Introduction to Non-Linear Elasticity and Non-Euclidean Plates · 2013-09-17 · Introduction to...

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Introduction to Non-Linear Elasticity and Non-Euclidean Plates John Gemmer Shankar Venkataramani Program in Applied Mathematics University of Arizona Tucson, AZ 85721 [email protected] September 16, 2011

Transcript of Introduction to Non-Linear Elasticity and Non-Euclidean Plates · 2013-09-17 · Introduction to...

Page 1: Introduction to Non-Linear Elasticity and Non-Euclidean Plates · 2013-09-17 · Introduction to Non-Linear Elasticity and Non-Euclidean Plates John Gemmer Shankar Venkataramani Program

Introduction to Non-Linear

Elasticity and Non-Euclidean

Plates

John GemmerShankar Venkataramani

Program in Applied MathematicsUniversity of Arizona

Tucson, AZ 85721

[email protected]

September 16, 2011

Page 2: Introduction to Non-Linear Elasticity and Non-Euclidean Plates · 2013-09-17 · Introduction to Non-Linear Elasticity and Non-Euclidean Plates John Gemmer Shankar Venkataramani Program

Strain

1. Strain: Strain is a measurement of the local change in length ofa deformed body.

2. Let p = (x1, x2, x3) denote a point on a solid 3-dimensionalmanifold Ω mapped to x(p) = (y 1, y 2, y 3).

3. Define the displacement field by ui = y i − x i .4. Then, dy i = dui + dx i = ∂ui

∂x jdx j + dx i and therefore

(dl ′)2 = dl2 + 2∂ui

∂x jdx idx j +

∂ui

∂uj

∂ui

∂xkdx jdxk .

5. The strain tensor is

γij =1

2

(∂ui

∂x j+∂uj

∂x i+∂uk

∂x i

∂uk

∂x j

)

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Elastic Energy

1. The elastic energy of a deformed object with internal stress fieldσij is

E3D [x] =

∫Ωσij(εij)εij dV .

2. For Hookean material (also called linear elastic)

σik =E

1 + ν

(γik +

ν

1− 2νγjjδik

)γik =

1 + ν

Eσik −

ν

Eσjjδik ,

with the constants

E ∼ Young’s modulus

ν ∼ Poisson’s ratio.

3. The elastic energy of a Hookean material is

E3D [x] = ‖γij‖22.

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Euler Rod

Page 5: Introduction to Non-Linear Elasticity and Non-Euclidean Plates · 2013-09-17 · Introduction to Non-Linear Elasticity and Non-Euclidean Plates John Gemmer Shankar Venkataramani Program

Euler Rod

Page 6: Introduction to Non-Linear Elasticity and Non-Euclidean Plates · 2013-09-17 · Introduction to Non-Linear Elasticity and Non-Euclidean Plates John Gemmer Shankar Venkataramani Program

Euler Rod

Page 7: Introduction to Non-Linear Elasticity and Non-Euclidean Plates · 2013-09-17 · Introduction to Non-Linear Elasticity and Non-Euclidean Plates John Gemmer Shankar Venkataramani Program

Euler Rod

Page 8: Introduction to Non-Linear Elasticity and Non-Euclidean Plates · 2013-09-17 · Introduction to Non-Linear Elasticity and Non-Euclidean Plates John Gemmer Shankar Venkataramani Program

Euler Rod

Page 9: Introduction to Non-Linear Elasticity and Non-Euclidean Plates · 2013-09-17 · Introduction to Non-Linear Elasticity and Non-Euclidean Plates John Gemmer Shankar Venkataramani Program

Buckling

1. The governing equation and boundary conditions for the rod are

Cd2θ

ds2−Mg cos(θ) = 0 and

θ(0) = π

2dθds

∣∣s=L

= 0.

This equation is non-linear. Uniqueness of solutions is not guaranteed!

2. One solution is simply θ1 = π2 with the energy

E [θ1] =

∫ L

0

Mg ds = MgL.

3. Assume θ2 = π2 + θ(s). This gives

Cd2θ

ds2+ Mg θ = 0 and

θ(0) = 0d θds

∣∣∣s=L

= 0.

One solution is θ(s) = sin

(√MgC s

)if M = π2

4LCg .

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Buckling Movie

Play Movie

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Two-Dimensional Buckling

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Crumpling

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Inhomogeneous swelling

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Hyrdrogels

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Target Metric Model

I We model the stress free configuration as a two dimensionalRiemannian manifold D with a specified “target metric” g.

I A configuration is a sufficiently differentiable map x : D → R3 and wetake the elastic energy to be given by

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

∫D

∥∥∇xT · ∇x− g∥∥2

dA+t2

∫D

(4H

2 − K)

dA,

where H and K are the mean curvature and Gaussian curvature of xand t is the thickness of the sheet.

I The tensor γ = DxTDx − g measures the in-plane strain.

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Toy Problem

1. In this talk we will focus on the disk of radius R when in polarcoordinates (ρ, θ), g is given by

g = dρ2 +1

−Ksinh2(

√−Kρ) dθ2.

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

E [x] =

∫D

∥∥∥∇xT · ∇x− g∥∥∥2

dA︸ ︷︷ ︸0 for isometries

+t2

∫D

(k2

1 + k22

)dA︸ ︷︷ ︸

0 for flat surfaces

.

3. The isometric immersion is selected by the bending energy.

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Small Slopes Approximation

I In the small slopes approximation we assume that ε =√−K R 1

and use the dimensionless variables

Rr = ρ, Ru = x and Rv = y .

I The target metric takes the form

g = R2dr 2 + R2

(r 2 + ε2 r 4

3

)dθ2

= R2

(1 +

v 2

3ε2

)du2 − R2 2uv

3ε2 dudv + R2

(1 +

u2

3ε2

)dv 2.

I To match the target metric to lowest order we assume that

x(u, v) = R(u + ε2χ(u, v), v + ε2ξ(u, v), εη(u, v)

).

I A necessary condition that the surface is an isometric immersion is that

[η, η] = ηxxηyy − η2xy = −1.

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Small Slopes Elastic Energy

I Assuming the dimensionless thickness t = hR 1 satisfies ε2 t ε

then in terms of the small parameter τ = tε we have that

E [x] =

∫D

((γ11 + γ22)2 + γ2

11 + 2γ212 + γ2

22

)dudv

+τ 2

∫D

((∆η)2 − [η, η]

)dudv .

I The elastic energy of an isometric immersion is simply

E [x] = τ 2

∫D

((∆η)2 + 1

)dudv .

I The minimizer of the energy over the class of immersions is

χ = −xy 2

3and ξ = −x2y

3and η = xy .

I This gives us the upper bound

infx∈W 2,2(D)

E [x] ≤ τ 2π.

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Convergence to Saddle

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Convergence to Saddle

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Periodic Isometric Immersions

I A one parameter 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

E [x] ≤ πτ 2(4 cot2(π/n) + 1

).

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Energy of Periodic Configurations

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Energy of Periodic Configurations

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Theorems!

Definition

An is the set of configurations x = Id + ε(0, 0, η) + ε2(χ, ξ, 0) such that η isperiodic in θ with period 2π

n .

Lemma

For a fixed τ > 0 and E0 > 0 if there exists x ∈ An such that E [x] ≤ τ 2E0

then there exists constants C1,C2 > 0 independent of η and n such that

S[x] ≥ C1 − C2E 2

0

n2.

TheoremThere exists constant C1,C2 > 0 such that

C1nτ 2 ≤ infy∈An

E [y] ≤ C2τ2(4 cot2(π/n) + 1

).

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Discussion