Shape Selection in the non-Euclidean Model of Elasticity · 1. Present the non-Euclidean model for swelling thin elastic sheets. 2. Study minimizers in the Foppl-von K arm an model.

Shape Selection in the

non-Euclidean Model of

Elasticity

John Gemmer

Program in Applied MathematicsUniversity of Arizona

Tucson, AZ 85721

[email protected]

April 16, 2012

Shoulders of Giants

Shaping Thin Elastic Sheets

Outline

1. Present the non-Euclidean model for swelling thin elastic sheets.

2. Study minimizers in the Foppl-von Karman model. This model isaccurate for a particular scaling of the thickness and curvature ofthe sheet.

3. Study minimizers in the Kirchhoff approximation. This model isaccurate for very thin sheets and corresponds to finding lowenergy local isometric immersions of two dimensionalRiemannian metrics.

4. Discuss the implications of this work.

Outline





Outline





Outline





Metric Induced by Swelling

Metric Induced by R3

The Effect of Bending

The Effect of Bending

Non-Euclidean Model

I The equilibrium configuration of a sheet of thickness t is a sufficientlysmooth map x : D → R3 that minimizes the energy

E [x] = S[x]+t2B[x] =

∫D

∥∥DxT · Dx− g∥∥2

dA︸︷︷︸stretching energy

+t2

∫D

(4H

2

1− ν− 2K

)dA︸︷︷︸

bending energy

,

where H and K are the mean and Gaussian curvatures of x.I In the vanishing thickness limit the equilibrium configurations converge

to a solution of DxT · Dx− g = 0, i.e. an isometric immersion.

13 / 52

Toy Problem

I D is a disk or an annulus with outer radius R and in polarcoordinates (ρ, θ),

g = dρ2 +1

|K |sinh2(

√|K |ρ) dθ2.

I Stretching free configurations correspond to (local) isometricimmersions of the hyperbolic plane H2.

I If the sheet is sufficiently thin the isometric immersion is selectedby minimizing the bending energy.

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Growing H2

Foppl - von Karman Ansatz

1. Assume for some small parameter ε� 1 that g = g0 + ε2g1. TheFoppl - von Karman ansatz (FvK) is

x = i + εη + ε2χ,

where χ ∈W 1,2(D,R2) and η ∈W 2,2(D,R2⊥). The admissible set A

is the set of such deformations.

2. The in-plane strain tensor γ is defined by

γ = (Dχ)T + Dχ+ DηTDη − g1.

3. If τ = tRε � 1 then the FvK elastic energy Eτ : A → R is

Eτ [x] =

∫D

((tr(γ))2

1− ν− 2 det(γ)

)dA︸︷︷︸

S[x]

+τ 2

∫D

((∆η)2

1− ν− 2[η, η]

)dA︸︷︷︸

B[x]

,

where for f , g ∈ C2, [f , g ] = 12

(∂2f∂x2

∂2g∂y2 + ∂2f

∂x2∂2g∂y2 − 2 ∂2f

∂x∂y∂2g∂x∂y

).



x = i + εη + ε2χ,


is the set of such deformations.2. The in-plane strain tensor γ is defined by



Eτ [x] =

∫D

((tr(γ))2

1− ν− 2 det(γ)

)dA︸︷︷︸

S[x]

+τ 2

∫D

((∆η)2

1− ν− 2[η, η]

)dA︸︷︷︸

B[x]

,

where for f , g ∈ C2, [f , g ] = 12

(∂2f∂x2

∂2g∂y2 + ∂2f

∂x2∂2g∂y2 − 2 ∂2f


).



x = i + εη + ε2χ,


is the set of such deformations.2. The in-plane strain tensor γ is defined by



Eτ [x] =

∫D

((tr(γ))2

1− ν− 2 det(γ)

)dA︸︷︷︸

S[x]

+τ 2

∫D

((∆η)2

1− ν− 2[η, η]

)dA︸︷︷︸

B[x]

,

where for f , g ∈ C2, [f , g ] = 12

(∂2f∂x2

∂2g∂y2 + ∂2f

∂x2∂2g∂y2 − 2 ∂2f


).

