Upper and lower bounds for constitutive relation of crack-weakened rock masses under dynamic compressive loads

A micro-mechanics-based model is proposed to investigate the rate-dependent constitutive relation for crack-weakened rock masses subjected to dynamic compressive loads. The present micro-mechanical model reveals that the nucleation, growth and coalescence of sliding cracks dominate the failure and macroscopic properties of crack-weakened rock masses subjected to dynamic compressive loads. The interactions among multiple parallel sliding cracks in crack-weakened rock masses subjected to dynamic compressive loads are examined asymptotically in an explicit and quantitative manner in order to reveal fully their so-called shielding and magnification effects on the stress–strain relation. Based on the micro-mechanical framework and the asymptotic analysis, analytical upper and lower bounds are proposed for the rate-relation for rock masses containing multiple rows of echelon cracks subjected to dynamic compressive loads. The factors that affect the rate-dependent properties of crack-weakened rock masses have been analyzed. The strain energy density factor approach, which is related to crack growth velocity and dynamic fracture toughness of rock material, is employed in the analysis. The rate-dependent constitutive relation of crack-weakened rock masses is derived from micro-mechanical framework and the asymptotic analysis. The closed-form explicit expression for the rate-dependent constitutive relation of rock masses containing echelon cracks subjected to dynamic compressive loads is obtained. Finally, the present model is used to analyze the complete stress–strain relation and strength for jointed rock masses at shiplock slope of the Three Gorges Dam.     

考虑裂隙间相互作用情况下围压卸荷过程应力应变关系

岩体的稳定性和变形特性主要决定于裂隙，同时裂隙间的相互作用对岩体的稳定和变形产生显著的影响。裂隙岩体在加载和卸荷条件下的力学特性有显著的区别。为此本文首次利用位错模型法结合叠加原理研究在围压卸荷条件下裂隙间的相互作用对岩体的变形的影响问题。文中推导了考虑裂隙间的相互作用情况下裂隙岩体围压卸荷过程的应力应变关系及应力强度因子表达式，且进行了数值计算。     

Zonal disintegration of deep crack-weakened rock masses: A non-Euclidean model

Rock masses are characterized by the existence of distributed joints and fractures. One of behaviors of the deep rock masses is high in situ stresses. The internal space of rock-like materials subjected to high in situ stresses after deformation is treated as a non-Euclidean one. The incompatible deformation of the deep rock masses is induced by high in situ stresses within the framework of non-Euclidean geometric space. A non-Euclidean model in which effects of cracks on zonal disintegration phenomenon of the deep crack-weakened rock masses is taken into account is established. Based on the non-Euclidean model, the elastic stress-field distribution of the deep surrounding rock masses induced by compatible deformation of non-fractured zones and incompatible deformation of fractured zones is determined. The stress intensity factors at the tips of cracks is given out. The strain energy density factor is applied to investigate the occurrence of disintegration zones. It is observed from the numerical results that the magnitude and site of fractured zones depend on the value of in situ stress, mesomechanical parameters and non-Euclidean parameters.     

Rock burst of deep circular tunnels surrounded by weakened rock mass with cracks

Rock masses containing pre-existing cracks are considered as non-homogeneous geomaterials. During excavation of tunnels, pre-existing cracks may nucleate, grow and propagate through rock matrix, then secondary cracks may appear. The stress concentration at the tips of secondary cracks is comparatively large, which may lead to the unstable growth and coalescence of secondary cracks, and consequently the occurrence of fractured zones. For brittle rocks, the dissipative energy of slip and growth of pre-existing cracks and secondary crack growth is small, but the elastic strain energy storing in rock masses may be larger than the dissipative energy of slip and growth of pre-existing macrocracks and secondary crack growth. The sudden release of the residual elastic strain energy may lead to rock burst in crack-weakened rock masses. Based on this understanding, the criteria of rock burst in crack-weakened rock masses are established. The influences of the in situ stresses, micromechanical parameters and physico-mechanical parameters on the distribution of rock burst zones and area of rock burst zones are investigated in detail.     

Triaxial compressive behavior of rock with mesoscopic heterogenous behavior: Strain energy density factor approach

The strain energy density factor approach is used in conjunction with a micromechanics model to investigate the condition and direction of shear failure for brittle rock subjected to triaxial compression. Moderate confinement in addition to localized deformation and damage are considered. Quantified are the effects of the various geometric and load parameters that involve the interaction of microcrack, friction and the confining pressure such that the path of the wing crack is taken into account. The influence of all microcracks with different orientations are introduced into the constitutive relation. The closed-form solution for the complete stress–strain relation of rock containing microcracks is obtained. It is shown that the complete stress–strain relationship includes linear, nonlinear hardening, rapid stress drop and strain softening effects. The theoretical results show that deviation of the direction of wing cracks from the line of the pre-existing crack decreases with increasing confinement pressure and friction coefficient. Theoretical predictions and experimental results show good agreement.     

