Yuqing Dong, Zheming Zhu, Li Ren*, Lei Zhou, Peng Ying, Meng Wng

a State Key Laboratory of Hydraulics and Mountain River Engineering, School of Architecture and Environment, Sichuan University, Chengdu, 610065, China

b MOE Key Laboratory of Deep Earth Science and Engineering, School of Architecture and Environment, Sichuan University, Chengdu, 610065, China

Keywords:Crack arrest Stress wave Fracture toughness Reflected wave Impact loading

A B S T R A C T Crack dynamic propagation and arrest behaviors have received extensive attention over the years.However, there still remain many questions, e.g. under what conditions will a running crack come to arrest? In this paper, drop weight impact (DWI) tests were conducted to investigate crack arrest mechanism using single cleavage triangle(SCT)rock specimens.Green sandstone was selected to prepare the SCT specimens. Dynamic stress intensity factors (DSIFs) were calculated by ABAQUS code, and the critical DSIFs were determined by crack propagation speeds and fracture time measured by crack propagation gauges (CPGs). The test results show that the critical DSIF at propagation decreases with crack propagation speed. Numerical simulation for SCT specimens under different loading waves was performed using AUTODYN code. The reflected compressive wave from the incident and transmitted plates can induce crack arrests during propagation, and the number of arrest times increases with the wave length.In order to eliminate the effect of the incident and transmitted plates,models consisting of only one SCT specimen without incident and transmitted plates were established, and the same trapezoid-shaped loading wave was applied to the SCT specimen. The results show that for the SCT specimen with transmitted boundary(analogous to an infinite plate),the trapezoid-shaped loading wave cannot induce crack arrest anymore. The numerical results can well describe the occurrence of crack arrest in the experiments.

1. Introduction

Rock dynamic fracture is frequently observed in many rock engineering applications,such as tunnel blasting,mineral mining,and rockbursts. Significant efforts have been made to investigate crack initiation and propagation (Zhang and Zhao, 2014), but less attention has been paid to crack arrest.The study of crack arrest and reinitiation is of practical significance in brittle fracture as it allows us to know how crack initiates and how a running crack can be stopped.Although many results concerning crack arrests have been achieved, the nature of a running crack being arrested and the governing condition that determines crack arrest and repropagation still remain unknown.

The specimen size used in experimental study of crack arrest should be large enough because it can affect crack dynamic propagation behavior.Ravi-Chandar and Knauss(1984)pointed out that the wave reflection and interaction in a small specimen were possible reasons accounting for crack arrests. Freund (1977) proposed that crack arrests were related to the specimen geometry and suggested that the reflection and interaction of stress wave significantly affected the crack arrest. Hoagland et al. (1972)deduced that crack arrest behavior was related to the energy stored in the specimen experienced by loading history. Kanninen(1974) pointed out that the kinetic energy in the specimen played a significant role in crack arrest.

Dynamic stress intensity factor(DSIF)is a significant parameter which could be used to predict crack dynamic behavior.Bradley and Kobayashi (1971) concluded that as the instantaneous stress intensity factor (SIF) was lower than the quasi-static initial SIF, the crack was arrested. Kalthoff et al. (1977) proposed that the SIF value,when the crack was arrested,was a constant which is lower than the initial SIF.In recent years,some progress has been made in the study of crack arrest properties of rocks or rock-like materials.Yang et al. (2016) pointed out that alternate tensile and compressive stresses could reduce the degree of stress concentration at the crack tip,which could induce crack arrest.Yang et al.(2015)found an abrupt crack stop when using split-Hopkinson pressure bar(SHPB) system and pointed out that re-initiation of crack is the result of second loading.Recent studies also show that crack would be arrested under medium-low speed impacts(Wang et al.,2017a;Dong et al.,2018;Zhou et al.,2018a;Lang et al.,2019).Wang et al.(2017a)used a V-shaped bottom to arrest the running crack in the impact tests.Li et al.(2018)found that empty holes have the arrest role on outgoing cracks under blasting loading, and the spacing of two holes plays an important role in arresting moving cracks.

