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https://docs.google.com/presentation/d/1QKmffsHjb7LEl937hDIw5NUbu5QYfLeW4eWzok8y5Ug/edit?usp=sharing

Thin sheets are omnipresent in nature, technology, and everyday life. Being much thinner in one dimension than in the other two, they exhibit an unparalleled, rich variety of deformations when subjected to external forces, spatial constraints, or intrinsic growth
They buckle easily and create striking examples of periodic, self-similar patterns along the edges of growing leaves, flowers, marine invertebrate, torn plastic sheets.

PAST

the kinds of problems i’m interested in are inspired by hyperbolic sheets we see in nature
what do we see in nature? - division process of cells along the edge naturally results in hyperbolic surfaces when growing due to this differential growth
thin sheet is very hard to stretch but very easy to bend - nondimensional measure of how easy to bend compared to stretching is given by the ratio of the thickness to the other lengths in the problem - if this ratio is small than it’s much easier to bend
rich variety of deformations when subjected to external forces, spatial constraints, or intrinsic growth
understand the interesting mechanical properties of these systems and perhaps exploit them to engineer active/smart/programmable materials and structures that predictably and functionally respond to various stimuli - optimal design for self-propelling sea-slug-like soft robots

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From Sea Slugs to Robots: Computation, Discrete Geometry, and Soft Mechanics in Non-Euclidean Elasticity

Ken Yamamoto^a and Shankar Venkataramani^b

^a Department of Mathematics, Southern Methodist University

^b Department of Mathematics and Program in Applied Mathematics, University of Arizona

USNC TAM 2022

#M323: Soft Matter Mechanics, Physics, and Devices

June 22, 2022

Supported by NSF RTG Grant DMS-184026

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Introduction and Motivation

Thin sheets are omnipresent in nature, technology, and everyday life. Being much thinner in one dimension than in the other two, they exhibit an unparalleled, rich variety of deformations when subjected to external forces, spatial constraints, or intrinsic growth
They buckle easily and create striking examples of periodic, self-similar patterns along the edges of growing leaves, flowers, marine invertebrate, torn plastic sheets.

PAST

the kinds of problems i’m interested in are inspired by hyperbolic sheets we see in nature
what do we see in nature? - division process of cells along the edge naturally results in hyperbolic surfaces when growing due to this differential growth
thin sheet is very hard to stretch but very easy to bend - nondimensional measure of how easy to bend compared to stretching is given by the ratio of the thickness to the other lengths in the problem - if this ratio is small than it’s much easier to bend
rich variety of deformations when subjected to external forces, spatial constraints, or intrinsic growth
understand the interesting mechanical properties of these systems and perhaps exploit them to engineer active/smart/programmable materials and structures that predictably and functionally respond to various stimuli - optimal design for self-propelling sea-slug-like soft robots

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Introduction and Motivation

E. Sharon, B. Roman, and H. L. Swinney. Geometrically driven wrinkling observed in free plastic sheets and leaves. Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), 75(4):046211, 2007.

Thin sheets are omnipresent in nature, technology, and everyday life. Being much thinner in one dimension than in the other two, they exhibit an unparalleled, rich variety of deformations when subjected to external forces, spatial constraints, or intrinsic growth
They buckle easily and create striking examples of periodic, self-similar patterns along the edges of growing leaves, flowers, marine invertebrate, torn plastic sheets.

PAST

the kinds of problems i’m interested in are inspired by hyperbolic sheets we see in nature
what do we see in nature? - division process of cells along the edge naturally results in hyperbolic surfaces when growing due to this differential growth
thin sheet is very hard to stretch but very easy to bend - nondimensional measure of how easy to bend compared to stretching is given by the ratio of the thickness to the other lengths in the problem - if this ratio is small than it’s much easier to bend
rich variety of deformations when subjected to external forces, spatial constraints, or intrinsic growth
understand the interesting mechanical properties of these systems and perhaps exploit them to engineer active/smart/programmable materials and structures that predictably and functionally respond to various stimuli - optimal design for self-propelling sea-slug-like soft robots

