Dynamic fragmentation of microwave irradiated rock

Shui Wng, Ying Xu,*, Kiwen Xi,b, Tinyng Tong

a State Key Laboratory of Hydraulic Engineering Simulation and Safety, School of Civil Engineering, Tianjin University, Tianjin, 300072, China

b Department of Civil and Mineral Engineering, University of Toronto, Toronto, M5S 1A4, Canada

c Key Laboratory of Ministry of Education on Safe Mining of Deep Metal Mines, Northeastern University, Shenyang,110819, China

Keywords:Microwave irradiation Split Hopkinson pressure bar (SHPB)Momentum-trap Dynamic compressive strength Fragment size distribution (FSD)Fangshan granite (FG)

ABSTRACT The microwave-assisted rock fragmentation has been proven to be a promising approach in reducing cutting tools wear and improving efficiency in rock crushing and excavation.Thus, understanding the influence of damage induced by microwave irradiation on rock fragmentation is necessary.In this context, cylindrical Fangshan granite (FG) specimens were exposed to microwave irradiation at a power of 6 kW for different durations up to 4.5 min.The damages of the specimens induced by irradiation were quantified by using both X-ray micro-CT scanning and ultrasonic wave measurement.The CT value and P-wave velocity decreased with increase of irradiation duration.The irradiated specimens were then tested using a split Hopkinson pressure bar (SHPB) system to simulate rock fragmentation.A momentum-trap technique was utilized to ensure single-pulse loading on the specimen in SHPB tests, enabling valid fragment size distribution (FSD) analysis.The dependence of dynamic uniaxial compressive strength(UCS) on the irradiation duration and loading rate was revealed.The dynamic UCS increased with increase of loading rate while decreased with increase of irradiation duration.Using the sieve analysis,three fragmentation types were proposed based on FSD, which were dictated by both loading rate and irradiation duration.In addition, an average fragment size was proposed to quantify FSD.The results showed that the average fragment size decreased with increase of loading rate.A loading rate range was identified, where a dramatic reduction of the average fragment size occurred.The dependence of fragmentation on the irradiation duration and loading rate was also discussed.

1.Introduction

Rock fragmentation, especially for hard rocks, is vital in the mining, tunneling and other rock engineering disciplines.Conventional blasting and mechanical rock-breaking methods have some disadvantages, including strong disturbance during blasting,serious damage to the surrounding rocks, and rapid wear of the cutting tools.To address these problems, external energy assisted rock fragmentation methods have been proposed.The external energy, including thermal source (e.g.Lauriello and Fritsch,1974;Rauenzahn and Tester, 1989; Wang et al., 2008), microwave (e.g.Hassani et al., 2008; Nekoovaght and Hassani, 2014; Nekoovaght et al., 2015; Lu et al., 2019a,b), and laser power (Boutinguiza et al., 2005) can significantly reduce the strength of rocks and thus improve the rock fragmentation efficiency.Specifically, the microwave irradiation is highlighted as a promising auxiliary method capable of reducing energy consumption and improving fragmentation efficiency (e.g.Haque, 1999; Kingman et al., 2000;Hassani et al., 2016).In addition, the microwave-assisted fragmentation has also been proven to be a more economical approach to facilitate ore comminution and valuable minerals liberation(Whittles et al., 2003; Kingman et al., 2004a; Nanthakumar et al.,2007;Omran et al., 2014).

Materials capable of absorbing microwave energy can be heated by irradiation.Over the past few decades, extensive studies have been conducted to investigate the heating characteristics of various natural minerals and inorganic compounds when subjected to microwave irradiation (e.g.Chen et al., 1984; Liu et al., 1990;Walkiewicz et al.,1998; Haque,1999; Kingman et al., 2000; Lovás et al., 2010; Lu et al., 2017; Zheng et al., 2020b).According to their responses to microwave,materials can be classified into three groups: transparent, conductor, and absorber.Only the absorber materials can be rapidly heated within a short duration ofirradiation, which is known as the phenomenon of selective heating of microwave (Haque,1999).Based on existing investigations,most sulphide and oxide minerals are highly effective in absorbing microwave energy whereas carbonate minerals are inert(Liu et al.,1990).Moreover, the existence of Fe is considered to contribute to the microwave absorption capacity of rock-forming minerals (Lu et al., 2017).

