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revision 1780 by jfenwick, Mon Aug 11 03:33:40 2008 UTC revision 1781 by jfenwick, Thu Sep 11 05:03:14 2008 UTC
# Line 54  T(x)= T\hackscore{0} e^{-\beta \|x-x^{c} Line 54  T(x)= T\hackscore{0} e^{-\beta \|x-x^{c}
54  with a given positive constant $\beta$ and location $x^{c}$ in the domain. Later in \Sec{MODELFRAME} we will use  with a given positive constant $\beta$ and location $x^{c}$ in the domain. Later in \Sec{MODELFRAME} we will use
55  $T$ from a time-dependent temperature diffusion problem as discussed in \Sec{DIFFUSION CHAP}.  $T$ from a time-dependent temperature diffusion problem as discussed in \Sec{DIFFUSION CHAP}.
56    
57  When we insert~\eqn{HEATEDBLOCK linear elastic} we get a second oder system of linear PDEs for the displacements $u$ which is called  When we insert~\eqn{HEATEDBLOCK linear elastic} we get a second order system of linear PDEs for the displacements $u$ which is called
58  the Lame equation\index{Lame equation}. We want to solve  the Lame equation\index{Lame equation}. We want to solve
59  this using the \LinearPDE class to this. For a system of PDEs and a solution with several components the \LinearPDE class  this using the \LinearPDE class to this. For a system of PDEs and a solution with several components the \LinearPDE class
60  takes PDEs of the form  takes PDEs of the form
# Line 75  $r$ and $q$ are each \RankOne. Line 75  $r$ and $q$ are each \RankOne.
75  We can easily identify the coefficients in~\eqn{LINEARPDE.SYSTEM.1 TUTORIAL}:  We can easily identify the coefficients in~\eqn{LINEARPDE.SYSTEM.1 TUTORIAL}:
76  \begin{eqnarray}\label{LINEARPDE ELASTIC COEFFICIENTS}  \begin{eqnarray}\label{LINEARPDE ELASTIC COEFFICIENTS}
77  A\hackscore{ijkl}=\lambda \delta\hackscore{ij} \delta\hackscore{kl} + \mu (  A\hackscore{ijkl}=\lambda \delta\hackscore{ij} \delta\hackscore{kl} + \mu (
78  +\delta\hackscore{ik} \delta\hackscore{jl}  \delta\hackscore{ik} \delta\hackscore{jl}
79  \delta\hackscore{il} \delta\hackscore{jk}) \\  + \delta\hackscore{il} \delta\hackscore{jk}) \\
80  X\hackscore{ij}=(\lambda+\frac{2}{3} \mu) \;  \alpha \; (T-T\hackscore{ref})\delta\hackscore{ij} \\  X\hackscore{ij}=(\lambda+\frac{2}{3} \mu) \;  \alpha \; (T-T\hackscore{ref})\delta\hackscore{ij} \\
81  \end{eqnarray}  \end{eqnarray}
82  The characteristic function $q$ defining the locations and components where constraints are set is given by:  The characteristic function $q$ defining the locations and components where constraints are set is given by:
# Line 104  The \LinearPDE class is notified of this Line 104  The \LinearPDE class is notified of this
104  After we have solved the Lame equation we want to analyse the actual stress distribution. Typically the von--Mises stress\index{von--Mises stress} defined by  After we have solved the Lame equation we want to analyse the actual stress distribution. Typically the von--Mises stress\index{von--Mises stress} defined by
105  \begin{equation}  \begin{equation}
106  \sigma\hackscore{mises} = \sqrt{  \sigma\hackscore{mises} = \sqrt{
107  \frac{1}{6} ((\sigma\hackscore{00}-\sigma\hackscore{11})^2+  \frac{1}{6} ((\sigma\hackscore{00}-\sigma\hackscore{11})^2
108                 (\sigma\hackscore{11}-\sigma\hackscore{22})^2              + (\sigma\hackscore{11}-\sigma\hackscore{22})^2
109                 (\sigma\hackscore{22}-\sigma\hackscore{00})^2)              + (\sigma\hackscore{22}-\sigma\hackscore{00})^2)
110  +  \sigma\hackscore{01}^2+\sigma\hackscore{12}^2+\sigma\hackscore{20}^2}  +  \sigma\hackscore{01}^2+\sigma\hackscore{12}^2+\sigma\hackscore{20}^2}
111  \end{equation}  \end{equation}
112  is used to detect material damage. Here we want to calculate the von--Mises and write the stress to a file for visualization.  is used to detect material damage. Here we want to calculate the von--Mises and write the stress to a file for visualization.

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