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Hyperbolic art from the “The Crochet Coral Reef Project” (M. Wertheim and C. Wertheim) and jewelry from “Floraform” (J. Louis-Rosenberg. http://n-e-r-v-o-u-s.com/blog/?p=6721)

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Dynamics: Morphogenesis, Biomechanics, and Shape Control

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Dimensionless energy functional with gravity

Differential growth encoded by 2-D Riemannian reference metric g̃ defined on mid-surface Ω ⊂ ℝ²
Equilibrium conformation of the sheet is immersion F : Ω → ℝ³ that minimizes energy functional

Y = Young’s modulus
t = thickness of sheet
R = radius/length of sheet
L = vertical deflection of sheet
⍴ = density of sheet
g ∼ 10 m/s² is standard acceleration due to gravity
H = mean curvature
K = Gaussian curvature

August Föppl and Theodore von Kármán. 1910.

E. Efrati, E. Sharon, and R. Kupferman. 2009.

we can idealize a thin sheet as a two-dimensional object - we encode that fact that the sheet is non-flat by a non-flat reference metric on the two-manifold
think of deformation that we see of these hyperbolic sheets as embeddings F from R^2 to R^3 that minimize an elastic energy
the elastic energy has 3 contributions
stretching penalizes differences in the metric computed from the immersion from the reference metric

foppl and von Karman looked at energy of flat plates
efrati et al extended the theory to non-flat plates

bending penalizes curvature and gravity we’ll use the potential energy of the vertical deflection
strain is difference from induced metric of immersion into R^3 and the intrinsic metric - this results in some quadratic form Q(gamma) - deviation from target metric - for thin sheets buckling would be preferred over stretching
for bending, the natural quadratic form turns out to be this quantity which is the sum of the square of the two principal curvatures
gravity is included as a prototypical example of a weak external force

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Physical scales and units

Y ∼ 10⁶
Y(t/R)² ∼ 10³
𝞺gL ∼ 10²

Y ∼ 10⁶ Pa
t ∼ 10^-3 m
R ∼ 3×10^-2 m
L ∼ 10^-2 m
𝞺 ∼ 10³ kg/m³
g ∼ 10 m/s²

Stretching is dominant.

Bending and Gravity are comparable.

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Constrained variational problem

Minimize bending and gravity subject to zero-stretching constraint.
For zero-stretching, the intrinsic and extrinsic Gauss curvatures must equal, by Gauss’s Theorema Egregium. So, we constrain ourselves to immersions that satisfy zero-stretching, i.e., isometric immersions.

Note: There are different asymptotic regimes for the problem. We assume the external force is weak and both bending/gravity scale similarly. This is more appropriate for free rather than stamped sheets. When external forces are strong, e.g., in stamping, there is a different asymptotic regime that looks at so-called asymptotic rather than exact isometries (B. Davidovitch, I. Tobasco et al.).

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Föppl-von Kármán ansatz

Zero-stretching constraint PDE: Monge-Ampère Equation

Small-slopes approximation for the immersion F : Ω⊂ ℝ² → ℝ³

Monge-Ampère Equation

Say we take the sheet to have constant intrinsic (i.e., target) Gauss curvature

K(g̃) = –ε²

Then, the zero-stretching constraint is...

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Föppl-von Kármán ansatz

Scaled dimensionless energy functional

Small-slopes approximation for the immersion F : Ω⊂ ℝ² → ℝ³

Bending

Gravity

Minimize this energy functional over isometric immersions

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Gravity vs Bending Dominant Scaling & C^1,1 vs C²

Mechanics of Static Hyperbolic Sheets

Orange = Same angle for up/down sectors | Blue = Optimized angular ratio

There is a significant energy gap between C^1,1 vs C²

surfaces in the gravity-dominant regime.

Branch points allow for dramatic decreases in gravity energy.

