An experimental and numerical study of fracture coalescence in pre-cracked specimens under uniaxial compression

This study presents crack initiation, propagation and coalescence at or near pre-existing open cracks or flaws in a specimen under uniaxial compression. The flaw geometry in the specimen was a combination of a horizontal flaw and an inclined flaw underneath. This flaw geometry is different from those reported in the previous studies, where a pair of parallel flaws was used. Three materials were used, PMMA (Poly Methyl MethAcrylate), Diastone (types of molded gypsum), and Hwangdeung granite. Crack initiation and propagation showed similar and different patterns depending on the material. In PMMA, tensile cracks initiated at the flaw tips and propagated to the tip of the other flaw in the bridge area. The cracks then coalesced at a point of the inclined flaw, which is affected by the flaw inclination angle. For Diastone and Hwangdeung granite, tensile cracks were observed followed by the initiation of shear cracks. Coalescence occurred mainly through the tensile cracks or tensile and shear cracks. Crack coalescence was classified according to the crack coalescence types of parallel flaws for overlapping flaw geometry in the past works. In addition, crack initiation and coalescence stresses in the double-flawed specimens were analyzed and compared with those in the single-flawed specimen. Numerical simulations using PFC^2D (Particle Flow Code in two dimensions) based on the DEM (Discrete Element Method) were carried out and showed a good agreement with the experimental results in the coalescence characteristics in Hwangdeung granite. These experimental and numerical results are expected to improve the understanding of the characteristics of cracking and crack coalescence and can be used to analyze the stability of rock and rock structures, such as the excavated underground openings or slopes, tunneling construction, where pre-existing cracks or fractures play a crucial role in the overall integrity of such structures.     

Influence of flaw inclination angle and loading condition on crack initiation and propagation

With reference to the experimental observation of crack initiation and propagation from pre-existing flaws in rock specimens under compression, the influences of pre-existing flaw inclination angle on the cracking processes were analyzed by means of finite element method (FEM) and non-linear dynamics method. FEM analysis on the stress field distribution induced by the presence of a pre-existing flaw provided better understanding for the influence of flaw inclination angle on the initiation position and initiation angle of the potential cracks. Numerical analysis based on the non-linear dynamics method was performed to simulate the cracking processes. The resultant crack types, crack initiation sequences and the overall crack pattern were different under different loading conditions. Under a relatively low loading rate or a small magnitude of maximum loading pressure, tensile cracks would tend to initiate prior to shear cracks. In contrast, under a relatively high loading rate and a large magnitude of maximum loading pressure, shear cracks would tend to initiate prior to tensile cracks instead.     

The effects of the rock bridge ligament angle and the confinement on crack coalescence in rock bridges: An experimental study and discrete element method

The present article investigates the influences of the rock bridge ligament angle, β, and the confinement on crack coalescence patterns by conducting laboratory and numerical tests on rock-like specimens. Laboratory tests show that no coalescence in the rock bridge occurred for low β. With an increase of β, tensile-shear coalescence and tensile coalescences subsequently occurred. In addition, the increase in the confinement first promoted shear coalescence and then restrained crack coalescence for low β, whereas the tensile coalescence was restrained by the increase in confinement for high β. The numerical results corroborate the laboratory tests in the coalescence patterns. In addition, the numerical study shows that tensile and shear cracks subsequently initiated near crack tips because of the concentrated tensile and shear stresses, respectively. Regarding the influence of β on crack coalescence, tensile or shear stress failed to concentrate in rock bridges for low β. Therefore, the cracks failed to coalesce, whereas with the increase in β, tensile and shear stress concentrations occurred in the bridge and led to either tensile shear or tensile coalescence. Regarding the influence of confinement on crack coalescence, the increase in confinement restrained the tensile stress concentrations and further hindered tensile crack coalescence in rock bridges for high values of β.     

