Mindlin–Reissner plate theory

Definition

Let a plate occupy $\Omega \times (-h/2, h/2)$ with $\Omega \subset \R^2$, made of an isotropic linear elastic material with Young's modulus $E$, Poisson's ratio $\nu$ and shear modulus $G$. Indices $\alpha, \beta, \gamma$ run over $1, 2$ and repeated indices are summed. Let $D = Eh^3/(12(1 - \nu^2))$ and let $\kappa > 0$ be a shear correction factor. Mindlin–Reissner plate theory is the plate theory with these assumptions, for the bending part.

1. The displacement is $u_\alpha(x, z) = -z \, \theta_\alpha(x)$ and $u_3(x, z) = w(x)$, with independent unknowns $w$, $\theta_1$, $\theta_2$ on $\Omega$.
2. The bending moments and transverse shear forces are $$M_{\alpha\beta} = -D \left[ \tfrac{1 - \nu}{2} (\partial_\beta \theta_\alpha + \partial_\alpha \theta_\beta) + \nu \, \delta_{\alpha\beta} \, \partial_\gamma \theta_\gamma \right], \qquad Q_\alpha = \kappa G h \, (\partial_\alpha w - \theta_\alpha).$$
3. Under a transverse load $q$ per unit area the equilibrium equations are $$\partial_\beta M_{\alpha\beta} - Q_\alpha = 0, \qquad \partial_\alpha Q_\alpha + q = 0.$$