femCalc - BenchmarkUP5

Benchmark UP5

Cylinder Torsion

Let us test the forces required to sustain a torsion of a cylinder. The cylinder is supposed to have length $l$ , diameter $D$ and to be made of the Neo-Hookean material. The cylinder radius is denoted by $a = \frac{D}{2}$ .

Parameters

$\begin{matrix} D & = & 1 m \\ l & = & 1 m \\ κ & = & 10^{7} Pa \\ μ & = & 280709 Pa \end{matrix}$

Theory

The cylinder lower base is fully fixed and the cylinder upper base is subjected to the surface load according to the theoretical formula given in [Taber, 2004]

\begin{matrix} σ_{zz} & = & - 2 θ^{2} \int_{r}^{a} \tilde{r} \frac{\partial Ψ_{iso}}{\partial I_{1}} |_{J = 1} d \tilde{r} \\ σ_{θ z} & = & 2 θ r \frac{\partial Ψ_{iso}}{\partial I_{1}} |_{J = 1} \end{matrix}

After exerting coresponding surface loading on the upper base of the cylinder, a value of $θ$ is numerically measured and compared with its assumed value.

Results

Theory	femCalc (BRICK24)		femCalc (BRICK24)
	K=4, N=960		K=6, N=3240
$θ [rad]$	$θ [rad]$	$Ratio [-]$	$θ [rad]$	$Ratio [-]$
$0.1$	$0.1007$	$1.007$	$0.0997$	$0.997$
$0.2$	$0.2019$	$1.010$	$0.1998$	$1.001$
$0.3$	$0.3035$	$1.012$	$0.3003$	$1.001$
$0.4$	$0.4053$	$1.013$	$0.4014$	$1.003$
$0.5$	$0.5072$	$1.014$	$0.5026$	$1.005$
$0.6$	$0.6082$	$1.014$	$0.6034$	$1.006$
$0.7$	$0.7071$	$1.010$	$0.7024$	$1.003$
$0.8$	$0.8021$	$1.003$	$0.7975$	$0.997$
$0.9$	$0.8912$	$0.990$	$0.8864$	$0.985$
$1.0$	$0.9723$	$0.972$	$0.9671$	$0.967$