Influence of Intersection Angle of Prefabricated Cracks on Impact Failure of Flattened Brazilian Disc

LIU Hao, ZHANG Yafang, YIN Guoqi, OU Chenggui, LU Juan, HUO Yongjie

(1. Shenzhen Municipal Design Research Institute Co. Ltd, Shenzhen 518029, China; 2. School of Civil Engineering, Guangzhou University, Guangzhou 510006, China)

Abstract: Both experimental and numerical approaches were used to study dynamic failure properties and patterns of flattened Brazilian discs, containing two prefabricated cracks intersected at a varying angle. Mechanism of crack initiation, propagation, and cut-through were scrutinized and influences of the intersection angle on specimen strength and acoustic emission performance were also studied. All primary cracks initialize near the middle or the tip points of the upper prefabricated crack, and they continue to develop along the load direction and finally cut through the specimen. The secondary cracks could be observed in directions almost horizontal or parallel to the directions of prefabricated cracks. Furthermore, it is found that stress intensity factor reaches its maximum for specimen with intersection angle of 0 degree.

Key words: crack; flattened brazilian disc; impact load; SHPB; failure process; acoustic emission

1 Introduction

Brazilian splitting test can be used to measure indirectly tensile strength of brittle rock or concrete specimens. A large number of tests have been carried out on Brazilian discs because of the operating convenience[1-3]. Due to end effect, cracks may not initialize from the center of specimen[4]. Wanget al[5,6]proposed to flatten the Brazil disc, so contacting area in both ends can be enlarged to reduce stress concentration, thus most failures can initiate from the center of the specimens. In these tests, splitting Hopkins pressure bar (SHPB) has been widely adopted to simulate dynamic load[7,8]. A series dynamic fracture studies on Brazilian discs had been carried out recently. A hole had been set in some marble disc by Zhanget al[9,10]. They used length and width of the prefabricated cracks/holes as variables, and found that stress intensity factor of the specimens under splitting load increases with the crack/hole width. Qiet al[11]studied both crack initiation and failure pattern of concrete discs by changing loading angle. He pointed out that for concrete discs, significant strain rate can be observed under dynamic load. Wanget al[12]compared dynamic stress intensity factors of Brazilian discs with a line prefabricated crack under different loading angles. Most of the relevant numerical simulations carried out so far are based on the assumption of uniform material distribution, which does not correspond to the actual non-uniform distribution of materials. At the same time, the study of the expansion process and failure mechanism of flattened Brazilian discs considering prefabricated cracks under dynamic loading is extremely complicated, so the studies on the kinetic performance of such problems based on the consideration of non-uniform material distribution are rare at home and abroad. Zhanget al[13,14]conducted a three-dimensional numerical simulation on specimens, using crack inclination angle and number of prefabricated cracks as variable, based on non-uniform material distribution in meso level.

In this paper, impact tests have been conducted by using SHPB, to study properties and mechanism on disc specimens with prefabricated cracks intersected at varied angles. Considering non-uniform material distribution of all phases material in meso level, a commercial program named Realistic Failure Process Analysis (RFPA)-Dynamic[15,16]has also been adopted to set up numerical models of mortar discs with corresponding intersection angles. Triangular dynamic stress wave has been applied on the disc as dynamic load input, thus complete failure process, stress distribution fields and AE behavior can be observed and analyzed in details.

2 Experiments with SHPB

2.1 Preparation of specimens

Disc specimens were made of M50 cement mortar. The mortar was mixed with following components, cement: sand: water = 1: 1.4: 0.4. Compressive strength of the mortar was measured to be 54.6 MPa.

Two cracks (labeled with PC-A & B, hereafter) with intersection angle of 0/30/60/90 degrees were prefabricated into each specimen. The center distance of the two cracks was 25 mm, see Fig.1. Diameter and thickness of each disc specimen were 100 mm and 50 mm, respectively. Flattened angle in both ends was set to be 20o(Fig.1(a)). The length and width of cracks were 20 mm and 2 mm, respectively. Centers of both prefabricated cracks and center of the disc were in one line. In addition, for convenience of description, the specimens were labeled as PS0, PS30, PS60, and PS90, respectively.

Fig.1 Disc specimens with prefabricated cracks intersected at varied angles

In order to prefabricate the expected cracks, a particular mold was designed. As shown in Fig.2(a), AutoCAD was used to draw outline of the chassis, and the mold chassis is engraved with Walrun Laser Engraving Machine XZ-9060. Two smoothed wooden sheets were inserted into chassis and stuck with strong glue. Then a 100 mm×50 mm circular column was assembled to the mold, see Fig.2(b), and had been placed for 24 h till the whole mold was tightly formed.

