Kirchhoff–Love plate theory

Definition

Let a plate occupy $\Omega \times (-h/2, h/2)$ with $\Omega \subset \R^2$. Let it be made of an isotropic linear elastic material with Young's modulus $E$ and Poisson's ratio $\nu$. Indices $\alpha, \beta$ run over $1, 2$, and repeated indices are summed. Kirchhoff–Love plate theory is the plate theory with these assumptions.

1. The displacement is $u_\alpha(x, z) = u^0_\alpha(x) - z \, \partial_\alpha w(x)$ and $u_3(x, z) = w(x)$ for $x \in \Omega$ and $\abs{z} < h/2$.
2. The stress is in a state of plane stress parallel to the mid-plane.
3. The bending moments are $M_{\alpha\beta} = \int_{-h/2}^{h/2} z \, \sigma_{\alpha\beta} \, dz$, which gives $$M_{\alpha\beta} = -D \left[ (1 - \nu) \, \partial_\alpha \partial_\beta w + \nu \, \delta_{\alpha\beta} \, \Delta w \right], \qquad D = \frac{E h^3}{12 (1 - \nu^2)}.$$
4. Under a transverse load $q$ per unit area, equilibrium reads $\partial_\alpha \partial_\beta M_{\alpha\beta} + q = 0$.