Dynamic fracture problems are complicated due to the nonnegligible inertia force and the moving crack surface. Many experimental tests have been conducted on crack dynamic propagation behavior(Bradley and Kobayashi,1971;Kalthoff et al.,1977;Reddish et al.,2005;Grégoire et al.,2009;Zhang et al.,2009;Yang et al., 2015, 2016; Wang et al., 2016, 2017a, b; Dong et al., 2018; Li et al., 2018; Zhou et al., 2018a; Lang et al., 2019). In addition, numerical studies were also performed on crack propagation behavior(Wang et al., 2008, 2009; Dehghan Banadaki and Mohanty, 2012),and a number of numerical techniques have been applied, such as finite element method(FEM)(Desai et al.,1984;Zhu et al.,2015;Xie et al., 2019), boundary element method (BEM) (Fedelinski et al.,1996), finite difference method (FDM) (Coates and Schoenberg,1995; Wang et al., 2011), and discrete element method (DEM)(Fakhimi and Lanari, 2014; Zhou et al., 2016; Imani et al., 2017).

Loading waveforms play a significant role in crack propagation behavior, and various numerical studies have been reported accordingly.Xia and Gong(2018)investigated the effects of loading waveforms on crack damage at mesoscopic scale. Cho and Kaneko(2004) investigated crack arrest after the peak stress for the arrival time of reflected stress,but they only considered the effect of reflected tensile stress wave. Panchadhara et al. (2017) conducted propellant-based simulation of a borehole using peridynamics and pointed out that crack is arrested due to insufficient energy applied by stress wave. Grégoire et al. (2009) showed a good agreement between experiments and extended FEM (X-FEM) simulations in crack arrest and restart,but the criterion they used is not accurate enough to interpret crack arrest. Zhu et al. (2007, 2008) and Zhu(2009) established finite difference models for rock under blasting loads by AUTODYN, and proposed rock fracture mechanism under blasting loads. In this study, AUTODYN code is utilized to investigate the crack arrest mechanism,which is suitable in solving nonlinear problems and has been generally used in the study of brittle material failure behavior(Ma et al.,1998,2011;Li and Wong,2012; Wong and Li, 2013).

This study aims to investigate the crack dynamic propagation properties and arrest mechanism under impact loading. Impact experiments were conducted using large-scale SCT specimens and drop weight impact (DWI) apparatus. The DSIFs during crack propagation were calculated by ABAQUS code. To investigate the arrest mechanism, numerical models utilizing AUTODYN were established based on the SCT specimens.The effects of wave length,loading rate and boundary conditions on crack initiation,arrest and re-initiation were also discussed.

2. Experiment setup

2.1. Preparation of SCT specimens

Large-scale SCT rock specimens have the characteristics of simple configuration and easy processing, and by using them, not only the crack propagation behavior can be easily detected,but also the effect of reflected waves on crack behavior can be minimized.The SCT specimen is a rectangle plate with an equilateral triangular opening at the top,as shown in Fig.1.The thickness of the specimen is 30 mm, which is the same as those of the incident and transmitted plates. The pre-crack emanates from a vertex of triangle having crack length of 30 mm and width of 1 mm,with machining precision of ±0.5 mm.

A dense and uniform green sandstone sample from Ziyang City,Sichuan Province of China was used to prepare the SCT specimens because its stable structural characteristics and uniform particle size of mineral components are suitable for dynamic tests. The density of this sandstone is 2265 kg/m3, the elastic modulus is 15.9 GPa, and the Poisson’s ratio is 0.18. The ingredient of the sandstone was measured by X-ray diffraction (XRD). The parameters are tabulated in Table 1.The impact test was performed at the room temperature of 25°C.

2.2. DWI apparatus

SHPB system is the most widely used method to study dynamic fracture behavior.Nevertheless,it has limitation as the diameters of the incident and transmitted bars are relatively small, and consequently the specimen size used for this system is small. Small specimen size is not conducive to investigate crack arrest behavior because the reflected stress from the boundary will significantly affect the crack propagation. Fig. 2 presents the DWI apparatus applied in the impact tests, which is suited to large-scale specimens.