Fig.2 Mold of specimens

The mortar were poured into the mold and vibrated on ZHJ-1 Concrete Shaking Table. Then the mold was removed two days later, covered with plastic film after watering, and had been cured for 28 days in a standard curing room. The cured specimens were then flattened to an angle of 20°.

2.2 Experimental process

SHPB (Model No. LWKJ-HPKS-Y100) was used to carry out the impact splitting test. The diameter of the bar is 100 mm. Vaseline was smeared to the contact ends between the specimen and the bar to reduce friction effect and they were kept in the same horizontal line to guarantee that a 1D stress wave could be generated. The bullet was accelerated against the incident bar by using preserved air pressure. A compression wave was then generated and transferred to the transmission bar through the specimen. Strain gauges had been stuck on all bars to capture the incident, reflected and transmitted waves, respectively.

2.3 Results and analysis

Fig.3 shows the final failure pattern of specimens with prefabricated cracks, which were intersected to each other with preset angles. As can be seen from the figure, a vertical primary cut-through crack appeared along the loading direction in each specimen, and the secondary cracks could be observed in directions almost horizontal or parallel to the directions of prefabricated cracks. The primary crack initiated generally from the middle zone of prefabricated crack PC-Aand extended upwards. However, the position of the primary crack is different to each other. The primary crack of specimens PS0 and PS60 extended downward from the left tip of PC-A to the left tip of PC-B below. For specimen PS30, the primary crack developed in the middle zone of PC-A and extended downwards to the left tip of PC-B. For specimen PS90, the two primary cracks started from both tips of crack PC-A and extended downwards to the upper tip of PC-B. Then, by crossing PC-B, the primary cracks of all specimens extended downwards and cut through the whole specimen.

Fig.3 Final failure pattern of specimens

In the specimen PS0, the two secondary cracks are oriented nearly horizontally, perpendicular to the direction of load loading, through the specimen. In the specimen PS30, one secondary crack was horizontally oriented and the other along the direction of the prefabricated crack. In the specimens PS60 and PS90, both secondary cracks extend diagonally from the left and right ends of the first prefabricated crack PC-A in a parallel direction towards the upper loading end, eventually penetrating the specimen.

Jianget al[17]used three-dimensional printed gypsum-like material to investigate the crack expansion pattern of Brazilian disc specimens with prefabricated cracks at exactly the same inclination as in this paper under dynamic conditions and found that the primary crack initiation point of the specimens was also at the crack tip of the prefabricated crack and grown in the loading direction, which is consistent with the laboratory experimental results in this paper. However, no secondary cracks were generated in the specimens due to the nearly uniform distribution of the three- dimensional printed gypsum-like material.

A plane stress flattened Brazilian disc with a thicknessHand a radiusRis shown in Fig.4. The disc is subjected to isotropic material and small deformation assumptions. A pair of uniformly distributed loadsPare applied in the opposite direction. The flattened angle is 20° as mentioned in Section 2.1 of this paper.

Fig.4 Diagram of loads applied on flattened Brazilian disc

Suet al[18]suggested that whenθ=0o, stress value in bothxandydirections at center of the flattened Brazilian disc could be described as followings:

Referred to calibration results shown in Wang[5,6], calibrated stressfat timetof the flattened Brazilian disc is:

where, σt=E0ε0,t,E0, A0& ε0,tare the elastic modulus, cross-sectional area of the SHPB, and detected real-time transmitted wave strain value, respectively.

The transmission wave value obtained in the test could be substituted into Eq.(3), thus the stress-time curves of the specimen can be obtained, see Fig.5. In the first 300 μs, the incident wave increased sharply and the transmitted wave increased correspondingly. At time of 300 μs, all reached their peak value. After that, the curves dropped, but there existed some local fluctuations before they hit zero. The peak strength values of all specimens are listed in Table 1, from which it can be seen that the peak strength of specimen PS0 is the maximum among all specimens, indicating that the impact load perpendicular to the horizontal crack has the least impact on the strength. On the other hand, the larger the crack intersection angle, the less influence the impact load in the vertical direction could apply on the specimen strength. This is because the intersection angle of cracks in the specimen could affect the propagation of stress wave[19,20]. Hence when the intersection angle increases, the influence on the transmission path of stress wave decreases. In order to understand the influence of crack intersection angle on specimen strength under impact load easily, the peak strength of other specimens relative to specimen PS0 are also listed in Table 1.

