An Improved True Triaxial Hoek–Brown Strength Criterion
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摘要: Hoek–Brown强度准则已经广泛应用于岩石与岩体力学分析、岩石边坡稳定性分析、岩石隧道设计等领域，但常规Hoek–Brown强度准则未考虑中间主应力对岩石强度的影响，应用上不能反映岩石的真三轴力学性能，因此有必要对Hoek–Brown强度准则进行改进。通过分析岩石的真三轴试验结果，明确了岩石中间主应力对岩石强度的影响存在增强和抑制两种特征，当中间主应力大于临界值
$\sigma_2^* $ 时，岩石强度被增强；当中间主应力小于临界值$\sigma_2^* $ 时，岩石强度被减弱。基于中间主应力的临界应力状态，引入中间主应力系数β，将一种2维Hoek–Brown强度准则改进为真三轴Hoek–Brown强度准则。采用非线性拟合方法确定了模型中的参数，通过对比8种岩石类型的真三轴试验数据，对改进后强度准则的精确性和普适性进行验证，并与其他3种真三轴强度准则做对比。结果表明：改进的真三轴强度准则的最佳参数m取值均能落在Hoek和Brown建议的取值区间，能够继承Hoek–Brown强度准则的优点。改进后的强度准则对8种岩石类型的均方根差（RMSE）均低于另外3种准则，能够准确预测岩石强度，且适用于大多数岩石类型，较好地体现了中间主应力对岩石强度的影响；Pan–Hudson准则总是在低σ2条件下高估岩石强度值，在高σ2条件下低估岩石强度值；Zhang–Zhu和Priest准则对岩石类型具有较强依赖性，描述精度较差。因此，改进的真三轴Hoek–Brown强度准则具有重要的应用推广价值。-
关键词:
- 强度准则 /
- Hoek–Brown强度准则 /
- 中间主应力 /
- 真三轴
Abstract: Hoek–Brown strength criterion has been widely used in rock and rock mechanics analysis, rock slope stability analysis, rock tunnel design, and other fields, but the conventional Hoek–Brown strength criterion does not consider the influence of intermediate principal stress on rock strength, and its application cannot reflect the true triaxial mechanical properties of rock. Therefore, it is necessary to improve the Hoek–Brown strength criterion. By analyzing the true triaxial test results of rock, the influence law of rock intermediate principal stress on rock strength was clarified. There are two characteristics of strengthening and restraining its influence: when it is greater than the critical value$\sigma_2^* $ , the rock strength is enhanced, and when it is less than the critical value$\sigma_2^* $ , the rock strength is weakened. Based on the critical stress state of the intermediate principal stress, the intermediate principal stress coefficient βwas introduced, and a two-dimensional Hoek–Brown strength criterion was improved to the true triaxial Hoek–Brown strength criterion. The parameters in the model were determined by the nonlinear fitting method. By comparing the true triaxial test data of 8 rock types, the accuracy and universality of the improved strength criterion were verified. and compared with the other three true triaxial strength criteria. The