Flexoelektricita I¶
Následující poznámky jsou sestaveny podle [1], [2], [3] a [4]
Základní vztahy¶
Tenzor deformace je definovaný podobně jako v klasické pružnosti
(1)¶\[S_{ij}=\frac{1}{2}\big(u_{i,j}+u_{j,i}\big).\]
Rovnice rovnováhy mají tvar totožný s rovnicemi z gradientní pružnosti
(2)¶\[T_{jk,j}-\hat{T}_{ijk,ij}+b_k=0,\]
kde \(T_{jk}\) je tenzor napětí, \(\hat{T}_{ijk}\) je tenzor napětí třetího řádu, \(b_k\) je vektor objemové síly a symbol \(\cdot_{,i}\) značí parciální derivaci podle souřadnice \(x_i\). Podobně jako v klasické elektrostatice, jsou v dielektriku tři vektorová pole - elektrické pole \(E_i\), elektrické posuvy \(D_i\) a polarizace \(P_i\). Tyto veličiny jsou navzájem svázány vztahem
(3)¶\[D_i=\varepsilon_0E_i+P_i,\]
kde \(\varepsilon\) je permitivita vakua. Elektrické pole \(E_i\) je záporný gradient potenciálu, tj. \(E_i=-\varphi_{,i}\), a protože v dielektriku není náboj, platí podle Maxwella
(4)¶\[D_{i,i}=-\varepsilon_0\varphi_{,ii}+P_{i,i}=0.\]
Diferenciální rovnice (2) a (4) s okrajovými podmínkami
(5)¶\[\begin{split}\begin{equation}
\begin{split}
&u_i=\overline{u}_i\quad\mathrm{na}\ \partial V_u \\
&n_j\big(T_{jk}-\hat{T}_{ijk,i}\big)-D_j^nn_i\hat{T}_{ijk} \\
&\quad-\big(D^n_pn_p\big)n_in_j\hat{T}_{ijk}=\overline{t}_k\quad\mathrm{na}\
\partial V_t, \\
&u_{i,j}n_j=\overline{v}_i\quad\mathrm{na}\ \partial V_v \\
&n_in_j\hat{T}_{ijk}=\overline{r}_k\quad\mathrm{na}\ \partial V_r, \\
&\varphi=\overline{\varphi}\quad\mathrm{na}\ \partial V_{\varphi}, \\
&n_iD_i=-\overline{\omega}\quad\mathrm{na}\ \partial V_D,
\end{split}
\end{equation}\end{split}\]
kde \(n_i\) je vnější normála k hranici oblasti a \(\boldsymbol{D}^n=\nabla-\boldsymbol{n}\boldsymbol{n}\cdot\nabla\) je povrchový gradient. U správně formulovaného okrajového problému se požaduje, aby vnější hranice oblasti \(V\) byla rozdělena tak, že platí
(6)¶\[\partial V=\partial V_u\cup\partial V_t=\partial V_v\cup\partial V_r=\partial
V_\varphi\cup\partial V_D,\]
přičemž
(7)¶\[\emptyset=\partial V_u\cap\partial V_t=\partial V_v\cap\partial V_r=\partial
V_\varphi\cap\partial V_D.\]
V případě izotropního materiálu jsou konstitutivní vztahy následující
(8)¶\[T_{ij}=\lambda S_{kk}\delta_{ij}+2\mu S_{ij},\]
(9)¶\[\hat{T}_{ijk}=\big(\lambda S_{pp,i}\delta_{jk}+2\mu S_{jk,i}\big)l^2 +
\big(f_1\delta_{jk}P_i+f_2\delta_{ij}P_k+f_2\delta_{ik}P_j\big),\]
(10)¶\[E_i=aP_i+f_1S_{kk,i}+2f_2S_{ij,j},\]
kde \(\lambda\) a \(\mu\) jsou Lamého konstanty, \(f_1\) a \(f_2\) jsou flexoelektrické konstanty, \(a\) je konstanta svázána s permitivitou dielektrika \(\varepsilon\) vztahem \(a^{-1}=\varepsilon-\varepsilon_0\). Derivováním (10) podle \(x_i\) se dostane
(11)¶\[E_{i,i}=aP_{i,i}+f_1S_{kk,ii}+2f_2S_{ij,ji}.\]
Z definice elektrického pole platí, že \(E_{i,i}=-\varphi_{,ii}\). Dále také podle (4) platí, že \(P_{i,i}=\varepsilon_0\varphi_{,ii}\). Dosazením těchto dvou vztahů a (1) do (11) a přepsáním sčítacích indexů v \(S_{ij,ji}\) se dostane
(12)¶\[\begin{split}\begin{equation}
\begin{split}
-\varphi_{,ii} &= a\varepsilon_0\varphi_{,ii} + f_1\frac{1}{2}\big(u_{k,kii} +
u_{k,kii}\big) + 2f_2\frac{1}{2}\big(u_{i,jji} +
u_{j,iji}\big)\quad\Rightarrow \\
0 &= a\big(a^{-1}+\varepsilon_0\big)\varphi_{,ii} + f_1u_{k,kii} +
f_2\big(u_{k,jjk}+u_{k,iki}\big).
\end{split}
\end{equation}\end{split}\]
Dosazením za \(a^{-1}=\varepsilon-\varepsilon_0\) a po drobných úpravách se dostane
(13)¶\[\nabla^2\big(a\varepsilon\varphi+fu_{k,k}\big)=0,\]
kde \(f=f_1+2f_2\). Tato rovnice se může přepsat do tvaru
(14)¶\[\nabla^2\big(a\varepsilon\varphi+f\nabla\cdot\boldsymbol{u}\big) = 0.\]
Divergence tenzoru (8) se může psát ve tvaru
(15)¶\[\begin{split}\begin{equation}
\begin{split}
T_{jk,j}=&\lambda S_{ii,j}\delta_{jk}+2\mu S_{jk,j} \\
=&\lambda u_{i,ik}+\mu\big(u_{j,kj}+u_{k,jj}\big) \\
=&(\lambda+\mu)u_{j,jk}+\mu u_{k,jj}.
\end{split}
\end{equation}\end{split}\]
Podobně pro divergence tenzoru (9) platí
(16)¶\[\begin{split}\begin{equation}
\begin{split}
\hat{T}_{ijk,ij}=&\big(\lambda S_{pp,iij}\delta_{jk} + 2\mu S_{jk,iij}\big)l^2 \\
&+ \big(f_1\delta_{jk}P_{i,ij} + f_2\delta_{ij}P_{k,ij} +
f_2\delta_{ik}P_{j,ij}\big) \\
=&\big(\lambda S_{pp,iik}+2\mu S_{jk,iij}\big)l^2 \\
&+ \big(f_1P_{i,ik} + f_2P_{k,ii} + f_2P_{j,kj}\big) \\
=&\big[\lambda
u_{p,piik}+\mu\big(u_{j,kiij}+u_{k,jiij}\big)\big]l^2 \\
&+ \big(f_1P_{i,ik} + f_2P_{k,ii} + f_2P_{j,kj}\big) \\
=&\big[(\lambda+\mu)u_{j,jkii}+\mu u_{k,jjii}\big]l^2 \\
&+ \big(f_1P_{i,ik} + f_2P_{k,ii} + f_2P_{j,kj}\big) \\
=&\big[(\lambda+\mu)\nabla^2u_{j,jk}+\mu\nabla^2u_{k,jj}\big]l^2 \\
&+ \big(f_1P_{i,ik} + f_2P_{k,ii} + f_2P_{j,kj}\big).
\end{split}
\end{equation}\end{split}\]
Klíčový je v předchozím vztahu výraz \(P_{k,ii}\), který se vyjádří pomocí (10) následovně
(17)¶\[\begin{split}\begin{equation}
\begin{split}
P_{k,ii}=&a^{-1}E_{k,ii} - a^{-1}f_1S_{jj,kii} - 2f_2a^{-1}S_{kj,jii} \\
=&-a^{-1}\varphi_{,kii} - a^{-1}f_1u_{j,jkii} - a^{-1}f_2\big(u_{k,jjii} +
u_{j,kjii}\big) \\
=&-a^{-1}\nabla^2\varphi_{,k} - a^{-1}f_1\nabla^2u_{j,jk} -
a^{-1}f_2\nabla^2u_{k,jj} - a^{-1}f_2\nabla^2u_{j,kj}.
\end{split}
\end{equation}\end{split}\]
Dosazením za \(a^{-1}=\varepsilon-\varepsilon_0\) a použití Maxwellova vztahu (4) se dostane
(18)¶\[\begin{split}\begin{equation}
\begin{split}
P_{k,ii}=&-\big(\varepsilon-\varepsilon_0\big)\nabla^2\varphi_{,k} -
a^{-1}f_1\nabla^2u_{j,jk} \\
& - a^{-1}f_2\nabla^2u_{k,jj} - a^{-1}f_2\nabla^2u_{j,kj} \\
=&-\varepsilon\nabla^2\varphi_{,k} + P_{i,ik} -
a^{-1}f_1\nabla^2u_{j,jk} \\
& - a^{-1}f_2\nabla^2u_{k,jj} - a^{-1}f_2\nabla^2u_{j,kj}.