Toy Problem

I If ε =√−K R � 1 and ρ = Rr then

g = dρ2 +1

−Ksinh2

(√−Kρ

)dθ2 = R2dr 2 + R2

(r 2 + ε2 r 4

3

)dθ2.

I Without loss of generality we will assume that D is an annulus withouter radius R = 1, inner radius r0, and K = −1.

I For n ∈ {2, 3 . . .}, we study minimizers over the admissible set ofn-periodic deformations An ⊂ A defined by x ∈ An if and only if inpolar coordinates (ρ, θ) the out-of-plane displacement η satisfies

I η is periodic in θ with period 2π/n,

I η vanishes along the lines θ = 0 and θ = π/n,

I η (θ − π/n) = −η(θ).

We call the lines θ = mπ/n, m ∈ {0, . . . , 2n − 1}, lines of inflection.

Toy Problem


g = dρ2 +1

−Ksinh2

(√−Kρ

)dθ2 = R2dr 2 + R2

(r 2 + ε2 r 4

3

)dθ2.





I η (θ − π/n) = −η(θ).


Toy Problem


g = dρ2 +1

−Ksinh2

(√−Kρ

)dθ2 = R2dr 2 + R2

(r 2 + ε2 r 4

3

)dθ2.





I η (θ − π/n) = −η(θ).


Foppl - von Karman Equations

1. The Euler-Lagrange equations are{1

2(1+ν) ∆2Φ + [η, η] = −1

[Φ, η] = τ 2

4(1−ν) ∆2η,

where Φ is the Airy stress potential defined by

D2Φ =

( ν1−ν γ11 + 1

1−ν γ22 −γ12

−γ121

1−ν γ11 + ν1−ν γ22

).

2. The natural boundary conditions are(∂2Φ∂y2 ,− ∂2Φ

∂x∂y

)· n∣∣∣∂D

= 0 and(− ∂2Φ∂x∂y ,

∂2Φ∂x2

)· n∣∣∣∂D

= 0

11−ν

∂∆η∂n − nT · D2η · n

∣∣∣∂D

= 0

11−ν∆η + ∂

∂t

(nT · D2η · t

)∣∣∣∂D

= 0,

where n is the outward normal and t is the tangent vector to ∂D.

Flat Solutions

1. Define the admissible set of flat deformations by

Af = {x ∈ An : η = 0}.

2. Assuming radial symmetry, Φ satisfies

∂4Φ

∂r 4+

2

r

∂3Φ

∂r 3− 1

r 2

∂2Φ

∂r 2+

1

r 3

∂Φ

∂r= −2(1 + ν),

∂Φ

∂r

∣∣∣∣r=r0,1

= 0.

3. The general solution is

Φ = c1r 2 + c2r 2 ln(r)− 1 + ν

32r 4.

4. Define F = infx∈Af

Eτ [x].

“Isometric Immersions”

1. Define the set of FvK isometric immersions by

A0 = {x : An : γ = 0}.

If x ∈ A0 and extremizes the energy then η satisfies

[η, η] = −1 and ∆2η = 0.

.2. If an isometric immersion exists we have that

infx∈AEτ [x] ≤ min{F , τ 2C2}.

Theorem

Let A′ = {x ∈ A : γ = 0}. If A′ 6= ∅ then,

limτ→0

τ−2 infx∈AEτ [x] = inf

x∈A′B[x].

Convergence to Saddle

I The elastic energy of an isometric immersion is simply

Eτ [x] = τ 2

∫D

(1

2(1− ν)(∆η)2 + 2

)dxdy .

I The function η = xy is harmonic and satisfies [η, η] = −1. Thus, theglobal minimum of the elastic energy over the class of isometricimmersions is obtained by

η = xy =r 2

2sin(2θ).

TheoremLet x∗τ ∈ A be a sequence with corresponding out-of-plane displacement η∗τsuch that infx∈A Eτ [x] = Eτ [x∗τ ]. Then, there exists a subsequence η∗τk andx∗ ∈ A0 such that η∗τk ⇀ η∗. Moreover, there exists A ∈ SO(2) and b ∈ Rsuch that η∗(A(x , y)) + b = xy.

