1. Thin Plate (Solid Mechanics)¶

1.1. Problem Setup¶

We will solve thin plate equations to find displacement and stress distribution of thin plate

Based on Kirchhoff-Love plate theory, three hypotheses were used
- straight lines normal to the mid-surface remain straight after deformation
- straight lines normal to the mid-surface remain normal to the mid-surface after deformation
- the thickness of the plate does not change during a deformation

A non-uniform stretching force is applied to square elastic plate

Only one quarter of the plate is considered since the geometry and in-plane forces are symmetric (yellow domain)

Problem properties

$$ E = 50 \operatorname{Mpa}, \quad \nu = 0.3, \quad \omega = 20 \operatorname{mm}, \quad h = 1 \operatorname{mm}, \quad f = 1 \operatorname{Mpa} $$

Governing equations (Föppl–von Kármán equations) for the isotropic elastic plate:

$$ \begin{align*} &{E \over 1 - \nu^2}\left({\partial^2 u \over \partial x^2} + {1 - \nu \over 2}{\partial^2 u \over \partial y^2} + {1 + \nu \over 2}{\partial^2 v \over \partial x \partial y} \right) = 0\\\\ &{E \over 1 - \nu^2}\left({\partial^2 v \over \partial y^2} + {1 - \nu \over 2}{\partial^2 v \over \partial x^2} + {1 + \nu \over 2}{\partial^2 x \over \partial x \partial y} \right) = 0 \end{align*} $$

Two Dirichlet boundary conditions at $x = 0,\, y = 0\; (B.C.①, B.C.②)$:

$$ v(x,y) = 0 \qquad \text{at} \quad y = 0\\\\ u(x,y) = 0 \qquad \text{at} \quad x = 0 $$

Two free boundary conditions at $y = \omega / 2\; (B.C.③)$:

$$ \sigma_{yy} = 0,\quad \sigma_{yx} = 0 $$

Free boundary condition and in-plane force boundary condition at $x = \omega / 2\; (B.C.④)$:

$$ \sigma_{xx} = P \centerdot h,\quad \sigma_{xy} = 0 $$

Make a neural network and loss funcitons like below:

1.2. Numerical Solution¶

Numerical solution of this problem is illustrated in below figures
- $x, y$ direction displacement and stress $u$, $v$, $\sigma_{xx}$, $\sigma_{yy}$, respectively

Solve this problem using PINN and then compare with a numerical solution