For heterogeneous rocks containing absorber and inert minerals, the thermal stress is induced by the expansion mismatch in different minerals when subjected to irradiation(Zeng et al.,2019;Zheng et al., 2020a).Microcracks are generated if the microwave induced thermal stress exceeds the tensile strength between minerals(Ali and Bradshaw,2009;Hartlieb et al.,2016).According to existing studies, the microcracks caused by microwave irradiation could be experimentally observed, often appearing along the grain boundaries(Ali and Bradshaw,2009;Guo et al.,2011;Hassani et al.,2016).For granites,the microwave induced cracks are mostly distributed along the grain boundaries or the quartz-feldspar cleavage planes (Hartlieb et al., 2016), while the basic chemical composition and total weight remain almost the same(Znamenácková et al., 2003).

The microwave induced damage can result in strength degradation and facilitate rock breakage.Researchers have investigated experimentally and numerically the influence of microwave irradiation on degradation of mechanical properties of rocks.Jones et al.(2007) employed a two-dimensional (2D) thermomechanical model to simulate the weakening processes induced by continuous and pulsed irradiations, respectively.The results indicated that the pulsed microwave treatment was more effective in reducing the original strength of pyrite ore.According to the mechanical test results of irradiated rocks,Hassani and Nekoovaght(2011) pointed out that the reduction of tensile strength was greater than that of compressive strength.In addition,the dynamic responses of microwave irradiated rocks (ores) were also studied.Kingman et al.(2004a) reported that the irradiated ore became more fragile when subjected to the impact loading.Rizmanoski(2011) investigated the impact resistance of irradiated rocks using the drop-weight impact tests.The results showed that irradiation could reduce the dynamic strength of rocks.Li et al.(2020)studied the effect of microwave irradiation on the properties of dynamic mode-I fracture, and declared the initial fracture toughness and fracture arrest toughness could be weakened by irradiation.

Effect of microwave irradiation on rock fragmentation has also been investigated.Kingman et al.(2004a) demonstrated microwave could also lead to a considerable reduction in the strength of ore specimens by point load test,and then discussed the influence of microwave power on ore breakage.The results of conventional ore comminution tests and liberation analysis indicated that the microwave treatment could also promote the rock fragmentation(Batchelor et al.,2017).Furthermore,it was demonstrated that the microwave pretreatment could significantly reduce the comminution energy and improve the grindability of hard rocks or ores(e.g.Kingman et al., 2004b; Jones et al., 2005; Hartlieb et al., 2016).

Although the influence of microwave irradiation on the static strength reduction and fragmentation has been extensively investigated, few studies have been conducted on the dynamic properties of irradiated rocks.During the operation of tunnel boring machines(TBMs),the cutters impact on the rock faces all the time.Thus,the rock failure is fundamentally dynamic during this process(Zhang, 2004; Sun et al., 2013; Shepel et al., 2018; Li et al., 2020).Therefore, for the microwave-assisted mechanical rock breakage using TBMs,it is critical to systematically investigate the influence of microwave irradiation on the dynamic properties and the fragmentation behaviors of rocks.This study can provide support to the design of TBMs for microwave-assisted rock breakage.

To achieve these goals,dynamic compression tests on irradiated Fangshan granite (FG) specimens were carried out using a split Hopkinson pressure bar (SHPB)system.New techniques,i.e.ultrahigh-speed camera and momentum-trap,were used in the tests to facilitate quantitative fragmentation analysis.