\end{split}
\end{equation}\end{split}\]
Dosazením (18) do (16) se dostane
(19)¶\[\begin{split}\begin{equation}
\begin{split}
\hat{T}_{ijk,ij}=&\big[(\lambda+\mu)\nabla^2u_{j,jk}+\mu\nabla^2u_{k,jj}\big]l^2 \\
&+ \big[(f_1+2f_2)P_{i,ik} - f_2\varepsilon\nabla^2\varphi_{,k}
- a^{-1}f_1f_2\nabla^2u_{j,jk} \\
&- a^{-1}f_2^2\nabla^2u_{k,jj} - a^{-1}f_2^2\nabla^2u_{j,kj}\big]
\\
=&\big[(\lambda+\mu)\nabla^2u_{j,jk}+\mu\nabla^2u_{k,jj}\big]l^2 \\
&+ \big(fP_{i,ik} - f_2\varepsilon\nabla^2\varphi_{,k}
- a^{-1}f_1f_2\nabla^2u_{j,jk} \\
&- a^{-1}f_2^2\nabla^2u_{k,jj} - a^{-1}f_2^2\nabla^2u_{j,kj}\big).
\end{split}
\end{equation}\end{split}\]
Z rovnice rovnováhy (13) a Maxwellovy rovnice (4) platí
(20)¶\[\varepsilon_0\nabla^2\varphi=P_{i,i}=-\frac{\varepsilon_0}{a\varepsilon}f\nabla^2u_{k,k},\]
takže po dosazení těchto rovností do (19) se dostane
(21)¶\[\begin{split}\begin{equation}
\begin{split}
\hat{T}_{ijk,ij}=&\big[(\lambda+\mu)\nabla^2u_{j,jk}+\mu\nabla^2u_{k,jj}\big]l^2 \\
&+ \bigg(-\frac{\varepsilon_0}{a\varepsilon}f^2\nabla^2u_{j,jk}
+ a^{-1}ff_2\nabla^2u_{j,jk} - a^{-1}f_1f_2\nabla^2u_{j,jk} \\
&- a^{-1}f_2^2\nabla^2u_{k,jj} - a^{-1}f_2^2\nabla^2u_{j,kj}\bigg).
\end{split}
\end{equation}\end{split}\]
Roznásobením \(ff_2\) se konečně dostane
(22)¶\[\begin{split}\begin{equation}
\begin{split}
\hat{T}_{ijk,ij}=&\big[(\lambda+\mu)\nabla^2u_{j,jk}+\mu\nabla^2u_{k,jj}\big]l^2 \\
&+ \bigg(-\frac{\varepsilon_0}{a\varepsilon}f^2\nabla^2u_{j,jk}
+ a^{-1}f_2^2\nabla^2u_{j,jk} - a^{-1}f_2^2\nabla^2u_{k,jj}\bigg)
\\
=&(\lambda+\mu)l_1^2\nabla^2u_{j,jk}+\mu l_2^2\nabla^2u_{k,jj},
\end{split}
\end{equation}\end{split}\]
kde
(23)¶\[l_1^2=l^2-\frac{\varepsilon_0f^2}{(\lambda+\mu)a\varepsilon}+\frac{f_2^2}{(\lambda+\mu)a},
\quad l_2^2=l^2-\frac{f_2^2}{a\mu}.\]
Dosazením (22) do rovnice rovnováhy a při zanedbání objemových sil \(b_k\) se dostane rovnice
(24)¶\[(\lambda+\mu)\big(1-l_1^2\nabla^2\big)u_{j,jk}+\mu\big(1-l_2^2\nabla^2\big)u_{k,jj}=0,\]
která se může přepsat do tvaru
(25)¶\[(\lambda+\mu)\big(1-l_1^2\nabla^2\big)\nabla(\nabla\cdot\boldsymbol{u})
+\mu\big(1-l_2^2\nabla^2\big)\nabla^2\boldsymbol{u}=0.\]
Aplikací divergence na levou stranu předchozí rovnice se dostane
(26)¶\[(\lambda+\mu)\big(1-l_1^2\nabla^2\big)u_{j,jkk}+\mu\big(1-l_2^2\nabla^2\big)u_{k,kjj}=0.\]
Odtud se dostane
(27)¶\[\begin{split}\begin{equation}
\begin{split}
&\big\{\lambda+\mu-\big[(\lambda+\mu)l^2-\varepsilon_0f^2+a^{-1}f_2^2\big]\nabla^2 \\
& \qquad +\mu-\big[\mu l^2-a^{-1}f_2^2\big]\nabla^2\big\}\nabla^2u_{k,k}=0.
\end{split}
\end{equation}\end{split}\]
Tato rovnice a (13) se může přepsat do tvaru
(28)¶\[(\lambda+2\mu)\big(1-l_0^2\nabla^2\big)\nabla^2(\nabla\cdot\boldsymbol{u}) = 0,\]
kde
(29)¶\[l_0=l^2-\frac{\varepsilon_0f^2}{\lambda+2\mu}.\]
Rovnice rovnováhy v polárních souřadnicích¶
Gradient divergence a Laplacián vektoru \(\boldsymbol{u}\) se v polárních souřadnicích zapíše následovně
(30)¶\[\begin{split}\begin{equation}
\begin{split}
\nabla(\nabla\cdot\boldsymbol{u})
=& \boldsymbol{s}_1=s_{1r}\boldsymbol{e}_r+s_{1\theta}\boldsymbol{e}_\theta \\
=& \Big(\boldsymbol{e}_r\partial_r+\boldsymbol{e}_\theta\frac{1}{r}\partial_\theta\Big)
\Bigg[\Big(\boldsymbol{e}_r\partial_r+\boldsymbol{e}_\theta\frac{1}{r}\partial_\theta\Big)
\cdot\big(u_r\boldsymbol{e}_r+u_\theta\boldsymbol{e}_\theta\big)\Bigg] \\
=&
\Big(\boldsymbol{e}_r\partial_r+\boldsymbol{e}_\theta\frac{1}{r}\partial_\theta\Big)
\Big[\partial_ru_r\boldsymbol{e}_r\cdot\boldsymbol{e}_r +
\partial_ru_\theta\boldsymbol{e}_r\cdot\boldsymbol{e}_\theta +
\frac{1}{r}\partial_\theta u_r\boldsymbol{e}_\theta\cdot\boldsymbol{e}_r \\
&+\frac{1}{r}u_r\boldsymbol{e}_\theta\cdot\boldsymbol{e}_\theta +
\frac{1}{r}\partial_\theta
u_\theta\boldsymbol{e}_\theta\cdot\boldsymbol{e}_\theta -
\frac{1}{r}u_\theta\boldsymbol{e}_\theta\cdot\boldsymbol{e}_r \Big] \\
=& \Big(\boldsymbol{e}_r\partial_r+\boldsymbol{e}_\theta\frac{1}{r}\partial_\theta\Big)
\Big[\partial_ru_r + \frac{1}{r}u_r + \frac{1}{r}\partial_\theta u_\theta\Big] \\
=& \partial_r\Big(\partial_ru_r + \frac{1}{r}u_r + \frac{1}{r}\partial_\theta
u_\theta\Big)\boldsymbol{e}_r \\
& +\frac{1}{r}\partial_\theta
\Big(\partial_ru_r + \frac{1}{r}u_r + \frac{1}{r}\partial_\theta
u_\theta\Big)\boldsymbol{e}_\theta
\end{split}
\end{equation}\end{split}\]
a
(31)¶\[\begin{split}\begin{equation}
\begin{split}
\nabla^2\boldsymbol{u}=&\boldsymbol{s}_2=s_{2r}\boldsymbol{e}_r+s_{2\theta}\boldsymbol{e}_\theta \\
=& \Big(\boldsymbol{e}_r\partial_r+\boldsymbol{e}_\theta\frac{1}{r}\partial_\theta\Big)\cdot
\Big(\boldsymbol{e}_r\partial_r+\boldsymbol{e}_\theta\frac{1}{r}\partial_\theta\Big)
\big(u_r\boldsymbol{e}_r+u_\theta\boldsymbol{e}_\theta\big) \\
=& \Big(\boldsymbol{e}_r\cdot\boldsymbol{e}_r\partial_{rr}
+\boldsymbol{e}_r\cdot\boldsymbol{e}_\theta\partial_{r\theta}
+\boldsymbol{e}_\theta\cdot\boldsymbol{e}_\theta\frac{1}{r}\partial_r
+\boldsymbol{e}_\theta\cdot\boldsymbol{e}_r\frac{1}{r}\partial_{r\theta} \\
& -\boldsymbol{e}_\theta\cdot\boldsymbol{e}_r\frac{1}{r^2}\partial_\theta
+\boldsymbol{e}_\theta\cdot\boldsymbol{e}_\theta\frac{1}{r^2}\partial_{\theta\theta}\Big)
\big(u_r\boldsymbol{e}_r+u_\theta\boldsymbol{e}_\theta\big) \\
=& \Big(\partial_{rr}+\frac{1}{r}\partial_r+\frac{1}{r^2}\partial_{\theta\theta}\Big)
\big(u_r\boldsymbol{e}_r+u_\theta\boldsymbol{e}_\theta\big) \\
=& \partial_{rr}u_r\boldsymbol{e}_r+\partial_{rr}u_\theta\boldsymbol{e}_\theta
+\frac{1}{r}\partial_ru_r\boldsymbol{e}_r+\frac{1}{r}\partial_ru_\theta\boldsymbol{e}_\theta \\
& +\frac{1}{r^2}\partial_\theta\big(\partial_\theta u_r\boldsymbol{e}_r
+u_r\boldsymbol{e}_\theta+\partial_\theta u_\theta\boldsymbol{e}_\theta
-u_\theta\boldsymbol{e}_r\big) \\
=& \partial_{rr}u_r\boldsymbol{e}_r+\partial_{rr}u_\theta\boldsymbol{e}_\theta
+\frac{1}{r}\partial_ru_r\boldsymbol{e}_r+\frac{1}{r}\partial_ru_\theta\boldsymbol{e}_\theta \\
& +\frac{1}{r^2}\big(\partial_{\theta\theta}u_r\boldsymbol{e}_r
+\partial_\theta u_r\boldsymbol{e}_\theta +\partial_\theta u_r\boldsymbol{e}_\theta
-u_r\boldsymbol{e}_r \\
& +\partial_{\theta\theta}u_\theta\boldsymbol{e}_\theta -\partial_\theta u_\theta\boldsymbol{e}_r
-\partial_\theta u_\theta\boldsymbol{e}_r-u_\theta\boldsymbol{e}_\theta\big) \\
=& \Big(\partial_{rr}u_r+\frac{1}{r}\partial_ru_r+\frac{1}{r^2}\partial_{\theta\theta}u_r
-\frac{1}{r^2}u_r-2\frac{1}{r^2}\partial_\theta u_\theta\Big)\boldsymbol{e}_r \\
& +\Big(\partial_{rr}u_\theta+\frac{1}{r}\partial_ru_\theta
+\frac{1}{r^2}\partial_{\theta\theta}u_\theta+2\frac{1}{r^2}\partial_\theta u_r
-\frac{1}{r^2}u_\theta\Big)\boldsymbol{e}_\theta \\
=& \Big[\partial_r\Big(\partial_{r}u_r+\frac{1}{r}u_r\Big)
+\frac{1}{r^2}\partial_\theta\big(\partial_\theta u_r
-2u_\theta\big)\Big]\boldsymbol{e}_r \\
& +\Big[\partial_r\Big(\partial_ru_\theta+\frac{1}{r}u_\theta\Big)
+\frac{1}{r^2}\partial_\theta\big(\partial_{\theta}u_\theta+2u_r\big)
\Big]\boldsymbol{e}_\theta.
