A modified 3D mean strain energy density criterion for predicting shale mixed-mode Ⅰ/Ⅲ fracture toughness

The fracture toughness of rocks is a critical fracturing parameter in geo-energy exploitation playing a significant role in fracture mechanics and hydraulic fracturing.The edge-notched disk bending(ENDB)specimens are employed to measure the entire range of mixed-modeⅠ/Ⅲfracture toughness of Longmaxi shale.To theoretically interpret the fracture mechanisms,this research first introduces the detailed derivations of three established fracture criteria.By distinguishing the volumetric and distortional strain energy densities,an improved three-dimensional mean strain energy density(MSED)criterion is proposed.As the critical volumetric to distortional MSED ratio decreases,the transition from tensiondominated fracture to shear-dominated fracture is observed.Our results indicate that both peak load and applied energy increase significantly with the transition from pure mode I(i.e.,tension)to pure modeⅢ(i.e.,torsion or tearing)since mode-Ⅲcracking happens in a twisted manner and mode-Ⅰcracking occurs in a coplanar manner.The macroscopic fracture signatures are consistent with those of triaxial hydraulic fracturing.The average ratio of pure mode-Ⅲfracture toughness to pure mode-Ⅰfracture toughness is 0.68,indicating that the obtained mode-Ⅲfracture resistance for a tensionbased loading system is apparent rather than true.Compared to the three mainstream fracture criteria,the present fracture criterion exhibits greater competitiveness and can successfully evaluate and predict mixed-modeⅠ/Ⅲfracture toughness of distinct materials and loading methods.

Effect of specimen thickness on Mode Ⅱ fracture toughness of rock

Anti symmetric four point bending specimens with different thickness, without and with guiding grooves, were used to conduct Mode Ⅱ fracture test and study the effect of specimen thickness on Mode Ⅱ fracture toughness of rock. Numerical calculations show that the occurrence of Mode Ⅱ fracture in the specimens without guiding grooves (when the inner and outer loading points are moved close to the notch plane) and with guiding grooves is attributed to a favorable stress condition created for Mode Ⅱ fracture, i.e. tensile stress at the notch tip is depressed to be lower than the tensile strength or to be compressive stress, and the ratio of shear stress to tensile stress at notch tip is very high. The measured value of Mode Ⅱ fracture toughness K ⅡC decreases with the increase of the specimen thickness or the net thickness of specimen. This is because a thick specimen promotes a plane strain state and thus results in a relatively small fracture toughness.

Prediction of mode I fracture toughness of rock using linear multiple regression and gene expression programming

Prediction of mode I fracture toughness(KIC) of rock is of significant importance in rock engineering analyses. In this study, linear multiple regression(LMR) and gene expression programming(GEP)methods were used to provide a reliable relationship to determine mode I fracture toughness of rock. The presented model was developed based on 60 datasets taken from the previous literature. To predict fracture parameters, three mechanical parameters of rock mass including uniaxial compressive strength(UCS), Brazilian tensile strength(BTS), and elastic modulus(E) have been selected as the input parameters. A cluster of data was collected and divided into two random groups of training and testing datasets.Then, different statistical linear and artificial intelligence based nonlinear analyses were conducted on the training data to provide a reliable prediction model of KIC. These two predictive methods were then evaluated based on the testing data. To evaluate the efficiency of the proposed models for predicting the mode I fracture toughness of rock, various statistical indices including coefficient of determination(R2),root mean square error(RMSE), and mean absolute error(MAE) were utilized herein. In the case of testing datasets, the values of R2, RMSE, and MAE for the GEP model were 0.87, 0.188, and 0.156,respectively, while they were 0.74, 0.473, and 0.223, respectively, for the LMR model. The results indicated that the selected GEP model delivered superior performance with a higher R2value and lower errors.

Mode I rock fracture toughness with different types of brazilian disc

According to the results evaluated by researchers for mode I rock fracture toughness measurement, a series of comparison tests with different types Brazilian discs were conducted in order to search for the simplest geometry specimens by which accurate, comparable and consistent mode I rock fracture toughness could obtain.

