偏差第3不変量

$ J_3^{\bm T}=I_3^{\bm T}-\frac{\mathrm{tr}\bm I-2}{\mathrm{tr}\bm I}I_1^{\bm T}I_2^{\bm T}+\frac13\frac{\mathrm{tr}\bm I-2}{\mathrm{tr}\bm I}\frac{\mathrm{tr}\bm I-1}{\mathrm{tr}\bm I}\left(I_1^{\bm T}\right)^3

導出

$ J_3^{\bm T}:=I_3^{{\cal\pmb D}:\bm T}

$ =-\frac13\left(0^3-\mathrm{tr}\left(({\cal\pmb D}:\bm T)^3\right)\right)+0\cdot J_2^{\bm T}

$ =\frac13\mathrm{tr}(({\cal\pmb D}:\bm T)^3)

$ = \frac13{\rm tr}(\bm T^3)-\left(\frac{{\rm tr}\bm T}{{\rm tr}\bm I}\right){\rm tr}(\bm T^2)+\left(\frac{{\rm tr}\bm T}{{\rm tr}\bm I}\right)^2{\rm tr}\bm T-\frac13\left(\frac{{\rm tr}\bm T}{{\rm tr}\bm I}\right)^3{\rm tr}\bm I

$ \because({\cal\pmb D}:\bm T)^3=\bm T^3-3\left(\frac{{\rm tr}\bm T}{{\rm tr}\bm I}\right)\bm T^2+3\left(\frac{{\rm tr}\bm T}{{\rm tr}\bm I}\right)^2\bm T-\left(\frac{{\rm tr}\bm T}{{\rm tr}\bm I}\right)^3\bm I

$ = \frac13{\rm tr}(\bm T^3)-\left(\frac{{\rm tr}\bm T}{{\rm tr}\bm I}\right){\rm tr}(\bm T^2)+\frac23\frac{({\rm tr}\bm T)^3}{({\rm tr}\bm I)^2}

$ = \frac13\left(I_1^{\bm T}\right)^3+I_3^{\bm T}-I_1^{\bm T}I_2^{\bm T}-\left(\frac{I_1^{\bm T}}{{\rm tr}\bm I}\right)\left(\left(I_1^{\bm T}\right)^2-2I_2^{\bm T}\right)+\frac23\frac{\left(I_1^{\bm T}\right)^3}{({\rm tr}\bm I)^2}

$ \because\mathrm{tr}\left({\bm T}^2\right)=\left(I_1^{\bm T}\right)^2-2I_2^{\bm T}

$ \because \mathrm{tr}\left(\bm T^3\right)=\left(I_1^{\bm T}\right)^3+3I_3^{\bm T}-3I_1^{\bm T}I_2^{\bm T}

$ = I_3^{\bm T}-\left(\frac{\left(I_1^{\bm T}\right)^3}{{\rm tr}\bm I}\right)-I_1^{\bm T}I_2^{\bm T}+2\left(\frac{I_1^{\bm T}}{{\rm tr}\bm I}\right)I_2^{\bm T}+\frac13\left(1+\frac{2}{(\mathrm{tr}\bm I)^2}\right)\left(I_1^{\bm T}\right)^3

$ = I_3^{\bm T}-I_1^{\bm T}I_2^{\bm T}+2\left(\frac{I_1^{\bm T}}{{\rm tr}\bm I}\right)I_2^{\bm T}+\frac13\left(1-\frac3{\mathrm{tr}\bm I}+\frac2{(\mathrm{tr}\bm I)^2}\right)\left(I_1^{\bm T}\right)^3

$ = I_3^{\bm T}+\left(\frac{2}{{\rm tr}\bm I}-1\right)I_1^{\bm T}I_2^{\bm T}+\frac13\left(1-\frac3{\mathrm{tr}\bm I}+\frac2{(\mathrm{tr}\bm I)^2}\right)\left(I_1^{\bm T}\right)^3

$ = I_3^{\bm T}+\frac{2-\mathrm{tr}\bm I}{\mathrm{tr}\bm I}I_1^{\bm T}I_2^{\bm T}+\frac13\frac{\left(\mathrm{tr}\bm I\right)^2-3\mathrm{tr}\bm I+2}{\left(\mathrm{tr}\bm I\right)^2}\left(I_1^{\bm T}\right)^3

$ = I_3^{\bm T}+\frac{2-\mathrm{tr}\bm I}{\mathrm{tr}\bm I}I_1^{\bm T}I_2^{\bm T}+\frac13\frac{\left(\mathrm{tr}\bm I-2\right)\left(\mathrm{tr}\bm I-1\right)}{\left(\mathrm{tr}\bm I\right)^2}\left(I_1^{\bm T}\right)^3