Gravity Dominant

Bending Dominant

1.01

1.12

1.19

1.78

1.47

1.85

2.04

1.97

Up/Down Angular Ratio

all the curves converge to the same smooth saddle in the bending-dominant regime
in gravity-dominant regime, the sheet wants to lower it’s center of mass, e.g., flatten - but the smoothness of the isometric immersion greatly affects/limits how much the sheet can do that
as one might expect, relaxing the smoothness requirement expands the possible immersions to include those that can better minimize the energy, especially the gravity in the gravity-dominant regime
so smoothness matters - smoother class of isometries have more gravity energy than a slightly less smooth class of isometries - take advantage of mild non-smoothness to decrease the gravity energy

more

in gravity dominant there are non-smooth branch points that make only c1,1 solutions that cannot be approximated by c2 - c2 is a good approximation of minimizers in bending dominant regime
surfaces are floppy due to branch point singularities
allowance of c1,1 defects expands the space of possible geometries greatly
furthermore, many (even significantly different) c1,1 geometries turn out to be energetically similar - that is, we can introduce and move branch points to cause large changes in geometry without much change in energy
in particular, weak external forces are already strong enough to easily generate and manipulate these geometric defects since it costs relatively little energy to do so
this is the essence of floppiness, which is characterized by a relatively flat energy landscape, where large changes in geometry are possible with little changes in energy
hence, inherent floppiness of non-Euclidean elastic sheets is governed by and may, in turn, be quantified by these localized geometric defects

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Review: Smoothness classes

C²

Twice differentiable

Second derivative is continuous

C^1,1

Second derivative may be discontinuous but it’s bounded

C^0,1

Not even once differentiable

y = x²

y = |x| x

y = |x|

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Zero-stretching constraint PDE: Monge-Ampère Equation

Solutions are...

Monge-Ampère Equation

Say we take the sheet to have intrinsic (i.e., target) Gauss curvature K(g̃) = -1.

Then, the zero-stretching constraint is...

quadratic, saddle-shaped, and doubly-ruled surfaces.

such that

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Construction of C^1,1 Isometries

Red lines are inflection/asymptotic lines

Intersection is branch point

Cut and glue quadratic, saddle-shaped, and ruled surfaces

Gemmer et al. 2009

This surface is C²

This surface is C^1,1

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C^1,1 Isometries: Single branch point at origin

𝜃_+/-=𝜋/3 ⇒ number of waves = 6

𝜃_+/-=𝜋/4 ⇒ number of waves = 8

𝜃_+/-=𝜋/2 ⇒ number of waves = 4

This surface is C²

These surfaces are C^1,1

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Numerical investigation: Single branch point at the origin

Minimal Energy

Red = Total Energy

Blue = Bending

Green = Gravity

λ/R²= 0.0003 ⇒ Lowest energy = 26 wrinkles

Bending

Gravity

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Also Optimize Up/Down Angular Ratio

18 wrinkles

Total Energy = 0.4288

Bending = 0.1618 | Gravity = 0.2770

16 wrinkles

Up/Down angular ratio = 1.93

Total Energy = 0.4205

Bending = 0.1419 | Gravity = 0.2786

λ/R²= 0.0016

UPWARD Sectors LARGER due to GRAVITY

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Also Optimize Up/Down Angular Ratio

Orange = Same angle for up/down sectors | Blue = Optimized angular ratio

Tune Up/Down Angular Ratio to avoid additional Wrinkles

UPWARD Sectors LARGER due to GRAVITY

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C^1,1 Solutions to No-Stretching Constraint PDE: Multiple, offsetted branch points

Gemmer et al. 2016

Branch points

Inflection/Asymptotic lines

Graph Degree = 6

(6 asymptotic lines extend from it)

there’s nothing special about the origin - i go to any other point and do the same process relative to the local tangent plane and call that the origin - so i can do this multiple times - cut a piece out of a sector and do essentially what i did before with odd reflections

more

has tangent plane everywhere - further, the tangent is lipschitz - but the surface is not twice differentiable - since you’re patching a surface that curves up with one that curves down, the second derivative is not continuous although it’s bounded
the c1,1 surfaces are isometries because they are isometries on each patch and across the lines of inflection the metrics agree
general message is that we can make a new class of non-smooth isometric immersions - but, it’s a hard a combinatorial problem with two pieces - enumerate all the possible quadmeshes that give different topologies (and there’s a lot of them!) and there’s geometric data on top of that

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Optimize 6 Degrees of Freedom

Single Origin Branch Point

16 wrinkles

Energy = 0.4205

UPWARD Sectors LARGER due to GRAVITY

NO Branch Points placed in DOWNWARD Parent Sectors

5 Distributed Branch Points

16 wrinkles

Energy = 0.4004

λ/R²= 0.0016

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Gravity vs Bending Dominant Scaling & C^1,1 vs C²

Mechanics of Static Hyperbolic Sheets

Orange = Same angle for up/down sectors | Blue = Optimized angular ratio

There is a significant energy gap between C^1,1 vs C²

surfaces in the gravity-dominant regime.