Zonal fracturing mechanism in deep crack-weakened rock masses

The mechanical behaviors of deep crack-weakened rock masses are different from those of shallow crack-weakened rock masses. The surrounding rock in shallow crack-weakened rock mass engineering is classified into loose zone, plastic zone and elastic zone, while the surrounding rock in deep crack-weakened rock mass engineering is classified into fractured zone and non-fractured zone, which occur alternatively. It is assumed that the deep rock masses contain one joint set, in which the probability density function describing the distribution of sizes is assumed to follow the Rayleigh distribution, and the probability density function describing the distribution of spacing is assumed to follow the Weibull distribution. On the basis of strength criterion of deep rock mass, the near-field stress redistribution around circular opening induced by excavation is determined. The strong interaction among cracks is investigated by using the dislocation model. The nucleation, growth, interaction and coalescence of cracks were analyzed based on the strain energy density factor theory. When cracks coalesce, failure of deep crack-weakened rock masses occurs, fractured zone is formed. Then, size and quantity of fractured zone and non-fractured zone are given out. The size and quantity of fractured zone increase with decreasing strength of rock mass. The size and quantity of fractured zone increase with increasing in situ stress. Zonal fracturing phenomenon occurs once value of in situ stress is larger than the unaxial compressive strength of rock masses. The size and quantity of fractured zone decrease with increasing λ when p₂ > p₁. The size and quantity of fractured zone increase with increasing λ when p₂ < p₁.     

ZONAL DISINTEGRATION MECHANISM OF DEEP CRACK-WEAKENED ROCK MASSES UNDER DYNAMIC UNLOADING

Size and quantity of fractured zone and non-fractured zone are controlled by cracks contained in deep rock masses. Zonal disintegration mechanism is strongly dependent on the interaction among cracks. The strong interaction among cracks is investigated using stress superposition principle and the Chebyshev polynomials expansion of the pseudo-traction. It is found from numerical results that crack nucleation, growth and coalescence lead to failure of deep crack- weakened rock masses. The stress redistribution around the surrounding rock mass induced by unloading excavation is studied. The effect of the excavation time on nucleation, growth, interaction and coalescence of cracks was analyzed. Moreover, the influence of the excavation time on the size and quantity of fractured zone and non-fractured zone was given. When the excavation time is short, zonal disintegration phenomenon may occur in deep rock masses. It is shown from numerical results that the size and quantity of fractured zone increase with decreasing excavation time, and the size and quantity of fractured zone increase with the increasing value of in-situ geostress.     

Study on the coalescence mechanism of splitting failure of crack-weakened rock subjected to compressive loads

It is of important significance to study the coalescence mechanism of splitting failure of crack-weakened rock masses under compressive loads. In this paper, a simplified mechanism of crack propagation, in which the crack grows along the direction of maximum principal compressive stress, is proposed. Thus, only mode I is taken into account in the formulation and solution. On the basis of the near crack line analysis method, the elastic–plastic stress field near the crack line is analyzed, and the law that the length of the plastic zone along the crack line is varied with an external loads have been established by the matching condition of the elastic- plastic fields on the boundary, the coalescence stress and the strength properties of rock masses have been determined. The solution is a function of the geometry of the crack array. The results show that the crack coalescence depends on the crack interface friction coefficient, the sliding crack spacing, orientation of the cracks, and the crack half-length. The conclusions are of important significance for rock mass engineering.     

Constitutive relationship of brittle rock subjected to dynamic uniaxial tensile loads with microcrack interaction effects

A micromechanical model is proposed to describe both stable and unstable damage evolution in microcrack-weakened brittle rock material subjected to dynamic uniaxial tensile loads. The basic idea of the present model is to classify the constitution relationship of rock material subjected to dynamic uniaxial tensile loads into four stages including some of the stages of linear elasticity, pre-peak nonlinear hardening, rapid stress drop, and strain softening, and to investigate their corresponding micromechanical damage mechanisms individually. Special attention is paid to the transition from structure rearrangements on microscale to the macroscopic inelastic strain, to the transition from distribution damage to localization of damage and the transition from homogeneous deformation to localization of deformation. The influence of all microcracks with different sizes and orientations are introduced into the constitutive relation by using the statistical average method. Effects of microcrack interaction on the complete stress-strain relation as well as the localization of damage for microcrack-weakened brittle rock material are analyzed by using effective medium method. Each microcrack is assumed to be embedded in an approximate effective medium that is weakened by uniformly distributed microcracks of the statistically-averaged length depending on the actual damage state. The elastic moduli of the approximate effective medium can be determined by using the dilute distribution method. Micromechanical kinetic equations for stable and unstable growth characterizing the ‘process domains’ of active microcracks are taken into account. These ‘process domains’ together with ‘open microcrack domains’ completely determine the integration domains of ensemble averaged constitutive equations relating macro-strain and macro-stress. Theoretical predictions have shown to consistent with the experimental results.     