The SCT specimen was placed between the incident and transmitted plates. The contact surfaces between the specimen and plates were lubricated with grease to eliminate the effect of contact friction. A brass plate was placed on the top of incident plate as a wave shaper. The material parameters and geometry of the apparatus are tabulated in Table 2.During the tests,the impact plate hits the top of incident plate longitudinally, which induces the downward traveling P-wave (longitudinal compressive wave). Then the P-wave in the incident plate is reflected at the interface between the incident plate and specimen, and the remaining wave travels through the specimen and reaches the transmitted plate. The specimens are mainly subjected to the longitudinal compressive Pwave.

A pair of strain gauges was longitudinally stuck on the incident and transmitted plates to measure the strain signals. The stresses σi(t)and σt(t)acting on the top and bottom of the specimen can be obtained by

Fig.1. Dimension of SCT specimen (unit in mm).

Table 1 Material parameters of green sandstone.

Fig. 2. DWI system and crack propagation gauge (CPG) measuring system.

where Epand Apare the Young’s modulus and cross-sectional area of the incident and transmitted plates,respectively;As-topand As-botare the top and bottom cross-sectional areas of the specimen,respectively, which are different because there is an open triangle on the top; εi(t), εr(t) and εt(t) denote the incident, reflection and transmitted strains, respectively, as shown in Fig. 3a. One of the typical calculation results of loading stresses σiand σtare plotted in Fig. 3b.

It is worth noting that the stress on the top of the specimen is larger than that on the bottom because the area of the top is less than that of bottom for the SCT specimen (see Fig.1). The loading rate is defined as the slope of the elastic part of σi(t) versus time curve, as shown in Fig. 3b.

2.3. Crack propagation gauge (CPG) measuring system and measuring results

The CPGs (BKX3.5-10CY, resistance of 3.5 Ω) were utilized to measure the crack tip position during crack dynamic propagation.The CPG is sensitive and accurate in the determination of crack position (Wang et al., 2015) and is widely used in rock dynamic tests(Zhang and Zhao,2013;Dong et al.,2018).The CPG consists of a series of parallel resistance grids,as shown in Fig.2.The spacing between two adjacent grids is 2.2 mm. The CPG coverage range is 44 mm along the crack propagation path.

The grids will break in succession with crack propagation,which can cause voltage jumping change, as shown in Fig. 4a and b. The breaking time of the first grid is the crack initiation time.

3. Test results

3.1. Crack propagation path and speed

To prevent specimen bending, in the front and back of the specimen,two steel plates were connected through four steel screw bars to limit the specimen displacement.Fig.5 presents four typical failure patterns of sandstone SCT specimens under different impact speeds.The crack propagates straightly at early stage due to mode I fracturing,but later,the cracks shift,which may be induced by the heterogeneity of sandstone material. When the impact speed is 5.32 m/s(specimen#2),between the 13th and 14th grids in Fig.4a,the crack takes a very long time to travel, which means that the crack may arrest there and restart to propagate. The crack has not reached the bottom and stopped in the middle of the specimen.As the impact speed increases to 6.26 m/s, the crack has reached the specimen bottom.For specimen#9 under impact speed of 6.82 m/s,the CPG’s grids are broken consecutively as represented in Fig. 4b,and the breaking time between two adjacent grids is almost thesame in the CPG coverage zone.This indicates that the crack arrest phenomenon is more likely to occur at lower impact speeds.

Table 2 Parameters of DWI system.

Fig. 3. Histories of strain and impact load versus time.

All experimental results of SCT specimens are listed in Table 3.It can be seen that crack propagation speed increases with loading speed or loading rate, as shown in Fig. 6a. For specimen #1 under impact speed of 4.84 m/s, the average propagation speed is 83.65 m/s, whereas for specimen #10 under impact speed of 7.35 m/s, the average speed is 457.46 m/s, which represents 447%increase in crack speed as loading rate increases from 181.12 GPa/s to 743.61 GPa/s.

The maximum arrest time refers to as the maximum time spent between two adjacent grids.From Fig.6b,the maximum arrest time decreases with loading speed (or loading rate). For specimen #1,the maximum arrest time is 271.3 μs;while for specimen#10,it is 15.4 μs,which represents 94.2%decrease as impact speed increases from 4.84 m/s to 7.35 m/s.