2.Specimen preparation, irradiation, and characterization

2.1.Mineralogical analysis of FG and specimen preparation

Rocks containing minerals with different dielectric constants and thermal expansion coefficients can result in expansion mismatch when irradiated by microwave.The thermal expansion mismatch can induce microcracks in rocks,leading to degradations of physical and mechanical properties of rocks.Hence, rocks sensitive to microwave should contain both microwave absorber and inert minerals.Granite is sensitive to microwave, which is commonly encountered in excavation, drilling and crushing processes (Toifl et al., 2017; Shepel et al., 2018).Thus, FG from Beijing was selected as the studying rock material in this work.As depicted in Fig.1a, FG was shown to contain 59.4% albite, calcian-ordered,33.6% quartz, 4% annite, and 3% greenalite using the X-ray diffraction (XRD) analysis.Annite is a member of the biotite mica group,and greenalite is a serpentine group mineral.These minerals contain Fe, which enables them having stronger microwave absorbing capacity than albite and quartz do (Deer et al., 2013; Lu et al., 2017).In addition, the mineral map of the FG was obtained using the electron probe microanalyzer (see Fig.1b).The results show that FG is a medium-grained rock with an average albite size of 1.1 mm and an average quartz size of 0.95 mm.

Based on the suggested method for dynamic compression test of rock materials by the International Society for Rock Mechanics and Rock Engineering (ISRM) (Zhou et al., 2012), the cylindrical specimens with the slenderness ratio of 1:1 were prepared.Rock cores with 38 mm in diameter were drilled from the same FG block.The cores were then cut and polished to the height of 38 mm.After that,the prepared specimens were dried in an oven at the temperature of 105°C for 24 h to remove moisture.This is necessary because the existence of water in rocks will result in additional damage when exposed to microwaves (Hartlieb et al., 2018).

2.2.Microwave irradiation of FG

Many studies revealed that microwave irradiation could cause both transgranular and intergranular fractures (Song et al., 2013;Nicco et al., 2018) in heterogeneous rocks and thus considerably improve the efficiency of rock fragmentation and grinding (e.g.Walkiewicz et al., 1998; Kingman et al., 2000; Ferrari-John et al.,2016).

Generally, materials can be heated in both single mode and multimode cavity.Compared to single mode cavity,the microwave can be fully utilized using the multimode one and the heating inhomogeneity can be largely minimized (Lu et al., 2019a).The heating inhomogeneity in rock specimens may cause nonuniform damage, i.e.thermal shock and mineral melting.The nonuniform damage is undesirable in this study.Thus, the irradiation on rock specimens should be carefully controlled.A multimode cavity industrial microwave system at the Northeastern University of China was chosen to irradiate the FG specimens.This system was operated at the frequency of 2.45 GHz with power up to 6 kW.As can be seen in Fig.2, this microwave system consists of 6 microwave generators, a control panel, and a 490 mm × 490 mm × 490 mm closed metallic cavity.Each microwave generator can reach its peak power of 1 kW within 6 s.The generators are air-cooled using fans(Lu et al., 2019a).

Fig.1.(a) XRD spectra between the 2θ angles of 3° and 70° for FG powder using X’Pert3 Powder, where θ is the diffraction angle; and (b) Mineral map of FG.

Fig.2.The industrial microwave system at the Northeastern University of China: (a) Outside view of the system, and (b) Inside view of the cavity (1.control panel; 2.cavity; 3.microwave generators; 4.specimen; 5.bricks).

From existing studies, it is known that the microwave induced damage is mainly controlled by the microwave power and irradiation duration (Jones et al., 2005; Hassani et al., 2016).In the current study, FG specimens were irradiated by microwave with the power of 6 kW for different durations.To determine the appropriate range of irradiation duration,an FG specimen was exposed to microwave for 5 min in the preliminary trial.Fig.3a shows the lava eruption and solidification observed on the irradiated specimen.Many cracks were formed around the eruption point.The irradiated specimen with such drastic damage is inappropriate for the dynamic compression test.The upper limit of irradiation duration was thus set as 4.5 min, where the irradiated specimen remains intact without macroscopic damage (see Fig.3b).

Fig.3.Typical irradiated specimens with different durations: (a) Solidified lava observed on the specimen irradiated for 5 min, and (b) Specimens irradiated up to 4.5 min.Unit of the rulers is cm.

It has been proven that placing the specimen at the center of the cavity can facilitate achieving the maximum heating efficiency (Lu et al., 2019a).Therefore, the specimen was placed on the bricks at the same location for irradiation one by one to ensure consistency.The bricks are made of mullite (a material with weak microwave absorbing ability) which can withstand high temperature up to 1600°C (Lu et al., 2017).