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Periodic Isometric Immersions

I A one parameter family of isometric immersions are of the form

ηa =1

2

(ax2 − 1

ay 2

).

I By letting a = tan(π/2n) we can construct n-wave isometricimmersions through odd periodic extensions.

I This gives us the upper bound

Eτ [x] ≤ Cn2τ 2.

Periodic Deformations

Convergence to Minimizer

Scaling

Lemma

For a fixed τ > 0 and B0 > 0 if there exists x ∈ An such that B[x] ≤ B0

then there exists a constant C > 0 independent of η and n such that

S[x] ≥ F2− CB2

0

n2.

Theorem

Let n ∈ {2, 3 . . .} and τ > 0. There exists constant c,C > 0 independent ofn such that

min{cnτ 2,F/2} ≤ infx∈An

Eτ [x] ≤ min{F ,Cn2τ 2}

Crossover Regimes

Boundary Layers

Perturbations

The elastic energy of perturbations η = y(x − cot

(πn

)y)

+ η and Φ = Φ is

Eτ [x] = 2n

∫Bn

1

1 + ν

(∆Φ)2

dxdy

+2nτ 2

∫Bn

[1

1− ν

(∆η − 2 cot

(πn

))2

+4 cot(π

n

) ∂2η

∂x2+ 4

∂2η

∂x∂y− 2[η, η] + 2

]dxdy .

Outer Radius Layer

Rescale by

rout = τα(1− r), ηout = τβ η′out , Φout = τγΦ′out .

To lowest order the stretching and bending energies near the outer radius are

S[x]

2n= τ 2γ+4α

∫Bn

1

1 + ν

(∂2Φ′out∂ r 2

out

)2

dxdy ,

τ 2B[x]

2n= τ 2

∫Bn

[1

1− ν

(τβ+2α ∂

2η′out∂ r 2

out

− 2 cot(π

n

))2

+4τβ+2α cos(θ)(

cot(π

n

)+ sin(θ)

) ∂2η′out∂ r 2

out

+ 2

]dxdy .

The balance is α = −1

2, β = 1 and γ = 2.

Width of Boundary Layers

1. The width of the boundary layer in which the Gaussian curvature issignificantly reduced satisfies the following scaling

width(θ)ρ=R ∼ t12 |K0|−1/4 csc(π/n) cos(θ) cos(π/n − θ).

2. For n ≥ 3, the width of the boundary layer in which the meancurvature is significantly reduced satisfies the following scaling

width(θ)ρ=R ∼ t12 |K0|−1/4

√sin(π/n) sec(θ) sec(π/n − θ).

Bottom Boundary Layer

Rescale byybt = ταy , ηbt = τβ η′bt , Φbt = τγΦ′bt .

To lowest order the stretching and bending energies near the bottom of thesector are

S[x]

2n= τ 2γ+4α

∫Bn

1

1 + ν

(∂2Φ′bt∂y 2

bt

)2

dxdy ,

τ 2B[x]

2n= τ 2

∫Bn

[1

1− ν

(τβ+2α ∂

2η′bt∂y 2

bt

− 2 cot(π

n

))2

+ 2

]dxdy ,

The balance is α = −1

3, β =

2

3and γ =

5

3. The width of the boundary

layer is

width(ρ)y=0 = t13 ρ

13 |K0|−

16 .

Summary of FvK Model

1. With decreasing thickness minimizers converge to a two waveprofile.

2. The periodic profiles do agree qualitatively with experimentalobservations.

3. We proved rigorously that there is no “crossover” regime whichallows for refinement of the number of waves.

4. We showed that the minimum are perturbations of isometricimmersions with localized regions of stretching near the edge ofthe domain and along the lines of inflection.

Kirchhoff Model

I Define the admissible set AKi by

AKi = {x ∈W 2,2(D,R3) : (Dx)T · Dx = g}.

