SIAM 05 08 16

Geometric defects in thin elastic structures

SIAM MS16 May 8, 2016

Shankar Venkataramani

Eran Sharon

John Gemmer

Images courtesy Eran Sharon and John Gemmer

Morphology: growth and plastic deformation

Shaping Thin Elastic Sheets

Shaping Thin Elastic Sheets

Overview

•

Quick review of differential geometry

•

What is the puzzle? Some rigorous results

•

C

1,1

isometric immersions and branch points.

•

Construction of such surfaces -

Discrete differential geometry

.

N

(

p

)

·

d

r

(

p

) = 0

I

≡

ds

2

=

d

r

(

p

)

·

d

r

(

p

)

Geometry: The Gauss Normal map

p

x

y

u

v

w

Φ

Elastic energy of a thin sheet

E

=

!

∥

γ

∥

2

+

ϵ

2

∥

κ

∥

2

κ

= ˆ

n

·

D

2

Φ

γ

= (

D

Φ

)

T

·

D

Φ

−

g

Stretching energy

Bending energy

Elastic energy

E

t

[ ] =

S

[ ] +

t

2

B

[

H, K

]

=

Z

⌦

Q

( )

dxdy

+

t

2

Z

⌦

(4

H

2

2

K

)

dxdy,

Lewicka and Pakzad (2011).

–limit:

lim

t

!

0

t

2

E

t

=

(R

⌦

(4

H

2

2

K

)

dxdy

2

W

2

,

2

iso

+

1

otherwise

x

y

u

v

w

Immersion

:

⌦

!

R

3

of the center surface.

Reference Riemannian metric

g

.

Hyperbolic Monge Ampere equations

n

= (

w

x

, w

y

,

1)

N

=

k

n

n

k

.

dA

= (

w

xx

w

yy

w

xy

2

)

dx

^

dy

.

d

=

k

dA

n

k

dp

^

dq

=

K

F vK

(

x, y

)

dx

^

dy

.

dp

^

dq

=

K

k

(

n

x,y

k

)

dx

^

dy

For any domain on which the normal map is one-to-one, the area of the

spherical image cannot exceed 2

⇡

. No such restriction for the planar image.

Negatively curved sheets: Strip geometry

In the small slopes approximation, a necessary and sufficient

condition for the existence of strain free embeddings:

w

xx

w

yy

w

xy

2

=

f

yy

<

0

Translation invariant target metric:

g

=

dy

2

+

f

(y

)

2

dx

2

One parameter scaling

w

a

(

x, y

) =

a

1

w

(

ax, y

)

Scaling of the curvatures

Solutions of the Monge Ampere equation

w

xx

w

yy

w

xy

2

=

f

yy

<

0

Product solutions

w

=

Ae

y

cos(

x

)

,

f

(

y

) = 1 +

A

2

4

e

2

y

w

=

Ay

(x),

f

1

⇥

y

2

Theorem:

For all

f

such that

f <

0, there exists a smooth, periodic

solu-tion to the translasolu-tion invariant Hyperbolic Monge-Ampere equasolu-tion

w

w

w

=

f

(

y

) on a domain

S

times any closed interval on which

f

is bounded.

Formulation as an EDS

dw

=

pdx

+

qdy

dp

^

dx

+

dq

^

dy

= 0

dp

^

dq

+

dx

^

(

f

00

(

y

)

dy

) = 0

Let

=

p

f

00

(

y

)

, du

=

dy.

(dp

±

du)

^

(dx

⌥

dq/

) = 0.

Linear wave equation!

•

Variation of parameters gives linear first order equations

relating solutions for different f(y).

Rigidity of smooth solutions

Theorem:

If

w

(

x, y

) is a smooth solution of the Hyperbolic Monge-Ampere

equation

w

w

w

=

f

(

y

), then there exists a

C

vector field

v

=

w

w

2

f

(

y

)

⇤

⇤

x

+

⇤

⇤

y

whose integral curves foliate the domain and are transversal to the curves

y

=

constant. Further,

v

(

w

) =

w

w

w

w

2

⇥

(

y

)

w

= (

x, y

)

w

,

where

is a continuous function.

The lines of inflection cannot bifurcate

Negatively curved sheets: Disk geometry

Small slopes approximation:

det(

⇥⇥

w

) = 1

Solutions:

w

=

1

2

⇣

ax

2

y

a

⌘

.

Hyperbolic disks: Full geometry

By allowing non-smooth embeddings, we can decrease the

curvature, and the energy.

