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<!-- DOI: 10.1051/0004-6361/201014281 -->

<h2 class="sec">Online Material</h2>

<p></p><h2 class="sec"><a name="SECTION00080000000000000000"></a><A NAME="appendixA"></A>
Appendix A: Solution of the Duhamel-Neumann equation
</h2> 

<p>We outline the main steps needed to solve the Duhamel-Neumann
equation in Eq.&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#duhneueq2">18</a>) for our work. The temperature
field, which produces the thermal stresses, is assumed to have a
linearized form 
<!-- MATH: $T=T_{\rm av}+\Delta T$ -->
<IMG SRC="img32.png" ALT="$T= T_{\rm av}+\Delta T$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="80">
where <IMG SRC="img30.png" ALT="$\Delta T$" align="bottom" BORDER="0" HEIGHT="14" WIDTH="21">
is given
by Eq.&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#temperatureExpansion">9</a>).
The uniqueness of the solution arises from (i) the regularity in the
whole volume; and (ii)&nbsp;matching the free boundary conditions
(given by Eq.&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#bcond1">19</a>)) at the surface&nbsp;<i>r</i> = <i>R</i>.

</p><p>There are different ways in which we can decompose the
displacement vector <IMG SRC="img63.png" ALT="$\vec{u}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="12">
into spherical-harmonics-type
expansion (see, e.g., <A NAME="aaref33"></A><a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#thorne80">Thorne  1980</a>, for an insightful review).
Here we use a decomposition into spheroidal and toroidal components traditionally used in
geophysical analyses (e.g., <a name="tex2html50" href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#kaula68">Kaula  1968</a>; <a name="tex2html51" href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#bullen75">Bullen  1975</a>). With this
approach, related to what <a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#thorne80">Thorne (1980)</a> calls pure-spin vector
harmonics, we have 
<br></p><p></p>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
\vec{u} = \sum\limits_{n=0}^\infty\sum\limits_{k=-n}^n\left(\vec{u}^{\rm
S}_{nk}+\vec{u}^{\rm T}_{nk}  \right)~\exp(\imath k\omega t),
\end{equation} -->
<TABLE ALIGN="CENTER" WIDTH="100%">
<tbody><tr VALIGN="MIDDLE"><td ALIGN="CENTER" nowrap="nowrap"><A NAME="uVec"></A><IMG SRC="img241.png" ALT="\begin{displaymath}\vec{u} = \sum\limits_{n=0}^\infty\sum\limits_{k=-n}^n\left(\...
...m
S}_{nk}+\vec{u}^{\rm T}_{nk} \right)~\exp(\imath k\omega t),
\end{displaymath}" HEIGHT="80" WIDTH="216">
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.1)
</td>
</tr>
</table></div><br clear="all"><p></p>
with<br>
<DIV ALIGN="CENTER"><A NAME="sph12"></A><A NAME="tor"></A>
<!-- MATH: \begin{eqnarray}
\vec{u}^{\rm S}_{nk} &=& \left(\begin{array}{c} U_{nk}(r) \\V_{nk}(r)~\frac{\partial }{\partial \theta} \\V_{nk}(r)~\frac{1}{\sin\theta}\frac{\partial }{\partial \phi} \end{array}
\right) Y_{nk}(\theta,\phi)\; ,\\
\vec{u}^{\rm T}_{nk} &=& W_{nk}(r)\left(\begin{array}{c} 0 \\-\frac{1}{\sin\theta}\frac{\partial }{\partial \phi} \\\frac{\partial }{\partial \theta} \end{array}
\right) Y_{nk}(\theta,\phi)\; .
\end{eqnarray} -->
<TABLE ALIGN="CENTER" cellpadding="0" WIDTH="100%">
<tbody><tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <IMG SRC="img242.png" ALT="$\displaystyle \vec{u}^{\rm S}_{nk}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="35" WIDTH="24">
</td>
<td ALIGN="CENTER" nowrap="nowrap">=
</td>
<td ALIGN="LEFT" nowrap="nowrap"><IMG SRC="img243.png" ALT="$\displaystyle \left(\begin{array}{c} U_{nk}(r) \\  V_{nk}(r)~\frac{\partial }{\...
...heta}\frac{\partial }{\partial \phi} \end{array}\right) Y_{nk}(\theta,\phi)\; ,$" ALIGN="MIDDLE" BORDER="0" HEIGHT="74" WIDTH="164">
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.2)
</td>
</tr>
<tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap"><IMG SRC="img244.png" ALT="$\displaystyle \vec{u}^{\rm T}_{nk}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="35" WIDTH="24">
</td>
<td ALIGN="CENTER" nowrap="nowrap">=
</td>
<td ALIGN="LEFT" nowrap="nowrap"><IMG SRC="img245.png" ALT="$\displaystyle W_{nk}(r)\left(\begin{array}{c} 0 \\  -\frac{1}{\sin\theta}\frac{...
...\  \frac{\partial }{\partial \theta} \end{array}\right) Y_{nk}(\theta,\phi)\; .$" ALIGN="MIDDLE" BORDER="0" HEIGHT="74" WIDTH="176">
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.3)
</td>
</tr>
</table></div><br clear="all"><p></p>
The first component in Eq.&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#uVec">A.1</a>), 
<!-- MATH: $\vec{u}^{\rm S}_{nk}$ -->
<IMG SRC="img246.png" ALT="$\vec{u}^{\rm S}_{nk}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="30" WIDTH="21">,
is
actually two separate spheroidal terms, that we call&nbsp;S1 and&nbsp;S2 in
Sect.&nbsp;<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#tdpart">3.1.2</a>, characterized with radial-profile amplitudes
<i>U</i><sub><i>nk</i></sub>(<i>r</i>) and <i>V</i><sub><i>nk</i></sub>(<i>r</i>). The second component in Eq.&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#uVec">A.1</a>), 
<!-- MATH: $\vec{u}^{\rm T}_{nk}$ -->
<IMG SRC="img247.png" ALT="$\vec{u}^{\rm T}_{nk}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="30" WIDTH="22">,
is the toroidal component.

<p>The spheroidal character of our source (temperature) term the
Duhamel-Neumann equation implies two simplifications. First, the
toroidal part of the displacement vector becomes negligible and we
have <i>W</i><sub><i>nk</i></sub>=0. Second, we can restrict the summation over
degrees <i>n</i> in Eq.&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#uVec">A.1</a>) to the dipole and higher-order
terms only, ignoring the monopole <i>n</i>=0. This is because the
monopole part would correspond to purely radial temperature field,
such as has been considered, for instance, in the previous works
on our topic (e.g., <a name="tex2html52" href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#tambovtseva99">Tambovtseva &amp; Shestakova  1999</a>; <a name="tex2html53" href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#shestakova97">Shestakova &amp; Tambovtseva  1997</a>; <a name="tex2html54" href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#kuehrt84">Kuehrt  1984</a>).
Our temperature representation 
<!-- MATH: $T=T_{\rm av}+\Delta T$ -->
<IMG SRC="img32.png" ALT="$T= T_{\rm av}+\Delta T$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="80">
does not
contain a non-trivial, purely radial profile<A NAME="tex2html20" HREF="#foot2185"><sup><IMG ALT="[*]" SRC="/icons/foot_motif.png" align="bottom" BORDER="1"></sup></A>
and the only viable free monopole term must have 
<!-- MATH: $U_{00}=W_{00}=0$ -->
<i>U</i><sub>00</sub>=<i>W</i><sub>00</sub>=0to match the boundary conditions. Finally, we note that we also
anticipated the Fourier-development structure in Eq.&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#uVec">A.1</a>)
as it follows from the source (<IMG SRC="img30.png" ALT="$\Delta T$" align="bottom" BORDER="0" HEIGHT="14" WIDTH="21">
development).

