# Deflection of a membrane

I am currently working on a project which is described as the deflection of a circular membrane. What I am trying to model is the deflection of a piece of plastic film (E=200MPa,v=0.5) when placed under pressure. I would like some recomendations on the initial displacement field (w(r)) and the horizontal displacement field (u(r)). Keeping in mind that this membrane is not of a small size, but can vary from 5 cm upto 20 cm in diameter. It has a thinkness of 0.05 mm.

Queit honestly I am not exactly sure what the difference is between theory of using a plate to model this and using a membrane.

I hope to start a discussion this way and receive some input with which I can further my study. Experimental investigation has been completed however the theoretical model is still in progress and I need some help.

I hope to hear more soon! If anything is unclear or I have forgotten something, please let me know.

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If you are talking about small displacements, where $\sin\theta\approx\theta$ is a reasonable approximation for the angle from horizontal, then I suggest looking at the linear model. Start with the one-dimensional case, which is a string suspended between two points, uniform load $p$ force/length, and tension $T$. Then a small piece $\Delta x$ long has vertical force $p\Delta x$ due to the load, and on the left end $T\sin\theta\approx T\tan\theta= T\frac{dw}{dx}$, and same expression with opposite sign on the right end. The vertical force balance is $$-T\frac{dw}{dx}(x)+T\frac{dw}{dx}(x+\Delta x) = p\Delta x$$ Divide by $\Delta x$ and take the limit to get $$Tw''=p$$ Once you are ok with this, you can do the membrane version, which requires Green's theorem to work out, but the physical ideas are the same, giving $$\nabla^2 w = p/T$$The difference between membrane and plate is that a membrane does not have any resistance to bending. The linear plate equation involves fourth derivatives.