# 客观应力率

• 柯西应力张量${\displaystyle {\boldsymbol {\sigma }}}$的特鲁斯德尔（Truesdell）应力率
${\displaystyle {\overset {\circ }{\boldsymbol {\sigma }}}={\dot {\boldsymbol {\sigma }}}-{\boldsymbol {l}}\cdot {\boldsymbol {\sigma }}-{\boldsymbol {\sigma }}\cdot {\boldsymbol {l}}^{T}+{\text{tr}}({\boldsymbol {l}})~{\boldsymbol {\sigma }}}$

（其中${\displaystyle {\boldsymbol {l}}}$为速度梯度张量）

• 柯西应力张量${\displaystyle {\boldsymbol {\sigma }}}$的格林－纳厄迪（Green-Naghdi）应力率
${\displaystyle {\overset {\square }{\boldsymbol {\sigma }}}={\dot {\boldsymbol {\sigma }}}+{\boldsymbol {\sigma }}\cdot {\boldsymbol {\Omega }}-{\boldsymbol {\Omega }}\cdot {\boldsymbol {\sigma }}}$

（其中${\displaystyle {\boldsymbol {\Omega }}={\dot {\boldsymbol {R}}}\cdot {\boldsymbol {R}}^{T}}$${\displaystyle {\boldsymbol {R}}}$为转动张量）

• 柯西应力张量${\displaystyle {\boldsymbol {\sigma }}}$的耀曼（Jaumann）应力率
${\displaystyle {\overset {\triangle }{\boldsymbol {\sigma }}}={\dot {\boldsymbol {\sigma }}}+{\boldsymbol {\sigma }}\cdot {\boldsymbol {w}}-{\boldsymbol {w}}\cdot {\boldsymbol {\sigma }}}$

（其中${\displaystyle {\boldsymbol {w}}}$为自旋张量，即速度梯度张量的反对称部分）

## 参考文献

1. ^ M.E. Gurtin, E. Fried and L. Anand (2010). "The mechanics and thermodynamics of continua". Cambridge University Press