Numerical study on coalescence of two pre-existing coplanar flaws in rock

Crack propagation and coalescence processes are the fundamental mechanisms leading to progressive failure processes in rock masses, in which parallel non-persistent rock joints are commonly involved. The coalescence behavior of the latter, which are represented as pre-existing coplanar flaws (cracks), is numerically investigated in the present study. By using AUTODYN as the numerical tool, the present study systematically simulates the coalescence of two pre-existing coplanar flaws in rock under compression. The cumulative damage failure criterion is adopted in the numerical models to simulate the cumulative damage process in the crack initiation and propagation. The crack types (shear or tensile) are identified by analyzing the mechanics information associated with the crack initiation and propagation processes. The simulation results, which are generally in a good accordance with physical experimental results, indicate that the ligament length and the flaw inclination angle have a great influence on the coalescence pattern. The coalescence pattern is relatively simple for the flaw arrangements with a short ligament length, which becomes more complicated for those with a long ligament length. The coalescence trajectory is composed of shear cracks only when the flaw inclination angle is small (such as β ? 30°). When the pre-existing flaws are steep (such as β ? 75°), the coalescence trajectory is composed of tensile cracks as well as shear cracks. When the inclination angle is close to the failure angle of the corresponding intact rock material, and the ligament length is not long (such as L ? 2a), the direct shear coalescence is the more favorable coalescence pattern. In the special case that the two pre-existing flaws are vertical, the model will have a direct tensile coalescence pattern when the ligament length is short (L ? a), while the coalescence between the two inner flaw tips is not easy to achieve if the ligament length is long (L ? 2a).     

45钢柱壳爆炸膨胀断裂的SPH模拟分析

金属柱壳爆炸膨胀断裂存在拉伸、剪切及拉剪混合等多种断裂模式,目前其物理机制及影响因素还不清晰。本文中采用光滑粒子流体动力学方法（smoothed particle hydrodynamics, SPH）对45钢柱壳在JOB-9003及RHT-901不同装药条件下的外爆实验进行了数值模拟,探讨柱壳在不同装药条件下发生的剪切断裂、拉剪混合断裂模式及其演化过程,模拟结果与实验结果一致。SPH数值模拟结果表明:在爆炸加载阶段,随着冲击波在柱壳内、外壁间来回反射形成二次塑性区,沿柱壳壁厚等效塑性应变演化呈凸形分布,壁厚中部区域等效塑性应变较内、外壁大;在较高爆炸压力（JOB-9003）作用下,柱壳断裂发生在爆轰波加载阶段,损伤裂纹从塑性应变积累较大的壁厚中部开始沿剪切方向向内、外壁扩展,形成剪切型断裂模式;而在RHT-901空心炸药加载下,虽然裂纹仍从壁厚中部开始沿剪切方向扩展,但随后柱壳进入自由膨胀阶段,未断区域处于拉伸应力状态,柱壳局部发生结构失稳,形成类似“颈缩”现象,裂纹从剪切方向转向沿颈缩区向外扩展,呈现拉剪混合断裂模式。拉伸裂纹占截面的比例与柱壳结构失稳时刻相关。可见,柱壳断裂演化是一个爆炸冲击波与柱壳结构相互作用的过程,不能简单将其作为一系列膨胀拉伸环处理。     

Finite Element study of fracture initiation in flaws subject to internal fluid pressure and vertical stress

Hydraulic fracturing is a method used routinely in oil and gas exploitation and in engineered geothermal systems. While used frequently, there are many aspects of hydraulic fracturing, such as the direction of propagation of the newly-created fractures, which are not very well understood. Even though it is known that the local stress field plays a fundamental role in the orientation of the new fractures, there may be other factors, such as the geometry of the existing fractures and the magnitude of the hydraulic pressure applied, that may play a major role in the path that a new fracture follows when pressurized.The main goal of this study is to numerically analyze the effect of the ratio between a vertical load, or stress, and the hydraulic pressure applied in existing flaws on the stress field in the vicinity of the flaw tips. For that purpose, a double flaw geometry 2a-30-30 was modeled in the Finite Element code ABAQUS, and different vertical loads and internal flaw pressures were applied to the model. The variation of the maximum principal stresses and maximum shear stresses around the flaw tips were analyzed and related to fracture initiation.The study showed that the ratio between the water pressure applied in the flaws and the vertical load/stress (WP/VL) plays a crucial role in the magnitude and shape of the stress field around a flaw tip, and therefore in the location of tensile and shear fracture initiation. As WP/VL increases, the location of initiation of new tensile fractures shifts from the upper face of the studied flaw towards the region right ahead of the flaw tip; simultaneously, the location of initiation of new shear fractures shifts from the region ahead of the flaw tip to the upper face of the analyzed tip.     