THE CONSTITUTIVE RELATION OF CRACK-WEAKENED ROCK MASSES UNDER AXIAL-DIMENSIONAL UNLOADING

An accurate and efficient numerical method for solving the crack-crack interaction problem is presented. The method is mainly by means of the dislocation model, stress superposition principle and Chebyshev polynomial expansion of the pseudo-traction. This method can be applied to compute the stress intensity factors of multiple kinked cracks and multiple rows of periodic cracks as well as the overall strains of rock masses containing multiple kinked cracks under complex loads. Many complex computational examples are given. The dependence of the crack-crack interaction on the crack configuration, the geometrical and physical parameters, and loads pattern, is investigated. By comparison with numerical results under confining pressure unloading, it is shown that the crack-crack interaction under axial-dimensional unloading is weaker than those under confining pressure unloading. Numerical results for single faults and crossed faults show that the single faults are more unstable than the crossed faults. It is found from numerical results for different crack lengths and different crack spacing that the interaction among kinked cracks decreases with an increase in length of the kinked cracks and the crack spacing under axial-dimensional unloading.     

Analysis of the localization of damage and the complete stress-strain relation for mesoscopic heterogeneous brittle rock subjected to compressive loads

A micromechanics-based model is established. The model takes the interaction among sliding cracks into account, and it is able to quantify the effect of various parameters on the localization condition of damage and deformation for brittle rock subjected to compressive loads. The closed-form explicit expression for the complete stress-strain relation of rock containing microcracks subjected to compressive loads was obtained. It is showed that the complete stress-strain relation includes linear elasticity, nonlinear hardening, rapid stress drop and strain softening. The behavior of rapid stress drop and strain softening is due to localization of deformation and damage. Theoretical predictions have shown to be consistent with the exoerimental results.     

Analysis of the localization of deformation and the complete stress–strain relation for mesoscopic heterogeneous brittle rock under dynamic uniaxial tensile loading

Stress redistribution induced by excavation results in the tensile zone in parts of the surrounding rock mass. It is significant to analyze the localization of deformation and damage, and to study the complete stress–strain relation for mesoscopic heterogeneous rock under dynamic uniaxial tensile loading. On the basis of micromechanics, the complete stress–strain relation including linear elasticity, nonlinear hardening, rapid stress drop and strain softening is obtained. The behaviors of rapid stress drop and strain softening are due to localization of deformation and damage. The constitutive model, which analyze localization of deformation and damage, is distinct from the conventional model. Theoretical predictions have shown to consistent with the experimental results.     

岩石单轴压缩作用下变形局部化的梯度塑性解

采用梯度塑性理论研究单轴压缩作用下岩石变形局部化,得到了单轴压缩作用下岩石变形局部化带宽度的一维、二维解析解,为实验测定内部材料长度参数提供了理论依据.     

考虑裂纹闭合效应的岩石损伤本构关系

岩石中的预存裂纹只有在一定的法向压应力即裂纹闭合应力的作用下才可能闭合，其闭合过程与其方位和外加应力场有关，并且，即使对于裂纹已经完全闭合的岩石，如果裂纹闭合应力不同，则岩石的应力应交关系也不相同。本文建立了考虑裂纹闭合效应的岩石细观损伤力学模型，分析了裂纹闭合应力对岩石损伤演化过程和应力应变关系的影响。数值结果表明裂纹闭合应力显著地改变岩石的应力应变关系，表现为随裂纹闭合应力的增加，岩石的轴向应变变化较小，侧向应变和体积应变则大为增加。     

A fracture damage model for Jointed rock masses and its application to the stability analysis of dam abutments

Based on continuum damage mechanics, for jointed rock masses, a fracture damage model is presented in this paper. First, the damage tensors are defined through the elastic-flexibility of intact rock and the equivalent elastic-damage flexibility for rock mass. Then, by the self-consistent principle of solid mechanics, the equivalent elastic-damage flexibility tensors involving the interaction between multicracks are deduced. The damage evolution law is proposed involving the mechanism of crack propagation process: frictional sliding, crack kinking, growing of branched tension cracks, interlinking of the microcracks near branched crack tips leading to the breakthrough of macro-cracks and finally the failure of rock mass. Thus the evolution of damage variables reasonably unified with the process of crack propagation is given. Finally, a plastic-brittle damage constitutive relation including brittle coupled strain rate, developed and applied to the stability analysis of complicated rock foundation of a dam in China, is described in this paper.     

