## Benchmark UP4

### Simple Shear

In this subsection, we test the forces required to sustain simple shear of a homogenous block. Let us set the block’s height and width to $1\mathrm{m}$. The depth is chosen so that the block has just one element across the depth, supposing that each element is a brick with equally long edges. The number of elements is than $N×N×1$ and the block depth is $1}{N}$ meters.

Materials:

• Gent

##### Parameters

$\begin{array}{ccc}\hfill \kappa & \hfill =\hfill & {10}^{8}\mathrm{Pa}\hfill \\ \hfill \mu & \hfill =\hfill & 280709\mathrm{Pa}\hfill \\ \hfill {J}_{m}& \hfill =\hfill & 1.1575\hfill \end{array}$

##### Theory

The surface load, which must be applied to sustain the block’s simple shear, is given by [Taber, 2004]

$\begin{array}{ccc}\hfill {\sigma }_{11}& \hfill =\hfill & -2{k}^{2}\frac{\partial {\Psi }_{\text{iso}}}{\partial {I}_{1}}{|}_{J=1}\hfill \\ \hfill {\sigma }_{12}& \hfill =\hfill & {\sigma }_{21}=-2k\frac{\partial {\Psi }_{\text{iso}}}{\partial {I}_{1}}{|}_{J=1}\hfill \\ \hfill {\sigma }_{22}& \hfill =\hfill & 0\hfill \end{array}$

After exerting coresponding surface loading on the block (for detailed formulae see [Stembera, 2013]), a value of $k$ is numerically measured and compared with its assumed value.

##### Results

Theory femCalc (BRICK24) femCalc (BRICK60) femCalc (BRICK81)
$k\text{\hspace{0.17em}}\left[\mathrm{-}\right]$ $k\text{\hspace{0.17em}}\left[\mathrm{-}\right]$ $\text{Ratio}\text{\hspace{0.17em}}\left[\mathrm{-}\right]$ $k\text{\hspace{0.17em}}\left[\mathrm{-}\right]$ $\text{Ratio}\text{\hspace{0.17em}}\left[\mathrm{-}\right]$ $k\text{\hspace{0.17em}}\left[\mathrm{-}\right]$ $\text{Ratio}\text{\hspace{0.17em}}\left[\mathrm{-}\right]$
$0.1$ $0.1000$ $1.000$ $0.1000$ $1.000$ $0.1000$ $1.000$
$0.2$ $0.2002$ $1.001$ $0.2001$ $1.001$ $0.2001$ $1.001$
$0.3$ $0.3005$ $1.002$ $0.3005$ $1.002$ $0.3006$ $1.002$
$0.4$ $0.4022$ $1.006$ $0.4022$ $1.006$ $0.4022$ $1.006$
$0.5$ $0.5063$ $1.013$ $0.5064$ $1.013$ $0.5064$ $1.025$
$0.6$ $0.6144$ $1.024$ $0.6147$ $1.025$ $0.6149$ $1.025$