Gradientní pružnost¶
Základní vztahy¶
Makroposuvy \(u_{i}(x_{i})\) jsou funkcí prostorových souřadnic \(x_{i}\). Mikroposuvy \(u_{i}^{\text{'}}(x_{i},x_{i}^{'})\) závisí na prostorových souřadnicích \(x_{i}\) a jako lineární vektorová funkce na prostorových souřadnicích \(x_{i}^{\text{'}}\), které jsou vázány na mikroobjem, tj.
(1)¶\[u_{j}^{\text{'}}=x_{k}^{\text{'}}\psi_{kj}(x_{i}).\]
Tedy, tenzor mikrodeformace \(\psi_{ij}\) závisí pouze na souřadnicích \(x_{i}\). Pro deformaci platí standartní vztah
(2)¶\[\varepsilon_{ij}=\frac{1}{2}\left(\partial_{i}u_{j}+\partial_{j}u_{i}\right),\]
pro mikrodeformaci z (1) plyne
(3)¶\[\psi_{ij}=\partial_{i}^{'}u_{j}^{'}.\]
Dál se definuje relativní deformace jako rozdíl mezi deformací a mikrodeformací
(4)¶\[\gamma_{ij}=\partial_{i}u_{j}-\psi_{ij}\]
a gradient mikrodeformace
(5)¶\[\kappa_{ijk}=\partial_{i}\psi_{jk}.\]
Potenciální energie tedy závisí na všech těchto složkách deformace
(6)¶\[W=W(\varepsilon_{ij},\gamma_{ij},\kappa_{ijk})\]
a odpovídající tenzory pro Cuachyho, relativní a dvojice napětí
(7)¶\[\begin{split}\tau_{ij}=\tau_{ji} &= \frac{\partial W}{\partial\varepsilon_{ij}}, \\
\sigma_{ij} &= \frac{\partial W}{\partial\gamma_{ij}}, \\
\mu_{ijk} &= \frac{\partial W}{\partial\kappa_{ijk}}.\end{split}\]
Takže
(8)¶\[\begin{split}\delta W &= \tau_{ij}\delta\varepsilon_{ij}+\sigma_{ij}\delta\gamma_{ij}+\mu_{ijk}\delta\kappa_{ijk} \\
&= \tau_{ij}\partial_{i}\delta u_{j}+\sigma_{ij}\left(\partial_{i}\delta u_{j}
-\delta\psi_{ij}\right)+\mu_{ijk}\partial_{i}\delta\psi_{ijk} \\
&= \partial_{i}\left[\left(\tau_{ij}+\sigma_{ij}\right)\delta u_{j}\right]
-\partial_{i}\left(\tau_{ij}+\sigma_{ij}\right)\delta u_{j}
-\sigma_{ij}\delta\psi_{ij}+\partial_{i}\left(\mu_{ijk}\delta\psi_{jk}\right)-\partial_{i}\mu_{ijk}\delta\psi_{jk}\end{split}\]
Podle Gauss-Ostrogradského se může psát
(9)¶\[\begin{split}\delta\mathscr{W} &= \int_{V}\delta W\mathrm{d}V=-\int_{V}\partial_{i}\left(\tau_{ij}+\sigma_{ij}\right)\delta u_{j}\mathrm{d}V
-\int_{V}\left(\partial_{i}\mu_{ijk}+\sigma_{jk}\right)\delta\psi_{jk}\mathrm{d}V \\
&+ \int_{S}n_{i}\left(\tau_{ij}+\sigma_{ij}\right)\delta u_{j}\mathrm{d}S
+\int_{S}n_{i}\mu_{ijk}\delta\psi_{jk}\mathrm{d}S.\end{split}\]
Práce vnějších sil se může zapsat ve tvaru
(10)¶\[\delta\mathscr{W}_{1} = \int_{V}f_{j}\delta u_{j}\mathrm{d}V+\int_{V}\Phi_{jk}\delta\psi_{jk}\mathrm{d}V
+ \int_{S}t_{j}\delta u_{j}\mathrm{d}S+\int_{S}T_{jk}\delta\psi_{jk}\mathrm{d}S,\]
kde \(f_{j}\) a \(t_{j}\) jsou objemová a plošná síla v klasickém pojetí, \(\Phi_{jk}\) a \(T_{jk}\) jsou objemové a plošné dvojice sil. Diagonální prvky \(\Phi_{jk}\) a \(T_{jk}\) jsou bez momentů, nediagonální prvky naopak s momentem. U obou \(\Phi_{jk}\) a \(T_{jk}\) první index značí orientaci ramene momentu dvojice sil a druhý index orientaci sil. U plochy s vnější normálou v kladném směru je síla na kladném konci ramene kladná (jako “kladný” se myslí směr kladné osy souřadnic rovnoběžné s ramenem momentu nebo síly). V případě normály v záporném směru jsou všechny orientace naopak. Z rovnováhy mezi potenciální eneregií a vnějším zatížením plyne
(11)¶\[\begin{split}\int_{V}\left(\partial_{i}\tau_{ij}+\partial_{i}\sigma_{ij}+f_{j}\right)\delta u_{j}\mathrm{d}V
+\int_{V}\left(\partial_{i}\mu_{ijk}+\sigma_{jk}+\Phi_{jk}\right)\delta\psi_{jk}\mathrm{d}V & \\
+\int_{S}\left[t_{j}-n_{i}\left(\tau_{ij}+\sigma_{ij}\right)\right]\delta u_{j}\mathrm{d}S
+\int_{S}\left(T_{jk}-n_{i}\mu_{ijk}\right)\delta\psi_{jk}\mathrm{d}S &=0.\end{split}\]
Odtud plyne soustava rovnic rovnováhy
(12)¶\[\begin{split}\partial_{i}\left(\tau_{ij}+\sigma_{ij}\right)+f_{j} &=0, \\
\partial_{i}\mu_{ijk}+\sigma_{jk}+\Phi_{jk} &=0\end{split}\]
a Neumannovy okrajové podmínky
(13)¶\[\begin{split}t_{j} &= n_{i}\left(\tau_{ij}+\sigma_{ij}\right), \\
T_{jk} &= n_{i}\mu_{ijk}.\end{split}\]
Z pohledu konstitutivních rovnic je potenciální energie kvadratickou formou proměnných \(\varepsilon_{ij}\), \(\gamma_{ij}\) a \(\kappa_{ijk}\),
(14)¶\[\begin{split}W &= \frac{1}{2}c_{ijkl}\varepsilon_{ij}\varepsilon_{kl}+\frac{1}{2}b_{ijkl}\gamma_{ij}\gamma_{kl}
+\frac{1}{2}a_{ijklmn}\kappa_{ijk}\kappa_{lmnk} \\
&+ d_{ijklm}\gamma_{ij}\kappa_{klm}+f_{ijklm}\kappa_{ijk}\varepsilon_{lm}+g_{ijkl}\gamma_{ij}\varepsilon_{kl}.\end{split}\]
Z toho je \(\frac{1}{2}\times42\times43=903\) nezávislých koeficientů. V případě isotropního materiálu dochází k značné redukci koeficientů,
(15)¶\[\begin{split}W &= \frac{1}{2}\lambda\varepsilon_{ii}\varepsilon_{jj}+\mu\varepsilon_{ij}\varepsilon_{ij} \\
&+ \frac{1}{2}b_{1}\gamma_{ii}\gamma_{jj}+\frac{1}{2}b_{2}\gamma_{ij}\gamma_{ij}
+\frac{1}{2}b_{3}\gamma_{ij}\gamma_{ji} \\
&+ g_{1}\gamma_{ii}\varepsilon_{jj}+g_{2}\left(\gamma_{ij}+\gamma_{ji}\right)\varepsilon_{ij} \\
&+ a_{1}\kappa_{iik}\kappa_{kjj}+a_{2}\kappa_{iik}\kappa_{jkj}+\frac{1}{2}a_{3}\kappa_{iik}\kappa_{jjk}
+\frac{1}{2}a_{4}\kappa_{ijj}\kappa_{ikk}+a_{5}\kappa_{ijj}\kappa_{kik} \\
&+ \frac{1}{2}a_{8}\kappa_{iji}\kappa_{kjk}
+\frac{1}{2}a_{10}\kappa_{ijk}\kappa_{ijk}+a_{11}\kappa_{ijk}\kappa_{jki}
+\frac{1}{2}a_{13}\kappa_{ijk}\kappa_{ikj} \\
&+ \frac{1}{2}a_{14}\kappa_{ijk}\kappa_{jik}
+\frac{1}{2}a_{15}\kappa_{ijk}\kappa_{kji}.\end{split}\]
Odtud pro jednotlivá napětí platí
(16)¶\[\begin{split}\tau_{pq} &= \lambda\delta_{pr}\varepsilon_{ii}+2\mu\varepsilon_{pq}+g_{1}\delta_{pq}\gamma_{ii}
+g_{2}\left(\gamma_{pq}+\gamma_{qp}\right), \\
\sigma_{pq} &= b_{1}\delta_{pq}\gamma_{ii}+b_{2}\gamma_{pq}+b_{3}\gamma_{qp}+g_{1}\delta_{pq}\varepsilon_{ii}
+2g_{2}\varepsilon_{pq}, \\
\mu_{pqr} &= a_{1}\left(\delta_{qr}\kappa_{iip}+\delta_{pq}\kappa_{rii}\right)
+a_{2}\left(\delta_{pr}\kappa_{iiq}+\delta_{pq}\kappa_{iri}\right)
+a_{3}\delta_{pq}\kappa_{iir}+a_{4}\delta_{qr}\kappa_{pii} \\
&+ a_{5}\left(\delta_{qr}\kappa_{ipi}+\delta_{qr}\kappa_{ipi}\right)
+a_{8}\delta_{pr}\kappa_{iqi}+a_{10}\kappa_{pqr} \\
&+ a_{11}\left(\kappa_{qrp}+\kappa_{rpq}\right)+a_{13}\kappa_{prq}
+a_{14}\kappa_{qpr}+a_{15}\kappa_{rqp}.\end{split}\]
Zjednodušené formy I a II¶
Forma I¶
Za jistých předpokladů lze brát úvahu následující
(17)¶\[\sigma_{(ij)}=0,\]
(18)¶\[b_{2}-b_{3}\rightarrow\infty,\quad\gamma_{[ij]}\rightarrow0,\]
kde indexy v kulatých resp. hranatých závorkách symbolizují symterické resp. antisymetrické části tenzorů. Takže
(19)¶\[\begin{split}\tau_{pq} &= \lambda\delta_{pr}\varepsilon_{ii}+2\mu\varepsilon_{pq}+g_{1}\delta_{pq}\gamma_{ii}+2g_{2}\gamma_{(pq)}, \\\end{split}\]
(20)¶\[\begin{split}\sigma_{(pq)} &= \frac{1}{2}\left(\sigma_{pq}+\sigma_{qp}\right) \\
&= \frac{1}{2}\left(g_{1}\delta_{pq}\varepsilon_{ii}+2g_{2}\varepsilon_{pq}
+b_{1}\delta_{pq}\gamma_{ii}+b_{2}\gamma_{pq}+b_{3}\gamma_{qp}\right. \\
&\left. +g_{1}\delta_{qp}\varepsilon_{ii}+2g_{2}\varepsilon_{qp}+b_{1}\delta_{qp}\gamma_{ii}
+b_{2}\gamma_{qp}+b_{3}\gamma_{pq}\right) \\
&= g_{1}\delta_{pq}\varepsilon_{ii}+2g_{2}\varepsilon_{pq}+b_{1}\delta_{pq}\gamma_{ii}
+\left(b_{2}+b_{3}\right)\gamma_{(pq)},\end{split}\]
(21)¶\[\begin{split}\sigma_{[pq]} &= \frac{1}{2}\left(\sigma_{pq}-\sigma_{qp}\right) \\
&= \frac{1}{2}\left(g_{1}\delta_{pq}\varepsilon_{ii}+2g_{2}\varepsilon_{pq}
+b_{1}\delta_{pq}\gamma_{ii}+b_{2}\gamma_{pq}+b_{3}\gamma_{qp}\right. \\
&\left. -g_{1}\delta_{qp}\varepsilon_{ii}-2g_{2}\varepsilon_{qp}-b_{1}\delta_{qp}\gamma_{ii}
-b_{2}\gamma_{qp}-b_{3}\gamma_{pq}\right) \\
&= \left(b_{2}-b_{3}\right)\gamma_{[pq]}.\end{split}\]
Podle (18) a (21) je \(\sigma_{[pq]}\) neurčitý výraz a z podmínky (17) a rovnic (20) pro \(p=q=1,2\) a \(3\) plyne
(22)¶\[\begin{split}0 &= g_{1}\varepsilon_{ii}+2g_{2}\varepsilon_{11}+b_{1}\gamma_{ii}
+\left(b_{2}+b_{3}\right)\frac{1}{2}\left(\gamma_{11}+\gamma_{11}\right), \\
0 &= g_{1}\varepsilon_{ii}+2g_{2}\varepsilon_{22}+b_{1}\gamma_{ii}
+\left(b_{2}+b_{3}\right)\frac{1}{2}\left(\gamma_{22}+\gamma_{22}\right), \\
0 &= g_{1}\varepsilon_{ii}+2g_{2}\varepsilon_{33}+b_{1}\gamma_{ii}
+\left(b_{2}+b_{3}\right)\frac{1}{2}\left(\gamma_{33}+\gamma_{33}\right).\end{split}\]
Sečtením rovnic se dostane
(23)¶\[0=3g_{1}\varepsilon_{ii}+2g_{2}\varepsilon_{ii}+3b_{1}\gamma_{ii}+\left(b_{2}+b_{3}\right)\gamma_{ii},\]
odkud plyne
(24)¶\[\gamma_{ii}=\frac{3g_{1}+2g_{2}}{3b_{1}+b_{2}+b_{3}}\varepsilon_{ii}.\]
Dosazením tohoto výrazu zpět do (20) se dostane
(25)¶\[\gamma_{(pq)}=-\alpha\delta_{pq}\varepsilon_{ii}+\left(1-\beta\right)\varepsilon_{pq},\]
kde
(26)¶\[\alpha=\frac{1}{b_{2}+b_{3}}\left[g_{1}-\frac{b_{1}\left(3g_{1}+2g_{2}\right)}{3b_{1}+b_{2}+b_{3}}\right]
,\quad\beta=1+\frac{2g_{2}}{b_{2}+b_{3}}.\]
Protože
(27)¶\[\gamma_{pq}=\partial_{p}u_{q}-\psi_{pq}\]
platí pro symetrickou a antisymetrickou část \(\gamma_{pq}\) následující
(28)¶\[\gamma_{(pq)}=\varepsilon_{pq}-\psi_{(pq)}\quad\mathrm{a\quad}\gamma_{[pq]}=\omega_{pq}-\psi_{[pq]}.\]
Podle druhé limity v (18) a (25) platí
(29)¶\[\psi_{[pq]}=\omega_{pq}\]
a
(30)¶\[\psi_{(pq)}=\alpha\delta_{pq}\varepsilon_{ii}+\beta\varepsilon_{pq}.\]
Z definice gradientu mikrodeformace
(31)¶\[\kappa_{ijk}=\partial_{i}\psi_{jk}=\partial_{i}\psi_{(jk)}+\partial_{i}\psi_{[jk]}\]
plyne
(32)¶\[\begin{split}\kappa_{ijk}\rightarrow & \alpha\delta_{jk}\partial_{i}\varepsilon_{ll}+\beta\partial_{i}\varepsilon_{jk}