MECHANISM ANALYSIS OF THICKNESS EFFECT ON MIXED MODE Ⅰ/Ⅱ FRACTURE OF LC4-CS ALUMINUM ALLOY

Mixed mode Ⅰ/Ⅱ fracture erperiments of LC4-CS aluminum alloy were conductedby using tension--shear specimens with thicknesses of 2, 4, 8 and 14mm. Fracturemechanisms of thickness effect on mixed mode Ⅰ/Ⅱ fracture were first examined fromfracture surface morphology to correlate with the macroscopic fracture behavior andstress state. It is found that specimen thickness has a strong influence on mixed modefracture. As thickness varies from thin to thick the macroscopic fracture surfacesappear the characteristics of plane stress state (2mm, 4mm--thick specimen), three--dimensional stress state (8mm--thick specimens), and plane strain state (14mm--thickspecimens), respectively. The specimens of all kinds of thicknesses are typical of ten-sile type failure under mode Ⅰ loading condition and shear type failure under mode Ⅱloading condition. Two distinct features coexist on the fracture surfaces under mixedmode loading conditions, and the corresponding proportion varies with loading mix-ity. Void--growth processes are the failure mechanism in both predominately tensile-and shears--type fractures. The size and depth of dimples on the fracture surface varygreatly with thickness. Therefore, it is extraordinary necessary to take into accountthe thickness effect when a mixed mode fracture criterion is being established.

Tensile and tear-type fracture toughness of gypsum material:Direct and indirect testing methods

In this context,four specimens,i.e.(i)circumferentially notched cylindrical torsion(CNCT),(ii)circum-ferentially notched cylindrical direct tension(CNCDT),(iii)edge notch disc bend(ENDB)and(iv)three-point bend beam(3PBB),were utilized to measure the modesⅠandⅢfracture toughness values of gypsum.While the CNCT specimen provides pure modeⅢloading in a direct manner,this pure mode condition is indirectly produced by the ENDB specimen.The ENDB specimen provided lower KⅢc and a non-coplanar(i.e.twisted)fracture surface compared with the CNCT specimen,which showed a planar modeⅢfracture surface.The ENDB specimen is also employed for conducting pure modeⅠ(with different crack depths)and mixed modeⅠ/Ⅲtests.KIc value was independent of the notch depth,and it was consistent with the RILEM and ASTM standard methods.But the modeⅢfracture results were highly sensitive to the notch depth.While the fracture resistance against modeⅢwas significantly lower than that of modeⅠ,the greater work of fracture under modeⅢwas noticeable.

Fracture Toughness of Glass-Carbon (0/90)_sFiber Reinforced Polymer Composite – An Experimental and Numerical Study

Mode-I fracture behavior of glass-carbon fiber reinforced hybrid polymer composite was investigated based on experimental and finite element analysis. The compact tension (CT) specimen was employed to conduct mode-I fracture test using special loading fixtures as per ASTM standards. Fracture toughness was determined experimentally for along and across the fiber orientation of the specimen. Results indicated that the cracked specimens are tougher along the fiber orientations as compared with across the fiber orientations. A similar fracture test was simulated using finite element analysis software ANSYS. Critical stress intensity factor (K) was calculated at fracture/failure using displacement extrapolation method, for both along and across the fiber orientations. The fractured surfaces of the glasscarbon epoxy composite under mode-I loading condition was examined by electron microscope.

Simultaneously measuring initiation toughness of pure mode Ⅰ and mode Ⅱ cracks subjected to impact loads

In order to simultaneously measure the initiation toughness of pure mode Ⅰ and mode Ⅱ cracks in one specimen,a large-size double-cracked concave-convex plate(DCCP)specimen configuration was proposed.Impacting tests were implemented in the drop plate impact device.Strain gauges were employed to measure impact loads and crack initiation time.The corresponding numerical model was established by using the dynamic finite difference program AUTODYN,and the experimental-numerical method and ABAQUS code were utilized to obtain the initial fracture toughness of the crack.Using experiments and numerical research,we concluded that the DCCP specimen is suitable for measuring the initial fracture toughness of pure mode Ⅰ and mode Ⅱ cracks at the same time;the dynamic initiation toughness increases with the increase of loading rate and the crack initiation time decreases with increasing loading rate;the initiation toughness of mode Ⅱ crack is 0.5 times that of mode Ⅰ crack when subjected to the same loading rate.For the pre-crack in the vicinity of the bottom of a sample,when its length increases from 20 to 100 mm,the dynamic initiation toughness of the pure mode Ⅰ crack gradually decreases,and the longer the lower crack length is,the easier the crack would initiate,but the dynamic initiation toughness of pure mode Ⅱ crack varies little.