results show that the best parameterm of the improved true triaxial strength criterion can fall within the range suggested by Hoek and Brown, and can inherit the advantages of the Hoek–Brown strength criterion. The predicted root mean square difference (RMSE) values of the improved strength criterion for the eight rock types are lower than those of the other three criteria, which can accurately predict the rock strength, is suitable for most rock types, and better reflects the influence of the intermediate principal stress on the rock strength. The Pan–Hudson criterion always overestimates the rock strength value under low σ2 conditions and underestimates the rock strength value under high σ2 conditions. The Zhang–Zhu and Priest criteria have a strong dependence on the rock type and the description accuracy is relatively poor. The improved true triaxial Hoek–Brown strength criterion has an important application and promotion value. -
岩石工程中，岩石材料的力学参数、本构模型及强度理论一直是学术界的研究热点之一。在过去几十年中，国内外研究学者提出许多岩石强度准则描述岩石的强度特征。Hoek–Brown强度准则是1980年由Hoek和Brown结合几百组岩石三轴试验结果和工程实践提出的，被广泛应用于岩石与岩体力学分析、岩石边坡稳定性分析、岩石隧道设计等领域，能够反映岩石和岩体的非线性强度特点[1-2]。 1992年，Hoek等[3]对Hoek–Brown强度准则进行改进，修正了岩体在低应力水平下的强度包络线。此后，通过工程应用发现，修正的Hoek–Brown强度准则在应用于质量良好的岩体时过于保守，为此， Hoek等[4-5]先后对原有成果进一步进行修改，提出满足不同质量岩体的广义Hoek–Brown强度准则。自2002年起，Hoek等[5-8]全面讨论了Hoek–Brown准则参数取值关系，引入了扰动因子D描述施工扰动，如爆破损伤等对岩体力学性质的影响等，提出基于地质强度指标（GSI）参数mb、s、a研究的新方法。后续，国内外学者对Hoek–Brown强度准则进行了很多改进，例如：李斌等[9-10]提出基于岩石破坏临界理论改进的Hoek–Brown强度准则；Saroglou等[11]针对各向异性岩石引入新的系数进行准则改进。但这些改进皆未考虑岩石的中间主应力效应。
近年来，随着Hoek–Brown强度准则的广泛应用，很多学者将Hoek–Brown强度准则推广至真三轴Hoek–Brown强度准则。例如：Pan等[12]在Hoek–Brown强度准则基础上，提出一种真三轴Hoek–Brown强度准则，但其不能简化为2维Hoek–Brown强度准则，且预测值较试验值高；Singe等[13]利用中间主应力和最小主应力的平均值代替了Hoek–Brown强度准则最小主应力，从而提出一种真三轴的Hoek–Brown强度准则；Priest[14]将Hoek–Brown强度准则与真三轴的Drucker–Prager准则组合成了一种新的准则，提高了岩石强度的预测精度；Zhang等[15]基于Hoek–Brown强度准则和Mogi强度准则，提出一个新的准则，该准则既继承了Hoek–Brown强度准则的优点，又考虑了岩石中间主应力的影响，但该准则屈服面上有一个明显的凸点，不能完全满足屈服面外凸的要求；Jiang等[16]借鉴了混凝土Ottsosen强度准则表达式，提出一种简便的真三轴Hoek–Brown强度准则。这些真三轴的Hoek–Brown强度准则考虑了中间主应力σ2，能在一定程度上反映σ2 对岩石强度的影响，但其普适性有待验证。
一般来讲，将Hoek–Brown强度准则扩展到真三轴Hoek–Brown强度准则的方法，一种是基于Hoek–Brown强度准则计算拉伸和压缩子午线半径比，然后选择合适的函数；另一种是对已有的表达式进行修正，构建出一个新的表达式。本文将中间主应力系数β引入到一种常规三轴Hoek–Brown强度准则中，并结合8种岩石真三轴试验数据，分析中间主应力对岩石强度的影响；然后，对比不同强度准则对真三轴岩石强度的拟合精度，讨论不同强度准则的普适性。
