I think a better way is to animate it with bones.

Create a mesh that is as simple as possible.
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[![enter image description here][1]][1]


Set the bones to match the seams of the mesh.
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[![enter image description here][2]][2]

 1. The adjustment of this parametrization of the item transform tab at this time is correct, and the adjustment is made by the correct combination of the mets.

The enlargement of the pump is done by cutting
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[![enter image description here][3]][3]

Assign vertices to the bones according to the vertices you want to move.
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[![enter image description here][4]][4]


The animation key will look like this.
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[![enter image description here][5]][5]

The trick is to composite the start and end of the animation for each bone at the same time.
This will make the bending look more natural.


On the roundness of a square
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[![enter image description here][6]][6]

I used a hevel and a WELD modifier for this video.

By the looks of it, it's working.
If you want to run it with just the armature modifier, you can use the
If you want to run with only the armature modifier, you can assign more bones for rounding between the bones, or
If you want to use only the armature modifier, you will need to assign more rounding bones between the bones or make the vertices more complex to distribute the weight.

And the final result
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[![enter image description here][7]][7]


  [1]: https://i.sstatic.net/XWyt3.jpg
  [2]: https://i.sstatic.net/rUS8D.jpg
  [3]: https://i.sstatic.net/rthsP.jpg
  [4]: https://i.sstatic.net/bYE1X.gif
  [5]: https://i.sstatic.net/P4wg1.gif
  [6]: https://i.sstatic.net/6hJVF.png
  [7]: https://i.sstatic.net/Oitbd.gif