From 597f731e4417173db7db1ea10fbc6101a3b33ff8 Mon Sep 17 00:00:00 2001
From: Wenqing Wang <wenqing.wang@ufz.de>
Date: Mon, 29 Oct 2018 15:59:47 +0100
Subject: [PATCH] [web/BGRa] Corrected two typos and made a consistency of one
symbol within the context
---
web/content/docs/benchmarks/creepbgra/CreepBRGa.pandoc | 10 +++++-----
1 file changed, 5 insertions(+), 5 deletions(-)
diff --git a/web/content/docs/benchmarks/creepbgra/CreepBRGa.pandoc b/web/content/docs/benchmarks/creepbgra/CreepBRGa.pandoc
index 4d9b0d5b3b8..3d643ea60bf 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.
--
GitLab