIn this section, we cannot use the full apparatus of differential geometry (climb

on Gauss shoulders), since we have no compelling argument—only guesses—allowingto characterize the solution minimizing the FvK energy when this is not a smoothdevelopable surface. We shall base our guess on the following remark: as it is written,the FvK energy is certainly made smaller if the surface is as close as possible to adevelopable surface on as much as possible of its area. Accordingly, it is better to



Figure 8. A cut by the plane y = 0.5 of the previous singular surface (2.7 a). Note theswallowtail which prooves that the developable surface intersects itself.

have a non-developable surface near points or perhaps lines, and to have a smoothdevelopable surface elsewhere. This brings to mind two possibilities (that are notwithout connection).

(i) One makes a kind of Maxwell construction in order to make almost invertiblethe map f(.) (that is not one-to-one in the situation we are considering). Let uschoose on [f(s1), f(s2)] a point of coordinates sT that will be the tip of a conemerging smoothly with the rest of the surface defined by f(s). Let sα and sβ be twoextreme points (out of three) such that sT = f(sα) = f(sβ). The surface that weconsider is made of the generatrices (s, f(s)), with s outside of the arc [sα, sβ]. Thissurface is made of two pieces merging at sT. Then, for s in the arc [sα, sβ], we decidethat the image f(s) is always at sT. This surface is smooth everywhere but at sTon Γ , and is made of a cone with a tip at sT on Γ , and of two smooth pieces ofdevelopable surface. This surface can be mapped globally on a piece of plane and itis developable, in this (weak) sense, but it has infinite curvature (in general) at sTon Γ , something that can be regularized by introducing the flexural energy (exactlyas done in some details for the d-cone below). This kind of Maxwell construction isnot the possibility we are going to consider now because it depends on the details ofthe shape of the curve Γ , and so would not be as general as the other possibility weare just going to look at.

(ii) Contrary to what we just did, we can change of philosophy and try to builda developable surface on entirely new principles; that is without relying upon aMaxwell construction as in (i). Our idea is the following one: consider a cone, thatis, a bundle of half lines merging in R3 at a single point P . In general, this conehas locally a vanishing Gaussian curvature, but it cannot be mapped smoothly andglobally on a piece of plane by conserving the distances; let us draw a circle on the


Crumpled paper 749

surface of the cone with its centre at the tip of the cone. In general, the perimeter ofthis circle will not be 2π times its radius, unless this cone has something special. Weshall call these ‘special’ cones d(evelopable)-cones. The link between these d-conesand the solutions of the FvK equations will be explained later on, but it is relevant topoint out at the present stage that it has to do with the solution of equation (2.1 b)at large distances; if one assumes that the plate is infinite, the solution of equation(2.1 b) can be written quite simply by the Green’s function method where the term−γ(r) = −[ξ, ξ] appears as a source term.

When γ(r) is not zero, one can calculate the asymptotic behaviour of the Airypotential, or better, of its Laplacian using Green’s functions. When γ(r) is localizednear a singular point (at r = 0), the large distance behaviour of the solution of (2.1 b)(r′ ∼= L h) is obtained by a standard multipolar expansion. The dominant orderis given by the monopole contribution:

∆χ′(r′) = −∫

Ωdxdy γ(r) log |r − r′| (3.7 a)

≈ − log(r′)∫

Ωdxdy γ(r) +

∫Ω

dxdy γ(r)rr′

r′2. (3.7 b)

This shows that two necessary conditions need to be met for a negligible extensionalenergy. The first integral in (3.7 b), once put in equation (2.5 a) or (2.5 b), givesan extensional energy which behaves like L2 log(L) times the square of the Gauss–Bonnet integral. This energy needs to be suppressed to minimize the FvK energy,so we get the condition of vanishing of the Gauss–Bonnet integral. The second in-tegral behaves like 1/r′, which yields an energy like log(L). This is the dominantcontribution if the first one cancels out. Since near singular points like the tip of thecone, the Gaussian curvature γ(r) is not zero; in principle, one needs to satisfy thesetwo conditions (3.7 a, 3.7 b). Clearly, a ‘good’ cone (a d-cone) cannot contribute tosuch an important contribution to the extensional energy. So, a d-cone will satisfya condition of vanishing of the Gauss–Bonnet integral in the tip area which will beexpressed quantitatively in the next paragraph.

4. Construction of a d-cone

(a ) Gaussian curvature of a cone, near the tipHere, polar coordinates are used for the calculation of the conic singularity:

z = ξ(r, θ) = rg(θ). (4.1)

For a cone of revolution, g(θ) = tan(Φ), where Φ is half the angle at the tip. Sincewe expect a Dirac distribution for g(r, θ), we calculate the Gauss–Bonnet integral:

I =∫

Ωr dr

∫dθγ(r, θ) =

∫Ωr dr

∫dθ[ξ, ξ].

