[cig-commits] r14624 - doc/snac

echoi at geodynamics.org echoi at geodynamics.org
Wed Apr 8 11:36:52 PDT 2009 

Author: echoi
Date: 2009-04-08 11:36:52 -0700 (Wed, 08 Apr 2009)
New Revision: 14624

Modified:
   doc/snac/snac.lyx
Log:
Expanded "Constitutive Update" section by adding "Hardening/Softening" subsection.



Modified: doc/snac/snac.lyx
===================================================================
--- doc/snac/snac.lyx	2009-04-08 04:30:34 UTC (rev 14623)
+++ doc/snac/snac.lyx	2009-04-08 18:36:52 UTC (rev 14624)
@@ -1390,7 +1390,6 @@
 \end_inset
 
 .
- 
 \end_layout
 
 \begin_layout Standard
@@ -1475,10 +1474,6 @@
 
 \end_layout
 
-\begin_layout Plain Layout
-
-\end_layout
-
 \end_inset
 
 
@@ -1649,10 +1644,6 @@
 
 \end_layout
 
-\begin_layout Plain Layout
-
-\end_layout
-
 \end_inset
 
 
@@ -1731,7 +1722,7 @@
 \begin_inset Formula $\beta$
 \end_inset
 
- is the magnitude of the plastic flow.
+ is a proortionality constant.
  
 \begin_inset Formula $\beta$
 \end_inset
@@ -1750,6 +1741,26 @@
 \end_inset
 
  is the elastic moduli tensor.
+ This condition is called the 
+\emph on
+consistency condition
+\emph default
+ and the parameter 
+\begin_inset Formula $\beta$
+\end_inset
+
+ is thus called the 
+\emph on
+consistency parameter
+\emph default
+ 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "SimoHugh2004"
+
+\end_inset
+
+.
  In the principal component representation, 
 \begin_inset Formula $\sigma_{A}=a_{AB}^{e}e_{B}$
 \end_inset
@@ -1803,6 +1814,350 @@
 
