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:
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:
$$
U= frac 12int(nablavec u)^2dA,
$$
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
asked Nov 19 at 14:56
minmax
48518
add a comment |
up vote
2
down vote
favorite
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:
$$
U= frac 12int(nablavec u)^2dA,
$$
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
asked Nov 19 at 14:56
minmax
48518
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
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:
$$
U= frac 12int(nablavec u)^2dA,
$$
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
asked Nov 19 at 14:56
minmax
48518
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:
$$
U= frac 12int(nablavec u)^2dA,
$$
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
asked Nov 19 at 14:56
minmax
48518
asked Nov 19 at 14:56
minmax
48518
asked Nov 19 at 14:56
minmax
48518
asked Nov 19 at 14:56
minmax
48518
asked Nov 19 at 14:56
minmax
48518
48518
add a comment |
add a comment |
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