3.2. Calculation method of critical DSIFs

For a crack at initiation or arrest,v=0,and k(0)=1.For a crack running with Rayleigh wave speed, v = cR, and k (v) = 0, which indicates that the crack SIF is zero.Therefore,the universal function decreases with crack speed in the range from 0 to cR(cR=1457.6 m/s in this study).

The finite element code ABAQUS is utilized to calculate the DSIFs of SCT rock specimens under impact,as shown in Fig.7.The model was discretized by quadrilateral elements CPS8 and singular elements CPS6 near the crack tip.The dynamic loading stress σiand σtare calculated by Eq. (1). The SIFcan be obtained by

where rOBdenotes the distance from node B to node O (the crack tip),rOB=4rOA,in which rOAis the distance from node A to node O,as shown in Fig. 7.

3.3. Results of critical DSIFs

For specimen #9, when the crack is initiated, the DSIFs calculated by ABAQUS code are depicted in Fig.8a.The crack is closed at the initial stage due to compressive stress wave; therefore, at this moment,the DSIF of mode I is negative. The crack was initiated at the time 244.5 μs according to the experimental results of specimen#9,and the corresponding critical DSIF,i.e.the initiation toughness,is 7.11 MPa m1/2calculated from the curve depicted in Fig. 8a.

Fig. 4. The voltage signals recorded by the CPGs and the crack propagation length of specimens #2 (impact speed of 5.32 m/s) and #9 (impact speed of 6.82 m/s).

For propagating cracks, the DSIFs vary with crack length. If the crack extended to the 3rd grid position,where the crack length was 34.4 mm (crack extended 4.4 mm), we established the numerical model with crack length of 34.4 mm under impact loading.Fig.8b presents the time histories of the SIFs for the stationary crackand the dynamic DSIFs

The experimental results show that the 3rd and 4th grids were broken at 260.5 μs and 266.1 μs, respectively. The crack length between these two grids is 2.2 mm; thus in between the 3rd and 4th grids, the propagation speed is 392.85 m/s. The universal function is k(v)=0.7939 according to Eq.(3);then from Eq.(2),one can obtain the curve of the dynamic SIFversus time, i.e. the solid curve in Fig.8b.According to the breaking time of 260.5 μs in the 3rd grids and the solid curve in Fig. 8b, the propagation toughness is 5.3 MPa m1/2.The propagation toughness at different grid positions can be determined using the same procedure. For specimens #2 and #9, the curves of the critical DSIFs versus crack propagation length are presented in Fig. 9a.

In between grids 13 and 14 of specimen#2,the crack speed is only 12.9 m/s. It is well below the average speed, and the crack may be arrested there.According to Eq.(3),the universal function is k(12.9)=0.9937 ≈1.Therefore,the corresponding critical DSIF of 9.21 MPa m1/2could be considered to be the arrest toughness.

For specimens #2 and #9 shown in Fig. 9a, the critical DSIFs at initiation are highest. The critical DSIFs at propagation are lower than those at initiation and are not constant. The critical DSIF at arrest is slightly higher than that at propagation.

Fig. 5. Four typical failure patterns under different impact speeds.

Table 3 Test results for SCT specimens.

Fig. 6. Test results of average crack propagation speed and maximum arrest time.

Fig. 7. Mesh of an SCT specimen.

For all the 10 specimens,the test results of critical DSIFs versus the ratio of crack speed to cRare presented in Fig.9b.It can be seen that generally, the critical DSIF decreases with increasing crack propagation speed, indicating that the critical DSIF is not an independent material parameter and it is related to crack speeds. For sandstone studied in this paper, the critical DSIFcan be expressed as

where v/cRis in the range of 0-0.58,and the correspondingis in the range of 2.46-7.11 MPa m1/2.

4. Numerical model and validation

Numerical studies have advantages in investigating crack arrest mechanism that all the factors affecting crack dynamic behavior could be considered in numerical models, whereas experimental studies can only discover partial properties of crack behavior due to the limit of experimental condition and the cost in experiments. In this context, the finite difference code AUTODYN was applied.

Fig. 8. Curves of DSIF versus time.

