# 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/pylith-debug/lib/python3.12/site-packages/pylith/apps/PyLithApp.py:79:main
 -- info (application-flow)
 -- Running on 1 process(es).

# -- many lines omitted --

 >> src/cig/pylith/libsrc/pylith/problems/TimeDependent.cc:473:void pylith::problems::TimeDependent::solve()
 -- info (application-flow)
 -- Component 'timedependent.problem': Solving equations.
0 TS dt 0.001 time 0.
    0 SNES Function norm 1.817939142477e+01
      Linear solve converged due to CONVERGED_ATOL iterations 3
    1 SNES Function norm 8.280473801872e-08
    Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
1 TS dt 0.001 time 0.001
 >> software/pylith-debug/lib/python3.12/site-packages/pylith/problems/Problem.py:222:finalize
 -- info (application-flow)
 -- 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 the `pylith_viz` utility to visualize the x displacement field.

```{code-block} console
---
caption: Visualize PyLith output using `pylith_viz`.
---
pylith_viz --filenames=output/step03_sheardisptract-domain.h5 warp_grid --component=x
```

:::{figure-md} fig:example:box:2d:step03:solution
<img src="figs/step03-solution.*" alt="Solution for Step 3. The colors indicate the x displacement, and the deformation is exaggerated by a factor of 1000." width="400px"/>

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