Typical irradiated specimens with different durations (1.5 min,2 min, 2.5 min, 3 min, 3.5 min, 4 min, and 4.5 min) are shown in Fig.3b.The specimens without irradiation were dark gray,whereas the specimens irradiated for 4.5 min were light gray.It can be seen that the longer the irradiation duration,the lighter the rock color.It is worth noticing that the damage in the specimen was uniform after microwave pretreatment due to the use of the multimode cavity.Furthermore, two ends of the irradiated specimen can still meet the dynamic testing conditions according to the measurement.

2.3.Damage quantification of irradiated specimens

Many methods are available for assessing the damage of rocks(e.g.Nicco et al., 2018).Among these methods, X-ray micro-CT ishighlighted as a promising technique capable of nondestructively detecting internal flaws of a rock specimen.Micro-CT scanning of the FG specimens with different irradiation durations was conducted using the Sanying nano Voxel 2000 micro-CT scanning system,which was operated at a voltage of 150 kV with an electric current of 60 μA.Due to the tradeoff between specimen size and resolution, the resolution was set as 32.5 μm when the entire specimen was scanned.In order to minimize the random error,three specimens with the same irradiation duration were scanned for damage quantification.

Fig.4 shows the typical CT images of irradiated FG specimens with different durations.There were no visible differences between them.This is probably due to the fact that the opening displacements of microcracks are below the resolution of the micro-CT system.The CT value, also known as Hounsfield radiological density, was used to quantify the rock damage.The CT value has been successfully employed to quantify the damage induced by thermal treatment or stress in rocks(Feng et al.,2004;Huang and Xia,2015;Yao et al., 2016).

The CT value (Hrm) can be determined by the mass absorbing coefficient (Feng et al., 2004) as

The ultrasonic method, another commonly used measurement to quantify rock damage, was also employed in this work to measure the P-wave velocities of the specimens with different irradiation durations(Yao et al.,2017).The variations of P-wave velocities with the different irradiation durations for FG are also plotted in Fig.5.It can be seen that both the P-wave velocity and the CT value decreased with increase of the irradiation duration.However, the exact trends for these two physical parameters were different.The P-wave velocity reduced to 3525 m/s at 2 min, which was approximately 80.7% of that of the specimen without irradiation,and then it decreased to 2874 m/s at 4.5 min(about 65.8%of that of the original specimen).The reduction of P-wave velocity induced by irradiation was also found in other rocks (Hartlieb et al., 2012,2018).This scenario can be explained by the increase of microcracks due to irradiation (Azhari and Hassani, 2013; Saroglou and Kallimogiannis, 2017).A similar decreasing trend can be observed for the CT values.The CT values, however, were featured with a drastic reduction in the first 2 min(2301 Hu at 0 min and 1767 Hu at 2 min).After that,the value slowly reduced to 1618 Hu at 4.5 min,which was approximately 70.3% of the original value.

Fig.4.Typical CT images of irradiated FG specimens with different durations.

Fig.5.P-wave velocities and CT values of irradiated FG specimens.

3.Dynamic loading apparatus and diagnostic systems

3.1.SHPB system

SHPB system has been widely used in investigation of dynamic response of various materials at both intermediate and high loading rates (Chen and Song, 2011).SHPB system has been suggested by ISRM as a standard facility for determining dynamic properties(Zhou et al., 2012).Fig.6 illustrates the configuration of the SHPB system used in this study.This system consisted of a striker bar,an incident bar(3000 mm in length),and a transmitted bar(1800 mm in length) with identical diameters of 50 mm.These bars were made of high-strength steel with Young’s modulus of 211 GPa,and one-dimensional (1D) P-wave velocity of 5201 m/s.Two sets of strain gages were mounted on the incident and transmitted bars,respectively.These gages were connected to an oscilloscope through a Wheatstone bridge and a dynamic strain meter.The amplified strain pulses associated with stress waves in the incident and transmitted bars can then be recorded.

