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