# Step 3: Shear Displacement and Tractions

% Meatadata extracted from parameter files
```{include} step03_sheardisptract-synopsis.md
```

## Simulation parameters

In Step 3 we replace the Dirichlet (displacement) boundary conditions on the +y and -y boundaries with equivalent Neumann (traction) boundary conditions.
In order to maintain symmetry and prevent rigid body motion, we constrain both the x and y displacements on the +x and -x boundaries.
The solution matches that in Step 2.
{numref}`fig:example:box:2d:step03:diagram` shows the boundary conditions on the domain.
The parameters specific to this example are in `step03_sheardisptract.cfg`.

:::{figure-md} fig:example:box:2d:step03:diagram
<img src="figs/step03-diagram.*" alt="" scale="75%">

Boundary conditions for shear deformation.
We constrain the x and y displacements on the +x and -x boundaries.
We apply tangential (shear) tractions on the +y and -y boundaries.
:::

The tractions are uniform on each of the two boundaries, so we use a `UniformDB`.
In PyLith the direction of the tangential tractions in 2D is defined by the cross product of the +z direction and the outward normal on the boundary.
On the +y boundary a positive tangential traction is in the -x direction, and on the -y boundary a positive tangential traction is in the +x direction.
We want tractions in the opposite direction as shown by the arrows in {numref}`fig:example:box:2d:step03:diagram`, so we apply negative tangential tractions.

```{code-block} cfg
---
caption: Specifying the boundary conditions for Step 3. We only show the detailed settings for the -x and -y boundaries.
---
[pylithapp.problem]
bc = [bc_xneg, bc_yneg, bc_xpos, bc_ypos]
bc.bc_xneg = pylith.bc.DirichletTimeDependent
bc.bc_xpos = pylith.bc.DirichletTimeDependent
bc.bc_yneg = pylith.bc.NeumannTimeDependent
bc.bc_ypos = pylith.bc.NeumannTimeDependent

[pylithapp.problem.bc.bc_xneg]
# Degrees of freedom (dof) 0 and 1 correspond to the x and y displacements. 
constrained_dof = [0, 1]
label = boundary_xneg
db_auxiliary_field = spatialdata.spatialdb.SimpleDB
db_auxiliary_field.description = Dirichlet BC -x boundary
db_auxiliary_field.iohandler.filename = sheardisp_bc_xneg.spatialdb
db_auxiliary_field.query_type = linear

[pylithapp.problem.bc.bc_yneg]
label = boundary_yneg
db_auxiliary_field = spatialdata.spatialdb.UniformDB
db_auxiliary_field.description = Neumann BC -y boundary
db_auxiliary_field.values = [initial_amplitude_tangential, initial_amplitude_normal]
db_auxiliary_field.data = [-4.5*MPa, 0*MPa]
```

## Running the simulation

```{code-block} console
---
caption: Run Step 3 simulation
---
$ pylith step03_sheardisptract.cfg

# The output should look something like the following.
 >> /software/unix/py39-venv/pylith-debug/lib/python3.9/site-packages/pylith/meshio/MeshIOObj.py:44:read
 -- meshioascii(info)
 -- Reading finite-element mesh
 >> /src/cig/pylith/libsrc/pylith/meshio/MeshIO.cc:94:void pylith::meshio::MeshIO::read(topology::Mesh *)
 -- meshioascii(info)
 -- Component 'reader': Domain bounding box:
    (-6000, 6000)
    (-16000, -0)

# -- many lines omitted --

 >> /software/unix/py39-venv/pylith-debug/lib/python3.9/site-packages/pylith/problems/TimeDependent.py:139:run
 -- timedependent(info)
 -- Solving problem.
0 TS dt 0.01 time 0.
    0 SNES Function norm 6.059797141590e-03 
    Linear solve converged due to CONVERGED_ATOL iterations 1
    1 SNES Function norm 2.140441363908e-18 
  Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
1 TS dt 0.01 time 0.01
 >> /software/unix/py39-venv/pylith-debug/lib/python3.9/site-packages/pylith/problems/Problem.py:201:finalize
 -- timedependent(info)
 -- Finalizing problem.
```

As expected, the output written to the terminal is nearly identical to what we saw for Steps 1 and 2.

## Visualizing the results

In {numref}`fig:example:box:2d:step03:solution` we use ParaView to visualize the x displacement field using the `viz/plot_dispwarp.py` Python script.
As in Step 2 we override the default name of the simulation file with the name of the current simulation before running the `viz/plot_dispwarp.py` Python script.

```{code-block} python
---
caption: Set the simulation in the ParaView Python Shell.
---
>>> SIM = "step03_sheardisptract"
```

:::{figure-md} fig:example:box:2d:step03:solution
<img src="figs/step03-solution.*" alt="Solution for Step 3. The colors indicate the magnitude of the displacement, and the deformation is exaggerated by a factor of 1000." width="100%"/>

Solution for Step 3.
The colors of the shaded surface indicate the magnitude of the x displacement, and the deformation is exaggerated by a factor of 1000.
The undeformed configuration is show by the gray wireframe.
The solution matches the one from Step 2.
:::