\end{split}
\end{equation}\end{split}\]
Takže platí
(32)¶\[\begin{split}\begin{equation}
\begin{split}
s_{1r} =& \partial_r\Big(\partial_ru_r + \frac{1}{r}u_r + \frac{1}{r}\partial_\theta
u_\theta\Big) \\
s_{1\theta} =& \frac{1}{r}\partial_\theta
\Big(\partial_ru_r + \frac{1}{r}u_r + \frac{1}{r}\partial_\theta
u_\theta\Big)
\end{split}
\end{equation}\end{split}\]
a
(33)¶\[\begin{split}\begin{equation}
\begin{split}
s_{2r} =& \partial_r\Big(\partial_{r}u_r+\frac{1}{r}u_r\Big)
+\frac{1}{r^2}\partial_\theta\big(\partial_\theta u_r
-2u_\theta\big) \\
s_{2\theta} =& \partial_r\Big(\partial_ru_\theta+\frac{1}{r}u_\theta\Big)
+\frac{1}{r^2}\partial_\theta\big(\partial_{\theta}u_\theta+2u_r\big)
\end{split}
\end{equation}\end{split}\]
Bez problémů se dá ukázat, že
(34)¶\[\begin{split}\begin{equation}
\begin{split}
s_r =& s_{1r}+(1-2\nu)s_{2r} \\
=& 2(1-\nu)\partial_r\Big(\partial_ru_r
+\frac{1}{r}u_r+\frac{1}{r}\partial_\theta u_\theta\Big) \\
& -(1-2\nu)\frac{1}{r}\partial_\theta
\Big(\partial_ru_\varphi-\frac{1}{r}\partial_\theta u_r
+\frac{1}{r}u_\theta\Big), \\
s_\theta =& s_{1\theta}+(1-2\nu)s_{2\theta} \\
=& (1-2\nu)\partial_r\Big(\partial_ru_\theta
+\frac{1}{r}u_\theta-\frac{1}{r}\partial_\theta u_r\Big) \\
& +2(1-\nu)\frac{1}{r}\partial_\theta
\Big(\partial_ru_r+\frac{1}{r}u_r
+\frac{1}{r}\partial_\theta u_\theta\Big),
\end{split}
\end{equation}\end{split}\]
což jsou výrazy z [4] odpovídající případu izotropního gradientního materiálu. Rovnice (25) se pak může přepsat do tvaru
(35)¶\[\begin{split}\begin{equation}
\begin{split}
s_r -& l_1^2\Big(\nabla^2s_{1r}-\frac{1}{r^2}s_{1r}
-2\frac{1}{r^2}\partial_\theta s_{1\theta}\Big) \\
-& (1-2\nu)l_2^2\Big(\nabla^2s_{2r}-\frac{1}{r^2}s_{2r}
-2\frac{1}{r^2}\partial_\theta s_{2\theta}\Big)=0 \\
s_\theta -& l_1^2\Big(\nabla^2s_{1\theta}-\frac{1}{r^2}s_{1\theta}
+2\frac{1}{r^2}\partial_\theta s_{1r}\Big) \\
-& (1-2\nu)l_2^2\Big(\nabla^2s_{2\theta}-\frac{1}{r^2}s_{2\theta}
+2\frac{1}{r^2}\partial_\theta s_{2r}\Big)=0.
\end{split}
\end{equation}\end{split}\]
Okrajové podmínky v polárních souřadnicích¶
Zde budou rozepsány do polárních souřadnic výrazy na levé straně některých okrajových podmínek (5). Pro normálu \(\boldsymbol{n}=\boldsymbol{e}_rn_r(r,\theta)+\boldsymbol{e}_\theta n_\theta(r,\theta)\) platí
(36)¶\[n_r^2+n_\varphi^2=1.\]
Pak pro \(\boldsymbol{\nu}\) platí
(37)¶\[\begin{split}\boldsymbol{\nu} &= \mathrm{grad}(\boldsymbol{u})\cdot\boldsymbol{n}\\
&= (u_{r,r}\boldsymbol{e}_r\boldsymbol{e}_r
+\frac{1}{r}(u_{r,\varphi}-u_\varphi)\boldsymbol{e}_r\boldsymbol{e}_\varphi
+u_{\varphi,r}\boldsymbol{e}_\varphi\boldsymbol{e}_r
+\frac{1}{r}(u_r+u_{\varphi,\varphi})\boldsymbol{e}_\varphi\boldsymbol{e}_\varphi)
\cdot(\boldsymbol{e}_rn_r
+\boldsymbol{e}_{\varphi}n_{\varphi})\\
&= u_{r,r}\boldsymbol{e}_r\boldsymbol{e}_r\cdot\boldsymbol{e}_rn_r
+\frac{1}{r}(u_{r,\varphi}-u_\varphi)\boldsymbol{e}_r\boldsymbol{e}_\varphi
\cdot\boldsymbol{e}_rn_r
+u_{\varphi,r}\boldsymbol{e}_\varphi\boldsymbol{e}_r\cdot\boldsymbol{e}_rn_r
+\frac{1}{r}(u_r+u_{\varphi,\varphi})\boldsymbol{e}_\varphi\boldsymbol{e}_\varphi
\cdot\boldsymbol{e}_rn_r\\
&+ u_{r,r}\boldsymbol{e}_r\boldsymbol{e}_r\cdot\boldsymbol{e}_\varphi n_\varphi
+\frac{1}{r}(u_{r,\varphi}-u_\varphi)\boldsymbol{e}_r\boldsymbol{e}_\varphi
\cdot\boldsymbol{e}_\varphi n_\varphi
+u_{\varphi,r}\boldsymbol{e}_\varphi\boldsymbol{e}_r\cdot\boldsymbol{e}_\varphi n_\varphi
+\frac{1}{r}(u_r+u_{\varphi,\varphi})\boldsymbol{e}_\varphi\boldsymbol{e}_\varphi
\cdot\boldsymbol{e}_\varphi n_\varphi\\
&= u_{r,r}\boldsymbol{e}_rn_r+u_{\varphi,r}\boldsymbol{e}_\varphi n_r
+\frac{1}{r}(u_{r,\varphi}-u_\varphi)\boldsymbol{e}_rn_\varphi
+\frac{1}{r}(u_r+u_{\varphi,\varphi})\boldsymbol{e}_\varphi n_\varphi\\
&= (u_{r,r}n_r+\frac{1}{r}(u_{r,\varphi}-u_\varphi)n_\varphi)\boldsymbol{e}_r\\
&+ (u_{\varphi,r}n_r+\frac{1}{r}(u_r+u_{\varphi,\varphi})n_\varphi)\boldsymbol{e}_{\varphi}.\end{split}\]
Pro další okrajovou podmínku platí
(38)¶\[\begin{split}\boldsymbol{t} &= \boldsymbol{T}\cdot\boldsymbol{n}-\mathrm{div}(\hat{\boldsymbol{T}})\cdot\boldsymbol{n}
-\mathrm{div}(\hat{\boldsymbol{T}}\cdot\boldsymbol{n})
+(\mathrm{grad}(\hat{T}\cdot\boldsymbol{n})\cdot\boldsymbol{n})\cdot\boldsymbol{n} \\
&+ [\mathrm{div}(\boldsymbol{n})-(\mathrm{grad}(\boldsymbol{n})\cdot\boldsymbol{n})
\cdot\boldsymbol{n}]((\hat{\boldsymbol{T}}
\cdot\boldsymbol{n})\cdot\boldsymbol{n}),\end{split}\]
kde se první tři součiny mohou rozepsat následovně
(39)¶\[\begin{split}\boldsymbol{T}\cdot\boldsymbol{n} &= (T_{rr}\boldsymbol{e}_{r}\boldsymbol{e}_{r}
+T_{r\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}
+T_{\varphi r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}
+T_{\varphi\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi})
\cdot(\boldsymbol{e}_{r}n_{r}+\boldsymbol{e}_{\varphi}n_{\varphi}) \\
&= T_{rr}\boldsymbol{e}_{r}\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{r}n_{r}
+T_{r\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{r}n_{r}
+T_{\varphi r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{r}n_{r}
+T_{\varphi\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{r}n_{r} \\
&+ T_{rr}\boldsymbol{e}_{r}\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{\varphi}n_{\varphi}
+T_{r\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi}
+T_{\varphi r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi}
+T_{\varphi\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi} \\
&= T_{rr}\boldsymbol{e}_{r}n_{r}+T_{\varphi r}\boldsymbol{e}_{\varphi}n_{r}
+T_{r\varphi}\boldsymbol{e}_{r}n_{\varphi}
+T_{\varphi\varphi}\boldsymbol{e}_{\varphi}n_{\varphi} \\
&= (T_{rr}n_{r}+T_{r\varphi}n_{\varphi})\boldsymbol{e}_{r}
+(T_{\varphi r}n_{r}+T_{\varphi\varphi}n_{\varphi})\boldsymbol{e}_{\varphi},\end{split}\]
(40)¶\[\begin{split}\mathrm{div}(\hat{\boldsymbol{T}})\cdot\boldsymbol{n}
&= [(\hat{T}_{rrr,r}+\frac{1}{r}\hat{T}_{rrr}+\frac{1}{r}\hat{T}_{rr\varphi,\varphi}
-\frac{1}{r}\hat{T}_{r\varphi\varphi}
-\frac{1}{r}\hat{T}_{\varphi r\varphi})\boldsymbol{e}_{r}\boldsymbol{e}_{r} \\
&+ (\hat{T}_{r\varphi r,r}+\frac{1}{r}\hat{T}_{rr\varphi}+\frac{1}{r}\hat{T}_{r\varphi r}
