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@nicholasmr
nicholasmr committed Jan 21, 2024
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12 changes: 5 additions & 7 deletions cpo-dynamics-orthotropic/index.html
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Expand Up @@ -154,21 +154,19 @@ <h2 id="problem">Problem</h2>
\\
b({\bf x},t,\theta,\phi)=\sum_{l=0}^{L}\sum_{m=-l}^{l}b_{l}^{m}({\bf x},t) Y_{l}^{m}(\theta,\phi) \quad\text{(distribution of slip directions)},
\]</div>
<p>CPO evolution can be written as a matrix problem involving the (block) state vector</p>
<div class="arithmatex">\[
{\bf s} = \begin{bmatrix} {\bf s}_n \\ {\bf s}_b \end{bmatrix} \quad\text{(state vector)},
\]</div>
<p>where</p>
<p>CPO evolution can be written as two independent matrix problems involving the CPO state vector fields</p>
<div class="arithmatex">\[
{\bf s}_n = [n_0^0,n_2^{-2},n_2^{-1},n_2^{0},n_2^{1},n_2^{2},n_4^{-4},\cdots,n_4^{4},\cdots,n_L^{-L},\cdots,n_L^{L}]^{\mathrm{T}} \quad\text{($n$ state vector)},
\\
{\bf s}_b = [b_0^0,b_2^{-2},b_2^{-1},b_2^{0},b_2^{1},b_2^{2},b_4^{-4},\cdots,b_4^{4},\cdots,b_L^{-L},\cdots,b_L^{L}]^{\mathrm{T}} \quad\text{($b$ state vector)},
\]</div>
<p>such that </p>
<div class="arithmatex">\[
\frac{\mathrm{D}{\bf s}}{\mathrm{D} t} = {\bf M} \cdot {\bf s} \quad\text{(state evolution)},
\frac{\mathrm{D}{\bf s}_n}{\mathrm{D} t} = {\bf M}_n \cdot {\bf s}_n \quad\text{($n$ state evolution)},
\\
\frac{\mathrm{D}{\bf s}_b}{\mathrm{D} t} = {\bf M}_b \cdot {\bf s}_b \quad\text{($b$ state evolution)},
\]</div>
<p>where the operator (matrix) <span class="arithmatex">\({\bf M}\)</span> represents the effect of a given CPO process, which may depend on stress, strain-rate, temperature, etc.</p>
<p>where the operators (matrices) <span class="arithmatex">\({\bf M}_n\)</span> and <span class="arithmatex">\({\bf M}_b\)</span> represents the effect of a given CPO process, which may depend on stress, strain-rate, temperature, etc.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The distributions may also be understood as the mass density fraction of grains with a given slip-plane-normal and slip-direction orientation.
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12 changes: 6 additions & 6 deletions cpo-idealized/index.html
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Expand Up @@ -126,15 +126,15 @@
<div class="section" itemprop="articleBody">

