Reference solution
=====================


Calculation method
-----------------

It is an analytical solution. Taking into account the boundary conditions, displacements can be obtained from the analytical resolution of the equation for the conservation of momentum.

Since Poisson's Ratio :math:`\nu` is zero, the problem is one-dimensional according to :math:`y`. The stress tensor is uniform across the domain: :math:`\sigma =Eϵ=E\ast {ϵ}_{\mathit{yy}}{e}_{y}\otimes {e}_{y}`.

Or :math:`\text{Div}(\sigma )=0` so :math:`\frac{\partial {ϵ}_{\mathit{yy}}}{\partial y}=0`. Based on the boundary conditions applied :math:`{ϵ}_{\mathit{yy}}=\frac{{u}_{\text{y, impo}}}{L}`

Finally, :math:`\sigma =E\ast \frac{{u}_{\text{y, impo}}}{L}{e}_{y}\otimes {e}_{y}={\sigma }_{\text{yy}}{e}_{y}\otimes {e}_{y}`.

At the crack level, :math:`{e}_{r}=-\mathrm{sin}(\theta ){e}_{\text{x}}+\mathrm{cos}(\theta ){e}_{\text{y}}` and :math:`{e}_{\theta }=-\mathrm{cos}(\theta ){e}_{\text{x}}-\mathrm{sin}(\theta ){e}_{\text{y}}`.

At a point in the crack with coordinates :math:`(R,\theta )`, if there was no crack, we would have:

:math:`\sigma \mathrm{.}{e}_{\text{r}}=({e}_{\text{r}}\mathrm{.}\sigma \mathrm{.}{e}_{\text{r}}){e}_{\text{r}}+({e}_{\text{r}}\mathrm{.}\sigma \mathrm{.}{e}_{\theta }){e}_{\theta }={\sigma }_{\text{yy}}\ast [{({e}_{y}\mathrm{.}{e}_{r})}^{2}{e}_{\text{r}}+({e}_{y}\mathrm{.}{e}_{r})x({e}_{y}\mathrm{.}{e}_{\theta }){e}_{\theta }]`

:math:`\sigma \mathrm{.}{e}_{\text{r}}={\sigma }_{\text{yy}}\ast (\mathit{cos²}(\theta ){e}_{\text{r}}-\mathrm{sin}(\theta )\mathrm{cos}(\theta ){e}_{\theta })`

Finally the solution to the problem is:

:math:`\sigma ={\sigma }_{\text{yy}}{e}_{y}\otimes {e}_{y}`

:math:`u(x,y)=\frac{{u}_{\text{y, impo}}}{L}\ast (\frac{L}{2}+y)`


Reference quantities and results
-----------------------------------

We test the value of the constraints SIGMAXX and SIGMAYY on the whole column.


.. csv-table::

    "Quantities tested", "Reference type", "Reference value"
    "SIGMAXX (MPa)", "'ANALYTIQUE'", "0.0"
    "SIGMAYY (MPa)", "'ANALYTIQUE'", "-5800.0"



Uncertainty about the solution
----------------------------

None, the solution is analytical.