引入非线性
元件 |
改进的西原
正夫模型[58] |
 |
\( \varepsilon =\left\{\begin{aligned}&\frac{\sigma }{2{E}_{1}}+\frac{\sigma }{2{E}_{2}}\left(1-{{\mathrm{exp}}}\left({-\frac{{E}_{2}}{{\eta }_{1}}t}\right)\right),\sigma < {\sigma }_{\text{s}}\\&\frac{\sigma }{2{E}_{1}}+\frac{\sigma }{2{E}_{2}}\left(1-{{\mathrm{exp}}}\left({-\frac{{E}_{2}}{{\eta }_{1}}t}\right)\right)+\\&\qquad \frac{\sigma -{\sigma }_{\text{s}}}{2{\eta }_{2}}\left(\frac{1}{3}{t}^{3}-\frac{1}{2}\frac{B}{A}{t}^{2}+\frac{C}{A}t\right), \sigma \geqslant {\sigma }_{\text{s}}\end{aligned}\right. \)  |
可描述瞬时蠕变、
稳定蠕变、
加速蠕变 |
非线性蠕变
模型[59] |
 |
\( \varepsilon =\left\{\begin{aligned}&\frac{{\eta }_{1}}{{E}_{2}}\dot{\varepsilon }+\varepsilon -\frac{{E}_{1}+{E}_{2}}{{E}_{1}{E}_{2}}\sigma =0, \sigma < {\sigma }_{\text{s}}\\&\ddot{\varepsilon }+\frac{{E}_{2}\left(\varepsilon \right)}{{\eta }_{1}}\dot{\varepsilon }-\frac{{E}_{2}\left(\varepsilon \right)}{{\eta }_{1}{\eta }_{2}}\left(\sigma -{\sigma }_{\text{s}}\right), \sigma \geqslant {\sigma }_{\text{s}}\end{aligned}\right. \)  |
可分析三阶段蠕变、
稳定性及判据 |
| 河海模型[61] |
 |
\(\begin{aligned}&\varepsilon =\frac{\sigma }{{E}_{1}}+\frac{\sigma }{{E}_{2}}\left(1-{{\mathrm{exp}}}\left({-\frac{{E}_{2}}{{\eta }_{1}}t}\right)\right)+\\&\qquad \frac{\sigma }{{E}_{3}}\left(1-{{\mathrm{exp}}}\left({-\frac{{E}_{3}}{{\eta }_{2}}t}\right)\right)+\frac{\sigma -{\sigma }_{\text{s}}}{{\eta }_{3}}{t}^{n}\end{aligned} \)  |
可描述绿片岩
三阶段蠕变，
但元件多 |
分数阶蠕变
模型[63] |
 |
\( \varepsilon =\left\{\begin{aligned}&\frac{\sigma }{{E}_{1}}+\frac{\sigma }{{\eta }_{1}}\frac{{t}^{\beta }}{\mathrm{\varGamma }\left(\beta +1\right)},\sigma < {\sigma }_{\text{s}}\\&\frac{\sigma }{{E}_{1}}+\frac{\sigma }{{\eta }_{1}}\frac{{t}^{\beta }}{\mathrm{\varGamma }\left(\beta +1\right)}+\\&\qquad \frac{\sigma -{\sigma }_{{\mathrm{s}}}}{{\eta }_{2}}\frac{{t}^{\varGamma }}{\mathrm{\varGamma }\left(\varGamma +1\right)}, \sigma \geqslant {\sigma }_{\text{s}}\end{aligned}\right. \)  |
可描述盐岩
瞬时蠕变、
稳定蠕变 |
非线性分数
阶蠕变模型[64] |
 |
\( \varepsilon =\left\{\begin{aligned}&\frac{\sigma }{E}+\frac{\sigma }{{\eta }_{1}}t+\frac{\sigma }{{\eta }_{2}^{\beta }}\frac{{t}^{\beta }}{\mathrm{\varGamma }\left(\beta +1\right)},\sigma < {\sigma }_{{\mathrm{s}}}\\&\frac{\sigma }{E}+\frac{\sigma }{{\eta }_{1}}t+\frac{\sigma }{{\eta }_{2}^{\beta }}\frac{{t}^{\beta }}{\mathrm{\varGamma }\left(\beta +1\right)}+\\&\qquad \frac{\sigma -{\sigma }_{{\mathrm{s}}}}{{\eta }_{3}}H\left(t-{t}_{{\mathrm{F}}}\right), \sigma \geqslant {\sigma }_{\text{s}}\end{aligned}\right. \)  |