Piecewise quadratic surfaces

w

(

x, y

) =

(

xy

y

2

cot(

✓

+

))

0



✓



✓

+

xy

+

y

2

cot(

✓

))

✓



✓



0

Let us consider solutions of det(

D

2

w

) =

1

w

is

C

1,1

.

All the straight lines

through any point

Geodesics and Asymptotic lines: C

2

surfaces

Lelieuvre formulae

N

·

r

u

= 0 =

N

·

r

v

Asymptotic parameterization:

=

r(

u, v

)

N

=

r

u

⇥

r

v

|

r

u

⇥

r

v

|

N

u

·

r

u

= 0 =

N

v

·

r

v

,

N

u

·

r

v

=

N

v

·

r

u

r

u

=

⇢

N

⇥

N

u

,

r

v

=

⇢

N

⇥

N

v

N

u

⇥

N

v

=

1

⇢

2

r

u

⇥

r

v

=

)

K

=

⇢

Consistency and discretization

Two independent first order equations for

r

(

u, v

).

r

uv

=

r

vu

=

)

N

⇥

(2

⇢

N

uv

+

⇢

v

N

u

+

⇢

u

N

v

) = 0

Scaled Normal:

⌫

=

p

⇢

N

Moutard equation:

⌫

uv

k

⌫

Discrete differential geometry

r

r

=

⌫

⇥

⌫

,

r

r

=

⌫

⇥

⌫

(⌫

+

⌫

)

⇥

(⌫

+

⌫

) = 0

,

k

⌫

k

= [

K

(

r

)]

Given

⌫

n,

₀

,

⌫

₀

,m

and

r

₀

,

₀

we have as many equations as unknowns.

Quad-mesh

r

n

+1

,m

r

n,m

=

⌫

n,m

⇥

⌫

n

+1

,m

,

r

n,m

+1

r

n,m

=

⌫

n,m

⇥

⌫

n,m

+1

(⌫

+

⌫

)

⇥

(⌫

+

⌫

) = 0

,

k

⌫

k

= [

K

(

r

)]

Although we have used the notation

for a mapping

r

:

Z

2

!

R

3

, this makes

sense as a map

r

:

G

!

R

3

, where

G

⇢

R

is a connected planar graph where every vertex has even degree.

By construction, the Normal vector is Lipschitz so the

resulting limiting surface is C

1,1

r

:

G

!

R

is a

discrete Asymptotic net. Every star in the image is planar

DDG for small-slopes

r

= (x, y

),

p

= (w

x

, w

y

). We want to solve

w

xx

w

yy

w

xy

2

=

⇢

2

.

r

u

=

⇢

p

?

u

,

r

v

=

⇢

p

?

v

z

u

=

⇢

p

·

p

?

u

,

z

v

=

⇢

p

·

p

?

v

Consistency for

r

implies consistency for

z

.

Discretization.

⌫

=

p

⇢

p

r

n+1,m

r

n,m

=

p

⇢

n+1,m

⌫

n+1,m

?

p

⇢

n,m

⌫

n,m

?

,

r

n,m+1

r

n,m

=

p

⇢

n,m+1

⌫

n,m+1

?

+

p

⇢

n,m

⌫

n,m

?

,

z

n+1,m

z

n,m

=

⌫

n,m

·

⌫

n+1,m

?

,

z

v

=

⌫

n,m

·

⌫

n,m+1

?

Surfaces with branch points

By construction, the normal is “Lipschitz” and the

4

see Fig. 4(e-f). Note that, if a hyperbolic surface is C2, every

point is locally a (regular) saddle (as in Fig. 4(a)) and

there-fore cannot contain branch points. Non-C2 immersions are

therefore qualitatively di↵erent from C2 immersions in that

they admit 3-saddles (“monkey saddles”) and higher order saddles, which can mediate a local refinement of the buckling wavelength (See Fig. 5).

FIG. 4. (a-b) Small slope isometric immersions w0

4(x1, x2) and

w0₄(x₁, x₂) for constant Gaussian curvature K = 1. w0₄(x₁, x₂) is

con-structed by taking odd periodic reflections of the piece of w0

4(x1, x2)

bounded between the green lines. The mesh on both of these sur-faces correspond to their asymptotic lines. (c-d) Projection of the

asymptotic lines of w0

4(x1, x2) and w04(x1, x2) onto the x1, x2 plane.

(e-f) Direction of the gradient rw along circles centered at the

ori-gin. The regular saddle in (a) corresponds to a gradient field with winding number -1, so the gradient map is 1 to 1. The 4-saddle in (b) has winding number -3, so the gradient map is a 3 sheeted covering near the origin.

Multiple branch points can be introduced on the surface by replicating the above process at any point, not just the origin.