</p><p>Substituting the spheroidal-vector representation of <IMG SRC="img63.png" ALT="$\vec{u}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="12">
into
the Duhamel-Neumann Eq.&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#duhneueq2">18</a>) we obtain the
following system of equations for the radial profile of the
amplitude functions <i>U</i><sub><i>nk</i></sub>(<i>r</i>) and <i>V</i><sub><i>nk</i></sub>(<i>r</i>) (<IMG SRC="img248.png" ALT="$n \geq 1$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="33">):

</p><p></p><DIV ALIGN="CENTER"><A NAME="DuhNeuEQ3"></A>
<!-- MATH: \begin{eqnarray}
&& (\lambda+2~\mu)~r^2 \frac{{\rm d}^2}{{\rm d}r^2}U_{nk}(r) +
2(\lambda+2~\mu)~r \frac{{\rm d}}{{\rm d}r}U_{nk}(r) \nonumber \\
 & & \qquad -  n (n+1)(\lambda+\mu)~r \frac{\rm d}{{\rm d}r}V_{nk}(r)+
  n(n+1)(\lambda+3~\mu)~V_{nk}(r) \nonumber \\
 &&  \qquad - \left[ ~2(\lambda+2\mu)+\mu~n(n+1)-k^2\omega^2r^2\rho \right]~
  U_{nk}(r) \nonumber \\
 &&\quad= \alpha~(3\lambda+2\mu)~r^2~\frac{{\rm d}}{{\rm d}r}T_{nk}(r),
\end{eqnarray} -->
<TABLE ALIGN="CENTER" cellpadding="0" WIDTH="100%">
<tbody><tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap">&nbsp;
</td>
<td>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
&nbsp; &nbsp; &nbsp;&nbsp;
</td>
<td ALIGN="LEFT" nowrap="nowrap"><IMG SRC="img249.png" ALT="$\displaystyle (\lambda+2~\mu)~r^2 \frac{{\rm d}^2}{{\rm d}r^2}U_{nk}(r) +
2(\lambda+2~\mu)~r \frac{{\rm d}}{{\rm d}r}U_{nk}(r)$" ALIGN="MIDDLE" BORDER="0" HEIGHT="52" WIDTH="282">
</td>
<td ALIGN="RIGHT" WIDTH="10">
&nbsp;
</td>
</tr>
<tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap">&nbsp;
</td>
<td>&nbsp;
</td>
<td ALIGN="LEFT" nowrap="nowrap"><IMG SRC="img250.png" ALT="$\displaystyle \qquad - n (n+1)(\lambda+\mu)~r \frac{\rm d}{{\rm d}r}V_{nk}(r)+
n(n+1)(\lambda+3~\mu)~V_{nk}(r)$" ALIGN="MIDDLE" BORDER="0" HEIGHT="47" WIDTH="362">
</td>
<td ALIGN="RIGHT" WIDTH="10">
&nbsp;
</td>
</tr>
<tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap">&nbsp;
</td>
<td>&nbsp;
</td>
<td ALIGN="LEFT" nowrap="nowrap"><IMG SRC="img251.png" ALT="$\displaystyle \qquad - \left[ ~2(\lambda+2\mu)+\mu~n(n+1)-k^2\omega^2r^2\rho \right]~
U_{nk}(r)$" ALIGN="MIDDLE" BORDER="0" HEIGHT="36" WIDTH="308">
</td>
<td ALIGN="RIGHT" WIDTH="10">
&nbsp;
</td>
</tr>
<tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap">&nbsp;
</td>
<td>&nbsp;
</td>
<td ALIGN="LEFT" nowrap="nowrap"><IMG SRC="img252.png" ALT="$\displaystyle \quad= \alpha~(3\lambda+2\mu)~r^2~\frac{{\rm d}}{{\rm d}r}T_{nk}(r),$" ALIGN="MIDDLE" BORDER="0" HEIGHT="49" WIDTH="183">
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.4)
</td>
</tr>
</table></div><br clear="all"><p></p>
<p>and

</p><p></p><DIV ALIGN="CENTER"><A NAME="DuhNeuEQ4"></A>
<!-- MATH: \begin{eqnarray}
&& \mu~r^2\frac{{\rm d}^2}{{\rm d}r^2}V_{nk}(r) +2\mu~r\frac{\rm d}{{\rm d}r}V_{nk}(r)
+(\lambda+\mu)~r\frac {\rm d}{{\rm d}r}U_{nk}(r) \nonumber \\
 &&  \qquad -  \left[ n(n+1)(\lambda+2\mu)-k^2\omega^2r^2\rho \right]~V_{nk}(r)
  \nonumber \\
 & & \qquad+ 2(\lambda+2\mu)~U_{nk}(r)=~\alpha~(3\lambda+2\mu)~r~T_{nk}(r).
\end{eqnarray} -->
<TABLE ALIGN="CENTER" cellpadding="0" WIDTH="100%">
<tbody><tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap">&nbsp;
</td>
<td>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
</td>
<td ALIGN="LEFT" nowrap="nowrap"><IMG SRC="img253.png" ALT="$\displaystyle \mu~r^2\frac{{\rm d}^2}{{\rm d}r^2}V_{nk}(r) +2\mu~r\frac{\rm d}{{\rm d}r}V_{nk}(r)
+(\lambda+\mu)~r\frac {\rm d}{{\rm d}r}U_{nk}(r)$" ALIGN="MIDDLE" BORDER="0" HEIGHT="52" WIDTH="311">
</td>
<td ALIGN="RIGHT" WIDTH="10">
&nbsp;
</td>
</tr>
<tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap">&nbsp;
</td>
<td>&nbsp;
</td>
<td ALIGN="LEFT" nowrap="nowrap"><IMG SRC="img254.png" ALT="$\displaystyle \qquad - \left[ n(n+1)(\lambda+2\mu)-k^2\omega^2r^2\rho \right]~V_{nk}(r)$" ALIGN="MIDDLE" BORDER="0" HEIGHT="36" WIDTH="266">
</td>
<td ALIGN="RIGHT" WIDTH="10">
&nbsp;
</td>
</tr>
<tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap">&nbsp;
</td>
<td>&nbsp;
</td>
<td ALIGN="LEFT" nowrap="nowrap"><IMG SRC="img255.png" ALT="$\displaystyle \qquad+ 2(\lambda+2\mu)~U_{nk}(r)=~\alpha~(3\lambda+2\mu)~r~T_{nk}(r).$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="292">
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.5)
</td>
</tr>
</table></div><br clear="all"><p></p>
<p>Here we found it useful to separate the <i>r</i>- and <i>t</i>-dependences
of the 
<!-- MATH: $t_{nk}(r,t)$ -->
<i>t</i><sub><i>nk</i></sub>(<i>r</i>,<i>t</i>) amplitudes of the <IMG SRC="img30.png" ALT="$\Delta T$" align="bottom" BORDER="0" HEIGHT="14" WIDTH="21">
development in
Eq.&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#temperatureExpansion">9</a>) and introduce pure radial parts
<i>T</i><sub><i>nk</i></sub>(<i>r</i>) such that 
<!-- MATH: $t_{nk}(r,t)=T_{nk}(r) \exp(\imath k\omega
t)$ -->
<IMG SRC="img256.png" ALT="$t_{nk}(r,t)=T_{nk}(r) \exp(\imath k\omega
t)$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="147">.
These represent source terms in Eqs.&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#DuhNeuEQ3">A.4</a>) and&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#DuhNeuEQ4">A.5</a>).

