diff --git a/web/content/docs/benchmarks/creepbgra/CreepBRGa.pandoc b/web/content/docs/benchmarks/creepbgra/CreepBRGa.pandoc index 4d9b0d5b3b830f030e8e4475afc725528019d268..3d643ea60bfdfcb976b56f722e9d97dec1e76419 100644 --- a/web/content/docs/benchmarks/creepbgra/CreepBRGa.pandoc +++ b/web/content/docs/benchmarks/creepbgra/CreepBRGa.pandoc @@ -38,12 +38,12 @@ The creep strain rate is then expressed as $$\begin{gathered} \dot { \mathbf \epsilon}^c ({ \sigma})= {\dfrac{\partial g^c}{\partial {\bar\sigma}}} {\dfrac{\partial { \bar\sigma}}{\partial { \mathbf \sigma}}} -=\sqrt{{\frac{2}{3}}}{\dfrac{\partial g^c}{\partial {\bar \sigma}}}\dfrac{{\mathbf s}}{{\left\Vert{\mathbf s}\right\Vert}} +=\sqrt{{\frac{3}{2}}}{\dfrac{\partial g^c}{\partial {\bar \sigma}}}\dfrac{{\mathbf s}}{{\left\Vert{\mathbf s}\right\Vert}} \end{gathered}$$ The above creep strain rate expression must be valid for problems independent from the Euclidean dimension. Applying the creep rate equation to a uniaxial stress state ${\mathbf \sigma} = \mathrm{diag}[\sigma_1, 0, 0]$, - which gives ${\mathbf s} = \mathrm{diag}[\frac{2}{3}\sigma_1, -\frac{2}{3}\sigma_1, -\frac{2}{3}\sigma_1]$, + which gives ${\mathbf s} = \mathrm{diag}[\frac{2}{3}\sigma_1, -\frac{1}{3}\sigma_1, -\frac{1}{3}\sigma_1]$, we have $$\begin{gathered} \dot { \epsilon_1}^c = {\dfrac{\partial g^c}{\partial { \bar\sigma}}}=Ae^{-Q/R_uT}\left(\dfrac{{ \sigma_1}}{{ \sigma}_f}\right)^m @@ -55,7 +55,7 @@ $$\begin{gathered} Therefore, the creep strain rate for multi-dimensional problems can be derived as $$\begin{gathered} - \dot { \mathbf \epsilon}^c ({ \sigma})={\color{red} {\sqrt{\frac{3}{2}}}}Ae^{-Q/R_uT}\left(\dfrac{{ \sigma}}{{ \sigma}_f}\right)^m\dfrac{{\mathbf s}}{{\left\Vert{\mathbf s}\right\Vert}} + \dot { \mathbf \epsilon}^c ({ \sigma})={\color{red} {\sqrt{\frac{3}{2}}}}Ae^{-Q/R_uT}\left(\dfrac{{\bar \sigma}}{{ \sigma}_f}\right)^m\dfrac{{\mathbf s}}{{\left\Vert{\mathbf s}\right\Vert}} \end{gathered}$$ Stress integration @@ -78,8 +78,8 @@ $$\begin{gathered} (\dot { \mathbf \epsilon}- \dot { \mathbf \epsilon}^T- \dot { \mathbf \epsilon}^c) \end{gathered}$$ 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, +$\mathbf{C} = \lambda \mathcal{I} + 2G \mathbf I \otimes \mathbf I$ +with $\mathcal{I}$ the fourth order identity, $\mathbf I$ the second order identity, $\lambda$ the Lamé constant, $G$ the shear modulus, and $\otimes$ the tensor product notation.