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

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

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

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

I am trying to derive the Thin Plate Energy Functional. Given a Thin Plate $z = z(x,y)$, how does one derive the energy function easily:

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

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

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