工程科学与技术2026，Vol.58Issue(1)：46-56,11.DOI:10.12454/j.jsuese.202500274
考虑中主应力影响的修正摩尔-库仑硬岩强度准则
Modified Mohr-Coulomb Hard Rock Strength Criterion Considering the Influence of Intermediate Principal Stress
摘要
Abstract
Objective Accurate prediction of the strength of deeply buried hard rock is crucial for ensuring the safety and stability of deep rock engineering.Currently,two key factors influencing the strength prediction of deep-buried hard rock are high stress and the intermediate principal stress.High stress produces nonlinear deformation characteristics in deeply buried hard rock,and the intermediate principal stress substantially affects the de-formation and failure behavior of hard rock.This study proposes a modified Mohr-Coulomb strength criterion for hard rock,which simultane-ously considers the nonlinear strength characteristics under large-range variations in confining pressure and the influence of the intermediate prin-cipal stress on rock strength. Methods First,an analysis of Mohr's stress circle under true triaxial conditions was conducted.Based on the theory of Mohr's stress circle,it was determined that rock strength comprised cohesive and frictional strength components.The intermediate principal stress significantly influ-enced the cohesive strength component,while its effect on the frictional strength component remained negligible.A generalized functional form of the Mohr-Coulomb(M-C)strength criterion under true triaxial stress conditions was proposed,which attributed the influence of the intermedi-ate principal stress on rock strength to an intermediate principal stress effect function.The rationality of this functional form was preliminarily validated using true triaxial test data from four types of hard rock.Then,an analysis was conducted on commonly used rock strength criteria that reduced to the M-C criterion under certain conditions,including the D-P criterion,the Mogi criterion,and the modified Lade criterion.The re-sults revealed that the significant deviations in their strength predictions primarily resulted from inaccuracies in the mathematical characterization of the intermediate principal stress effect function.Therefore,the specific forms of the intermediate principal stress effect functions corresponding to the D-P,Mogi,and modified Lade criteria were examined in detail.On this basis,a parameter η,which represented the difference between the tensile and compressive strengths of the rock,and a parameter γ,which characterized the influence of the intermediate principal stress(σ2)on rock strength under constant minimum principal stress(σ3),were introduced.A new intermediate principal stress effect function was constructed.The classical M-C criterion was modified,establishing a nonlinear M-C strength criterion that accounted for the influence of the intermediate principal stress by integrating the critical state characteristics of rock.Then,methods for determining the parameters were provided.Finally,the applicability of the new criterion was validated using true triaxial test data from six types of hard rock,and a comparative analysis was conducted with four traditional strength criteria. Results and Discussions Predictions from this criterion were compared to true triaxial test data from six types of hard rock,along with a com-parative analysis against four traditional strength criteria to validate the applicability of the proposed modified M-C strength criterion that consid-ered the influence of the intermediate principal stress.The results demonstrated that the proposed criterion effectively captured the observed trend in which rock strength initially increased and then decreased with the rise of the intermediate principal stress,as well as the significant strength differences exhibited under the conditions of σ2=σ3 and σ2=σ1.Under different confining pressure σ3 conditions,the variation patterns captured by this strength criterion remained consistent.However,as the ∂σ1/∂σ3 ratio gradually decreased,the increase in rock strength became more moder-ated with rising σ3,indicating that the enhancing effect of the intermediate principal stress on rock strength exhibited a decaying characteristic with increasing stress levels.When σ2=σ3,the criterion reduces to a nonlinear M-C strength criterion that accounts for the critical state of rock.With increasing σ3,the lateral deformation constraint effect continuously strengthened,which led to a persistent increase in the curvature of the strength envelope and reflected the transition of rock mechanical behavior from brittle to ductile.When σ3 ³ σrc,the strength envelope approached horizontal,indicating that the rock entered a fully ductile state.In addition,for all six types of hard rock,the goodness of fit R2 of the proposed criterion exceeded 0.94,which demonstrated its high accuracy in strength prediction.Based on the true triaxial test data from these six hard rock types,it was found that the D-P criterion exhibited the largest prediction errors.Although the M-MC true triaxial criterion captured the trend of rock strength initially increasing and then decreasing with the rise of the intermediate principal stress and showed high prediction accuracy under the condition of σ2=σ3,its prediction deviation became significant when σ2=σ1.The Mogi criterion,while relatively accurate,failed to reflect the difference between generalized triaxial compression and tensile strength,which contradicted experimental observations.The modified Lade crite-rion performed well under low σ3 conditions,but its prediction deviation gradually increased as σ3 rose.In comparison,the root mean square error(RMSE)and mean absolute percentage error(MAPE)of the proposed criterion remained lower than those of the other four traditional strength cri-teria.In addition,it more accurately described the influence of the intermediate principal stress on rock strength under large-range variations of σ3.These results demonstrated the rationality and feasibility of the proposed criterion. Conclusions The proposed strength criterion exhibits a clear formulation with parameters that are simple to determine and possess explicit physi-cal significance.It effectively captures the influence of intermediate principal stress on hard rock strength and represents the nonlinear mechani-cal response under high stress conditions.Its high predictive accuracy for various hard rocks confirms the criterion's broad applicability under complex stress states and reinforces its value for practical engineering applications.关键词
深部岩石/强度准则/中主应力/临界围压/非线性Key words
deep rock/strength criteria/intermediate principal stress/critical confining pressure/nonlinearity分类
建筑与水利引用本文复制引用
俞缙,周永盛,张超,常旭,蔡燕燕..考虑中主应力影响的修正摩尔-库仑硬岩强度准则[J].工程科学与技术,2026,58(1):46-56,11.基金项目
国家自然科学基金项目(52374090) （52374090）
中国博士后科学基金资助项目(2023MD744275) （2023MD744275）
贵州省基础研究计划(黔科合基础MS[2025]037) （黔科合基础MS[2025]037）