+\frac{1}{r}\hat{T}_{r\varphi\varphi,\varphi}
-\frac{1}{r}\hat{T}_{\varphi\varphi\varphi})\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi} \\
&+ (\hat{T}_{\varphi rr,r}+\frac{1}{r}\hat{T}_{rr\varphi}
+\frac{1}{r}\hat{T}_{\varphi rr}+\frac{1}{r}\hat{T}_{\varphi r\varphi,\varphi}
-\frac{1}{r}\hat{T}_{\varphi\varphi\varphi})\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r} \\
&+ (\hat{T}_{\varphi\varphi r,r}
+\frac{1}{r}\hat{T}_{r\varphi\varphi}+\frac{1}{r}\hat{T}_{\varphi r\varphi}
+\frac{1}{r}\hat{T}_{\varphi\varphi r}
+\frac{1}{r}\hat{T}_{\varphi\varphi\varphi,\varphi})\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}]
\cdot(\boldsymbol{e}_{r}n_{r}+\boldsymbol{e}_{\varphi}n_{\varphi})\\
&= (\hat{T}_{rrr,r}+\frac{1}{r}\hat{T}_{rrr}+\frac{1}{r}\hat{T}_{rr\varphi,\varphi}
-\frac{1}{r}\hat{T}_{r\varphi\varphi}
-\frac{1}{r}\hat{T}_{\varphi r\varphi})\boldsymbol{e}_{r}\boldsymbol{e}_{r}
\cdot\boldsymbol{e}_{r}n_{r}\\
&+ (\hat{T}_{r\varphi r,r}+\frac{1}{r}\hat{T}_{rr\varphi}+\frac{1}{r}\hat{T}_{r\varphi r}
+\frac{1}{r}\hat{T}_{r\varphi\varphi,\varphi}
-\frac{1}{r}\hat{T}_{\varphi\varphi\varphi})\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{r}n_{r}\\
&+ (\hat{T}_{\varphi rr,r}+\frac{1}{r}\hat{T}_{rr\varphi}+\frac{1}{r}\hat{T}_{\varphi rr}
+\frac{1}{r}\hat{T}_{\varphi r\varphi,\varphi}
-\frac{1}{r}\hat{T}_{\varphi\varphi\varphi})\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}
\cdot\boldsymbol{e}_{r}n_{r}\\
&+ (\hat{T}_{\varphi\varphi r,r}+\frac{1}{r}\hat{T}_{r\varphi\varphi}
+\frac{1}{r}\hat{T}_{\varphi r\varphi}+\frac{1}{r}\hat{T}_{\varphi\varphi r}
+\frac{1}{r}\hat{T}_{\varphi\varphi\varphi,\varphi})\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{r}n_{r}\\
&+ (\hat{T}_{rrr,r}+\frac{1}{r}\hat{T}_{rrr}+\frac{1}{r}\hat{T}_{rr\varphi,\varphi}
-\frac{1}{r}\hat{T}_{r\varphi\varphi}
-\frac{1}{r}\hat{T}_{\varphi r\varphi})\boldsymbol{e}_{r}\boldsymbol{e}_{r}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi}\\&+(\hat{T}_{r\varphi r,r}
+\frac{1}{r}\hat{T}_{rr\varphi}+\frac{1}{r}\hat{T}_{r\varphi r}
+\frac{1}{r}\hat{T}_{r\varphi\varphi,\varphi}
-\frac{1}{r}\hat{T}_{\varphi\varphi\varphi})\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi}\\&+(\hat{T}_{\varphi rr,r}
+\frac{1}{r}\hat{T}_{rr\varphi}+\frac{1}{r}\hat{T}_{\varphi rr}
+\frac{1}{r}\hat{T}_{\varphi r\varphi,\varphi}
-\frac{1}{r}\hat{T}_{\varphi\varphi\varphi})\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi}\\
&+ (\hat{T}_{\varphi\varphi r,r}+\frac{1}{r}\hat{T}_{r\varphi\varphi}
+\frac{1}{r}\hat{T}_{\varphi r\varphi}+\frac{1}{r}\hat{T}_{\varphi\varphi r}
+\frac{1}{r}\hat{T}_{\varphi\varphi\varphi,\varphi})\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi}\\
&= [(\hat{T}_{rrr,r}+\frac{1}{r}\hat{T}_{rrr}+\frac{1}{r}\hat{T}_{rr\varphi,\varphi}
-\frac{1}{r}\hat{T}_{r\varphi\varphi}-\frac{1}{r}\hat{T}_{\varphi r\varphi})n_{r}\\
&+ (\hat{T}_{r\varphi r,r}+\frac{1}{r}\hat{T}_{rr\varphi}+\frac{1}{r}\hat{T}_{r\varphi r}
+\frac{1}{r}\hat{T}_{r\varphi\varphi,\varphi}
-\frac{1}{r}\hat{T}_{\varphi\varphi\varphi})n_{\varphi}]\boldsymbol{e}_{r}\\
&+ [(\hat{T}_{\varphi rr,r}+\frac{1}{r}\hat{T}_{rr\varphi}+\frac{1}{r}\hat{T}_{\varphi rr}
+\frac{1}{r}\hat{T}_{\varphi r\varphi,\varphi}
-\frac{1}{r}\hat{T}_{\varphi\varphi\varphi})n_{r}\\
&+ (\hat{T}_{\varphi\varphi r,r}+\frac{1}{r}\hat{T}_{r\varphi\varphi}
+\frac{1}{r}\hat{T}_{\varphi r\varphi}+\frac{1}{r}\hat{T}_{\varphi\varphi r}
+\frac{1}{r}\hat{T}_{\varphi\varphi\varphi,\varphi})n_{\varphi}]\boldsymbol{e}_{\varphi}\end{split}\]
(41)¶\[\begin{split}\mathrm{div}(\hat{\boldsymbol{T}}\cdot\boldsymbol{n})
&= \mathrm{div}([\hat{T}_{rrr}\boldsymbol{e}_{r}\boldsymbol{e}_{r}\boldsymbol{e}_{r}
+\hat{T}_{rr\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}
+\hat{T}_{r\varphi r}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}
+\hat{T}_{r\varphi\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}\\
&+ \hat{T}_{\varphi rr}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{r}
+\hat{T}_{\varphi r\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}
+\hat{T}_{\varphi\varphi r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}\\
&+ \hat{T}_{\varphi\varphi\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}]
\cdot(\boldsymbol{e}_{r}n_{r}+\boldsymbol{e}_{\varphi}n_{\varphi}))\\
&= \mathrm{div}([\hat{T}_{rrr}\boldsymbol{e}_{r}\boldsymbol{e}_{r}\boldsymbol{e}_{r}
\cdot\boldsymbol{e}_{r}n_{r}
+\hat{T}_{rr\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{r}n_{r}
+\hat{T}_{r\varphi r}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}
\cdot\boldsymbol{e}_{r}n_{r}\\
&+ \hat{T}_{r\varphi\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{r}n_{r}
+\hat{T}_{\varphi rr}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{r}
\cdot\boldsymbol{e}_{r}n_{r}
+\hat{T}_{\varphi r\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{r}n_{r}\\
&+ \hat{T}_{\varphi\varphi r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}
\cdot\boldsymbol{e}_{r}n_{r}
+\hat{T}_{\varphi\varphi\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{r}n_{r}\\
&+ \hat{T}_{rrr}\boldsymbol{e}_{r}\boldsymbol{e}_{r}\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{\varphi}n_{\varphi}
+\hat{T}_{rr\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi}
+\hat{T}_{r\varphi r}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi}\\
&+ \hat{T}_{r\varphi\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi}
+\hat{T}_{\varphi rr}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{r}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi}
+\hat{T}_{\varphi r\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi}\\
&+ \hat{T}_{\varphi\varphi r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi}
+\hat{T}_{\varphi\varphi\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi})\\
&= \mathrm{div}((\hat{T}_{rrr}n_{r}+\hat{T}_{rr\varphi}n_{\varphi})\boldsymbol{e}_{r}\boldsymbol{e}_{r}
+(\hat{T}_{r\varphi r}n_{r}
+\hat{T}_{r\varphi\varphi}n_{\varphi})\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}\\
&+ (\hat{T}_{\varphi rr}n_{r}
