In this benchmark example we consider a linear elastic small deformation problem. More specifically, a plate with a central hole that is put under tension on its top boundary is simulated. By exploiting symmetries, below we evaluate this problem just for the top right quarter of this disc.
Doing this it is important to define boundary conditions for the regarded section of the plate. For the bottom boundary ($x$-axis, $\theta=-90°$) and the left boundary ($y$-axis, $\theta=0°$) we prescribe Dirichlet boundary conditions that constrain the normal displacement along the edge to be zero. On the top, where a tensile traction is applied to the plate, Neumann boundary conditions are prescribed.
For the description of the plate and its load, some dimensions need to be defined. The quarter disc under consideration can be seen as a square with an edge length of $30\,\text{mm}$. The radius of the circular hole is $a = 4\,\text{mm}$ and the applied tension on the top boundary has a value of $\sigma = 10\,\text{MPa}$.
To fully capture and understand the behaviour of stress and strain distributions around the hole, it is necessary to also define material properties. In case of isotropic linear elasticity, the relevant parameters are the Young's modulus and Poisson's ratio. The following parameters are chosen here:
$ E = 1\,\text{MPa} \qquad \nu = 0.3$
For verification of the numerical implementation, the numerical solution of the problem will be compared to the analytical solution.
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## Analytical solution
The overall stress distributions in the plate around the hole can be represented by Kirsch's Solution, which is expressed here in cylindrical coordinates. The following equations are valid for an infinitely extended plate. Since the hole is very small compared to the dimension of the plate, we can consider this specification as rendered. The parameter $\sigma$ stands for the applied tension whereas $a$ identifies the radius of the inner hole. $r$ and $\theta$ are the cylindrical coordinates and describe the distance of a point to the centre of the hole ($r$) and the angle ($\theta$) under which it is deflected with respect to the axis along which the traction is applied.
In the chosen OGS model, the traction is applied as mentioned before on the top boundary of the plate along the $y$-axis, so the $x$-axis is at $\theta = -90°$.
To visualise the problem, we plot the stress distribution along the $x$- and $y$-axes as well as along the diagonal ($\theta = -45°$).
As we can see below, the cavity causes characteristic stress distributions. We now want to take a closer look at those along the $x$-axis. For larger distances from the hole, the stresses are approximately distributed as if the plate was continuous. As the hole is approached, the tangential stress is increasing until it reaches its maximum value directly at the contour. Interestingly, that value is three times higher than the applied traction. It can be concluded, that the hole leads to a threefold stress concentration in the plate.