+\partial_{i}\omega_{jk} \\
&= \alpha\delta_{jk}\partial_{i}\partial_{l}u_{l}
+\beta\frac{1}{2}\partial_{i}\left(\partial_{j}u_{k}+\partial_{k}u_{j}\right)
+\frac{1}{2}\partial_{i}\left(\partial_{j}u_{k}-\partial_{k}u_{j}\right),\end{split}\]
nebo-li
(33)¶\[\kappa_{ijk}\rightarrow\alpha\delta_{jk}\tilde{\kappa}_{ill}\delta_{jk}+\frac{1}{2}\left(1+\beta\right)\tilde{\kappa}_{ijk}
+\frac{1}{2}\left(1-\beta\right)\tilde{\kappa}_{ikj},\]
kde
(34)¶\[\tilde{\kappa}_{ijk}\equiv\partial_{i}\partial_{j}u_{k}=\tilde{\kappa}_{jik}.\]
Dosazením vztahů pro \(\kappa_{ijk}\), \(\gamma_{ij}\), (17) a (18) do (15) se dostane
(35)¶\[\begin{split}W\rightarrow\tilde{W} &= \frac{1}{2}\tilde{\lambda}\varepsilon_{ii}\varepsilon_{jj}
+\tilde{\mu}\varepsilon_{ij}\varepsilon_{ij}+\tilde{a}_{1}\tilde{\kappa}_{iik}\tilde{\kappa}_{kjj}
+\tilde{a}_{2}\tilde{\kappa}_{ijj}\tilde{\kappa}_{ikk}+ \\
&+ \tilde{a}_{3}\tilde{\kappa}_{iik}\tilde{\kappa}_{jjk}
+\tilde{a}_{4}\tilde{\kappa}_{ijk}\tilde{\kappa}_{ijk}
+\tilde{a}_{5}\tilde{\kappa}_{ijk}\tilde{\kappa}_{kji},\end{split}\]
kde
(36)¶\[\begin{split}\tilde{\lambda}+2\tilde{\mu} &= \lambda+2\mu-\frac{8g_{2}^{2}}{3\left(b_{2}+b_{3}\right)}
-\frac{\left(3g_{1}+2g_{2}\right)^{2}}{2\left(3b_{1}+b_{2}+b_{3}\right)}, \\
\tilde{\mu} &= \mu-\frac{2g_{2}^{2}}{b_{2}+b_{3}}.\end{split}\]
Ostatní členy jsou děsná prasečina, viz. [1]. Můžou se tedy nadefinovat nová napětí
(37)¶\[\tilde{\tau}_{ij}=\frac{\partial\tilde{W}}{\partial\varepsilon_{ij}}=\tilde{\tau}_{ji},\]
(38)¶\[\tilde{\mu}_{ijk}=\frac{\partial\tilde{W}}{\partial\tilde{\kappa}_{ijk}}=\tilde{\mu}_{jik}.\]
V dalším bude třeba zavést Levi-Civitův symbol
(39)¶\[\begin{split}\varepsilon_{ijk}=
\begin{cases}
1 & \mathrm{pro}\,(i,j,k)=(1,2,3),(2,3,1)\,\mathrm{a\,}(3,1,2),\\
-1 & \mathrm{pro}\,(i,j,l)=(3,2,1),(1,3,2)\,\mathrm{a\,}(2,1,3),\\
0 & \mathrm{jinak.}
\end{cases}\end{split}\]
Důležitý je vztah
(40)¶\[\varepsilon_{ijk}\varepsilon_{imn}=\delta_{jm}\delta_{kn}-\delta_{jn}\delta_{km}\]
nebo po sudém přehození indexů
(41)¶\[\varepsilon_{jki}\varepsilon_{imn}=\delta_{jm}\delta_{kn}-\delta_{jn}\delta_{km}.\]
V dalším je totiž důležitý operátor (derivace ve směru tečny ke křivce s normálou \(n_{j}\) - povrchový gradient)
(42)¶\[D_{j}\equiv\left(\delta_{jl}-n_{j}n_{l}\right)\partial_{l}\]
a skalární vztah
(43)¶\[D_{j}v_{j}=\left(\delta_{jl}-n_{j}n_{l}\right)\partial_{l}v_{j}=\partial_{j}v_{j}-n_{j}n_{l}\partial_{l}v_{j},\]
kde \(v_{j}\) je vektor kolineární s normálou \(n_{j}\), tj. platí
(44)¶\[v_{j}=n_{k}v_{k}n_{j}.\]
Výraz \(n_{k}v_{k}\) je tedy vlastně velikost vektoru \(v_{j}\). Nejdříve první výraz na pravé straně roznásobíme \(n_{l}n_{l}\), což je jednička (velikost normály)
(45)¶\[D_{j}v_{j}=n_{l}n_{l}\partial_{j}v_{j}-n_{j}n_{l}\partial_{l}v_{j}\]
a pak od pravé strany odečteme výraz \(n_{l}v_{j}\partial_{j}n_{l}\), který i následně přičteme
(46)¶\[D_{j}v_{j}=n_{l}n_{l}\partial_{j}v_{j}-n_{j}n_{l}\partial_{l}v_{j}-n_{l}v_{j}\partial_{j}n_{l}+n_{l}v_{j}\partial_{j}n_{l}.\]
Do třetího výrazu na pravé straně se místo \(v_{j}\) dosadí (44) a tím se dostane
(47)¶\[D_{j}v_{j}=n_{l}n_{l}\partial_{j}v_{j}-n_{j}n_{l}\partial_{l}v_{j}-n_{k}v_{k}n_{l}n_{j}\partial_{j}n_{l}
+n_{l}v_{j}\partial_{j}n_{l}.\]
Pokračujeme v masáži. Předchozí výraz přeuspořádáme
(48)¶\[D_{j}v_{j}=\left(n_{l}v_{j}\partial_{j}n_{l}+n_{l}n_{l}\partial_{j}v_{j}-n_{j}n_{l}\partial_{l}v_{j}\right)
-n_{k}v_{k}n_{l}n_{j}\partial_{j}n_{l}.\]
Poslední přičtení a odečtení, tentokrát výrazu \(n_{j}v_{j}\partial_{l}n_{l}\), tedy
(49)¶\[D_{j}v_{j}=\left(n_{l}v_{j}\partial_{j}n_{l}+n_{l}n_{l}\partial_{j}v_{j}
-n_{j}v_{j}\partial_{l}n_{l}-n_{j}n_{l}\partial_{l}v_{j}\right)
+\left(n_{j}v_{j}\partial_{l}n_{l}-n_{k}v_{k}n_{l}n_{j}\partial_{j}n_{l}\right).\]
Závorky lze přepsat následovně
(50)¶\[D_{j}v_{j}=\left[n_{l}\partial_{j}\left(n_{l}v_{j}\right)-n_{j}\partial_{l}\left(n_{l}v_{j}\right)\right]
+n_{j}v_{j}\left(\partial_{l}n_{l}-n_{l}n_{k}\partial_{k}n_{l}\right).\]
Může se využít Kroneckerovo \(\delta_{ij}\),
(51)¶\[D_{j}v_{j}=\left[n_{l}\delta_{pj}\partial_{p}\left(n_{l}v_{j}\right)
-n_{j}\delta_{pl}\partial_{p}\left(n_{l}v_{j}\right)\right]
+n_{j}v_{j}\left(\delta_{lk}\partial_{k}n_{l}-n_{l}n_{k}\partial_{k}n_{l}\right),\]
které umožní vytknout výrazy s parciálními derivacemi,
(52)¶\[D_{j}v_{j}=\left[n_{q}\delta_{ql}\delta_{pj}-n_{q}\delta_{qj}\delta_{pl}\right]
\partial_{p}\left(n_{l}v_{j}\right)+n_{j}v_{j}\left(\delta_{lk}-n_{l}n_{k}\right)\partial_{k}n_{l}.\]
Dalším použitím \(\delta_{ij}\) lze z první závorky vytknout normálu
(53)¶\[D_{j}v_{j}=n_{q}\left[\delta_{ql}\delta_{pj}-\delta_{qj}\delta_{pl}\right]
\partial_{p}\left(n_{l}v_{j}\right)+n_{j}v_{j}\left(\delta_{lk}-n_{l}n_{k}\right)\partial_{k}n_{l}.\]
Podle (41) a (43) se pak dostane
(54)¶\[D_{j}v_{j}=n_{q}\varepsilon_{qpm}\varepsilon_{mlj}\partial_{p}\left(n_{l}v_{j}\right)+n_{j}v_{j}\left(D_{l}n_{l}\right).\]
Tuhle šílenost odvodil [2] a později vzorec (54) využijeme. Na základě definice napětí v (37), (38) a potenciálu (35) platí
(55)¶\[\begin{split}\tilde{\tau}_{pq} &= \tilde{\lambda}\delta_{pq}\varepsilon_{ii}+2\tilde{\mu}\varepsilon_{pq}, \\
\tilde{\mu}_{pqr} &= \frac{1}{2}\tilde{a}_{1}\left(\tilde{\kappa}_{iip}\delta_{qr}
+2\tilde{\kappa}_{rii}\delta_{pq}+\tilde{\kappa}_{iiq}\delta_{pr}\right)
+\tilde{a}_{2}\left(\tilde{\kappa}_{pii}\delta_{qr}+\tilde{\kappa}_{qii}\delta_{pr}\right) \\
&= +2\tilde{a}_{3}\tilde{\kappa}_{iir}\delta_{pq}+2\tilde{a}_{4}\tilde{\kappa}_{pqr}
+\tilde{a}_{5}\left(\tilde{\kappa}_{rqp}+\tilde{\kappa}_{rpq}\right).\end{split}\]
Variace hustoty potenciální energie je
(56)¶\[\begin{split}\delta\tilde{W} &= \tilde{\tau}_{ij}\delta\varepsilon_{ij}+\tilde{\mu}_{ijk}\delta\tilde{\kappa}_{ijk} \\
&= \tilde{\tau}_{ij}\partial_{i}\delta u_{j}+\tilde{\mu}_{ijk}\partial_{i}\partial_{j}\delta u{}_{k} \\
&= \partial_{j}\left[\left(\tilde{\tau}_{jk}-\partial_{i}\tilde{\mu}_{ijk}\right)\delta u_{k}\right]
-\partial_{j}\left(\tilde{\tau}_{jk}-\partial_{i}\tilde{\mu_{ijk}}\right)\delta u_{k}
+\partial_{i}\left(\tilde{\mu}_{ijk}\partial_{j}\delta u_{k}\right).\end{split}\]
Takže
(57)¶\[\int_{V}\delta\tilde{W}\mathrm{d}V=\int_{S}n_{j}\left(\tilde{\tau}_{jk}-\partial_{i}\tilde{\mu}_{ijk}\right)
\delta u_{k}\mathrm{d}S-\int_{V}\partial_{j}
\left(\tilde{\tau}_{jk}-\partial_{i}\tilde{\mu}_{ijk}\right)
\delta u_{k}\mathrm{d}V+\int_{S}n_{i}\tilde{\mu}_{ijk}\partial_{j}\delta u_{k}\mathrm{d}S.\]
Nyní je třeba se zbavit derivace \(\partial_{j}\) ve výrazu \(\partial_{j}\delta u_{k}\) třetího integrálu, protože je tato derivace závislá na variaci \(\delta u_{k}\). Nezávislá je pouze její normálová složka \(n_{i}\partial_{i}\delta u_{k}\) (povrchová síla - Neumennova okrajová podmínka). Na to odstranění se musí jít oklikou, takže nejdříve se třetí integrand přepíše do složitějšího tvaru
(58)¶\[n_{i}\tilde{\mu}_{ijk}\partial_{j}\delta u_{k}=n_{i}\tilde{\mu}_{ijk}D_{j}\delta u_{k}+n_{i}\tilde{\mu}_{ijk}n_{j}D\delta u_{k},\]
kde \(D_{j}\) je definováno výše vztahem (42) a
(59)¶\[D\equiv n_{l}\partial_{l}.\]
První výraz na levé straně (58) se může dále rozepsat
(60)¶\[n_{i}\tilde{\mu}_{ijk}D_{j}\delta u_{k}=D_{j}\left(n_{i}\tilde{\mu}_{ijk}\delta u_{k}\right)
-n_{i}D_{j}\tilde{\mu}_{ijk}\delta u_{k}
-\left(D_{j}n_{i}\right)\tilde{\mu}_{ijk}\delta u_{k}.\]
Poslední dva výrazy v (60) již neobsahují derivaci \(\delta u_{k}\). První výraz se může rozepsat podle (54), kde
(61)¶\[v_{j}=n_{i}\tilde{\mu}_{ijk}\delta u_{k},\]
(62)¶\[D_{j}\left(n_{i}\tilde{\mu}_{ijk}\delta u_{k}\right)=\left(D_{l}n_{l}\right)n_{j}n_{i}\tilde{\mu}_{ijk}\delta u_{k}
+n_{q}\varepsilon_{qpm}\partial_{p}
\left(\varepsilon_{mlj}n_{l}n_{i}\tilde{\mu}_{ijk}\delta u_{k}\right).\]
Podle Stokesovy věty je integrál z druhého výrazu na pravé straně (62) přes hladkou plochu nulový. Jestliže však tuto plochu rozdělíme na dvě, \(S_{1}\) a \(S_{2}\), křivkou \(C\), Stokes nám dá výraz
(63)¶\[\int_{S}n_{q}\varepsilon_{qpm}\partial_{p}\left(\varepsilon_{mlj}n_{l}n_{i}\tilde{\mu}_{ijk}\delta u_{k}\right)\mathrm{d}S
=\oint_{C}\big[\!\big[ s_{m}\varepsilon_{mlj}n_{l}n_{i}\tilde{\mu}_{ijk}\big]\!\big] \delta u_{k}\mathrm{d}s
=\oint_{C}\big[\!\big[ n_{i}m_{j}\tilde{\mu}_{ijk}\big]\!\big] \delta u_{k}\mathrm{d}s,\]
kde \(m_{j}=\varepsilon_{mlj}s_{m}n_{l}\) a \(s_{m}\) jsou složky jednotkového tečného vektoru a binormálového vektoru ke křivce \(C\). Dvojité hranaté závorky \([\![.]\!]\) značí, že hodnota uvitř je rozdílem mezi hodnotami na ploše \(S_{1}\) a \(S_{2}\). Takže suma sumárum, z výsledků (58), (60), (62) a (63) se pro \(\delta\tilde{W}\) dostane
(64)¶\[\begin{split}\int_{V}\delta\tilde{W}\mathrm{d}V &= -\int_{V}\partial_{j}\left(\tilde{\tau}_{jk}-\partial_{i}\tilde{\mu}_{ijk}\right)
\delta u_{k}\mathrm{d}V \\
&+ \int_{S}\left[n_{j}\tilde{\tau}_{jk}-n_{j}\partial_{i}\tilde{\mu}_{ijk}
+\left(D_{l}n_{l}\right)n_{j}n_{i}\tilde{\mu}_{ijk}-n_{i}D_{j}\tilde{\mu}_{ijk}
-\left(D_{j}n_{i}\right)\tilde{\mu}_{ijk}\right]\delta u_{k}\mathrm{d}S \\
&+ \int_{S}n_{i}n_{j}\tilde{\mu}_{ijk}D\delta u_{k}\mathrm{d}S
+\oint_{C}\big[\!\big[n_{i}m_{j}\tilde{\mu}_{ijk}\big]\!\big]\delta u_{k}\mathrm{d}s.\end{split}\]
Pro \(n_{j}\partial_{i}\tilde{\mu}_{ijk}\) platí