According to the 1D elastic wave theory,the dynamic stresses σ1and σ2on both ends of a specimen can be determined(Xia and Yao,2015) as

Fig.6.SHPB system with momentum-trap and ultra-high-speed camera (1.striker; 2.flange;3.mass;4.incident bar;5.fragment collector;6.transmitted bar;7.flashlamps;8.ultra-high-speed camera).

where t is the time;ASis the cross-sectional area of the specimen;Aband Ebare the cross-sectional areas and the Young’s modulus of bars, respectively; and εi（t）, εr（t）, and εt（t） are the strain histories associated with the incident, reflected, and transmitted waves,respectively.

3.2.Momentum-trap and pulse shaping technique

In order to carry out fragmentation analysis, controlling of the loading pulse is essential.In conventional SHPB tests, specimens generally experience multi-pulse loading(Xia et al.,2008),which is undesired for fragmentation analysis, because the amount of energy for fragmentation is not quantifiable.According to the 1D elastic wave theory, when the striker bar impacts on the incident bar,an incident pulse(compressive wave)is generated to propagate through the incident bar.This incident pulse is partially reflected back at the incident bar-specimen interface,resulting in a reflected pulse (tensile wave).This reflection pulse is reflected again at the impact end of the incident bar, changing from tensile wave to compressive wave,which loads up the specimen for a second time.In this manner, the specimen will unavoidably experience multipulse loading (Chen and Song, 2011).

To ensure the single-pulse loading, many different designs on trapping additional momentum have been developed (e.g.Chen and Song, 2011).The momentum-trap technique developed by Chen and Song (2011) was employed in this study, which is illustrated in Fig.6.The momentum-trap consisted of a flange plate and a rigid mass.The flange plate was installed near the impact end of the incident bar.The gap between the flange plate and the mass should be precisely controlled to ensure that only the first compressive pulse can pass through and load up the specimen(Xia et al.,2008;Chen and Song,2011).The gap between the flange plate and the mass(dgap)was set to the displacement of the impact end of the incident bar due to the incident pulse (Xia et al.,2008):

where C is the 1D P-wave velocity of the bar,and T is the duration of the incident pulse.

According to the method suggested by ISRM(Zhou et al.,2012),the pulse shaping technique is vital to the investigation of the dynamic behaviors of rocks.For the dynamic tests of rocks conducted by the SHPB system, careful implementation of pulse shaping can ensure a ramp pulse rather than a square pulse.The ramp pulse contributes to the maintenance of the stress equilibrium in the specimen.Several methods,e.g.conical striker,three-bar,and pulse shaper,were proposed to shape the loading pulse(Chen and Song,2011;Zhou et al.,2012;Xia and Yao,2015).In this study,the pulse shaper method was used and the shapers were made of copper with a diameter of 10 mm and a thickness of 1.5 mm.

3.3.Ultra-high-speed imaging system

High-speed imaging technique has been frequently used in dynamic material tests.In this study, a Kirana ultra-high-speed camera with two high-power flashlamps (Specialized Imaging AD500) was used to capture snapshots of the rock fragmentation process.The configurations of the camera and flashlamps are shown in Fig.6.Two flashlamps were simultaneously triggered by the incident wave to eliminate the shadow on the specimen.A trigger system was utilized to synchronize the camera, the lamps,and the SHPB system.The camera was operated at the framing rate of 100,000 images per second with the exposure time of 500 ns,at the spatial resolution of 924×768 pixels.

4.Results

4.1.Single-pulse loading

Given the d value in Eq.(4),the flange will be in contact with the mass after the passage of the incident pulse.As a result, when the reflected pulse arrived at the impact end of the bar,this end was no more a free boundary but approximately a fixed one.According to the stress wave theory,the second loading pulse was tensile in this case rather than compressive as observed in conventional SHPB tests.Fig.7 illustrates the strain signal recorded on the incident bar from a typical SHPB test with momentum-trap.It can be seen that the first loading (i.e.incident) pulse was compressive while the second loading pulse had been successfully changed to tensile using the momentum-trap.This tensile pulse will separate the incident bar from the specimen upon reaching the specimen,therefore achieving the single-pulse loading to the specimen.Single-pulse loading is the prerequisite to conduct the valid quantitative fragmentation analysis.

