Plate theory

Definition

Let $\Omega \subset \R^2$ be a bounded domain and $h > 0$ a thickness much smaller than the diameter of $\Omega$. A plate is a body occupying $\Omega \times (-h/2, h/2)$, made of a linear elastic material, with mid-plane $\Omega \times \set{0}$. A plate theory is a model of the plate in which:

1. The unknowns are fields on $\Omega$, such as the in-plane displacements $u_1, u_2$ and the deflection $w$ of the mid-plane. They may include the rotations of the normals.
2. The displacement of each point $(x, z)$ of the body, with $x \in \Omega$ and $\abs{z} < h/2$, is an assumed function of $z$ whose coefficients are these fields.
3. Integrating the stresses through the thickness gives stress resultants on $\Omega$: membrane forces $N_{\alpha\beta}$, bending moments $M_{\alpha\beta}$ and transverse shear forces $Q_\alpha$.
4. These resultants satisfy two-dimensional equilibrium equations, with a constitutive law derived from the three-dimensional one and from the assumption of clause 2.