(65)¶\[n_{j}\partial_{i}\tilde{\mu}_{ijk}=n_{j}\left(\delta_{il}-n_{i}n_{l}\right)\partial_{l}\tilde{\mu}_{ijk}
+n_{j}n_{i}n_{l}\partial_{l}\tilde{\mu}_{ijk}=n_{j}D_{i}\tilde{\mu}_{ijk}+n_{j}n_{i}D\tilde{\mu}_{ijk}.\]
Pak
(66)¶\[\begin{split}\int_{V}\delta\tilde{W}\mathrm{d}V &= -\int_{V}\partial_{j}\left(\tilde{\tau}_{jk}-\partial_{i}\tilde{\mu}_{ijk}\right)
\delta u_{k}\mathrm{d}V \\
&+ \int_{S}\left[n_{j}\tilde{\tau}_{jk}-n_{j}n_{i}D\tilde{\mu}_{ijk}
-2n_{j}D_{i}\tilde{\mu}_{ijk}
+\left(n_{i}n_{j}D_{l}n_{l}-D_{j}n_{i}\right)\tilde{\mu}_{ijk}\right]
\delta u_{k}\mathrm{d}S \\
&+ \int_{S}n_{i}n_{j}\tilde{\mu}_{ijk}D\delta u_{k}\mathrm{d}S
+\oint_{C}\big[\!\big[n_{i}m_{j}\tilde{\mu}_{ijk}\big]\!\big]\delta u_{k}\mathrm{d}s.\end{split}\]
Na základě tvaru variace potenciální energie vyjádříme variaci práce vnějších sil
(67)¶\[\delta\mathscr{W}_{1}=\int_{V}F_{k}\delta u_{k}\mathrm{d}V+\int_{S}\tilde{P}_{k}\delta u_{k}\mathrm{d}S
+\int_{S}\tilde{R}_{k}D\delta u_{k}\mathrm{d}S+\oint_{C}\tilde{E}_{k}\delta u_{k}\mathrm{d}s.\]
Z rovnosti
(68)¶\[\delta\mathscr{W}=\int_{V}\delta\tilde{W}\mathrm{d}V=\delta\mathscr{W}_{1}\]
plynou rovnice rovnováhy a okrajové podmínky
(69)¶\[\begin{split}\partial_{j}\left(\tilde{\tau}_{jk}-\partial_{i}\tilde{\mu}_{ijk}\right)+F_{k} &= 0, \\
n_{j}\tilde{\tau}_{jk}-n_{j}n_{i}D\tilde{\mu}_{ijk}-2n_{j}D_{i}\tilde{\mu}_{ijk}
+\left(n_{i}n_{j}D_{l}n_{l}-D_{j}n_{i}\right)\tilde{\mu}_{ijk} &= \tilde{P}_{k}, \\
n_{i}n_{j}\tilde{\mu}_{ijk} &= \tilde{R}_{k}, \\
\big[\!\big[ n_{i}m_{j}\tilde{\mu}_{ijk}\big]\!\big] &= \tilde{E}_{k}.\end{split}\]
Forma II¶
Derivace \(\partial_{i}\partial_{j}u_{k}\) může být vyjádřena jako tenzor
(70)¶\[\hat{\kappa}_{ijk}\equiv\partial_{i}\varepsilon_{jk}=
\frac{1}{2}\left(\partial_{i}\partial_{j}u_{k}+\partial_{i}\partial_{k}u_{j}\right)=\hat{\kappa}_{ijk}.\]
Hustotu potenciální energie lze vyjádřit pomocí hustoty potenciální energie popsané v části Forma I, ale nahrazením \(\tilde{\kappa}_{ijk}\) výrazem
(71)¶\[\tilde{\kappa}_{ijk}=\hat{\kappa}_{ijk}+\hat{\kappa}_{jki}-\hat{\kappa}_{kij}.\]
Tím se dostane
(72)¶\[W\rightarrow\hat{W}=\frac{1}{2}\tilde{\lambda}\varepsilon_{ii}\varepsilon_{jj}+\tilde{\mu}\varepsilon_{ij}\varepsilon_{ij}
+\hat{a}_{1}\hat{\kappa}{}_{iik}\hat{\kappa}_{kjj}+\hat{a}_{2}\hat{\kappa}_{ijj}\hat{\kappa}_{ikk}
+\hat{a}_{3}\hat{\kappa}_{iik}\hat{\kappa}_{jjk}+\hat{a}_{4}\hat{\kappa}_{ijk}\hat{\kappa}_{ijk}
+\hat{a}_{5}\hat{\kappa}_{ijk}\hat{\kappa}_{kji},\]
kde
(73)¶\[\begin{split}\hat{a}_{1} &= 2\tilde{a}_{1}-4\tilde{a}_{3}, \\
\hat{a}_{2} &= -\tilde{a}_{1}+\tilde{a}_{2}+\tilde{a}_{3}, \\
\hat{a}_{3} &= 4\tilde{a}_{3}, \\
\hat{a}_{4} &= 3\tilde{a}_{4}-\tilde{a}_{5}, \\
\hat{a}_{5} &= -2\tilde{a}_{4}+2\tilde{a}_{5}.\end{split}\]
Definice nových napětí
(74)¶\[\hat{\tau}_{ij}=\frac{\partial\hat{W}}{\partial\varepsilon_{ij}}=\hat{\tau}_{ji},\]
(75)¶\[\hat{\mu}_{ijk}=\frac{\partial\hat{W}}{\partial\hat{\kappa}_{ijk}}=\hat{\mu}_{ikj}\]
a
(76)¶\[\begin{split}\hat{\tau}_{pq} &= \tilde{\lambda}\delta_{pq}\varepsilon_{ii}+2\tilde{\mu}\varepsilon_{pq}, \\
\hat{\mu}_{pqr} &= \frac{1}{2}\hat{a}_{1}\left(\delta_{pq}\hat{\kappa}_{rii}+2\delta_{qr}\hat{\kappa}_{iip}
+\delta_{rp}\hat{\kappa}_{qii}\right)+2\hat{a}_{2}\delta_{qr}\hat{\kappa}_{pii} \\
&+ \hat{a}_{3}\left(\delta_{pq}\hat{\kappa}_{iir}+\delta_{pr}\hat{\kappa}_{iiq}\right)
+2\hat{a}_{4}\hat{\kappa}_{pqr}+\hat{a}_{5}\left(\hat{\kappa}_{rpq}+\hat{\kappa}_{qrp}\right).\end{split}\]
Variace hustoty potenciální energie
(77)¶\[\begin{split}\delta\hat{W} &= \hat{\tau}_{ij}\delta\varepsilon_{ij}+\hat{\mu}_{ijk}\delta\hat{\kappa}_{ijk}
=\hat{\tau}_{ij}\partial_{i}\delta u_{j}+\hat{\mu}_{ijk}\partial_{i}\partial_{j}\delta u_{k} \\
&= \partial_{j}\left[\left(\hat{\tau}_{ij}-\partial_{i}\hat{\mu}_{ijk}\right)\delta u_{k}\right]
-\partial_{j}\left(\hat{\tau}_{jk}-\partial_{i}\hat{\mu}_{ijk}\right)\delta u_{k}
+\partial_{i}\left(\hat{\mu}_{ijk}\partial_{j}\delta u_{k}\right).\end{split}\]
Variace potenciálu je uplně stejná jako v případě formy I, viz část Forma I, takže se stejným postupem dostanou stejné rovnice rovnováhy a okrajové podmínky kromě výrazu \(2n_{j}D_{i}\), který je nahrazen výrazem \(n_{j}D_{i}+n_{i}D_{j}\),
(78)¶\[\partial_{j}\left(\hat{\tau}_{jk}-\partial_{i}\hat{\mu}_{ijk}\right)+F_{k}=0,\]
(79)¶\[n_{j}\hat{\tau}_{jk}-n_{j}n_{i}D\hat{\mu}_{ijk}-\left(n_{j}D_{i}\hat{\mu}_{ijk}+n_{i}D_{j}\hat{\mu}_{ijk}\right)
+\left(n_{i}n_{j}D_{l}n_{l}-D_{j}n_{i}\right)\hat{\mu}_{ijk}=\hat{P}_{k},\]
(80)¶\[n_{i}n_{j}\hat{\mu}_{ijk}=\hat{R}_{k},\]
(81)¶\[\big[\!\big[ n_{i}m_{j}\hat{\mu}_{ijk}\big]\!\big] =\hat{E}_{k}.\]
Gradientní pružnost¶
Gradientní elasticita vychází z Mindlinovy formy II (detaily někdy jindy, viz také Forma II), a spočívá ve zjednodušení tvaru hustoty potenciální energie,
(82)¶\[W=\frac{1}{2}\lambda\varepsilon_{pp}\varepsilon_{qq}+\mu\varepsilon_{ij}\varepsilon_{ij}
+c\frac{1}{2}\lambda\left(\partial_{r}\varepsilon_{pp}\right)\left(\partial_{r}\varepsilon_{qq}\right)
+c\mu\left(\partial_{r}\varepsilon_{pq}\right)\left(\partial_{r}\varepsilon_{pq}\right),\]
kde \(c\) gradientní koeficient s rozměry \([d\acute{e}lka]^{2}\). Podle (74) a (75) platí
(83)¶\[\tau_{pq}=\frac{\partial W}{\partial\varepsilon_{pq}}=\lambda\delta_{pq}\varepsilon_{ii}+2\mu\varepsilon_{pq},\]
(84)¶\[m_{rpq}=\frac{\partial W}{\partial\left(\partial_{r}\varepsilon_{pq}\right)}
=c\partial_{r}\left(\lambda\delta_{pq}\varepsilon_{ii}+2\mu\varepsilon_{pq}\right)=c\partial_{r}\tau_{pq}\]
a při zanedbání objemových sil \(F_{k}\) a podle (65) a (78)–(81)
(85)¶\[\partial_{p}\left(\tau_{pq}-\partial_{r}m_{rpq}\right)=0,\]
(86)¶\[n_{p}\left(\tau_{pq}-\partial_{r}m_{rpq}\right)
-D_{p}\left(n_{r}m_{rpq}\right)+\left(D_{j}n_{j}\right)n_{r}n_{p}m_{rpq}=P_{q},\]
(87)¶\[n_{r}n_{p}m_{rpq}=R_{q}.\]
Rovinná gradientní pružnost¶
Hookeův zákon pro izotropní materiál v rovinných kartézských souřadnicích má podle (83) a (84) tvar,
(88)¶\[\tau_{xx}=\left(\lambda+2\mu\right)\varepsilon_{xx}+\lambda\varepsilon_{yy},\]
(89)¶\[\tau_{xy}=2\mu\varepsilon_{xy},\]
(90)¶\[\tau_{yy}=\left(\lambda+2\mu\right)\varepsilon_{yy}+\lambda\varepsilon_{xx},\]
(91)¶\[\begin{split}m_{xxx} &= c\partial_{x}\left[\left(\lambda+2\mu\right)\varepsilon_{xx}+\lambda\varepsilon_{yy}\right], \\
m_{xxy} &= 2c\mu\partial_{x}\varepsilon_{xy}, \\
m_{xyy} &= c\partial_{x}\left[\left(\lambda+2\mu\right)\varepsilon_{yy}+\lambda\varepsilon_{xx}\right], \\
m_{yxx} &= c\partial_{y}\left[\left(\lambda+2\mu\right)\varepsilon_{xx}+\lambda\varepsilon_{yy}\right], \\
m_{yxy} &= 2c\mu\partial_{y}\varepsilon_{xy}, \\
m_{yyy} &= c\partial_{y}\left[\left(\lambda+2\mu\right)\varepsilon_{yy}+\lambda\varepsilon_{xx}\right]. \\\end{split}\]
To stejné platí v polárních souřadnicích. Ukázka odvození vztahů pro napětí \(\tau_{ij}\) je v části Vektorová analýza. Takže platí
(92)¶\[\begin{split}\tau_{rr} &= \left(\lambda+2\mu\right)\varepsilon_{rr}+\lambda\varepsilon_{\varphi\varphi}, \\
\tau_{r\varphi} &= 2\mu\varepsilon_{r\varphi}, \\
\tau_{\varphi\varphi} &= \left(\lambda+2\mu\right)\varepsilon_{\varphi\varphi}+\lambda\varepsilon_{rr}, \\\end{split}\]
Pro momenty \(m_{ijk}\) jako derivaci \(\partial_{i}\tau_{jk}\) je jednodušší vyjít z triády
(93)¶\[\nabla\boldsymbol{\tau}=\left(\boldsymbol{e}_{r}\partial_{r}+\boldsymbol{e}_{\varphi}r^{-1}\partial_{\varphi}\right)
\otimes\left[\tau_{rr}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}
+\tau_{r\varphi}\left(\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}
+\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\right)
+\tau_{\varphi\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\right].\]
Teda pomocí vztahu (46) v části Vektorová analýza, kde se přehodí první a třetí index, se dotane
(94)¶\[\begin{split}\boldsymbol{m}=\nabla\boldsymbol{\tau}
&= \partial_{r}\tau_{rr}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}
+\partial_{r}\tau_{r\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}
+\partial_{r}\tau_{\varphi r}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}
+\partial_{r}\tau_{\varphi\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi} \\
&+ \frac{1}{r}\left(\partial_{\varphi}\tau_{rr}-\tau_{r\varphi}-\tau_{\varphi r}\right)
\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}
+\frac{1}{r}\left(\tau_{rr}-\tau_{\varphi\varphi}+\partial_{\varphi}\tau_{\varphi r}\right)
\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r} \\
&+ \frac{1}{r}\left(\tau_{rr}-\tau_{\varphi\varphi}+\partial_{\varphi}\tau_{r\varphi}\right)
\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}
+\frac{1}{r}\left(\tau_{r\varphi}+\tau_{\varphi r}+\partial_{\varphi}\tau_{\varphi\varphi}\right)
\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi},\end{split}\]
takže
(95)¶\[m_{rrr}=c\partial_{r}\tau_{rr},\quad m_{rr\varphi}=c\partial_{r}\tau_{r\varphi},
\quad m_{r\varphi\varphi}=c\partial_{r}\tau_{\varphi\varphi},\]
(96)¶\[m_{\varphi rr}=c\frac{1}{r}\left(\partial_{\varphi}\tau_{rr}-2\tau_{r\varphi}\right),
\quad m_{\varphi\varphi r}=c\frac{1}{r}\left(\partial_{\varphi}\tau_{\varphi r}+\tau_{rr}-\tau_{\varphi\varphi}\right),