4.1. Dynamic finite difference method

In the finite difference model, each material was assigned one sub-grid.For the impact test system,four sub-grids were designed.The interaction between two adjacent materials was transferred by a gap. The gap was set to be 1/10 of the smallest element size between two adjacent sub-grids (Century Dynamics Inc, 2003).As a node moves to the gap zone, a repulsive force will act on it.The repulsive force is proportional to the distance trapped into the gap.

The deviatoric stresses Sx, Syand Sxycan be expressed as

where n is the time step;G is the shear modulus;andare the strain rates; ˙e is the volumetric strain rate,andand Δt is the time increment.

The stresses can be expressed as

where P is the average pressure determined by the equation of state(EOS), which demonstrates the relationship between hydrostatic pressure and local density (or local specific volume) (Century Dynamics Inc., 2003). For sandstone,a kind of brittle material, the influence of temperature(or the entropy) is not very conspicuous,and the material behavior before failure can be described by a linear EOS:

where K is the bulk modulus;and ρ and ρ0denote the current and initial densities, respectively.

In the simulation,the principal stress failure criterion was used to describe the SCT specimen failure behavior.When the maximum tensile stress σ1or shear stress τmaxis larger than the dynamic tensile strength σTor shear strength τc, the material fails:

In order to save time,one can select a small domain instead of a large domain using a transmitting boundary, and the outgoing waves are not allowed to be reflected back at the transmitting boundary.

In AUTODYN code, the pressure at the transmitting boundary can be calculated by

where Unis the component of mean velocity normal to the boundary, and Prefand Urefare the reference values and they are normally set as zero. Thus we have

Given the velocity components (u1, v1) and (u2, v2) at the two boundary nodes (x1, y1) and (x2, y2) of boundary cell, Unis determined as follows:

As compressive stress waves reach a free boundary,the pressure can be calculated by Eq.(11).In order to eliminate the compressive stress wave so that no reflected tensile stress wave is developed on the boundary, the pressure P will be changed to be zero by imposing a negative P (i.e. -P) on the boundary. This boundary is called transmitting boundary where the reflection is not allowed.This type of boundary is generally applied in the simulation of a longitudinal wave traveling in an infinite elastic medium.

Fig. 9. Curves of critical DSIF versus crack propagation length and the ratio of crack speed to cR.

The numerical models for SCT specimen under DWI were established.The discretization of numerical model with the impact system is shown in Fig.10.

4.2. Comparisons between test and numerical results

(1) Delayed fracture time and maximum arrest time

Based on the numerical models established above, a series of impact tests with different loading rates is simulated. The test results of loading stress were applied on the top of the incident plate,and the end of the transmitted plate was set as transmitting boundary.The loading rate is defined as the slope of the elastic part of the σi(t) versus time curve shown in Fig. 3b. By changing the impact speed of the impact plate,various dynamic loading rates can be obtained numerically.

The test results for the relationship between delayed fracture time (the difference between the time of crack initiation and the time when the stress waves reach the crack tip)(Zhou et al.,2018b;Ying et al.,2019)and loading rate are shown in Fig.11a.One can find that the delayed fracture time decreases with the increase of loading rates,and the numerical results generally are in agreement with the test results.

Fig.10. Numerical model of the impact system.

The numerical results for the relationship of the maximum arrest time versus loading rate are compared with the test results in Fig.11b. The maximum arrest time decreases with the increase of loading rates. The difference between the numerical and test results may be caused by sandstone heterogeneity,but generally the numerical results agree with the test results.

(2) Crack arrest behavior

The test results of loading stress σiof specimens #2 and #9 presented in Fig.12 were applied on the top of the incident plate,and the end of the transmitted plate was set as transmitting boundary.For specimens#2 and#9,the numerical results of curves of crack propagation length versus time and the corresponding CPG voltage signals are presented in Fig.13.

It can be seen that the crack is arrested in the CPG coverage zone when the impact speed was 5.32 m/s for specimen#2 as shown in Fig.13a.For specimen#9,the impact speed of 6.82 m/s was higher than that for specimen #2, and the crack was arrested in the last wire of the CPG, which is almost consistent with the numerical results. The experiments can only investigate the crack arrest phenomenon in CPG coverage zone, while the numerical simulation by AUTODYN code would reflect the whole process of crack propagation.

