15 description of plate with hole in docs

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

@@ -0,0 +1,96 @@
+# Infinite linear elastic plate with hole
+
+We consider the case of an infinite plate with a circular hole with radius $a$ in the center. The plate is subjected to uniform tensile load $p$ 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 stress components at a point with polar coordinates $(r,\theta)\in\mathbb R_+ \times \mathbb R$. Assume that the infinite plate is loaded in $x$-direction with load $p$, then the polar stress components are given by
+
+$$
+    \begin{aligned}
+        \sigma_{rr}(r,\theta) &= \frac{p}{2}\left(1-\frac{a^2}{r^2}\right)+\frac{p}{2}\left(1-\frac{a^2}{r^2}\right)\left(1-\frac{3a^2}{r^2}\right)\cos(2\theta)\\
+        \sigma_{\theta\theta}(r,\theta) &=\frac{p}{2}\left(1+\frac{a^2}{r^2}\right) - \frac{p}{2}\left(1+\frac{3a^4}{r^4}\right)\cos(2\theta)\\
+        \sigma_{r\theta}(r,\theta) &= -\frac{p}{2}\left(1-\frac{a^2}{r^2}\right)\left(1+\frac{3a^2}{r^2}\right)\sin(2\theta)
+    \end{aligned}
+$$
+
+In order to write the stresses in a cartesian coordiante system, they need to be rotated by $\theta$, which results in
+
+$$
+    \begin{aligned}
+        \sigma_{xx} (r,\theta) &=  \frac{3 a^{4} p \cos{\left(4 \theta \right)}}{2 r^{4}} - \frac{a^{2} p \left(1.5 \cos{\left(2 \theta \right)} + \cos{\left(4 \theta \right)}\right)}{r^{2}} + p \\
+        \sigma_{yy} (r,\theta)&= - \frac{3 a^{4} p \cos{\left(4 \theta \right)}}{2 r^{4}} - \frac{a^{2} p \left(\frac{\cos{\left(2 \theta \right)}}{2} - \cos{\left(4 \theta \right)}\right)}{r^{2}}\\
+        \sigma_{xy} (r,\theta) &= \frac{3 a^{4} p \sin{\left(4 \theta \right)}}{2 r^{4}} - \frac{a^{2} p \left(\frac{\sin{\left(2 \theta \right)}}{2} + \sin{\left(4 \theta \right)}\right)}{r^{2}}
+    \end{aligned}
+$$
+
+with the full stress tensor solution given by
+
+$$
+\boldsymbol\sigma_\mathrm{analytical} (r,\theta)= \begin{bmatrix} \sigma_{xx} & \sigma_{xy}\\\ \sigma_{xy} & \sigma_{yy} \end{bmatrix}.
+$$
+
+or for a cartesion point $(x,y)\in \mathbb R_+^2$:
+
+$$
+\boldsymbol\sigma_\mathrm{analytical} (x,y)=\boldsymbol\sigma_\mathrm{analytical} \left(\sqrt{x^2 + y^2},\arccos\frac{x}{\sqrt{x^2+y^2}}\right). 
+$$
+
+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 with length $l$ and assuming symmetry conditions at the edges. Let 
+
+$$
+\Omega =[0,l]^2 \setminus \lbrace (x,y) \mid \sqrt{x^2+y^2}<a \rbrace
+$$ 
+
+be the domain of the benchmark example and
+
+$$
+\begin{aligned}
+\Gamma_\mathrm{D_1} &= \lbrace (x,y)\in \partial\Omega | y=0\rbrace \\
+\Gamma_\mathrm{D_2} &= \lbrace (x,y)\in \partial\Omega | x=0\rbrace \\
+\Gamma_\mathrm{N} &= \lbrace (x,y)\in \partial\Omega | x=l \lor y=l \rbrace
+\end{aligned}
+$$
+
+then the PDE is given by
+
+$$
+\begin{aligned}
+\mathrm{div}\boldsymbol{\sigma}(\boldsymbol{\varepsilon}(\boldsymbol{u})) &= 0 &\quad \text{ on } \Omega & \\
+\boldsymbol{\varepsilon}(\boldsymbol u) &= \frac{1}{2}\left(\nabla \boldsymbol u + (\nabla\boldsymbol u)^\top\right) &&\text{Infinitesimal strain}\\
+\boldsymbol{\sigma}(\boldsymbol{\varepsilon}) &= \frac{E}{1-\nu^2}\left((1-\nu)\boldsymbol{\varepsilon} + \nu \mathrm{tr}\boldsymbol{\varepsilon}\boldsymbol I_2\right) && \text{Plane stress law}\\
+\boldsymbol u_y &=0 & \text{ on } \Gamma_\mathrm{D_1}& \text{ Dirichlet BC}\\
+\boldsymbol u_x &=0 & \text{ on } \Gamma_\mathrm{D_2}& \text{ Dirichlet BC}\\
+\boldsymbol t &= \boldsymbol{\sigma}_\mathrm{analytical} \cdot \boldsymbol n & \text{ on } \Gamma_\mathrm{N} & \text{ Neumann BC}\\
+\end{aligned}
+$$
+
+with the material parameters $E,\nu$ -- the Youngs modulus and Poisson ratio.
+In the weak formulation of the problem, we want to find $\boldsymbol u$ such that
+
+$$
+B(\boldsymbol u,\delta\boldsymbol u) = f(\delta\boldsymbol u) \quad \forall \boldsymbol \delta u
+$$
+
+with a test function $\delta \boldsymbol u$ and 
+
+$$
+\begin{aligned}
+B(\boldsymbol u,\delta\boldsymbol u) &= \int_{\Omega} \boldsymbol\varepsilon(\delta\boldsymbol{u}) : \boldsymbol{\sigma}(\boldsymbol{\varepsilon}(\boldsymbol{u})) \mathrm{d}{\boldsymbol{x}} \\
+    f(\delta\boldsymbol u)&=\int_{\Gamma_{\mathrm{N}}} {\boldsymbol{t}}\cdot\delta\boldsymbol{u}\mathrm{d}{\boldsymbol{s}}.
+\end{aligned}
+$$
+
+## Table of parameters
+
+| Parameter    | Description                     |
+| ------------ | ------------------------------  |
+| $a$ in $m$   | Radius of the hole.             |
+| $l$ in $m$   | Length of the benchmark domain. |
+| $E$ in $Pa$  | Youngs modulus.                 |
+| $\nu$ in --  | Poisson ratio.                  |
+| $p$ in $Pa$  | Load at infinity.               |
+

docs/literature.bib

@@ -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={Zeitschrift des Vereines deutscher Ingenieure},
+  volume={42},
+  pages={797--807},
+  year={1898}
+}