From: frohne Date: Tue, 5 Feb 2013 03:47:26 +0000 (+0000) Subject: working on section 2 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=9d0aa027987bb2a5e7fbcc5b89019e97361362d1;p=dealii-svn.git working on section 2 git-svn-id: https://svn.dealii.org/trunk@28228 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-42/doc/intro-step-42.tex b/deal.II/examples/step-42/doc/intro-step-42.tex index 854946cd40..ba71b4afbd 100644 --- a/deal.II/examples/step-42/doc/intro-step-42.tex +++ b/deal.II/examples/step-42/doc/intro-step-42.tex @@ -28,41 +28,54 @@ dimensions, even with adaptive mesh refinement, we decided to use Trilinos and p4est to run our code in parallel, building on the framework of step-40 for the parallelization. +\begin{huge} +{distributed} +\end{huge} + \section{Classical formulation} The classical formulation of the problem possesses the following form: \begin{align*} \varepsilon(u) &= A\sigma + \lambda & &\quad\text{in } \Omega,\\ - \lambda(\tau - \sigma) &\geq 0\quad\forall\tau\text{ mit }\mathcal{F}(\tau)\leq 0 & &\quad\text{in } \Omega,\\ + \lambda(\tau - \sigma) &\geq 0\quad\forall\tau\text{ with + }\mathcal{F}(\tau)\leq 0 & &\quad\text{in } \Omega,\\ -\textrm{div}\ \sigma &= f & &\quad\text{in } \Omega,\\ u(\mathbf x) &= 0 & &\quad\text{on }\Gamma_D,\\ \sigma_t(u) &= 0,\quad\sigma_n(u)\leq 0 & &\quad\text{on }\Gamma_C,\\ \sigma_n(u)(u_n - g) &= 0,\quad u_n(\mathbf x) - g(\mathbf x) \leq 0 & &\quad\text{on } \Gamma_C \end{align*} -with $u\in H^2(\Omega)$. The vector valued function $u$ denotes the -displacement in the deformable body. The first two lines describe the elast-plastic -material behavior. Therein the equation shows the deformation $\varepsilon (u)$ as the additive -decomposition of the elastic part $A\sigma$ and the plastic part $\lambda$. $A$ is defined as -the compliance tensor of fourth order which contains some material constants and $\sigma$ as the +with $u\in H^2(\Omega),\Omega\subset\mathbb{R}^3$. The vector valued +function $u$ denotes the displacement in the deformable body. The first two lines describe the +elasto-plastic material behavior. Therein the equation shows the +strain of the deformation $\varepsilon (u)$ as the additive decomposition of the +elastic part $A\sigma$ and the plastic part $\lambda$. $A$ is defined as the compliance tensor of fourth order which contains some material constants and $\sigma$ as the symmetric stress tensor of second order. So we have to consider the inequality in the second row component-by-component and furthermore we have to distinguish two cases.\\ The continuous and convex function $\mathcal{F}$ denotes the von Mises flow function -$$\mathcal{F}(\tau) = \vert\tau^D\vert - \sigma_0$$ +$$\mathcal{F}(\tau) = \vert\tau^D\vert - \sigma_0,\quad \tau^D = \tau - +\dfrac{1}{3}tr(\tau)I$$ with $\sigma_0$ as yield stress. If there is no plastic deformation - that is $\lambda=0$ - this yields $\vert\sigma^D\vert < \sigma_0$ -and otherwise if $\lambda > 0$ it follows that $\vert\sigma^D\vert = \sigma_0$. That means if the stress is smaller as the yield stress -there are only elastic deformations. Therein the Index $D$ denotes the deviator part of the stress $\sigma$ which -is defined as -$$\sigma^D = \sigma - \dfrac{1}{3}tr(\sigma).$$ -It describes the hydrostatic part of the stress tensor in contrast to the volumetric part. For metal the hydrostatic -stress composes the main indicator for the plastic deformation.\\ -The second equation is called equilibrium condition with a force of areal density $f$ which we will neglect in our example. +and otherwise if $\lambda > 0$ it follows that $\vert\sigma^D\vert = \sigma_0$. +That means if the stress is smaller than the yield stress there are only elastic +deformations. Or to consider it the other way around. If the deviator stress is +in a norm bigger or equal than the yield stress there are plastic deformations +and $\lambda$ would be positiv.\\ +There the index $D$ denotes the deviator part of for example the stress where +$tr(.)$ is the trace of a tensor. The definition shows an additive decomposition +of the stress $\sigma$ into a hydrostatic part (or volumetric part) $\dfrac{1}{3}tr(\tau)I$ and the deviator +part $\sigma^D$. For metal the deviator stress composes the main indicator for +plastic deformations.\\ +The third equation is called equilibrium condition with a force of volume +density $f$ which we will neglect in our example. The boundary of $\Omega$ separates as follows $\Gamma=\Gamma_D\bigcup\Gamma_C$ and $\Gamma_D\bigcap\Gamma_C=\emptyset$. At the boundary $\Gamma_D$ we have zero Dirichlet conditions. $\Gamma_C$ denotes the potential contact boundary.\\ The last two lines decribe the so-called Signorini contact conditions. If there is no contact the normal stress $$ \sigma_n = \sigma n\cdot n$$ is zero with the outward normal $n$. If there is contact ($u_n = g$) the tangential stress $\sigma_t = \sigma\cdot n - \sigma_n n$ -vanishes, because we consider a frictionless situation and the normal stress is negative. +vanishes, because we consider a frictionless situation and the normal stress is +negative. The gap $g$ comes with the start configuration of the obstacle and the +deformable body. \section{Derivation of the variational inequality}