I In the Kirchhoff model the elastic energy EKi : AKi → R is given by

EKi [x] =Y

24(1 + ν)

∫D

[4H2

1− ν− 2K

]dAg.

I It is sufficient to study minimizers over the functional W : AKi → Rdefined by

W[x] =

∫D

(k2

1 + k22

)dAg.

Kirchhoff Model


AKi = {x ∈W 2,2(D,R3) : (Dx)T · Dx = g}.


EKi [x] =Y

24(1 + ν)

∫D

[4H2

1− ν− 2K

]dAg.


W[x] =

∫D

(k2

1 + k22

)dAg.

Kirchhoff Model


AKi = {x ∈W 2,2(D,R3) : (Dx)T · Dx = g}.


EKi [x] =Y

24(1 + ν)

∫D

[4H2

1− ν− 2K

]dAg.


W[x] =

∫D

(k2

1 + k22

)dAg.

Theorems

Theorem

Hilbert’s Theorem: There are no real analytic isometric immersionsof H2 into R3.

Theorem

Efimov’s Theorem: There are no C2 isometric immersions of H2

into R3.

Theorem

Nash Kuiper Theorem: There are C1 isometric immersions of H2

into R3

Theorem

Amsler/Bianchi: The points where a surface fails to be a smoothisometric immersion form analytic curves on the surface

Chebychev Nets

I A Chebychev Net is a configuration x(u, v) withmetric

g = du2 + cos(φ(u, v))dudv + dv 2.

I φ satisfies the sine-Gordon equation

∂2φ

∂u∂v= −K sin(φ).

I The bending energy can be put into the equivalentform

B[φ] ∼∫D

(tan2(φ/2) + cot2(φ/2)

)dA

=

∫D

1 + cos2(φ)

sin2(φ)dA ≥

∫D

1 + cos2(φ) dA.

L∞ Minimizers

I Numerically minimizers of ‖ tan2(φ/2) + cot2(φ/2)‖∞ satisfy

φ(u, v) = ψ(u + v).

I Assuming this ansatz, ψ′′ = −KR sin(ψ). This leads to followingconjecture

infx∈AKi∩C∞

max{|k1|, |k2|} ≥1

64exp(2R).

Amsler Surfaces

1. A surface of constant negative Gaussian curvature K and twostraight asymptotic curves is called an Amsler Surface.

2. The similarity solution ϕ(λ) = φ(u, v), λ = 2√

uv satisfies thefollowing Painleve III equation in trigonometric form:{

ϕ′′(λ) + 1λϕ′(λ)− sin(ϕ(λ)) = 0

ϕ(0) = πn and dϕ

dλ (0) = 0.

3. A key property of these surfaces is that the surface has two asymptotic

lines that intersect at the origin at an angle ϕ(0) = πn .

Periodic Amsler Surfaces

Energy of Amsler Surfaces

Energy of Amsler Surfaces

Summary of Kirchhoff Model

1. We provided numerical evidence that the maximum principalcurvatures of smooth isometric immersions grows exponentiallyin√−K0R. This is not capture by the FvK approximation.

2. We constructed piecewise smooth isometric immersions thathave the same symmetry as the experimentally observeddeformations. These periodic Amsler surfaces have bendingenergy then their smooth counterparts.

3. For each n ≥ 2 there is a radius Rn ∼ log(n) such that then-periodic Amsler surfaces only exist for 0 < R < Rn. This givesa natural mechanism for refinement of wavelength withincreasing diameter.

Future Directions

1. The full problem has multiple scales. Perhaps a combination ofdifferent reduced theories is needed in various regions of thedomain.

2. The periodic shapes in swelling hydrogels are the result ofdynamical processes. It may be more appropriate to model thistype of differential growth dynamically, perhaps as a gradientflow of the elastic energy. seem to describe the observedpatterns.

3. In the FvK approximation the deformation is perturbed around aflat surface. Perhaps it is more appropriate to perturb around anexact isometric immersion.

Shape Selection in the non-Euclidean Model of Elasticity · 1. Present the non-Euclidean model for swelling thin elastic sheets. 2. Study minimizers in the Foppl-von K arm an model.

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