# Viscous inclusions under simple and pure shear

*This section was contributed by Cedric Thieulot.*

The setup for this experiment originates in {cite:t}`halter:macherel:schmalholz:2022`.
In the paper the authors present a simple computer program to calculate the stress,
pressure, velocity and strain rate fields for two-dimensional (2D) viscous inclusion-matrix systems under pure
shear and simple shear.

We here focus on two of their experiments: a strong rectangular inclusion and a weak elliptical inclusion.
In both cases the domain is a Cartesian box of size $(L_x,L_y) = (1,1)$ and the magnitude of the applied
velocity on the boundaries is 0.5 (this model is formulated in terms of dimensionless quantities).
The viscosity of the matrix is set to $\eta_m=1$, while the viscosity of the inclusion is set to $\eta_i=1000$
for the rectangular inclusion and $\eta_i=0.001$ for the elliptical inclusion, see {numref}`fig:inclusion-visc`.
The setup for the elliptical inclusion is shown in {numref}`fig:inclusion-setup`.

```{figure-md} fig:inclusion-visc
<img src="viscosities.*" width="90%" />

Viscosity fields for the elliptical and rectangular inclusions.
```

```{figure-md} fig:inclusion-setup
<img src="setup.*" width="60%" />

Model configuration and boundary conditions. An elliptical inclusion of
different viscosity than its surrounding matrix is located in the center of a
square box of size $L_x$ by $L_y$. The ellipse has a semi-major axis $a$ and semi-minor
axis $b$ and is rotated by angle $\phi$ from the horizontal axis. Both simple (SS) and
pure (PS) shear boundary conditions can be applied.
Taken from {cite:t}`halter:macherel:schmalholz:2022`.
```

The velocity, pressure and strain rate fields for all four experiments
are shown in {numref}`fig:inclusion-results`.


```{figure-md} fig:inclusion-results
<img src="results1.*" width="100%" />

Results for both inclusion types and both boundary condition types. 'SS' stands
for simple shear and 'PS' stands for pure shear.
```

In the paper the authors also look at the case of a power-law rheology for the matrix.
Instead of trying to reproduce their results exactly, let us simply create a test case for such an experiment.
We then replace the simple material model by the Visco Plastic material model,
select the dislocation creep flow law, set the
activation volume and activation energy to zero, and set the cohesion to a very large value
so that plastic behavior is never triggered.
In the end the effective viscosity is then given by:

```{math}
\eta = \frac12 A^{-1/n} \dot\varepsilon_e
```

If $n=1$ then the viscosity is Newtonian, i.e. $\eta=\frac12 A^{-1}$, and by taking $A=500$ then
we ensure that the inclusion has a constant viscosity $\eta_i=0.001$ as before.
Concerning the matrix, we set $A=0.5$ so that when $n=1$ we recover $\eta_m=1$.
In {numref}`fig:inclusion-nonlinear` results obtained with $n=1,2,3$ are shown.

```{figure-md} fig:inclusion-nonlinear
<img src="results2.*" width="100%" />

Results for an elliptical inclusion under pure shear with a power-law rheology for the matrix with $n=1,2,3$.
```