I'm reading Howell's Applied Solid Mechanics to gain background for a research project. I'm struggling with the following derivation in the text that the authors refer to as a "lengthy exercise." The energy per unit area in a von Karman plate is $$\frac{D}{2}\left\{(\nabla^2w)^2-2(1-\nu)\left(\frac{\partial^2 w}{\partial x^2}\frac{\partial^2 w}{\partial y^2}-\left(\frac{\partial^2 w}{\partial x\partial y}\right)^2\right)\right\}\\+\frac{1}{2Eh}\left\{(T_{xx}+T_{yy})^2-2(1+\nu)(T_{xx}T_{yy}-T_{xy}^2)\right\}$$
where the integrated stress components are given by $$T_{xx}=\frac{Eh}{1-\nu^2}\left(\frac{\partial u}{\partial x}+\nu\frac{\partial v}{\partial y}\right)\\T_{xy}=T_{yx}=\frac{Eh}{1+\nu}\left(\frac{1}{2}\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)\right)\\T_{yy}=\frac{Eh}{1-\nu^2}\left(\nu\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)$$
The author then claims that "a lengthy exercise in the calculus of variations shows that when we consider virtual displacements $(u,v,w)$ that minimise the net strain energy in the plate, we do retrieve:" $$\frac{\partial^2\mathfrak A}{\partial y^2}\frac{\partial^2 w}{\partial x^2}-2\frac{\partial^2\mathfrak A}{\partial x\partial y}\frac{\partial^2w}{\partial x\partial y}+\frac{\partial^2\mathfrak A}{\partial x^2}\frac{\partial^2 w}{\partial y^2}-D\nabla^4w-\varsigma g=\varsigma\frac{\partial^2 w}{\partial t^2}\\\nabla^4\mathfrak A=-Eh\left\{\frac{\partial^2w}{\partial x^2}\frac{\partial^2w}{\partial y^2}-\left(\frac{\partial^2 w}{\partial x\partial y}\right)^2\right\}$$ where $\mathfrak A$ is an Airy stress function.
I'd like to add growth to the original energy per unit area and derive the analogous equations. I have very little background in this area so if someone can walk me through the derivation without growth, I think I can manage to do so when growth is present.