偏差tensorの不変量
$ ({\cal\pmb D}:\bm T):({\cal\pmb D}:\bm T)=\bm T:\bm T-\frac1{\mathrm{tr}\bm I}(\mathrm{tr}\bm T)^2
導出
$ ({\cal\pmb D}:\bm T):({\cal\pmb D}:\bm T)=\bm T:{\cal\pmb D}:{\cal\pmb D}:\bm T
$ \because{\cal\pmb D}^\top={\cal\pmb D}
$ = \bm T:{\cal\pmb D}:\bm T
$ \because{\cal\pmb D}:{\cal\pmb D}={\cal\pmb D}
$ = \bm T:\bm T-\frac1{\mathrm{tr}\bm I}(\mathrm{tr}\bm T)^2
他は主不変量$ J_1^{\bm{T}},J_3^{\bm{T}}と同じなので略
$ J_1^{\bm{T}}=\mathrm{tr}({\cal\pmb D}:\bm T)=0
偏差第2不変量$ J_2^{\bm T}=I_2^{\bm T}-\frac12\frac{\mathrm{tr}\bm I-1}{\mathrm{tr}\bm I}\left(I_1^{\bm T}\right)^2 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 $ J_2^{\bm{\sigma}}=-\frac32{\tau_{oct}}^2
$ \cos3\theta=\frac{3^\frac32}{2}\frac{J_3^{\bm{\sigma}}}{(-J_2^{\bm{\sigma}})^\frac32}
$ = \frac12\frac{J_3^{\bm{\sigma}}}{\left(-\frac13J_2^{\bm{\sigma}}\right)^\frac32}
$ = \frac12\frac{J_3^{\bm{\sigma}}}{\left(\frac12{\tau_{oct}}^2\right)^\frac32}
$ = \frac{2\sqrt2}2\frac{J_3^{\bm{\sigma}}}{{\tau_{oct}}^3}
$ = \frac{J_3^{\bm{\sigma}}}{{\tau_{oct}}^3}\sqrt2