Improvement of multi-step brittle-plastic approach

doi:10.3785/j.issn.1008-973X.2023.09.002

Journal of ZheJiang University (Engineering Science)

2023, Vol. 57

Issue (9): 1706-1717 DOI: 10.3785/j.issn.1008-973X.2023.09.002

Improvement of multi-step brittle-plastic approach

Jun-chao JIN1,2(

),Lai-hong JING1,2,Feng-wei YANG1,2,Zhi-yu SONG1,2,Peng-yang SHANG3

1. Yellow River Engineering Consulting Limited Company, Zhengzhou 450003, China
2. Key Laboratory of Water Management and Water Security for Yellow River Basin, Ministry of Water Resources (Under Construction), Zhengzhou 450003, China
3. China Water Resources Beifang Investigation, Design and Research Limited Company, Tianjin 300222, China

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Abstract

Aiming at the problem of multi-step brittle-plastic approach in stress-drop calculation, the defects of existing stress-drop calculation method based on the deviator stress dropping, the method based on the constant minor principal stress in the brittle-plastic process, the method based on the plastic potential theory and the method based on the invariant spherical stress were systematically analyzed in the principal stress space, combined with the deformation and the failure characteristics of feature points. Considering the Poisson’s effect in the brittle-plastic deformation and failure process, the method based on the plastic potential theory was improved. The corresponding stress update process was derived and embedded in the program Abaqus through the UMAT subroutine. The original stress-drop calculation method was replaced by the improved method, realizing the numerical simulation of elastic-plastic strain softening process. The calculation of full elastic-plastic deformation and failure process was further realized by introducing the plastic-strengthening algorithm, which is verified by several examples. The excavation simulation of Mine-by tunnel and auxiliary tunnels of a hydropower station diversion tunnel shows that the improved method can reasonably simulate the elastic-plastic deformation and failure phenomenon of surrounding rock.

Key words： strain softening brittle-plastic stress-drop numerical algorithm finite element

Received: 01 November 2022 Published: 16 October 2023

CLC:

TU 45

Fund: 中国博士后科学基金资助项目（2022M721299）；河南省重点研发与推广专项（232102320339）

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	Jun-chao JIN
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	Feng-wei YANG
	Zhi-yu SONG
	Peng-yang SHANG

Cite this article:

Jun-chao JIN,Lai-hong JING,Feng-wei YANG,Zhi-yu SONG,Peng-yang SHANG. Improvement of multi-step brittle-plastic approach. Journal of ZheJiang University (Engineering Science), 2023, 57(9): 1706-1717.

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https://www.zjujournals.com/eng/10.3785/j.issn.1008-973X.2023.09.002 OR https://www.zjujournals.com/eng/Y2023/V57/I9/1706

脆塑性迭代逼近算法的改进

针对应力跌落计算中脆塑性迭代逼近算法存在的问题，在主应力空间，结合特征点变形和破坏特征，通过理论推导，系统分析偏应力等比例跌落方法、最小主应力不变跌落方法、塑性位势跌落方法及应力球量不变跌落方法的缺陷. 考虑脆塑性变形和破坏过程中的泊松效应，提出改进的塑性位势跌落方法；推导应力跌落计算更新过程，编写UMAT子程序将该更新过程嵌入软件Abaqus，实现岩石弹塑性应变软化过程数值求解. 采用所提改进方法替换原有的应力跌落计算方法，实现弹塑性应变软化过程的数值模拟. 引入塑性强化算法，实现岩石弹塑性变形破坏全过程的数值计算，并进行多算例验证. 对不同地质条件的Mine-by试验洞及某水电站引水隧洞辅助洞进行开挖模拟，结果表明所提改进方法能够合理模拟工程中围岩弹塑性变形破坏现象.

