Stretched elastic band shape











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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:



enter image description here



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:
$$
U= frac 12int(nablavec u)^2dA,
$$
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 $vec u$,
$$
Deltavec u=0
$$
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.






pde elliptic-equations







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    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:



    enter image description here



    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:
    $$
    U= frac 12int(nablavec u)^2dA,
    $$
    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 $vec u$,
    $$
    Deltavec u=0
    $$
    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.






    pde elliptic-equations







    share|cite|improve this question
























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      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:



      enter image description here



      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:
      $$
      U= frac 12int(nablavec u)^2dA,
      $$
      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 $vec u$,
      $$
      Deltavec u=0
      $$
      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.






      pde elliptic-equations







      share|cite|improve this question













      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:



      enter image description here



      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:
      $$
      U= frac 12int(nablavec u)^2dA,
      $$
      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 $vec u$,
      $$
      Deltavec u=0
      $$
      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.






      pde elliptic-equations




      pde elliptic-equations






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