Fig.5 Stress vs. time, for specimens with prefabricated cracks intersected with diあerent angles

Table 1 Peak strength/relative peak strength for specimens with varied intersection angles

3 Numerical simulation

3.1 Numerical model

To reflect mesoscopic non-uniformity of each phase material, RFPA-Dynamic is adopted in this study to simulate the above mentioned experimental process. Weibull function has been introduced to describe both physical and mechanical properties of mesoscopic elements after discretization[14], which could be expressed as:

where,urepresents the mechanical properties of the material (such as strength, elastic modulus, Poisson’s ratio andetc.);u0represents the average value of the corresponding mechanical properties of the element;mis the shape parameter of the distribution density function, also known as the homogeneity of the material. It reflects the discreteness of mechanical parameter values. The larger the value ofmis, the more uniform the mechanical properties of the material.

According to Hamilton’s variation principle, the dynamic equation at timetcan be expressed as:

whereM,C, andKare the mass, damping, and stiあness matrix of the model, respectively. and are the acceleration vectors, velocity, and displacement, at timet, respectively. At timet+△t, according to Newmark method, the velocity vector and displacement vectors are:

where, γ, β are the integration coeきcients in Newmark method.

Hence, the dynamic equation at timet+Δtis:

Eq.(8) can be transformed and simplified into isoparametric Eq.(9):

where:

In RFPA, a diameter of 100 mm was set to the disc model, same as the experimental data. The discwas divided into square elements with size of 0.4 mm. To simulate mortar material, the following mechanical parameters were set to the model and are summarized in Table 2.

Table 2 Mechanical parameters of specimen elements

Shown in Fig.6, a triangular stress wave was used as load. As suggested by Pinget al[21], using triangular waveform could effectively reduce dispersion of stress waves, and can also better simulate the waveforms in the SHPB test. The stress wave lifting time is 80 μs, the peak value is 30 MPa, and the duration is 100 μs.

Fig.6 Triangular stress wave

3.2 Failure patten

Fig.7 shows the impact failure diagram of specimens with prefabricated crack intersected with varied angles in the numerical simulation. The failure process of specimens mainly experienced three stages, which are crack initiation, propagation and cut-through, respectively. In details, they are:

Stage 1: Referred to Figs.7-I, for all specimens, the crack initialized at about 33 to 34 μs, and the stress measured are ranged from 12.38 to 12.75 MPa. Starting point of crack in each specimen is near the middle point ofPC-A, which is consistent with the experiment (Fig.3).

Stage 2: As shown in Figs. 7-II, the corresponding stress value is between 18.75 and 21.38 MPa, when the impact time ranges between 50 and 57 μs. The primary crack continues to extend and connects the two prefabricated cracks. The first part of primary crack starts from PC-A extends upwards along the loading direction. A second part of primary crack appears at the left tip of PC-A and extends downwards to the left tip of PC-B, and it gradually connects with PC-B. Almost in the same time, another part of primary crack starts at the right tip of PC-B and gradually extends downwards to the disc end.

Stage 3: Referred to Figs.7-III, the loading drops to zero at time of 100μs. The primary crack parts of each specimen generated from cracks PC-A & PCB, respectively, penetrate to the upper and the lower flattened ends, respectively. The whole specimen is broken. The results of numerical simulation are basically consistent with the failure pattern of specimens obtained by SHPB impact test.

It is noted that the numerical simulation results of the specimen NS90, as shown in Fig.7(d)-III, have a good consistency with the results of SHPB impact test (Fig.3(d)).

Fig.7 Diagram of maximum shear stress in numerical specimens

According to Fenget al[22], the crack-tip stress concentration leads to the nucleation and the development of micro-cracks in the vicinity of the primary crack. Moreover, the propagation of the primary crack affects the positions, sizes, and orientations of other micro-cracks around the crack tip. The above findings provide a good insight into the process of initiation, propagation and cut-through of the primary crack in the flattened Brazilian disc containing prefabricated cracks of different intersected angles in this paper.