关键词： 应变软化, 脆塑性, 应力跌落, 数值算法, 有限元

Fig.1 Schematic of rock failure types

破坏类型	峰值应力	残余应力	屈服函数求解	合理性
单轴拉伸破坏	$\left[ \begin{array}{c} \sigma _1^{\rm{p} }\\ \sigma _2^{\rm{p} }\\ \sigma _3^{\rm{p} }\\ \end{array} \right] = \left[ \begin{array}{c} \sigma _1^{\rm{p} }\\ 0\\ 0\\\end{array} \right]$	$\left[ \begin{gathered} \sigma _1^{\rm{r} } \\ \sigma _2^{\rm{r} } \\ \sigma _3^{\rm{r} } \\\end{gathered} \right] = \left[ \begin{gathered} \dfrac{ {\left( {1+2\beta } \right)\sigma _1^{\rm{p} } } }{3} \\ \dfrac{ {\left( {1 - \beta } \right)\sigma _1^{\rm{p} } } }{3} \\ \dfrac{ {\left( {1 - \beta } \right)\sigma _1^{\rm{p} } } }{3} \\ \end{gathered} \right] $	$\left. {\begin{array}{*{20}{l} }\;\;\;{f_{\rm{p} }^{ {\rm{M} } - {\rm{C} } }\left( { {\sigma ^{\rm{p} } } } \right) = \sigma _1^{\rm{p} } - \dfrac{ {2{c _{\rm{p} } } \cdot \cos {\varphi _{\rm{p} } } } }{ {1 + \sin {\varphi _{\rm{p} } } } } = 0},\\\begin{array}{l}f _{\rm{r} }^{M - C}\left( { {\sigma ^{\rm{r} } } } \right) = \dfrac{ {\left( {1 + 2\beta } \right)\sigma _1^{\rm{p} } } }{3} - \dfrac{ {1 - \sin {\varphi_{\rm{r} } } } }{ {1 + \sin {\varphi_{\rm{r} } } } }\dfrac{ {\left( {1 - \beta } \right)\sigma _1^{\rm{p} } } }{3} - \dfrac{ {2{c _{\rm{r} } } \cdot \cos {\varphi_{\rm{r} } } } }{ {1 + \sin {\varphi_{\rm{r} } } } } = 0.\end{array}\end{array} } \right\}$	错误。除了主拉应力方向，横向应力方向也存在残余应力，与事实不符
单轴压缩破坏	$\left[ \begin{array}{c} \sigma _1^{\rm{p} }\\ \sigma _2^{\rm{p} }\\ \sigma _3^{\rm{p} }\\\end{array} \right] = \left[ \begin{array}{c} 0\\ 0\\ \sigma _3^{\rm{p} }\\ \end{array} \right]$	$\left[ \begin{gathered} \sigma _1^{\rm{r} } \\ \sigma _2^{\rm{r} } \\ \sigma _3^{\rm{r} } \\ \end{gathered} \right] = \left[ \begin{gathered} \dfrac{ {\left( {1 - \beta } \right)\sigma _3^{\rm{p} } } }{3} \\ \dfrac{ {\left( {1 - \beta } \right)\sigma _3^{\rm{p} } } }{3} \\ \dfrac{ {\left( {1+2\beta } \right)\sigma _3^{\rm{p} } } }{3} \\ \end{gathered} \right]$	