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.

Fracture behavior and mechanism of highly fragmented steel cylindrical shell under explosive loading

An in-depth understanding of the fracture behavior and mechanism of metallic shells under internal explosive loading can help develop material designs for warheads and regulate the quantity and mass distribution of the fragments formed.This study investigated the fragmentation performance of a new high-carbon silicon-manganese(HCSiMn)steel cylindrical shell through fragment recovery experiments.Compared with the conventional 45Cr steel shell,the number of small mass fragments produced by the HCSi Mn steel shell was significantly increased with a scale parameter of 0.57 g fitted by the Weibull distribution model.The fragmentation process of the HCSi Mn shell exhibited more brittle tensile fracture characteristics,with the microcrack damage zone on the outer surface being the direct cause of its high fragmentation.On the one hand,the doping of alloy elements resulted in grain refinement by forming metallographic structure of tempered sorbite,so that microscopic intergranular fracture reduces the characteristic mass of the fragments;on the other hand,the distribution of alloy carbides can exert a"pinning"effect on the substrate grains,causing more initial cracks to form and propagate along the brittle carbides,further improving the shell fragmentation.Although the killing power radius for light armored vehicles was slightly reduced by about 6%,the dense killing radius of HCSiMn steel projectile against personnel can be significantly increased by about 26%based on theoretical assessment.These results provided an experimental basis for high fragmentation warhead design,and to some extent,revealed the correlation mechanism between metallographic structure and shell fragmentation.

Dynamic Phase Field Description on Ductile Fracture Process

In comparison to discrete descriptions of fracture process,the recently proposed phase field methodology averts the numerical tracking strategy of discontinuities in solids,which enables the numerical implement simplification.An implicit finite element formulation based on the diffuse phase field is extended for stable and efficient analysis of complex dynamic fracture process in ductile solids.This exhibited formulation is shown to capture entire range of the characteristics of ductile material presenting J2-plasticity,embracing plasticization,cracks initiation,propagation,branching and merging while fulfilling the basic principle of thermodynamics.Herein,we implement a staggered time integration scheme of the dynamic elasto-plastic phase field method into the commercial finite element code.The numerical performance of the present advanced phase field model has been examined through several classic dynamic fracture benchmarks,and in all cases simulation results are in good agreement with the associated experimental data and other numerical results in previous literature.

Effect of dynamic loading orientation on fracture properties of surrounding rocks in twin tunnels

For expedited transportation,vehicular tunnels are often designed as two adjacent tunnels,which frequently experience dynamic stress waves from various orientations during blasting excavation.To analyze the impact of dynamic loading orientation on the stability of the twin-tunnel,a split Hopkinson pressure bar(SHPB)apparatus was used to conduct a dynamic test on the twin-tunnel specimens.The two tunnels were rotated around the specimen’s center to consider the effect of dynamic loading orientation.LS-DYNA software was used for numerical simulation to reveal the failure properties and stress wave propagation law of the twin-tunnel specimens.The findings indicate that,for a twin-tunnel exposed to a dynamic load from different orientations,the crack initiation position appears most often at the tunnel corner,tunnel spandrel,and tunnel floor.As the impact direction is created by a certain angle(30°,45°,60°,120°,135°,and 150°),the fractures are produced in the middle of the line between the left tunnel corner and the right tunnel spandrel.As the impact loading angle(a)is 90°,the tunnel sustains minimal damage,and only tensile fractures form in the surrounding rocks.The orientation of the impact load could change the stress distribution in the twin-tunnel,and major fractures are more likely to form in areas where the tensile stress is concentrated.

A Review of Experimental Research on the Mode I Fracture Behavior of Bamboo

Bamboo is an eco-friendly material with light weight,high strength,short growth cycle and high sustainability,which is widely used in building structures.Engineered bamboo has further promoted the development of modern bamboo structures due to its unrestricted size and shape.However,as a fiber-reinforced material,fracture damage,especially Mode I fracture damage,becomes the most likely damage mode of its structure,so Mode I fracture characteristics are an important subject in the research of mechanical properties of bamboo.This paper summarizes the current status of experimental research on the Mode I fracture properties of bamboo based on the three-point bending(TPB)method,the single-edge notched beam(SENB)method,the compact tension(CT)method and the double cantilever beam(DCB)method,compares the fracture toughness of different species of bamboo,analyzes the toughening mechanisms and fracture damage modes,discusses the applicability of different theoretical calculation methods,and makes suggestions for future research priorities,aiming to provide a reference for future research and engineering applications in related fields.

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.

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.

