Dislocation
Isometric immersions, geometric incompatibility and
strain induced shape formation
IMA Hot topics workshop
May 20, 2011
Shankar Venkataramani
Eran Sharon
Crumpled Paper
x
y
u
v
w
Φ
Elastic energy of thin sheets
κ
= ˆ
n
·
D
2
Φ
γ
= (
D
Φ)
T
·
D
Φ
−
I
E
=
!
!
γ
!
2
+
"
2
!
κ
!
2
(b)
(a)
a
a
Thin sheet clamped to a frame
•
l
= width of sheet.
•
σ
= thickness of sheet.
No stretching
E
s= 0
, E
b∼
σ
2l
a
.
With Stretching
E
s=?
, E
b=?
.
Minimal ridge revisited
Conti and Maggi, 2008.
2
!
w
x
"
(
x
)
Stretching ridges
E
b
∼
σ
2
!
1
w
"
2
lw
E
s
∼
#
w
l
$
4
lw
E
s
E
b
5
∼
C
E
s
+
E
b
∼
σ
5
/
3
l
1
/
3
Tom Witten, Alex Lobkosvky
(b)
(a)
a
a
Comparing the energies
E
s
+
E
b
∼
σ
2
l
a
E
s
+
E
b
∼
σ
5
/
3
l
1
/
3
K
∼
1
/r
E
b
∼
σ
2
log(
R/a
)
E
s
∼
a
2
a
∼
σ
E
∼
σ
2
log(
R/σ
)
a
∼
e
−
(R/σ)
1/3
Optimizing
Crossover
Vertex Energy (Cone Scaling)
a
Lower and Upper bounds
For the elastic energy in the small slopes approximation:
Lemma:
The stretching and the bending energies satisfy the inequality
E
sE
b5≥
C
!
1
−
C
!"
aE
bσ
2α
2l
#3
$2
and the upper bound
E ≤
min
!
C
α
7/3σ
5/3l
1/3"
1 +
#
σ
l
$
1/3log
+!
l
a
%&
, C
!α
2σ
2l
a
%
Theorem:
The elastic energy in the small slopes approximation of a sheet
clamped to a frame with smooth “corners” on a scale
a
satisfies the lower bound
E
≥
min
!
c
α
7/3σ
5/3l
1/3"
1 +
#
σ
l
$1
/3log
+!
l
a
%&
, c
!α
2σ
2l
a
%
Equipartition
Virial Theorem:
E
b
≈
5E
S
Local version
w
yy
≈
w
4
x
4
−
w
xx
Pointwise bounds on the ridge
Theorem 2.
[V.]
If the elastic energy
E
(
u, v, w
)
≤
E
¯
α
7
/
3
σ
5
/
3
l
1
/
3
for a configuration
(
u, v, w
)
,
∃
C
1
and
C
1
!
that only depend on
E
¯
such that
ρ
(x)
≤
σ
1
/
3
l
−
2
/
15
C
1
(l
2
−
x
2
)
2
/
5
+
C
1
!
a.
•
Ridge
necessarily
has non-trivial structure along its length.
•
Basic tool – bound the stretching energy using length scales
ρ
(
x
)
and
w
(
x,
0)
at a given point
x
.
•
Pinching argument
.
Multiple scale buckling
The observed curvature of the arcs when they are flattened is called the geodesic curvature along these lines— another property controlled by their metric. An important observation is that the geodesic curvature along the edges of the wavy leaf in Figure 8 is nearly constant. We do not see any big variations in this curvature that are correlated either with the vein struc-ture or with the waviness of the leaf. The tissue along the edge grew nearly uniformly, the growth law was uni-form, and the leaf grew as a simple leaf. Like the plastic sheets, it should have been a simple featureless leaf, but because of the geometrical limitations of space, it was forced to break the symmetry and to adopt a wavy shape.
