update

Author: ralfHielscher

images/SingleSlipModel.png

@@ -81,22 +81,21 @@
 {% endhighlight %}
 
 {% highlight matlab %}
-% We may visualize the orientation depedence of the spin tensor as a quiver
-% plot
-plot(Omega,'section','sigma')
-{% endhighlight %}
-<center>
-{% include inline_image.html file="SingleSlipModel_01.png" %}
-</center><p>or as the divergence of this vectorfield</p>
-{% highlight matlab %}
-plotSection(div(Omega),'sigma')
+% We may visualize the orientation depedence of the spin tensor by plotting
+% its divergence in sigma sections and on top of it the spin tensors as a
+% quiver plot
 
+plotSection(div(Omega),'sigma','noGrid')
 mtexColorMap blue2red
 mtexColorbar
+
+hold on
+plot(Omega,'add2all','linewidth',1,'color','k')
+hold off
 {% endhighlight %}
 <center>
-{% include inline_image.html file="SingleSlipModel_02.png" %}
-</center><p>The divergence plots can be read as follows. Negative (blue) regions indicate orientations that increase in volume, whereas positive regions indicate orientations that decrease in volume. Accordingly, we expect the texture to become more and more concentrated within the blue regions. In the example example illustrated above with only the second slip system beeing active, we would expect the c-axis to align more and more with the the z-direction.</p><h2 id="10">Solutions of the Continuity Equation</h2><p>The solutions of the continuity equation can be analytically computed and are available via the command <a href="SO3FunSBF.SO3FunSBF.html"><code class="language-plaintext highlighter-rouge">SO3FunSBF</code></a>. This command takes as input the specific slips system <code class="language-plaintext highlighter-rouge">sS</code> and the makroscopic strain tensor <code class="language-plaintext highlighter-rouge">E</code></p>
+{% include inline_image.html file="SingleSlipModel_01.png" %}
+</center><p>The divergence plots can be read as follows. Negative (blue) regions indicate orientations that increase in volume, whereas positive regions indicate orientations that decrease in volume. Accordingly, we expect the texture to become more and more concentrated within the blue regions. In the example example illustrated above with only the second slip system beeing active, we would expect the c-axis to align more and more with the the z-direction.</p><h2 id="9">Solutions of the Continuity Equation</h2><p>The solutions of the continuity equation can be analytically computed and are available via the command <a href="SO3FunSBF.SO3FunSBF.html"><code class="language-plaintext highlighter-rouge">SO3FunSBF</code></a>. This command takes as input the specific slips system <code class="language-plaintext highlighter-rouge">sS</code> and the makroscopic strain tensor <code class="language-plaintext highlighter-rouge">E</code></p>
 {% highlight matlab %}
 odf1 = SO3FunSBF(sSOli(1),E)
 odf2 = SO3FunSBF(sSOli(2),E)
@@ -130,43 +129,44 @@
 plotSection(odf2,'sigma')
 {% endhighlight %}
 <center>
-{% include inline_image.html file="SingleSlipModel_03.png" %}
-</center><p>Lets visualize these solution by their pole figures</p>
+{% include inline_image.html file="SingleSlipModel_02.png" %}
+</center><p>We observe exactly the concentration of the c-axis around z as predicted by the model. This can be seen even more clear when looking a the pole figures</p>
 {% highlight matlab %}
 h = Miller({1,0,0},{0,1,0},{0,0,1},csOli);
-plotPDF(odf1,h,'resolution',2*degree,'colorRange','equal')
+
+plotPDF(odf2,h,'resolution',2*degree,'colorRange','equal')
 mtexColorbar
 {% endhighlight %}
 <center>
-{% include inline_image.html file="SingleSlipModel_04.png" %}
-</center>
+{% include inline_image.html file="SingleSlipModel_03.png" %}
+</center><p>For completeness the pole figures of the other two basis functions.</p>
 {% highlight matlab %}
-plotPDF(odf2,h,'resolution',2*degree,'colorRange','equal')
+plotPDF(odf1,h,'resolution',2*degree,'colorRange','equal')
 mtexColorbar
 {% endhighlight %}
 <center>
-{% include inline_image.html file="SingleSlipModel_05.png" %}
+{% include inline_image.html file="SingleSlipModel_04.png" %}
 </center>
 {% highlight matlab %}
 plotPDF(odf3,h,'resolution',2*degree,'colorRange','equal')
 mtexColorbar
 {% endhighlight %}
 <center>
-{% include inline_image.html file="SingleSlipModel_06.png" %}
+{% include inline_image.html file="SingleSlipModel_05.png" %}
 </center><p>We observe that the pole figure with respect to \(n \times b\) is always uniform, where \(n\) is the slip normal and \(b\) is the slip direction.</p><p>Since in practice all three slip systems are active we can model the resulting ODF as a linear combination of the different basis functions</p>
 {% highlight matlab %}
 plotPDF(odf1 + odf2 + odf3,h,'resolution',2*degree,'colorRange','equal')
 mtexColorbar
 {% endhighlight %}
 <center>
-{% include inline_image.html file="SingleSlipModel_07.png" %}
-</center><h2 id="17">Checking the for steady state</h2><p>We may also check for which orientations an ODF is already in a steady state of the continous equation, i.e., the time derivative \(\text{div}(f \Omega) = 0\) is zero.</p>
+{% include inline_image.html file="SingleSlipModel_06.png" %}
+</center><h2 id="16">Checking the for steady state</h2><p>We may also check for which orientations an ODF is already in a steady state of the continous equation, i.e., the time derivative \(\text{div}(f \Omega) = 0\) is zero.</p>
 {% highlight matlab %}
 plotSection(div(odf2 .* Omega),'sigma')
 mtexColorMap blue2red
 mtexColorbar
 setColorRange(max(abs(clim))*[-1,1])
 {% endhighlight %}
 <center>
-{% include inline_image.html file="SingleSlipModel_08.png" %}
+{% include inline_image.html file="SingleSlipModel_07.png" %}
 </center></div></body></html>

images/SingleSlipModel_01.png

@@ -110,11 +110,11 @@ <h2 id="4">Properties of grain boundaries</h2><p>A variable of type grain bounda
 
   Bunge Euler angles in degree
      phi1     Phi    phi2
-  90.5949 86.0962 269.995
+  209.646 93.9045  210.23
 
 
-      plane parallel   direction parallel      fit
-(01-1-1) || (-110-1)   [10-11] || [10-1-1]     0.628°
+     plane parallel   direction parallel      fit
+(1-10-1) || (10-11)   [01-1-1] || [1-10-1]     0.48°
 {% endhighlight %}
 <p>Bases on the output above we may now define the special orientation relationship as</p>
 {% highlight matlab %}