Experimental study on time-dependent stress and strain of in-plane shear(ModeⅡ) fracture process of rock

Shear-box test with strain measurement was used to study time-dependent stress and strain of in-plane shear(Mode Ⅱ) fracture process of rock and to reveal the mechanism of Mode Ⅱ fracture.Numerical results show that the maximum shear stress τmax at the crack tip is much larger than the maximum tensile stress σ1 and the ratio of τmax/σ1 is about 5,which favors Mode Ⅱ fracture of rock.Test results indicate that the strain-time curve comprises three stages:the linear deformation stage,the micro-cracking stage and the macroscopic crack propagation.The strain in the direction of the original notch plane is negative,due to restraining effect of compressive loading applied to the original notch plane.Both σ1 and τmax are increased as the load increases,but the slope of τmax is larger than that of σ1 and the value of τmax is always larger than that of σ1.Therefore,τmax reaches its limited value at peak load before σ1 and results in Mode Ⅱ fracture of rock.Shear-box(i.e.compression-shear) test becomes a potential standard method for achieving the true Mode Ⅱ fracture and determining Mode Ⅱ fracture toughness of rock.

Experiment on Mode Ⅰ/Ⅱ Mixed Interfacial Fracture Characterization of Foam Core Sandwich Materials at Elevated Temperatures

Foam-cored sandwich materials have been widely used in the civil engineering due to their advantages such as lightweight,high strength,and excellent anti-corrosion ability. However,the interfacial bonding strength of foamcored sandwich materials is weakened at elevated temperatures. In practice,the effect of high temperature cannot be ignored,because the composites and foams are sensitive to the change of temperature in the environment. In this study,a series of single-leg bending beams were tested at different temperatures to evaluate the influences of high temperatures on Mode Ⅰ/Ⅱ mixed interfacial fracture of foam core sandwich materials. The temperature was from29 ℃ to 90 ℃,covered the glass transition temperature of composites and foam core,respectively. The Mode Ⅰ/Ⅱ mixed interfacial crack prorogation and its corresponding interfacial strain energy release rate were summarized.

EFFECT OF TEMPERATURE ON THE FRACTURE MODES OF BHW-35 STEEL UNDER MODE-Ⅱ LOADING

Finite element method(FEM)has been used to analyze the stress and strain fields and the stress tri-axial levels around the tip of the crack under mode-Ⅱ loading.The results show that: under mode-Ⅱ loading,the direction of the maximum tensile stress and that of the maximum tri-axial levels(R_ )exist at an angle of-75.3°from the original crack plane:the maximum shear stress and R_ =0 exist along the original crack plane. Mode-Ⅱ loading experiment using BHW-35 steel at different temperatures show that there are two kinds of fracture mode.opening mode(or tensile mode)and sliding mode(or shear mode).A de- crease in temperature causes the fracture mode to change from shear mode to tensile mode.For BHW- 35 steel,this critical temperature is about-90C.Actually.under any kind of loading mode(mode Ⅰ,mode Ⅱ,mode Ⅲ or mixed mode),there always exist several kinds of potenital fracture modes (for example,opening mode.sliding mode.tearing mode or mixed mode).The effect of temperature under mode-Ⅱ loading is actually related to the change of the elastic-plastic properties of the material.

Dynamic behavior and fracture mode of TiAl intermetallics with different microstructures at elevated temperatures

Experimental studies were conducted on the tensile behaviors and fracture modes of TiAl（Ti-46.5Al-2Nb-2Cr） alloys with near gamma（NG） equiaxed and near lamellar（NL） microstructures over a temperature range from room temperature to 840 ℃ and a strain rate range of 0.001-1 350 s-1.The results indicate that the alloys are both temperature and strain rate dependent and they have a similar dependence.The dynamic strength is higher than the quasi-static strength but almost insensitive to high strain rate range of 320-1 350 s-1.The brittle-to-ductile transition temperature（BDTT） increases with increasing strain rates.NG TiAl yields obviously,while NL TiAl does not.Below BDTT,as the temperature increases,the fracture modes of the two alloys change from planar cleavage fracture to a mixture of transgranular and intergranular fractures,and finally to totally intergranular fracture.

Transition of plasticity and fracture mode of Zr-Al-Ni-Cu bulk metallic glasses with network structures

Effect of network structure on plasticity and fracture mode of Zr？Al？Ni？Cu bulk metallic glasses （BMGs） was investigated. The microstructures of transversal and longitudinal sections were exposed by chemical etching and observed by scanning electron microscopy （SEM）. The mechanical properties were examined by room-temperature uniaxial compression test. The results show that both plasticity and fracture mode are significantly affected by the network structure and the alteration occurs when the size of the network structure reaches up to a critical value. When the cell size （dc） of the network structure is ~3μm, Zr-based BMGs characterize in plasticity that decreases with increasingdc. The fracture mode gradually transforms from single 45° shear fracture to double 45° shear fracture and then cleavage fracture with increasingdc. In addition, the mechanisms of the transition of the plasticity and the fracture mode for these Zr-based BMGs are also discussed.

