Title page:
Hello. My name is Martin Michael Müller. I am working at the University of Lorraine in 
France. Today I would like to talk about boundary conditions on free edges of elastic 
sheets. This is work in collaboration with Jemal Guven and Pablo Vazquez-Montejo. 

Slide 1:
Originally, I come from a biophysics background with an interest in fluid membranes and 
I use differential geometry to study the shapes and mechanics of these objects. 
A while ago I wondered together with my collegues whether one can not convey the tools 
that we use to objects like elastic sheets.

We have already heard some wonderful talks on the subject in the past few weeks of this 
program and I am happy to present some of my research in this context, too. So, as you 
know, one major difference between homogeneous fluid membranes and elastic sheets is 
that elastic sheets resist to shear. They crumple and exhibit singularities.
On the slide you see some examples such as soft growing tissues and defects in paper 
sheets. 

Slide 2:
One way to learn more about the associated shapes and stresses in these objects is to 
use a continuum description and model the elastic sheet as a two-dimensional surface. 
There is an energy associated to this sheet which is a surface integral over a scalar 
energy density. For elastic sheets, there is bending and stretching. 

Slide 3:
So, let me first remind you of the bending energy density. For an up-down symmetric 
sheet the leading-order term comprises the mean and the Gaussian curvature. I use K 
which is twice the mean curvature and KG for the Gaussian curvature. The associated 
moduli are the bending rigidity and the saddle-splay modulus. In the following, I 
will scale out the bending rigidity and introduce the ratio kG of both moduli. 

We will talk a lot about these curvatures in the following so let us review quickly 
how they are defined. 

The curvature of a curve at point P in the plane can be obtained as follows: draw a 
circle whose tangent corresponds to the tangent of the curve in the point P and make 
it touch the curve as closely as possible. The curvature of the curve in this point 
is the inverse of the radius of this circle. 

Slide 4:
In two dimensions the situation is more complicated but still tractable. There exists 
an infinite amount of curvatures that you can define in one point on the surface but 
if you look at it more carefully you find that there are two so-called principal 
curvatures which are sufficient if you want to know the curvature properties of the 
given point. 

Consider a point on the surface and an arbitrary surface curve through this point. The 
curvature of the curve can then be decomposed into a part which is due to the fact 
that the surface is curved in R3 and a part due to the fact that the curve itself is 
curved in R3 as well.
The former is called the normal curvature, the latter the geodesic curvature of the 
curve. To say something about the curvature properties of the surface, we thus have to 
consider normal curvatures looking at circles which lie in planes normal to the 
surface. Now you can choose any surface curve passing through the point and to each 
curve you can find the associated normal curvature. The extremal values are the 
principal curvatures, the associated directions the so-called principal directions. 

If you take the sum of the two principal curvatures c1 and c2 you get K, if you take 
the product you get KG. In this way, one also sees directly that the bending energy 
is a term quadratic in curvatures. 

Slide 5:
Now, let us see how we can describe the geometry of our sheet. I will use a position 
vector function X which depends on the local coordinates u1 and u2. With this 
parametrization you can define a local basis: Two tangent vectors e1 and e2 which 
point along the coordinate lines and the surface normal n which is the normalized 
cross product of the tangent vectors. 

With these vectors one can define the so-called first and second fundamental form, 
the metric gab whose components can be obtained via the scalar products of the tangent 
vectors and the extrinsic curvature tensor which measures the changes of the surface 
normal along the coordinate lines. Nabla_b indicates the covariant surface derivative 
and the indices a and b will always run from 1 to 2 since we study a two-dimensional 
surface. 

BTW, whenever you see an index popping up twice, once as a subscript and once as a 
superscript, you have to sum over this index from 1 to 2. Note that the mean and 
Gaussian curvatures can be obtained with the help of the fundamental forms. Even though 
KG is expressed in terms of K and Kab which are quantities extrinsic to the surface, 
Gauss has shown us that it is a quantity which is intrinsic to the surface and solely 
depends on the metric.

Slide 6:
Having said all this, we can now return to the energy of our elastic sheet. Typically, 
the thickness h of the sheet is much smaller than its lateral extensions. When looking 
at the two terms more closely, one finds that the bending term is proportional to h^3, 
whereas the stretching term is proportional to h. This was already mentioned in some 
of the previous talks and discussions. When you send the thickness to zero, stretching 
will focus in singularities and the rest of the sheet can be modeled as an unstretchable 
sheet. 
In mathematics, you say that your surface is isometric which means that your 
metric is fixed outside of the singularities no matter what bending deformation 
you apply. This implies that the Gaussian curvature will be fixed as well.
How can this be implemented in the energy?