雁形裂纹扩展的模型试验及断裂力学机制研究

通过相似材料模拟试验的方法研究了双轴压缩载荷作用下闭合雁形裂纹的起裂、扩展和岩桥的贯穿机理，得到了双轴压缩载荷作用下，不同方位雁形裂纹的开裂角、起裂载荷、岩桥贯通载荷及临界失稳载荷等重要的断裂力学参数，提出岩桥的破坏模式有剪切破坏、拉剪复合破坏和翼裂纹扩展三种     

An interacting micro-crack damage model for failure of brittle materials under compression

A model is developed for brittle failure under compressive loading with an explicit accounting of micro-crack interactions. The model incorporates a pre-existing flaw distribution in the material. The macroscopic inelastic deformation is assumed to be due to the nucleation and growth of tensile “wing” micro-cracks associated with frictional sliding on these flaws. Interactions among the cracks are modeled by means of a crack-matrix-effective-medium approach in which each crack experiences a stress field different from that acting on isolated cracks. This yields an effective stress intensity factor at the crack tips which is utilized in the formulation of the crack growth dynamics. Load-induced damage in the material is defined in terms of a scalar crack density parameter, the evolution of which is a function of the existing flaw distribution and the crack growth dynamics. This methodology is applied for the case of uniaxial compression under constant strain rate loading. The model provides a natural prediction of a peak stress (defined as the compressive strength of the material) and also of a transition strain rate, beyond which the compressive strength increases dramatically with the imposed strain rate. The influences of the crack growth dynamics, the initial flaw distribution, and the imposed strain rate on the constitutive response and the damage evolution are studied. It is shown that different characteristics of the flaw distribution are dominant at different imposed strain rates: at low rates the spread of the distribution is critical, while at high strain rates the total flaw density is critical.     

Loading rate and confining pressure effect on dilatancy,acoustic emission,and failure characteristics of fissured rock with two pre-existing flaws

Investigating the dilatancy, acoustic emission and failure characteristics of fissured rock are significant to ensure their geotechnical stability. In this paper, the uniaxial and triaxial compression experiments with AE monitoring under different loading rates were carried out on fissured rock specimens with the same geometrical distribution of two pre-existing flaws. The dilatancy and AE activity of these specimens were discussed, and the effects of the confining pressure and loading rate on the mechanical parameters and failure characteristics were analyzed. The results show that the exponential strength criterion is more suitable than the Mohr–Coulomb strength criterion to characterize the strength characteristics of fissured rock. The crack evolution and failure characteristics of fissured rock specimens are more complicated than those of intact rock specimens. The failure characteristics of the fissured rock follow the tensile shear coalescence model, crack branching occurs with increasing the loading rate, and the multi-section coalescence model is verified with increasing the confining pressure. The phenomena of stress drop and yield platform usually occur after the dilatancy onset, the specimen does not fail instantaneously, and the propagation and coalescence of cracks cause a sharp increase in the AE signals, circumferential strain, and volumetric strain.     

冲击载荷下两裂纹间的连通规律

脆性材料内部含有大量裂纹,当某一裂纹扩展时,其他裂纹会对扩展裂纹产生影响。为了研究冲击载荷下,脆性材料内两裂纹的相互影响、连通规律及裂纹尖端应力强度因子的变化规律,利用有机玻璃板制作了含非平行双裂纹的实验试件,利用落板冲击设备进行了中低速冲击实验,结合有限元分析软件ABAQUS计算出裂纹尖端应力强度因子,利用有限差分软件AUTODYN进行了动态数值模拟研究,并将其模拟结果与实验结果进行对比分析。实验及模拟结果表明:裂纹破坏形态与AUTODYN数值模拟破坏形态基本一致;试件的断裂形态随着两裂纹间距不同而不同;裂纹间的相互影响程度随着裂纹间间距增大而减小;裂纹尖端应力强度因子K_I随着裂纹间距的增大而减小,而K_II随着裂纹间距增大而增大。     