爆生气体作用下岩石裂纹的扩展机理

在爆生气体作用下 ,爆破近区的裂纹在气体驱动压力下扩展 ,而爆破中区的裂纹扩展是在气体膨胀压力场和原岩应力共同作用下发生的。基于岩石细观损伤断裂理论 ,认为裂纹扩展的过程就是裂纹尖端到周围岩石的逐渐损伤引起的损伤区移动过程 ;建立了这两个区域的损伤断裂准则和裂纹尖端的损伤局部化模型 ,可以更好地反映爆生气体作用下裂纹扩展的实际过程。     

巴西圆盘泥岩试件裂纹扩展及应变演化实验研究

为了研究准静态加载条件下岩石试件巴西劈裂裂纹扩展规律,采用MTS试验机进行准静态加载,同时用高速摄像机记录裂纹扩展过程。采用白光数字散斑处理软件对摄像机记录的照片进行处理,得到试件裂纹扩展过程中应变场的演化情况。通过实验和分析可以看出,由于端部效应及加载方式的原因,因此裂纹起裂点在底部加载部位;泥岩试件表面裂纹的平均扩展速度为252m/s;岩石的非均质性即内部微缺陷、微裂纹使得泥岩试样的开裂并不是沿着中心直径方向,而是偏离一定的角度,初始偏离角度约为17°。裂纹扩展过程可以划分为三个阶段:泥岩试件宏观变形阶段(宏观无裂纹)、宏观裂纹稳定扩展阶段、宏观裂纹动态张裂阶段。同时,在裂纹扩展过程中,表面第一主应变场、水平位移场等变化明显,在开裂部位第一主应变最大。通过对圆盘泥岩试件裂纹扩展实验研究,可为研究岩石破裂及其演化规律提供依据。     

分段装药爆炸应变场与裂隙场分布规律

为探究分段装药爆炸应变场与裂隙场分布规律,采用数字图像相关分析方法与电子计算机断层扫描实验方法,分析了孔内分段装药爆炸全场应变传播规律,建立了爆后“岩石—爆炸裂隙”的三维重构模型,描述了爆炸裂纹位置与形态的空间分布情况,得到岩石材料爆炸裂隙的分形维数与损伤度。研究结果表明：分段装药改变了连续装药对介质的全场应变形态,由一次应变改变为两次应变,在满足第一段炸药对介质的破坏作用下,同时加大了第二段炸药对介质的作用效应;上分段装药占比0.4时,下分段介质受爆炸作用应变峰值更大,更好满足工程实践中下半段岩体对爆炸能量的需求;相同装药系数下,连续装药结构爆炸裂纹没有贯穿试件整体,炮孔封堵段的爆炸裂纹较少,分段装药结构下,由于提高了炸药的位置,使得上部分岩体能够更好地利用炸药爆炸的能量破碎岩石;分段装药岩石整体损伤度较连续装药提高了23.5%,其中上分段岩石损伤差异较大,分段装药上分段损伤度比连续装药提高46.4%。     

The Three-Stage Model Based on Strain Strength Distribution for the Tensile Failure Process of Rock and Concrete Materials

A three-stage model is introduced to describe the tensile failure process of rock and concrete materials.Failure of the material is defined to contain three stages in the model,which include elastic deformation stage,body damage stage and localization damage stage.The failure mode change from uniform body damage to localization damage is expressed.The heterogeneity of material is described with strain strength distribution.The fracture factor and intact factor,defined as the distribution function of strain strength,are used to express the fracture state in the failure process.And the distributive parameters can be determined through the experimental stress-strain curve.     

卸荷速率对岩石强度影响的试验研究

通过MTS 815.02型电液伺服岩石力学试验系统,研究砂岩试样在常规三轴条件下卸 围压过程中的应力-应变关系,探讨不同卸荷速率对试样强度的影响情况,分析得 出卸荷速率与试样强度的变化规律;观察、测量破坏试样的破裂特征及角度,初步探究试 样在卸荷作用下的破坏机理;研究成果可为工程实践的岩体卸荷问题提供一定的理 论参考.