</p><p>While the solution of <i>U</i><sub><i>nk</i></sub>(<i>r</i>) and <i>V</i><sub><i>nk</i></sub>(<i>r</i>) is coupled by
means of Eqs.&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#DuhNeuEQ3">A.4</a>) and&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#DuhNeuEQ4">A.5</a>), the
fundamental implication of the Duhamel-Neumann equation linearity
is that amplitude terms of different degrees and orders in the
spherical harmonics development as well as the different Fourier
modes are not mixed and can be solved separately.

</p><p>Once we obtain <i>U</i><sub><i>nk</i></sub>(<i>r</i>) and <i>V</i><sub><i>nk</i></sub>(<i>r</i>), we can readily compute
components of the corresponding stress tensor

<!-- MATH: $\vec{\tau}^{\rm U}$ -->
<IMG SRC="img257.png" ALT="$\vec{\tau}^{\rm U}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="30" WIDTH="19">
arising from the displacement vector
field by using the Hook's law in Eq.&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#generalizedhookslaw">16</a>).
Given its linearity, we thus again have
<br></p><p></p>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
\vec{\tau}^{\rm
U} = \vec{\tau}(\vec{u}) =
\sum\limits_{n=1}^\infty\sum\limits_{k=-n}^n \vec{\tau}_{nk}~\exp(\imath k\omega t),
\end{equation} -->
<TABLE ALIGN="CENTER" WIDTH="100%">
<tbody><tr VALIGN="MIDDLE"><td ALIGN="CENTER" nowrap="nowrap"><A NAME="sTen"></A><IMG SRC="img258.png" ALT="\begin{displaymath}\vec{\tau}^{\rm
U} = \vec{\tau}(\vec{u}) =
\sum\limits_{n=1}...
...ty\sum\limits_{k=-n}^n \vec{\tau}_{nk}~\exp(\imath k\omega t),
\end{displaymath}" HEIGHT="80" WIDTH="218">
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.6)
</td>
</tr>
</table></div><br clear="all"><p></p>
where 
<!-- MATH: $\vec{\tau}_{nk}=\vec{\tau}(\vec{u}_{nk})$ -->
<IMG SRC="img259.png" ALT="$\vec{\tau}_{nk}=\vec{\tau}(\vec{u}_{nk})$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="72">.
Projecting components of the stress tensor onto the orthonormal
basis 
<!-- MATH: $(\vec{e}_{\rm r},\vec{e}_{\rm\theta},\vec{e}_{\rm\phi})$ -->
<IMG SRC="img160.png" ALT="$(\vec{e}_{\rm r},\vec{e}_{\rm\theta}, \vec{e}_{\rm\phi})$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="57">from Eq.&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#orto">33</a>), as outlined in Sect.&nbsp;<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#secExpressions">3.1</a>,
we obtain (for simplicity we dropped here the degree- and
order-indexes <i>n</i> and <i>k</i>)
<br>
<DIV ALIGN="CENTER"><A NAME="s_taurr"></A><A NAME="s_taurt"></A><A NAME="s_taurf"></A><A NAME="s_tautt"></A><A NAME="s_tauff"></A><A NAME="s_tautf"></A>
<!-- MATH: \begin{eqnarray}
\nonumber  \tau_{\rm rr} &=& \frac{1}{r}\left[ \left( \lambda+2\mu \right)~r~
\frac{{\rm d} U_{nk}}{{\rm d} r}+2\lambda U_{nk}- \lambda n(n+1)V_{nk} \right] Y_{nk},\\
 &&\\
 \tau_{\rm r\theta} &=& \frac{\mu}{r}\left[ U_{nk}+r~\frac{{\rm d} V_{nk}}{{\rm d} r}
  -V_{nk} \right]\frac{\partial}{\partial\theta}Y_{nk},\\
 \tau_{\rm r\phi} &=& \frac{\mu}{r}\left[ U_{nk}+r~\frac{{\rm d} V_{nk}}{{\rm d} r}
  -V_{nk} \right]\frac{\imath k}{\sin\theta} Y_{nk}\; ,\\
 \tau_{\rm\theta\theta} &=& \frac{1}{r}\Bigl[2(\lambda+\mu)U_{nk}-
  \lambda n(n+1)V_{nk} \nonumber \\
 & \quad +& \lambda~r~\frac{{\rm d} U_{nk}}{{\rm d} r}\Bigr]Y_{nk}+2\mu\frac{V_{nk}}{r}
  \frac{\partial}{\partial\theta}Y_{nk},\\
 \tau_{\rm\phi\phi} &=& \frac{1}{r}\Bigl[2(\lambda+\mu)U_{nk}
  -n(n+1)(\lambda+2\mu)V_{nk} \nonumber \\
 &\quad +& \lambda~r~\frac{{\rm d} U_{nk}}{{\rm d} r}\Bigr]Y_{nk}
  -2\mu~\frac{V_{nk}}{r}\frac{\partial^2 }{\partial \theta^2}Y_{nk},\\
 \tau_{\rm\theta\phi} &=& 2\mu\frac{V_{nk}}{r}\frac{\imath
  k}{\sin\theta} \biggl(\frac{\partial}{\partial\theta}-\frac{\cos\theta}{\sin\theta}
  \biggr)Y_{nk}.
\end{eqnarray} -->
<TABLE ALIGN="CENTER" cellpadding="0" WIDTH="100%">
<tbody><tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap">&nbsp;
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<IMG SRC="img122.png" ALT="$\displaystyle \tau_{\rm rr}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="28" WIDTH="19">
</td>
<td ALIGN="CENTER" nowrap="nowrap">=
</td>
<td ALIGN="LEFT" nowrap="nowrap"><IMG style="border: 0px solid ; width: 304px; height: 53px;" SRC="img260.png" ALT="$\displaystyle \frac{1}{r}\left[ \left( \lambda+2\mu \right)~r~" ALIGN="MIDDLE">
</td>
<td ALIGN="RIGHT" WIDTH="10">
&nbsp;
</td>
</tr>
<tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap">&nbsp;
</td>
<td>&nbsp;
</td>
<td>&nbsp;
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.7)
</td>
</tr>
<tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap"><IMG SRC="img124.png" ALT="$\displaystyle \tau_{\rm r\theta}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="28" WIDTH="21">
</td>
<td ALIGN="CENTER" nowrap="nowrap">=
</td>
<td ALIGN="LEFT" nowrap="nowrap"><IMG SRC="img261.png" ALT="$\displaystyle \frac{\mu}{r}\left[ U_{nk}+r~\frac{{\rm d} V_{nk}}{{\rm d} r}
-V_{nk} \right]\frac{\partial}{\partial\theta}Y_{nk},$" ALIGN="MIDDLE" BORDER="0" HEIGHT="53" WIDTH="191">
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.8)
</td>
</tr>
<tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap"><IMG SRC="img262.png" ALT="$\displaystyle \tau_{\rm r\phi}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="28" WIDTH="22">
</td>
<td ALIGN="CENTER" nowrap="nowrap">=
</td>
<td ALIGN="LEFT" nowrap="nowrap"><IMG SRC="img263.png" ALT="$\displaystyle \frac{\mu}{r}\left[ U_{nk}+r~\frac{{\rm d} V_{nk}}{{\rm d} r}
-V_{nk} \right]\frac{\imath k}{\sin\theta} Y_{nk}\; ,$" ALIGN="MIDDLE" BORDER="0" HEIGHT="53" WIDTH="208">
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.9)
</td>
</tr>
<tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap"><IMG SRC="img126.png" ALT="$\displaystyle \tau_{\rm\theta\theta}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="28" WIDTH="22">