Numerical study of failure behaviour of pre-cracked rock specimens under conventional triaxial compression

Macroscopic pre-existing flaws play an important role in evaluating the strength and the failure modes of a heterogeneous rock mass. Crack initiation, propagation and coalescence from macroscopic pre-existing flaws are considered in a 3-D numerical model (RFPA3D) to investigate their effects on the underlying failure modes of rock. A feature of the code RFPA3D is that it can numerically simulate the evolution of cracks in three-dimensional space, as well as the heterogeneity of the rock mass. Three types of flaw geometries were evaluated numerically against experimental results: Type A for intact specimen, and Types B and C for flawed cylindrical specimens with different macroscopic pre-existing flaws, respectively. The effect of confining pressure on the fracture evolution was also considered. Numerical results showed that both the ligament angle and the flaw angle of two pre-existing cracks can affect the uniaxial compressive strength of the specimen and the mechanism of fracture evolution. In addition, both the uniaxial compressive strength and the accumulated acoustic emission increase with increasing heterogeneity.     

A 3-D ellipsoidal flaw model for brittle fracture in compression

Engineering materials are rarely free of flaws. Mode I cracking from pre-existing flaws is the major cause of the brittle fracture in compression of materials such as concrete and rock. A 3-D ellipsoidal flaw model is used to show the significant influence of flaw geometry on crack initiation in uniform uniaxial, biaxial and triaxial compression. The model shows that the governing criterion for crack initiation may change from energy to stress with increasing crack size, and that for voids of similar size a spherical void is the most critical shape for crack initiation. The model thus provides a basis for a better understanding of both the phenomenon and the mechanism of brittle fracture in compression.     

Modeling dynamic brittle behavior of materials with circular flaws or pores

Compressive failure of brittle materials is driven primarily by crack growth from pre-existing flaws in the material. These flaws, such as grain boundaries, pores, preexisting cracks, inclusions and missing grains, are randomly spaced and have a range of possible shapes and sizes. The current work proposes a micromechanics-based model for compressive dynamic failure of brittle materials with circular pore flaws, which incorporates both the number density and the size distribution of flaws. Results show that the distribution of flaw sizes is very important, particularly at moderate strain rate, since analyses based solely on the mean flaw size overpredict strength. Therefore, in order to increase dynamic strength at low to moderate strain rates, it is most effective to control the presence of large flaws. At very high strain rates, however, crack growth is activated even in small flaws and therefore controlling the total number density rather than the size of the flaws is effective for increasing dynamic strength. Finally, the model shows that neglecting very small flaws in the pore population may not have significant effects on the results in many cases, suggesting that the model is a useful tool for identifying a minimum resolution required for experimental characterization of microstructure.     

Environmentally assisted initiation and growth of multiple surface cracks

The initial stages of stress corrosion on an amorphous polymer is investigated. This is done by exposing stressed specimens of polycarbonate to an acetone and water solution. The surface develops two distinct features of degradation that appear on different length scales when subjected to tensile stress. Small pits form on the surface and make it rough. These pits are in the order of micrometers, and are found to be randomly distributed. They occur even without load and seem to slightly increase in number with increasing stress. In the millimeter domain, visible to the bare eye, surface cracks are formed transverse to the direction of loading. The occurrence of cracks is seen to have a positive stress threshold value, exceeding which, a linear increase of number of cracks with stress is found. The manners in which the cracks grow and coalesce on the surface are examined. It is seen that they do not meet crack tip to crack tip. Instead, they avoid each other initially and coalesce crack tip to crack side. The results are discussed in the light of mechanical considerations. A stress analysis for a few configurations of meeting cracks supports the experimental observations. With assumptions of stress corrosion crack growth and coalescence, a simulation of cracks growing from randomly distributed initiation sites is performed. Similar crack patterns as obtained in the experiments are found.     

