or as the divergence of this vectorfield
-{% 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 %}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.
The solutions of the continuity equation can be analytically computed and are available via the command SO3FunSBF. This command takes as input the specific slips system sS and the makroscopic strain tensor E
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.
The solutions of the continuity equation can be analytically computed and are available via the command SO3FunSBF. This command takes as input the specific slips system sS and the makroscopic strain tensor E
Lets visualize these solution by their pole figures
+{% include inline_image.html file="SingleSlipModel_02.png" %} +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
{% 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 %}For completeness the pole figures of the other two basis functions.
{% highlight matlab %} -plotPDF(odf2,h,'resolution',2*degree,'colorRange','equal') +plotPDF(odf1,h,'resolution',2*degree,'colorRange','equal') mtexColorbar {% endhighlight %}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.
Since in practice all three slip systems are active we can model the resulting ODF as a linear combination of the different basis functions
{% highlight matlab %} plotPDF(odf1 + odf2 + odf3,h,'resolution',2*degree,'colorRange','equal') mtexColorbar {% endhighlight %}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.
+{% include inline_image.html file="SingleSlipModel_06.png" %} +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.
{% highlight matlab %} plotSection(div(odf2 .* Omega),'sigma') mtexColorMap blue2red @@ -168,5 +168,5 @@ setColorRange(max(abs(clim))*[-1,1]) {% endhighlight %}