\quad m_{\varphi\varphi\varphi}=c\frac{1}{r}\left(\partial_{\varphi}\tau_{\varphi\varphi}+2\tau_{r\varphi}\right).\]
Deformace se může zapsat pomocí operátoru nabla následovně
(97)¶\[\varepsilon_{ij}=\frac{1}{2}\left(\boldsymbol{u}\nabla+\nabla\boldsymbol{u}\right).\]
Takže v kartézském souřadnicovém systému, kde \(\nabla=\boldsymbol{e}_{x}\partial_{x}+\boldsymbol{e}_{y}\partial_{y}\), platí
(98)¶\[\varepsilon_{xx}=\partial_{x}u_{x},
\quad\varepsilon_{xy}=\frac{1}{2}\left(\partial_{y}u_{x}+\partial_{x}u_{y}\right),
\quad\varepsilon_{yy}=\partial_{x}u_{y}.\]
Derivace deformace má v kartézském souřadnicovém systému jednoduchý tvar
(99)¶\[\partial_{x}\varepsilon_{xx}=\partial_{xx}u_{x},
\ \partial_{x}\varepsilon_{xy}=\frac{1}{2}\left(\partial_{xy}u_{x}+\partial_{xx}u_{y}\right),
\ \partial_{x}\varepsilon_{yy}=\partial_{xy}u_{y},\]
(100)¶\[\partial_{y}\varepsilon_{xx}=\partial_{yx}u_{x},
\ \partial_{y}\varepsilon_{xy}=\frac{1}{2}\left(\partial_{yy}u_{x}+\partial_{yx}u_{y}\right),
\ \partial_{y}\varepsilon_{yy}=\partial_{yy}u_{y}\]
a Hookeův zákon má tvar
(101)¶\[\begin{split}\tau_{xx} &= \left(\lambda+2\mu\right)\partial_{x}u_{x}+\lambda\partial_{y}u_{y}, \\
\tau_{xy} &= \mu\left(\partial_{y}u_{x}+\partial_{x}u_{y}\right), \\
\tau_{yy} &= \left(\lambda+2\mu\right)\partial_{y}u_{y}+\lambda\partial_{x}u_{x},\end{split}\]
(102)¶\[\begin{split}m_{xxx} &= c\partial_{x}\left[\left(\lambda+2\mu\right)\partial_{x}u_{x}+\lambda\partial_{y}u_{y}\right], \\
m_{xxy} &= c\mu\partial_{x}\left(\partial_{y}u_{x}+\partial_{x}u_{y}\right), \\
m_{xyy} &= c\partial_{x}\left[\left(\lambda+2\mu\right)\partial_{y}u_{y}+\lambda\partial_{x}u_{x}\right], \\
m_{yxx} &= c\partial_{y}\left[\left(\lambda+2\mu\right)\partial_{x}u_{x}+\lambda\partial_{y}u_{y}\right], \\
m_{yxy} &= c\mu\partial_{y}\left(\partial_{y}u_{x}+\partial_{x}u_{y}\right), \\
m_{yyy} &= c\partial_{y}\left[\left(\lambda+2\mu\right)\partial_{y}u_{y}+\lambda\partial_{x}u_{x}\right].\end{split}\]
V polárních souřadnicích je podle (38) v části Vektorová analýza a (97)
(103)¶\[\varepsilon_{rr}=\partial_{r}u_{r},
\quad\varepsilon_{r\varphi}=\frac{1}{2}\left[\frac{1}{r}\left(\partial_{\varphi}u_{r}-u_{\varphi}\right)
+\partial_{r}u_{\varphi}\right],
\quad\varepsilon_{\varphi\varphi}=\frac{1}{r}\left(\partial_{\varphi}u_{\varphi}+u_{r}\right).\]
Svou roli hraje i gradient deformace v polárních souřadnicích, viz (94),
(104)¶\[\begin{split}\nabla\boldsymbol{\varepsilon} &= \partial_{r}\varepsilon_{rr}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}
+\partial_{r}\varepsilon_{r\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}
+\partial_{r}\varepsilon_{\varphi r}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}
+\partial_{r}\varepsilon_{\varphi\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi} \\
&+ \frac{1}{r}\left(\partial_{\varphi}\varepsilon_{rr}-\varepsilon_{r\varphi}-\varepsilon_{\varphi r}\right)
\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}
+\frac{1}{r}\left(\varepsilon_{rr}-\varepsilon_{\varphi\varphi}+\partial_{\varphi}\varepsilon_{\varphi r}\right)
\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r} \\
&+ \frac{1}{r}\left(\varepsilon_{rr}-\varepsilon_{\varphi\varphi}+\partial_{\varphi}\varepsilon_{r\varphi}\right)
\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}
+\frac{1}{r}\left(\varepsilon_{r\varphi}+\varepsilon_{\varphi r}+\partial_{\varphi}\varepsilon_{\varphi\varphi}\right)
\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi},\end{split}\]
(105)¶\[\begin{split}\tau_{rr} &= \left(\lambda+2\mu\right)\partial_{r}u_{r}+\lambda\frac{1}{r}\left(u_{r}+\partial_{\varphi}u_{\varphi}\right), \\
\tau_{r\varphi} &= \mu\left[\frac{1}{r}\left(\partial_{\varphi}u_{r}-u_{\varphi}\right)+\partial_{r}u_{\varphi}\right], \\
\tau_{\varphi\varphi} &= \left(\lambda+2\mu\right)\frac{1}{r}\left(u_{r}+\partial_{\varphi}u_{\varphi}\right)
+\lambda\partial_{r}u_{r},\end{split}\]
(106)¶\[\begin{split}m_{rrr} &= c\partial_{r}\left[\left(\lambda+2\mu\right)\partial_{r}u_{r}
+\lambda\frac{1}{r}\left(u_{r}+\partial_{\varphi}u_{\varphi}\right)\right], \\
m_{rr\varphi} &= c\mu\partial_{r}\left[\frac{1}{r}\left(\partial_{\varphi}u_{r}-u_{\varphi}\right)
+\partial_{r}u_{\varphi}\right], \\
m_{r\varphi\varphi} &= c\partial_{r}\left[\left(\lambda+2\mu\right)\frac{1}{r}\left(u_{r}
+\partial_{\varphi}u_{\varphi}\right)+\lambda\partial_{r}u_{r}\right], \\
m_{\varphi rr} &= c\frac{1}{r}\left[\partial_{\varphi}\left[\left(\lambda+2\mu\right)\partial_{r}u_{r}
+\lambda\frac{1}{r}\left(u_{r}+\partial_{\varphi}u_{\varphi}\right)\right]\right. \\
&\left. -2\mu\left[\frac{1}{r}\left(\partial_{\varphi}u_{r}-u_{\varphi}\right)+\partial_{r}u_{\varphi}\right]\right], \\
m_{\varphi\varphi r} &= c\frac{1}{r}\mu\left[\partial_{\varphi}\left[\frac{1}{r}\left(\partial_{\varphi}u_{r}-3u_{\varphi}\right)
+\partial_{r}u_{\varphi}\right]\right. \\
&\left. +2\left(\partial_{r}u_{r}-\frac{1}{r}u_{r}\right)\right], \\
m_{\varphi\varphi\varphi} &= c\frac{1}{r}\left[\partial_{\varphi}\left[\left(\lambda+2\mu\right)
\frac{1}{r}\left(u_{r}+\partial_{\varphi}u_{\varphi}\right)
+\lambda\partial_{r}u_{r}\right]\right. \\
&\left. +2\mu\left[\frac{1}{r}\left(\partial_{\varphi}u_{r}-u_{\varphi}\right)+\partial_{r}u_{\varphi}\right]\right].\end{split}\]
Totální napětí v kartézských souřadnicích se určí pro rovinu s normálou \(\boldsymbol{n}=\left(0,\pm1\right)\)
(107)¶\[\begin{split}t_{yx}\equiv P_{x} &= \pm\left(\tau_{yx}-\partial_{x}m_{xyx}-\partial_{y}m_{yyx}-\partial_{x}m_{yxx}\right), \\
t_{yy}\equiv P_{y} &= \pm\left(\tau_{yy}-\partial_{x}m_{xyy}-\partial_{y}m_{yyy}-\partial_{x}m_{yxy}\right). \\\end{split}\]
V případě polárních souřadnic je výhodnější rovnici (86) přepsat do tvaru
(108)¶\[\boldsymbol{P}=\boldsymbol{n}\cdot\left(\boldsymbol{\tau}-\nabla\cdot\boldsymbol{m}\right)
-\overset{\boldsymbol{s}}{\nabla}\cdot\left(\boldsymbol{n}\cdot\boldsymbol{m}\right)
+\left(\overset{\boldsymbol{s}}{\nabla}\cdot\boldsymbol{n}\right)
\left(\boldsymbol{n}\boldsymbol{n}\colon\boldsymbol{m}\right),\]
kde
(109)¶\[\overset{\boldsymbol{s}}{\nabla}=\left(\boldsymbol{I}-\boldsymbol{n}\boldsymbol{n}\right)\cdot\nabla\]
je operátor povrchového gradientu, který se v případě kartézských souřadnic zapíše jako (42). Jestiže v případě polárních souřadnic
(110)¶\[\nabla=\boldsymbol{e}_{r}\partial_{r}+r^{-1}\boldsymbol{e}_{\varphi}\partial_{\varphi}
\quad\mathrm{a}\quad\boldsymbol{n}=\boldsymbol{e}_{\varphi},\]
pak má povrchový gradient tvar
(111)¶\[\begin{split}\overset{\boldsymbol{s}}{\nabla} &= \boldsymbol{I}\cdot\nabla-\boldsymbol{n}\boldsymbol{n}\cdot\nabla \\
&= \boldsymbol{e}_{r}\left(\boldsymbol{e}_{r}\cdot\nabla\right)
+\boldsymbol{e}_{\varphi}\left(\boldsymbol{e}_{\varphi}\cdot\nabla\right)
-\boldsymbol{e}_{\varphi}\left(\boldsymbol{e}_{\varphi}\cdot\nabla\right) \\
&= \boldsymbol{e}_{r}\partial_{r}+\boldsymbol{e}_{\varphi}\frac{1}{r}\partial_{\varphi}
-\boldsymbol{e}_{\varphi}\frac{1}{r}\partial_{\varphi} \\
&= \boldsymbol{e}_{r}\partial_{r}.\end{split}\]
Jednotlivé části vztahu (108), se vzetím v potaz (32) z části Vektorová analýza a nulovosti ostatních derivací, mohou se vyjádřit následovně
(112)¶\[\boldsymbol{n}\cdot\boldsymbol{m}=m_{\varphi rr}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}
+m_{\varphi r\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}
+m_{\varphi\varphi r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}
+m_{\varphi\varphi\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi},\]
a podle (49) z části Vektorová analýza
(113)¶\[\begin{split}\boldsymbol{n}\cdot\left(\nabla\cdot\boldsymbol{m}\right) &= \left(\partial_{r}m_{r\varphi r}+\frac{1}{r}m_{r\varphi r}
+\frac{1}{r}m_{\varphi rr}+\frac{1}{r}\partial_{\varphi}m_{\varphi\varphi r}
-\frac{1}{r}m_{\varphi\varphi\varphi}\right)\boldsymbol{e}_{r} \\
&+ \left(\partial_{r}m_{r\varphi\varphi}+\frac{1}{r}m_{r\varphi\varphi}+\frac{1}{r}m_{\varphi r\varphi}
+\frac{1}{r}m_{\varphi\varphi r}+\frac{1}{r}\partial_{\varphi}m_{\varphi\varphi\varphi}\right)\boldsymbol{e}_{\varphi}.\end{split}\]
Dále
(114)¶\[\begin{split}\overset{\boldsymbol{s}}{\nabla}\cdot\left(\boldsymbol{n}\cdot\boldsymbol{m}\right)
&= \partial_{r}m_{\varphi rr}\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}
+\partial_{r}m_{\varphi r\varphi}\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi} \\
&+ \partial_{r}m_{\varphi\varphi r}\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}
+\partial_{r}m_{\varphi\varphi\varphi}\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi} \\
&= \partial_{r}m_{\varphi rr}\boldsymbol{e}_{r}+\partial_{r}m_{\varphi r\varphi}\boldsymbol{e}_{\varphi},\end{split}\]
kde
(115)¶\[\overset{\boldsymbol{s}}{\nabla}\cdot\boldsymbol{n}=\boldsymbol{e}_{r}\cdot\partial_{r}\boldsymbol{e}_{\varphi}=0.\]
Potom pro (108) platí
(116)¶\[\begin{split}\boldsymbol{P} &= \boldsymbol{n}\cdot\left(\boldsymbol{\tau}-\nabla\cdot\boldsymbol{m}\right)
-\overset{\boldsymbol{s}}{\nabla}\cdot\left(\boldsymbol{n}\cdot\boldsymbol{m}\right), \\
&= \tau_{rr}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}
+\tau_{r\varphi}\left(\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}
+\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\right)
