The Mathematics of Wrinkles and Folds
Robert V. Kohn
Courant Institute, NYU
NYU Mathematics Society, 2/2/2012
Robert V. Kohn
Wrinkles and Folds
Wrinkles and folds
We’ll be talking about
inextensible sheets (like paper)
extensible sheets (rubber or cloth)
thin crystalline films on substrates
a little geometry and mechanics
a lot of calculus of variations
Robert V. Kohn
Wrinkles and Folds
Paper, deformed smoothly
Paper is (almost) inextensible. So we can describe deformations of a
flat piece of paper using
g : R2 → R3
such that (Dg)T Dg = I.
Explanation:
|Dg · v |2 = 1 ⇔ hv , (Dg)T Dg v i = 1;
and this must hold for all unit vectors v in R2 .
Facts from geometry:
A surface in R3 has 2 principal curvatures κ1 , κ2 (eigenvalues of
quadratic form associated to quadratic approxn).
For image of a smooth isometry, κ1 κ2 = 0 pointwise.
Image of a smooth isometry is a developable surface.
Robert V. Kohn
Wrinkles and Folds
Paper, deformed smoothly
Essential mechanics of paper: it resists bending.
If midline is isometric but image is curved, then lines above/below
middle are stretched/shrunk. If thickness is h and curvature is κ, then
Z L Z h/2
Z L
2 2
3
unhappiness =
z κ dz dx = ch
κ2 dx.
0
−h/2
0
Basic model of paper:
Z
min
κ21 + κ22 dA
g(Ω)
where Ω ⊂ R2 , g : Ω → R3 is an isometry, and κ1 , κ2 are the principal
curvatures of the image.
Not necessarily easy to solve (what is the shape of a Mobius band?).
Boundary conditions matter; sometimes gravity matters too.
Robert V. Kohn
Wrinkles and Folds
Paper, with singularities
Actually: smooth isometries are not sufficient. Depending on loads
and bdry conds, we easily see
formation of point singularities
(“d-cones”)
formation of line singularities
(“crumpling”)
Why? Because there is no isometry with finite bending energy.
Heuristic calculation: consider a “perfectly conical d-cone”: use
Ω = {r < 1} and g(r , θ) = r ϕ(θ) where ϕ(θ) is a curve on S 2 with the
right length (2π). Then
curvature at radius r ∼
so
Z
2
Z
κ dA ∼
0
1
1
rdr
r2
Robert V. Kohn
1
r
is divergent.
Wrinkles and Folds
Paper, with singularities
How to model d-cones and crumpling? Main idea:
Z
Z
Eh = min |(Dg)T Dg − I|2 dA + h2
(curvature)2 dA
Ω
image
First term (“membrane energy”) prefers isometry. It is nonconvex (for
paper: many smooth minimizers).
Second term (“bending energy”) resists bending, but has h2 in front.
If bc permit smooth def with curvature in L2 then Eh ∼ h2 as h → 0.
For d-cone and crumpling, Eh h2 as h → 0. (How does Eh behave
as h → 0? We have conjectures, but few theorems.)
Robert V. Kohn
Wrinkles and Folds
Digression: folding paper flat
Problem: NYC subway map is difficult to fold “correctly.”
Solution: The Miura map is easy to fold “correctly.”
Is there math here? Well, a little bit:
folding paper flat ⇔ g : R2 → R2 such that Dg ∈ O(2).
So g is locally a rotation (preserving or reversing orientation).
When 4 creases meet, opposite angles
must add to π (to permit folding flat).
For arbitary “creases” (violating angle condition) there is no way
to fold paper flat using them.
For rectangular creases there are many ways. For Miura pattern
there is essentially one way.
Robert V. Kohn
Wrinkles and Folds
Wrinkling
In many settings we see wrinkles rather than creases or point
singularities. Some examples:
hanging drapes
stretched sheets
water drop on sheet
floating on water
As h → 0, scale of wrinkling → 0. (Easiest to see for drapes.)
Mathematically: Eh = stretching energy + h2 bending energy. But
minimizing stretching term requires infinite bending energy.
Robert V. Kohn
Wrinkles and Folds
Patterns vs scaling laws
Eh = stretching energy + h2 bending energy.
Min stretching requires infinite bending.
Problem: It is difficult to describe a pattern (even a well-organized
one, as in the figures), let alone explain why it occurs.
Solution: Focus instead on the scaling law of the minimum energy,
i.e. find asymptotics of Eh as h → 0:
lower bound ≤ Eh ≤ upper bound
Upper bound can be obtained by guessing form of solution.
(Nature often gives us a hint.)
Lower bound is different: it must consider for any pattern (even
those not seen in nature).
If bounds are similar than we have (more or less) nailed it.