</td>
<td ALIGN="CENTER" nowrap="nowrap">=
</td>
<td ALIGN="LEFT" nowrap="nowrap"><IMG SRC="img264.png" ALT="$\displaystyle \frac{1}{r}\Bigl[2(\lambda+\mu)U_{nk}-
\lambda n(n+1)V_{nk}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="49" WIDTH="189">
</td>
<td ALIGN="RIGHT" WIDTH="10">
&nbsp;
</td>
</tr>
<tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap">&nbsp;
</td>
<td ALIGN="CENTER" nowrap="nowrap"><IMG SRC="img128.png" ALT="$\textstyle \quad +$" align="bottom" BORDER="0" HEIGHT="15" WIDTH="29">
</td>
<td ALIGN="LEFT" nowrap="nowrap"><IMG SRC="img265.png" ALT="$\displaystyle \lambda~r~\frac{{\rm d} U_{nk}}{{\rm d} r}\Bigr]Y_{nk}+2\mu\frac{V_{nk}}{r}
\frac{\partial}{\partial\theta}Y_{nk},$" ALIGN="MIDDLE" BORDER="0" HEIGHT="50" WIDTH="183">
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.10)
</td>
</tr>
<tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap"><IMG SRC="img130.png" ALT="$\displaystyle \tau_{\rm\phi \phi}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="28" WIDTH="24">
</td>
<td ALIGN="CENTER" nowrap="nowrap">=
</td>
<td ALIGN="LEFT" nowrap="nowrap"><IMG SRC="img266.png" ALT="$\displaystyle \frac{1}{r}\Bigl[2(\lambda+\mu)U_{nk}
-n(n+1)(\lambda+2\mu)V_{nk}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="49" WIDTH="232">
</td>
<td ALIGN="RIGHT" WIDTH="10">
&nbsp;
</td>
</tr>
<tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap">&nbsp;
</td>
<td ALIGN="CENTER" nowrap="nowrap"><IMG SRC="img128.png" ALT="$\textstyle \quad +$" align="bottom" BORDER="0" HEIGHT="15" WIDTH="29">
</td>
<td ALIGN="LEFT" nowrap="nowrap"><IMG SRC="img267.png" ALT="$\displaystyle \lambda~r~\frac{{\rm d} U_{nk}}{{\rm d} r}\Bigr]Y_{nk}
-2\mu~\frac{V_{nk}}{r}\frac{\partial^2 }{\partial \theta^2}Y_{nk},$" ALIGN="MIDDLE" BORDER="0" HEIGHT="54" WIDTH="193">
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.11)
</td>
</tr>
<tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap"><IMG SRC="img268.png" ALT="$\displaystyle \tau_{\rm\theta\phi}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="28" WIDTH="23">
</td>
<td ALIGN="CENTER" nowrap="nowrap">=
</td>
<td ALIGN="LEFT" nowrap="nowrap"><IMG SRC="img269.png" ALT="$\displaystyle 2\mu\frac{V_{nk}}{r}\frac{\imath
k}{\sin\theta} \biggl(\frac{\partial}{\partial\theta}-\frac{\cos\theta}{\sin\theta}
\biggr)Y_{nk}.$" ALIGN="MIDDLE" BORDER="0" HEIGHT="53" WIDTH="178">
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.12)
</td>
</tr>
</table></div><br clear="all"><p></p>
The partial derivatives of the spherical functions <i>Y</i><sub><i>nk</i></sub> are
computed using
<br>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{eqnarray}
\frac{\partial }{\partial \theta}Y_{nk} &=& \frac{1}{2}\Bigl[\sqrt{(n-k)(n+k+1)}~
e^{-\imath\phi}~Y_{nk+1} \nonumber \\
 & \quad-&  \sqrt{(n+k)(n-k+1)}~e^{\imath\phi}~Y_{nk-1}\Bigr],\\
 \frac{\partial^2 }{\partial \theta^2}Y_{nk} &=& \left[ \frac{k^2}{\sin^2\theta}-n(n+1) \right]
  Y_{nk}-\frac{\cos\theta}{\sin\theta}\frac{\partial }{\partial \theta}Y_{nk}.
\end{eqnarray} -->
<TABLE ALIGN="CENTER" cellpadding="0" WIDTH="100%">
<tbody><tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<IMG SRC="img270.png" ALT="$\displaystyle \frac{\partial }{\partial \theta}Y_{nk}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="50" WIDTH="42">
</td>
<td ALIGN="CENTER" nowrap="nowrap">=
</td>
<td ALIGN="LEFT" nowrap="nowrap"><IMG SRC="img271.png" ALT="$\displaystyle \frac{1}{2}\Bigl[\sqrt{(n-k)(n+k+1)}~
e^{-\imath\phi}~Y_{nk+1}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="49" WIDTH="204">
</td>
<td ALIGN="RIGHT" WIDTH="10">
&nbsp;
</td>
</tr>
<tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap">&nbsp;
</td>
<td ALIGN="CENTER" nowrap="nowrap"><IMG SRC="img132.png" ALT="$\textstyle \quad -$" align="bottom" BORDER="0" HEIGHT="15" WIDTH="29">
</td>
<td ALIGN="LEFT" nowrap="nowrap"><IMG SRC="img272.png" ALT="$\displaystyle \sqrt{(n+k)(n-k+1)}~e^{\imath\phi}~Y_{nk-1}\Bigr],$" ALIGN="MIDDLE" BORDER="0" HEIGHT="45" WIDTH="191">
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.13)
</td>
</tr>
<tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap"><IMG SRC="img273.png" ALT="$\displaystyle \frac{\partial^2 }{\partial \theta^2}Y_{nk}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="54" WIDTH="47">
</td>
<td ALIGN="CENTER" nowrap="nowrap">=
</td>
<td ALIGN="LEFT" nowrap="nowrap"><IMG SRC="img274.png" ALT="$\displaystyle \left[ \frac{k^2}{\sin^2\theta}-n(n+1) \right]
Y_{nk}-\frac{\cos\theta}{\sin\theta}\frac{\partial }{\partial \theta}Y_{nk}.$" ALIGN="MIDDLE" BORDER="0" HEIGHT="54" WIDTH="233">
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.14)
</td>
</tr>
</table></div><br clear="all"><p></p>
Equations&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#s_taurr">A.7</a>) to&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#s_tautf">A.12</a>) yield components of
the stress tensor that explicitly depend on the displacement
vector <IMG SRC="img63.png" ALT="$\vec{u}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="12">.
The part 
<!-- MATH: $\vec{\tau}^{\rm T}= -\alpha(3
\lambda + 2 \mu)~ \Delta T~ {\bf 1}$ -->
<IMG SRC="img70.png" ALT="$\vec{\tau}^{\rm T}= -\alpha(3\lambda+2\mu)~ \Delta T~
{\bf 1}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="30" WIDTH="136">,
which explicitly depends on
the temperature (see the generalized Hook's law in Eq.&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#generalizedhookslaw">16</a>), should be added separately to the
total stress-tensor field. Because of the explicit analytical solution for <IMG SRC="img30.png" ALT="$\Delta T$" align="bottom" BORDER="0" HEIGHT="14" WIDTH="21">,
this is achieved at no computational expense.

