# Stretched elastic band shape

I have written a python code to obtain the minimum energy of a network of beads connected to one another via springs (all with elastic constants $$k=1$$). The rest length of each spring is zero, to simplify the code. The initial shape of the band has rectangular with a triangular mesh. Each bead is connected to 6 nearest neighbors, except for those at the lateral borders (mostly with 5 nearest neighbors). The bottom and top lines are held fixed.

The initial configuration:

and the final configuration:

My question is: How to obtain an analytical expression for the lateral curve?

What I did was the discrete version of a homogeneous elastic band numerically. I think one should minimize the elastic energy of the homogeneous band: $$U= \frac 12\int(\nabla\vec u)^2dA,$$ where $$\vec u$$ is the displacement from equilibrium. By minimizing the elastic potential energy functional, one should get the Laplacian equation for each component of $$\vec u$$, $$\Delta\vec u=0$$ The boundary conditions are such that the bottom and top margins are held fixed (Dirichlet B.C.). I think that on the lateral margins the normal displacement (or stress) $$u_n=0$$ in equilibrium (Neumann B.C.). So we have a mixed boundary conditions problem.