$\left. {\begin{array}{*{20}{l} }\;\;\;{f_{\rm{p} }^{ {\rm{M} } - {\rm{C} } }\left( { {\sigma ^{\rm{p} } } } \right) = \left( {\sin {\varphi _{\rm{p} } } - 1} \right)\sigma _3^{\rm{p} } - 2{c _{\rm{p} } } \cdot \cos {\varphi _{\rm{p} } } = 0},\\\begin{array}{l}f _{\rm{r} }^{ {\rm{M} } - {\rm{C} } }\left( { {\sigma ^{\rm{r} } } } \right) = \dfrac{ {\left( {1 - \beta } \right)\sigma _3^{\rm{p} } } }{3} - \dfrac{ {1 - \sin {\varphi_{\rm{r} } } } }{ {1 + \sin {\varphi_{\rm{r} } } } }\dfrac{ {\left( {1 + 2\beta } \right)\sigma _3^{\rm{p} } } }{3} - \dfrac{ {2{c _{\rm{r} } } \cdot \cos {\varphi_{\rm{r} } } } }{ {1 + \sin {\varphi_{\rm{r} } } } } = 0.\end{array}\end{array} } \right\}$	错误。除了主压应力方向，横向应力方向也存在残余应力，与事实不符
二向纯剪破坏	$\left[ \begin{array}{c} \sigma _1^{\rm{p} }\\ \sigma _2^{\rm{p} }\\ \sigma _3^{\rm{p} }\\\end{array} \right] = \left[ \begin{array}{c} \sigma _1^{\rm{p} }\\ 0\\ - \sigma _1^{\rm{p} }\\ \end{array} \right]$	$\left[ \begin{array}{c} \sigma _1^{\rm{r} } \\ \sigma _2^{\rm{r} } \\ \sigma _3^{\rm{r} } \\ \end{array} \right] = \left[ \begin{array}{c} \beta \sigma _1^{\rm{p} } \\ 0 \\ - \beta \sigma _1^{\rm{p} } \\\end{array} \right]$	$\left. {\begin{array}{*{20}{l} } {f_{\rm{p} }^{ {\rm{M} } - {\rm{C} } }\left( { {\sigma ^{\rm{p} } } } \right) = \sigma _1^{\rm{p} } - \dfrac{ {1 - \sin {\varphi_{\rm{r} } } }}{ {1+\sin {\varphi_{\rm{r} } } }}\left( { - \sigma _1^{\rm{p} } } \right) - \dfrac{ {2{c _{\rm{p} } } \cdot \cos {\varphi _{\rm{p} } } }}{ {1+\sin {\varphi _{\rm{p} } } }} = 0}, \\ {f _{\rm{r} }^{ {\rm{M} } - {\rm{C} } }\left( { {\sigma ^{\rm{r} } } } \right) = \beta \sigma _1^{\rm{p} } - \dfrac{ {1 - \sin {\varphi_{\rm{r} } } }}{ {1+\sin {\varphi_{\rm{r} } } }}\left( { - \beta \sigma _1^{\rm{p} } } \right) - \dfrac{ {2{c _{\rm{r} } } \cdot \cos {\varphi_{\rm{r} } } }}{ {1+\sin {\varphi_{\rm{r} } } }} = 0}.\end{array} } \right\}$	正确。残余阶段满足二向纯剪的应力状态，与事实相符

