 Isotropic thin-plate bending energy

For a homogeneous isotropic plate of thickness $h$, Young's modulus $E$ and Poisson's ratio $\nu$, the bending stiffness is $D=E{h}^3/[12(1-{\nu }^2)]$. Its quadratic bending energy isIt follows by integrating the strain energy density $\frac{1}{2}{\tau }_{ij}{e}_{ij}$ through the thickness: linear bending strains are proportional to $z$, and ${\int }_{-h/2}^{h/2}{z}^{2}dz=h^3/12$. Taking a first variation and integrating by parts twice gives $D{\Delta }^{2}w=0$ without applied transverse load. Clamped boundary variations have both zero value and zero normal derivative.