Triaxiality function 的推导

Triaxiality function \(R_v\) 常被用于描述构件在多轴状态下的情形。在各向同性情况下，其表达式为：

\[ R_v = \dfrac{2(1+\nu)}{3} + 3(1-2\nu) \left(\dfrac{\sigma_m}{\sigma_{eq}} \right)^2 \]

以下为该表达式的简单推导。

各向同性损伤应变能释放率

\[ Y=\frac{1}{2}\varepsilon^e:\frac{\partial E_D}{\partial D}:\varepsilon^e \]

由 \(E_D = (1-D)E \Rightarrow \dfrac{\partial E_D}{\partial D}=-E ,\quad \sigma= (1-D)E:\varepsilon\),

\[ -Y=\frac{1}{1-D}\frac{1}{2}\varepsilon^e:\sigma \]

由 \(\varepsilon^e_{ij} = e^e_{ij}+\varepsilon^e_m \delta_{ij} ,\quad \sigma = s_{ij}+\sigma_m \delta_{ij}\)，将应力和应变进行分解

\[ -Y = \frac{1}{(1-D)} \left( e^e_{ij} +\varepsilon^e_{m}\delta_{ij} \right) \left( s_{ij} +\sigma_m\delta_{ij} \right) \]

由偏应力偏应变的关系 \(s_{ij} =2 \mu e^e_{ij} \Rightarrow e^e_{ij} = \dfrac{1+\nu}{E_D}s_{ij}\)、球应力和球应变关系 \(\sigma_{kk} = 3K\varepsilon^e_m \Rightarrow \varepsilon^e_m = \dfrac{1-2\nu}{E_D} \sigma_m\)

\[ \begin{aligned} -Y &= \frac{1}{2 (1-D)^2}\left( \frac{1+\nu}{E}s_{ij} + \frac{1-2\nu}{E}\sigma_m\delta_{ij} \right) \left( s_{ij} +\sigma_m\delta_{ij} \right) \\ &= \frac{1}{2 (1-D)^2} \left( \frac{1+\nu}{E}s_{ij} s_{ij} + \frac{1+\nu}{E} \sigma_ms_{ij} \delta_{ij} + \frac{1-2\nu}{E}\sigma_m s_{ij} \delta_{ij} + \frac{1-2\nu}{E}\sigma^2_m\delta_{ij}\delta_{ij} \right) \\ &= \frac{1}{2 (1-D)^2} \left( \frac{1+\nu}{E}s_{ij} s_{ij} + \frac{3(1-2\nu)}{E}\sigma^2_m \right) \end{aligned} \]

由等效应力 \(\sigma_{eq} = \sqrt{\dfrac{3}{2} s_{ij} s_{ij}} \Rightarrow s_{ij}s_{ij} = \dfrac{2}{3} \sigma^2_{eq}\)

\[ \begin{aligned} -Y &= \frac{\sigma_{eq}^2}{2E(1-D)^2} \left[ \frac{2(1+\nu)}{3} + 3(1-2\nu) \left(\frac{\sigma_m}{\sigma_{eq}} \right)^2 \right] \\ &= \frac{\sigma_{eq}^2}{2E(1-D)^2} R_v \end{aligned} \]

其中\(R_v = \dfrac{2(1+\nu)}{3} + 3(1-2\nu) \left(\dfrac{\sigma_m}{\sigma_{eq}} \right)^2\) 则为 Triaxiality function。

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参考文献：

Zhang W, Cai Y. Continuum damage mechanics and numerical applications[M]. Springer Science & Business Media, 2010. (Sec 3.6）