+\tau_{\varphi\varphi}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi} \\
&- \left(\partial_{r}m_{r\varphi r}+\frac{1}{r}m_{r\varphi r}+\frac{1}{r}m_{\varphi rr}
+\frac{1}{r}\partial_{\varphi}m_{\varphi\varphi r}-\frac{1}{r}m_{\varphi\varphi\varphi}\right)\boldsymbol{e}_{r} \\
&- \left(\partial_{r}m_{r\varphi\varphi}+\frac{1}{r}m_{r\varphi\varphi}+\frac{1}{r}m_{\varphi r\varphi}
+\frac{1}{r}m_{\varphi\varphi r}
+\frac{1}{r}\partial_{\varphi}m_{\varphi\varphi\varphi}\right)\boldsymbol{e}_{\varphi} \\
&- \partial_{r}m_{\varphi rr}\boldsymbol{e}_{r}+\partial_{r}m_{\varphi r\varphi}\boldsymbol{e}_{\varphi} \\
&= \left(\tau_{r\varphi}-\partial_{r}m_{r\varphi r}-\frac{1}{r}m_{r\varphi r}-\frac{1}{r}m_{\varphi rr}
-\frac{1}{r}\partial_{\varphi}m_{\varphi\varphi r}+\frac{1}{r}m_{\varphi\varphi\varphi}
-\partial_{r}m_{\varphi rr}\right)\boldsymbol{e}_{r} \\
&+ \left(\tau_{\varphi\varphi}-\partial_{r}m_{r\varphi\varphi}-\frac{1}{r}m_{r\varphi\varphi}
-\frac{1}{r}m_{\varphi r\varphi}-\frac{1}{r}m_{\varphi\varphi r}
-\frac{1}{r}\partial_{\varphi}m_{\varphi\varphi\varphi}
+\partial_{r}m_{\varphi r\varphi}\right)\boldsymbol{e}_{\varphi}.\end{split}\]
Po rozepsání do složek
(117)¶\[t_{\varphi r}\equiv P_{r}=\tau_{r\varphi}-\partial_{r}m_{r\varphi r}-\frac{1}{r}m_{r\varphi r}
-\frac{1}{r}m_{\varphi rr}-\frac{1}{r}\partial_{\varphi}m_{\varphi\varphi r}
+\frac{1}{r}m_{\varphi\varphi\varphi}-\partial_{r}m_{\varphi rr}\]
a
(118)¶\[\begin{split}t_{\varphi\varphi} &\equiv P_{\varphi}=\tau_{\varphi\varphi}-\partial_{r}m_{r\varphi\varphi}
-\frac{1}{r}m_{r\varphi\varphi}-\frac{1}{r}m_{\varphi r\varphi}-\frac{1}{r}m_{\varphi\varphi r}
-\frac{1}{r}\partial_{\varphi}m_{\varphi\varphi\varphi}+\partial_{r}m_{\varphi r\varphi} \\
&= \tau_{\varphi\varphi}-\partial_{r}m_{r\varphi\varphi}-\frac{1}{r}m_{r\varphi\varphi}-\frac{2}{r}m_{\varphi r\varphi}
-\frac{1}{r}\partial_{\varphi}m_{\varphi\varphi\varphi}+\partial_{r}m_{\varphi r\varphi}.\end{split}\]
Podobně jako pro totální napětí, také rovnice rovnováhy je výhodné přepsat do tvaru s vektorovými operátory
(119)¶\[\boldsymbol{\tau}-\nabla\cdot\boldsymbol{m}=0.\]
Dosazením (94) se rovnice (119) převede do tvaru
(120)¶\[\boldsymbol{\tau}-c\nabla\cdot\nabla\boldsymbol{\tau}=\boldsymbol{\tau}-c\nabla^{2}\boldsymbol{\tau}=0.\]
Hookeův zákon (83) lze pomocí (97) napsat ve tvaru
(121)¶\[\boldsymbol{\tau}=\frac{1}{2}\lambda\boldsymbol{I}\left[\boldsymbol{I}\colon\left(\boldsymbol{u}\nabla
+\nabla\boldsymbol{u}\right)\right]
+\mu\left(\boldsymbol{u}\nabla+\nabla\boldsymbol{u}\right)\]
a po roznásobení výrazu v hranaté závorce
(122)¶\[\boldsymbol{\tau}=\lambda\left(\nabla\cdot\boldsymbol{u}\right)\boldsymbol{I}
+\mu\left(\boldsymbol{u}\nabla+\nabla\boldsymbol{u}\right).\]
Dosazením tohoto výrazu do (120) se dostane
(123)¶\[\begin{split}0 &= \lambda\left(\nabla\cdot\boldsymbol{u}\right)\boldsymbol{I}
+\mu\left(\boldsymbol{u}\nabla+\nabla\boldsymbol{u}\right) \\
&- c\nabla^{2}\left[\lambda\left(\nabla\cdot\boldsymbol{u}\right)\boldsymbol{I}
+\mu\left(\boldsymbol{u}\nabla+\nabla\boldsymbol{u}\right)\right]\end{split}\]
Místo Lamého konstant \(\lambda\) a \(\mu\) je lepší použít Youngův modul \(E\) a Poissonovo číslo \(\nu\),
(124)¶\[\lambda=\frac{E\nu}{\left(1+\nu\right)\left(1-2\nu\right)},\quad\mu=\frac{E}{2\left(1+\nu\right)}.\]
Pak se teda dostane
(125)¶\[\begin{split}0 &= \frac{E\nu}{\left(1+\nu\right)\left(1-2\nu\right)}\left(\nabla\cdot\boldsymbol{u}\right)\boldsymbol{I}
+\frac{E}{2\left(1+\nu\right)}\left(\boldsymbol{u}\nabla+\nabla\boldsymbol{u}\right) \\
&- c\nabla^{2}\left[\frac{E\nu}{\left(1+\nu\right)\left(1-2\nu\right)}\left(\nabla\cdot\boldsymbol{u}\right)\boldsymbol{I}
+\frac{E}{2\left(1+\nu\right)}\left(\boldsymbol{u}\nabla+\nabla\boldsymbol{u}\right)\right]\end{split}\]
Po úpravě se dotane
(126)¶\[\begin{split}0 &= 2\nu\left(\nabla\cdot\boldsymbol{u}\right)\boldsymbol{I}
+\left(1-2\nu\right)\left(\boldsymbol{u}\nabla+\nabla\boldsymbol{u}\right) \\
&- c\nabla^{2}\left[2\nu\left(\nabla\cdot\boldsymbol{u}\right)\boldsymbol{I}
+\left(1-2\nu\right)\left(\boldsymbol{u}\nabla+\nabla\boldsymbol{u}\right)\right].\end{split}\]
Dále se skalárně rovnice vynásobí operátorem \(\nabla\),
(127)¶\[\begin{split}0 &= 2\nu\left(\nabla\cdot\boldsymbol{u}\right)\nabla\cdot\boldsymbol{I}
+\left(1-2\nu\right)\left(\nabla\cdot\boldsymbol{u}\nabla+\nabla\cdot\nabla\boldsymbol{u}\right) \\
&- c\nabla^{2}\left[2\nu\left(\nabla\cdot\boldsymbol{u}\right)\nabla\cdot\boldsymbol{I}
+\left(1-2\nu\right)\left(\nabla\cdot\boldsymbol{u}\nabla+\nabla\cdot\nabla\boldsymbol{u}\right)\right],\end{split}\]
nebo-li
(128)¶\[\begin{split}0 &= 2\nu\left(\nabla\cdot\boldsymbol{u}\right)\nabla
+\left(1-2\nu\right)\left(\nabla\cdot\boldsymbol{u}\nabla+\nabla^{2}\boldsymbol{u}\right) \\
&- c\nabla^{2}\left[2\nu\left(\nabla\cdot\boldsymbol{u}\right)\nabla
+\left(1-2\nu\right)\left(\nabla\cdot\boldsymbol{u}\nabla+\nabla^{2}\boldsymbol{u}\right)\right].\end{split}\]
Nakonec se výrazy \(2\nu\) odečtou,
(129)¶\[\begin{split}0 &= \nabla\cdot\boldsymbol{u}\nabla+\left(1-2\nu\right)\nabla^{2}\boldsymbol{u} \\
&- c\nabla^{2}\left(\nabla\cdot\boldsymbol{u}\nabla+\left(1-2\nu\right)\nabla^{2}\boldsymbol{u}\right)\end{split}\]
a výraz v závorce se vytkne
(130)¶\[\left(1-c\nabla^{2}\right)\left[\left(1-2\nu\right)\nabla^{2}\boldsymbol{u}
+\nabla\left(\nabla\cdot\boldsymbol{u}\right)\right]=0.\]
V polárních souřadnicích se (130) napíše jako
(131)¶\[\left(1-c\nabla^{2}\right)\left[s_{r}\boldsymbol{e_{r}}+s_{\varphi}\boldsymbol{e}_{\varphi}\right]=0\]
(132)¶\[\begin{split}&\Rightarrow \left[1-c\left(\boldsymbol{e}_{r}\partial_{r}+\frac{1}{r}\boldsymbol{e}_{\varphi}\partial_{\varphi}\right)
\cdot\left(\boldsymbol{e}_{r}\partial_{r}
+\frac{1}{r}\boldsymbol{e}_{\varphi}\partial_{\varphi}\right)\right] \\
&\times \left[\left(1-2\nu\right)\left(\boldsymbol{e}_{r}\partial_{r}
+\frac{1}{r}\boldsymbol{e}_{\varphi}\partial_{\varphi}\right)
\cdot\left(\boldsymbol{e}_{r}\partial_{r}+\frac{1}{r}\boldsymbol{e}_{\varphi}\partial_{\varphi}\right)
\left(u_{r}\boldsymbol{e}_{r}+u_{\varphi}\boldsymbol{e}_{\varphi}\right)\right. \\
&\left. +\left(\boldsymbol{e}_{r}\partial_{r}+\frac{1}{r}\boldsymbol{e}_{\varphi}\partial_{\varphi}\right)
\left[\left(\boldsymbol{e}_{r}\partial_{r}+\frac{1}{r}\boldsymbol{e}_{\varphi}\partial_{\varphi}\right)
\cdot\left(u_{r}\boldsymbol{e}_{r}+u_{\varphi}\boldsymbol{e}_{\varphi}\right)\right]\right]=0 \\
&\Rightarrow \left[1-c\left(\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{r}\partial_{rr}
+\frac{1}{r}\boldsymbol{e}_{\varphi}\cdot\partial_{\varphi}\boldsymbol{e}_{r}\partial_{r}
+\frac{1}{r}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{r}\partial_{\varphi r}
+\frac{1}{r^{2}}\boldsymbol{e}_{\varphi}
\cdot\partial_{\varphi}\boldsymbol{e}_{\varphi}\partial_{\varphi}
+\frac{1}{r^{2}}\boldsymbol{e}_{\varphi}
\cdot\boldsymbol{e}_{\varphi}\partial_{\varphi\varphi}\right)\right] \\
&\times \left[\left(1-2\nu\right)\left(\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{r}\partial_{rr}
+\frac{1}{r}\boldsymbol{e}_{\varphi}\cdot\partial_{\varphi}\boldsymbol{e}_{r}\partial_{r}
+\frac{1}{r}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{r}\partial_{\varphi r}
\right.\right. \\
&+ \left.\frac{1}{r^{2}}\boldsymbol{e}_{\varphi}\cdot\partial_{\varphi}\boldsymbol{e}_{\varphi}\partial_{\varphi}
+\frac{1}{r^{2}}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{\varphi}\partial_{\varphi\varphi}\right)
\left(u_{r}\boldsymbol{e}_{r}+u_{\varphi}\boldsymbol{e}_{\varphi}\right) \\
&+ \left(\boldsymbol{e}_{r}\partial_{r}+\frac{1}{r}\boldsymbol{e}_{\varphi}\partial_{\varphi}\right)
\left[\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{r}\partial_{r}u_{r}+u_{r}\boldsymbol{e}_{r}\cdot\partial_{r}\boldsymbol{e}_{r}
+\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{\varphi}\partial_{r}u_{\varphi}
+u_{\varphi}\boldsymbol{e}_{r}\cdot\partial_{r}\boldsymbol{e}_{\varphi}\right. \\
&\left.\left. +\frac{1}{r}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{r}\partial_{\varphi}u_{r}
+\frac{1}{r}u_{r}\boldsymbol{e}_{\varphi}\cdot\partial_{\varphi}\boldsymbol{e}_{r}
+\frac{1}{r}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{\varphi}\partial_{\varphi}u_{\varphi}
+\frac{1}{r}u_{\varphi}\boldsymbol{e}_{\varphi}\cdot\partial_{\varphi}\boldsymbol{e}_{\varphi}\right]\right]=0 \\
&\Rightarrow \left[1-c\left(\partial_{rr}+\frac{1}{r}\partial_{r}+\frac{1}{r^{2}}\partial_{\varphi\varphi}\right)\right]
\left[\left(1-2\nu\right)\left(\partial_{rr}+\frac{1}{r}\partial_{r}
+\frac{1}{r^{2}}\partial_{\varphi\varphi}\right)
\left(u_{r}\boldsymbol{e}_{r}+u_{\varphi}\boldsymbol{e}_{\varphi}\right)\right. \\
&\left. +\left(\boldsymbol{e}_{r}\partial_{r}+\frac{1}{r}\boldsymbol{e}_{\varphi}\partial_{\varphi}\right)
\left(\partial_{r}u_{r}+\frac{1}{r}u_{r}+\frac{1}{r}\partial_{\varphi}u_{\varphi}\right)\right]=0 \\
&\Rightarrow \left[1-c\left(\partial_{rr}+\frac{1}{r}\partial_{r}+\frac{1}{r^{2}}\partial_{\varphi\varphi}\right)\right]
\left[\left(1-2\nu\right)\left(\boldsymbol{e}_{r}\partial_{rr}u_{r}
+\boldsymbol{e}_{\varphi}\partial_{rr}u_{\varphi}
