# Diff of /trunk/doc/user/Models.tex

revision 2437 by gross, Wed May 20 08:43:43 2009 UTC revision 2438 by gross, Tue May 26 02:26:52 2009 UTC
# Line 401  D\hackscore{ij}^{q'}=\frac{1}{2 \eta^{q} Line 401  D\hackscore{ij}^{q'}=\frac{1}{2 \eta^{q}
401  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
402  betwee the the strain in material $q$  betwee the the strain in material $q$
403  \label{IKM-EQU-5b}  \label{IKM-EQU-5b}
404  \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)^{1-n^{q}}
405  \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}}
406
407  for a given power law coefficients $n^{q}\ge1$ and transition stresses $\tau\hackscore{t}^q$, see~\cite{Muhlhaus2005}.  for a given power law coefficients $n^{q}\ge1$ and transition stresses $\tau\hackscore{t}^q$, see~\cite{Muhlhaus2005}.
# Line 515  In fact we have Line 515  In fact we have
515  As  As
516  \label{IKM-EQU-47}  \label{IKM-EQU-47}
517  \left(\frac{1}{\eta^{q}} \right)'  \left(\frac{1}{\eta^{q}} \right)'
518  = \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{n^{q}-1}{\eta^{q}\hackscore{N}} \cdot \frac{\tau^{n^{q}-2}}{(\tau\hackscore{t}^q)^{n^{q}-1}}
519  = \frac{1-\frac{1}{n^{q}}}{\eta^{q}}\frac{1}{\tau}  = \frac{n^{q}-1}{\eta^{q}}\cdot\frac{1}{\tau}
520
521  we have  we have
522  \label{IKM-EQU-48}  \label{IKM-EQU-48}
523  \Theta' = \frac{1}{\tau} \omega \mbox{ with } \omega = \sum\hackscore{q}\frac{1-\frac{1}{n^{q}}}{\eta^{q}}  \Theta' = \frac{1}{\tau} \omega \mbox{ with } \omega = \sum\hackscore{q}\frac{n^{q}-1}{\eta^{q}}
524