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Introduction to Non-Linear
Elasticity and Non-Euclidean
Plates
John GemmerShankar Venkataramani
Program in Applied MathematicsUniversity of Arizona
Tucson, AZ 85721
jgemmer@math.arizona.edu
September 16, 2011
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
)
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.
Euler Rod
Euler Rod
Euler Rod
Euler Rod
Euler Rod
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 .
Buckling Movie
Play Movie
Two-Dimensional Buckling
Crumpling
Inhomogeneous swelling
Hyrdrogels
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.
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.
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.
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.
Convergence to Saddle
Convergence to Saddle
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
).
Energy of Periodic Configurations
Energy of Periodic Configurations
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
).
Discussion
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