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Author: srosenbu

docs/benchmarks/linear elasticity/plate-with-hole.md

@@ -0,0 +1,31 @@
+# Infinite linear elastic plate with hole
+
+We consider the case of an infinite plate with a circular hole in the center. The plate is subjected to uniform tensile load at infinity. The analytical solution for the stress field has been derived by Kirsch in 1898 [@Kirsch1898].
+<!-- include an svg picture here-->
+![Infinite linear elastic plate with hole](plate-with-hole.svg)
+
+The solution is given in polar coordinates. Assume that the infinite plate is loaded in $x$-direction, then at
+a point  with polar coordinates $(r,\theta)\in\mathbb R_+ \times \mathbb R$, the polar stress components are given by
+
+\[
+    \begin{aligned}
+        \sigma_r &= \frac{p}{2}\left(1-\frac{a^2}{r^2}\right)+\frac{p}{2}\left(1-\frac{a^2}{r^2}\right)\left(1-\frac{3a^2}{r^2}\right)\cos(2\theta)\\
+        \sigma_\theta &=\frac{p}{2}\left(1+\frac{a^2}{r^2}\right) - \frac{p}{2}\left(1+\frac{3a^4}{r^4}\right)\cos(2\theta)\\
+        \sigma_{r\theta} &= -\frac{p}{2}\left(1-\frac{a^2}{r^2}\right)\left(1+\frac{3a^2}{r^2}\right)\sin(2\theta)
+    \end{aligned}
+\]
+In order to transform this into a practical benchmark, we consider a rectangular subdomain
+of the infinite plate around the hole. The boundary conditions of the subdomain are determined
+from the analytical solution. The example is further reduced by only simulating one quarter
+of the rectangular domain and assuming symmetry conditions at the edges. Let $\Omega =[0,l]^2 \setminus\left\{(x,y) \mid \sqrt{x^2+y^2}<a \right\}$ be the domain of the benchmark example, then the PDE is given by
+
+\[
+    \begin{aligned}
+\mathrm{div}\boldsymbol{\sigma}(\boldsymbol{\varepsilon}(\boldsymbol{u})) &= 0 &\quad \boldsymbol u \in \Omega & \\
+\boldsymbol{\varepsilon}(\boldsymbol u) &= \frac{1}{2}\left(\nabla \boldsymbol u + (\nabla\boldsymbol u)^\top\right) &&\text{Infinitesimal strain}\\
+\boldsymbol{\sigma}(\boldsymbol{\varepsilon}) &= \frac{E}{1-\nu^2}\left((1-\nu)\boldsymbol{\varepsilon} + \nu \mathrm{tr}\boldsymbol{\varepsilon}\boldsymbol I_2\right) && \text{Plane stress law}\\
+\boldsymbol u_y &=0 & y=0& \text{Dirichlet BC}\\
+\boldsymbol u_x &=0 & x=0& \text{Dirichlet BC}\\
+boldsymbol{n}
+\end{aligned}
+\]

docs/benchmarks/linear elasticity/plate-with-hole.svg

@@ -19,4 +19,12 @@ @Book{Simo1998
   language  = {en},
   url       = {https://www.springer.com/gp/book/9780387975207},
   urldate   = {2020-06-23},
+}
+@article{Kirsch1898,
+  title={Die Theorie der Elastizität und die Bedürfnisse der Festigkeitslehre},
+  author={Kirsch, Ernst Gustav},
+  journal={Zeitshrift des Vereines deutscher Ingenieure},
+  volume={42},
+  pages={797--807},
+  year={1898}
 }