femCalc - BenchmarkUP4

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 $1 m$ . 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 $N \times N \times 1$ and the block depth is $\frac{1}{N}$ meters.

Materials:

Gent

Parameters

$\begin{matrix} κ & = & 10^{8} Pa \\ μ & = & 280709 Pa \\ J_{m} & = & 1.1575 \end{matrix}$

Theory

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

\begin{matrix} σ_{11} & = & - 2 k^{2} \frac{\partial Ψ_{iso}}{\partial I_{1}} |_{J = 1} \\ σ_{12} & = & σ_{21} = - 2 k \frac{\partial Ψ_{iso}}{\partial I_{1}} |_{J = 1} \\ σ_{22} & = & 0 \end{matrix}

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)
$k [-]$	$k [-]$	$Ratio [-]$	$k [-]$	$Ratio [-]$	$k [-]$	$Ratio [-]$
$0.1$	$0.1000$	$1.000$	$0.1000$	$1.000$	$0.1000$	$1.000$
$0.2$	$0.2002$	$1.001$	$0.2001$	$1.001$	$0.2001$	$1.001$
$0.3$	$0.3005$	$1.002$	$0.3005$	$1.002$	$0.3006$	$1.002$
$0.4$	$0.4022$	$1.006$	$0.4022$	$1.006$	$0.4022$	$1.006$
$0.5$	$0.5063$	$1.013$	$0.5064$	$1.013$	$0.5064$	$1.025$
$0.6$	$0.6144$	$1.024$	$0.6147$	$1.025$	$0.6149$	$1.025$