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:

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and the final configuration: enter image description here

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.



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