An idea with which we came up a long time ago, is the following: one can use a Lagrange 
multiplier Tab to fix the metric to some reference metric gab0. The Tab are local 
tensors which, as we will see, encode the tangential stresses ensuring that the surface 
is not stretched. How can one proceed further? How can we get the shapes and stresses 
in equilibrium for the sheet?

Slide 7:
Well, let us look at a variation deltaX of the position vector X. The energy H will 
respond in a certain way which we will try to determine. In equilibrium, the first 
variation of H has to be zero. In general, this variation comprises an integral over 
the bulk of the surface and a boundary term. Let's look at the bulk term first. In 
general, this term can be written as a surface divergence of the stress tensor fa 
dotted into the variation delta X. fa indicates which stresses are acting in the surface 
and can have components in the normal direction as well. 

If we decompose the variation deltaX into its tangential and normal parts for our case, 
we obtain this expression. Epsilon0 contains the terms due to bending which enter the 
normal variation whereas there are terms due to the isometric constraint in both the 
normal and tangential variations. In equilibrium, the prefactors have to be zero which 
yields the shape equations. 
This set of nonlinear partial differential equations has to be solved to determine the 
shape of the sheet.

Up to this point I could and I actually have given this talk already 13 years ago. But 
in our work of that time we never considered the boundary conditions in detail. This 
is something we only did recently and I would like to tell you a bit more about that.

Slide 8:
Let us look more carefully at the geometry of a boundary curve to say more about the 
boundary term of the energy variation. For the boundary we will use a Darboux frame 
which is adapted to the surface. The tangent T points along the boundary, the vector n 
is the surface normal, and the conormal L is perpendicular to the two. Note that 
tangential vectors of the Darboux frame and the tangential vectors e1 and e2 do not 
necessarily coincide at the boundary. One can nevertheless write L and T in terms of 
the surface frame: L = La ea and T=Ta ea, where La and Ta are the components of L and 
T in this frame.

The motion of the Darboux frame along the curve can be expressed with the help of 
three geometrical quantities. Here, ell is the arc-length of the boundary curve and 
we will use dots to indicate a derivative wrt ell. The first two quantities are the 
geodesic and normal curvatures of the curve. As mentioned before, the geodesic 
curvature kappag is an intrinsic property of the curve which will become important in 
the following. Finally, we have the so-called geodesic torsion which tells you how 
the frame rotates around the tangent T when moving along the boundary. Note that taug 
corresponds to the off-diagonal term of the extrinsic curvature tensor in the Darboux 
frame. 

Slide 9:
With the geometry in place, we can now return to the variation of the energy. 

We have shown in the past that the bondary term can be written as a combination of 
a term containing the variation of the position deltaX and a term which captures the 
variation of the surface normal at the boundary. The second term thus concerns the 
balance of torques at the boundary.

The prefactors in the variation are the stress tensor fa for which you can see the 
full expression here, whereas Hab is the functional derivate of the energy density 
wrt the extrinsic curvature tensor Kab which you can see here. Note that Hab contains 
a term due to the Gaussian curvature.

Even though I use integrals over a closed boundary in this talk, the results can also 
be applied to sheets with more complex boundaries.

Let us now decompose our variation deltaX wrt the Darboux frame. 

The resulting boundary variation contains the three projections of delta X onto the 
Darboux frame and an additional variation associated to the derivative of the normal 
variation perpendicular to the boundary curve. Note that I have used the superscript 
D to indicate that we look at components of Tab in the Darboux frame. 

Naively, the four terms of the variation can be varied independently on free 
boundaries. However, consider the balance of torques captured by K + kG kappan=0. For 
a flat sheet the Gaussian curvature is zero and the corresponding term in the energy 
vanishes. 
Our equation thus tells us that the mean curvature has to be zero at the boundary. This 
does not make sense since this together with the vanishing of the Gaussian curvature 
implies that both principal curvatures and thus all normal surface curvatures have to 
vanish at a free boundary. This cannot be true. 

Slide 10:
So, where does the problem come from? In fact, the three components of the variation 
deltaX are not independent. When doing the variation one has to take care that the 
geometrical properties of the boundary curve, which are invariant under isometries, 
are conserved. What are the properties that have to be fixed? The arc-length is 
obvious but is there anything else? It turns out that one also has to fix the geodesic 
curvature kappag of the curve to get consistent boundary conditions. 

This leads us to the idea how this can be done in practice. We gonna fix the intrinsic 
geometry of the boundary curve with the help of Lagrange multipliers, T for the 
constraint on the arc-length and Lambda for the constraint on the geodesic curvature. 
This allows to treat the three components of the variation deltaX as independent 
variables. 

We finally obtain the following boundary integral on the free edge of an infinitely 
thin sheet. Since this looks rather complicated, let me guide you through this 
expression. 

