[Web] Added a documentation about BGRa creep model

web/content/docs/benchmarks/CreepBGRa/CreepBRGa.md

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+date = "2018-07-19T15:15:45+01:00"
+title = "BGRa creep model"
+weight = 171
+project = "ThermoMechanics/CreepBGRa/SimpleAxisymmetricCreep/SimpleAxisymmetricCreepWithAnalyticSolution.prj"
+author = "Wenqing Wang"
+
+[menu]
+  [menu.benchmarks]
+    parent = "thermo-mechanics"
+
++++
+
+{{< data-link >}}
+
+The BGRa stationary creep model defines the uniaxial creep strain
+increment as
+$$\Delta{ \epsilon}^c=Ae^{-Q/R_uT}\left(\dfrac{{ \sigma}}{{ \sigma}_f}\right)^m \Delta t
+$$
+ where $A$, $m$ and $Q$ are parameters determined
+by experiments, $Q$ is called activation energy,
+$R_u=8.314472 \mbox{J/(Kmol)}$ is the universal gas constant, and
+${ \sigma}_f$ is a stress scaling factor.
+
+Creep potential and creep strain
+================================
+
+Assume there is a creep potential $g^c$ and the creep induced strain
+rate is given by the flow rule
+$$\dot { \mathbf \epsilon}^c ({ \sigma})= \frac{2}{3}{\dfrac{\partial g^c}{\partial { \mathbf \sigma}}}
+$$ where ${ \sigma}$ is the equivalence stress
+defined by
+${ \sigma}={\sqrt{{\frac{3}{2}}}}{\left\Vert{\mathbf s}\right\Vert}$
+with ${\mathbf s}= { \mathbf \sigma}-\frac{1}{3}{ \sigma}{\mathbf I}$,
+the deviatoric stress. The creep strain rate is then expressed as
+$$\begin{gathered}
+\dot { \mathbf \epsilon}^c ({ \sigma})= \frac{2}{3} {\dfrac{\partial g^c}{\partial { \sigma}}}{\dfrac{\partial { \sigma}}{\partial { \mathbf \sigma}}}=\sqrt{{\frac{2}{3}}}{\dfrac{\partial g^c}{\partial { \sigma}}}\dfrac{{\mathbf s}}{{\left\Vert{\mathbf s}\right\Vert}}={\dfrac{\partial g^c}{\partial { \sigma}}}\dfrac{{\mathbf s}}{\sigma}
+\end{gathered}$$
+ The above creep strain rate expression
+must be valid for the problems with different dimension. Applying
+the creep rate equation to one dimensional leads to
+ $$\begin{gathered}
+\dot { \epsilon}^c = {\dfrac{\partial g^c}{\partial { \sigma}}}{\color{red} {\frac{2}{3}}}=Ae^{-Q/R_uT}\left(\dfrac{{ \sigma}}{{ \sigma}_f}\right)^m
+\end{gathered}$$ which says 
+
+$$\begin{gathered}
+ {\dfrac{\partial g^c}{\partial { \sigma}}}={\color{red}{\frac{3}{2}}}Ae^{-Q/R_u T}\left(\dfrac{{ \sigma}}{{ \sigma}_f}\right)^m
+\end{gathered}$$
+
+Therefore, the creep strain rate for multi-dimensional problems can be
+derived as $$\begin{gathered}
+ \dot { \mathbf \epsilon}^c ({ \sigma})={\color{red} {\sqrt{\frac{3}{2}}}}Ae^{-Q/R_uT}\left(\dfrac{{ \sigma}}{{ \sigma}_f}\right)^m\dfrac{{\mathbf s}}{{\left\Vert{\mathbf s}\right\Vert}}
+\end{gathered}$$
+
+Stress integration
+==================
+
+The above creep strain rate expression can be simplified as $$\begin{gathered}
+  \dot { \mathbf \epsilon}^c ({ \sigma}) = b {\left\Vert{\mathbf s}\right\Vert}^{m-1} {\mathbf s}
+\end{gathered}$$
+with
+$$b={\left(\dfrac{3}{2}\right)^{(m+1)/2}} \dfrac{Ae^{-Q/R_uT}}{{{ \sigma}_f}^m}$$
+
+The strain rate under creeping condition can be expressed as
+$$\begin{gathered}
+\dot { \mathbf \epsilon}= \dot { \mathbf \epsilon}^e + \dot { \mathbf \epsilon}^c
+\end{gathered}$$ Due to Hook’s law, the stress rate is
+given by 
+$$\begin{gathered}
+\dot { \mathbf \sigma}= \mathbf{C} \dot { \mathbf \epsilon}^e = \mathbf{C} (\dot { \mathbf \epsilon}- \dot { \mathbf \epsilon}^c) 
+\end{gathered}$$ 
+where
+$\mathbf{C}:= \lambda \mathcal{J} + 2G \mathbf I \otimes \mathbf I  $ 
+with $\mathcal{J}$ the forth order identity, $\mathbf I$ the second order identity,
+$\lambda$ the Lame constant,  $G$ the shear module, and $\otimes$  the tensor
+product notation.
