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I'm trying to model, in MATLAB, a membrane in 2D which has an initial triangle-like shape. enter image description here

This membrane has pressure acting on it from the inside. There are also some pulling forces in random directions on the inner side. I want to analyse the behavior of this membrane over time. The way I want to model this is as follows:

I have the set of membrane points. I define the total energy of the membrane in terms of the pressure and the forces acting on it. Since this is in 2D, my total energy is actually energy per some unit length. So, the pressure energy can be modeled as $-\text{p}(\text{area under the membrane})$, and I can use trapezoidal rule to write area as $\sum_{i=1}^{n-1}(y_i+y_{i+1})(x_{i+1}-x_i)$, where $(x_i,y_i)$ is $i$th point of the membrane, and $n$ is the total number of points of the membrane. (Although it doesn't look like it, the base)

I am unaware how to model the Bending Energy of the membrane in a similar way. It has to be in the form of $x_i,y_i$ and some constant. A reference to an appropriate bending theory would also be helpful. I seem to remember some formula like $\int \chi H^2 dA$, but I can't really find a source online. Once I have the total energy of the membrane, I can solve some differential equations to get $x_i(t),y_i(t)$. How do I model the bending energy term?

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The usual definition of "a membrane" is "something sufficiently thin that the bending energy can be ignored". If you don't want to ignore it, you are talking about a "thin shell," not a "membrane".

The theory of thin shells is well known, but too long to describe in an answer here. Any good textbook on solid mechanics, or the finite element method, should have a section on the subject.

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