Wrapping Up
Flowers, like leaves, form complex buckled shapes. Geometrically, the main difference between the two is that
leaves form essentially from long, free-standing strips, whereas flowers have more complex geometries; the central tube of a daffodil, for example, closes on itself like a cylinder. What happens to such a cylinder or tube when we ap-ply to it a metric that increases toward its edge? Just as the leaf grows from the center, we can think about “growing” such cylinders starting from a ring of cells and adding rings on top of one another. If the rings all have the same number of cells, they will have the same diameter and will form a cylin-der. However, as the number of cells that form a ring grows exponentially upward, the metric of the cylinder in-creases also, leading to an increasing diameter of the cylinder in its upper part and to a trumpet-like shape.
As the metric of the flower increases, the edge of the flower splays outward more and more. Eventually, it splays out so much that the edge of the flower
is perpendicular to the direction of the stem along which it is growing. It forms a circle with a radius we’ll call R. That marks the end of this phase of flower growth. If cells continue to at-tach to the end of the flower, causing the metric to grow at an ever-steeper rate as the flower grows sideways, the perimeter of the edge will have to be longer than 2!R. This is known to be impossible in our Euclidean space without breaking the axial symmetry. The edge of the flower must buckle.
In Figure 9a we show the result of an experimental study using thin tubes made of polyacrylamide gel. This gel changes its volume depending on its environment. It swells in water, but shrinks in acetone. We used this proper-ty to change the metric of the tube. First, we dipped the tube in acetone, causing it to shrink uniformly. Next we dipped one end of the tube in water, allowing the water to diffuse into the tube. As a
2004 May–June 259 www.americanscientist.org
© 2004 Sigma Xi, The Scientific Research Society. Reproduction with permission only. Contact [email protected].
Figure 7. Can a leaf that is normally flat be induced to become wavy? Here the growth hormone auxin is applied to the edge of a normally flat leaf from an eggplant, causing enhanced growth only near its margins. This growth imposes a negative Gaussian curvature on the leaf, simi-lar to that in the torn plastic sheets in Figures 5 and 6. After 10 days of such a treatment, waves have developed; at 12 and 14 days the waves have grown bigger, and waves within waves become discernible. (Photographs courtesy of the authors.)
after 12 days after 14 days
before after 10 days
Images Courtesy Eran Sharon
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
−
βy
cos(
β
x
)
,
f
(
y
) = 1 +
A
2
4
e
−
2
βy
w
=
Ay
−
α
φ
α
(
x
)
,
f
−
1
∼
y
−
2
α
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
xx
w
yy
−
w
xy
2
=
−
f
!!
(
y
) on a domain
S
1
times any closed interval on which
f
!!
is bounded.
The solutions are constructed by a contraction mapping argument.
No refinement!
The product solutions are “stable”
Rigidity of smooth solutions
Theorem:
If
w
(
x, y
) is a smooth solution of the Hyperbolic Monge-Ampere
equation
w
xx
w
yy
−
w
xy
2
=
−
f
!!
(
y
), then there exists a
C
1
vector field
v
=
−
w
xy
w
yy
2
f
!!
(
y
)
∂
∂
x
+
∂
∂
y
whose integral curves foliate the domain and are transversal to the curves
y
=
constant. Further,
v
(
w
xx
) =
w
xy
w
xyy
−
w
yy
w
xxy
2
β
2
(
y
)
w
xx
=
α
(
x, y
)
w
xx
,
where
α
is a continuous function.
The lines of inflection cannot bifurcate
Full “nonlinear” geometry
For any bounded domain, there will exist isometric immersions for
negatively curved surfaces.
Shikin and Poznyak, Han and Hong.
Question:
Do there exist immersions of the metric
g
=
dy
2
+ (1 +
f
(
y
))
2
dx
2
into R^3 which have the form
u
=
x
+
ξ
(x, y
), v
=
ζ
(x, y), w
=
η
(x, y) where
ξ
,
ζ
and
η
are periodic in the
first argument.
If
f
(
y
)
∼
!
2
e
−
2
βy
, there exist order by order isometric immersions with this
property for all orders.
Eran Sharon and S.V.