Branch points allow for dramatic decreases in gravity energy.

Gravity Dominant

Bending Dominant

1.01

1.12

1.19

1.78

1.47

1.85

2.04

1.97

Up/Down Angular Ratio

all the curves converge to the same smooth saddle in the bending-dominant regime
in gravity-dominant regime, the sheet wants to lower it’s center of mass, e.g., flatten - but the smoothness of the isometric immersion greatly affects/limits how much the sheet can do that
as one might expect, relaxing the smoothness requirement expands the possible immersions to include those that can better minimize the energy, especially the gravity in the gravity-dominant regime
so smoothness matters - smoother class of isometries have more gravity energy than a slightly less smooth class of isometries - take advantage of mild smoothness to decrease the gravity energy

more

in gravity dominant there are non-smooth branch points that make only c1,1 solutions that cannot be approximated by c2 - c2 is a good approximation of minimizers in bending dominant regime
surfaces are floppy due to branch point singularities
allowance of c1,1 defects expands the space of possible geometries greatly
furthermore, many (even significantly different) c1,1 geometries turn out to be energetically similar - that is, we can introduce and move branch points to cause large changes in geometry without much change in energy
in particular, weak external forces are already strong enough to easily generate and manipulate these geometric defects since it costs relatively little energy to do so
this is the essence of floppiness, which is characterized by a relatively flat energy landscape, where large changes in geometry are possible with little changes in energy
hence, inherent floppiness of non-Euclidean elastic sheets is governed by and may, in turn, be quantified by these localized geometric defects

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Gravity vs Bending Dominant Scaling & C^1,1 vs C²

Mechanics of Static Hyperbolic Sheets

Orange = Same angle for up/down sectors | Blue = Optimized angular ratio

Gravity Dominant

Bending Dominant

2.04

1.85

1.47

1.78

1.19

1.01

C^1,1 isometries allow for wrinkles to form in the gravity-dominant regime and substantial decreases in gravity energy.

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What is a hyperbolic metric?

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DDG: Small-slopes Isometry Constraint

Monge-Ampère Equation

Small-slopes approximation of a constant Gauss curvature metric.

This is a nonlinear hyperbolic PDE ⇒ Solve along characteristics.

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DDG: No-stretching Constraint

KEY IDEAS

Discrete versions of these equations retain compatibility at any discretization size.
Displacement boundary data along the u and v characteristics do not “talk to” each other. This allows patching together of sectors along characteristics where each sector may be constructed independently.

small-slopes isometry constraint (monge ampere) is hyperbolic, so there are 2 characteristics - along these characteristics, the pde reduces to odes - we discretize that ode to compute the x and y coordinates of the characteristics in the solution surface - then, we solve for the corresponding vertical coordinates along these characteristics using the trapezoidal rule
there is a ddg for this system where we solve odes along the u and v characteristics given boundary data - at least we’ve shown it for solving for out-of-plane displacement as a function of material coordinates (x,y) - we actually find that there’s even a ddg for the in-plane displacements as well
and there’s also a required compatibility condition that ensures that mixed partial derivatives equal to each other - in particular our discrete solution scheme should build the same surface whether we go along one side of the quadrilateral or the other
remarkable that compatibility is preserved at all discretization sizes - the method of characteristics allows solving for the solution locally in sectors
characteristics is flow of info | tracking how surface builds up using preexsting info of the already built surface | piece by piece and patching is possible by partition info flow / preserves structure that is important

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Asymptotic Skeleton: Quadmesh

Shearman 2017

Smooth Saddle (C²)