<p>Because of the linearity of the Duhamel-Neumann equation, a
general solution is expressed in terms of a linear superposition
of (i) a solution of the homogeneous system; and (ii) a particular
solution of the inhomogeneous system. The next two sections
discuss the two cases separately.

</p><p></p><h3 class="sec2"><a name="SECTION00081000000000000000"></a><A NAME="homoDN"></A>
A.1 Solution of the homogeneous Duhamel-Neumann equation
</h3> 

<p>Equations&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#DuhNeuEQ3">A.4</a>) and&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#DuhNeuEQ4">A.5</a>)
with zero right-hand sides represent the homogeneous
Duhamel-Neumann equation broken into parts corresponding to the
individual spheroidal modes. Its solution is quite complicated,
but may be significantly simplified in our case. This is because
for the range of material parameters, sizes and rotation
frequencies that apply for meteoroids we always have<A NAME="tex2html21" HREF="#foot2208"><sup><IMG ALT="[*]" SRC="/icons/foot_motif.png" align="bottom" BORDER="1"></sup></A>

<!-- MATH: $\omega^2r^2\rho\ll \mu \sim \lambda+2\mu$ -->
<IMG SRC="img280.png" ALT="$\omega^2r^2\rho\ll \mu \sim \lambda+2\mu$" ALIGN="MIDDLE" BORDER="0" HEIGHT="30" WIDTH="118">.
With these we may
neglect the troublesome term 
<!-- MATH: $k^2\omega^2r^2\rho$ -->
<IMG SRC="img281.png" ALT="$k^2\omega^2r^2\rho$" ALIGN="MIDDLE" BORDER="0" HEIGHT="30" WIDTH="49">
in
Eqs.&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#DuhNeuEQ3">A.4</a>) and&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#DuhNeuEQ4">A.5</a>). A major implication
of this is then that the system of solutions of the homogeneous
Duhamel-Neumann equation become degenerate in the <i>k</i>&nbsp;(order)
index of the spherical-harmonics representation.

</p><p>Adopting the aforementioned approximation, the homogeneous system
of Eqs.&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#DuhNeuEQ3">A.4</a>) and&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#DuhNeuEQ4">A.5</a>) now has a form of
Euler equations. As such, it has a fundamental system of power-law
solutions 
<!-- MATH: $U^i_{nk}=Q_i~r^{m_i}$ -->
<i>U</i><sup><i>i</i></sup><sub><i>nk</i></sub>=<i>Q</i><sub><i>i</i></sub>&nbsp;<i>r</i><sup><i>m</i><sub><i>i</i></sub></sup> and 
<!-- MATH: $V^i_{nk} = r^{m_i}$ -->
<i>V</i><sup><i>i</i></sup><sub><i>nk</i></sub> = <i>r</i><sup><i>m</i><sub><i>i</i></sub></sup> with

<!-- MATH: $i=1, \ldots, 4$ -->
<IMG SRC="img282.png" ALT="$i=1, \ldots, 4$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="66">,
real-valued exponents <i>m</i><sub><i>i</i></sub> and amplitudes <i>Q</i><sub><i>i</i></sub>.
After a straightforward algebra, we obtain
<br>
</p><DIV ALIGN="CENTER">

<!-- MATH: \begin{eqnarray}
U^1_{nk}(r) &=& (n+1)~\frac{n~\lambda+(n-2)~\mu}{(n+3)~\lambda+(n+5)
~\mu}~r^{n+1}, \\
 V^1_{nk}(r) &=& r^{n+1}, \\
 U^2_{nk}(r) &=& n~r^{n-1},\quad V^2_{nk}(r) = r^{n-1}, \\
 U^3_{nk}(r) &=& -(n+1)~r^{-n+2},\quad V^3_{nk}(r) = r^{-n+2}, \\
 U^4_{nk}(r) &=& n~\frac{(n+1)~\lambda+(n+3)~\mu}{(n-2)~\lambda+
  (n-4)~\mu}~r^{-n}, \\
 V^4_{nk}(r) &=& r^{-n}.
\end{eqnarray} -->
<TABLE ALIGN="CENTER" cellpadding="0" WIDTH="100%">
<tbody><tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap"><i>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; U</i><sup>1</sup><sub><i>nk</i></sub>(<i>r</i>)
</td>
<td ALIGN="CENTER" nowrap="nowrap">=
</td>
<td ALIGN="LEFT" nowrap="nowrap"><IMG SRC="img283.png" ALT="$\displaystyle (n+1)~\frac{n~\lambda+(n-2)~\mu}{(n+3)~\lambda+(n+5)
~\mu}~r^{n+1},$" ALIGN="MIDDLE" BORDER="0" HEIGHT="49" WIDTH="210">
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.15)
</td>
</tr>
<tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap"><i>V</i><sup>1</sup><sub><i>nk</i></sub>(<i>r</i>)
</td>
<td ALIGN="CENTER" nowrap="nowrap">=
</td>
<td ALIGN="LEFT" nowrap="nowrap"><i>r</i><sup><i>n</i>+1</sup>,
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.16)
</td>
</tr>
<tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap"><i>U</i><sup>2</sup><sub><i>nk</i></sub>(<i>r</i>)
</td>
<td ALIGN="CENTER" nowrap="nowrap">=
</td>
<td ALIGN="LEFT" nowrap="nowrap"><IMG SRC="img284.png" ALT="$\displaystyle n~r^{n-1},\quad V^2_{nk}(r) = r^{n-1},$" ALIGN="MIDDLE" BORDER="0" HEIGHT="35" WIDTH="146">
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.17)
</td>
</tr>
<tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap"><i>U</i><sup>3</sup><sub><i>nk</i></sub>(<i>r</i>)
</td>
<td ALIGN="CENTER" nowrap="nowrap">=
</td>
<td ALIGN="LEFT" nowrap="nowrap"><IMG SRC="img285.png" ALT="$\displaystyle -(n+1)~r^{-n+2},\quad V^3_{nk}(r) = r^{-n+2},$" ALIGN="MIDDLE" BORDER="0" HEIGHT="35" WIDTH="203">
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.18)
</td>
</tr>
<tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap"><i>U</i><sup>4</sup><sub><i>nk</i></sub>(<i>r</i>)
</td>
<td ALIGN="CENTER" nowrap="nowrap">=
</td>
<td ALIGN="LEFT" nowrap="nowrap"><IMG SRC="img286.png" ALT="$\displaystyle n~\frac{(n+1)~\lambda+(n+3)~\mu}{(n-2)~\lambda+
(n-4)~\mu}~r^{-n},$" ALIGN="MIDDLE" BORDER="0" HEIGHT="49" WIDTH="171">
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.19)
</td>
</tr>
<tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap"><i>V</i><sup>4</sup><sub><i>nk</i></sub>(<i>r</i>)
</td>
<td ALIGN="CENTER" nowrap="nowrap">=
</td>
<td ALIGN="LEFT" nowrap="nowrap"><i>r</i><sup>-<i>n</i></sup>.
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.20)
</td>
</tr>
</table></div><br clear="all"><p></p>
The last two modes, 3 and 4, diverge at the center <i>r</i>=0 and
therefore must be excluded. We are thus left with the first two
modes, 1 and&nbsp;2, that produce the spheroidal modes 
<!-- MATH: $\vec{u}^{\rm
S1}=\vec{u}^{\rm S} (U^1,V^1)$ -->
<IMG SRC="img287.png" ALT="$\vec{u}^{\rm
S1}=\vec{u}^{\rm S} (U^1,V^1)$" ALIGN="MIDDLE" BORDER="0" HEIGHT="30" WIDTH="100">
and 
<!-- MATH: $\vec{u}^{\rm S2}=\vec{u}^{\rm
S}(U^2,V^2)$ -->
<IMG SRC="img288.png" ALT="$\vec{u}^{\rm S2}=\vec{u}^{\rm
S}(U^2,V^2)$" ALIGN="MIDDLE" BORDER="0" HEIGHT="30" WIDTH="99">
in Sect.&nbsp;<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#tdpart">3.1.2</a> and whose associated stress
field was given in
Eqs.&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#eqHomSol1begin">44</a>)-(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#eqHomSol1end">49</a>) and
Eqs.&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#eqHomSol2begin">50</a>)-(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#eqHomSol2end">55</a>). Obviously, they
have also been used to obtain the stationary part of the stress
field discussed in Sect.&nbsp;<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#tipart">3.1.1</a>. We note that the second
spheroidal mode represents a pure shear with no volumic changes
(compression or expansion) because 
<!-- MATH: $\nabla\cdot\vec{u}^{\rm S2}=0$ -->
<IMG SRC="img178.png" ALT="$\nabla\cdot \vec{u}^{\rm S2}=0$" ALIGN="MIDDLE" BORDER="0" HEIGHT="30" WIDTH="64">.