Negatively curved sheets: Disk geometry
Small slopes approximation:
det(
∇∇
w
) =
−
1
Theorem:
(J. Gemmer, S.V.) The elastic energy, in the small slopes
approx-imation, of the optimal
n
-periodic configuration of a hyperbolic disk satisfies the
inequalities
Hyperbolic disks: Full geometry
Chebyshev net:
g
=
du
2
+ 2 cos(
φ
(
u, v
))
dudv
+
dv
2
Sine-Gordon equation:
∂
2
∂
u∂
v
φ
=
κ
2
sin(
φ
)
Quantitative Hilbert’s theorem:
Hilbert’s theorem
: There is no smooth (or
C
3
) isometric immersion of
H
2
into
R
3
.
Holmgren, 1904
A
=
{
φ
: [0
,
1]
×
[0
,
1]
→
R
|
φ
uv
=
λ
sin
φ
}
φ
p
= arg min
φ
∈A
!
cot
2
φ
!
p
φ
∞
(
u, v
) =
f
(
u
+
v
)
min
max
cot
2
φ
=
1
64
exp[2
√
Hyperbolic disks: Full geometry
By allowing non-smooth embeddings, we can decrease the
curvature, and the energy.
It is often enough to allow points where the embedding is not C^2,
but is still C^{1,1}.
Pogorelov, Zalgaller, Shikin, Poznyak,...
The key (and obvious) point here is that the limit functional ¯
E
is independent
of the small parameter
!
.
Γ
–convergence
•
Convergence of functionals implies convergence of minimizers.
Γ
−
lim
!
→
0
E
!
= ¯
E
, implies that ¯
E
has a minimizer ¯
u
which is a good
approx-imation to the minimizer(s)
u
!
of
E
!
for small
!
.
(
X, d
) is a metric space and
E
!
,
!
>
0 is a family of functionals defined on
(subsets) of
X
. Then the family
E
!
Γ
converges to ¯
E
if
•
x
!
→
x
implies that ¯
E
(
x
)
≤
lim inf
!
→
0
E
!
(
x
!
).
•
For all
x
∈
X
, there is are sequencec
x
n
→
x,
!
n
→
0 such that
E
!
(
x
!
)
→
¯
Problems with multiple small parameters
Find
u
!
,
δ
that minimizes
E
!
,
δ
[
u
] =
!
F
1
[
u
] +
δ
F
2
[
u
].
Even if
F
1
and
F
2
are convex and have smooth unique minimizers,
generi-cally,
minimizer of
F
1
= lim
!
→
0
δ
lim
→
0
u
!
,
δ
!
= lim
δ
→
0
!
lim
→
0
u
!
,
δ
= minimizer of
F
2
•
Di
ff
erent scaling regimes:
!
!
δ
,
!
∼
δ
and
!
#
δ
.
•
The behavior of the minimizers is di
ff
erent in these regions.
Problems with multiple small parameters
•
The general situation is when
E
is non-convex, and the variational limits
for
F
1
and
F
2
have singular minimizers that live on di
ff
erent function
spaces.
•
Possibility of multiple scaling regions and “global bifurcations” where the
character of the minimizers changes dramatically as the parameters are
varied.
•
A single limiting function,
independent of the small parameters
, cannot
Rigorous “multi-parameter” scaling laws
E
!
,
δ
is defined on an admissible set
A
!
,
δ
. A rigorous scaling law
E
!
,
δ
∼
E
(!
,
δ)
is given by a function
E
(!
,
δ) such that:
•
Ansatz free lower bounds:
There is a constant
c >
0 such that for all
admissible test functions
u
∈
A
!
,
δ
, we have
E
!
,
δ
[u]
≥
cE
(
!
,
δ
)
E
!
,
δ
[
u
∗
]
≤
CE
(!
,
δ
)
•
Upper bounds:
There is a constant
C >
0 such that for some admissible
test functions
u
∗
∈
A
!
,
δ
, we have
Rampant speculation
Image courtesy U. Md.
Chaos Group
Crumpling experiments:
Future directions
•
Multi-parameter scaling laws – use for rigorous derivation of reduced
mod-els which are valid in multiple scaling regimes.
•
Analysis of solutions of the equation given by “appproximate” local
equipar-tition. Does this yield test function satisfies the scaling law?
•
Use this to define low-energy states, dynamics, entropy?
•
Borrow methods from dynamical systems – Dimension spectra for
mea-sures, for studying global bifurcations in variational problems.