+\hat{T}_{\varphi r\varphi}n_{\varphi})\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}
+(\hat{T}_{\varphi\varphi r}n_{r}
+\hat{T}_{\varphi\varphi\varphi}n_{\varphi})\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi})\\
&= [(\hat{T}_{rrr}n_{r}+\hat{T}_{rr\varphi}n_{\varphi})_{,r}
+\frac{1}{r}(\hat{T}_{rrr}n_{r}+\hat{T}_{rr\varphi}n_{\varphi})\\
&+ \frac{1}{r}(\hat{T}_{r\varphi r}n_{r}+\hat{T}_{r\varphi\varphi}n_{\varphi})_{,\varphi}
-\frac{1}{r}(\hat{T}_{\varphi\varphi r}n_{r}+\hat{T}_{\varphi\varphi\varphi}n_{\varphi})]\boldsymbol{e}_{r}\\
&+ [(\hat{T}_{\varphi rr}n_{r}+\hat{T}_{\varphi r\varphi}n_{\varphi})_{,r}
+\frac{1}{r}(\hat{T}_{r\varphi r}n_{r}+\hat{T}_{r\varphi\varphi}n_{\varphi})\\
&+ \frac{1}{r}(\hat{T}_{\varphi rr}n_{r}+\hat{T}_{\varphi r\varphi}n_{\varphi})
+\frac{1}{r}(\hat{T}_{\varphi\varphi r}n_{r}
+\hat{T}_{\varphi\varphi\varphi}n_{\varphi})_{,\varphi}]\boldsymbol{e}_{\varphi}.\end{split}\]
Čtvrtý výraz v (38) je obsáhlejší. Pro rozepsání se dostane
(42)¶\[\begin{split}(\mathrm{grad}(\hat{T}\cdot\boldsymbol{n})\cdot\boldsymbol{n})\cdot\boldsymbol{n}
&= [\mathrm{grad}((\hat{T}_{rrr}\boldsymbol{e}_{r}\boldsymbol{e}_{r}\boldsymbol{e}_{r}
+\hat{T}_{rr\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}
+\hat{T}_{r\varphi r}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}
+\hat{T}_{r\varphi\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}\\
&+ \hat{T}_{\varphi rr}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{r}
+\hat{T}_{\varphi r\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}
+\hat{T}_{\varphi\varphi r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}\\
&+ \hat{T}_{\varphi\varphi\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi})
\cdot(\boldsymbol{e}_{r}n_{r}
+\boldsymbol{e}_{\varphi}n_{\varphi}))\cdot(\boldsymbol{e}_{r}n_{r}
+\boldsymbol{e}_{\varphi}n_{\varphi})]\cdot(\boldsymbol{e}_{r}n_{r}
+\boldsymbol{e}_{\varphi}n_{\varphi}).\end{split}\]
Výpočtem gradientu se výraz upraví na
(43)¶\[\begin{split}(\mathrm{grad}(\hat{T}\cdot\boldsymbol{n})\cdot\boldsymbol{n})\cdot\boldsymbol{n}
&= [\mathrm{grad}(\hat{T}_{rrr}\boldsymbol{e}_{r}\boldsymbol{e}_{r}\boldsymbol{e}_{r}
\cdot\boldsymbol{e}_{r}n_{r}
+\hat{T}_{rr\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{r}n_{r}
+\hat{T}_{r\varphi r}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}
\cdot\boldsymbol{e}_{r}n_{r} \\
&+ \hat{T}_{r\varphi\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{r}n_{r}
+\hat{T}_{\varphi rr}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{r}
\cdot\boldsymbol{e}_{r}n_{r}
+\hat{T}_{\varphi r\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{r}n_{r} \\
&+ \hat{T}_{\varphi\varphi r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}
\cdot\boldsymbol{e}_{r}n_{r}
+\hat{T}_{\varphi\varphi\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{r}n_{r}
+\hat{T}_{rrr}\boldsymbol{e}_{r}\boldsymbol{e}_{r}\boldsymbol{e}_{r}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi} \\
&+ \hat{T}_{rr\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi}
+\hat{T}_{r\varphi r}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi}
+\hat{T}_{r\varphi\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi}\\
&+ \hat{T}_{\varphi rr}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{r}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi}
+\hat{T}_{\varphi r\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi}
+\hat{T}_{\varphi\varphi r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi}\\
&+ \hat{T}_{\varphi\varphi\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi})\cdot(\boldsymbol{e}_{r}n_{r}
+\boldsymbol{e}_{\varphi}n_{\varphi})]\cdot(\boldsymbol{e}_{r}n_{r}
+\boldsymbol{e}_{\varphi}n_{\varphi})\\
&= [\mathrm{grad}((\hat{T}_{rrr}n_{r}+\hat{T}_{rr\varphi}n_{\varphi})\boldsymbol{e}_{r}\boldsymbol{e}_{r}
+(\hat{T}_{r\varphi r}n_{r}
+\hat{T}_{r\varphi\varphi}n_{\varphi})\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}\\
&+ (\hat{T}_{\varphi rr}n_{r}
+\hat{T}_{\varphi r\varphi}n_{\varphi})\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r} \\
&+ (\hat{T}_{\varphi\varphi r}n_{r}
+\hat{T}_{\varphi\varphi\varphi}n_{\varphi})\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi})
\cdot(\boldsymbol{e}_{r}n_{r}+\boldsymbol{e}_{\varphi}n_{\varphi})]\cdot(\boldsymbol{e}_{r}n_{r}
+\boldsymbol{e}_{\varphi}n_{\varphi})\\
&= [(\hat{T}_{rrr}n_{r}
+\hat{T}_{rr\varphi}n_{\varphi})_{,r}\boldsymbol{e}_{r}\boldsymbol{e}_{r}\boldsymbol{e}_{r}\\
&+ \frac{1}{r}((\hat{T}_{rrr}n_{r}+\hat{T}_{rr\varphi}n_{\varphi})_{,\varphi}
-(\hat{T}_{r\varphi r}n_{r}+\hat{T}_{r\varphi\varphi}n_{\varphi})
-(\hat{T}_{\varphi rr}n_{r}+\hat{T}_{\varphi r\varphi}n_{\varphi}))
\boldsymbol{e}_{r}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}\\
&+ (\hat{T}_{r\varphi r}n_{r}+\hat{T}_{r\varphi\varphi}n_{\varphi})_{,r}
\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}\\
&+ \frac{1}{r}((\hat{T}_{rrr}n_{r}+\hat{T}_{rr\varphi}n_{\varphi})
-(\hat{T}_{\varphi\varphi r}n_{r}+\hat{T}_{\varphi\varphi\varphi}n_{\varphi})
+(\hat{T}_{r\varphi r}n_{r}+\hat{T}_{r\varphi\varphi}n_{\varphi})_{,\varphi})
\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}\\
&+ (\hat{T}_{\varphi rr}n_{r}+\hat{T}_{\varphi r\varphi}n_{\varphi})_{,r}
\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{r}\\
&+ \frac{1}{r}((\hat{T}_{rrr}n_r+\hat{T}_{rr\varphi}n_\varphi)
+(\hat{T}_{\varphi rr}n_r+\hat{T}_{\varphi r\varphi}n_\varphi)_\varphi
-(\hat{T}_{\varphi\varphi r}n_{r}+\hat{T}_{\varphi\varphi\varphi}n_{\varphi}))
\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}\\
&+ (\hat{T}_{\varphi\varphi r}n_{r}
+\hat{T}_{\varphi\varphi\varphi}n_{\varphi})_{,r}
\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}\\
&+ \frac{1}{r}((\hat{T}_{r\varphi r}n_{r}+\hat{T}_{r\varphi\varphi}n_{\varphi})
+(\hat{T}_{\varphi rr}n_{r}+\hat{T}_{\varphi r\varphi}n_{\varphi}) \\
&+ (\hat{T}_{\varphi\varphi r}n_{r}
+\hat{T}_{\varphi\varphi\varphi}n_{\varphi})_{,\varphi})
\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}]
\cdot(\boldsymbol{e}_{r}n_{r}+\boldsymbol{e}_{\varphi}n_{\varphi})
\cdot(\boldsymbol{e}_{r}n_{r}+\boldsymbol{e}_{\varphi}n_{\varphi}).\end{split}\]