For example, consider the surface w0₂(x₁, x₂) = x₁x₂ which is

ruled by the asymptotic lines x₁, x₂ = const. A branch point

can be added at (x₁, x₂) = (1/ p2, 1/ p2) by removing the

sec-tor x₁, x₂ 1/ p2 and in this region fitting three rotated and

translated copies of w0₆(x₁, x₂) = x₂(x₁ p3x₂) so that the

resulting surface has continuous partial derivatives across the

cut; see Fig 5(a). Three more branch points b₂,1, b2,2, b2,3

at a radial distance of 1/4 from b_1,1 can be added along rays

emanating from b₁,1 that bisect the lines of inflection; see Fig

5(b). This construction can be continued so that at the n-th

it-eration 3n new branch points are added at a radial distance of

(1/2)n from the previous branch points. The surface w(x₁, x₂)

formed in the limit n _{! 1} is a fractal with an infinite number

of subwrinkles in the region x₁ 0, x₂ 0, x2₁ + x2₂ 1, and

it satisfies [w, w] = 1. The solution can be extended by odd

periodic reflections to give a small-slopes isometric

immer-sion of the unit disk with K = 1. To illustrate the wrinkling

behavior near the edge we map w to a strip geometry through

a conformal map h[x + iy] = w[ex+iy]; see Figs. 5(c-d).

FIG. 5. Finite bending energy solutions to the Monge-Ampere

equa-tion [w0_,w0_{] = 1. (a) Three subwrinkle solution created by}

insert-ing three rotated and translated copies of the solution w0

6(x1, x2) =

x₂(x1 p3x2) onto the solution w0₂(x1, x2) = x1x2 at a branch point.

(b) Nine subwrinkle solution created by inserting nine copies of

w0

12(x1, x2) = x2(x1 (2 +

p

3)x2) at three branch points added onto

the three subwrinkle solution. (c) Extension of the nine subwrinkle solution to the full circular domain. (d) The nine subwrinkle solution mapped to the strip geometry by a conformal map.

The existence of self-similar isometric immersions has im-plications to the modeling of non-Euclidean elastic sheets.

As for the strip with = 1, the solution w0₂(x₁, x₂) is

har-monic yet the extension of w0₂(x₁, x₂) to an exact

isomet-ric immersion has divergent bending energy for R ' 1.25

with the bending content concentrated near the singular point

x₁ = x₂ ⇡ 1.25/ p2 [22]. We can isometrically immerse disks

with larger R by a global refinement of the wavelength i.e

taking n > 2. These solutions increase the bending energy

globally. An energetically favorable alternative might be to

introduce a branch point in the n = 2 solution near the

singu-lar point, and locally refining the wavelength instead. Indeed,

numerics for = 1/3 in the strip geometry indicate that, even

within the small slopes approximation, localized self similar wrinkling profiles may be energetically preferred over global refinement of the wavelength [2, 23].

Crumpled sheets have an energy scale t5/3 _{which is}

inter-mediate between the stretching and bending energies [32, 33].

In contrast, the existence of W2,2 _{isometric immersions for}

Index of a branch point

C

2

isometries are not

dense in

W

2

,

2

isometries!

What does all this mean?

•

Free sheets:

the relevant singularities are

branch points

and

lines of inflection

.

The tangent plane is continuous and the

normal is Lipschitz.

•

Unique —

they

do not concentrate elastic energy

in the

vanishing thickness limit. They are not picked out by the

Gamma-limit.

•

Degeneracy:

The construction with branch points is very

flexible. These sheets have lots of isometries with finite

bending energy, so thin free sheets should be very floppy.

References:

1. Gemmer, J. A. & Venkataramani, S. C. Shape selection in non-Euclidean plates. Physica D:

Nonlinear Phenomena 240, 1536–1552 (2011).

2. Klein, Y., Venkataramani, S. & Sharon, E. Experimental Study of Shape Transitions and

Energy Scaling in Thin Non-Euclidean Plates. Phys. Rev. Lett. 106, 118303 (11 Mar. 2011).

3. Gemmer, J. A. & Venkataramani, S. C. Defects and boundary layers in non-Euclidean plates.

Nonlinearity 25, 3553–3581 (2012).

4. Gemmer, J. A. & Venkataramani, S. C. Shape transitions in hyperbolic non-Euclidean plates.

Soft Matter 9, 8151–8161 (2013).

5. Gemmer, J., Sharon, E., Shearman, T. & Venkataramani, S. C. Isometric immersions, energy

minimization and self-similar buckling in Non-Euclidean elastic sheets,

accepted

, Europhys.

Lett., arXiv:1601.06863 (2016).