--- trunk/doc/user/Models.tex 2009/05/12 03:33:49 2414 +++ trunk/doc/user/Models.tex 2009/05/13 02:48:39 2415 @@ -401,7 +401,7 @@ % \section{Drucker Prager Model} \section{Isotropic Kelvin Material \label{IKM}} -As proposed by Kelvin~\cite{Muhlhaus2005} material strain $D\hackscore{ij}=v\hackscore{i,j}+v\hackscore{j,i}$ can be decomposed into +As proposed by Kelvin~\cite{Muhlhaus2005} material strain $D\hackscore{ij}=\frac{1}{2}(v\hackscore{i,j}+v\hackscore{j,i})$ can be decomposed into an elastic part $D\hackscore{ij}^{el}$ and visco-plastic part $D\hackscore{ij}^{vp}$: $$\label{IKM-EQU-2} D\hackscore{ij}=D\hackscore{ij}^{el}+D\hackscore{ij}^{vp} @@ -465,12 +465,12 @@$$ and $$\label{IKM-EQU-2b} -D\hackscore{ij}'=\left(\frac{1}{2 \eta^{vp}} + \frac{1}{2 \mu dt}\right) \sigma\hackscore{ij}'+\frac{1}{2 \mu dt } \sigma\hackscore{ij}^{-'} +D\hackscore{ij}'=\left(\frac{1}{2 \eta^{vp}} + \frac{1}{2 \mu dt}\right) \sigma\hackscore{ij}'-\frac{1}{2 \mu dt } \sigma\hackscore{ij}^{-'}$$ where $\sigma\hackscore{ij}^{-}$ is the stress at the precious time step. With $$\label{IKM-EQU-2c} -\dot{\gamma} = \sqrt{ 2 \left( D\hackscore{ij}' - -\frac{1}{ 2 \mu \; dt} \sigma\hackscore{ij}^{'-}\right)^2} +\dot{\gamma} = \sqrt{ 2 \left( D\hackscore{ij}' + +\frac{1}{ 2 \mu \; dt} \sigma\hackscore{ij}^{-'}\right)^2}$$ we have $$@@ -490,14 +490,14 @@$$ The upper bound $\eta\hackscore{max}$ makes sure that yield condtion~\ref{IKM-EQU-8c} holds. With this setting the eqaution \ref{IKM-EQU-2b} takes the form $$\label{IKM-EQU-10} -\sigma\hackscore{ij}' = 2 \eta\hackscore{eff} \left( D\hackscore{ij}' - +\sigma\hackscore{ij}' = 2 \eta\hackscore{eff} \left( D\hackscore{ij}' + \frac{1}{ 2 \mu \; dt} \sigma\hackscore{ij}^{'-}\right)$$ After inserting~\ref{IKM-EQU-10} into~\ref{IKM-EQU-1} we get $$\label{IKM-EQU-1ib} -\left(\eta\hackscore{eff} (v\hackscore{i,j}+ v\hackscore{i,j}) -\right)\hackscore{,j}+p\hackscore{,i}=F\hackscore{i}- -\frac{\eta\hackscore{eff}}{\mu dt } \sigma\hackscore{ij,j}^{'-} +\right)\hackscore{,j}+p\hackscore{,i}=F\hackscore{i}+ + \left(\frac{\eta\hackscore{eff}}{\mu dt } \sigma\hackscore{ij}^{'-} \right)\hackscore{,j}$$ Combining this with the incomressibilty condition~\ref{IKM-EQU-2} we need to solve a Stokes problem as discussed in section~\ref{STOKES SOLVE} in each time step.