[ξ, ξ] vanishes far from the tip (the order of magnitude of the region where theGaussian curvature is not zero is given by h, the plate thickness, as shown later).According to equation (2.2), [ξ, ξ] can be written as the divergence of a vector P ,so the 2D integral I can be transformed. We split the quadrature domain into twoparts by a small circle of radius ε, Ωε containing the tip. So,

I =∫

Ωεr dr

∫dθ[ξ, ξ] +

∫Ω−Ωε

r dr∫

dθ[ξ, ξ].



The second integral vanishes because γ(r, θ) vanishes everywhere for a cone, exceptat the tip. Let us transform the first integral into a contour integral as done byLandau & Lifchitz (1990):

I =12

∫r=ε

ε dθ∂ξ

∂r

[−2 cos(θ) sin(θ)

∂2ξ

∂x∂y+ sin2(θ)

∂2ξ

∂x2 + cos2(θ)∂2ξ

∂y2

]+

12

∫r=ε

dθ∂ξ

∂θ

[cos(θ) sin(θ)

∂2ξ

∂x2 −∂2ξ

∂y2

+ (sin2(θ)− cos2(θ))

∂2ξ

∂x∂y

].

Finally, one gets

I =12

∫r=ε

ε dθ∂ξ

∂r

[1r2

∂2ξ

∂θ2 +1r

∂ξ

∂r

]+

12

∫r=ε

dθ∂ξ

∂θ

[1r2

∂ξ

∂θ− 1r

∂2ξ

∂r∂θ

].

For a cone such that: ξ(r, θ) = rg(θ), one obtains

I =12

∫r=ε

dθ g(θ)[g′′(θ) + g(θ)],

which gives the following compact expression for the Gaussian curvature of a coneat its tip:

γ(r, θ) =12δ(r)rg(θ)[g′′(θ) + g(θ)], (4.2)

where δ(r) is a ‘Dirac’ distribution defined in such a way that∫ ∞0

dr δ(r)φ(r) = φ(0),

where φ(r) is an aribtrary smooth function of the radius.

(b ) Minimization of the extensional elastic energyFrom the explicit form of the Gaussian curvature of a conic singularity, we can

calculate the extensional energy due to such a singularity. Knowing the Green’sfunction of the bilaplacian which appears both in equation (2.1 a) and (2.1 b), we canfind the general solutions of (2.1 b):

χ′(r′) = −∫

Ωr dr dθ γ(r, θ)[|r − r′|2 log |r − r′| − 1] + Φ(r′), (4.3)

Φ(r′) being an arbitrary degree of freedom which satisfies ∆2Φ(r′) = 0.Once the singular Gaussian curvature γ(r, θ) given in (4.2) is put into (4.3), one

finds the condition for the Airy potential to vanish everywhere:∫dθ g(θ)[g′′(θ) + g(θ)] = 0. (4.4)

This is equivalent to the condition of vanishing of the Gauss–Bonnet integral:∫ds [ξ, ξ] = 0, (4.5)

where s is the arclength along a contour enclosing the singularity at the tip. Equation(4.4) or equation (4.5) can be easily satisfied. Representing g(θ) by its Fourier series

g(θ) =∑n

(aneinθ + a∗ne−inθ),


Crumpled paper 751

one gets that the condition (4.4) is equivalent to

2∑m

ama∗m(m2 − 1) = (a0 + a∗0)2. (4.6)

From (4.4), we deduce that a cone of revolution is never a good candidate to minimizethe extensional energy, as it would correspond to a0 non-zero only out of the Fouriercoefficients. The simplest d-cone which can be found is given by

g(θ) = g0(1± ( 23)1/2 cos(2θ)),

g0 being an arbitrary number (see in §4 c below the discussion showing that thisnumber should be small ). A less symmetrical solution is given by

g(θ) = g0[1± 12 cos(3θ)],

but, of course, equation (4.4) can be satisfied by many others combinations, and soby an infinity of possible shapes.

(c ) Selection of the d-coneAs we have seen, the d-cone is not unique. Given Γ , condition (4.4) restricts

the choice of the tip of the d-cone to a 2-parameter manifold in R3 that might beeventually empty. Its precise location can be found by energy consideration (if onesticks to the principle of minimization of the elastic energy, which might be irrelevantfor the concrete problem, where the limit yield of the material is usually exceedednear the tip of the cone). We begin with a naive analysis by scaling laws. Near thetip of the d-cone, which we will call the inner region or Ωh (for a reason which willbe clearer thereafter), we perform a boundary layer analysis: all the terms in theFvK equations (equations (2.1 a) and (2.1 b)) have the same strength. From equation(2.1 b), we deduce that χ is of order Eξ2 while, from equation (2.1 a), one deducesthat ξ is order h. No length scale enters explicitly into the FvK equations but thesmallest one, and this is h itself. The bending energy localized in Ωh is of order Eh3

and, by assumption, of the same order of magnitude as the extensional energy. In Ωh,the two kinds of elastic energy—the in-plane and the off-plane ones—do contributeequally. Out of Ωh, in the outer region, the d-cone solution is assumed. We havealready noticed that there is no extensional energy in this case (χ ≈ 0), but thetotal bending energy is of order Eh3(ξ2/L2) = Eh3(L2/R2