5. Numerical study of crack arrest mechanism

The experimental results show that crack arrest phenomenon,i.e. crack initiation arrests temporarily during propagation, is frequently encountered.However,the reason causing crack arrest is still not clear yet. Some researchers (e.g. Freund, 1977; Ravi-Chandar and Knauss, 1984) pointed out that crack arrests are related to the specimen geometry in which the reflected stress wave affects crack dynamic behavior, and some researchers (e.g.Hoagland et al., 1972; Kanninen, 1974; Panchadhara et al., 2017)considered that the energy stored in the specimen is an important factor affecting crack arrest.Yang et al.(2015)pointed out that crack arrest and restart may be induced by the second impact loading.

Fig.11. Comparison of test and numerical results.

Fig.12. Test results of loading stress of specimens #2 and #9.

5.1. Effect of wave length on crack arrest

In this section, the effects of the length and the terrace length of the trapezoid-shaped wave on crack arrest were investigated.The loading curves of the test results in Fig.12 can be simplified as a trapezoidal wave, and the trapezoid-shaped stress waves in Fig.14a are applied to the top of the incident plate.The peak stress of the incident wave was kept as a constant of 40 MPa,which was close to the experimental results. The P-wave speed of the sandstone, cP, is 2563 m/s, and the wave length Lbis expressed as the product of the travel time t and the wave speed cP,i.e.Lb=tcP.The loading rate is the slope of linear rising region of the loading stress curve.Both the rising and descending times of loading wave were kept as a constant of 100 μs,thus the loading rate was a constant of 400 GPa/s. The wave length Lband the top terrace length Ltopchanged equally.

The numerical results of the curves of crack propagation length versus time for different wave lengths Lbare shown in Fig.14b.One can observe that for all the waves,the delayed initiation time is the same(223.7 μs).This is because the wave front parts are the same for all trapezoid-shaped waves with different lengths.

Fig.13. Numerical results of curves of crack propagation length versus time.

Fig.14. (a) Trapezoid-shaped loading wave with constant rising and descending times of 100 μs, and (b) the numerical results of crack propagation length.

Fig.15. The wave positions at the moments of crack initiation, arrest and re-initiation (loading rate of 400 GPa/s).

For the wave length Lb≤300cP, in the CPG coverage zone(44 mm), no crack arrest phenomenon occurred, but beyond the CPG coverage zone, the crack was arrested. When Lb= 300cP,between 403.4 μs and 546.9 μs, the crack was arrested there for 143.5 μs.

For the wave length Lb≥400cP,in the CPG coverage zone(44 mm),crackarrest phenomenacan beobserved,asshowninFig.14b.Beyond the CPG zone,crack arrest phenomena could also occur.For the cases of Lb= 400cPand 500cP, totally the crack was arrested twice. For Lb≥600cP,three times of arrest phenomena were observed.

Fig.16. Particle velocity vectors at the moments of crack initiation, arrest and re-initiation under the trapezoid-shaped wave.

From the above simulation results, one can conclude that crack arrest behavior is influenced by the wave length significantly.As the wave length is less than a certain value,crack can only be arrested one time,and the arrest occurs at late time.As the wave length is larger than a certain value, the crack is arrested early and the number of arrest times increases with the wave length.This canwell explain that in the impact tests, sometimes in the CPG zone, crack arrest phenomenon can be observed,and sometimes no crack arrest phenomenon occurs.

We select the trapezoid-shaped wave with the length Lb=300cPas an example to illustrate how the crack initiates, arrests and reinitiates. The crack arrests one time, as illustrated in Fig.14b, and the positions of the incident and reflected waves at the moments of crack initiation,arrest and re-initiation are illustrated in Fig.15.At 223.7 μs, the crack is initiated immediately before the reflected wave reaches the crack tip,as shown in Fig.15a.The corresponding particles are moving down, as shown in Fig.16a.

At 403.4 μs,the crack has propagated a length of 74 mm.The 2nd reflected wave is moving down from the top,as shown in Fig.15b,but the 1st reflected wave is moving up,which is larger(23.5 MPa)than the 2nd reflected wave (16.7 MPa) in amplitude, so that the particles are moving up, as illustrated in Fig. 16b. Therefore, the crack was arrested at this moment.