+\boldsymbol{e}_{r}\frac{1}{r}\partial_{r}u_{r}
+\boldsymbol{e}_{\varphi}\frac{1}{r}\partial_{r}u_{\varphi}\right.\right. \\
&\left. +\frac{1}{r^{2}}\partial_{\varphi}\left(\boldsymbol{e}_{r}\partial_{\varphi}u_{r}
+u_{r}\partial_{\varphi}\boldsymbol{e}_{r}
+\boldsymbol{e}_{\varphi}\partial_{\varphi}u_{\varphi}
+u_{\varphi}\partial_{\varphi}\boldsymbol{e}_{\varphi}\right)\right) \\
&+ \boldsymbol{e}_{r}\left(\partial_{rr}u_{r}-\frac{1}{r^{2}}u_{r}+\frac{1}{r}\partial_{r}u_{r}
-\frac{1}{r^{2}}\partial_{\varphi}u_{\varphi}+\frac{1}{r}\partial_{r\varphi}u_{\varphi}\right) \\
&\left. +\boldsymbol{e}_{\varphi}\left(\frac{1}{r}\partial_{\varphi r}u_{r}+\frac{1}{r^{2}}\partial_{\varphi}u_{r}
+\frac{1}{r^{2}}\partial_{\varphi\varphi}u_{\varphi}\right)\right]=0 \\
&\Rightarrow \left[1-c\left(\partial_{rr}+\frac{1}{r}\partial_{r}+\frac{1}{r^{2}}\partial_{\varphi\varphi}\right)\right]
\left[\left(1-2\nu\right)\left(\boldsymbol{e}_{r}\partial_{rr}u_{r}
+\boldsymbol{e}_{\varphi}\partial_{rr}u_{\varphi}+\boldsymbol{e}_{r}\frac{1}{r}\partial_{r}u_{r}
+\boldsymbol{e}_{\varphi}\frac{1}{r}\partial_{r}u_{\varphi}\right.\right. \\
&+ \frac{1}{r^{2}}\left(\boldsymbol{e}_{\varphi}\partial_{\varphi}u_{r}+\boldsymbol{e}_{r}\partial_{\varphi\varphi}u_{r}
+\partial_{\varphi}u_{r}\boldsymbol{e}_{\varphi}
+u_{r}\partial_{\varphi}\boldsymbol{e}_{\varphi}\right. \\
&\left.\left. -\boldsymbol{e}_{r}\partial_{\varphi}u_{\varphi}+\boldsymbol{e}_{\varphi}\partial_{\varphi\varphi}u_{\varphi}
-\partial_{\varphi}u_{\varphi}\boldsymbol{e}_{r}-u_{\varphi}\partial_{\varphi}\boldsymbol{e}_{r}\right)\right) \\
&+ \boldsymbol{e}_{r}\left(\partial_{rr}u_{r}-\frac{1}{r^{2}}u_{r}+\frac{1}{r}\partial_{r}u_{r}
-\frac{1}{r^{2}}\partial_{\varphi}u_{\varphi}+\frac{1}{r}\partial_{r\varphi}u_{\varphi}\right) \\
&\left. +\boldsymbol{e}_{\varphi}\left(\frac{1}{r}\partial_{\varphi r}u_{r}+\frac{1}{r^{2}}\partial_{\varphi}u_{r}
+\frac{1}{r^{2}}\partial_{\varphi\varphi}u_{\varphi}\right)\right]=0 \\
&\Rightarrow \left[1-c\left(\partial_{rr}+\frac{1}{r}\partial_{r}+\frac{1}{r^{2}}\partial_{\varphi\varphi}\right)\right]
\left[\boldsymbol{e}_{r}\left(\partial_{rr}u_{r}-\frac{1}{r^{2}}u_{r}+\frac{1}{r}\partial_{r}u_{r}
-\frac{1}{r^{2}}\partial_{\varphi}u_{\varphi}
+\frac{1}{r}\partial_{r\varphi}u_{\varphi}\right.\right. \\
&+ \left(1-2\nu\right)\partial_{rr}u_{r}+\left(1-2\nu\right)\frac{1}{r}\partial_{r}u_{r} \\
&\left. +\left(1-2\nu\right)\frac{1}{r^{2}}\partial_{\varphi\varphi}u_{r}
-\left(1-2\nu\right)\frac{1}{r^{2}}\partial_{\varphi}u_{\varphi}
-\left(1-2\nu\right)\frac{1}{r^{2}}\partial_{\varphi}u_{\varphi}
-\left(1-2\nu\right)\frac{1}{r^{2}}u_{r}\right) \\
&+ \boldsymbol{e}_{\varphi}\left(\frac{1}{r}\partial_{\varphi r}u_{r}+\frac{1}{r^{2}}\partial_{\varphi}u_{r}
+\frac{1}{r^{2}}\partial_{\varphi\varphi}u_{\varphi}+\left(1-2\nu\right)
\partial_{rr}u_{\varphi}+\left(1-2\nu\right)\frac{1}{r}\partial_{r}u_{\varphi}\right. \\
&\left.\left. +\frac{1}{r^{2}}\left(1-2\nu\right)\partial_{\varphi}u_{r}+\left(1-2\nu\right)
\frac{1}{r^{2}}\partial_{\varphi}u_{r}+\left(1-2\nu\right)\frac{1}{r^{2}}\partial_{\varphi\varphi}u_{\varphi}
-\left(1-2\nu\right)\frac{1}{r^{2}}u_{\varphi}\right)\right]=0 \\
&\Rightarrow \left[1-c\left(\partial_{rr}+\frac{1}{r}\partial_{r}+\frac{1}{r^{2}}\partial_{\varphi\varphi}\right)\right]
\left[\boldsymbol{e}_{r}\left[-\left(1-2\nu\right)\frac{1}{r}\partial_{\varphi}
\left(\partial_{r}u_{\varphi}-\frac{1}{r}\partial_{\varphi}u_{r}
+\frac{1}{r}u_{\varphi}\right)\right.\right. \\
&\left. +2\left(1-\nu\right)\partial_{r}\left(\partial_{r}u_{r}+\frac{1}{r}u_{r}
+\frac{1}{r}\partial_{\varphi}u_{\varphi}\right)\right] \\
&\left. +\boldsymbol{e}_{\varphi}\left[2\left(1-\nu\right)\frac{1}{r}\partial_{\varphi}
\left(\partial_{r}u_{r}+\frac{1}{r}u_{r}+\frac{1}{r}\partial_{\varphi}u_{\varphi}\right)
+\left(1-2\nu\right)\partial_{r}
\left(\partial_{r}u_{\varphi}+\frac{1}{r}u_{\varphi}
-\frac{1}{r}\partial_{\varphi}u_{r}\right)\right]\right]=0,\end{split}\]
nebo-li
(133)¶\[\left[1-c\left(\partial_{rr}+\frac{1}{r}\partial_{r}+\frac{1}{r^{2}}\partial_{\varphi\varphi}\right)\right]
\left(s_{r}\boldsymbol{e}_{r}+s_{\varphi}\boldsymbol{e}_{\varphi}\right)=0\]
(134)¶\[\begin{split}&\Rightarrow s_{r}\boldsymbol{e}_{r}-c\left[\partial_{rr}s_{r}\boldsymbol{e}_{r}+\frac{1}{r}\partial_{r}s_{r}\boldsymbol{e}_{r}
+\frac{1}{r^{2}}\partial_{\varphi}\left(\partial_{\varphi}s_{r}\boldsymbol{e}_{r}
+s_{r}\boldsymbol{e}_{\varphi}\right)\right] \\
&+ s_{\varphi}\boldsymbol{e}_{\varphi}-c\left[\partial_{rr}s_{\varphi}\boldsymbol{e}_{\varphi}
+\frac{1}{r}\partial_{r}s_{\varphi}\boldsymbol{e}_{\varphi}
+\frac{1}{r^{2}}\partial_{\varphi}
\left(\partial_{\varphi}s_{\varphi}\boldsymbol{e}_{\varphi}
-s_{\varphi}\boldsymbol{e}_{r}\right)\right]=0 \\
&\Rightarrow s_{r}\boldsymbol{e}_{r}-c\left[\partial_{rr}s_{r}\boldsymbol{e}_{r}+\frac{1}{r}\partial_{r}s_{r}\boldsymbol{e}_{r}
+\frac{1}{r^{2}}\left(\partial_{\varphi\varphi}s_{r}\boldsymbol{e}_{r}
+\partial_{\varphi}s_{r}\boldsymbol{e}_{\varphi}
+\partial_{\varphi}s_{r}\boldsymbol{e}_{\varphi}
-s_{r}\boldsymbol{e}_{r}\right)\right] \\
&+ s_{\varphi}\boldsymbol{e}_{\varphi}-c\left[\partial_{rr}s_{\varphi}\boldsymbol{e}_{\varphi}
+\frac{1}{r}\partial_{r}s_{\varphi}\boldsymbol{e}_{\varphi}
+\frac{1}{r^{2}}\left(\partial_{\varphi\varphi}s_{\varphi}\boldsymbol{e}_{\varphi}
-\partial_{\varphi}s_{\varphi}\boldsymbol{e}_{r}
-\partial_{\varphi}s_{\varphi}\boldsymbol{e}_{r}
-s_{\varphi}\boldsymbol{e}_{\varphi}\right)\right]=0.\end{split}\]
Takže nakonec se dostanou dvě rovnice ve směru \(\boldsymbol{e}_{r}\) a \(\boldsymbol{e}_{\varphi}\),
(135)¶\[s_{r}-c\left(\nabla^{2}s_{r}-\frac{1}{r^{2}}s_{r}-2\frac{1}{r^{2}}\partial_{\varphi}s_{\varphi}\right)=0\]
a
(136)¶\[s_{\varphi}-c\left(\nabla^{2}s_{\varphi}-\frac{1}{r^{2}}s_{\varphi}+2\frac{1}{r^{2}}\partial_{\varphi}s_{r}\right)=0,\]
kde
(137)¶\[s_{r}=2\left(1-\nu\right)\partial_{r}\left(\partial_{r}u_{r}+\frac{1}{r}u_{r}+\frac{1}{r}\partial_{\varphi}u_{\varphi}\right)
-\left(1-2\nu\right)\frac{1}{r}\partial_{\varphi}
\left(\partial_{r}u_{\varphi}-\frac{1}{r}\partial_{\varphi}u_{r}+\frac{1}{r}u_{\varphi}\right),\]
(138)¶\[s_{\varphi}=\left(1-2\nu\right)\partial_{r}\left(\partial_{r}u_{\varphi}+\frac{1}{r}u_{\varphi}
-\frac{1}{r}\partial_{\varphi}u_{r}\right)
+2\left(1-\nu\right)\frac{1}{r}\partial_{\varphi}\left(\partial_{r}u_{r}
+\frac{1}{r}u_{r}+\frac{1}{r}\partial_{\varphi}u_{\varphi}\right).\]
Obě rovnice (135) a (136) při \(c\rightarrow0\) přejdou k Navier-Cauchy rovnicím z klasické pružnosti.
Literatura
Rovnice rovnováhy kubického materiálu v kartézských souřadnicích¶
Pro složky tenzoru napětí \(\boldsymbol{\tau}\) kubického materiálu platí
(139)¶\[\begin{split}\begin{equation}
\begin{split}
\tau_{xx} =& c_{11}\varepsilon_{xx} + c_{12}\varepsilon_{yy}, \\
\tau_{xy} =& 2c_{44}\varepsilon_{xy}, \\
\tau_{yy} =& c_{12}\varepsilon_{xx}+c_{11}\varepsilon_{yy}, \\
\end{split}
\end{equation}\end{split}\]
kde
(140)¶\[\varepsilon_{xx}=\partial_xu_x,\quad
\varepsilon_{xy}=\frac{1}{2}\left(
\partial_xu_y+\partial_yu_x
\right),\quad
\varepsilon_{yy}=\partial_yu_y.\]
Pro složky divergence tenzoru napětí \(\nabla\cdot\boldsymbol{\tau}\) pak platí
(141)¶\[\begin{split}\begin{equation}
\begin{split}
s_x =& \big(\nabla\cdot\boldsymbol{\tau}\big)_x
= \partial_x\tau_{xx}+\partial_y\tau_{xy} \\
=& \partial_x\left(
c_{11}\varepsilon_{xx}
+c_{12}\varepsilon_{yy}
\right)
+2c_{44}\partial_y\varepsilon_{xy} \\
=& \partial_x\left(
c_{11}\partial_xu_x
+c_{12}\partial_yu_y
\right)
+c_{44}\partial_y\left(
\partial_xu_y
+\partial_yu_x
\right) \\
=& c_{11}\partial_{xx}u_x+c_{44}\partial_{yy}u_x
+\left(
c_{12}+c_{44}
\right)\partial_{xy}u_y,
\end{split}
\end{equation}\end{split}\]
(142)¶\[\begin{split}\begin{equation}
\begin{split}
s_y =& \big(\nabla\cdot\boldsymbol{\tau}\big)_y
= \partial_x\tau_{xy}+\partial_y\tau_{yy} \\
=& 2c_{44}\partial_x\varepsilon_{xy}
+\partial_y\left(
c_{12}\varepsilon_{xx}
+c_{11}\varepsilon_{yy}
\right) \\
=& c_{44}\partial_x\left(
\partial_xu_y
+\partial_yu_x
\right)
+\partial_y\left(
c_{12}\partial_xu_x
+c_{11}\partial_yu_y
\right) \\
=& c_{44}\partial_{xx}u_y+c_{11}\partial_{yy}u_y
+\left(
c_{12}+c_{44}
\right)\partial_{xy}u_x,
\end{split}
\end{equation}\end{split}\]
Další částí rovnice rovnováhy jsou složky divergence Laplaciánu tenzoru napětí \(\nabla\cdot\nabla^2\boldsymbol\tau\), pro které platí
(143)¶\[\begin{split}\begin{equation}
\begin{split}
\left(
\nabla\cdot\nabla^2\boldsymbol{\tau}
\right)_x =&
\partial_x\nabla^2\tau_{xx}
+\partial_y\nabla^2\tau_{xy} \\
=& \nabla^2\left(
\partial_x\tau_{xx}
+\partial_y\tau_{xy}
\right) \\
=& \nabla^2 s_x
\end{split}
\end{equation}\end{split}\]
(144)¶\[\begin{split}\begin{equation}
\begin{split}
\left(
\nabla\cdot\nabla^2\boldsymbol{\tau}
\right)_y =&
\partial_x\nabla^2\tau_{xy}
+\partial_y\nabla^2\tau_{yy} \\
=& \nabla^2\left(
\partial_x\tau_{xy}
+\partial_y\tau_{yy}
\right) \\
=& \nabla^2 s_y
\end{split}
\end{equation}\end{split}\]
Pomocí vztahů (141), (142), (143) a (144) se rovnice rovnováhy mohou napsat ve tvaru
(145)¶\[\begin{split}\begin{equation}
\begin{split}
s_x-c\nabla^2s_x &= 0, \\
s_y-c\nabla^2s_y &= 0.