Robert V. Kohn
Wrinkles and Folds
A 1D example
1
Z
min
v (0)=v (1)=0
2
(vx2 − 1)2 + ε2 vxx
+ αv 2 dx
0
When ε = 0, α > 0, min value is 0, not attained. Min sequence has
“vx = ±1 with prob 1/2 each.”
When ε > 0, min scales like ε2/3 α1/3 , since for sawtooth with N teeth,
value is about ∼ εN + αN −2 . Best N ∼ (α/ε)1/3 .
Case ε > 0, α = 0 is different: just one tooth; min value is c0 ε.
Robert V. Kohn
Wrinkles and Folds
2D is richer than 1D
Z
(vy2 − 1)2 + vx2 + ε2 |∇∇v |2
min
v =0 at x=0
[0,L]×[0,1]
Like 1D example (in y ), but microstructure is required by bdry cond
rather than a lower-order term
Microstructural length scale ` depends on x (finer near x = 0). In fact,
`(x) ∼ ε1/3 x 2/3 .
Min energy has scaling law ε2/3 L1/3 .
Studied 20 years ago (Kohn-Müller, Conti); motivated by patterns seen
in martensitic phase transformation.
Schematic top view, and photo showing mechanism of refinement:
Robert V. Kohn
Wrinkles and Folds
A real example
Recent work with Hoai-Minh Nguyen. Patterns seen in a thin, stiff
layer compressed by a thick, soft substrate.
stretch a polymer layer
deposit the film
release the polymer
film buckles to avoid compression
Commonly seen pattern: herringbone
silicon on pdms
Robert V. Kohn
gold on pdms
Wrinkles and Folds
Explanation of the herringbone pattern
We use a “small-slope” (von Karman) version
of elasticity, writing (w1 , w2 , u3 ) for the elastic
displacement. The energy per unit area Eh
has three terms:
(1) Stretching term captures fact that film’s natural length is larger than that
of the substrate:
Z
αm h |e(w) + 21 ∇u3 ⊗ ∇u3 − ηI|2 dx dy
(2) Bending term captures resistance to bending:
Z
h3 |∇∇u3 |2 dx dy
(3) Substrate term captures fact that substrate acts as a “spring”, tending to
keep film flat. Cheating a bit (to simplify), it behaves like
Z
1/2
√
2
αs η
u3 dx dy
Robert V. Kohn
Wrinkles and Folds
Explanation of the herringbone, cont’d
To permit spatial averaging, we assume
periodicity on some (large) scale L, and we
focus on the energy per unit area:
Eh
=
αm h
L2
+
Z
[0,L]2
h3
L2
|e(w) + 12 ∇u3 ⊗ ∇u3 − ηI|2 dx dy
Z
|∇∇u3 |2 dx dy +
[0,L]2
1/2
√ Z
αs η
u32 dx dy
L
Summary of results:
2/3
min Eh ∼ min{αm η 2 h, αs ηh}
energy of unbuckled state w = u3 = 0 is αm η 2 h
2/3
energy of herringbone pattern is Cαs ηh.
herringbone achieves the optimal scaling law in the “far from critical”
2/3
regime αs αm η.
Robert V. Kohn
Wrinkles and Folds
Sketch of the upper bound
Energy of unbuckled state is αm η 2 h
2/3
Energy of herringbone is Cαs ηh
2/3
So min Eh ≤ min{αm η 2 h, Cαs ηh}
Eh
=
αm h
L2
+
Z
[0,L]2
h3
L2
|e(w) + 12 ∇u3 ⊗ ∇u3 − ηI|2 dx dy
Z
|∇∇u3 |2 dx dy +
[0,L]2
1/2
√ Z
αs η
u32 dx dy
L
Key features of herringbone:
stretching term is negligible
√
typical slope is |∇u3 | ∼ η
if scale of wrinkling is ` then bending term is h3 η/`2 and substrate term
−1/3
2/3
is αs η`. Optimal ` ∼ hαs
gives value ∼ αs ηh.
Note resemblance to our 1D example
Z 1
2
(vx2 − 1)2 + ε2 vxx
+ αv 2 dx
0
Robert V. Kohn
Wrinkles and Folds
Sketch of the lower bound
2/3
Claim: For any pattern, energy must be at least C min{αm η 2 h, αs ηh}.
Let’s take L = 1, η = 1 for simplicity. All we need about stretching term is:
Z
stretching term ≥ αm h |∂x w1 + 12 |∂x u3 |2 − 1|2 dx dy .
Recall:
bending term = h3 k∇∇u3 k2L2 ;
substrate term = αs ku3 kL2 .
Also: (w, u3 ) is periodic.