Monkey Saddle (C^1,1)

Center node is a branch point and has

Graph Degree = 6

(6 asymptotic lines extend from it)

Asymptotic skeleton describes the network of branch points and lines of inflection in potentially non-smooth hyperbolic surfaces

we’re computing on a lattice - the discrete equations relates points on quadrilaterals - so we can use any underlying quadmesh - this allows us to defined meshes with nontrivial topology - there are points that are not special with four neighbors - but this point is actually different; it has six neighbors
diff asymp meshes ⇒ diff surfaces in the continuous limit due to nontrivial topology built in to the surface
idea from hyperbolic problems that there’s not information transfer across a characteristic - so you can patch things together so long as your putting a characteristic with a characteristic
monkey saddle is not smooth because any c^2 surface can only have two local asymptotic directions - the monkey saddle has branch point with 3 local asymptotic directions
on a computer, how do you decide if something is smooth - topology of quad mesh corresponds to the smoothness of the corresponding continuous solution
(high level ddg - quads and closing/compatibility)
asymptotic/topological skeleton tells you where are the points of excess degree are and how they are distributed - then there’s also geometric boundary data that tells you how the surface is along the characteristics (i.e., asymptotic lines)
framework where we can put this branch-point structure in / whereas with other discretization we cannot do it - more faithful geometric representation / like appropriating a corner with smooth function / parsimonious - many people do other discretization / but philosophically we try to delineate and characterize branch point and asymptotic - non-smoothness is important in our problem / want discretization that allows us to characterize/quantify the branch points and non-smoothness and the degrees of the non-smoothness and we want to a way to characterize and represent them

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Evolution of Curved Asymptotic Lines in Minimal-Energy C² Sheets

Gravity Dominant

Bending Dominant

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Gravity vs Bending Dominant Scaling & C^1,1 vs C²

Mechanics of Static Hyperbolic Sheets

Orange = Same angle for up/down sectors | Blue = Optimized angular ratio

There is a significant energy gap between C^1,1 vs C²

surfaces in the gravity-dominant regime.

Branch points allow for dramatic decreases in gravity energy.

Gravity Dominant

Bending Dominant

1.01

1.12

1.19

1.78

1.47

1.85

2.04

1.97

Up/Down Angular Ratio

all the curves converge to the same smooth saddle in the bending-dominant regime
in gravity-dominant regime, the sheet wants to lower it’s center of mass, e.g., flatten - but the smoothness of the isometric immersion greatly affects/limits how much the sheet can do that
as one might expect, relaxing the smoothness requirement expands the possible immersions to include those that can better minimize the energy, especially the gravity in the gravity-dominant regime
so smoothness matters - smoother class of isometries have more gravity energy than a slightly less smooth class of isometries - take advantage of mild smoothness to decrease the gravity energy

more

in gravity dominant there are non-smooth branch points that make only c1,1 solutions that cannot be approximated by c2 - c2 is a good approximation of minimizers in bending dominant regime
surfaces are floppy due to branch point singularities
allowance of c1,1 defects expands the space of possible geometries greatly
furthermore, many (even significantly different) c1,1 geometries turn out to be energetically similar - that is, we can introduce and move branch points to cause large changes in geometry without much change in energy
in particular, weak external forces are already strong enough to easily generate and manipulate these geometric defects since it costs relatively little energy to do so
this is the essence of floppiness, which is characterized by a relatively flat energy landscape, where large changes in geometry are possible with little changes in energy
hence, inherent floppiness of non-Euclidean elastic sheets is governed by and may, in turn, be quantified by these localized geometric defects

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DDG: Small-slopes Isometry Constraint

Monge-Ampère Equation

Small-slopes approximation of a constant Gauss curvature metric.

This is a nonlinear hyperbolic PDE ⇒ Solve along characteristics.

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Isometry condition ⇒ Specific forcings as a function of the out-of-plane displacement w.
First solve for w that satisfies the isometry condition (i.e., Monge Ampère equation) ⇒

All strain entries can be made to equal zero exactly.
There exists a DDG to solve PDE system locally quadrilateral-by-quadrilateral.