4.2.Dynamic stress balance and determination of the loading rate

According to Eqs.(2)and(3),the dynamic compressive stresses on both ends of the specimen can be determined and those from a typical test are shown in Fig.8a.It indicates that the stresses (σ1and σ2) on both ends of the specimen are almost identical during the loading period.With careful pulse shaping,the dynamic stress balance in the specimen can be achieved (Dai et al., 2010).

The loading rate is one of the parameters to characterize the loading condition.The loading rate determination method suggested by the ISRM (Zhou et al., 2012) was followed in this study.Fig.8b depicts the dynamic stress history of the specimen for the typical compression test.It shows that the stress is almost linearly increasing from approximately 113-160 μs.The data in this region can be fitted by a straight line as shown in Fig.8b, with slope identified as the loading rate.For the typical test, the loading rate was determined as 2465 GPa/s.The peak of the stress was identified as the rock dynamic uniaxial compressive strength (UCS), which was 240 MPa in this case.Further, the stress equilibrium can be satisfied for all the specimens regardless of irradiation durations.

4.3.Dynamic UCS

The dynamic UCS values of FG specimens for different irradiation durations are shown in Fig.9.It indicates that the dynamic UCS values of FG specimens almost increased linearly with theincreasing loading rate for all durations.Thus,the dynamic UCS of both original and irradiated FG specimens exhibited a dependence on the loading rate.Moreover, the dynamic UCS value decreased with increase of durations for a given loading rate, and the microwave irradiation can significantly reduce the dynamic UCS.For example, the strength of the FG specimens irradiated for 2 min increased by 77.6%with the loading rate varying from 1512 GPa/s to 5400 GPa/s.For a given loading rate level(around 4100 GPa/s),the FG specimens with 4.5 min of irradiation had a decrease of 20.6%in its strength compared with that without irradiation.

Fig.7.Incident signals from a compression test conducted on the SHPB system with a momentum-trap.

Fig.8.(a) Stress equilibrium and (b) determination of loading rate for a typical dynamic test (The symbols In., Re., Tr.are referred to as incident, reflected and transmitted,respectively).

The XRD analysis(see Fig.1)shows that FG contains four kinds of minerals,among which annite and greenalite are more sensitive to the microwave irradiation than the other two minerals because of the presence of Fe (Lu et al., 2017).This leads to the irradiation damage as explained in Section 1.In addition, the longer the irradiation duration, the greater the damage in FG specimens as demonstrated by the variations of the P-wave velocities and the CT values (see Fig.5).Therefore, the dynamic UCS decreased with increase of irradiation duration.

5.Dynamic fragmentation analysis

5.1.Snapshots of typical tests

Fig.9.Dynamic UCS values of FG specimens irradiated for various durations.

Fig.10 shows the fragmenting sequences of specimens (irradiated for 2 min) tested at three different loading rates.Time zero corresponds to the time instance when the incident wave arrived at the specimen.As can be seen from Fig.10,higher loading rate leads to faster rock deformation and more cracks.Regardless of the loading rate,the basic failure mode is axial splitting,confirming the uniaxial loading condition in the SHPB tests(Jaeger et al.,2007;Dai et al.,2010).Besides,similar failure modes can also be observed in both original and irradiated specimens.The validity of single-pulse loading can be also confirmed through the snapshots of the specimens.

5.2.Classification of fragmentation types

The fragmented specimens were collected by a fragment collector made of polymethyl methacrylate (PMMA) (see Fig.6).The collector is a transparent box having two holes on its two sides.These holes allow the bars to go through the box.Based on the fragment characteristics, three types of fragmentations were proposed: type I, type II and type III.Representative photos of recovered fragments corresponding to these three types are shown in Fig.11.For type I,the specimen was split into several large slender pieces, accompanied by some small pieces and a little bit of fines.There exists at least one piece of fragment with length equal to that of the intact specimen.For type II, granular fragments were generated, and the lengths of slender fragments were shorter compared with those in the type I.Notably, there is no single fragment with length equal to that of the intact specimen.For type III,specimens were almost pulverized.The granular fragments and fines were dominant and there were no large slender fragments.It is worth noticing that the fragments of both original and irradiated specimens after being impacted can be classified into aforementioned three types.As will be discussed in Section 5.3, this classification is not purely qualitative.The fragmentation type is closely related to the shape of the fragment size distribution (FSD) curve.

