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revision 2251 by gross, Fri Feb 6 06:50:39 2009 UTC revision 2252 by gross, Fri Feb 6 07:51:28 2009 UTC
# Line 404  D\hackscore{ij}=D\hackscore{ij}^{el}+D\h Line 404  D\hackscore{ij}=D\hackscore{ij}^{el}+D\h
404  \end{equation}  \end{equation}
405  with the elastic strain given as  with the elastic strain given as
406  \begin{equation}\label{IKM-EQU-3}  \begin{equation}\label{IKM-EQU-3}
407  D\hackscore{ij}'^{el}=\frac{1}{2 \mu} \dot{\sigma}\hackscore{ij}'  D\hackscore{ij}^{el'}=\frac{1}{2 \mu} \dot{\sigma}\hackscore{ij}'
408  \end{equation}  \end{equation}
409  where $\sigma'\hackscore{ij}$ is the deviatoric stress (Notice that $\sigma'\hackscore{ii}=0$).  where $\sigma'\hackscore{ij}$ is the deviatoric stress (Notice that $\sigma'\hackscore{ii}=0$).
410  If the material is composed by materials $q$ the visco-plastic strain can be decomposed as  If the material is composed by materials $q$ the visco-plastic strain can be decomposed as
411  \begin{equation}\label{IKM-EQU-4}  \begin{equation}\label{IKM-EQU-4}
412  D\hackscore{ij}'^{vp}=\sum\hackscore{q} D\hackscore{ij}'^{q}  D\hackscore{ij}^{vp'}=\sum\hackscore{q} D\hackscore{ij}^{q'}
413  \end{equation}  \end{equation}
414  where $D\hackscore{ij}^{q}$ is the strain in material $q$ given as  where $D\hackscore{ij}^{q}$ is the strain in material $q$ given as
415  \begin{equation}\label{IKM-EQU-5}  \begin{equation}\label{IKM-EQU-5}
416  D\hackscore{ij}'^{q}=\frac{1}{2 \eta^{q}} \sigma'\hackscore{ij}  D\hackscore{ij}^{q'}=\frac{1}{2 \eta^{q}} \sigma'\hackscore{ij}
417  \end{equation}  \end{equation}
418  where $\eta^{q}$ is the viscosity of material $q$. We assume the following  where $\eta^{q}$ is the viscosity of material $q$. We assume the following
419  betwee the the strain in material $q$  betwee the the strain in material $q$
# Line 421  betwee the the strain in material $q$ Line 421  betwee the the strain in material $q$
421  \eta^{q}=\eta^{q}\hackscore{N} \left(\frac{\tau}{\tau\hackscore{t}^q}\right)^{\frac{1}{n^{q}}-1}  \eta^{q}=\eta^{q}\hackscore{N} \left(\frac{\tau}{\tau\hackscore{t}^q}\right)^{\frac{1}{n^{q}}-1}
422  \mbox{ with } \tau=\sqrt{\frac{1}{2}\sigma'\hackscore{ij} \sigma'\hackscore{ij}}  \mbox{ with } \tau=\sqrt{\frac{1}{2}\sigma'\hackscore{ij} \sigma'\hackscore{ij}}
423  \end{equation}  \end{equation}
424  for a given power law coefficients $n^{q}$ and transition stresses $\tau\hackscore{t}^q$, see~\ref{POERLAW}.  for a given power law coefficients $n^{q}\ge1$ and transition stresses $\tau\hackscore{t}^q$, see~\ref{POERLAW}.
425  Notice that $n^{q}=1$ gives a constant viscosity.  Notice that $n^{q}=1$ gives a constant viscosity.
426  After inserting equation~\ref{IKM-EQU-5} into equation \ref{IKM-EQU-4} one gets:  After inserting equation~\ref{IKM-EQU-5} into equation \ref{IKM-EQU-4} one gets:
427  \begin{equation}\label{IKM-EQU-6}  \begin{equation}\label{IKM-EQU-6}
# Line 484  After inserting~\ref{IKM-EQU-10} into~\r Line 484  After inserting~\ref{IKM-EQU-10} into~\r
484  \right)\hackscore{,j}+p\hackscore{,i}=F\hackscore{i}+  \right)\hackscore{,j}+p\hackscore{,i}=F\hackscore{i}+
485  \frac{\eta\hackscore{eff}}{\mu dt } \sigma\hackscore{ij,j}^{'-}  \frac{\eta\hackscore{eff}}{\mu dt } \sigma\hackscore{ij,j}^{'-}
486  \end{equation}  \end{equation}
487  Together with the incomressibilty condition~\ref{IKM-EQU-2} we need to solve a problem with a form almost identical  Combining this with the incomressibilty condition~\ref{IKM-EQU-2} we need to solve a
488  to the Stokes problem discussed in section~\ref{STOKES SOLVE} but with the difference that $\eta\hackscore{eff}$ is depending on the solution. Analog to the iteration scheme~\ref{SADDLEPOINT iteration 2} we can run  Stokes problem as discussed in section~\ref{STOKES SOLVE} in each time step.