<h1 id="idealized-cpos">Idealized CPOs</h1>
<p>Three types of idealized CPO states can be said to exist:</p>
<p>If concerned with the distribution of a <em>single</em> crystallographic axis, three types of idealized CPO states can be said to exist:</p>
<ul>
<li><strong>Unidirectional CPO</strong>: crystallographic axes are perfectly aligned, i.e. perfect single maximum.</li>
<li><strong>Planar CPO</strong>: crystallographic axes are perfectly distributed on a plane, i.e. a great circle on <span class="arithmatex">\(S^2\)</span>.</li>
<li><strong>Circle CPO</strong>: crystallographic axes are perfectly distributed on a small circle on <span class="arithmatex">\(S^2\)</span>.</li>
<li><strong>Unidirectional CPO</strong>: all axes are perfectly aligned, i.e. perfect single maximum.</li>
<li><strong>Planar CPO</strong>: all axes are perfectly distributed on a plane, i.e. a great circle on <span class="arithmatex">\(S^2\)</span>.</li>
<li><strong>Circle CPO</strong>: all axes are perfectly distributed on a small circle on <span class="arithmatex">\(S^2\)</span>.</li>
</ul>
<p>Each of these can be expanded in terms of spherical harmonics by using the sifting property of the delta function, <span class="arithmatex">\(\delta({\hat {\bf r}})\)</span>.</p>
<p>Each of these can be expanded as a spherical harmonics series by using the sifting property of the delta function <span class="arithmatex">\(\delta({\hat {\bf r}})\)</span>.</p>
<h2 id="unidirectional">Unidirectional</h2>
<p>Consider the case where grains are perfectly aligned with <span class="arithmatex">\({{\bf m}}\)</span> such that <span class="arithmatex">\(n({\hat {\bf r}}) = \delta({\hat {\bf r}}-{{\bf m}})\)</span>.
<p>Consider the case where slip-plane normals are perfectly aligned with <span class="arithmatex">\({{\bf m}}\)</span> such that <span class="arithmatex">\(n({\hat {\bf r}}) = \delta({\hat {\bf r}}-{{\bf m}})\)</span>.
The corresponding expansion coefficients follow from the usual overlap integral:</p>
<div class="arithmatex">\[
n_l^m
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4 changes: 2 additions & 2 deletions cpo-representation/index.html
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Expand Up @@ -176,8 +176,7 @@ <h1 id="cpo-representation">CPO representation</h1>
</div>
<div class="admonition note">
<p class="admonition-title">Olivine</p>
<p>For orthotropic grains such as olivine, both <span class="arithmatex">\(n(\theta,\phi)\)</span> and <span class="arithmatex">\(b(\theta,\phi)\)</span> distributions must be tracked to represent the CPO.
Note that <span class="arithmatex">\(n(\theta,\phi)\)</span> and <span class="arithmatex">\(b(\theta,\phi)\)</span> represent the distributions of particular crystallographic axes (<span class="arithmatex">\({\bf m}'_i\)</span>) depending on fabric type (A&mdash;E type).</p>
<p>For orthotropic grains such as olivine, both <span class="arithmatex">\(n(\theta,\phi)\)</span> and <span class="arithmatex">\(b(\theta,\phi)\)</span> distributions must be tracked to represent the CPO.</p>
<table>
<thead>
<tr>
Expand All @@ -192,6 +191,7 @@ <h1 id="cpo-representation">CPO representation</h1>
</tr>
</tbody>
</table>
<p>Note that <span class="arithmatex">\(n(\theta,\phi)\)</span> and <span class="arithmatex">\(b(\theta,\phi)\)</span> represent the distributions of particular crystallographic axes (<span class="arithmatex">\({\bf m}'_i\)</span>) depending on fabric type (A&mdash;E type).</p>
</div>
<h2 id="odf">ODF</h2>
<p>The orientation distribution function (ODF) of a given slip-system axis (crystallographic axis) <span class="arithmatex">\(f\in \lbrace n,b\rbrace\)</span> is defined as the normalized distribution</p>
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2 changes: 1 addition & 1 deletion index.html
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Expand Up @@ -252,5 +252,5 @@ <h2 id="initialize">Initialize</h2>

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32 changes: 16 additions & 16 deletions sitemap.xml
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5 changes: 4 additions & 1 deletion wavepropagation-elastic/index.html
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Expand Up @@ -252,9 +252,12 @@ <h3 id="example-for-olivine">Example for olivine</h3>

### Physical parameters (SI units)
rho = 3355 # density of olivine
alpha = 1 # Voigt--Reuss weight; only alpha=1 supported for now
# Note the below ordering of elastic parameters assume an A-type fabric; that is, (blm,nlm,vlm) refer to the distibutions of (m1',m2',m3') axes, respectively.
# If concerned with another fabric type, you will need to re-order the components of Lame_grain accordingly.
Cij = (320.5e9, 196.5e9, 233.5e9, 64.0e9, 77.0e9, 78.7e9, 76.8e9, 71.6e9, 68.15e9) # Abramson (1997) parameters (C11,C22,C33,C44,C55,C66,C23,C13,C12)
Lame_grain = sf.Cij_to_Lame_orthotropic(Cij) # Lame parameters (lam11,lam22,lam33, lam23,lam13,lam12, mu1,mu2,mu3)
alpha = 1 # Voigt--Reuss weight; only alpha=1 supported for now


### Propagation directions of interest
theta, phi = np.deg2rad([90,70,]), np.deg2rad([0,10,]) # wave-vector directions (theta is colatitude, phi is longitude)
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