At 546.9 μs,the positions of the 2nd,3rd and 4th reflected waves are shown in Fig.15c,and the particle velocity is shown in Fig.16c.It can be seen that at this moment,the particle velocity was inclined and the horizontal component of the stress was tensile which caused the crack re-initiation.

5.2. Effect of loading rate on crack arrest

To investigate the effect of the loading rate on crack arrest, the wave length Lbwas kept as a constant of 300cP,and the peak stress of the trapezoid-shaped wave was kept as a constant of 40 MPa.The top terrace length Ltopis designed in the range from 0 to 300cP, as shown in Fig.17a, thus the loading rate is increasing with the top terrace length.

The curves of crack propagation length versus time for the SCT specimen with different loading rates are presented in Fig.17b. It can be seen that:

(1) When the loading rate is 267 GPa/s and the corresponding top terrace length is zero, which is a triangle-shaped wave,the crack propagation length is very short.As the loading rate increases, the crack propagation length increases.

(2) The crack arrest occurs in all the cases, but the crack arrest and re-initiation times are different.

(3) The larger the loading rate is,the earlier the crack initiation is.

To further investigate the effect of loading rate on crack dynamic behavior, we analyzed the positions of the incident and reflected waves with the loading rate of 267 GPa/s,as shown in Fig.18.When the loading rate is 267 GPa/s and top terrace length is Ltop=0,it is a triangle-shaped wave. The positions of the incident and reflected waves at the moments of crack initiation, arrest, and re-initiation are illustrated in Fig. 18. The corresponding particle velocity vectors are shown in Fig.19.It can be seen that when Ltop=0,the crack was initiated at 252.3 μs, which is later than that of Ltop= 100cPwith the loading rate of 400 GPa/s (the crack was initiated at 223.7 μs), as shown in Fig.15. This is because for the same wave length,the peak stress of 40 MPa for the case of Ltop=100cParrived earlier than that for the case of Ltop= 0.

At 316 μs,the incident wavewas decreasing and the 2nd reflected wave was moving down from the top,but the 1st reflected wave was moving up(see Fig.18b).At this moment,the particles were moving up,as shown in Fig.19b,thus the crack was arrested at this moment.

At 539.4 μs, the positions of the 2nd, 3rd and 4th reflected waves are shown in Fig. 18c, and at this moment, the particles were moving down, as shown in Fig.19c, thus the crack was reinitiated.

5.3. Crack propagation in an infinite plane

To investigate whether the crack arrest is merely related to the incident and transmitted plates,numerical models consisting of the specimen without incident and transmitted plates were established.Three edges of the SCT specimen,i.e.the left edge,right edge and bottom,are set as transmitting boundaries.This means that no stress wave can be reflected back from those boundaries, and the specimen can be considered as an infinite plate.

The peak stress of loading wave was designed as a constant of 160 MPa, which is larger than the test results because the transmitting boundary can eliminate the action of reflected stress wave.Also, the simulation results showed that as the peak stress of the wave was less than 160 MPa, the crack could not initiate or the propagation distance was small so that the crack arrest mechanism was difficult to be investigated.

Fig.17. (a) A trapezoid-shaped loading wave with a constant wave length of 300cP, and (b) the numerical results of crack propagation length.

Fig.18. The wave positions at the moments of crack initiation, arrest and re-initiation (loading rate of 267 GPa/s).

Fig.19. Particle velocity vectors at the moments of crack initiation, arrest and re-initiation under the triangle-shaped wave.

Fig. 20. (a) Trapezoid-shaped loading wave with constant rising and descending times of 100 μs, and (b) numerical results of crack propagation length.

The trapezoid-shaped loading waves with different wave lengths Lb,as shown in Fig.20a,were applied to the top of the SCT specimen. Both the rising time from 0 to the top terrace and descending time from the top terrace to 0 are 100 μs, thus the loading rate was kept constant at 1600 GPa/s.The wave length was designed from 300cPto 900cP. The top terrace length and wave length changed equally.

