Mindlin's Plate Equations (1950s)

Mindlin's Plate Equations (1950s)

Mindlin generalized Timoshenko's Equations to 2D.

The new dependent variables are (height) and $(\psi_{x},\psi_{y})$ (angles).
plate thickness is
Can be written as a system of eight PDEs:

$\displaystyle \rho h\frac{\partial v}{\partial t}$ $\displaystyle =$ $\displaystyle \frac{\partial q_{x}}{\partial x}+\frac{\partial q_{y}}{\partial y}$

$\displaystyle \frac{1}{\kappa^{2}Gh}\frac{\partial q_{x}}{\partial t}$ $\displaystyle =$ $\displaystyle \frac{\partial v}{\partial x}+\omega_{x}$

$\displaystyle \frac{1}{\kappa^{2}Gh}\frac{\partial q_{y}}{\partial t}$ $\displaystyle =$ $\displaystyle \frac{\partial v}{\partial y}+\omega_{y}$

$\displaystyle \frac{\rho h^{3}}{12}\frac{\partial \omega_{x}}{\partial t}$ $\displaystyle =$ $\displaystyle \frac{\partial m_{x}}{\partial x}+\frac{\partial m_{xy}}{\partial y}-q_{x}$

$\displaystyle \frac{\rho h^{3}}{12}\frac{\partial \omega_{y}}{\partial t}$ $\displaystyle =$ $\displaystyle \frac{\partial m_{xy}}{\partial x}+\frac{\partial m_{y}}{\partial y}-q_{y}$

$\displaystyle \frac{1}{D}\frac{\partial m_{x}}{\partial t}$ $\displaystyle =$ $\displaystyle \frac{\partial \omega_{x}}{\partial x}+\nu\frac{\partial \omega_{y}}{\partial y}$

$\displaystyle \frac{1}{D}\frac{\partial m_{y}}{\partial t}$ $\displaystyle =$ $\displaystyle \frac{\partial \omega_{y}}{\partial y}+\nu\frac{\partial \omega_{x}}{\partial x}$

$\displaystyle \frac{2}{D(1-\nu)}\frac{\partial m_{xy}}{\partial t}$ $\displaystyle =$ $\displaystyle \frac{\partial \omega_{y}}{\partial x}+\frac{\partial \omega_{x}}{\partial y}$

with:

$\displaystyle \omega_{x} = \frac{\partial \psi_{x}}{\partial t}\hspace{0.5in}\omega_{y} = \frac{\partial \psi_{y}}{\partial t}\hspace{0.5in}v = \frac{\partial w}{\partial t}$

$\displaystyle D = \frac{Eh^{3}}{12(1-\nu^{2})}\hspace{0.5in}\nu = {\rm Poisson's ratio}$ (4)
A 2D ``parallel plate'' connected to a 5-variable system