Roznásobením výrazu v hranaté závorce a normály \(\boldsymbol{n}\) se dále dostane
(44)¶\[\begin{split}(\mathrm{grad}(\hat{T}\cdot\boldsymbol{n})\cdot\boldsymbol{n})\cdot\boldsymbol{n}
&= [(\hat{T}_{rrr}n_{r}+\hat{T}_{rr\varphi}n_{\varphi})_{,r}\boldsymbol{e}_{r}\boldsymbol{e}_{r}n_{r}\\
&+ (\hat{T}_{r\varphi r}n_{r}+\hat{T}_{r\varphi\varphi}n_{\varphi})_{,r}
\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}n_{r}\\
&+ (\hat{T}_{\varphi rr}n_{r}+\hat{T}_{\varphi r\varphi}n_{\varphi})_{,r}
\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}n_{r}\\
&+ (\hat{T}_{\varphi\varphi r}n_{r}+\hat{T}_{\varphi\varphi\varphi}n_{\varphi})_{,r}
\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}n_{r}\\
&+ \frac{1}{r}((\hat{T}_{rrr}n_{r}+\hat{T}_{rr\varphi}n_{\varphi})_{,\varphi}
-(\hat{T}_{r\varphi r}n_{r}+\hat{T}_{r\varphi\varphi}n_{\varphi})
-(\hat{T}_{\varphi rr}n_{r}+\hat{T}_{\varphi r\varphi}n_{\varphi}))
\boldsymbol{e}_{r}\boldsymbol{e}_{r}n_{\varphi}\\
&+ \frac{1}{r}((\hat{T}_{rrr}n_{r}+\hat{T}_{rr\varphi}n_{\varphi})
-(\hat{T}_{\varphi\varphi r}n_{r}+\hat{T}_{\varphi\varphi\varphi}n_{\varphi})
+(\hat{T}_{r\varphi r}n_{r}+\hat{T}_{r\varphi\varphi}n_{\varphi})_{,\varphi})
\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}n_{\varphi}\\
&+ \frac{1}{r}((\hat{T}_{rrr}n_{r}+\hat{T}_{rr\varphi}n_{\varphi})
+(\hat{T}_{\varphi rr}n_{r}+\hat{T}_{\varphi r\varphi}n_{\varphi})_{\varphi}
-(\hat{T}_{\varphi\varphi r}n_{r}+\hat{T}_{\varphi\varphi\varphi}n_{\varphi}))
\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}n_{\varphi}\\
&+ \frac{1}{r}((\hat{T}_{r\varphi r}n_{r}+\hat{T}_{r\varphi\varphi}n_{\varphi})
+(\hat{T}_{\varphi rr}n_{r}+\hat{T}_{\varphi r\varphi}n_{\varphi})
+(\hat{T}_{\varphi\varphi r}n_{r}+\hat{T}_{\varphi\varphi\varphi}n_{\varphi})_{,\varphi})
\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}n_{\varphi}]
\cdot(\boldsymbol{e}_{r}n_{r}+\boldsymbol{e}_{\varphi}n_{\varphi}).\end{split}\]
A konečně, roznásobením posledního skalárního součinu s normálou \(\boldsymbol{n}\) se dostane
(45)¶\[\begin{split}(\mathrm{grad}(\hat{T}\cdot\boldsymbol{n})\cdot\boldsymbol{n})\cdot\boldsymbol{n}
&= (\hat{T}_{rrr}n_{r}+\hat{T}_{rr\varphi}n_{\varphi})_{,r}\boldsymbol{e}_{r}n_{r}n_{r}\\
&+ (\hat{T}_{\varphi rr}n_{r}+\hat{T}_{\varphi r\varphi}n_{\varphi})_{,r}
\boldsymbol{e}_{\varphi}n_{r}n_{r}\\
&+ \frac{1}{r}((\hat{T}_{rrr}n_{r}+\hat{T}_{rr\varphi}n_{\varphi})_{,\varphi}
-(\hat{T}_{r\varphi r}n_{r}+\hat{T}_{r\varphi\varphi}n_{\varphi})
-(\hat{T}_{\varphi rr}n_{r}+\hat{T}_{\varphi r\varphi}n_{\varphi}))
\boldsymbol{e}_{r}n_{\varphi}n_{r}\\
&+ \frac{1}{r}((\hat{T}_{rrr}n_{r}+\hat{T}_{rr\varphi}n_{\varphi})
+(\hat{T}_{\varphi rr}n_{r}+\hat{T}_{\varphi r\varphi}n_{\varphi})_{,\varphi}
-(\hat{T}_{\varphi\varphi r}n_{r}+\hat{T}_{\varphi\varphi\varphi}n_{\varphi}))
\boldsymbol{e}_{\varphi}n_{\varphi}n_{r}\\
&+ (\hat{T}_{r\varphi r}n_{r}+\hat{T}_{r\varphi\varphi}n_{\varphi})_{,r}
\boldsymbol{e}_{r}n_{r}n_{\varphi}\\
&+ (\hat{T}_{\varphi\varphi r}n_{r}+\hat{T}_{\varphi\varphi\varphi}n_{\varphi})_{,r}
\boldsymbol{e}_{\varphi}n_{r}n_{\varphi}\\
&+ \frac{1}{r}((\hat{T}_{rrr}n_{r}+\hat{T}_{rr\varphi}n_{\varphi})
-(\hat{T}_{\varphi\varphi r}n_{r}+\hat{T}_{\varphi\varphi\varphi}n_{\varphi})
+(\hat{T}_{r\varphi r}n_{r}+\hat{T}_{r\varphi\varphi}n_{\varphi})_{,\varphi})
\boldsymbol{e}_{r}n_{\varphi}n_{\varphi}\\
&+ \frac{1}{r}((\hat{T}_{r\varphi r}n_{r}+\hat{T}_{r\varphi\varphi}n_{\varphi})
+(\hat{T}_{\varphi rr}n_{r}+\hat{T}_{\varphi r\varphi}n_{\varphi})
+(\hat{T}_{\varphi\varphi r}n_{r}+\hat{T}_{\varphi\varphi\varphi}n_{\varphi})_{,\varphi})
\boldsymbol{e}_{\varphi}n_{\varphi}n_{\varphi}.\end{split}\]
S využitím symetrie
(46)¶\[\hat{T}_{ijk}=\hat{T}_{jik}\]
se může konečně psát
(47)¶\[\begin{split}(\mathrm{grad}(\hat{T}\cdot\boldsymbol{n})\cdot\boldsymbol{n})\cdot\boldsymbol{n}
&= [(\hat{T}_{rrr}n_{r}+\hat{T}_{rr\varphi}n_{\varphi})_{,r}n_{r}n_{r}\\
&+ \frac{1}{r}((\hat{T}_{rrr}n_{r}+\hat{T}_{rr\varphi}n_{\varphi})_{,\varphi}
-2\hat{T}_{r\varphi r}n_{r}-2\hat{T}_{r\varphi\varphi}n_{\varphi})n_{\varphi}n_{r}\\
&+ (\hat{T}_{r\varphi r}n_{r}+\hat{T}_{r\varphi\varphi}n_{\varphi})_{,r}n_{r}n_{\varphi}\\
&+ \frac{1}{r}((\hat{T}_{rrr}n_{r}+\hat{T}_{rr\varphi}n_{\varphi})-(\hat{T}_{\varphi\varphi r}n_{r}
+\hat{T}_{\varphi\varphi\varphi}n_{\varphi})+(\hat{T}_{r\varphi r}n_{r}
+\hat{T}_{r\varphi\varphi}n_{\varphi})_{,\varphi})n_{\varphi}n_{\varphi}]\boldsymbol{e}_{r}\\
&+ [(\hat{T}_{\varphi rr}n_{r}+\hat{T}_{\varphi r\varphi}n_{\varphi})_{,r}n_{r}n_{r}\\
&+ \frac{1}{r}((\hat{T}_{rrr}n_{r}+\hat{T}_{rr\varphi}n_{\varphi})+(\hat{T}_{\varphi rr}n_{r}
+\hat{T}_{\varphi r\varphi}n_{\varphi})_{,\varphi}
-(\hat{T}_{\varphi\varphi r}n_{r}+\hat{T}_{\varphi\varphi\varphi}n_{\varphi}))n_{\varphi}n_{r}\\
&+ (\hat{T}_{\varphi\varphi r}n_{r}+\hat{T}_{\varphi\varphi\varphi}n_{\varphi})_{,r}n_{r}n_{\varphi}\\
&+ \frac{1}{r}(2\hat{T}_{r\varphi r}n_{r}+2\hat{T}_{r\varphi\varphi}n_{\varphi}
+(\hat{T}_{\varphi\varphi r}n_{r}+\hat{T}_{\varphi\varphi\varphi}n_{\varphi})_{,\varphi})
n_{\varphi}n_{\varphi}]\boldsymbol{e}_{\varphi}.\end{split}\]
Pro poslední výraz v (38) je
(48)¶\[\begin{split}[\mathrm{div}(\boldsymbol{n})
&- (\mathrm{grad}(\boldsymbol{n})\cdot\boldsymbol{n})\cdot\boldsymbol{n}]
((\hat{\boldsymbol{T}}\cdot\boldsymbol{n})\cdot\boldsymbol{n})
=[n_{r,r}+\frac{1}{r}n_{r}+\frac{1}{r}n_{\varphi,\varphi} \\
&- [(n_{r,r}\boldsymbol{e}_{r}\boldsymbol{e}_{r}+\frac{1}{r}(n_{r,\varphi}
-n_{\varphi})\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}
+n_{\varphi,r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r} \\
&+ \frac{1}{r}(n_{r}+n_{\varphi,\varphi})\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi})
\cdot(\boldsymbol{e}_{r}n_{r}+\boldsymbol{e}_{\varphi}n_{\varphi})]
\cdot(\boldsymbol{e}_{r}n_{r}+\boldsymbol{e}_{\varphi}n_{\varphi})] \\
&\times [[(\hat{T}_{rrr}\boldsymbol{e}_{r}\boldsymbol{e}_{r}\boldsymbol{e}_{r}
+\hat{T}_{rr\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}
+\hat{T}_{r\varphi r}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}