\end{split}
\end{equation}\end{split}\]
Rovnice rovnováhy kubického materiálu v polárních souřadnicích¶
Na základě transformačních vztahů pro tenzor napětí \(\tau_{rr}\), \(\tau_{r\varphi}\) a \(\tau_{\varphi\varphi}\) a inverzních transformačních vztahů pro tenzor deformace \(\varepsilon_{rr}\), \(\varepsilon_{r\varphi}\) a \(\varepsilon_{\varphi\varphi}\) v polárních souřadnicích
(146)¶\[\begin{split}\begin{equation}
\begin{split}
\tau_{rr} =& \tau_{xx}\cos^2\varphi+2\tau_{xy}\cos\varphi\sin\varphi+\tau_{yy}\sin^2\varphi, \\
\tau_{r\varphi} =&
-\tau_{xx}\cos\varphi\sin\varphi+\tau_{xy}\big(\cos^2\varphi-\sin^2\varphi\big)
+\tau_{yy}\cos\varphi\sin\varphi, \\
\tau_{r\varphi} =& \tau_{xx}\sin^2\varphi-2\tau_{xy}\cos\varphi\sin\varphi+\tau_{yy}\cos^2\varphi,
\end{split}
\end{equation}\end{split}\]
(147)¶\[\begin{split}\begin{equation}
\begin{split}
\varepsilon_{xx} =& \varepsilon_{rr}\cos^2\varphi-2\varepsilon_{r\varphi}\cos\varphi\sin\varphi
+\varepsilon_{\varphi\varphi}\sin^2\varphi, \\
\varepsilon_{xy} =&
\varepsilon_{rr}\cos\varphi\sin\varphi+\varepsilon_{r\varphi}\big(\cos^2\varphi-\sin^2\varphi\big)
-\varepsilon_{\varphi\varphi}\cos\varphi\sin\varphi, \\
\varepsilon_{xy} =& \varepsilon_{rr}\sin^2\varphi+2\varepsilon_{r\varphi}\cos\varphi\sin\varphi
+\varepsilon_{\varphi\varphi}\cos^2\varphi,
\end{split}
\end{equation}\end{split}\]
a Hookeova zákona (139) pro složky tenzoru napětí \(\tau_{rr}\), \(\tau_{r\varphi}\) a \(\tau_{\varphi\varphi}\) v kubickém materiálu platí
(148)¶\[\begin{split}\begin{equation}
\begin{split}
\tau_{rr}=&
\varepsilon_{rr}\big[
c_{11}\sin^4(\varphi)+c_{11}\cos^4(\varphi) \\
& +2c_{12}\sin^2(\varphi)\cos^2(\varphi) \\
& +4c_{44}\sin^2(\varphi)\cos^2(\varphi)
\big] \\
& +\varepsilon_{r\varphi}\big[
2c_{11}\sin^3(\varphi)\cos(\varphi)-2c_{11}\sin(\varphi)\cos^3(\varphi) \\
& -2c_{12}\sin^3(\varphi)\cos(\varphi)+2c_{12}\sin(\varphi)\cos^3(\varphi) \\
& -4c_{44}\sin^3(\varphi)\cos(\varphi)+4c_{44}\sin(\varphi)\cos^3(\varphi)
\big] \\
& +\varepsilon_{\varphi\varphi}\big[
2c_{11}\sin^2(\varphi)\cos^2(\varphi) \\
& +c_{12}\sin^4(\varphi)+c_{12}\cos^4(\varphi) \\
& -4c_{44}\sin^2(\varphi)\cos^2(\varphi)
\big],
\end{split}
\end{equation}\end{split}\]
(149)¶\[\begin{split}\begin{equation}
\begin{split}
\tau_{r\varphi}=&
\varepsilon_{rr}\big[
c_{11}\sin^3(\varphi)\cos(\varphi)
-c_{11}\sin(\varphi)\cos^3(\varphi) \\
& -c_{12}\sin^3(\varphi)\cos(\varphi)
+c_{12}\sin(\varphi)\cos^3(\varphi) \\
& -2c_{44}\sin^3(\varphi)\cos(\varphi)
+2c_{44}\sin(\varphi)\cos^3(\varphi)
\big] \\
&+\varepsilon_{r\varphi}\big[
4c_{11}\sin^2(\varphi)\cos^2(\varphi)
-4c_{12}\sin^2(\varphi)\cos^2(\varphi) \\
& +2c_{44}\sin^4(\varphi)
-4c_{44}\sin^2(\varphi)\cos^2(\varphi)
+2c_{44}\cos^4(\varphi)]
\big] \\
&+\varepsilon_{\varphi\varphi}\big[
-c_{11}\sin^3\varphi\cos(\varphi)
+c_{11}\sin(\varphi)\cos^3(\varphi) \\
& +c_{12}\sin^3(\varphi)\cos(\varphi)
-c_{12}\sin(\varphi)\cos^3(\varphi) \\
& +2c_{44}\sin^3(\varphi)\cos(\varphi)
-2c_{44}\sin(\varphi)\cos^3(\varphi)
\big],
\end{split}
\end{equation}\end{split}\]
(150)¶\[\begin{split}\begin{equation}
\begin{split}
\tau_{\varphi\varphi}=&
\varepsilon_{rr}\big[
2c_{11}\sin^2(\varphi)\cos^2(\varphi) \\
& +c_{12}\sin^4(\varphi)+c_{12}\cos^4(\varphi) \\
& -4c_{44}\sin^2(\varphi)\cos^2(\varphi)
\big] \\
& +\varepsilon_{r\varphi}\big[
-2c_{11}\sin^3(\varphi)\cos(\varphi)
+2c_{11}\sin(\varphi)\cos^3(\varphi) \\
& +2c_{12}\sin^3(\varphi)\cos(\varphi)
-2c_{12}\sin(\varphi)\cos^3(\varphi) \\
& +4c_{44}\sin^3(\varphi)\cos(\varphi)
-4c_{44}\sin(\varphi)\cos^3(\varphi)
\big] \\
& +\varepsilon_{\varphi\varphi}\big[
c_{11}\sin^4(\varphi)+c_{11}\cos^4(\varphi) \\
& +2c_{12}\sin^2(\varphi)\cos^2(\varphi)
+4c_{44}\sin^2(\varphi)cos^2(\varphi)
\big],
\end{split}
\end{equation}\end{split}\]
kde
(151)¶\[\begin{split}\begin{equation}
\begin{split}
& \varepsilon_{rr}=\partial_ru_r \\
& \varepsilon_{r\varphi}=\frac{1}{2}\left[
\frac{1}{r}\left(
\partial_\varphi u_r-u_\varphi
\right)+\partial_ru_\varphi
\right] \\
& \varepsilon_{\varphi\varphi}=\frac{1}{r}\left(
\partial_\varphi u_\varphi+u_r
\right).
\end{split}
\end{equation}\end{split}\]
Vztahy (148)-(150) se mohou upravit do tvaru
(152)¶\[\begin{split}\begin{equation}
\begin{split}
\tau_{rr} =& c_{11}\varepsilon_{rr}+c_{12}\varepsilon_{\varphi\varphi}
-\frac{1}{2}c_a\big(\varepsilon_{rr}-\varepsilon_{\varphi\varphi}\big)
-c_b\varepsilon_{r\varphi}, \\
\tau_{r\varphi} =& 2c_{44}\varepsilon_{r\varphi}
-\frac{1}{2}c_b\big(\varepsilon_{rr}-\varepsilon_{\varphi\varphi}\big)
+c_a\varepsilon_{r\varphi}, \\
\tau_{\varphi\varphi} =& c_{12}\varepsilon_{rr}+c_{11}\varepsilon_{\varphi\varphi}
+\frac{1}{2}c_a\big(\varepsilon_{rr}-\varepsilon_{\varphi\varphi}\big)
+c_b\varepsilon_{r\varphi},
\end{split}
\end{equation}\end{split}\]
kde
(153)¶\[\begin{split}\begin{equation}
\begin{split}
c_a =& \big(c_{11}-c_{12}-2c_{44}\big)\sin^22\varphi, \\
c_b =& \big(c_{11}-c_{12}-2c_{44}\big)\cos2\varphi\sin2\varphi.
\end{split}
\end{equation}\end{split}\]
Dále se budou hodit derivace vztahů (153)
(154)¶\[\begin{split}\begin{equation}
\begin{split}
c_a^\prime =& \big(c_{11}-c_{12}-2c_{44}\big)4\cos2\varphi\sin2\varphi=4c_b \\
c_b^\prime =& \big(c_{11}-c_{12}-2c_{44}\big)\big(2\cos^22\varphi-2\sin^22\varphi\big) \\
=& \big(c_{11}-c_{12}-2c_{44}\big)2\big(1-2\sin^22\varphi\big) \\
=& 2\big(c_{11}-c_{12}-2c_{44}\big)-2c_a.
\end{split}
\end{equation}\end{split}\]
Pro složky divergence tenzoru napětí \(\nabla\cdot\boldsymbol{\tau}\) platí
(155)¶\[\begin{split}\begin{equation}
\begin{split}
\big(\nabla\cdot\boldsymbol{\tau}\big)_r
=& \partial_r\tau_{rr}+\frac{1}{r}\tau_{rr}
-\frac{1}{r}\tau_{\varphi\varphi}+\frac{1}{r}\partial_\varphi\tau_{r\varphi} \\
=& \partial_r\left(
c_{rr}^{rr}\varepsilon_{rr}
+c_{rr}^{r\varphi}\varepsilon_{r\varphi}
+c_{rr}^{\varphi\varphi}\varepsilon_{\varphi\varphi}
\right) \\
& +\frac{1}{r}\left(
c_{rr}^{rr}\varepsilon_{rr}
+c_{rr}^{r\varphi}\varepsilon_{r\varphi}
+c_{rr}^{\varphi\varphi}\varepsilon_{\varphi\varphi}
\right) \\
& -\frac{1}{r}\left(
c_{\varphi\varphi}^{rr}\varepsilon_{rr}
+c_{\varphi\varphi}^{r\varphi}\varepsilon_{r\varphi}
+c_{\varphi\varphi}^{\varphi\varphi}\varepsilon_{\varphi\varphi}
\right) \\
& +\frac{1}{r}\partial_\varphi\left(
c_{r\varphi}^{rr}\varepsilon_{rr}
+c_{r\varphi}^{r\varphi}\varepsilon_{r\varphi}
+c_{r\varphi}^{\varphi\varphi}\varepsilon_{\varphi\varphi}
\right)\\
=& \frac{1}{r}\left(
c_{rr}^{rr}-c_{\varphi\varphi}^{rr}+c_{r\varphi}^{rr\prime}
\right)\partial_ru_r \\
& +\frac{1}{2r}\left(
c_{rr}^{r\varphi}-c_{\varphi\varphi}^{r\varphi}+c_{r\varphi}^{r\varphi\prime}
\right)\left(
\frac{1}{r}\partial_\varphi u_r-\frac{1}{r}u_\varphi+\partial_ru_\varphi
\right) \\
& +\frac{1}{r^2}\left(
c_{rr}^{\varphi\varphi}-c_{\varphi\varphi}^{\varphi\varphi}
+c_{r\varphi}^{\varphi\varphi\prime}
\right)\left(
\partial_\varphi u_\varphi+u_r
\right) \\
& +c_{rr}^{rr}\partial_{rr}u_r \\
& +\frac{1}{2}c_{rr}^{r\varphi}\left(
-\frac{1}{r^2}\left(
\partial_\varphi u_r-u_\varphi
\right)
+\frac{1}{r}\partial_r\left(
\partial_\varphi u_r-u_\varphi
\right)+\partial_{rr}u_\varphi
\right) \\
& +\frac{1}{r}c_{rr}^{\varphi\varphi}\left(
-\frac{1}{r^2}\left(
\partial_\varphi u_\varphi+u_r
\right)
+\partial_r\left(
\partial_\varphi u_\varphi+u_r
\right)
\right) \\
& +\frac{1}{r}c_{r\varphi}^{rr}\partial_{\varphi r}u_r \\
& +\frac{1}{2r}c_{r\varphi}^{r\varphi}\left(
\frac{1}{r}\partial_\varphi\left(
\partial_\varphi u_r
-u_\varphi
\right)
+\partial_{\varphi r}u_\varphi
\right) \\
& +\frac{1}{r^2}c_{r\varphi}^{\varphi\varphi}
\partial_\varphi\left(
\partial_{\varphi}u_\varphi+u_r
\right)
\end{split}
\end{equation}\end{split}\]
(156)¶\[\begin{split}\begin{equation}
\begin{split}
\big(\nabla\cdot\boldsymbol{\tau}\big)_\varphi
=& \partial_r\tau_{r\varphi}+\frac{2}{r}\tau_{r\varphi}
+\frac{1}{r}\partial_\varphi\tau_{\varphi\varphi} \\
=& \partial_r\left(
c_{r\varphi}^{rr}\varepsilon_{rr}
+c_{r\varphi}^{r\varphi}\varepsilon_{r\varphi}
+c_{r\varphi}^{\varphi\varphi}\varepsilon_{\varphi\varphi}
\right) \\
& +\frac{2}{r}\left(
c_{r\varphi}^{rr}\varepsilon_{rr}
+c_{r\varphi}^{r\varphi}\varepsilon_{r\varphi}
+c_{r\varphi}^{\varphi\varphi}\varepsilon_{\varphi\varphi}
\right) \\
& +\frac{1}{r}\partial_\varphi\left(
c_{\varphi\varphi}^{rr}\varepsilon_{rr}
+c_{\varphi\varphi}^{r\varphi}\varepsilon_{r\varphi}
+c_{\varphi\varphi}^{\varphi\varphi}\varepsilon_{\varphi\varphi}
\right) \\
=& \frac{1}{r}\left(
2c_{r\varphi}^{rr}+c_{\varphi\varphi}^{rr\prime}
\right)\partial_ru_r \\