Tab.1 Defect of existing calculation method based on deviator stress dropping

破坏类型	峰值应力	残余应力	屈服函数求解	合理性
单轴拉伸破坏	$\left[ \begin{array}{c} \sigma _1^{\rm{p} } \\ \sigma _2^{\rm{p} } \\ \sigma _3^{\rm{p} } \\ \end{array} \right] = \left[ \begin{array}{c} \sigma _1^{\rm{p} } \\ 0 \\ 0 \\ \end{array} \right]$	$\left[ \begin{array}{c} \sigma _1^{\rm{r} } \\ \sigma _2^{\rm{r} } \\ \sigma _3^{\rm{r} } \\ \end{array} \right] = \left[ \begin{array}{c} \sigma _1^{\rm{p} } \\ 0 \\ 0 \\ \end{array} \right]$	$\left. {\begin{array}{*{20}{l} } {f_{\rm{p} }^{ {\rm{M} } - {\rm{C} } }\left( { {\sigma ^{\rm{p} } } } \right) = \sigma _1^{\rm{p} } - \dfrac{ {2{c _{\rm{p} } } \cdot \cos {\varphi _{\rm{p} } } } }{ {1+\sin {\varphi _{\rm{p} } } } } = 0}, \\ {f _{\rm{r} }^{ {\rm{M} } - {\rm{C} } }\left( { {\sigma ^{\rm{r} } } } \right) = \sigma _1^{\rm{p} } - \dfrac{ {2{f _{\rm{r} } } \cdot \cos { {\varphi _{\rm{r} } } } } }{ {1+\sin { {\varphi _{\rm{r} } } } } } = 0}. \end{array} } \right\}$	错误。残余强度面屈服函数无解
单轴压缩破坏	$\left[ \begin{array}{c} \sigma _1^{\rm{p} } \\ \sigma _2^{\rm{p} } \\ \sigma _3^{\rm{p} } \\ \end{array} \right] = \left[ \begin{array}{c} 0 \\ 0 \\ \sigma _3^{\rm{p} } \\ \end{array} \right]$	$\left[ \begin{array}{c} \sigma _1^{\rm{r} } \\ \sigma _2^{\rm{r} } \\ \sigma _3^{\rm{r} } \\ \end{array} \right] = \left[ \begin{array}{c} 0 \\ 0 \\ \sigma _3^{\rm{r} } \\ \end{array} \right]$	$\left. {\begin{array}{*{20}{l} } {f_{\rm{p} }^{ {\rm{M} } - {\rm{C} } }\left( { {\sigma ^{\rm{p} } } } \right) = - \dfrac{ {1 - \sin { {\varphi _{\rm{r} } } } } }{ {1+\sin { {\varphi _{\rm{r} } } } } }\sigma _3^{\rm{p} } - \dfrac{ {2{c _{\rm{p} } } \cdot \cos {\varphi _{\rm{p} } } } }{ {1+\sin {\varphi _{\rm{p} } } } } = 0}, \\ {f _{\rm{r} }^{ {\rm{M} } - {\rm{C} } }\left( { {\sigma ^{\rm{r} } } } \right) = - \dfrac{ {1 - \sin { {\varphi _{\rm{r} } } } } }{ {1+\sin { {\varphi _{\rm{r} } } } } }\sigma _3^{\rm{r} } - \dfrac{ {2{f _{\rm{r} } } \cdot \cos { {\varphi _{\rm{r} } } } } }{ {1+\sin { {\varphi _{\rm{r} } } } } } = 0}. \end{array} } \right\}$	正确。残余阶段满足单轴压缩的应力状态，与事实相符
二向纯剪破坏	$\left[ \begin{array}{c} \sigma _1^{\rm{p} } \\ \sigma _2^{\rm{p} } \\ \sigma _3^{\rm{p} } \\\end{array} \right] = \left[ \begin{array}{c} \sigma _1^{\rm{p} } \\ 0 \\ - \sigma _1^{\rm{p} } \\\end{array} \right]$	$\left[ \begin{array}{c} \sigma _1^{\rm{r} } \\ \sigma _2^{\rm{r} } \\ \sigma _3^{\rm{r} } \\ \end{array} \right] = \left[ \begin{array}{c} \sigma _1^{\rm{p} } \\ 0 \\ \sigma _3^{\rm{r} } \\ \end{array} \right]$	$\left. {\begin{array}{*{20}{l} } {f_{\rm{p} }^{ {\rm{M} } - {\rm{C} } }\left( { {\sigma ^{\rm{p} } } } \right) = \sigma _1^{\rm{p} }+\dfrac{ {1 - \sin {\varphi _{\rm{p} } } } }{ {1+\sin {\varphi _{\rm{p} } } } }\sigma _1^{\rm{p} } - \dfrac{ {2{c _{\rm{p} } } \cdot \cos {\varphi _{\rm{p} } } } }{ {1+\sin {\varphi _{\rm{p} } } } } = 0}, \\ {f _{\rm{r} }^{ {\rm{M} } - {\rm{C} } }\left( { {\sigma ^{\rm{r} } } } \right) = \sigma _1^{\rm{p} } - \dfrac{ {1 - \sin { {\varphi _{\rm{r} } } } } }{ {1+\sin { {\varphi _{\rm{r} } } } } }\sigma _3^{\rm{r} } - \dfrac{ {2{f _{\rm{r} } } \cdot \cos { {\varphi _{\rm{r} } } } } }{ {1+\sin { {\varphi _{\rm{r} } } } } } = 0}.\end{array} } \right\}$	错误。残余阶段不满足二向纯剪的应力状态，与事实不符