Dynamic ModeⅡfracture behavior of rocks under hydrostatic pressure using the short core in compression(SCC)method

The shear failure of rocks under both a static triaxial stress and a dynamic disturbance is common in deep underground engineering and it is therefore essential for the design of underground engineering to quantitively estimate the dynamic ModeⅡfracture toughness KⅡCof rocks under a triaxial stress state.However,the method for determining the dynamic KⅡCof rocks under a triaxial stress has not been developed yet.With an optimal sample preparation,the short core in compression(SCC)method was designed and verified in this study to measure the dynamic KⅡCof Fangshan marble(FM)subjected to different hydrostatic pressures through a triaxial dynamic testing system.The formula for calculating the dynamic KⅡCof the rock SCC specimen under hydrostatic pressures was obtained by using the finite element method in combination with secondary cracks.The experimental results indicate that the failure mode of the rock SCC specimen under a hydrostatic pressure is the shear fracture and the KⅡCof FM increases as the loading rate.In addition,at a given loading rate the dynamic rock KⅡCis barely affected by hydrostatic pressures.Another important observation is that the dynamic fracture energy of FM enhances with loading rates and hydrostatic pressures.

A MIXED MODE FRACTURE CRITERION BASED ON THE MAXIMUM TANGENTIAL STRESS IN BRITTLE INCLUSION

A closed-form solution for predicting the tangential stress of an inclusion located in mixed mode Ⅰ and Ⅱ crack tip field was developed based on the Eshelby equivalent inclusion theory. Then a mixed mode fracture criterion, including the fracture direction and the critical load, was established based on the maximum tangential stress in the inclusion for brittle inclusioninduced fracture materials. The proposed fracture criterion is a function of the inclusion fracture stress, its size and volume fraction, as well as the elastic constants of the inclusion and the matrix material. The present criterion will reduce to the conventional one as the inclusion having the same elastic behavior as the matrix material. The proposed solutions are in good agreement with detailed finite element analysis and measurement.

Rock fracture under anti-plane shear (Mode Ⅲ) loading

Anti-plane punch-through shear test and anti-planefour-point bending test are used to study the crack initiation and propagation under anti-plane shear (Mode Ⅲ) loading. The tensile and shear stresses at the crack tip are calculated by finite element method. The results show that under Mode Ⅲ loading the maximum principal stress σ1 at crack tip is smaller or a little larger than the maximum shear stress τmax. Since the tensile strength of brittle rock is much lower than its shear strength, σ1 is easy to reach its critical value before τmax reaches its critical value and thus results in Mode I fracture. The fracture trajectory is helicoid and the normal direction of tangential plane with the fractured helicoid is along the predicted direction of the maximum principal stress at the notch tip. It is further proved that Mode Ⅰ instead of Mode Ⅲ fracture occurs in brittle rock under Mode Ⅲ loading. The fracture mode depending on the fracture mechanism must be distinguished from the loading form.

MODE II AND MIXED MODE I-II ROCK FRACTURE RESEARCH

of PhD thesis For the mode I rock fracture toughness measurement,three standard methods have been recommended by the ISRM,but there has not been a standard method for the determination of mode II and mixed mode I-II rock fracture toughness. However mode II and mixed mode I-II fracturing of rock structures is more commonly observed than mode I in various geological and structural engineering settings. So it is of great important to thoroughly research these rock fracture problems and establish a standard method for determining the mode II or mixed mode I-II fracture toughness for rock materials. Based on the progress made for mode I rock fracture research,the cracked chevron notched Brazilian disk (CCNBD) specimen was also introduced for mode II and mixed mode I-II rock fracture toughness measurement. When the crack is orientated at an angle with respect to the diametrical loading,the crack of the CCNBD specimen is exposed to the mode II or mixed mode I-II stress distribution conditions. The solutions for stress intensity factors in the vicinity of the crack tip have been evaluated by the stepwise superimposition technique. In order to make sure that the theoretical analysis is correct,numerical calculation method has been employed to calibrate the theoretical results. It has been proved that the theoretical results yielded by the dislocation method are correct and reliable. According to the characteristic that the propagation of the crack in the CCNBD specimen is in its own plane and application of the energy superposition principle,the stress intensity factor of the mixed mode I-II has been defined in dimensionless terms as 212II2Imix])()[(***+=YYY. It was found that the curve of *mixY was concave. There exists a lowest point which corresponds to the maximum external load and indicates the crack has reached its critical state. Since the values of ***IIImix and YYY, are only dependent on the specimen geometry (qaaa and 10B,,),the critical values of ***IIImix and YYY, can be to known as long as the CCNBD specimen is prepared ready. It is only necessary to record the maximum load during the fracture tests. The fracture locus is very useful to know whether the crack in a rock structure has reached its critical condition. According to the amount of practical fracture testing data obtained,the rock fracturing locus was found to be 123IICII23ICI=+KKKK and the S-critical criterion was found to be more suitable for rock mixed mode I-II fracturing assessment.

