Finish description

Author: srosenbu

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

@@ -1,25 +1,41 @@
 # 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].
+We consider the case of an infinite plate with a circular hole 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 coordinates. Assume that the infinite plate is loaded in $x$-direction, then at
+The solution is given in polar coordinates. Assume that the infinite plate is loaded in $x$-direction with load $p$, then at
 a point  with polar coordinates $(r,\theta)\in\mathbb R_+ \times \mathbb R$, the polar stress components are given by
 
 $$
     \begin{aligned}
-        \sigma_r &= \frac{p}{2}\left(1-\frac{a^2}{r^2}\right)+\frac{p}{2}\left(1-\frac{a^2}{r^2}\right)\left(1-\frac{3a^2}{r^2}\right)\cos(2\theta)\\
-        \sigma_\theta &=\frac{p}{2}\left(1+\frac{a^2}{r^2}\right) - \frac{p}{2}\left(1+\frac{3a^4}{r^4}\right)\cos(2\theta)\\
-        \sigma_{r\theta} &= -\frac{p}{2}\left(1-\frac{a^2}{r^2}\right)\left(1+\frac{3a^2}{r^2}\right)\sin(2\theta)
+        \sigma_{rr}(r,\theta) &= \frac{p}{2}\left(1-\frac{a^2}{r^2}\right)+\frac{p}{2}\left(1-\frac{a^2}{r^2}\right)\left(1-\frac{3a^2}{r^2}\right)\cos(2\theta)\\
+        \sigma_{\theta\theta}(r,\theta) &=\frac{p}{2}\left(1+\frac{a^2}{r^2}\right) - \frac{p}{2}\left(1+\frac{3a^4}{r^4}\right)\cos(2\theta)\\
+        \sigma_{r\theta}(r,\theta) &= -\frac{p}{2}\left(1-\frac{a^2}{r^2}\right)\left(1+\frac{3a^2}{r^2}\right)\sin(2\theta)
     \end{aligned}
 $$
 
-* What is $p$
-* What are $E,\nu$
-* Cartesian coordiantes
-* Neumann BCs
-* write Dirichlet with set
+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 $(x,y)$:
+
+$$
+\boldsymbol\sigma_\mathrm{analytical} (x,y)=\boldsymbol\sigma_\mathrm{analytical} \left(\sqrt{x^2 + y^2},\arccos\frac{x}{\sqrt{x^2+y^2}}\right) 
+$$
 
 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
@@ -28,10 +44,20 @@ of the rectangular domain and assuming symmetry conditions at the edges. Let $\O
 
 $$
     \begin{aligned}
-\mathrm{div}\boldsymbol{\sigma}(\boldsymbol{\varepsilon}(\boldsymbol{u})) &= 0 &\quad \boldsymbol u \in \Omega & \\
+\mathrm{div}\boldsymbol{\sigma}(\boldsymbol{\varepsilon}(\boldsymbol{u})) &= 0 &\quad \text{ on } \Omega & \\
 \boldsymbol{\varepsilon}(\boldsymbol u) &= \frac{1}{2}\left(\nabla \boldsymbol u + (\nabla\boldsymbol u)^\top\right) &&\text{Infinitesimal strain}\\
 \boldsymbol{\sigma}(\boldsymbol{\varepsilon}) &= \frac{E}{1-\nu^2}\left((1-\nu)\boldsymbol{\varepsilon} + \nu \mathrm{tr}\boldsymbol{\varepsilon}\boldsymbol I_2\right) && \text{Plane stress law}\\
-\boldsymbol u_y &=0 & y=0& \text{Dirichlet BC}\\
-\boldsymbol u_x &=0 & x=0& \text{Dirichlet BC}\\
+\boldsymbol u_y &=0 & \text{ on } \lbrace (x,y)\in \partial\Omega | y=0\rbrace& \text{ Dirichlet BC}\\
+\boldsymbol u_x &=0 & \text{ on } \lbrace (x,y)\in \partial\Omega | x=0\rbrace& \text{ Dirichlet BC}\\
+\boldsymbol t &= \boldsymbol{\sigma}_\mathrm{analytical} \cdot \boldsymbol n&\text{ on }\lbrace (x,y)\in \partial\Omega | x=l \lor y=l \rbrace& \text{ Neumann BC}
 \end{aligned}
 $$
+
+The weak form is
+
+$$
+\int_{\Omega} \boldsymbol\varepsilon(\delta\boldsymbol{u}) : \boldsymbol{\sigma} \mathrm{d}{\boldsymbol{x}} =
+    \int_{\Gamma_{\mathrm{N}}} {\boldsymbol{t}}\cdot\delta\boldsymbol{u}\mathrm{d}{\boldsymbol{s}}
+$$
+
+with a test function $\delta \boldsymbol u$