Aspen-20

On leaves, flowers and sea-slugs

Aspen Winter conference

Low-dimensional solids in hard and soft condensed matter: mechanics, thermodynamics, and electrons.

February 5, 2020

Shankar Venkataramani

University of Arizona, Mathematics

Or

Overview

•

Quick review of elasticity/differential geometry.

•

C

1,1

isometric immersions, branch points.

•

Hyperbolic Origami

•

Applications: mechanics, dynamics.

Physics of thin sheets

•

Bending is “easy”

•

Stretching is “hard”

Theorema Egregium III

Mapping the Hyperbolic plane

Flowers and hyperbolic

Geometry

Every layer of cells creates

a “daughter” layer with more cells

x

y

u

v

w

Φ

Elastic energy of a thin sheet

E

=

!

!

γ

!

2

+

"

2

!

κ

!

2

κ

= ˆ

n

·

D

2

Φ

γ

= (

D

Φ

)

T

·

D

Φ

−

g

N

(

p

)

·

d

r

(

p

) = 0

I

≡

ds

2

=

d

r

(

p

)

·

d

r

(

p

)

Geometry: The Gauss Normal map

p

The prescribed Gauss curvature equation

Theorema Egregium:

I

=

g

)

det(

II

)

det(

I

)

=

K

[

g

].

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d

⌦

=

KdA

=

Kd

(

i

N

(

dV

))

.

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In (Eulerian) coordinates:

w

xx

w

yy

w

xy

2

(1 +

|

r

w

|

2

)

2

=

K

.

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K <

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0 gives a Hyperbolic Monge-Ampere equation.

Local models:

u

xx

u

yy

u

2_xy

=

±

1. Solutions:

u

=

1₂

[

x y

]

Q

[

x y

]

T

with

det(

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Q

) =

±

1.

Hyperbolic surfaces: A quadratic saddle

These surfaces are “doubly ruled”.

w

=

1

2

(

ax

2

+ 2

bxy

+

cy

2

) =

Negatively curved sheets: Disk geometry

Small slopes approximation:

det(

⇥⇥

w) = 1

Solutions:

w

=

1

2

⇣

ax

2

y

a

2

⌘

.

w

= 0 for

y

=

±

ax

. Pick

a

= cot(

/n

).

Resulting surface is C

1,1

. Piecewise smooth and patched across

“Lines of inflection”.

Defects : Scales of smoothness.

Fold or corner

Inflection

Flat sheets

Hyperbolic sheets

Crumpling

Undulating

Lipschitz (C0,1) but not di↵erentiable (C1).

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Origami vs. “hyperbolic origami”

Edges and corners.

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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 w0

2(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}

Topology: Index of a branch point

Consequences for Numerics?

Energy gap?

Branched isometries cannot

be approximated by smooth

Geodesics and Asymptotic lines: C

2

surfaces

Quad-Graphs: Geometry of Characteristics

The idea of discretizing hyperbolic surfaces through asymptotic

parameterization is very fruitful.

Sauer, Wunderlich, Rozendorn, Efimov, Bobenko,

Pinkall,Toda, Dorfmeister, Brander…

Smooth saddle

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Monkey saddle

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Branched isometry of a hyperbolic disk

Nudibranch “Spanish dancer”

Video used with permission from the copyright holders: Wavelength Reef Cruises, Port Douglas, Australia

Ru

ffl

ed

A spinning “hyperbolic coin”

The formulation of the mechanics requires 3 natural frames!

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Three frames are better than two

Dynamics is driven by a Goldstone mode

Asymptotic

)

Lagrangian:

(

x

+

iy

) =

e

i⌦t

(

u

+

iv

)

Asymptotic

)

Eulerian:

(

X

+

iY, Z

) =

e

i

↵

t

(

u

+

iv

)

,

1

2

(1 +

uv

)

Exotic continua

Hyperbolic thin sheets are prototypical

Exotic continua

–

local signatures of non-trivial geometry/topology

in

defect structure

.

Strongly nonlinear and spatially inhomogeneous response to stress

Combinatorial non-uniqueness/degeneracy in undulating 3D morphologies for a

given intrinsic hyperbolic geometry –

novel Statistical mechanics

Degeneracy of low energy states and responsiveness to stresses –

potential

bio-mechanical

and

technological

applications

Thank you for your attention!

John Gemmer

Toby Shearman

Ken Yamamoto