基于ABAQUS平台考虑T应力的Ⅰ型裂纹扩展模拟开发

工程结构在制造工艺过程中或使用期间会产生裂纹,对结构断裂路径的预测和研究是防治工程安全问题发生的重要手段。在考虑裂纹尖端应力场常数项T应力的基础上对传统的最大周向应力准则(Maximum tangential stress criterion,MTS)和最小应变能密度因子准则(Minimum strain energy density criterion,SED)进行修正,采用Python语言对ABAQUS的前、后处理和有限元计算模块进行二次开发,通过计算最优解的粒子群算法(Particle swarm optimization,PSO)将修正后的准则编入裂纹自动扩展程序脚本中。利用上述二次开发程序对初始纯Ⅰ型裂纹的扩展路径进行模拟,结果表明:采用ABAQUS脚本程序模拟结果与相关文献实验结果吻合,表明了程序的有效性,进而实现考虑T应力的多种断裂准则对裂纹扩展路径的预测;当T应力值处于一定范围内时,修正的MTS准则无法预测裂纹发生的偏转现象,扩展路径呈直线,此时可采用修正的SED准则进行预测。

Is crack branching under shear loading caused by shear fracture? ——A critical review on maximum circumferential stress theory

When a crack is subjected to shear force, crack branching usually occurs. Theoretical study shows that the crack branching under shear loading is caused by tensile stress, but not caused by shear fracture. The co plane shear fracture could be obtained if compressive stress with given direction is applied to the specimen, subsequently, calculated shear fracture toughness, K ⅡC , is larger than K ⅠC . A prerequisite of possible occurrence of mode Ⅱ fracture was proposed. The study of shear fracture shows that the maximum circumferential stress theory considered its criterion as a parametric equation of a curve in K Ⅰ, K Ⅱ plane is incorrect; the predicted ratio K ⅡC / K ⅠC =0.866 is incorrect too. [

Tests and Simulation Analysis on Fracture Performance of Concrete

To obtain the fracture parameters of concrete, fracture tests were conducted with three-point bending beam method aiming at 30 concrete beams with different sizes and different intensity. The concrete specimen with prefabricated crack to determine the fracture parameters of concrete were conducted and the fracture performance of the specimen was analyzed. The test results show that, initial fracture toughness is unrelated to the size of specimens; while unstable fracture toughness is related to the size of specimens. As for specimens of bastard size, when concrete intensity is relatively low, unstable fracture toughness increases along with the increase of intensity; when concrete intensity is relatively high, unstable fracture toughness will decrease; when concrete intensity increases continuously, unstable fracture toughness will further increase somewhat. As for specimens of standard size, unstable fracture toughness will increase along with the increase of intensity. Aiming at concrete beam specimens, we conducted two-dimensional non-linear finite element analysis, obtained the stress intensity factor, and carried out contrastive analysis with the experimental results.

Anisotropy in shear-sliding fracture behavior of layered shale under different normal stress conditions

Shear-sliding mode(mode Ⅱ)fracture of rocks is a vital failure form in deep underground engineering.To gain deep insight into the anisotropic shear fracture behaviors of a typical shale under high normal stress conditions,a series of direct shear tests were conducted on double-notched specimens in three typical bedding orientations(i.e.,the arrester,divider,short-transverse orientations)and under five normal stresses.The modeⅡfracture toughness(K_(Ⅱc))is found to exhibit a significant 3D anisotropy.The maximum K_(Ⅱc)is obtained in the divider orientation,followed by those in the arrester and short-transverse orientations.In contrast,the 3D anisotropy in the critical modeⅡenergy release rate(G_(Ⅱc))is not as significant as that in K_(Ⅱc),and G_(Ⅱc)in the arrester orientation is quite close to that in the divider orientation.The anisotropy in the prepeak input energy accumulated during shearing is found to be exactly consistent with that in G_(Ⅱc),which has not been noted before.Furthermore,the anisotropies in the modeⅡfracture resistances will,unexpectedly,not be weakened by the high normal stress.Owing to the layered structures,tensile cracks are involved during the modeⅡfracture process,resulting in the formation of rough fracture surfaces.

Application of split Hopkinson tension bar technique to the study of dynamic fracture properties of materials

A novel approach is proposed in determining dy- namic fracture toughness （DFT） of high strength steel, using the split Hopkinson tension bar （SHTB） apparatus, com- bined with a hybrid experimental-numerical method. The center-cracked tension specimen is connected between the bars with a specially designed fixture device. The fracture initiation time is measured by the strain gage method, and dynamic stress intensity factors （DSIF） are obtained with the aid of 3D finite element analysis （FEA）. In this approach, the dimensions of the specimen are not restricted by the connec- tion strength or the stress-state equilibrium conditions, and hence plane strain state can be attained conveniently at the crack tip. Through comparison between the obtained results and those in open publication, it is concluded that the ex- perimental data are valid, and the method proposed here is reliable. The validity of the obtained DFT is checked with the ASTM criteria, and fracture surfaces are examined at the end of paper.

