diff --git a/web/content/docs/benchmarks/creepbgra/CreepBRGa.md b/web/content/docs/benchmarks/creepbgra/CreepBRGa.md index f2d0f8fbfec290daec48d721ecbc35c1df9bd2ad..76bd21ac23c81bec8570649874c7470c4be02512 100644 --- a/web/content/docs/benchmarks/creepbgra/CreepBRGa.md +++ b/web/content/docs/benchmarks/creepbgra/CreepBRGa.md @@ -151,8 +151,35 @@ straightforward as $$\begin{gathered} { \epsilon}=-\dfrac{{ \sigma}_0}{E}-Ae^{-Q/RT}{ \sigma}_0^m t \end{gathered}$$ - The comparison of the results + +The problem is solved as axisymmterical one. Therefore +$$\begin{gathered} +{ \epsilon_z}=-\dfrac{{ \sigma}_0}{E}-Ae^{-Q/RT}{ \sigma}_0^m t +\end{gathered}$$ + +Because of +$$\begin{gathered} +\lambda\epsilon_v^e+2G\epsilon_r^e=\lambda\epsilon_v^e+2G\epsilon_t^e=0 +\end{gathered}$$ + we have +$$\begin{gathered} +\epsilon_r^e=\epsilon_t^e +\end{gathered}$$ + and +$$\begin{gathered} +\sigma_z=2G(\epsilon_z^e-\epsilon_r^e)=2G(\epsilon_z^e-\epsilon_t^e) +\end{gathered}$$ +with $\epsilon^e=\epsilon-\epsilon^c$, +where superscript $e$ means elasticity and $c$ means creep. With the creep strain +formulas that are presented in the above context, the analytical solutions +of the other main strains are + +$$\begin{gathered} +{ \epsilon_r}={ \epsilon_t}=\frac{\sigma_0}{2G}+\epsilon_z+ +\frac{3}{2}A e^{-Q/RT}\sigma_0^m t +\end{gathered}$$ + + The comparison of the result of $\epsilon_z$ obtained by the present multidimensional scheme with the analytical solution is shown in the following figure: {{< img src="../bgra0.png" >}} -