489    In oder to perform step~\ref{IKM iteration 2} we need to calculate the $\eta\hackscore{eff}$ which
490    is a function of $\sigma\hackscore{ij}$ via $\tau$.  To get $\tau$ and $\eta\hackscore{eff}$ we need to solve the
491    non-linear equation
492  \begin{equation}  \begin{equation}
493  \begin{array}{rcl}  \tau = \eta\hackscore{eff} \cdot \dot{\gamma}\hackscore{total} \mbox{ with }
494  -\left(\eta\hackscore{eff}(dv\hackscore{i,j}+ dv\hackscore{i,j}  \dot{\gamma}\hackscore{total} = \sqrt{ 2 \left( D\hackscore{ij}' +
 )\right)\hackscore{,j} & = & F\hackscore{i}+ \sigma\hackscore{ij,j}'-p\hackscore{,i} \\  
 \frac{1}{\eta\hackscore{eff}} dp & = & - v\hackscore{i,i}^+  
 \end{array}  
 \label{IKM iteration 2}  
 \end{equation}  
 where $v^+=v+dv$. As this problem is non-linear the Jacobi-free Newton-GMRES method is used with the norm  
 \begin{equation}  
 \|(v, p)\|^2= \int\hackscore{\Omega} v\hackscore{i,j}^2 + \frac{1}{\bar{\eta}^2\hackscore{eff}} p^2 \; dx  
 \label{IKM iteration 3}  
 \end{equation}  
 where  $\bar{\eta}\hackscore{eff}$ is the caracteristic viscosity, for instance:  
 \begin{equation}  
 \frac{1}{\bar{\eta}\hackscore{eff}} = \frac{1}{\tau^{-}}+\sum\hackscore{q}  \frac{1}{\eta^{q}\hackscore{N}}  
 \label{IKM iteration 4}  
 \end{equation}  
 In oder to perform step~\ref{IKM iteration 2} we need to calculate the $\eta\hackscore{eff}$ as well as $\sigma\hackscore{ij}'$ while via $\tau$ the first is a function of the latter. The priority is the  
 calculation of $\eta\hackscore{eff}$ with the Newton-Raphson scheme. This value can then be used to calculate  
 $\sigma\hackscore{ij}'$ via~\ref{IKM-EQU-10}. We need to solve  
 \begin{equation}  
 \tau = \eta\hackscore{eff} \cdot \epsilon \mbox{ with }  
 \epsilon = \sqrt{ 2 \left( D\hackscore{ij}' +  
495  \frac{1}{  2 \mu \; dt} \sigma\hackscore{ij}^{'-}\right)^2}  \frac{1}{  2 \mu \; dt} \sigma\hackscore{ij}^{'-}\right)^2}
496  \label{IKM iteration 5}  \label{IKM iteration 5}
497  \end{equation}  \end{equation}
498  The Newton scheme takes the form  The Newton scheme takes the form
499  \begin{equation}  \begin{equation}
500  \tau\hackscore{n+1} = \min(\tau\hackscore{n} - \frac{\tau\hackscore{n} - \eta\hackscore{eff}  \cdot \epsilon}{1 - \eta\hackscore{eff}'  \cdot  \epsilon}, \tau\hackscore{Y} + \beta \; p)  \tau\hackscore{n+1} = \min(\tau\hackscore{n} - \frac{\tau\hackscore{n} - \eta\hackscore{eff}  \cdot \dot{\gamma}\hackscore{total}}{1 - \eta\hackscore{eff}'  \cdot  \dot{\gamma}\hackscore{total}}, \tau\hackscore{Y} + \beta \; p)
 = \min(\frac{\eta\hackscore{eff} - \tau\hackscore{n}  \eta\hackscore{eff}'}  
 {1 - \eta\hackscore{eff}'  \cdot  \epsilon}, \frac{\tau\hackscore{Y} + \beta \; p}{\epsilon}) \epsilon  
501  \label{IKM iteration 6}  \label{IKM iteration 6}
502  \end{equation}  \end{equation}
503  where $\eta\hackscore{eff}'$ denotes the derivative of $\eta\hackscore{eff}$ with respect of $\tau$. The second term in $\min$ is droped of $\tau\hackscore{Y} + \beta \; p<0$ or $\epsilon=0$. In fact we have  where $\eta\hackscore{eff}'$ denotes the derivative of $\eta\hackscore{eff}$ with respect of $\tau$. The second term in $\min$ is droped of $\tau\hackscore{Y} + \beta \; p<0$ (?)). We have
504  \begin{equation}  \begin{equation}
505  \eta\hackscore{eff}' = - \eta\hackscore{eff}^2 \left(\frac{1}{\eta\hackscore{eff}}\right)'  \eta\hackscore{eff}' = - \eta\hackscore{eff}^2 \left(\frac{1}{\eta\hackscore{eff}}\right)'
506  \mbox{ with }  \mbox{ with }
# Line 530  where $\eta\hackscore{eff}'$ denotes the Line 510  where $\eta\hackscore{eff}'$ denotes the
510  \begin{equation}\label{IKM-EQU-5XX}  \begin{equation}\label{IKM-EQU-5XX}
511  \left(\frac{1}{\eta^{q}} \right)'  \left(\frac{1}{\eta^{q}} \right)'
512  = \frac{1-\frac{1}{n^{q}}}{\eta^{q}\hackscore{N}} \frac{\tau^{-\frac{1}{n^{q}}}}{(\tau\hackscore{t}^q)^{1-\frac{1}{n^{q}}}}  = \frac{1-\frac{1}{n^{q}}}{\eta^{q}\hackscore{N}} \frac{\tau^{-\frac{1}{n^{q}}}}{(\tau\hackscore{t}^q)^{1-\frac{1}{n^{q}}}}
513  = \frac{1-\frac{1}{n^{q}}}{ \tau \eta^{q}}  = \frac{1-\frac{1}{n^{q}}}{\tau}\frac{1}{\eta^{q}}
514  \end{equation}  \end{equation}
515  Notice that allways $\eta\hackscore{eff}'\le 0$ which makes the denomionator in~\ref{IKM iteration 6}  Notice that allways $\eta\hackscore{eff}'\le 0$ which makes the denomionator in~\ref{IKM iteration 6}
516  positive.  positive.
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