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I am trying to derive the "thin plate energy functional". Given a thin plate $z = z(x,y)$, how does one derive easily the energy functional

$$\iint_{\mathbb{R}^2} \,\left[\left(\frac{\partial ^2z}{\partial x^2}\right)^2+2\left(\frac{\partial^2 z}{\partial x\partial y}\right)^2+\left(\frac{\partial^2 z}{\partial y^2}\right)^2\right]\mathrm{d}x\,\mathrm{d}y\quad ?$$

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It very much depends on what you consider a "derivation" and what you consider an "easy" one. You can start assuming that the energy density depends on $(\kappa_1^2+\kappa_2^2)g^{\frac 12}$, where $\kappa_i$ are the principal curvatures, which can be considered quite natural from a physical point of view (in one dimension this corresponds to assuming that the energy density of a 1D elastic continuum depends quadratically on the curvature, and it leads to the non-linear theory of the Euler Elastica). Then you can deduce your integral functional as a linearized form of the problem, as is done in most textbooks (and also sketched in the wiki page you linked).

If instead you want more something like a "microscopic" derivation, then you'll need more sophisticated tools to make sense of the limit functional of a suitable sequence of 3D Cauchy-continuum energy functionals, when one of the dimensions goes to zero (and the coefficients are rescaled accordingly). A standard reference (in my view a very good one, as it covers the mathematical aspects in full rigor) is: Ciarlet, P. G. (1997). Mathematical Elasticity: Volume II: Theory of Plates. Elsevier.

Hope this helps.

Btw: I would not say "Given a thin plate $z=z(x,y)$", as $z$ is in fact the placement function.

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