<p></p><h3 class="sec2"><a name="SECTION00082000000000000000"></a><A NAME="inhomoDN"></A>
A.2 Particular solution of the Duhamel-Neumann equation
</h3>

<p>We next find a particular solution of the
inhomogeneous Duhamel-Neumann equation with the thermal source

<!-- MATH: $T_{nk}(r)\neq 0$ -->
<IMG SRC="img289.png" ALT="$T_{nk}(r)\neq 0$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="61">.
We divide this task into a discussion of the
stationary case (<i>k</i>=0) and time-dependent case (<IMG SRC="img150.png" ALT="$k\neq 0$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="33">). In
both cases, we again use the approximation of neglecting the

<!-- MATH: $\propto (k\omega r \rho)^2$ -->
<IMG SRC="img290.png" ALT="$\propto (k\omega r \rho)^2$" ALIGN="MIDDLE" BORDER="0" HEIGHT="30" WIDTH="59">
terms in Eqs.&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#DuhNeuEQ3">A.4</a>) and&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#DuhNeuEQ4">A.5</a>).

</p><p></p><h4 class="sec3"><a name="SECTION00082100000000000000"</a>
A.2.1 Time-independent part</h4>

<p>The stationary temperature field is given by 
<!-- MATH: $\Delta
T=\sum_{n=1}^\infty~ C_{n0}(\theta_0)~r^n~Y_{n0}(\theta,\phi)$ -->
<IMG SRC="img291.png" ALT="$\Delta
T=\sum_{n=1}^\infty~ C_{n0}(\theta_0)~r^n~Y_{n0}(\theta,\phi)$" ALIGN="MIDDLE" BORDER="0" HEIGHT="27" WIDTH="179">(Eqs.&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#temperatureExpansion">9</a>) and&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#tno">10</a>)) and thus

<!-- MATH: $T_{n0}(r)=C_{n0}(\theta_0)~r^n$ -->
<IMG SRC="img292.png" ALT="$T_{n0}(r)=C_{n0}(\theta_0)~r^n$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="109">.
We again search the fundamental
system of solutions in a power-law form 
<!-- MATH: $U^{\rm P}_{n0}=Q_{\rm
U}~r^{m_{\rm U}}$ -->
<IMG SRC="img293.png" ALT="$U^{\rm P}_{n0}=Q_{\rm
U}~r^{m_{\rm U}}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="30" WIDTH="82">
and 
<!-- MATH: $V^{\rm P}_{n0}=Q_{\rm V} r^{m_{\rm V}}$ -->
<IMG SRC="img294.png" ALT="$V^{\rm P}_{n0}=Q_{\rm V} r^{m_{\rm V}}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="30" WIDTH="76">with some real-valued exponents 
<!-- MATH: $(m_{\rm U},m_{\rm V})$ -->
<IMG SRC="img295.png" ALT="$(m_{\rm U},m_{\rm V})$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="54">
and
amplitudes 
<!-- MATH: $(Q_{\rm U},Q_{\rm V})$ -->
<IMG SRC="img296.png" ALT="$(Q_{\rm U},Q_{\rm V})$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="54">.
After a brief algebraic
derivation, we obtain
<br>
</p><DIV ALIGN="CENTER">

<!-- MATH: \begin{eqnarray}
U_{n0}^{\rm P} &=& \frac{n+2}{2(2n+3)}\frac{\alpha(3\lambda+2\mu)}{
\lambda+2\mu}C_{n0}(\theta_0)~r^{n+1}, \\
 V_{n0}^{\rm P} &=& \frac{1}{2(2n+3)}\frac{\alpha(3\lambda+2\mu)}{
  \lambda+2\mu}C_{n0}(\theta_0)~r^{n+1}.
\end{eqnarray} -->
<TABLE ALIGN="CENTER" cellpadding="0" WIDTH="100%">
<tbody><tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <IMG SRC="img297.png" ALT="$\displaystyle U_{n0}^{\rm P}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="35" WIDTH="26">
</td>
<td ALIGN="CENTER" nowrap="nowrap">=
</td>
<td ALIGN="LEFT" nowrap="nowrap"><IMG SRC="img298.png" ALT="$\displaystyle \frac{n+2}{2(2n+3)}\frac{\alpha(3\lambda+2\mu)}{
\lambda+2\mu}C_{n0}(\theta_0)~r^{n+1},$" ALIGN="MIDDLE" BORDER="0" HEIGHT="49" WIDTH="214">
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.21)
</td>
</tr>
<tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap"><IMG SRC="img299.png" ALT="$\displaystyle V_{n0}^{\rm P}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="35" WIDTH="24">
</td>
<td ALIGN="CENTER" nowrap="nowrap">=
</td>
<td ALIGN="LEFT" nowrap="nowrap"><IMG SRC="img300.png" ALT="$\displaystyle \frac{1}{2(2n+3)}\frac{\alpha(3\lambda+2\mu)}{
\lambda+2\mu}C_{n0}(\theta_0)~r^{n+1}.$" ALIGN="MIDDLE" BORDER="0" HEIGHT="49" WIDTH="214">
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.22)
</td>
</tr>
</table></div><br clear="all"><p></p>
We note that this mode has the same radial profile as the

<!-- MATH: $\vec{u}^{\rm S1}$ -->
<IMG SRC="img154.png" ALT="$\vec{u}^{\rm
S1}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="30" WIDTH="21">
spheroidal model found above.