+\hat{T}_{r\varphi\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}
+\hat{T}_{\varphi rr}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{r} \\
&+ \hat{T}_{\varphi r\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}
+\hat{T}_{\varphi\varphi r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}
+\hat{T}_{\varphi\varphi\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi})
\cdot(\boldsymbol{e}_{r}n_{r}+\boldsymbol{e}_{\varphi}n_{\varphi})]
\cdot(\boldsymbol{e}_{r}n_{r}+\boldsymbol{e}_{\varphi}n_{\varphi})].\end{split}\]
Roznásobením skalárních součinů se dostane
(49)¶\[\begin{split}[\mathrm{div}(\boldsymbol{n})
&- (\mathrm{grad}(\boldsymbol{n})\cdot\boldsymbol{n})\cdot\boldsymbol{n}]
((\hat{\boldsymbol{T}}\cdot\boldsymbol{n})\cdot\boldsymbol{n})
=[n_{r,r}+\frac{1}{r}n_{r}+\frac{1}{r}n_{\varphi,\varphi} \\
&- [n_{r,r}\boldsymbol{e}_{r}\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{r}n_{r}
+\frac{1}{r}(n_{r,\varphi}-n_{\varphi})\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{r}n_{r}+n_{\varphi,r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{r}n_{r} \\
&+ \frac{1}{r}(n_{r}+n_{\varphi,\varphi})\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{r}n_{r}
+n_{r,r}\boldsymbol{e}_{r}\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{\varphi}n_{\varphi} \\
&+ \frac{1}{r}(n_{r,\varphi}-n_{\varphi})\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi}+n_{\varphi,r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi} \\
&+ \frac{1}{r}(n_{r}+n_{\varphi,\varphi})\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi}]\cdot(\boldsymbol{e}_{r}n_{r}+\boldsymbol{e}_{\varphi}n_{\varphi})] \\
&\times [[\hat{T}_{rrr}\boldsymbol{e}_{r}\boldsymbol{e}_{r}\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{r}n_{r}
+\hat{T}_{rr\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{r}n_{r}
+\hat{T}_{r\varphi r}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{r}n_{r} \\
&+ \hat{T}_{r\varphi\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{r}n_{r}
+\hat{T}_{\varphi rr}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{r}n_{r}
+\hat{T}_{\varphi r\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{r}n_{r} \\
&+ \hat{T}_{\varphi\varphi r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}
\cdot\boldsymbol{e}_{r}n_{r}
+\hat{T}_{\varphi\varphi\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{r}n_{r}
+\hat{T}_{rrr}\boldsymbol{e}_{r}\boldsymbol{e}_{r}\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{\varphi}n_{\varphi} \\
&+ \hat{T}_{rr\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{\varphi}n_{\varphi}
+\hat{T}_{r\varphi r}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi}
+\hat{T}_{r\varphi\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi} \\
&+ \hat{T}_{\varphi rr}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{r}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi}
+\hat{T}_{\varphi r\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi}
+\hat{T}_{\varphi\varphi r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi} \\
&+ \hat{T}_{\varphi\varphi\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi}]\cdot(\boldsymbol{e}_{r}n_{r}+\boldsymbol{e}_{\varphi}n_{\varphi})]\end{split}\]
a ještě jednou
(50)¶\[\begin{split}[\mathrm{div}(\boldsymbol{n})
&- (\mathrm{grad}(\boldsymbol{n})\cdot\boldsymbol{n})\cdot\boldsymbol{n}]
((\hat{\boldsymbol{T}}\cdot\boldsymbol{n})\cdot\boldsymbol{n})
= [n_{r,r}+\frac{1}{r}n_{r}+\frac{1}{r}n_{\varphi,\varphi} \\
&- [n_{r,r}\boldsymbol{e}_{r}n_{r}\cdot\boldsymbol{e}_{r}n_{r}
+n_{\varphi,r}\boldsymbol{e}_{\varphi}n_{r}\cdot\boldsymbol{e}_{r}n_{r}
+\frac{1}{r}(n_{r,\varphi}-n_{\varphi})\boldsymbol{e}_{r}n_{\varphi}\cdot\boldsymbol{e}_{r}n_{r} \\
&+ \frac{1}{r}(n_{r}+n_{\varphi,\varphi})\boldsymbol{e}_{\varphi}n_{\varphi}\cdot\boldsymbol{e}_{r}n_{r}
+n_{r,r}\boldsymbol{e}_{r}n_{r}\cdot\boldsymbol{e}_{\varphi}n_{\varphi}
+n_{\varphi,r}\boldsymbol{e}_{\varphi}n_{r}\cdot\boldsymbol{e}_{\varphi}n_{\varphi} \\
&+ \frac{1}{r}(n_{r,\varphi}-n_{\varphi})\boldsymbol{e}_{r}n_{\varphi}\cdot\boldsymbol{e}_{\varphi}n_{\varphi}
+\frac{1}{r}(n_{r}+n_{\varphi,\varphi})\boldsymbol{e}_{\varphi}n_{\varphi}\cdot\boldsymbol{e}_{\varphi}n_{\varphi}]] \\
&\times [\hat{T}_{rrr}\boldsymbol{e}_{r}\boldsymbol{e}_{r}n_{r}\cdot\boldsymbol{e}_{r}n_{r}
+\hat{T}_{r\varphi r}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}n_{r}\cdot\boldsymbol{e}_{r}n_{r}
+\hat{T}_{\varphi rr}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}n_{r}\cdot\boldsymbol{e}_{r}n_{r} \\
&+ \hat{T}_{\varphi\varphi r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}n_{r}\cdot\boldsymbol{e}_{r}n_{r}
+\hat{T}_{rr\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{r}n_{\varphi}\cdot\boldsymbol{e}_{r}n_{r}
+\hat{T}_{r\varphi\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}n_{\varphi}\cdot\boldsymbol{e}_{r}n_{r} \\
&+ \hat{T}_{\varphi r\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}n_{\varphi}\cdot\boldsymbol{e}_{r}n_{r}
+\hat{T}_{\varphi\varphi\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}n_{\varphi}
\cdot\boldsymbol{e}_{r}n_{r}
+\hat{T}_{rrr}\boldsymbol{e}_{r}\boldsymbol{e}_{r}n_{r}\cdot\boldsymbol{e}_{\varphi}n_{\varphi} \\
&+ \hat{T}_{r\varphi r}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}n_{r}\cdot\boldsymbol{e}_{\varphi}n_{\varphi}
+\hat{T}_{\varphi rr}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}n_{r}\cdot\boldsymbol{e}_{\varphi}n_{\varphi}
+\hat{T}_{\varphi\varphi r}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}n_{r}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi} \\
&+ \hat{T}_{rr\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{r}n_{\varphi}\cdot\boldsymbol{e}_{\varphi}n_{\varphi}
+\hat{T}_{r\varphi\varphi}\boldsymbol{e}_{r}\boldsymbol{e}_{\varphi}n_{\varphi}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi}
+\hat{T}_{\varphi r\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{r}n_{\varphi}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi} \\
&+ \hat{T}_{\varphi\varphi\varphi}\boldsymbol{e}_{\varphi}\boldsymbol{e}_{\varphi}n_{\varphi}
\cdot\boldsymbol{e}_{\varphi}n_{\varphi}].\end{split}\]