1. Hoek–Brown强度准则
对于完整岩样，Hoek–Brown强度准则表达式如下：
$$ {\sigma _1} - {\sigma _3}{\text{ = }}{\sigma _{\text{c}}}{\left( {\frac{{m{\sigma _3}}}{{{\sigma _{\text{c}}}}} + 1} \right)^{0.5}} $$ (1) 式中，σ1为最大主应力，σ3为最小主应力，σc为岩石单轴抗压强度，m为岩石量纲统一的经验参数。
李斌等[10]根据前人的研究结果，假设岩石存在某一临界围压，当围压超过该临界围压时，岩石会进入脆性向延性的转化，据此，改进的Hoek–Brown强度准则为：
$${\;\;\;\;\;\;\;\;\; {\sigma _1} - {\sigma _3}{\text{ = }}{\sigma _{\text{c}}}{\left( {\frac{{m{\sigma _3}}}{{{\sigma _{\text{c}}}}} + 1} \right)^{0.5}} - \frac{m}{{4{\sigma _{\text{c}}}\sqrt {m + 1} }}\sigma _3^2 }$$ (2) 研究结果显示，由式（2）改进的Hoek–Brown强度准则可更准确地预测完整岩石的三轴强度，尤其是仅通过岩石在低围压下的强度即可在小误差范围内成功预测岩石在高围压下的强度，但其未考虑到岩石中间主应力对岩石强度的影响[9]。
Pan等[12]提出一个真三轴Hoek–Brown强度准则，表达式如下：
$$ {\;\;\;\;\;\;\;\;\; s{\sigma _{\text{c}}} = \frac{3}{{{\sigma _{\text{c}}}}}{J_2} + \frac{{\sqrt 3 }}{2}{m_{\text{b}}}\sqrt {{J_2}} - {m_{\text{b}}}\sqrt {{J_2}} - {m_{\text{b}}}\frac{{{I_1}}}{3} } $$ (3) $$ {I_1} = {\sigma _1}{\text{ + }}{\sigma _2} + {\sigma _3} $$ (4) $${\;\;\;\;\;\;\;\;\; {J_2}{\text{ = }}\frac{{{{\left( {{\sigma _1} - {\sigma _{\text{2}}}} \right)}^2}{\text{ + }}{{\left( {{\sigma _2} - {\sigma _{\text{3}}}} \right)}^2}}}{6}{\text{ + }}\frac{{{{\left( {{\sigma _3} - {\sigma _{\text{1}}}} \right)}^2}}}{6}} $$ (5) 式（3）～（5）中：I1为第一应力不变量；J2为第二偏应力不变量；
$ {\sigma _2} $ 为中间主应力，MPa；s、$ {m_{\text{b}}} $ 为岩石的材料参数。Priest[14]将2维Hoek–Brown强度准则与真三轴的Drucker–Prager准则组合成了一个新的准则：
$$ \begin{aligned}[b] {\sigma _1} = &3\left[ {w{\sigma _2}{\text{ + }}\left( {1{{ - }}w} \right){\sigma _3}} \right] + \\& {\sigma _{\text{c}}}{\left[ {{m_{\text{b}}}\frac{{w{\sigma _2}{\text{ + }}\left( {1{{ - }}w} \right){\sigma _3}}}{{{\sigma _{\text{c}}}}} + s} \right]^a} - \left( {{\sigma _2} + {\sigma _3}} \right) \end{aligned}$$ (6) $$ w = \partial \sigma _3^\beta $$ (7) 式（6）～（7）中，
$ \partial $ 、$ w $ 、β、a为岩石的材料参数，对于大多数岩石，Priest建议$ \partial $ =β=0.15。Zhang等[15,17]基于Mogi提出的考虑岩石中间主应力的强度准则，提出一个新的准则，称为Zhang–Zhu准则：