2D-SiC_(f)/SiC复合材料层间Ⅰ型断裂试验及表征

二维编织碳化硅纤维增强碳化硅复合材料(2D-SiC_(f)/SiC)在航空领域中得到广泛使用,然而该材料层间强度低,使其易萌生层间裂纹,引起分层破坏。为此,本工作采用楔形双悬臂梁法(W-DCB)和悬臂梁法(DCB)开展了层间Ⅰ型断裂试验,获得了2D-SiC_(f)/SiC的层间裂纹驱动的加载数据,得到了其裂纹端口张开力及张开位移变形曲线。在试验加载过程,通过光学显微镜监测了视觉裂纹扩展过程,探究了2D-SiC_(f)/SiC的层间I型裂纹扩展规律。结合理论分析和裂纹视觉特征解释了加载曲线拐点及其他特征点的断裂力学含义。利用扫描电子显微镜分析了2D-SiC_(f)/SiC的层间断面特征,揭示了断面分层裂纹扩展机制。结果表明:W-DCB方法测量的2D-SiC_(f)/SiC层间Ⅰ型初始能量释放率与DCB方法等效;2D-SiC_(f)/SiC层间Ⅰ型断裂过程中,裂纹端口变形曲线的多峰性不符合经典线弹性断裂力学预测的加载峰后特征,反映了2D-SiC_(f)/SiC层间约束关系的复杂性;层间断面为结构性非完全损伤,发生了局部纤维桥连现象。

3D打印CFRP-Ⅰ型层间断裂韧性的断层替换测试法及参数化分析

为实现3D打印碳纤维增强树脂基复合材料(carbon fiber reinforced polymer,CFRP)Ⅰ型层间断裂韧性的测试分析,并量化评价打印参数对该型断裂韧性的影响规律,推进3D打印CFRP技术在桥梁工程中的应用,分别从试验及仿真角度开展了相关探索。首先,提出了一种新型的3D打印CFRP层间预制裂纹制备方法-断层替换法,利用该方法开展了系统性的试验研究,分析了两类关键打印参数对Ⅰ型层间断裂韧性的影响规律。其次,基于内聚区理论及surface-based cohesive关系,构建了各类打印工况下的预制裂纹试件双悬臂梁(double cantilever beam,DCB)试验的仿真模型,并完成了仿真计算与试验测试的对比分析。结果表明:3D打印CFRP的Ⅰ型层间断裂韧性随着打印温度及速度的提高而增大,且打印温度对断裂韧性的影响更显著,当温度由245℃升至285℃时,断裂韧性提高了294.6%;当速度由20 mm/s增至60 mm/s时,断裂韧性提高了27.6%;当打印温度与速度提高时,试件在打印过程中热量散失更小,Ⅰ型层间力学性能得到提高;但当打印速度过大时,印材在打印过程中无法完全沉积并实现较好的黏结,反而对Ⅰ型层间力学性能造成负面影响。同时,仿真结果与试验数据的相对误差均控制在15%以内,表明提出的断层替换法在I型层间断裂韧性试验数据获取方面是合理且稳定的。因此,该方法可为后续量化评价3D打印CFRP桥梁工程构件的层间力学性能提供试验技术支撑。

加载速率对高强钢40Cr和30CrMnSiNi2AⅠ型动态断裂韧性的影响

采用实验-数值方法对40Cr和30CrMnSiNi2A两种高强钢三点弯曲试样在不同加载速率的冲击载荷作用下进行了动态断裂韧性的测试,并对其率相关性进行研究．实验在Hopkinson压杆系统上完成,试样的起裂时间采用应变片法测得．结合有限元三维动态模拟,得到了不同加载速率下试样动态应力强度因子的时间历程并由实测的起裂时间确定材料动态断裂韧性．结果表明,在本工作加载速率范围内(10~6MPa·m^(1/2)/s),40Cr钢为解理型断裂,其动态断裂韧性随加载速率增加的变化趋势不明显；而30CrMnSiNi2A钢则在较大程度上表现为延性断裂的特征,其动态断裂韧性随加载速率的增加呈明显上升趋势．并对上述结果进行了宏观和微观机理分析．