Tab.2 Defects of existing calculation method based on constant minor principal stress in brittle-plastic process

破坏类型	峰值应力	应力跌落过程的主应力增量	合理性
单轴拉伸破坏	$\left[ \begin{array}{c} \sigma _1^{\rm{p} } \\ \sigma _2^{\rm{p} } \\ \sigma _3^{\rm{p} } \\ \end{array} \right] = \left[ \begin{array}{c} \sigma _1^{\rm{p} } \\ 0 \\ 0 \\ \end{array} \right]$	$ \;\;\;\; \left[ \begin{array}{c} \Delta {\sigma _1} \\ \Delta {\sigma _2} \\ \Delta {\sigma _3} \\ \end{array} \right] = \left[ {\begin{array}{*{20}{c}} {\lambda +2G}&\lambda &\lambda \\ \lambda &{\lambda +2G}&\lambda \\ \lambda &\lambda &{\lambda +2G} \end{array}} \right] \left[ \begin{array}{c} - \Delta \varepsilon _1^{\rm{p}}\\0\\0\end{array} \right] = \left[ \begin{array}{c}- \left( {\lambda +2G} \right)\Delta \varepsilon _1^{\rm{p}}\\ - \lambda \Delta \varepsilon _1^{\rm{p}}\\ - \lambda \Delta \varepsilon _1^{\rm{p}}\end{array} \right]$	错误。除了主拉伸方向应力发生改变之外，横向应力也发生改变，与事实不符
单轴压缩破坏	$\left[ \begin{array}{c} \sigma _1^{\rm{p} } \\ \sigma _2^{\rm{p} } \\ \sigma _3^{\rm{p} } \\ \end{array} \right] = \left[ \begin{array}{c} 0 \\ 0 \\ \sigma _3^{\rm{p} } \\\end{array} \right]$	$ \;\;\;\;\left[ \begin{array}{c} \Delta {\sigma _1} \\ \Delta {\sigma _2} \\ \Delta {\sigma _3} \\ \end{array} \right] = \left[ {\begin{array}{*{20}{c}} {\lambda +2G}&\lambda &\lambda \\ \lambda &{\lambda +2G}&\lambda \\ \lambda &\lambda &{\lambda +2G} \end{array}} \right]\left[ \begin{array}{c} \quad 0 \\\quad 0 \\ - \Delta \varepsilon _3^{\rm{p}} \\ \end{array} \right] = \left[ \begin{array}{c}\quad - \lambda \Delta \varepsilon _3^{\rm{p}} \\ \quad - \lambda \Delta \varepsilon _3^{\rm{p}} \\ - \left( {\lambda +2G} \right)\Delta \varepsilon _3^{\rm{p}} \\ \end{array} \right] $	错误。除了主压缩方向应力发生改变之外，横向应力也发生改变，与事实不符
二向纯剪破坏	$\left[ \begin{array}{c} \sigma _1^{\rm{p} } \\ \sigma _2^{\rm{p} } \\ \sigma _3^{\rm{p} } \\\end{array} \right] = \left[ \begin{array}{c} \sigma _1^{\rm{p} } \\ 0 \\ - \sigma _1^{\rm{p} } \\\end{array} \right]$	$ \left[ \begin{array}{c} \Delta {\sigma _1} \\ \Delta {\sigma _2} \\ \Delta {\sigma _3} \\ \end{array} \right] = \left[ {\begin{array}{*{20}{c}} {\lambda +2G}&\lambda &\lambda \\ \lambda &{\lambda +2G}&\lambda \\ \lambda &\lambda &{\lambda +2G} \end{array}} \right]\left[ \begin{array}{c} \Delta \varepsilon _1^{\rm{p}} \\ 0 \\ - \Delta \varepsilon _1^{\rm{p}} \\ \end{array} \right] = \left[ \begin{array}{c} 2G\Delta \varepsilon _1^{\rm{p}} \\ 0 \\ - 2G\Delta \varepsilon _1^{\rm{p}} \\ \end{array} \right] $	正确。只有1和3方向应力发生变化，与事实相符