$$ {\;\;\;\;\;\;\;\; s{\sigma _{\text{c}}} = \frac{9}{{2{\sigma _{\text{c}}}}}\tau _{{\text{oct}}}^2 + \frac{3}{{2\sqrt 2 }}{m_{\text{b}}}{\tau _{{\text{oct}}}} - {m_{\text{b}}}\frac{{{\sigma _1} + {\sigma _3}}}{2} } $$ (8) $${\;\;\;\;\;\;\;\; {\tau _{{\text{oct}}}}{\text{ = }}\frac{{{{\left[ {{{\left( {{\sigma _1}{{ - }}{\sigma _{\text{2}}}} \right)}^2}{\text{ + }}{{\left( {{\sigma _2}{{ - }}{\sigma _{\text{3}}}} \right)}^2}{\text{ + }}{{\left( {{\sigma _3}{{ - }}{\sigma _{\text{1}}}} \right)}^2}} \right]}^{0.5}}}}{3}} $$ (9) 式中，τoct为八面体剪应力。
值得注意的是：Pan–Hudson准则不能简化为Hoek–Brown强度准则的一般表达式，严格意义上不算真三轴Hoek–Brown强度准则。Zhang–Zhu准则屈服面上有一个明显的凸点，不能完全满足屈服面外凸的要求。简化的Priest准则有额外的参数
$ w $ 描述中间主应力对岩石强度的影响。综上，由于式（2）具有较高的描述精度，本文将其改进为真三轴强度准则，并对比其优势，在后文与其他3种真三轴强度准则进行对比。
2. 改进的真三轴Hoek–Brown强度准则
Mogi[18]先后分别对多种岩石做了真三轴试验，在Dunham白云岩、Solnhofen石灰岩及稻田花岗岩上都观察到σ2对岩石强度的影响，肯定了中间主应力效应的存在。图1显示了Mizuho粗面岩的真三轴试验结果。由图1可知，σ2在σ1到σ3的范围内变化，对于恒定的σ3，岩石强度随着σ2的增加先增加到峰值，然后随着σ2的增加逐渐减小。Takahashi等[19]对砂岩和页岩进行了真三轴试验；Chang[20-22]和Oku[23]等对Westerly花岗岩、KTB角闪岩、粉砂岩和角页岩进行了真三轴试验，都证明了岩石中间主应力效应对岩石强度的影响。所有证据皆表明，恒定σ3下，岩石强度随着中间主应力的增加先增加后减小，即：当σ2较低时，σ2和σ3一样，能抑制岩石的破坏进程和提高岩石强度；当σ2较高时，能加速岩石的破坏。
图 1${\boldsymbol{\sigma}} _{\boldsymbol{2}}$ 对Mizuho粗面岩岩石强度的影响Fig. 1 Effect of${\boldsymbol{\sigma}} _{\boldsymbol{2}}$ on rock strength of Mizuho trachyte
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为了表征中间主应力对岩石强度的影响规律，引用了文献[24]岩石中间主应力系数β，其定义式为：
$$ \beta {\text{ = }}\frac{{{\sigma _2}{{ - }}{\sigma _3}}}{{{\sigma _1}{{ - }}{\sigma _3}}},0 \leqslant \beta \leqslant 1 $$ (10) 根据式（10），当σ2=σ3时，β=0；当σ2=σ1时，β=1。
图2为在一定的围压下，Mizuho粗面岩的岩石强度随着中间主应力系数的变化趋势，能够较好地描述岩石强度的中间主应力效应，中间主应力系数β在0～1之间总存在一个临界值β*，当中间主应力系数β小于此临界值β*时，随着中间主应力σ2的增加，强度σ1增加；当大于临界值β*时，随着中间主应力σ2的增加，强度σ1减小。
图 2 β对岩石强度的影响Fig. 2 Influence of β on rock strength下载:
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根据中间主应力对岩石强度的影响规律，随着中间主应力σ2的增加，也存在某一临界值
$ \sigma _2^ * $ [25]，当$ {\sigma _3} \leqslant {\sigma _2} \leqslant \sigma _2^ * $ 时，中间主应力能抑制岩石破坏进程，中间主应力σ2与最小主应力σ3的综合影响将提高岩石的强度，岩石中间主应力系数β在0～1之间总能使：$$ \frac{{\beta {\sigma _2}{\text{ + }}{\sigma _3}}}{{1{\text{ + }}\beta }} \gt {\sigma _3} $$ (11) 此时，若
