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benchmark_elasticity
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benchmark_elasticity
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## Description of the example
In this example, we consider a solenoid conductor with finite thickness and infinite length.
This allow us to ignore the *z* components in our equations.
We admit that there is only a radial expansion.
## Geometry
The conductor $\Omega$ consists in a rectangular cross section torus.
The geometry also contains an external domain which is an approximation of $\mathbf{R}^3/\Omega$.
| Name | Description | Value | Unit |
| -------- | -------------------------------- | -------------------------------- | ------------------ |
| $r_1$ | internal radius | $1.10^{-3}$ | m |
| $r_2$ | external radius | $2.10^{-3}$ | m |
| dz~ | height | $2.10^{-1}$ | m |
## Input parameters
current density: $\textbf{j}$ in $A/m^2$.
magnetic field: $\textbf{b}$ in $T$.
## Model & Toolbox
- From the momentum conservation equation, we have:
\[
div\sigma+\textbf{j}\times\textbf{b}=0
\]
With the hypothesys specified in introduction, this equation may be rewritten :
\[
-\sigma_{\theta}+\frac{\partial}{\partial r}(r\sigma_{r})=-rj_{\theta}b_{z}
\]
For a solenoid conductor with finite thickness and infinite length, we have for a constant current density
$j_{\theta}$:
\[
- b_{z} = ...
\]
With these hypothesys, we can show that (see [REF002]):
\[
- u_{r} = ...
\]
- **toolbox**: elasticity
### Materials
|Name |Description | Value | Unit |
|$E$ |Young modulus||$128.10^{9}$|$Pa=kg.m^{-1} .s^{-2}$|
|$\nu$|Poisson's ratio|0.33|- |
### Boundary conditions
- entry/exit: $\textbf{u} \dot \texbf{n} = 0$, displacements only in the perpendical plane,
- top:bottom: clamped
## Outputs
The output is describe the output set of the example
### Fields
add scalar vectorial and matricial fields to be visualized
### Measures
add measures, scalar quantities, mean values, performance metrics
## Benchmark
Describe Benchmark type:
[X] Verification
[] Validation
[] Performance
The computed values of the displacements on r-axis in the solenoidal mid-plane are compared
with the analytical expression given by Montgomery.
## References (articles, papers, reports...)
- https://github.com/feelpp/hifimagnet/blob/develop/MagnetTools/docs/Stress/stress.tex
- [REF001] Montgomery, Solenoid Magnet Design, Wiley-Interscience, New-York. 1969.
- [REF002] M N. Wilson, Superconducting Magnets, Oxford University Press, London, 1987.