A po úpravách se dostane
(51)¶\[\begin{split}[\mathrm{div}(\boldsymbol{n})
&- (\mathrm{grad}(\boldsymbol{n})\cdot\boldsymbol{n})\cdot\boldsymbol{n}]
((\hat{\boldsymbol{T}}\cdot\boldsymbol{n})\cdot\boldsymbol{n})
=[n_{r,r}+\frac{1}{r}n_{r}+\frac{1}{r}n_{\varphi,\varphi} \\
&- [n_{r,r}n_{r}n_{r}+\frac{1}{r}(n_{r,\varphi}-n_{\varphi})n_{\varphi}n_{r}
+n_{\varphi,r}n_{r}n_{\varphi}+\frac{1}{r}(n_{r}
+n_{\varphi,\varphi})n_{\varphi}n_{\varphi}]] \\
&\times [\hat{T}_{rrr}\boldsymbol{e}_{r}n_{r}n_{r}
+\hat{T}_{\varphi rr}\boldsymbol{e}_{\varphi}n_{r}n_{r}
+\hat{T}_{rr\varphi}\boldsymbol{e}_{r}n_{\varphi}n_{r} \\
&+ \hat{T}_{\varphi r\varphi}\boldsymbol{e}_{\varphi}n_{\varphi}n_{r}
+\hat{T}_{r\varphi r}\boldsymbol{e}_{r}n_{r}n_{\varphi}
+\hat{T}_{\varphi\varphi r}\boldsymbol{e}_{\varphi}n_{r}n_{\varphi} \\
&+ \hat{T}_{r\varphi\varphi}\boldsymbol{e}_{r}n_{\varphi}n_{\varphi}
+\hat{T}_{\varphi\varphi\varphi}\boldsymbol{e}_{\varphi}n_{\varphi}n_{\varphi}] \\
&= [n_{r,r}(n_{r}^{2}+n_{\varphi}^{2})+\frac{1}{r}n_{r}(n_{r}^{2}
+n_{\varphi}^{2})+\frac{1}{r}n_{\varphi,\varphi}(n_{r}^{2}+n_{\varphi}^{2})
-n_{r,r}n_{r}^{2} \\
&- \frac{1}{r}n_{r,\varphi}n_{\varphi}n_{r}+\frac{1}{r}n_{\varphi}^{2}n_{r}
-n_{\varphi,r}n_{r}n_{\varphi}-\frac{1}{r}n_{r}n_{\varphi}^{2}
-\frac{1}{r}n_{\varphi,\varphi}n_{\varphi}^{2}] \\
&\times [(\hat{T}_{rrr}n_{r}n_{r}+\hat{T}_{rr\varphi}n_{\varphi}n_{r}
+\hat{T}_{r\varphi r}n_{r}n_{\varphi}
+\hat{T}_{r\varphi\varphi}n_{\varphi}n_{\varphi})\boldsymbol{e}_{r} \\
&+ (\hat{T}_{\varphi rr}n_{r}n_{r}+\hat{T}_{\varphi r\varphi}n_{\varphi}n_{r}
+\hat{T}_{\varphi\varphi r}n_{r}n_{\varphi}
+\hat{T}_{\varphi\varphi\varphi}n_{\varphi}n_{\varphi})\boldsymbol{e}_{\varphi}] \\
&= (n_{r,r}n_{\varphi}^{2}+\frac{1}{r}n_{r}^{3}+\frac{1}{r}n_{r}n_{\varphi}^{2}
+\frac{1}{r}n_{\varphi,\varphi}n_{r}^{2}-\frac{1}{r}n_{r,\varphi}n_{\varphi}n_{r}
-n_{\varphi,r}n_{r}n_{\varphi}) \\
&\times [(\hat{T}_{rrr}n_{r}^{2}+(\hat{T}_{rr\varphi}+\hat{T}_{r\varphi r})n_{r}n_{\varphi}
+\hat{T}_{r\varphi\varphi}n_{\varphi}^{2})\boldsymbol{e}_{r} \\
&+ (\hat{T}_{\varphi rr}n_{r}^{2}+(\hat{T}_{\varphi r\varphi}
+\hat{T}_{\varphi\varphi r})n_{r}n_{\varphi}
+\hat{T}_{\varphi\varphi\varphi}n_{\varphi}^{2})\boldsymbol{e}_{\varphi}]\end{split}\]
Takže nakonec se pro složky vektoru napětí \(\boldsymbol{t}\) v polárních souřadnících může psát
(52)¶\[\begin{split}t_{r} &= T_{rr}n_{r}+T_{r\varphi}n_{\varphi}-(\hat{T}_{rrr,r}+\frac{1}{r}\hat{T}_{rrr}
+\frac{1}{r}\hat{T}_{rr\varphi,\varphi}-\frac{2}{r}\hat{T}_{r\varphi\varphi})n_{r}\\
&- (\hat{T}_{r\varphi r,r}+\frac{1}{r}\hat{T}_{rr\varphi}+\frac{1}{r}\hat{T}_{r\varphi r}
+\frac{1}{r}\hat{T}_{r\varphi\varphi,\varphi}
-\frac{1}{r}\hat{T}_{\varphi\varphi\varphi})n_{\varphi}\\
&- (\hat{T}_{rrr}n_{r}+\hat{T}_{rr\varphi}n_{\varphi})_{,r}n_{\varphi}^{2}
+\frac{1}{r}(\hat{T}_{rrr}n_{r}+\hat{T}_{rr\varphi}n_{\varphi})_{,\varphi}n_{r}n_{\varphi}
-\frac{1}{r}(\hat{T}_{rrr}n_{r}+\hat{T}_{rr\varphi}n_{\varphi})n_{r}^{2}\\
&+ (\hat{T}_{r\varphi r}n_{r}+\hat{T}_{r\varphi\varphi}n_{\varphi})_{,r}n_{r}n_{\varphi}
-\frac{1}{r}(\hat{T}_{r\varphi r}n_{r}+\hat{T}_{r\varphi\varphi}n_{\varphi})_{,\varphi}n_{r}^{2}
-\frac{2}{r}(\hat{T}_{r\varphi r}n_{r}+\hat{T}_{r\varphi\varphi}n_{\varphi})n_{r}n_{\varphi}\\
&+ \frac{1}{r}(\hat{T}_{\varphi\varphi r}n_{r}+\hat{T}_{\varphi\varphi\varphi}n_{\varphi})n_{r}^{2}\\
&+ (n_{r,r}n_{\varphi}^{2}+\frac{1}{r}n_{r}^{3}+\frac{1}{r}n_{r}n_{\varphi}^{2}
+\frac{1}{r}n_{r}^{2}n_{\varphi,\varphi}-\frac{1}{r}n_{r,\varphi}n_{r}n_{\varphi}
-n_{r}n_{\varphi}n_{\varphi,r})\\
&\times (\hat{T}_{rrr}n_{r}^{2}+(\hat{T}_{rr\varphi}+\hat{T}_{r\varphi r})n_{r}n_{\varphi}
+\hat{T}_{r\varphi\varphi}n_{\varphi}^{2}),\end{split}\]
a
(53)¶\[\begin{split}t_{\varphi} &= T_{\varphi r}n_{r}+T_{\varphi\varphi}n_{\varphi}-(\hat{T}_{r\varphi r,r}
+\frac{1}{r}\hat{T}_{rr\varphi}+\frac{1}{r}\hat{T}_{r\varphi r}
+\frac{1}{r}\hat{T}_{r\varphi\varphi,\varphi}-\frac{1}{r}\hat{T}_{\varphi\varphi\varphi})n_{r}\\
&- (\hat{T}_{\varphi\varphi r,r}+\frac{2}{r}\hat{T}_{r\varphi\varphi}
+\frac{1}{r}\hat{T}_{\varphi\varphi r}
+\frac{1}{r}\hat{T}_{\varphi\varphi\varphi,\varphi})n_{\varphi}\\
&- (\hat{T}_{r\varphi r}n_{r}+\hat{T}_{r\varphi\varphi}n_{\varphi})_{,r}n_{\varphi}^{2}
+\frac{1}{r}(\hat{T}_{r\varphi r}n_{r}
+\hat{T}_{r\varphi\varphi}n_{\varphi})_{,\varphi}n_{r}n_{\varphi}
-\frac{2}{r}(\hat{T}_{r\varphi r}n_{r}+\hat{T}_{r\varphi\varphi}n_{\varphi})n_{r}^{2}\\
&+ (\hat{T}_{\varphi\varphi r}n_{r}+\hat{T}_{\varphi\varphi\varphi}n_{\varphi})_{,r}n_{r}n_{\varphi}
-\frac{1}{r}(\hat{T}_{\varphi\varphi r}n_{r}
+\hat{T}_{\varphi\varphi\varphi}n_{\varphi})_{,\varphi}n_{r}^{2}
-\frac{1}{r}(\hat{T}_{\varphi\varphi r}n_{r}
+\hat{T}_{\varphi\varphi\varphi}n_{\varphi})n_{r}n_{\varphi}\\
&+ \frac{1}{r}(\hat{T}_{rrr}n_{r}+\hat{T}_{rr\varphi}n_{\varphi})n_{r}n_{\varphi}\\
&+ (n_{r,r}n_{\varphi}^{2}+\frac{1}{r}n_{r}^{3}+\frac{1}{r}n_{r}n_{\varphi}^{2}
+\frac{1}{r}n_{r}^{2}n_{\varphi,\varphi}-\frac{1}{r}n_{r,\varphi}n_{r}n_{\varphi}
-n_{\varphi,r}n_{r}n_{\varphi})\\
&\times (\hat{T}_{r\varphi r}n_{r}^{2}+(\hat{T}_{r\varphi\varphi}
+\hat{T}_{\varphi\varphi r})n_{r}n_{\varphi}+\hat{T}_{\varphi\varphi\varphi}n_{\varphi}^{2}).\end{split}\]
Další okrajová podmínka je
(54)¶\[\begin{split}\boldsymbol{r} &= (\hat{\boldsymbol{T}}\cdot\boldsymbol{n})\cdot\boldsymbol{n}\\
&= (\hat{T}_{rrr}n_{r}^{2}+(\hat{T}_{rr\varphi}+\hat{T}_{r\varphi r})n_{r}n_{\varphi}
+\hat{T}_{r\varphi\varphi}n_{\varphi}^{2})\boldsymbol{e}_{r}\\
&+ (\hat{T}_{r\varphi r}n_{r}^{2}+(\hat{T}_{r\varphi\varphi}
+\hat{T}_{\varphi\varphi r})n_{r}n_{\varphi}
+\hat{T}_{\varphi\varphi\varphi}n_{\varphi}^{2})\boldsymbol{e}_{\varphi}.\end{split}\]
Literatura
| [1] | Mao, S. Continuum and Computational Modeling of Flexoelectricity.
Dissertation, University of Pennsylvania, 2016. |
| [2] | Mao, S., Purohit, P. K. Insights into Flexoelectric Solids from Strain-Gradient
Elasticity. Journal of Applied Mechanics, Vol. 81, 2014, pp. 081004-1-081004-10. |
| [3] | Mao, S., Purohit, P. K. Defects in flexoelectric solids. Journal of the
Mechanics and Physics of Solids, 84, 2015, pp. 95-115. |