Tab.3 Defect of existing calculation method based on plastic potential theory

破坏类型	峰值应力	残余应力	屈服函数求解	合理性
单轴拉伸破坏	$\left[\begin{array}{c} \sigma _1^{\rm{p} }\\ \sigma _2^{\rm{p} }\\ \sigma _3^{\rm{p} }\\\end{array} \right] = \left[ \begin{array}{c} \sigma _1^{\rm{p} } \\ 0 \\ 0 \\ \end{array} \right]$	$ \left. {\begin{array}{*{20}{c}} {\dfrac{{\sigma _1^{\rm{r}}}}{2} = \dfrac{{\sigma _1^{\rm{p}}}}{2}}, \\ {\dfrac{{\sigma _1^{\rm{r}}}}{{\sigma _1^{\rm{r}}}} = \dfrac{{\sigma _1^{\rm{p}}}}{{\sigma _1^{\rm{p}}}}} .\end{array}} \right\} $	$ \left. {\begin{array}{*{20}{l}} {f_{\rm{p}}^{{\rm{M}} - {\rm{C}}}\left( {{\sigma^{\rm{p}}}} \right) = \sigma _1^{\rm{p}}- \dfrac{{2{c _{\rm{p}}} \cdot \cos {\varphi ^{\rm{p}}}}}{{1+\sin {\varphi ^{\rm{p}}}}} = 0}, \\ {f_{\rm{r}}^{{\rm{M}} - {\rm{C}}}\left( {{\sigma ^{\rm{r}}}} \right) = \sigma _1^{\rm{p}}- \dfrac{{2{c _{\rm{r}}} \cdot \cos {{\varphi _{\rm{r}}}}}}{{1+\sin {{\varphi _{\rm{r}}}}}} = 0}. \end{array}} \right\} $	错误。残余强度面屈服函数无解
单轴压缩破坏	$\left[ \begin{array}{c} \sigma _1^{\rm{p} }\\ \sigma _2^{\rm{p} }\\ \sigma _3^{\rm{p} }\\\end{array} \right] = \left[ \begin{array}{c} 0 \\ 0\\ \sigma _3^{\rm{p} }\\\end{array} \right]$	$ \left. {\begin{array}{*{20}{c}} {\dfrac{{ - \sigma _3^{\rm{r}}}}{2} = \dfrac{{ - \sigma _3^{\rm{p}}}}{2}}, \\ {\dfrac{0}{{ - \sigma _3^{\rm{r}}}} = \dfrac{0}{{ - \sigma _3^{\rm{p}}}}}. \end{array}} \right\} $	$ \left. {\begin{array}{*{20}{l}} {f_{\rm{p}}^{{\rm{M}} - {\rm{C}}}\left( {{\sigma^{\rm{p}}}} \right) = - \dfrac{{1 - \sin {{\varphi _{\rm{r}}}}}}{{1+\sin {{\varphi _{\rm{r}}}}}}\sigma _3^{\rm{p}}- \dfrac{{2{c _{\rm{p}}} \cdot \cos {\varphi ^{\rm{p}}}}}{{1+\sin {\varphi ^{\rm{p}}}}} = 0}, \\ {f_{\rm{r}}^{{\rm{M}} - {\rm{C}}}\left( {{\sigma ^{\rm{r}}}} \right) = - \dfrac{{1 - \sin {{\varphi _{\rm{r}}}}}}{{1+\sin {{\varphi _{\rm{r}}}}}}\sigma _3^{\rm{p}}- \dfrac{{2{c _{\rm{r}}} \cdot \cos {{\varphi _{\rm{r}}}}}}{{1+\sin {{\varphi _{\rm{r}}}}}} = 0}. \end{array}} \right\} $	错误。残余强度面屈服函数无解
二向纯剪破坏	$\left[ \begin{array}{c} \sigma _1^{\rm{p} }\\ \sigma _2^{\rm{p} }\\ \sigma _3^{\rm{p} }\\\end{array} \right] = \left[ \begin{array}{c} \sigma _1^{\rm{p}}\\ 0 \\ - \sigma _1^{\rm{p}}\\ \end{array} \right]$	$ \left. {\begin{array}{*{20}{c}} {\dfrac{{\sigma _1^{\rm{r}} - \sigma _3^{\rm{r}}}}{2} = \dfrac{{\sigma _1^{\rm{p}}- \left( { - \sigma _1^{\rm{p}}} \right)}}{2}}, \\ {\dfrac{{\sigma _1^{\rm{r}}}}{{\sigma _1^{\rm{r}} - \sigma _3^{\rm{r}}}} = \dfrac{{\sigma _1^{\rm{p}}}}{{\sigma _1^{\rm{p}}- \left( { - \sigma _1^{\rm{p}}} \right)}}} .\end{array}} \right\} $	$ \left. {\begin{array}{*{20}{l}} {f_{\rm{p}}^{{\rm{M}} - {\rm{C}}}\left( {{\sigma^{\rm{p}}}} \right) = \sigma _1^{\rm{p}}+\dfrac{{1 - \sin {\varphi ^{\rm{p}}}}}{{1+\sin {\varphi ^{\rm{p}}}}}\sigma _1^{\rm{p}}- \dfrac{{2{c _{\rm{p}}} \cdot \cos {\varphi ^{\rm{p}}}}}{{1+\sin {\varphi ^{\rm{p}}}}} = 0} ,\\ {f_{\rm{r}}^{{\rm{M}} - {\rm{C}}}\left( {{\sigma ^{\rm{r}}}} \right) = \sigma _1^{\rm{p}}- \dfrac{{1 - \sin {{\varphi _{\rm{r}}}}}}{{1+\sin {{\varphi _{\rm{r}}}}}}\sigma _3^{\rm{p}}- \dfrac{{2{c _{\rm{r}}} \cdot \cos {{\varphi _{\rm{r}}}}}}{{1+\sin {{\varphi _{\rm{r}}}}}} = 0}. \end{array}} \right\} $	错误。残余强度面屈服函数无解

