From f8b9914f2108836443cb0e375a99e1a335d12cb8 Mon Sep 17 00:00:00 2001 From: bangerth Date: Mon, 18 Feb 2013 00:15:45 +0000 Subject: [PATCH] More markup fixes. git-svn-id: https://svn.dealii.org/trunk@28446 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-42/doc/intro.dox | 13 ++++++------- 1 file changed, 6 insertions(+), 7 deletions(-) diff --git a/deal.II/examples/step-42/doc/intro.dox b/deal.II/examples/step-42/doc/intro.dox index d0a868d5c9..fdcfe129df 100644 --- a/deal.II/examples/step-42/doc/intro.dox +++ b/deal.II/examples/step-42/doc/intro.dox @@ -139,17 +139,17 @@ which yields with the second inequality:\\ Find the displacement $u\in V^+$ with @f{gather*}\left(P_{\Pi}(C\varepsilon(u)),\varepsilon(\varphi) - \varepsilon(u)\right) \geq 0,\quad \forall \varphi\in V^+,@f} with the projection: -@f{gather*}P_{\Pi}(\tau):=@f{cases} +@f{gather*}P_{\Pi}(\tau):=\begin{cases} \tau, & \text{if }\vert\tau^D\vert \leq \sigma_0 + \gamma\xi,\\ \hat\alpha\dfrac{\tau^D}{\vert\tau^D\vert} + \dfrac{1}{3}tr(\tau), & \text{if }\vert\tau^D\vert > \sigma_0 + \gamma\xi, - @f}@f} + \end{cases}@f} with the radius @f{gather*}\hat\alpha := \sigma_0 + \gamma\xi .@f} With the relation $\xi = \vert\varepsilon(u) - A\sigma\vert$ it is possible to eliminate $\xi$ inside the projection $P_{\Pi}$:\\ -@f{gather*}P_{\Pi}(\tau):=@f{cases} +@f{gather*}P_{\Pi}(\tau):=\begin{cases} \tau, & \text{if }\vert\tau^D\vert \leq \sigma_0,\\ \alpha\dfrac{\tau^D}{\vert\tau^D\vert} + \dfrac{1}{3}tr(\tau), & \text{if }\vert\tau^D\vert > \sigma_0, - @f}@f} + \end{cases}@f} @f{gather*}\alpha := \sigma_0 + \dfrac{\gamma}{2\mu+\gamma}\left(\vert\tau^D\vert - \sigma_0\right) ,@f} with a further material parameter $\mu>0$ called shear modulus. We refer that this only possible for isotropic plasticity. @@ -196,13 +196,13 @@ semi-linearform $a(.;.)$ at the point $u^i$ is @f{gather*}a'(u^i;\psi,\varphi) = (I(x)\varepsilon(\psi),\varepsilon(\varphi)),\quad x\in\Omega,@f} @f{gather*} -I(x) := @f{cases} +I(x) := \begin{cases} 2\mu\left(\mathbb{I} - \dfrac{1}{3} I\otimes I\right) + \kappa I\otimes I, & \quad \vert \tau^D \vert \leq \sigma_0\\ \dfrac{\alpha}{\vert\tau^D\vert}2\mu\left(\mathbb{I} - \dfrac{1}{3} I\otimes I - \dfrac{\tau^D\otimes\tau^D}{\vert\tau^D\vert}\right) + \kappa I\otimes I, &\quad \vert \tau^D \vert > \sigma_0 -@f} +\end{cases} @f} with @f{gather*}\tau^D := C\varepsilon^D(u^i).@f} @@ -347,4 +347,3 @@ motivation in Chinese. If your audience is Japanese, please see the other entry for motivation. This is a word in Japanese and Korean, but it means "motive power" or "kinetic energy" (without the motivation meaning that you are probably looking for)".) - -- 2.39.5