$ \dfrac{{\beta {\sigma _2}{\text{ + }}{\sigma _3}}}{{1{\text{ + }}\beta }} $ 表征了中间主应力对岩石强度的增强效应，使$ \dfrac{{\beta {\sigma _2}{\text{ + }}{\sigma _3}}}{{1{\text{ + }}\beta }} = {\sigma _3} $ ，将其代入式（2）中，则式（2）变为：$$\begin{aligned}[b] {\sigma _1}{\text{ = }}& \frac{{\beta {\sigma _2}{\text{ + }}{\sigma _3}}}{{1{\text{ + }}\beta }}{\text{ + }}{\sigma _{\text{c}}}{\left[ {\frac{m}{{{\sigma _{\text{c}}}}}\cdot {\frac{{\beta {\sigma _2}{\text{ + }}{\sigma _3}}}{{1{\text{ + }}\beta }}} + 1} \right]^{0.5}} - \\& \frac{m}{{4{\sigma _{\text{c}}}\sqrt {m + 1} }}\cdot {\left( {\frac{{\beta {\sigma _2}{\text{ + }}{\sigma _3}}}{{1{\text{ + }}\beta }}} \right)^2},{\sigma _3} \leqslant {\sigma _2} \leqslant \sigma _2^ * \end{aligned}$$ (12) 当
$ \sigma _2^ * \leqslant {\sigma _2} \leqslant {\sigma _1} $ 时，中间主应力能促进岩石破坏进程，岩石的强度被降低，岩石中间主应力系数β总能使：$$ \frac{{\beta {\sigma _2}{\text{ + }}{\sigma _1}}}{{1{\text{ + }}\beta }} \lt {\sigma _1} $$ (13) 此时，若
$ \dfrac{{\beta {\sigma _2}{\text{ + }}{\sigma _1}}}{{1{\text{ + }}\beta }} $ 表征了中间主应力对岩石强度的降低效应，使$ \dfrac{{\beta {\sigma _2}{\text{ + }}{\sigma _1}}}{{1{\text{ + }}\beta }} = {\sigma _1} $ ， 将其代入式（2）中，则式（2）变为：$$ \begin{aligned}[b]& \frac{{\beta {\sigma _2}{\text{ + }}{\sigma _1}}}{{1{\text{ + }}\beta }}{\text{ = }}{\sigma _{\text{c}}}{\left( {m\frac{{{\sigma _3}}}{{{\sigma _{\text{c}}}}} + 1} \right)^{0.5}} - \frac{m}{{4{\sigma _{\text{c}}}\sqrt {m + 1} }}\cdot \sigma _3^2{\text{ + }} {\sigma _3},\\& \qquad \qquad\qquad\quad \sigma _2^ * \leqslant {\sigma _2} \leqslant {\sigma _1} \\[-10pt]\end{aligned} $$ (14) 则式（12）和（14）就组成了一种改进的真三轴Hoek–Brown强度准则。中间主应力临界值
$ \sigma _2^ * $ 可由式（12）和（14）中的σ1相等，联立求出$ \sigma _2^ * $ ，作为式（12）和（14）的选择条件。3. 强度准则的对比分析
为了对比本文强度准则与Zhang–Zhu准则、Priest准则及Pan–Hudson准则，验证本文强度准则的适用性，从已发表的文献中选取了8种岩石的真三轴试验数据。其中，Mizuho粗面岩、Solnhofen石灰岩和Dunham白云岩的数据来自Mogi[18,26]的试验研究，Shirahama砂岩和Yuubari板岩的数据取自Colmenares和Zoback[27]的研究，KTB角闪岩、Westerly花岗岩的数据取自Chang[22]和Haimson[28]等的研究，致密大理岩的数据取自Michelis[29-30]的研究。通过计算σ1在不同σ2和σ3情况下的值，分析试验数据与预测数据之间的误差，以比较强度准则的准确性。
采用均方根差RMSE最小为拟合目标来比较强度准则参数的拟合效果和准则预测效果。其定义式为：
$${\;\;\;\;\;\;\;\;\; {\text{RMSE = }}\sqrt {\frac{1}{N}\sum\limits_{i = 1}^N {{{\left( {\sigma _{1,i}^{{\text{cal}}} - \sigma _{1,i}^{{\text{test}}}} \right)}^2}} } }$$ (15) 式中，
$ \sigma _{1,i}^{{\text{cal}}} $ 为岩石强度准则预测值，$ \sigma _{1,i}^{{\text{test}}} $ 为岩石强度的试验值，N为试验数据的组数。表1为改进后的强度准则式（13）和（14）对8种岩石的拟合参数结果，最佳参数m取值基本都落在Hoek和Brown建议取值区间内[31-32]。
表 1 改进的强度准则拟合参数汇总Table 1 Summary of fitting parameters of improved strength criterion图3为本文强度准则对8种岩石强度的拟合结果。由图3可知：修正的强度准则能够较好地描述真三轴条件下的岩石强度变化规律。其中：Yuubari板岩的拟合效果最好，RMSE=7.1；Solnhofen石灰岩RMSE相对较高，为21.65。