5.3.Sieve analysis and fragment size distribution (FSD)

In order to quantitatively evaluate the fragments, sieve tests were carried out.Following the standard method by ASTM(2014),a stack of sieves with aperture sizes of 40 mm,20 mm,10 mm,5 mm,2 mm,1 mm, 0.5 mm, 0.25 mm, and 0.075 mm was adopted.The sieves were stacked in a descending order(from largest size at the top to the smallest at the bottom).A container was placed under the stack to collect the fines.The recovered fragments of a tested specimen were loaded into the top sieve.The stack was placed on top of a mechanical shaker.The loaded stack was then shaken for 5 min.Finally, the mass of fragments retained in each sieve was measured by a high precision balance.The FSD of tested FG specimens are summarized in Fig.12.

As can be seen from Fig.12 for a given irradiation duration,FSD varies with the change of loading rate.Additionally, based on the fragmentation classification criterion defined in Section 5.2, thefragmentation types of specimens were labeled in the bracket following the loading rates in the legends of Fig.12.In the following, we related the fragmentation type to the shape of the FSD curves.As shown in Fig.12,for a given irradiation duration,the FSD curves were concaved upward at low loading rates, and downward at high loading rates.The concaved downward FSD curves correspond to type I fragmentation by visual inspection of fragments, while those concaved upward are of type III.Type II fragmentations correspond to FSD curves that are quasi-linear.As summarized in Fig.13, the fragmentation type is well correlated with the loading rate and the shape of the FSD curve.But it should be highlighted that the boundary FSD curves of each type are still hard to accurately determine based on the available data.

Fig.10.Dynamic fragmentation processes of specimens tested at different loading rates.

For a given irradiation duration,we defined a transition loading rate corresponding to the straight FSD curve.The results of FSD show that accurate determination of transition loading rates for different durations is difficult, because none of the FSD curves shown in Fig.12 is strictly linear.Thus, a method to estimate the transition loading rate is desired.According to Fig.13,FSD for all the irradiation durations can be fitted by the following power function:

where x is the fragment size(mm);F is the cumulative percentage(%);n is a fitting parameter,which is a function of the loading rate.The transition loading rate corresponds to n value of 1.

Taking data with irradiation duration of 4 min from Fig.12 as an example,the fitting curves with the data are shown in Fig.14a.The fitting parameter(n)was plotted against the loading rate in Fig.14b.From the figure, the loading rate corresponding to n ＝1 can be determined.Using this method, the transition loading rates for all durations were obtained and plotted in Fig.15.

Fig.11.Three different fragmentation types:(a)Type I,(b)Type II,and(c)Type III(unit of the ruler is cm).

The transition loading rates with corresponding irradiation durations are shown in Fig.15.It can be seen that the transition loading rate decreased with increase of irradiation duration.Furthermore, the transition loading rate decreased at a relatively slow rate in the first 1.5 min.Thus there may exist an irradiation duration threshold (about 1.5 min), beyond which the transition loading rate decreased much faster.Before the threshold value,the transition loading rate will not change much because the microwave irradiation has not induced sufficient microcracks.This threshold of duration is consistent with the CT values in Fig.5,where a sharp reduction of the CT values was observed at the irradiation duration of 1.5 min.Besides, it is interesting that the transition loading rate slowly decreased again when the specimens were irradiated for more than 3.5 min.Based on the mechanism of irradiation-induced damage reviewed in Section 1, prolonging the irradiation duration can cause more microcracks to the microwave absorbing rock, hence theoretically facilitating fragmentation and the strength reduction for a given loading rate.Considering that FSD curves rely on both loading rate and irradiation duration,there is a tradeoff between these two factors to achieve a given FSD curve,i.e.the transition line.As discussed above, the transition line not only indicates a critical state of FSD curves but also probably provides a promising alternative approach to evaluate the effect of irradiation duration and loading condition on the fragmentation.Therefore,it can be seen from Fig.15 that the required input energy provided by the stress wave can be greatly reduced when the irradiation duration varies from 1.5 min to 3.5 min.