0), with L smaller than (orequal to) R0 itself, in order to keep the limit of validity of the FvK equations. So itappears that the elastic energy stored in the outer region can be of the same orderas in the inner region.This kind of consideration may appear rather naive becauseit does not take into account logarithmic contributions which appear for the d-cone.For evaluating the elastic energy, we will assume that Ωh is a small circle centred onthe tip of the d-cone and with a radius h R∗, such as it is located in the matchingzone between the inner and outer regions. In the outer region, the bending energyEb is given by

Eb =Eh3

24(1− σ2)

∫Ω−Ωh

r dr dθ (∇2ξ)2

=Eh3

24(1− σ2)

∫dθ [g(θ) + g′′(θ)]2[log(R(θ))− log(hR∗)].

In the limit of a vanishing h, the main contribution to Eb comes from log(hR∗),



but we cannot forget log(R(θ)) which is of order log(L) when L is a macroscopiclength (the typical radius of curvature of Γ , for instance). So, this calculation showsthat, in the limit L h, the dominant contribution to the elastic energy is of order

Eh3

24(1− σ2)log(L/h)

∫dθ [g(θ) + g′′(θ)]2,

where L/h is the ratio of the large scale to the thickness of the plate. There is noneed to specify more accurately the L, since that would amount to changing L bya constant factor, that would not change the logarithm, but by a constant amount,irrelevant for the dominant order. The d-cone that is the solution of the problemunder consideration for a given Γ will be such that it minimizes the integral∫

dθ [g(θ) + g′′(θ)]2. (4.7)

Note that this derivation of the energy assumes that the d-cone is smooth enoughto make this last integral (4.7) converge, which is a stronger condition for the smooth-ness of the function g(θ) than the condition of convergence of the integral in equation(4.4). In particular, this problem appears when the surface is polyhedral and whenstraight ridges on the surface merge at the tip of the d-cone. We shall come back tothis question in a future publication and content ourselves to assume that the func-tion g(θ) is smooth enough to make the integral that defines the energy converge.

It is not necessary to have a good representation of the inner region for getting thedominant contribution to the elastic energy; this one is given by the bending part inthe outer solution. Let us explain, however, how to find consistently the shape of thed-cone near its tip that depends on the solution of the inner problem. Indeed, thesimple d-cone solution cannot be relevant near the tip; there, the mean curvaturewould become singular and give an infinite flexural energy, although in a limitedarea. In order to avoid this, some rounding is necessary near the tip of the d-cone,precisely in the inner region to be described now. We notice first that the simpled-cone solution is a solution of the second FvK equation, but not of the first one.Actually, it is almost so, but for large distances. At r much larger than h, one canshow that the first term in (2.1 a) is negligible when one puts into it the value ofξ given by the d-cone solution. So, the d-cone solution is the beginning of a seriesexpansion in inverse powers of r of a solution of the full Fvk equations. Let us sketchthe derivation of the next term (that is the one just after the d-cone solution). Letus calculate the next order for ξ and χ. To do this, we insert the first (flexion) termof equation (2.1 a) into the d-cone solution (the dominant one at large r), and theresulting contribution has to be balanced by the first correction to χ, say χ1 insertedinto [ξ, χ]. This gives the following equation for χ1:

Eh2

12(1− σ2)1r3 [g(θ) + 2g′′(θ) + g′′′′(θ)] =

1r

∂2χ1

∂r2 (g(θ) + g′′(θ)),

that is solved as

χ1 = − Eh2

12(1− σ2)α(θ) log(r) + β(θ)r + η(θ) ,

with α(θ) = [1 + (g′′(θ) + g′′′′(θ))/(g(θ) + g′′(θ))]; β(θ) and η(θ) being arbitrary. Forthe same reason as mentioned in §3 d (that is, it is better to eliminate an energycontribution of order log(L)), one can put β(θ) = 0 and η(θ) is not significant in the


Crumpled paper 753

asymptotics. Putting this estimate of χ1 in equation (2.1 b), one gets the followingequation for the deviation of ξ from the d-cone solution:

1r

∂2ξ1

∂r2 (g(θ) + g′′(θ)) =h2

12(1− σ2)∆[

log(r)r2 α′′(θ)

].

The solution of this equation expands at large r like

ξ1 ≈ h2 log(r)r

.