Meso-mechanical anisotropy and fracture evolution of reef limestones from the Maldives Islands and the South China Sea

Reef limestone is a biogenic sedimentary rock widely distributed in coral reef areas, acting as an important foundation for coast construction. Due to its special biogenic origin, reef limestone is different from conventional rocks both in terms of rock structure and mechanical properties. In this study, mesoscale uniaxial compression experiments with five different loading directions were conducted on two kinds of reef limestones from the Maldives Islands and the South China Sea, respectively. The real-time high-resolution videos and images of failure processes were recorded simultaneously to investigate the fracture evolution and fracture surface roughness of reef limestones. It demonstrated that the reef limestones belonged to extremely soft to soft rocks, and their uniaxial compressive strength (UCS) values fluctuated with high discreteness. The mesoscale mechanical properties of reef limestones were highly anisotropic and mainly controlled by pore structure. The occurrence of dissolution pores in reef limestone tended to intensify mechanical anisotropy. With the integration of the fracture initiation and propagation features of reef limestones, it is supposed that the intrinsic mechanism of anisotropy was probably attributed to the differences in coral growth direction and dissolution. Furthermore, the quantified fracture surface roughness was revealed to have a good consistency with density and UCS for the reef limestones from the South China Sea. The findings are helpful for providing theoretical and experimental references for engineering construction in coral reef areas.

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.

EXPERIMENTAL AND THEORETICAL STUDY OF STATIC-DYNAMIC MIXED MODE BRITTLE FRACTURE OF ROCK

A new criterion of static dynamic state mixed mode brittle fracture initiation in rock is suggested, i.e. the distortion strain energy criterion.When t →∞, the static state distortion strain energy criteria and the formula for determiming static state fracture angle are obtained. The new static state criterion, suggested by authors, is in good agreement with the author′s experimental results. In addition, the relative accuracy of three current fracture theories and the new criterion in predicting static state mixed mode initiation fracture loading and fracture angle in rock have been evaluated experimentally.

Visualization of hydraulic fracture interacting with pre-existing fracture

Hydraulic fracturing is considered the main stimulation method to develop shale gas reservoirs. Due to its strong heterogeneity, the shale gas formation is typically embedded with geological discontinuities such as bedding planes and natural fractures. Many researchers realized that the interaction between natural fractures and hydraulic fractures plays a crucial role in generating a complex fracture network. In this paper, true tri-axial hydraulic fracturing tests were performed on polymethyl methacrylate (PMMA), on which pre-existing fracture was pre-manufactured to simulate natural fracture. A cohesive model has been developed to verify the results of the experimental tests. The key findings demonstrate that the experimental results agreed well with the numerical simulation outcomes where three main interaction modes were observed: crossing;being arrested by opening the pre-existing fracture;being arrested without dilating the pre-existing fracture. Crossing behavior is more likely to occur with the approaching angle, horizontal stress difference, and injection rate increase. Furthermore, the higher flow rate might assist in reactivating the natural fractures where both sides of the pre-existing fractures were reactivated as the injection rate increased from 5 to 20 mL/min. Additionally, hydraulic fractures show a tendency to extend vertically rather than along the direction of maximum horizontal stress when they are first terminated at the interface. This research may contribute to the field application of hydraulic fracturing in shale gas formation.

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.

Mode II fracture analysis of double edge cracked circular disk subjected to different diametral compression

A detailed analysis of mode II stress intensity factors(SIFs) for the double edge cracked Brazilian disk subjected to different diametral compression is presented using a weight function method. The mode II SIFs at crack tips can be obtained by simply calculating an integral of the product of mode II weight function and the shear stress on the prospective crack faces of uncracked disk loaded by a diametral compression. A semi-analytical formula for the calculation of normalized mode II SIF, f _Ⅱ, is derived for different crack lengths (from 0.1 to 0.7) and inclination angles (from 10° to 75°) with respect to loading direction. Comparison between the obtained results and finite element method solutions shows that the weight function method is of high precision. Combined with the authors previous work on mode I fracture analysis, the new specimen geometry can be used to study fracture through any combination of mode I and mode II loading by a simple alignment of the crack relative to the diameter of compression loading, and to obtain pure mode II crack extension. Another advantage of this specimen geometry is that it is available directly from rock core and is also easy to fabricate.

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.

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.