 \end_layout
 
+\begin_layout Paragraph
+Hardening/softening
+\end_layout
+
+\begin_layout Standard
+If strain hardening (including softening as a negative hardening) is considered,
+ a more general return maping is required.
+ For completeness, we review the 
+\emph on
+cutting-plane
+\emph default
+ algorithm and then discuss the simplified version for (piecewise-)linear
+ hardening implemented in SNAC.
+ For more detailed discussion, readers are referred to 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "SimoHugh2004"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Here are some preliminary definitions:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{equation}
+\dot{\epsilon}_{i}^{p}=\dot{\beta}\frac{\partial G}{\partial\sigma_{i}}\label{eq:plastic strain rate}\end{equation}
+
+\end_inset
+
+We define an internal variable such that plastic parameters are defined
+ as a function of that variable.
+ Our particular choice is
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{equation}
+\epsilon^{p*}=\sqrt{\frac{1}{2}\left\{ (\epsilon_{1}^{p}-\bar{\epsilon^{p}})^{2}+(\epsilon_{2}^{p}-\bar{\epsilon^{p}})^{2}+(\epsilon_{3}^{p}-\bar{\epsilon^{p}})^{2}+\bar{\epsilon^{p}}^{2}\right\} }\label{eq:internal variable}\end{equation}
+
+\end_inset
+
+where 
+\begin_inset Formula $\bar{\epsilon^{p}}=\frac{1}{3}(\epsilon_{1}^{p}+\epsilon_{2}^{p}+\epsilon_{3}^{p})$
+\end_inset
+
+.
+ The time rate of change of the internal variable is then
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{align}
+\dot{\epsilon}^{p*} & =\frac{\partial\epsilon^{p*}}{\partial\epsilon_{i}}\dot{\epsilon}_{i}^{p}\nonumber \\
+ & =\dot{\beta}\frac{\partial\epsilon^{p*}}{\partial\epsilon_{i}^{p}}\frac{\partial G(\mathbf{\sigma},\epsilon^{p*})}{\partial\sigma_{i}}\label{eq:internal variable rate}\\
+ & =-\dot{\beta}r(\mathbf{\sigma},\epsilon^{p*})\nonumber \end{align}
+
+\end_inset
+
+For the simpler notation, a new function 
+\begin_inset Formula $r(\mathbf{\sigma},\epsilon^{p*})$
+\end_inset
+
+ has been defined above.
+ Finally, we require that the updated stress stay on the yield surface (discrete
+ consistency condition): i.e., 
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{equation}
+\bar{F}(\Delta\beta)\equiv F(\mathbf{\sigma}(\Delta\beta),\epsilon^{p*}(\Delta\beta))=0,\label{eq:discrete consistency condition}\end{equation}
+
+\end_inset
+
+where 
+\begin_inset Formula $\Delta\beta$
+\end_inset
+
+ is the increment of the consistency paramter during the interval between
+ 
+\begin_inset Formula $t_{n}$
+\end_inset
+
+and 
+\begin_inset Formula $t_{n+1}$
+\end_inset
+
+.
+ For later uses, we further derive the derivative of 
+\begin_inset Formula $\bar{F}$
+\end_inset
+
+ with respect to 
+\begin_inset Formula $\beta$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{equation}
+\frac{d\bar{F}}{d\Delta\beta}=\frac{\partial F}{\partial\sigma_{i}}\cdot\frac{\partial\sigma_{i}}{\partial\Delta\beta}+\frac{\partial F}{\partial\epsilon^{P*}}\frac{\partial\epsilon^{p*}}{\partial\Delta\beta}\label{eq:yield function derivative w.r.t. internal variable increment}\end{equation}
+
+\end_inset
+
+Since 
+\begin_inset Formula $\sigma_{n+1}=\mathbf{a}^{e}\cdot(\epsilon_{n+1}-\epsilon_{n+1}^{p})=\mathbf{a}^{e}\cdot(\epsilon_{n+1}-(\epsilon_{n}^{p}+\Delta\beta\partial_{\sigma}G(\sigma_{n+1},\epsilon_{n+1}^{p*}))$
+\end_inset
+
+ and 
+\begin_inset Formula $\epsilon_{n+1}$
+\end_inset
+
+ and 
+\begin_inset Formula $\epsilon_{n}^{p}$
+\end_inset
+
+ is constant during an time interval,
