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