能描述软岩
三阶段蠕变 |
引入损伤力学和
断裂力学等理论 |
变参数蠕变
损伤模型[65] |
 |
\( \varepsilon =\left\{\begin{aligned}&\frac{\sigma }{\left(1-{D}_{t}\right){E}_{1}}+\frac{\sigma }{{\left(1-{D}_{t}\right)\eta }_{1}}+\\&\qquad \frac{\sigma }{\left(1-{D}_{t}\right){E}_{2}}\left(1-{{\mathrm{exp}}}\left({-\frac{{E}_{2}}{{\eta }_{2}}t}\right)\right), \sigma < {\sigma }_{\text{s}}\\&\frac{\sigma }{\left(1-{D}_{t}\right){E}_{1}}+\frac{\sigma }{{\left(1-{D}_{t}\right)\eta }_{1}}+\\&\qquad \frac{\sigma }{\left(1-{D}_{t}\right){E}_{2}}\left(1-{{\mathrm{exp}}}\left({-\frac{{E}_{2}}{{\eta }_{2}}t}\right)\right)+{\varepsilon }^{p}, \sigma \geqslant {\sigma }_{\text{s}}\end{aligned}\right. \)  |
能描述软岩瞬时
蠕变、稳定蠕变及
损伤劣化效应 |
BNMC蠕变
损伤模型[66] |
 |
损伤变量\( D=1-{\left(1-t/{t}_{{\mathrm{R}}}\right)}^{1/\left(\gamma +1\right)} \) 
黏聚力损伤方程\( c\left(t\right)={c}_{0}(1-D) \) 
内摩擦角损伤方程\( \varphi \left(t\right)={\varphi }_{0}(1-D) \)  |
能描述二辉
橄榄岩三阶段蠕变 |
非线性蠕变
损伤模型[67] |
 |
\( \varepsilon =\left\{\begin{aligned}&\frac{\sigma }{{E}_{1}}+\frac{\sigma }{{\eta }_{1}}t{{\mathrm{exp}}}({\lambda })+\frac{\sigma }{{E}_{2}}\left(1-{{\mathrm{exp}}}\left({-\frac{{E}_{2}}{{\eta }_{2}}t}\right)\right),\sigma < {\sigma }_{\text{s}}\\&\frac{\sigma }{{E}_{1}}+\frac{\sigma }{{\eta }_{1}}t{{\mathrm{exp}}}({\lambda })+\frac{\sigma }{{E}_{2}}\left(1-{{\mathrm{exp}}}\left({-\frac{{E}_{2}}{{\eta }_{2}}t}\right)\right)+\\&\qquad \frac{\sigma -{\sigma }_{\text{s}}}{{\eta }_{3}}t,\sigma \geqslant {\sigma }_{\text{s}},\varepsilon \leqslant {\varepsilon }^{*}\\&\frac{\sigma }{{E}_{1}}+\frac{\sigma }{{\eta }_{1}}t{{\mathrm{exp}}}({\lambda })+\frac{\sigma }{{E}_{2}}\left(1-{{\mathrm{exp}}}\left({-\frac{{E}_{2}}{{\eta }_{2}}t}\right)\right)+\\&\qquad \frac{\sigma -{\sigma }_{\text{s}}}{{\eta }_{3}}t{{\mathrm{exp}}}{\left[\beta {(t-{t}^{*})}^{m{D}_{{\mathrm{ini}}}+n}\right]},\sigma \geqslant {\sigma }_{\text{s}},\varepsilon > {\varepsilon }^{*}\end{aligned}\right. \)  |
可描述砂岩
三阶段蠕变 |
非线性蠕变
模型[69] |
 |
损伤变量\( D=1-E(\sigma ,t)/{E}_{0} \) 
非线性蠕变方程\( \varepsilon =\sigma {{\mathrm{exp}}}({\alpha t})/{E}_{0}+B{\sigma }^{(1-r)}{t}^{\beta } \)  |
可描述胶结充填体
三阶段蠕变 |
![高应力强扰动作用下巷道围岩时效变形破坏、结构失稳[3,5]与蠕变型冲击地压[6]](/data/tmp/figure/cc2532ba864a39ee12b173a7dc6f91f2.jpg)
![T-k等面积源型图[32]](/data/tmp/figure/270f4d1965139951d6a8bfec09534cc5.jpg)