The numerical simulation results in this paper are in good agreement with the results of the numerical simulation analysis study carried out by Zhanget al[23], using the extended finite element method (XFEM) on two sets of specimens with different prefabricated crack inclination angles. However, due to the assumption of uniformity of the material adopted by Zhanget al[23], no secondary cracks were developed when the specimens were damaged. The present study is based on the assumption of meso level non-uniformity distribution of each phase of the materials, which is compatible with the actual concrete materials. The numerical simulation analysis achieves the simulation of the whole process of secondary cracks initiating from the edge of the disc and extending towards the prefabricated cracks, which makes up for the lack of direct observation of the crack initiation point in experiments and is an effective supplement to laboratory studies.

In particularly, as the numerical simulation analysis in this paper takes into account the non-uniformity of the material, which is in line with the actual material properties, some secondary cracks can be observed in Fig.7, and the direction of extension of these secondary cracks, which are also close to the experimental results. The dynamic load is a vertical downward compressive stress wave, which eventually produces vertical cut-through cracks, indicating that the specimens are mainly affected by transverse splitting tensile stresses causing damage and correspondingly produce tensile cracks.

3.3 AE performance

Due to damage and destruction of structural elements within the brittle material, elastic waves were generated because of dissipated energy. Such transient elastic wave phenomenon of rapid release of energy in local areas is named acoustic emission (AE)[24]. According to the theory that AE or damage units of concrete have a consistent nature[25], the acoustic emission of each specimen is studied in the numerical simulation and computational analysis in this paper.

Fig.8 shows the accumulative AE counts versus time for specimens with different intersection angles. It can be seen that accumulative AE counts curves of each specimen have following characteristics:

AE curve was close to zero before 33 μs, indicating that only few elements were damaged at this stage. Along with the propagation of stress wave, the stress increased continuously, the dynamic load reached the tensile or shear strength of weak elements. Hence damage of these elements occurred continuously, and the accumulative AE curve started to rise. During the period from 85 to 100 μs, events of damage and destruction were more active, and the AE curve rose at a higher speed. The failure element of the specimen was gradually penetrated to form macroscopic cracks, and the crack growth rate was gradually accelerated. At this time, AE counts increased at a higher rate, and the curve sloped up significantly compared with previous stage b. It could also be seen from Fig.8 that the total AE counts of specimen NS0 was the least among all specimens, which indicated that the number of damaged elements of this specimen was also the least in the process of dynamic loading.

Fig.8 Accumulated AE counts of specimens

Relative peak strength obtained from both experimental and numerical simulation are shown in Fig.9. The data in the figure demonstrates that they are in good agreement with each other. Only for specimens with prefabricated cracks intersected at 30o, the difference is a little larger than other specimens. Both experimental and numerical simulation results show that for the specimen with intersection angle of 0o, the peak strength reaches the maximum. This conclusion is consistent with the research results in Hadiet al[26].

Fig.9 Relative peak strengths

4 Conclusions

Both SHPB impact experiment and RFPA numerical simulation were carried out for flattened Brazilian disc with prefabricated cracks intersected at varied angles. Effects of the angle change on the failure pattern, stress field distribution, and stress field of specimens have been analyzed. Furthermore, AE characteristics which reveal crack initiation and developing pattern have also been studied in details. Based on these analysis, the following conclusions can be drawn:

a) The initiation, development and failure pattern of the primary crack in both experimental and numerical simulation are consistent to each other. Almost all primary cracks in the upper zone initiated from the area near the middle point of crack PC-A and develop upwards. The cracks in the middle zone of the specimen start from the two tip ends of crack PC-A and extends down to connect with crack PC-B. All cracks in the lower zone, initiate from the lower tip of PC-B, and then develop downwards, except for the specimen PS0 or NS0. In specimen PS0 or NS0, the lower cracks started from the middle point of crack PC-A. The cracks then penetrate throughout the whole specimen along with the impact time.

b) In the impact experiment, primary cracks spread throughout in all specimens with prefabricated cracks of different intersection angles. In addition, two secondary cracks appeared in all specimens. However, positions of these secondary cracks are different to each other. For specimens with intersection angle of 0° and 30°, the secondary cracks initiate from the crack PC-A and PC-B, and tend to develop in the direction of paralleled to this prefabricated crack. For specimens of 60° and 90°, the secondary cracks appear on both ends of the crack PC-A, and develops in the direction vertical to the loading direction. The fracture distribution in the numerical simulation is similar to that in the experiment.

c) In both experimental and numerical tests, stress intensity of specimen with intersection angle of 0° is the maximum. In the numerical simulation, the accumulated AE counts of specimen NS0 is the minimum, which indicates that the number of damaged elements of this specimen is relatively less in the process of dynamic loading.