图 3 改进的强度准则对8种岩石强度的拟合曲线Fig. 3 Fitting curves of 8 kinds of rock strength with the improved strength criterion下载:
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图4总结了Zhang–Zhu准则、Priest准则、Pan–Hudson准则与本文强度准则对8种岩石强度的拟合曲线。由图4可知，本文的强度准则的预测值随围压的变化曲线与试验值曲线几乎完全重合，能够较好地表征中间主应力对岩石强度的影响；Pan–Hudson准则在低σ2条件下高估了岩石强度值，在高σ2条件下低估了岩石强度值，给出了最差的预测值；Zhang–Zhu和Priest准则的拟合效果稍好，对岩石的类型具有较强的依赖性，而本文的强度准则表现出较好的描述效果，说明其普适性最好。
图 4 4种强度准则对8种岩石强度的拟合曲线Fig. 4 Fitting curves of four strength criteria for eight kinds of rocks下载:
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为进一步对比4种强度准则，图5给出了4种强度准则对8种岩石的预测误差。由图5可以看出：本文强度准则和Zhang–Zhu准则对岩石的三轴试验强度预测的RMSE值较小，远小于其他2种强度准则，说明这2种强度准则对不同类型岩石强度均预测准确度均比较高；另外，本文的强度准则对8种岩石预测的平均RMSE值小于Zhang–Zhu、Pan–Hudson和Priest准则，说明本文修改的强度准则精度高于其他3种强度准则。
图 5 8种岩石类型的4个强度准则的预测误差Fig. 5 Prediction errors of four strength criteria for eight kinds of rocks下载:
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4. 结 论
中间主应力系数β能够较好地描述中间主应力对真三轴岩石强度的影响规律，结合一种常规三轴Hoek–Brown强度准则，引入中间主应力系数β，提出一个改进的Hoek–Brown真三轴强度准则。选取了8种真三轴岩石强度试验结果，对比分析了Priest准则、Zhang–Zhu准则和Pan–Hudson准则及本文准则的拟合精度及适应性，结果显示，本文准则对8种岩石都有较好的普适性，而Priest准则、Zhang–Zhu准则及Pan–Hudson准则对岩石类型都有较强的依赖性，普适性较差。
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图 1
${\boldsymbol{\sigma}} _{\boldsymbol{2}}$ 对Mizuho粗面岩岩石强度的影响Fig. 1 Effect of
${\boldsymbol{\sigma}} _{\boldsymbol{2}}$ on rock strength of Mizuho trachyte下载:
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图 2 β对岩石强度的影响
Fig. 2 Influence of β on rock strength
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图 3 改进的强度准则对8种岩石强度的拟合曲线
Fig. 3 Fitting curves of 8 kinds of rock strength with the improved strength criterion
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图 4 4种强度准则对8种岩石强度的拟合曲线
Fig. 4 Fitting curves of four strength criteria for eight kinds of rocks
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图 5 8种岩石类型的4个强度准则的预测误差
Fig. 5 Prediction errors of four strength criteria for eight kinds of rocks
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表 1 改进的强度准则拟合参数汇总
Table 1 Summary of fitting parameters of improved strength criterion
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