**PINN with Data
**

**Solid Mechanics Example
**

By Prof. Seungchul Lee

http://iai.postech.ac.kr/

Industrial AI Lab at POSTECH

http://iai.postech.ac.kr/

Industrial AI Lab at POSTECH

**Table of Contents**

- 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 (FÃ¶pplâ€“von KÃ¡rmÃ¡n 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 $x = 0,\, y = 0\; (B.C.â‘ , B.C.â‘¡)$:

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

- Two free boundary conditions at $y = \omega / 2\; (B.C.â‘¢)$:

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

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

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

- Make a neural network and loss funcitons like below:

- 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