The blue terms are due to the bending term, the red ones due to the unstretchability 
of the sheet. These terms were already in place in the first naive version of the 
boundary integral. Additionally, we now have terms due to the arc-length constraint 
(in green) and the constraint on the geodesic curvature (in brown). 

Slide 11:
Absorbing the material constant k_G in our Lagrange multiplier Lambda yields the final 
set of boundary conditions on the free edge of an unstretchable sheet. For a given 
problem one can now solve the shape equations subject to these boundary conditions.

For a boundary for which the normal curvature kappan does not vanish one obtains 
Lambda as the ratio of -K over kappan. This allows to express T as a function of the 
boundary geometry via Eq. (3). Finally, we can obtain the projections of the tangential 
stresses due to isometry in the Darboux frame as well.

Let us look at a simple example: the conical geometry.

Slide 12: 
A developable cone can be expressed with the help of a position vector function given 
by a radial coordinate r times a unit vector u which moves along a closed curve Gamma 
on a unit sphere. u depends solely on the arc-length s of Gamma. The apex of the cone 
lies in the center of the sphere. The extrinsic curvature tensor in this coordinate 
system is simple and contains only one nonvanishing component, the curvature kappa 
times r. 

The shape equation simplifies as well. One obtains an differential equation which only 
depends on the arc-length s. 

Slide 13:
Let's look at the conical defect. We have heard the nice talk given by Marcelo Dias 
in the second week of our program where he has presented his results on different 
types of cones. To say something about boundary conditions, let us look at a cone 
with a circular boundary at r=R. Since the energy is rotationally invariant, Cparallel 
has to be constant as well. But this implies that one does not have to solve the 
other shape equation, the conservation law of Tab, to determine the equiblibrium 
geometry. A fact we have exploited quite heavily in the past!

Slide 14:
Now, let's be crazy and do it now: the solution of the conservation law in terms of 
the components of Tab projected onto the surface basis is comparatively complicated 
but simplifies when Cparallel is a constant. 13 years ago, we have written down 
these equations but were not able to determine the functions Cparallelperp and Cperp 
explicitly. Thanks to the boundary conditions we can do it now!

This is the full set of boundary conditions we have just derived. In our case they 
simplify quite dramatically. For a circular cone the boundary normal curvature 
kappan coincides with the curvature K (which is twice the mean curvature). 

This implies that Lambda equals -1. Moreover, K just equals kappa over r. This 
yields T equal to -1/R implying that the off-diagonal element Tperpparallel in the 
Darboux frame is zero. Tperp in the Darboux frame is simply 1/R^2 times kappa^2/2 +1. 

This allows to rewrite the expressions of Tab in the surface frame until we finally 
obtain this result. Tperp depends on the distance R of the boundary from the appex 
but simplifies to -Tparallel when the circular boundary is sent to infinity.   

 
Slide 15:
For the complete tangential stress, one has to add the bending contribution and gets 
the complete tangential stresses in the circular cone. In the case of a simple 
icecream cone, the curvature kappa is a constant and the stresses only depend 
on 1/r^2. Thus, the cone is under tension along circles of constant r and under 
equal compression along its rulings. Interestingly, the stresses do not depend on the 
deficit angle that you choose. As simple as this example might be, the 
distribution of stress could not have been determined without the correct boundary 
conditions on the rim.

Slide 16:
One can look at more complicated configurations such as cones with a non-circular 
boundary. We have done this perturbatively for a cone with an elliptic boundary. 
Already for this simple example, the determination of the shape no longer decouples 
from the boundary. However, one finds that the correction to the bulk geometry and 
to the stress distribution only depends on the elliptical boundary conditions at 
second order of the ellipticity. On the picture you see the tangential stresses 
that result. 

Slide 17:
We have introduced a framework which allows to analyze the stress distribution in 
isometric sheets. In particular, there is no need to introduce boundary layers at the 
free edges of the sheet. 

The introduction of two local constraints on the boundary curve, namely arc-length 
and geodesic curvature, ensures that boundary deformations are consistent with isometry.

The equilibrium shapes of conical geometries are remarkably insensitive to boundary 
conditions, a fact which we have used in the past. 

The framework can easily be extended to other geometries. 

In terms of questions for the following discussion I would like to reiterate the 
fact that we do not need boundary layers in this approach. I am also looking forward 
to discuss with you about the relevance of the isometric limit and any other question 
you might have. 

Slide 18:
That beeing said, I would like to again thank my collaborators on this subject, 
Jemal Guven and Pablo Vazquez-Montejo. Here, you can see us a while ago, when we 
were still innocent not worrying about boundary conditions yet. 

I would also like to thank my funding and you for your attention.