The results using above numerical model with different lengths of trapezoid-shaped loading waves are shown in Fig.20b.One can find that the crack propagation length increases with the wave lengths.This is because the wave acting time on the crack increases with the wave length. No crack arrest phenomenon can be observed in the whole propagation process.

Fig. 20b shows when the wave length is 300cP(the top terrace was 100cP), the delayed initiation time is 248.9 μs, and when the wave length is larger than 300cP, all the delayed initiation time is 263.2 μs. This is because the compressive wave has two aspects of functions:forcing the particles inside the specimen to move down,and inducing horizontal compressive deformation which can resist crack propagation.When the wave length is shorter,the top terrace is short,thus the wave starts to decrease early,as shown in Fig.21a,which means that the resistance induced by the horizontal deformation (caused by the compressive wave) decreases early. Therefore, the crack initiates earlier as compared to the long trapezoidshaped wave,such as 500cP(see Fig. 21b).

Fig. 21. The trapezoid-shaped wave location at the moment of crack initiation.

In Fig. 20b, the crack propagation speeds are small in the early stage when the wave length is larger than 300cP.The crack starting acceleration time increases with the trapezoidal wave lengths.Fig. 22 shows when the wave length Lbis 500cP, the crack starting acceleration time is 451.7 μs,and when the wave length Lbis 700cP,the crack starting acceleration time is 658.1 μs,which is about 200 μs (i.e. 658.1 μs - 451.7 μs = 206.4 μs), later than that when the wave length is 500cP. This is because the horizontal deformation induced by the compressive wave resists the crack propagation.The longer wave length will resist crack propagation for a longer time.The compressive stress decreases earlier when the wave length is shorter, thus the acceleration starts earlier.

6. Discussion

Generally,in the SHPB impact systems,the forces at two ends of specimens are assumed to be balanced and the stresses are evenly distributed in the whole specimen. The specimens used in SHPB tests are small,and before failure,the stress wave could have been reflected and transmitted for several times. However, rock materials are often subjected to unidirectional dynamic loads, and the stress wave is one-way traveling, such as the waves induced by earthquakes and explosions, thus the stress balance condition is extremely rare in reality.

Fig. 22. The trapezoid-shaped wave location at the moment of crack starting acceleration.

The DWI test system is designed for large specimens, and it consists of an incident plate,a transmitted plate and a drop weight plate.The maximum specimen size is 300 mm×300 mm×30 mm.Although the large-scale specimen used in this study is not infinite,the reflected stress wave will take a longer time to travel back as compared with small-size specimen in SHPB tests. From Fig. 15a,one can find that when the wave is reflected back, the crack is initiated,and at this moment,the stress is not balanced at the two ends of the specimen.

The quasi-static stress analysis method in SHPB tests cannot be used in the DWI test because force balance at the two ends is not satisfied.The experimental-numerical method is used in this study to calculate the critical DSIFs, which does not need to satisfy the force balance condition.

7. Conclusions

In order to investigate crack propagation properties and arrest mechanism, experimental tests and numerical studies by utilizing SCT specimen were performed in this paper. The following conclusions are obtained:

(1) During crack dynamic propagation, crack speed varies and crack may stop and restart; and the critical DSIF generally decreases with crack propagation speed.

(2) The reflected compressive stress waves caused by the transmission and incident plates play a key role in crack arrest;without transmissionplate,the crack arrest phenomenonwill not occur for the SCT specimen under a trapezoid-shaped loading wave.When a crack is arrested,the particle velocity vector is in the opposite direction with the running crack.

(3) For the SCT specimen contacting with the incident and transmission plates under a trapezoid-shaped loading wave,as the wave length is less than a certain value,the crack can only arrest one time, and the arrest occurs comparatively later; as the wave length is larger than a certain value, the crack arrests earlier and the number of arrest times increases with wave length.

(4) Loading rates affect crack propagation behavior significantly.As the loading rate is low,the crack is more likely to arrest,and the larger the loading rate is,the earlier the crack initiation is.

(5) In an infinite plate (realized by transmitting boundary), under a trapezoid-shaped loading wave, no crack arrest phenomena occur.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was financially supported by the National Natural Science Foundation of China (Grant Nos. U19A2098,11672194 and 11702181), and the Fundamental Research Funds for the Central Universities.