Mathematical expression of energy storage I'm trying to develop an idea which is as follows.
Put simply, imagine a flat sheet of material which, when distorted (I.e. curved in the third dimension) stores energy. Now, by calculating the components of the Riemann Curvature Tensor to describe the actual distortion, how would you mathematically express the energy stored in the distorted area in relation to the components of the curvature tensor?
 A: I may be over-simplifying the problem a bit, but the laws of motion of a flexible sheet are very similar to that of a classical field. And the classical field is described by a Lagrangian density of 
$$\mathcal{L}(\phi;x,y,t) = \frac{1}{2} \left[\left(\frac{\partial \phi}{\partial t}\right)^2 - \left( \frac{\partial \phi}{\partial x}\right)^2 - \left( \frac{\partial \phi}{\partial y}\right)^2\right] - V(\phi)$$
up to a few constants to make units work out. And since we have no time variation (I'm assuming you're discussing the membrane not being in motion), we can assume $\frac{\partial \phi}{\partial t}$ to be 0. And we'll change the sign just to keep the math a little simpler.
So now we have a Lagrangian density of:
$$\mathcal{L}(\phi;x,y) = \frac{1}{2} \left[\left( \frac{\partial \phi}{\partial x}\right)^2 + \left( \frac{\partial \phi}{\partial y}\right)^2\right].$$
And since we're looking for energy, we'll compute a Hamiltonian density from the Lagrangian, which Should be $\mathcal{H} = \Sigma_i p_i \dot{q}_i - \mathcal{L}(\phi;x,y)$. Since we're no longer dealing with time coordinates, we replace the $\dot{q}$ by a $\frac{\partial \phi}{\partial x}$ and $\frac{\partial \phi}{\partial y}$. And our momentum will also be something similar (up to constants to fix our units). So the Hamiltonian density should be something proportional to
$$\mathcal{H} = \frac{1}{2} \left[\left( \frac{\partial \phi}{\partial x}\right)^2 + \left( \frac{\partial \phi}{\partial y}\right)^2\right].$$
I hope that helps some. I apologize for not including the constants for units, but whatever they are, they should have units of energy per unit area, since $\phi$ is essentially the elevation or depression of the membrane above its equilibrium position, and taking a derivative of position with respect to position, gives a unitless number, and this is a Hamiltonian density which must be integrated over its area.
I'm not too sure that all of that is exactly correct as far as the math goes, but it seemed like the most sensible route to me. Please comment if you have any corrections you'd apply.