In-plane Displacements: Discrete Differential Geometry

General forcing (f₁₁, f₁₂, f₂₂) ⇒ Minimization is elliptic linear elasticity problem in the in-plane displacements (ξ, η).

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In-plane Displacements: Discrete Differential Geometry

Compute in-plane displacements (ξ, η) quadrilateral-by-quadrilateral, but...

The solutions for (ξ, η) contain linear and constant terms for in-plane rotation and rigid translation. These need to be set properly for all quadrilaterals to patch together to make a continuous global surface.

Out-of-plane w

Rotate/translate each quad to patch

Compute in-plane (ξ, η) locally (quad-by-quad)

⇒

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In-plane Displacements: Discrete Differential Geometry

Out-of-plane displacement w(x,y) as a function of material coordinates (x,y)

The surface in full physical coordinates with in-plane displacements (ξ, η) applied to minimize stretching

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In-plane Displacements: Curved Asymptotic Lines

Out-of-plane w

Physical (DDG)

Physical (DSM)

fem is a one tool for all jobs | the ddg is more tailored - different methods/discretization based on preserving different properties - method respects the geometry / knows the asymptotic line / not fixed mesh for all problem | mesh changes / build surface locally rather than giant global - cannot just triangulate the domain from the beginning
while it has been shown that there exists a DDG for the full, nonlinear geometry of a hyperbolic surface, it is not assured that a corresponding DDG exists for the small-slopes projection, since some information is lost in the projection
it’s remarkable that the ddg exists in the small-slopes setting not just at the level of solving for the out-of-plane displamcents (i.e., a single monge ampere pde) but also for the in-plane displacements (i.e., a system of 3 nonlinear pdes)

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Experiments: Forces in Hyperbolic hydrogels

Thanks to Prof. Eran Sharon and Ido Levin

vertical extent of hydrogel as a function of number of wrinkles n

buckling transitions of hyperbolic sheets undergoing spatial confinement
compressed a hyperbolic sample in between two glass plates, measuring the force exerted by the sample on the plates and observing its morphological changes
each additional wrinkle, created when compressing the sample further, was accompanied by a sharp decrease in the force against the plates
nonconservative forces are significant (e.g., friction, adhesion) - there’s also probably stretching so isometries isn’t really accurate
minima force curves we expect this is correlated with where the transition occurs - this is a natural candidate
samples are not purely hyperbolic - strong compresion… is asymptotic regime is even valid? - friction and adhesion - just a start to quantify how floppy these things are
although the force curves and wrinkle transitions were not exactly the same as in the forward direction due to hysteresis and friction, repeating the experiment backwards, i.e., increasing the displacement between the plates, qualitatively reproduced the same observations in reverse: the hyperbolic gel transitioned from a plane-stress to a highly wrinkled state, before losing wrinkles one at a time, and the force trend was retraced backwards
work became even more exciting when I began to analyze and model this experiment mathematically which, in turn, prompted me to collect additional data and try other tests - his back-and-forth between theoretical and experimental investigations was a scientific thrill that I had not experienced before - collected a lot of good data that I will continue to analyze, as I seek to derive a theoretical model that reproduces the forces and buckling transitions of confined sheets
it’s thick enough to not be able to ignore stretching - but it’s thin enough that we can do no stretching with boundary layers seems promising

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Inverse Problem: Given Geometry (i.e., w)… Find λ

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Inverse Problem: MCMC Results

TRUE λ

Mean of MCMC Result

Future: Extract asymptotic skeleton from noisy profilometric data for real-world sheets.

TRUE SURFACE

MCMC RESULT

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Dynamics: Morphogenesis, Biomechanics, and Shape Control

A saddle surface whose asymptotic lines are rotating with respect to the material points satisfying a no-slip contact condition with the table below (left). The material coordinates represented by colored sectors are rotating at a slower rate than the frame in which the shape of the surface is fixed. The asymptotic frame is indicated by the white ball which rolls on the surface to remain at the minimum. A “mathematical” sea slug (right) with merging and splitting branch points is a cartoon for the motion of a true sea slug (middle).

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Inverse Problem: MCMC Results

TRUE λ

Mean of MCMC Result