$ \underline{= I_3^{\bm T}-\frac{\mathrm{tr}\bm I-2}{\mathrm{tr}\bm I}I_1^{\bm T}I_2^{\bm T}+\frac13\frac{\mathrm{tr}\bm I-2}{\mathrm{tr}\bm I}\frac{\mathrm{tr}\bm I-1}{\mathrm{tr}\bm I}\left(I_1^{\bm T}\right)^3\quad}_\blacksquare

$ J_3^{\bm T}= I_3^{\bm T}-\frac{\mathrm{tr}\bm I-2}{\mathrm{tr}\bm I}I_1^{\bm T}\left(I_2^{\bm T}-\frac13\frac{\mathrm{tr}\bm I-1}{\mathrm{tr}\bm I}\left(I_1^{\bm T}\right)^2\right)

$ = I_3^{\bm T}-\frac{\mathrm{tr}\bm I-2}{\mathrm{tr}\bm I}I_1^{\bm T}\left(J_2^{\bm T}+\frac12\frac{\mathrm{tr}\bm I-1}{\mathrm{tr}\bm I}\left(I_1^{\bm T}\right)^2-\frac13\frac{\mathrm{tr}\bm I-1}{\mathrm{tr}\bm I}\left(I_1^{\bm T}\right)^2\right)

$ = I_3^{\bm T}-\frac{\mathrm{tr}\bm I-2}{\mathrm{tr}\bm I}I_1^{\bm T}\left(J_2^{\bm T}+\frac16\frac{\mathrm{tr}\bm I-1}{\mathrm{tr}\bm I}\left(I_1^{\bm T}\right)^2\right)

$ J_3^{\bm T}= I_3^{\bm T}-\frac{\mathrm{tr}\bm I-2}{\mathrm{tr}\bm I}I_1^{\bm T}\left(I_2^{\bm T}-\frac13\frac{\mathrm{tr}\bm I-1}{\mathrm{tr}\bm I}\left(I_1^{\bm T}\right)^2\right)

$ = I_3^{\bm T}-(I_2^{\bm T}-J_2^{\bm T})I_1^{\bm T}\left(I_2^{\bm T}-\frac13(I_2^{\bm T}-J_2^{\bm T})\right)

$ =I_3^{\bm T}-(I_2^{\bm T}-J_2^{\bm T})I_1^{\bm T}I_2^{\bm T}+\frac13\left(I_2^{\bm T}-J_2^{\bm T}\right)^2

n次元表示

$ =I_3^{\bm T}-\frac13I_1^{\bm T}J_2^{\bm T}-\frac1{27}\left(I_1^{\bm T}\right)^3

$ ({\cal\pmb D}:\bm T)\cdot({\cal\pmb D}:\bm T)=\left(\bm T-\frac1{{\rm tr}\bm I}({\rm tr}\bm T)\bm I\right)\cdot\left(\bm T-\frac1{{\rm tr}\bm I}({\rm tr}\bm T)\bm I\right)

$ = \bm T^2-\frac2{{\rm tr}\bm I}({\rm tr}\bm T)\bm T+\frac1{({\rm tr}\bm I)^2}({\rm tr}\bm T)^2\bm I

$ = \bm T^2-2\left(\frac{{\rm tr}\bm T}{{\rm tr}\bm I}\right)\bm T+\left(\frac{{\rm tr}\bm T}{{\rm tr}\bm I}\right)^2\bm I

$ ({\cal\pmb D}:\bm T)\cdot({\cal\pmb D}:\bm T)\cdot({\cal\pmb D}:\bm T)=\bm T^3-3\left(\frac{{\rm tr}\bm T}{{\rm tr}\bm I}\right)\bm T^2+3\left(\frac{{\rm tr}\bm T}{{\rm tr}\bm I}\right)^2\bm T-\left(\frac{{\rm tr}\bm T}{{\rm tr}\bm I}\right)^3\bm I

$ I_3^{\bm T}=-\frac13\left(\left(I_1^{\bm T}\right)^3-\mathrm{tr}\left(\bm T^3\right)\right)+I_1^{\bm T}I_2^{\bm T}

$ \iff 3I_3^{\bm T}=-\left(I_1^{\bm T}\right)^3+\mathrm{tr}\left(\bm T^3\right)+3I_1^{\bm T}I_2^{\bm T}

$ \iff\mathrm{tr}\left(\bm T^3\right)=\left(I_1^{\bm T}\right)^3+3I_3^{\bm T}-3I_1^{\bm T}I_2^{\bm T}