<p></p><h4 class="sec3"><a name="SECTION00082200000000000000"</a>
A.2.2 Time-dependent part</h4>

<p>The time-dependent temperature field is given by 
<!-- MATH: $\Delta T=
\sum_{n=1}^\infty\sum_{k=-n}^n~C_{nk}(\theta_0)~j_n(z_k)~Y_{nk}
(\theta,\phi)~\exp(\imath k\omega t)$ -->
<IMG SRC="img301.png" ALT="$\Delta T=
\sum_{n=1}^\infty\sum_{k=-n}^n~C_{nk}(\theta_0)~j_n(z_k)~Y_{nk}
(\theta,\phi)~\exp(\imath k\omega t)$" ALIGN="MIDDLE" BORDER="0" HEIGHT="25" WIDTH="292">
with <IMG SRC="img302.png" ALT="$k\ne 0$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="33">(Eqs.&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#temperatureExpansion">9</a>) and&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#tnk">11</a>)) and thus

<!-- MATH: $T_{nk}=C_{nk}(\theta_0)~j_n(z_k)$ -->
<IMG SRC="img303.png" ALT="$T_{nk}=C_{nk}(\theta_0)~j_n(z_k)$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="114">.
We assume that the particular
solution has a form 
<!-- MATH: $\vec{u}^{\rm P}=\nabla \Phi$ -->
<IMG SRC="img304.png" ALT="$\vec{u}^{\rm P}=\nabla \Phi$" ALIGN="MIDDLE" BORDER="0" HEIGHT="30" WIDTH="53">.
Substituting
this ans&#xe4;tz to the Duhamel-Neumann equation
Eq.&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#duhneueq2">18</a>), we obtain
<br></p><p></p>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
(\lambda+2\mu)(\nabla\cdot\nabla)\Phi-\rho\frac{\partial^2 }{\partial t^2}\Phi
= \alpha(3\lambda+2\mu) \Delta T,
\end{equation} -->
<TABLE ALIGN="CENTER" WIDTH="100%">
<tbody><tr VALIGN="MIDDLE"><td ALIGN="CENTER" nowrap="nowrap"><A NAME="DuhNeuEQpar"></A><IMG SRC="img305.png" ALT="\begin{displaymath}(\lambda+2\mu)(\nabla\cdot\nabla)\Phi-\rho\frac{\partial^2 }{\partial t^2}\Phi
= \alpha(3\lambda+2\mu) \Delta T,
\end{displaymath}" HEIGHT="69" WIDTH="272">
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.23)
</td>
</tr>
</table></div><br clear="all"><p></p>
where we have suitably assumed that the arbitrary constant on the
right-hand side canceled the monopole (constant) temperature part.
This is an inhomogeneous wave equation on a sphere that, however,
takes a simple form because of the spherical-harmonic and Fourier
structure of the source term <IMG SRC="img30.png" ALT="$\Delta T$" align="bottom" BORDER="0" HEIGHT="14" WIDTH="21">
on the right-hand side.
Assuming thus a separation
<p></p>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
\Phi =
\sum_{n=1}^\infty\sum_{k=-n}^n \Gamma_{nk} \Sigma_{nk}(r)
Y_{nk}(\theta,\phi)\exp(\imath k\omega t),
\end{equation} -->
<TABLE ALIGN="CENTER" WIDTH="100%">
<tbody><tr VALIGN="MIDDLE"><td ALIGN="CENTER" nowrap="nowrap"><A NAME="phisep"></A><IMG SRC="img306.png" ALT="\begin{displaymath}\Phi =
\sum_{n=1}^\infty\sum_{k=-n}^n \Gamma_{nk} \Sigma_{nk}(r)
Y_{nk}(\theta,\phi)\exp(\imath k\omega t),
\end{displaymath}" HEIGHT="80" WIDTH="253">
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.24)
</td>
</tr>
</table></div><br clear="all"><p></p>
we obtain
<p></p>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
\Sigma_{nk}(r) = j_n(z_k),
\end{equation} -->
<TABLE ALIGN="CENTER" WIDTH="100%">
<tbody><tr VALIGN="MIDDLE"><td ALIGN="CENTER" nowrap="nowrap"><A NAME="sigma"></A><IMG SRC="img307.png" ALT="\begin{displaymath}\Sigma_{nk}(r) = j_n(z_k),
\end{displaymath}" HEIGHT="49" WIDTH="95">
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.25)
</td>
</tr>
</table></div><br clear="all"><p></p>
and
<p></p>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
\Gamma_{nk} = \frac{\alpha}{\rho}\frac{3\lambda+2\mu}{k^2\omega^2
+\imath k(v_{\rm P}/\ell_{\rm d})^2}~ C_{nk}(\theta_0)\; ,
\end{equation} -->
<TABLE ALIGN="CENTER" WIDTH="100%">
<tbody><tr VALIGN="MIDDLE"><td ALIGN="CENTER" nowrap="nowrap"><A NAME="gamma"></A><IMG SRC="img308.png" ALT="\begin{displaymath}\Gamma_{nk} = \frac{\alpha}{\rho}\frac{3\lambda+2\mu}{k^2\omega^2
+\imath k(v_{\rm P}/\ell_{\rm d})^2}~ C_{nk}(\theta_0)\; ,
\end{displaymath}" HEIGHT="70" WIDTH="219">
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.26)
</td>
</tr>
</table></div><br clear="all"><p></p>
where 
<!-- MATH: $v_{\rm P}=\sqrt{(\lambda+2\mu)/\rho}$ -->
<IMG SRC="img276.png" ALT="$v_{\rm P}=\sqrt{(\lambda+2\mu)/\rho}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="33" WIDTH="103">
is the elastic
P-wave velocity as above. Adopting again the approximation

