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‎images/SingleSlipModel.png

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‎pages/documentation_matlab/SingleSlipModel.html

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This HTML was auto-generated from MATLAB code.
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To make changes, update the MATLAB code and republish this document.
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--><title>Sigle Slip Model</title><link rel="schema.DC" href="http://purl.org/dc/elements/1.1/"><meta name="DC.source" content="script_SingleSlipModel.m"></head><body><font size="2"><a href="https://github.com/mtex-toolbox/mtex/blob/develop/doc/Plasticity/SingleSlipModel.m">
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edit page</a></font><div><!--introduction--><p>Details to this model can be found in</p><div><ul><li><a href="https://doi.org/10.1093/gji/ggy442">An analytical finite-strain parametrization for texture evolution in deforming olivine polycrystals</a>, Geoph. J. Intern. 216, 2019.</li></ul></div><!--/introduction--><h2 id="1">The Continuity Equation</h2><p>The evolution of the orientation distribution function (ODF) \(f(g)\) with respect to a crystallopgraphic spin \(\Omega(g)\) is governed by the continuity equation</p><p>\[\frac{\partial}{\partial t} f + \nabla f \cdot \Omega + f \text{div} \Omega = 0\]</p><p>The solution of this equation depends on the initial texture \(f_0(g)\) at time zero and the crystallographic spin \(\Omega(g)\). In this model we assume the initial texture to be isotrope, i.e., \(f_0 = 1\) and the crystallopgraphic spin be associated with a single slip system. The full ODF will be later modelled as a superposition of the single slip models.</p><p>In this example we consider Olivine with has orthorhombic symmetry</p>
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edit page</a></font><div><!--introduction--><p>Details to this model can be found in</p><div><ul><li><a href="https://doi.org/10.1093/gji/ggy442">An analytical finite-strain parametrization for texture evolution in deforming olivine polycrystals</a>, Geoph. J. Intern. 216, 2019.</li></ul></div><!--/introduction--><h2 id="1">The Continuity Equation</h2><p>The evolution of the orientation distribution function (ODF) \(f(g)\) with respect to a crystallopgraphic spin \(\Omega(g)\) is governed by the continuity equation</p><p>\[\frac{\partial}{\partial t} f + \nabla f \cdot \Omega + f \text{ div } \Omega = 0\]</p><p>The solution of this equation depends on the initial texture \(f_0(g)\) at time zero and the crystallographic spin \(\Omega(g)\). In this model we assume the initial texture to be isotrope, i.e., \(f_0 = 1\) and the crystallopgraphic spin be associated with a single slip system. The full ODF will be later modelled as a superposition of the single slip models.</p><p>In this example we consider Olivine with has orthorhombic symmetry</p>
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{% highlight matlab %}
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cs = crystalSymmetry('222',[4.779 10.277 5.995],'mineral','olivine');
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{% endhighlight %}
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0 0 0
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0 0 -1
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{% endhighlight %}
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<p>via minimizing the square difference</p><p>\[\int_{SO(3)} \sum_{i,j} (e_{i,j}(g) - E_{i,j})^2 dg \to \text{min}\]</p><p>the orientation dependent strain rate tensor is identified as</p><p>\[e_{i,j}(g) = 10/3 \left&lt; S(g), E \right&gt; S(g)\]</p><p>and the corresponding crystallographic spin tensor as</p><p>\[\Omega_i(g) = \epsilon_{ijk} e_{jk}(g)\]</p><p>This can be modeled in MTEX via</p>
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<p>via minimizing the square difference</p><p>\[\int_{SO(3)} \sum_{i,j} (e_{i,j}(g) - E_{i,j})^2 dg \to \text{min}\]</p><p>the orientation dependent strain rate tensor is identified as</p><p>\[e(g) = 10/3 \left&lt; S(g), E \right&gt; S(g)\]</p><p>and the corresponding crystallographic spin tensor as</p><p>\[\Omega_i(g) = \epsilon_{ijk} e_{jk}(g)\]</p><p>This can be modeled in MTEX via</p>
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{% highlight matlab %}
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% explicite version
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% Omega = @(ori) 0.5 * EinsteinSum(tensor.leviCivita,[1 -1 -2],(ori * S(1) : E) * (ori * S(1)),[-1 -2])
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Omega = @(ori) vector3d(spinTensor(((ori * S(1)) : E) .* (ori * S(1))));
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%Omega = @(ori) vector3d(spinTensor((S(1) : (inv(ori) * E)) .* S(1)));
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Omega = SO3VectorFieldHarmonic.quadrature(Omega,cs)
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% other versions
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% Omega = @(ori) 0.5 * EinsteinSum(tensor.leviCivita,[1 -1 -2],(ori * S(1) : E) * (ori * S(1)),[-1 -2])
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% Omega = @(ori) vector3d(spinTensor((S(1) : (inv(ori) * E)) .* S(1)));
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% Omega = SO3VectorFieldHarmonic.quadrature(Omega,cs)
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{% endhighlight %}
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{% highlight plaintext %}
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Omega = SO3VectorFieldHarmonic (olivine → xyz)
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isReal: true
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bandwidth: 3
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weights: [-1.9e-18 1.4e-18 2.5e-18]
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{% endhighlight %}
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<p>We may visualize the orientation depedence of the spin tensor via</p>
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{% highlight matlab %}
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{% include inline_image.html file="SingleSlipModel_01.png" %}
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</center><p>or the divergence of this vectorfield</p>
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{% highlight matlab %}
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plot(div(Omega),'sigma')
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plot(div(SO3VectorFieldHarmonic(Omega)))
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{% endhighlight %}
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<center>
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{% include inline_image.html file="SingleSlipModel_02.png" %}
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{% endhighlight %}
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<center>
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{% include inline_image.html file="SingleSlipModel_06.png" %}
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</center><h2 id="15">Checking the Continuity Equation</h2><p>We may now check wether the continuity equation is satisfied. In a stable state the time difference will be zero and hence \(f \text{div} \Omega\)</p>
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</center><h2 id="15">Checking the Continuity Equation</h2><p>We may now check wether the continuity equation is satisfied. In a stable state the time difference will be zero and hence \(\text{div}(f \Omega) = 0\)</p><p>TODO: this is currently not working and we do not know why!</p>
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{% highlight matlab %}
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figure(1)
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plot(odf1 .* div(Omega),'sigma')
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plot(div(SO3VectorFieldHarmonic(Omega .* odf1)))
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mtexColorbar('location','south')
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{% endhighlight %}
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<center>
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{% include inline_image.html file="SingleSlipModel_07.png" %}
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</center><p>should be the negative of the inner product \(\nabla f \cdot \Omega\)</p>
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{% highlight matlab %}
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figure(2)
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plot(dot(grad(odf1),Omega),'sigma')
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{% endhighlight %}
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<center>
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{% include inline_image.html file="SingleSlipModel_08.png" %}
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</center></div></body></html>

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