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{equation}
+\frac{\partial\sigma_{n+1}}{\partial\Delta\beta}=-\mathbf{a}^{e}\cdot\frac{\partial G}{\partial\sigma_{n+1}}\label{eq:stress derivative w.r.t. consistency parameter}\end{equation}
+
+\end_inset
+
+From 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:internal variable rate"
+
+\end_inset
+
+,
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{equation}
+\frac{\partial\Delta\epsilon^{p*}}{\partial\Delta\beta}=-r\label{eq:internal var increment derivative w.r.t. consistency parameter increment}\end{equation}
+
+\end_inset
+
+By substituting 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:stress derivative w.r.t. consistency parameter"
+
+\end_inset
+
+
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:internal var increment derivative w.r.t. consistency parameter increment"
+
+\end_inset
+
+ into 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:yield function derivative w.r.t. internal variable increment"
+
+\end_inset
+
+, we get
+\begin_inset Formula \begin{equation}
+\frac{d\bar{F}}{d\Delta\beta}=\frac{\partial F}{\partial\mathbf{\sigma}}\cdot\left(-\mathbf{a}^{e}\cdot\frac{\partial G}{\partial\mathbf{\sigma}}\right)+\frac{\partial F}{\partial\epsilon^{P*}}(-r)\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Now we present the algorithm of updating stress, internal variable and consisten
+cy parameter iteratively in case of a general non-linear hardening.
+ 
+\end_layout
+
+\begin_layout Enumerate
+Initialization
+\begin_inset Formula \begin{align*}
+\epsilon_{n+1}^{p(0)} & =\epsilon_{n}^{p}\\
+\epsilon_{n+1}^{p*(0)} & =\epsilon_{n}^{p*}\\
+\Delta\beta_{n+1}^{(0)} & =0\end{align*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Update stresses and evaluate 
+\begin_inset Formula $F$
+\end_inset
+
+:
+\begin_inset Formula \begin{align*}
+\sigma_{n+1}^{(k)} & =\mathbf{a}\cdot(\epsilon_{n+1}-\epsilon_{n+1}^{p(k)})\\
+F_{n+1}^{(k)} & =F(\sigma_{n+1}^{(k)},\epsilon_{n+1}^{p*(k)})\end{align*}
+
+\end_inset
+
+If 
+\begin_inset Formula $F_{n+1}^{(k)}\leq0$
+\end_inset
+
+? exit; otherwise go to step 3.
+\end_layout
+
+\begin_layout Enumerate
+Incremental upate for 
+\begin_inset Formula $k+1$
+\end_inset
+
+:
+\begin_inset Formula \begin{align*}
+\Delta^{2}\beta & =\frac{F_{n+1}^{(k)}}{-\frac{\partial F}{\partial\Delta\beta}}=\frac{F_{n+1}^{(k)}}{\frac{\partial F}{\partial\sigma}\cdot\mathbf{a}\cdot\frac{\partial G}{\partial\sigma}+\frac{\partial F}{\partial\epsilon^{p*}}r}\\
+\epsilon_{n+1}^{p(k+1)} & =\epsilon_{n+1}^{p(k)}+\Delta^{2}\beta\frac{\partial G}{\partial\sigma}(\sigma_{n+1}^{(k)},\epsilon_{n+1}^{p*(k)})\\
+\epsilon_{n+1}^{p*(k+1)} & =\epsilon_{n+1}^{p*(k)}-\Delta^{2}\beta r(\sigma_{n+1}^{(k)},\epsilon_{n+1}^{p*(k)})\\
+ & =\epsilon_{n+1}^{p*(k)}+\Delta^{2}\beta\left[\left(\frac{\partial\epsilon^{p*}}{\partial\epsilon^{p}}\right)_{n+1}^{(k)}\cdot\frac{\partial G}{\partial\sigma}(\sigma_{n+1}^{(k)},\epsilon_{n+1}^{p*(k)})\right]\\
+\Delta\beta_{n+1}^{(k+1)} & =\Delta\beta_{n+1}^{(k)}+\Delta^{2}\beta\end{align*}
+
+\end_inset
+
+Go back to step 2.
+\end_layout
+
+\begin_layout Standard
+This algorithm is just a standard Newton method applied to the discrete
+ consistency equation (
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:discrete consistency condition"
+
+\end_inset
+
+).
+ Thus, it should be obvious that the plastic correction is accomplished
+ by the single step update if the yield function, 
+\begin_inset Formula $F$
+\end_inset
+
+, is a linear function of the consistency variable.