<!-- MATH: $v_{\rm P}\gg \omega R \geq \omega\ell_{\rm d}$ -->
<IMG SRC="img309.png" ALT="$v_{\rm P}\gg \omega R \geq \omega\ell_{\rm d}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="92">,
we may
neglect the first term in the denominator of Eq.&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#gamma">A.26</a>).
Translating this solution into the amplitude-functions of the
spheroidal-field representation&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#sph12">A.2</a>), we finally obtain
<br>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{eqnarray}
U^{\rm P}_{nk} &=& -\alpha\frac{3\lambda+2\mu}{\lambda+2\mu}
C_{nk}(\theta_0)~\frac{\ell_{\rm d}}{\sqrt{-\imath k}}
  \frac{{\rm d}~ j_n(z_k)}{{\rm d}z},\\
 V^{\rm P}_{nk} &=& -\alpha\frac{3\lambda+2\mu}{\lambda+2\mu}
  C_{nk}(\theta_0)~\frac{\ell_{\rm d}}{\sqrt{-\imath k}}
  \frac{j_n(z_k)}{z_k}.
\end{eqnarray} -->
<TABLE ALIGN="CENTER" cellpadding="0" WIDTH="100%">
<tbody><tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <IMG SRC="img310.png" ALT="$\displaystyle U^{\rm P}_{nk}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="35" WIDTH="27">
</td>
<td ALIGN="CENTER" nowrap="nowrap">=
</td>
<td ALIGN="LEFT" nowrap="nowrap"><IMG SRC="img311.png" ALT="$\displaystyle -\alpha\frac{3\lambda+2\mu}{\lambda+2\mu}
C_{nk}(\theta_0)~\frac{\ell_{\rm d}}{\sqrt{-\imath k}}
\frac{{\rm d}~ j_n(z_k)}{{\rm d}z},$" ALIGN="MIDDLE" BORDER="0" HEIGHT="49" WIDTH="214">
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.27)
</td>
</tr>
<tr VALIGN="MIDDLE"><td ALIGN="RIGHT" nowrap="nowrap"><IMG SRC="img312.png" ALT="$\displaystyle V^{\rm P}_{nk}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="35" WIDTH="24">
</td>
<td ALIGN="CENTER" nowrap="nowrap">=
</td>
<td ALIGN="LEFT" nowrap="nowrap"><IMG SRC="img313.png" ALT="$\displaystyle -\alpha\frac{3\lambda+2\mu}{\lambda+2\mu}
C_{nk}(\theta_0)~\frac{\ell_{\rm d}}{\sqrt{-\imath k}}
\frac{j_n(z_k)}{z_k}.$" ALIGN="MIDDLE" BORDER="0" HEIGHT="49" WIDTH="203">
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.28)
</td>
</tr>
</table></div><br clear="all"><p></p>
The corresponding stress tensor is expressed by  Eqs.&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#eqParSolbegin">58</a>)-(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#eqParSolend">63</a>).

<p></p><h3 class="sec2"><a name="SECTION00083000000000000000"></a><A NAME="Secbc"></A>
A.3 Complete expression of the thermal stress tensor
</h3>

<p>The complete solution of the Duhamel-Neumann
equation is a linear combination of the free-spheroidal modes

<!-- MATH: $\vec{u}^{\rm S1}$ -->
<IMG SRC="img154.png" ALT="$\vec{u}^{\rm
S1}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="30" WIDTH="21">
and 
<!-- MATH: $\vec{u}^{\rm S2}$ -->
<IMG SRC="img155.png" ALT="$\vec{u}^{\rm S2}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="30" WIDTH="22">
from Sect.&nbsp;<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#homoDN">A.1</a>
and the particular mode 
<!-- MATH: $\vec{u}^{\rm
P}=\vec{u}(U^{\rm P},V^{\rm P})$ -->
<IMG SRC="img314.png" ALT="$\vec{u}^{\rm
P}=\vec{u}(U^{\rm P},V^{\rm P})$" ALIGN="MIDDLE" BORDER="0" HEIGHT="30" WIDTH="89">
from Sect.&nbsp;<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#inhomoDN">A.2</a>.
In the individual spherical harmonics modes, we have

<!-- MATH: $\vec{u}_{nk}=Q^1_{nk}\vec{u}^{\rm S1}_{nk}+Q^2_{nk} \vec{u}^{\rm
S2}_{nk}+\vec{u}^{\rm P}_{nk}$ -->
<IMG SRC="img315.png" ALT="$\vec{u}_{nk}=Q^1_{nk}\vec{u}^{\rm S1}_{nk}+Q^2_{nk} \vec{u}^{\rm
S2}_{nk}+\vec{u}^{\rm P}_{nk}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="30" WIDTH="165">,
where <i>Q</i><sup>1</sup><sub><i>nk</i></sub> and <i>Q</i><sup>2</sup><sub><i>nk</i></sub> are
some coefficients. We have to choose them to satisfy the surface
boundary condition&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#bcond1">19</a>), namely 
<!-- MATH: $\vec{\tau}\cdot
\vec{e}_{\rm r}=0$ -->
<IMG SRC="img316.png" ALT="$\vec{\tau}\cdot
\vec{e}_{\rm r}=0$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="53">
at <i>r</i>=<i>R</i>. Here the total stress tensor is given&nbsp;by
<br></p><p></p>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
\vec{\tau} = \vec{\tau}(\vec{u}) +
\vec{\tau}^{\rm T},
\end{equation} -->
<TABLE ALIGN="CENTER" WIDTH="100%">
<tbody><tr VALIGN="MIDDLE"><td ALIGN="CENTER" nowrap="nowrap"><A NAME="sgeneral1"></A><IMG SRC="img151.png" ALT="\begin{displaymath}\vec{\tau} = \vec{\tau}(\vec{u}) +
\vec{\tau}^{\rm T},
\end{displaymath}" HEIGHT="51" WIDTH="90">
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.29)
</td>
</tr>
</table></div><br clear="all"><p></p>
<p>or again in the spherical harmonics modes
<br></p><p></p>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
\vec{\tau}_{nk} = \vec{\tau}(\vec{u}_{nk}) +
\vec{\tau}^{\rm T}_{nk}.
\end{equation} -->
<TABLE ALIGN="CENTER" WIDTH="100%">
<tbody><tr VALIGN="MIDDLE"><td ALIGN="CENTER" nowrap="nowrap"><A NAME="sgeneral2"></A><IMG SRC="img317.png" ALT="\begin{displaymath}\vec{\tau}_{nk} = \vec{\tau}(\vec{u}_{nk}) +
\vec{\tau}^{\rm T}_{nk}.
\end{displaymath}" HEIGHT="53" WIDTH="115">
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.30)
</td>
</tr>
</table></div><br clear="all"><p></p>
The truly active and independent conditions are 
<!-- MATH: $\tau_{\rm rr}=0$ -->
<IMG SRC="img318.png" ALT="$\tau_{\rm rr}=0$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="40">and 
<!-- MATH: $\tau_{\rm r\theta}=0$ -->
<IMG SRC="img319.png" ALT="$\tau_{\rm r\theta}=0$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="42">,
from which the two constants
<i>Q</i><sup>1</sup><sub><i>nk</i></sub> and <i>Q</i><sup>2</sup><sub><i>nk</i></sub> follow. One  easily checks that the
third condition, 
<!-- MATH: $\tau_{\rm r\phi}=0$ -->
<IMG SRC="img320.png" ALT="$\tau_{\rm r\phi}=0$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="42">,
is always linearly  dependent
on 
<!-- MATH: $\tau_{\rm r\theta}=0$ -->
<IMG SRC="img319.png" ALT="$\tau_{\rm r\theta}=0$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="42">
(Eqs.&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#s_taurt">A.8</a>) and&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#s_taurf">A.9</a>))
and thus we do not need to consider it.

<p>We were able to carry out all necessary algedraic manipulations
and obtain a close form of the resulting formulae for the case of
the stationary (zonal, <i>k</i>=0) part of the stress field. These are
given in Eqs.&nbsp;(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#eqTSstat1">36</a>)-(<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#eqTSstat2">39</a>)
(Sect.&nbsp;<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#tipart">3.1.1</a>). In the case of the time-dependent part of
the stress field, the algebra is more involved and we could not
reach as simple and compact results as for the time-independent
part. We thus confine ourselves to provide formulae for the
stress-tensor components of the individual components and those
for the integration constants <i>Q</i><sup>1</sup><sub><i>nk</i></sub> and <i>Q</i><sup>2</sup><sub><i>nk</i></sub>(Sect.&nbsp;<a href="/articles/aa/full_html/2010/11/aa14281-10/aa14281-10.html#tdpart">3.1.2</a>).

</p><p></p>
</div></body></html>