& +\frac{1}{2r}\left(
2c_{r\varphi}^{r\varphi}+c_{\varphi\varphi}^{r\varphi\prime}
\right)\left(
\frac{1}{r}\partial_\varphi u_r-\frac{1}{r}u_\varphi+\partial_ru_\varphi
\right) \\
& +\frac{1}{r^2}\left(
2c_{r\varphi}^{\varphi\varphi}
+c_{\varphi\varphi}^{\varphi\varphi\prime}
\right)\left(
\partial_\varphi u_\varphi+u_r
\right) \\
& +c_{r\varphi}^{rr}\partial_{rr}u_r \\
& +\frac{1}{2}c_{r\varphi}^{r\varphi}\left(
-\frac{1}{r^2}\left(
\partial_\varphi u_r-u_\varphi
\right)
+\frac{1}{r}\partial_r\left(
\partial_\varphi u_r-u_\varphi
\right)+\partial_{rr}u_\varphi
\right) \\
& +\frac{1}{r}c_{r\varphi}^{\varphi\varphi}\left(
-\frac{1}{r^2}\left(
\partial_\varphi u_\varphi+u_r
\right)
+\partial_r\left(
\partial_\varphi u_\varphi+u_r
\right)
\right) \\
& +\frac{1}{r}c_{\varphi\varphi}^{rr}\partial_{\varphi r}u_r \\
& +\frac{1}{2r}c_{\varphi\varphi}^{r\varphi}\left(
\frac{1}{r}\partial_\varphi\left(
\partial_\varphi u_r
-u_\varphi
\right)
+\partial_{\varphi r}u_\varphi
\right) \\
& +\frac{1}{r^2}c_{\varphi\varphi}^{\varphi\varphi}
\partial_\varphi\left(
\partial_{\varphi}u_\varphi+u_r
\right)
\end{split}
\end{equation}\end{split}\]
V dalším se budou potřebovat následující pomocné výrazy. Jestliže \(f(r,\varphi)\) je skalární funkce, pak platí
(157)¶\[\nabla^2f=\partial_{rr}f+\frac{1}{r}\partial_rf+\frac{1}{r^2}\partial_{\varphi\varphi}f\]
(158)¶\[\begin{split}\begin{equation}
\begin{split}
\nabla^2\left(
\partial_rf
\right)
=& \frac{1}{r}\partial_{rr}f
+\partial_{rrr}f
+\frac{1}{r^2}\partial_{\varphi\varphi r}f \\
\partial_r\left(
\nabla^2f
\right)
=& -\frac{1}{r^2}\partial_rf
+\partial_{rrr}f
-\frac{2}{r^3}\partial_{\varphi\varphi}f
+\frac{1}{r^2}\partial_{\varphi\varphi r}f
+\frac{1}{r}\partial_{rr}f \\
=& \nabla^2\left(
\partial_rf
\right)
-\frac{1}{r^2}\partial_rf
-\frac{2}{r^3}\partial_{\varphi\varphi}f
\end{split}
\end{equation}\end{split}\]
(159)¶\[\begin{split}\begin{equation}
\begin{split}
\nabla^2\left(
\frac{1}{r}\partial_\varphi f
\right)
=& \frac{1}{r}\partial_r\left(
\frac{1}{r}\partial_\varphi f
\right)
+\partial_{rr}\left(
\frac{1}{r}\partial_\varphi f
\right)
+\frac{1}{r^2}\partial_{\varphi\varphi}\left(
\frac{1}{r}\partial_\varphi f
\right) \\
=& -\frac{1}{r^3}\partial_\varphi f
+\frac{1}{r^2}\partial_{\varphi r}f
+\frac{2}{r^3}\partial_\varphi f
-\frac{1}{r^2}\partial_{\varphi r}f \\
& -\frac{1}{r^2}\partial_{\varphi r}f
+\frac{1}{r}\partial_{\varphi rr}f
+\frac{1}{r^3}\partial_{\varphi\varphi\varphi}f \\
=& \frac{1}{r^3}\partial_\varphi f
-\frac{1}{r^2}\partial_{\varphi r}f
+\frac{1}{r}\partial_{\varphi rr}f
+\frac{1}{r^3}\partial_{\varphi\varphi\varphi}f \\
\frac{1}{r}\partial_\varphi\left(
\nabla^2f
\right)
=& \frac{1}{r^2}\partial_{r\varphi}f
+\frac{1}{r}\partial_{rr\varphi}
+\frac{1}{r^3}\partial_{\varphi\varphi\varphi}f \\
=& \nabla^2\left(
\frac{1}{r}\partial_\varphi f
\right)
-\frac{1}{r^3}\partial_\varphi f
+\frac{2}{r^2}\partial_{\varphi r}f
\end{split}
\end{equation}\end{split}\]
(160)¶\[\begin{split}\begin{equation}
\begin{split}
\nabla^2\left(
\frac{1}{r}f
\right)
=& \frac{1}{r}\partial_r\left(
\frac{1}{r}f
\right)
+\partial_{rr}\left(
\frac{1}{r}f
\right)
+\frac{1}{r^2}\partial_{\varphi\varphi}\left(
\frac{1}{r}f
\right) \\
=& -\frac{1}{r^3}f
+\frac{1}{r^2}\partial_rf
+\frac{2}{r^3}f
-\frac{1}{r^2}\partial_rf \\
& -\frac{1}{r^2}\partial_rf
+\frac{1}{r}\partial_{rr}f
+\frac{1}{r^3}\partial_{\varphi\varphi}f \\
=& \frac{1}{r^3}f
-\frac{1}{r^2}\partial_rf
+\frac{1}{r}\partial_{rr}f
+\frac{1}{r^3}\partial_{\varphi\varphi}f \\
\frac{1}{r}\partial_\varphi\left(
\nabla^2f
\right)
=& \frac{1}{r^2}\partial_rf
+\frac{1}{r}\partial_{rr}f
+\frac{1}{r^3}\partial_{\varphi\varphi}f \\
=& \nabla^2\left(
\frac{1}{r}f
\right)
+\frac{2}{r^2}\partial_rf-\frac{1}{r^3}f
\end{split}
\end{equation}\end{split}\]
Pomocí vždy druhého vztahu v (158)-(160) pro složky divergence Laplaciánu tenzoru napětí \(\nabla\cdot\nabla^2\boldsymbol\tau\) pak platí
(161)¶\[\begin{split}\begin{equation}
\begin{split}
\left(
\nabla\cdot\nabla^2\boldsymbol{\tau}
\right)_r =&
\partial_r\nabla^2\tau_{rr}+\frac{1}{r}\nabla^2\tau_{rr}
-\frac{1}{r}\nabla^2\tau_{\varphi\varphi}+\frac{1}{r}\partial_\varphi\nabla^2\tau_{r\varphi} \\
=& \nabla^2\left(
\partial_r\tau_{rr}
+\frac{1}{r}\tau_{rr}
-\frac{1}{r}\tau_{\varphi\varphi}
+\frac{1}{r}\partial_\varphi\tau_{r\varphi}
\right) \\
& -\frac{1}{r^2}\partial_r\tau_{rr}
-\frac{2}{r^3}\partial_{\varphi\varphi}\tau_{rr}
+\frac{2}{r^2}\partial_r\tau_{rr}
-\frac{1}{r^3}\tau_{rr} \\
& -\frac{2}{r^2}\partial_r\tau_{\varphi\varphi}
+\frac{1}{r^3}\tau_{\varphi\varphi}
+\frac{2}{r^2}\partial_{\varphi r}\tau_{r\varphi}
-\frac{1}{r^3}\partial_\varphi\tau_{r\varphi} \\
=& \nabla^2\left(
\partial_r\tau_{rr}
+\frac{1}{r}\tau_{rr}
-\frac{1}{r}\tau_{\varphi\varphi}
+\frac{1}{r}\partial_\varphi\tau_{r\varphi}
\right) \\
& -\frac{1}{r^2}\left(
\partial_r\tau_{rr}
+\frac{1}{r}\tau_{rr}
-\frac{1}{r}\tau_{\varphi\varphi}
+\frac{1}{r}\partial_\varphi\tau_{r\varphi}
\right) \\
& +\frac{2}{r^2}\partial_r\left(
\tau_{rr}
-\tau_{\varphi\varphi}
\right)
+\frac{2}{r^2}\partial_\varphi\left(
\partial_r\tau_{r\varphi}
-\frac{1}{r}\partial_\varphi\tau_{rr}
\right)
\end{split}
\end{equation}\end{split}\]
(162)¶\[\begin{split}\begin{equation}
\begin{split}
\left(
\nabla\cdot\nabla^2\boldsymbol{\tau}
\right)_\varphi
=& \partial_r\nabla^2\tau_{r\varphi}
+\frac{2}{r}\nabla^2\tau_{r\varphi}
+\frac{1}{r}\partial_\varphi\nabla^2\tau_{\varphi\varphi} \\
=& \nabla^2\left(
\partial_r\tau_{r\varphi}
+\frac{2}{r}\tau_{r\varphi}
+\frac{1}{r}\partial_\varphi\tau_{\varphi\varphi}
\right) \\
& -\frac{1}{r^2}\partial_r\tau_{r\varphi}
-\frac{2}{r^3}\partial_{\varphi\varphi}\tau_{r\varphi}
+\frac{4}{r^2}\partial_r\tau_{r\varphi} \\
& -\frac{2}{r^3}\tau_{r\varphi}
+\frac{2}{r^2}\partial_{\varphi r}\tau_{\varphi\varphi}
-\frac{1}{r^3}\partial_\varphi\tau_{\varphi\varphi} \\
=& \nabla^2\left(
\partial_r\tau_{r\varphi}
+\frac{2}{r}\tau_{r\varphi}
+\frac{1}{r}\partial_\varphi\tau_{\varphi\varphi}
\right) \\
& -\frac{1}{r^2}\left(
\partial_r\tau_{r\varphi}
+\frac{2}{r}\tau_{r\varphi}
+\frac{1}{r}\partial_\varphi\tau_{\varphi\varphi}
\right) \\
& -\frac{2}{r^2}\partial_\varphi\left(
\frac{1}{r}\partial_\varphi\tau_{r\varphi}
+\partial_r\tau_{\varphi\varphi}
\right)
-\frac{2}{r^3}\tau_{r\varphi}
\end{split}
\end{equation}\end{split}\]
Označením vztahů (155) a (156) na \(s_r\) a \(s_\varphi\) a s pomocí vztahů (148)-(150) a (1) se rovnice rovnováhy mohou napsat ve tvaru
(163)¶\[\begin{split}\begin{equation}
\begin{split}
0 =& s_r-c\left(
\nabla^2s_r
-\frac{1}{r^2}s_r
\right) \\
& -\frac{2}{r^2}c\partial_r\left(
\left(
c_{rr}^{rr}-c_{\varphi\varphi}^{rr}
\right)\left[
\partial_ru_r
-\frac{1}{r}\left(
\partial_\varphi u_\varphi+u_r
\right)
\right]
+c_{rr}^{r\varphi}\left[
\frac{1}{r}\left(
\partial_\varphi u_r-u_\varphi
\right)+\partial_ru_\varphi
\right]
\right) \\
& -\frac{2}{r^2}c\partial_{\varphi r}\left(
c_{r\varphi}^{rr}\left[
\partial_ru_r
-\frac{1}{r}\left(
\partial_\varphi u_\varphi+u_r
\right)
\right]
+c_{r\varphi}^{r\varphi}
\frac{1}{2}\left[
\frac{1}{r}\left(
\partial_\varphi u_r-u_\varphi
\right)+\partial_ru_\varphi
\right]
\right) \\
& +\frac{2}{r^3}c\partial_{\varphi\varphi}\left(
c_{rr}^{rr}\partial_ru_r
+c_{rr}^{r\varphi}
\frac{1}{2}\left[
\frac{1}{r}\left(
\partial_\varphi u_r-u_\varphi
\right)+\partial_ru_\varphi
\right]
-c_{rr}^{\varphi\varphi}\frac{1}{r}
\left(
\partial_\varphi u_\varphi+u_r
\right)
\right)
\end{split}
\end{equation}\end{split}\]
(164)¶\[\begin{split}\begin{equation}
\begin{split}
0 =& s_\varphi-c\left(
\nabla^2s_\varphi
-\frac{1}{r^2}s_\varphi
\right) \\
& +\frac{2}{r^3}c\left(
c_{r\varphi}^{rr}\left[
\partial_ru_r
-\frac{1}{r}\left(
\partial_\varphi u_\varphi+u_r
\right)
\right]
+c_{r\varphi}^{r\varphi}\left[
\frac{1}{r}\left(
\partial_\varphi u_r-u_\varphi
\right)+\partial_ru_\varphi
\right]
\right) \\
& +\frac{2}{r^3}c\partial_{\varphi\varphi}\left(
c_{r\varphi}^{rr}\left[
\partial_ru_r
-\frac{1}{r}\left(
\partial_\varphi u_\varphi+u_r
\right)
\right]
+c_{r\varphi}^{r\varphi}
\frac{1}{2}\left[
\frac{1}{r}\left(
\partial_\varphi u_r-u_\varphi
\right)+\partial_ru_\varphi
\right]
\right) \\
& +\frac{2}{r^2}c\partial_{\varphi r}\left(
c_{\varphi\varphi}^{rr}\partial_ru_r
+c_{\varphi\varphi}^{r\varphi}
\frac{1}{2}\left[
\frac{1}{r}\left(
\partial_\varphi u_r-u_\varphi
\right)+\partial_ru_\varphi
\right]
-c_{\varphi\varphi}^{\varphi\varphi}\frac{1}{r}
\left(
\partial_\varphi u_\varphi+u_r
\right)
\right)
\end{split}
\end{equation}\end{split}\]