R
2
CASE 1: If |∇u3 | is small, stretching ≥ αm h, since ∂x w1 has mean 0.
R
2
CASE 2: If |∇u3 | is large, use the “interpolation inequality”
k∇u3 k2L2 ≤ k∇∇u3 kL2 ku3 kL2
to see that
Bending + substrate terms
=
≥
h3 k∇∇u3 k2 + 21 αs ku3 k + 12 αs ku3 k
1/3
2/3
= Chαs .
C h3 k∇∇u3 k2 αs2 ku3 k2
using arith mean/geom mean inequality.
Robert V. Kohn
Wrinkles and Folds
Sketch of the lower bound
2/3
Claim: For any pattern, energy must be at least C min{αm η 2 h, αs ηh}.
Let’s take L = 1, η = 1 for simplicity. All we need about stretching term is:
Z
stretching term ≥ αm h |∂x w1 + 12 |∂x u3 |2 − 1|2 dx dy .
Recall:
bending term = h3 k∇∇u3 k2L2 ;
substrate term = αs ku3 kL2 .
Also: (w, u3 ) is periodic.
R
2
CASE 1: If |∇u3 | is small, stretching ≥ αm h, since ∂x w1 has mean 0.
R
2
CASE 2: If |∇u3 | is large, use the “interpolation inequality”
k∇u3 k2L2 ≤ k∇∇u3 kL2 ku3 kL2
to see that
Bending + substrate terms
=
≥
h3 k∇∇u3 k2 + 21 αs ku3 k + 12 αs ku3 k
1/3
2/3
= Chαs .
C h3 k∇∇u3 k2 αs2 ku3 k2
using arith mean/geom mean inequality.
Robert V. Kohn
Wrinkles and Folds
Sketch of the lower bound
2/3
Claim: For any pattern, energy must be at least C min{αm η 2 h, αs ηh}.
Let’s take L = 1, η = 1 for simplicity. All we need about stretching term is:
Z
stretching term ≥ αm h |∂x w1 + 12 |∂x u3 |2 − 1|2 dx dy .
Recall:
bending term = h3 k∇∇u3 k2L2 ;
substrate term = αs ku3 kL2 .
Also: (w, u3 ) is periodic.
R
2
CASE 1: If |∇u3 | is small, stretching ≥ αm h, since ∂x w1 has mean 0.
R
2
CASE 2: If |∇u3 | is large, use the “interpolation inequality”
k∇u3 k2L2 ≤ k∇∇u3 kL2 ku3 kL2
to see that
Bending + substrate terms
=
≥
h3 k∇∇u3 k2 + 21 αs ku3 k + 12 αs ku3 k
1/3
2/3
= Chαs .
C h3 k∇∇u3 k2 αs2 ku3 k2
using arith mean/geom mean inequality.
Robert V. Kohn
Wrinkles and Folds
Stepping back
Variational viewpoint, with thickness as a small parameter: the bending
term is formally smaller than the stretching term by a factor of h2 .
Singularities, wrinkling patterns, or both form if minimization of the
stretching term requires infinite bending.
Focus on energy scaling law is convenient: at least the question makes
sense.
We get insight also about patterns, e.g. whether they achieve the
optimal law. (But note: patterns in nature can be local minima.)
Analysis is useful, but so is simulation. Simulation permits exploring
local minima, and how patterns form.
Robert V. Kohn
Wrinkles and Folds
Stepping back, cont’d
For more about the herringbone pattern: preprint with Hoai-Minh Nguyen,
Analysis of a compressed thin film bonded to a compliant substrate: the
energy scaling law, to be posted shortly at www.math.nyu.edu/faculty/kohn.
Other current projects:
1
stretch-induced wrinkling in membranes (Peter Bella)
2
hanging drapes (Peter Bella)
3
confinement-induced wrinkling in floating sheets (Hoai-Minh Nguyen)
4
d-cones (Jeremy Brandman, Hoai-Minh Nguyen)
5
debonding of a compressed film from its substrate (Jacob Bedrossian)
Robert V. Kohn
Wrinkles and Folds
Image credits
S. Conti and F. Maggi, Arch Rational Mech Anal 187
(2008) 1-48
E. Cerda and L. Mahadevan, Proc R Soc A 461
(2005) 671-700
E. Cerda and L. Mahadevan, Phys Rev Lett 90 (2003)
074302
H. Vandeparre et al, Phys Rev Lett 106 (2011) 224301
J. Huang et al, Science 317 (2007) 650–653
X. Chen and J. Hutchinson, Scripta Materialia 50
(2004) 797–801
J. Song et al, J Appl Phys 103 (2008) 014303
Funding from NSF is gratefully acknowledged.
Robert V. Kohn
Wrinkles and Folds