+ In this case, eq.
+ [
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:flow parameter for shear failure"
+
+\end_inset
+
+,
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:flow parameter for tensile failure"
+
+\end_inset
+
+] are immediately retrieved.
+\end_layout
+
+\begin_layout Standard
+For reference, we further derive the explicit form of the function 
+\begin_inset Formula $r$
+\end_inset
+
+ in terms of principal plastic strains and stresses by working out the partial
+ derivative, 
+\begin_inset Formula $\partial\epsilon^{p*}/\partial\epsilon^{p}$
+\end_inset
+
+.
+ From eq.
+ 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:internal variable"
+
+\end_inset
+
+,
+\begin_inset Formula \begin{align*}
+\frac{\partial\epsilon^{p*}}{\partial\epsilon_{i}^{p}} & =\frac{1}{2\epsilon^{p*}}\frac{1}{2}\left\{ \sum_{j=1}^{3}2(\epsilon_{j}^{p}-\bar{\epsilon^{p}})(\delta_{ij}-\frac{1}{3})+\frac{2}{3}\bar{\epsilon^{p}}\right\} \\
+\Longrightarrow\frac{\partial\epsilon^{p*}}{\partial\epsilon_{1}^{p}} & =\frac{1}{2\epsilon^{p*}}\left\{ \frac{2}{3}(\epsilon_{1}^{p}-\bar{\epsilon^{p}})-\frac{1}{3}(\epsilon_{2}^{p}-\bar{\epsilon^{p}})-\frac{1}{3}(\epsilon_{3}^{p}-\bar{\epsilon^{p}})+\frac{1}{3}\bar{\epsilon^{p}}\right\} \\
+\frac{\partial\epsilon^{p*}}{\partial\epsilon_{2}^{p}} & =\frac{1}{2\epsilon^{p*}}\left\{ -\frac{1}{3}(\epsilon_{1}^{p}-\bar{\epsilon^{p}})+\frac{2}{3}(\epsilon_{2}^{p}-\bar{\epsilon^{p}})-\frac{1}{3}(\epsilon_{3}^{p}-\bar{\epsilon^{p}})+\frac{1}{3}\bar{\epsilon^{p}}\right\} \\
+\frac{\partial\epsilon^{p*}}{\partial\epsilon_{3}^{p}} & =\frac{1}{2\epsilon^{p*}}\left\{ -\frac{1}{3}(\epsilon_{1}^{p}-\bar{\epsilon^{p}})-\frac{1}{3}(\epsilon_{2}^{p}-\bar{\epsilon^{p}})+\frac{2}{3}(\epsilon_{3}^{p}-\bar{\epsilon^{p}})+\frac{1}{3}\bar{\epsilon^{p}}\right\} \end{align*}
+
+\end_inset
+
+With 
+\begin_inset Formula $\partial G/\partial\sigma$
+\end_inset
+
+ being 
+\begin_inset Formula $[1,0,-N_{\psi}]$
+\end_inset
+
+, the function 
+\begin_inset Formula $r(\sigma,\epsilon^{p*})$
+\end_inset
+
+ is given by
+\begin_inset Formula \begin{align*}
+r(\sigma,\epsilon^{p*}) & =-\left(\frac{\partial\epsilon^{p*}}{\partial\epsilon^{p}}\right)\cdot\left(\frac{\partial G}{\partial\sigma}\right)\\
+ & =-\frac{1}{2\epsilon^{p*}}\left[\frac{1}{3}\left(2+N_{\psi}\right)(\epsilon_{1}^{p}-\bar{\epsilon^{p}})-\frac{1}{3}\left(1-N_{\psi}\right)(\epsilon_{2}^{p}-\bar{\epsilon^{p}})-\frac{1}{3}\left(1+2N_{\psi}\right)(\epsilon_{3}^{p}-\bar{\epsilon^{p}})+\frac{1}{3}\left(1-N_{\psi}\right)\bar{\epsilon^{p}}\right]\end{align*}
+
+\end_inset
+
+Note that when 
+\begin_inset Formula $\psi\simeq0^{\circ}$
+\end_inset
+
+, 
+\begin_inset Formula $N_{\psi}\simeq1$
+\end_inset
+
+ and 
+\begin_inset Formula $\bar{\epsilon^{p}}\simeq0$
+\end_inset
+
+.
+ Then, the value of 
+\begin_inset Formula $r$
+\end_inset
+
+ becomes 
+\begin_inset Formula $\sim$
+\end_inset
+
+1.
+ This analysis and the strain weakening only through piecewise linear variation
+ in cohesion (i.e., 
+\begin_inset Formula $F$
+\end_inset
+
+ is linear w.rt.
+ the consistency paramter) justify the consistency parameter computaiton
+ currenctly implemented in SNAC.
+ However, if any non-linearity is introduced such as friction angle varying
+ with the internal variable and cohesion as a non-linear function of the
+ internal variable, the interation presented above must be performed.
+\end_layout
+
 \begin_layout Subsection
 Remeshing
 \end_layout

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