Tab.4 Defect of existing calculation method based on invariant spherical stress

Fig.2 Uniaxial tension test of red sandstone^[22]

破坏类型	峰值应力	应力跌落过程的主应力增量	合理性
单轴拉伸破坏	$\left[ \begin{array}{c} \sigma _1^{\rm{p} } \\ \sigma _2^{\rm{p} } \\ \sigma _3^{\rm{p} } \\\end{array} \right] = \left[ \begin{array}{c} \sigma _1^{\rm{p} }\\ 0 \\ 0 \\ \end{array} \right]$	$ \left[ \begin{array}{c} \Delta {\sigma _1} \\ \Delta {\sigma _2} \\ \Delta {\sigma _3} \\ \end{array} \right] = \left[ {\begin{array}{*{20}{c}} {\lambda +2G}&\lambda &\lambda \\ \lambda &{\lambda +2G}&\lambda \\ \lambda &\lambda &{\lambda +2G} \end{array}} \right]\left[\begin{array}{c} - \Delta \varepsilon _1^{\rm{p}} \\ v\Delta \varepsilon _1^{\rm{p}} \\ v\Delta \varepsilon _1^{\rm{p}} \\ \end{array} \right] = \left[ \begin{array}{c} - E\Delta \varepsilon _1^{\rm{p}} \\ 0 \\ 0 \\ \end{array} \right] $	正确。仅主拉伸方向应力发生改变与事实相符
单轴压缩破坏	$\left[ \begin{array}{c} \sigma _1^{\rm{p} } \\ \sigma _2^{\rm{p} } \\ \sigma _3^{\rm{p} } \\\end{array} \right] = \left[ \begin{array}{c} 0 \\ 0 \\ \sigma _3^{\rm{p} } \\\end{array} \right]$	$\left[ \begin{array}{c} \Delta {\sigma _1} \\ \Delta {\sigma _2} \\ \Delta {\sigma _3} \\ \end{array} \right] = \left[ {\begin{array}{*{20}{c} } {\lambda +2G}&\lambda &\lambda \\ \lambda &{\lambda +2G}&\lambda \\ \lambda &\lambda &{\lambda +2G} \end{array} } \right] \left[\begin{array}{c} v\Delta \varepsilon _3^{\rm{p} } \\ v\Delta \varepsilon _3^{\rm{p} } \\ - \Delta \varepsilon _3^{\rm{p} } \\ \end{array} \right] = \left[ \begin{array}{c}- \lambda \Delta \varepsilon _3^{\rm{p} } \\- \lambda \Delta \varepsilon _3^{\rm{p} }\\- \left( {\lambda +2G} \right)\Delta \varepsilon _3^{\rm{p} } \end{array} \right]$	正确。仅主压缩方向应力发生改变，与事实相符
二向纯剪破坏	$\left[ \begin{array}{c} \sigma _1^{\rm{p} } \\ \sigma _2^{\rm{p} } \\ \sigma _3^{\rm{p} } \\ \end{array} \right] = \left[ \begin{array}{c} \sigma _1^{\rm{p} } \\ 0 \\ - \sigma _1^{\rm{p} } \\\end{array} \right]$	$ \left[ \begin{array}{c} \Delta {\sigma _1} \\ \Delta {\sigma _2} \\ \Delta {\sigma _3} \\ \end{array} \right] = \left[ {\begin{array}{*{20}{c}} {\lambda +2G}&\lambda &\lambda \\ \lambda &{\lambda +2G}&\lambda \\ \lambda &\lambda &{\lambda +2G} \end{array}} \right] \left[ \begin{array}{c} \left( {1 - v} \right)\Delta \varepsilon _1^{\rm{p}} \\ 0 \\ - \left( {1 - v} \right)\Delta \varepsilon _1^{\rm{p}} \\\end{array} \right] = \left[\begin{array}{c} 2\left( {1 - v} \right)G\Delta \varepsilon _1^{\rm{p}} \\ 0 \\ - 2\left( {1 - v} \right)G\Delta \varepsilon _1^{\rm{p}} \\ \end{array} \right] $	正确。只有1和3方向应力发生变化，与事实相符

Tab.5 Rationality of improved calculation method based on plastic potential theory considering Poisson’s effect

Fig.3 Comparison of simulated stress-strain curves of red sandstone under uniaxial tension

Fig.4 Comparison of simulated stress-strain curves of granite under triaxial compression

Tab.6 Simulated residual stress of two stress-drop calculation method

Fig.5 Comparison of simulated stress-strain curves of sandstone under compression and shear

Fig.6 Numerical procedures of full elasto-plastic deformation and failure process

Fig.7 Experimental simulation of triaxial compression for Tennessee marble and Sanxia granite

Fig.8 Example of circular tunnel excavation and FEM

Fig.9 Comparisons of numerical and theoretical results of circular tunnel excavation

Fig.10 Numerical model of Mine-by tunnel

Tab.7 Model parameters of surrounding rock of Mine-by tunnel

Fig.11 Simulated plastic and failure zones by proposed method

Fig.12 Monitored plastic and failure zones^[29]

Fig.13 Plastic zone at section AK10+900^[31]

Fig.14 Numerical model of auxiliary tunnel

Tab.8 Model parameters of surrounding rock of auxiliary tunnel

Fig.15 Simulated results by proposed method


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