Experimental investigation on strength and failure behavior of pre-cracked marble under conventional triaxial compression

Fractures in natural rocks have an important effect on the strength and failure behavior of rock mass, which are often evaluated in rock engineering practice. The theoretical evaluation of mechanical behavior of fractured rock mass has no satisfactory answer due to the role of confining pressure and crack geometry. Therefore, in this paper, conventional triaxial compression experiments were carried out to study the strength and failure behavior of marble samples with two pre-existing closed cracks in non-overlapping geometry. Based on the experimental results of a number of triaxial compression tests, the effect of crack coalescence on the axial supporting capacity and deformation property were investigated with different confining pressures. The results show that intact samples and flawed samples (marble with pre-existing cracks) have different deformation properties after peak stress, which change from brittleness to plasticity and ductility with the increase of confining pressure. The peak strength and failure mode are found depending not only on the geometry of flaw, but also on the confining pressure. The strength of flawed samples shows distinct non-linear behavior, which is in a better agreement with non-linear Hoek–Brown criterion than linear Mohr–Coulomb criterion. For a kind of rock that has been evaluated as a Hoek–Brown material, a new evaluation criterion is put forward by adopting optimal approximation polynomial theory, which can be used to confirm more precisely the strength parameters (cohesion and internal friction angle) of flawed samples. For intact samples, the marble leads to typical shear failure mode with a single fracture surface under different confining pressures, while for flawed samples, under uniaxial compression and a lower confining pressure (σ₃ = 10 MPa), tests for coarse and medium marble (the coarse and medium refer to the grain size) exhibit three basic failure modes, i.e., tensile mode, shear mode, and mixed mode (tensile and shear). Shear mode is associated with lower strength behavior. However, under higher confining pressures (σ₃ = 30 MPa), for coarse marble, the axial supporting capacity is not related to the geometry of flaw. The friction among crystal grains determines the strength behavior of coarse marble. For medium marble, the failure mode and deformation behavior are dependent on the crack coalescence in the sample. The present research provides increased understanding of the fundamental nature of rock failure under conventional triaxial compression.     

ADVANCES IN RESEARCH OF HYDRAULIC FRACTURES IN UNCONSOLIDATED SANDS

疏松砂岩水力压裂裂缝的典型特征是大量分支缝、微裂缝和一定钝度的裂缝尖端及剪切带的形成等. 主要通过对相关文献的调研,分析了剪切作用、拉伸作用等对裂缝形态特征（起裂、转向、延伸）等力学行为所产生的影响,通过针对一些疏松砂岩的压裂实验研究发现,其滤失过程主要受到缝尖区域的影响,而且滤失一般发生在裂缝扩展之前. 裂缝缝尖周围的局部应力往往是影响裂缝起裂和裂缝开度的主导参数,裂缝的形态特征则是应力、渗透率、流态等综合作用的结果,对疏松砂岩裂缝形态特征的形成做了进一步的解释.     

The simulation of dynamic crack propagation using the cohesive segments method

The cohesive segments method is a finite element framework that allows for the simulation of the nucleation, growth and coalescence of multiple cracks in solids. In this framework, cracks are introduced as jumps in the displacement field by employing the partition of unity property of finite element shape functions. The magnitude of these jumps are governed by cohesive constitutive relations. In this paper, the cohesive segments method is extended for the simulation of fast crack propagation in brittle solids. The performance of the method is demonstrated in several examples involving crack growth in linear elastic solids under plane stress conditions: tensile loading of a block; shear loading of a block and crack growth along and near a bi-material interface.     

Damage evolution and crack propagation in rocks with dual elliptic flaws in compression

To give an insight into the understanding of damage evolution and crack propagation in rocks,a series of uniaxial and biaxial compression numerical tests are carried out.The investigations show that damage evolution occurs firstly in the weak rock,the area around the flaw and the area between the flaw and the neighboring rock layer.Cracks mostly generate as tensile cracks under uniaxial compression and shear cracks under biaxial compression.Crack patterns are classified and divided.The relationship between the accumulated lateral displacement and the short radius(b)is fitted,and the equation of crack path is also established.