+
+is a forth order tensor. Substituting equation and the expression of $C$
+into the stress rate expression, equation , we have
+ $$\begin{gathered}
+\dot { \mathbf \sigma}=  \mathbf{C} \dot { \mathbf \epsilon}- 2bG {\left\Vert{\mathbf s}\right\Vert}^{m-1} 
+{\mathbf s}
+\end{gathered}$$
+
+The stress of the present time step, $n+1$, is then can be expressed as
+$$\begin{gathered}
+ { \mathbf \sigma}^{n+1} = { \mathbf \sigma}^{n} + \mathbf{C} \Delta { \mathbf \epsilon}-
+ 2bG \Delta t {\left\Vert{\mathbf s}^{n+1}\right\Vert}^{m-1} {\mathbf s}^{n+1}
+\end{gathered}$$
+ To solve the stress, the
+Newton-Raphson is applied to .
+ Let $$\begin{gathered}
+ \mathbf{r}= { \mathbf \sigma}^{n+1} -
+ { \mathbf \sigma}^{n} - \mathbf{C} \Delta
+ { \mathbf \epsilon}+ 2bG \Delta t {\left\Vert{\mathbf s}^{n+1}\right\Vert}^{m-1}
+ {\mathbf s}^{n+1}
+\end{gathered}$$ 
+
+be the residual. The Jacobian
+of is derived as
+ $$\begin{gathered}
+ \mathbf{J}_{{ \mathbf \sigma}}= {\dfrac{\partial \mathbf{r}}{\partial { \mathbf \sigma}^{n+1}}}  =  \mathcal{I}  + 2bG\Delta t {\left\Vert{\mathbf s}^{n+1}\right\Vert}^{m-1} 
+ \left(\mathcal{I} - \frac{1}{3} \mathbf{I} \otimes \mathbf{I} + (m-1){\left\Vert{\mathbf s}^{n+1}\right\Vert}^{-2} {\mathbf s}^{n+1}\otimes {\mathbf s}^{n+1}\right)
+\end{gathered}$$
+
+Consistent tangent
+==================
+
+Once the converged stress integration is obtained, the tangential of
+stress with respect to strain can be obtained straightforwardly by
+applying the partial derivative to equation with respect to strain
+increment as 
+$$\begin{aligned}
+  {\dfrac{\partial { \mathbf \sigma}^{n+1}}{\partial \Delta { \mathbf \epsilon}}} =   \mathbf{C}  - {\dfrac{\partial \left(2bG \Delta t {\left\Vert{\mathbf s}^{n+1}\right\Vert}^{m-1} {\mathbf s}^{n+1}\right)}{\partial \Delta { \mathbf \epsilon}}}
+ \end{aligned}$$ 
+We see that $$\begin{aligned}
+  {\dfrac{\partial { \mathbf \sigma}^{n+1}}{\partial { \mathbf \epsilon}^{n+1}}}  = \mathbf{J}_{{ \mathbf \sigma}}^{-1} \mathbf{C}
+  \end{aligned}$$
+
+If the model is used for the thermo-mechanical problems and the problems
+are solved by the monolithic scheme, the displacement-temperature block
+of the global Jacobian can be derived as
+ $$\begin{aligned}
+ - 2G\dfrac{Q}{R} {{\int}_{\Omega} \dfrac{b}{T^2} {\left\Vert{\mathbf s}^{n+1}\right\Vert}^{m-1}  \mathbf{B}^T  {\mathbf s}^{n+1} \mathrm{d} \Omega}
+ \end{aligned}$$
+
+*Note*: The above rate form of stress integration is implemented in ogs6.
+ Alternatively, one can use a absolute stress integration form, which can be found in the attached 
+ [PDF](../doku_BGRa.pdf).
+
+Example
+=======
+
+We verify the scheme by a one dimensional extension with constant
+pressure of ${ \sigma}_0=5 ~\mbox{MPa}$. The values of the parameters are
+given as $A=0.18 \mbox{d}^{-1}$, $m=5$, $Q=54 ~\mbox{kJ/mol}$,
+$E=25000 ~\mbox{MPa}$, and the temperature is constant everywhere of
+100 $^{\circ}\mbox{C}$. The analytical solution of the strain is given
+straightforward as 
+$$\begin{gathered}
+{ \epsilon}=-\dfrac{{ \sigma}_0}{E}-Ae^{-Q/RT}{ \sigma}_0^m t
+\end{gathered}$$
+ The comparison of the results
+obtained by the present multidimensional scheme with the analytical
+solution is shown in the following figure:
+{{< img src="../bgra0.png" >}}
+