This, together with the property that the FvK equation can be written in a pa-rameterless form by scaling any length with h and χ as Eh2, suggests an uniformexpansion for ξ(ρ = r/h, θ) of the form

ξ(ρ, θ) = hΦ(ρ, θ) = hΦ(r/h, θ).

This form is valid everywhere, if Φ(ρ, θ) behaves asymptotically like ρg(θ), so thatthe beginning of the expansion of ξ(ρ, θ) at large r is such that

ξ(ρ, θ) ≈ hρg(θ) + h(log(ρ)/ρ) ≈ h(r/h)g(θ) + h2 log(r/h)r

(ρ→∞).

A rather long formal calculation that we shall not reproduce here shows that thesolution of the FvK equations can be expanded in this way at large r, so that thehigher term vanish faster and faster at large r. This correponds (up to logarithmiccontributions) to an expansion of Φ(ρ, θ) at large ρ in inverse powers of ρ, again bydiscarding logarithms. Finally, at finite ρ, no approximation can be made becausethe full FvK equations may be written in a parameterless form for r of order h. Thereis, however, a restriction to this, coming from the basic assumption for the validityof the FvK equations: the curvature has to remain everywhere much less than 1/h.Otherwise, one would have to solve the full three dimensional elasticity problemin the bulk of the plate. In the present case, this imposes that the dimensionlessfunction g(θ) should be small. However, this does not lead to a simplification of theFvK equations relevant for the inner solution near the tip. Let g0 be the order ofmagnitude of g(θ), a small number. It happens that this number can be eliminatedfrom the formulation of the inner problem. The FvK equations are invariant underthe transformation r → r/g0, χ → χ, ξ → ξ. This eliminates the small parametercoming from the outer solution ξ(r, θ) ≈ rg(θ), and it leaves the FvK equationsunaffected, which shows our point. This proves also that the size of the inner regiongrows like 1/g0 when g0 tends to zero.

Finally, let us come back to the question that was mentionned in the introduction;that is, the focusing of the stress near the tip of the d-cone. This is discussed bylooking at the order of magnitude of the gradient of the Airy potential far from thetip of the cone. The Airy potential is dominated by the contribution that we havejust computed:

χ1 = − Eh2

12(1− σ2)α(θ) log(r) + η(θ) .

Taking any second derivative of χ1 with respect to the x and y coordinate, one getsthat the stress in the plate far from the tip is of order

σαβ,outer ≈ Eh2 1r2 .



Do not confuse the stress tensor, denoted traditionnally by σαβ with the Poissonratio, usually denoted by the same Greek letter. In the neighbourhood of the tipof the d-cone,the scaling for the stress is the one of the inner problem. This yieldsa stress of order E in principle but, because of the remarks just made, this stressis actually multiplied by the small dimensionless quantity g0 measuring the angularopening of the d-cone. Therefore, the stress in the inner region is of order

σαβ,inner ≈ Eg20 ;

already a very large stress in physical terms. This explains the statement made in theintroduction about the permanent scars localized at the tip of the d-cones, becausethe limit yield of the material is overcome there, something that is of course notdescribed in the present Hookean approach of elasticity.

5. Summary and perspectives

We have shown how to solve the FvK equations in order to represent the problemof minimum of elastic energy for a bended plate. By considering a model problem, theone of a plate bounded by a given curve in space, we have been lead to introduce theidea that the solution of the minimization problem is not always a smooth surface,but can be what we have called a d-cone; that is, a surface which satisfies the theoremaegregium almost everywhere except at the tip of the cone, where a region of highcurvature exists, with a size depending on the thickness of the plate. There, thestress is focused in such a way that, in practical applications, the limit yield of theelastic material can be overcome, and permanent scars appear. In the future, weplan to consider the more complicated situation when more than one d-cone existson the same plate. Then, for obvious geometrical reason, the surface needs to haveplanar parts, merging along straight ridges. These ridges end up at the tip of thed-cones. In this situation, the geometrical problem is already much harder than theone considered here. Although we have not yet explored this possibility, it couldbe that the present idea of d-cone has applications to general relativity. After all,Einstein’s equations aim at minimizing some functional. They are finally of a higherorder in the derivation than the integrand of the functionnal. They could leave someroom for the occurence of singularities like the tip of the d-cone, that are in somesense weaker solutions of the minimization problem, but do not satisfy formallythe Euler–Lagrange equation everywhere because of the lack of derivability at thesingularities.It is a pleasure to thank Dr L. Mahadevan for enlightening discussions.

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Received 22 March 1996; accepted 29 March 1996


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Crumpled paper - École Normale Supérieure › ~benamar › paper › crumpling.pdf · Crumpled paper By M. Ben Amar1 and Y. Pomeau2 1Laboratoire de Physique Statistique, associ