5.4.Evaluation of degree of fragmentation based on average fragment size

In order to further investigate the effects of irradiation duration and loading rate on fragmentation, an average fragment size (D)was adopted.D can be calculated by the following equation (Deng et al., 2018; Holdich, 2002; Hong et al., 2009):

where diis the average fragment size of a given grain grade and riis the mass of fragments with an average size of di.The value of dican be determined by averaging the apertures sizes of two adjacentsieves.The average fragment sizes of all irradiation durations against the corresponding loading rates are plotted in Fig.16.It is interesting to see that regardless of the irradiation duration, the average fragment size follows roughly the same trend.Furthermore, the average fragment size correlated well with the fragmentation types:type I featured average fragment size greater than approximately 24 mm,type III less than about 10 mm,and type II in between.

Fig.12.FSDs of irradiated specimens at different loading rates.

Fig.13.Correlations between the fragmentation type, the loading rate and the shape of the FSD curve(1.low loading rate range;2.intermediate loading rate range;3.high loading rate range).

It can also be seen that between the loading rate of about 3000 GPa/s and 4500 GPa/s, there is a sharp reduction of the average fragment size.Below and above this range,the reduction of the average fragment size with increase of the loading rate is much less significant.It is worth noticing that almost all the specimens featuring type II fragmentation (see Fig.16) were dynamically loaded within this loading rate range.However,the certain loadingrate range for each irradiation group can be hardly determined,because the data are insufficient.Fortunately,given the analysis in Section 5.3, it can be concluded that significant reductions of average fragment size for all specimens approximately occurred at the intermediate loading rate ranges.Both low and high loading rates should be avoided to achieve an effective fragmentation in consideration of the cost in practice.

Fig.14.Estimation of transition loading rate: (a) Fitting curves of FSD, and (b) Determination of the transition loading rate (the irradiation duration is 4 min in this example).

Fig.15.Transition loading rate and irradiation duration for FG specimens.

In addition, the trend line in Fig.16 indicates that the average fragment size decreases with increase of the loading rate in a sigmoidal manner regardless of the irradiation duration.Thus, the Boltzmann function was employed in this study, given by

where S and ˙σ are the average fragment size and the loading rate,respectively; and a, b, c, and d are the fitting parameters that depend on the irradiation duration.The fitting results are shown in Fig.17 and Table 1.

Fig.16.Relation between average fragment size and loading rate.

As depicted in Fig.17,the Boltzmann function agreed well with the testing data, which can be also quantitatively verified by the coefficient of determination(R2)listed in Table 1.Besides,the value of parameter a decreased with irradiation duration,while the value of b increased.Meanwhile, the parameter c decreased in the first 3.5 min and then increased with increase of the duration.However,the parameter d seemed to be unrelated to the duration.

In the Boltzmann function, the meanings of a and b are the initial and final values,respectively.Based on this definition,these meanings can be extended in this study to represent the average fragment size at some loading rates.Therefore, the parameter a is defined as the initial size, and similarly, b means the final size.Mathematically,the values of a and b are the limits of Eq.(7)when the variable approaches to the negative and positive infinity,respectively.But the loading rate usually varies in a definite range in practical applications.It is thus essential to validate the feasibility of using these parameters to evaluate the average fragment size at some loading rates.Considering that the loading rate ranges from approximately 1500 GPa/s to 6500 GPa/s in this study, the differentials between the values calculated by Eq.(7) (e.g.S(1500),S(6500)) and the two fitting parameters (e.g.a, b) are given in Table 2.The results indicate that the relative errors are less than 4%for all the durations.Thus, the parameter a and b can be used to represent the initial and final average fragment sizes, respectively.

Fig.17.Fitting results of the relation between average fragment size and loading rate using the Boltzmann function.

Table 1Values of the fitting parameters.

Table 2Comparison between calculated values and fitting parameters.

Further,in view of the results shown in Fig.16 and Table 1,it can be seen that the microwave irradiation can facilitate the fragmentation of FG specimens at low loading rates.In contrast,the average fragment size increased with decrease of irradiation at high loading rates.This scenario indicates that the FG specimens with more irradiation induced damage seems to generate more granular fragments rather than fines when dynamically loaded at high loading rates.

As discussed, the microwave-assisted rock fragmentation provides an alternative and promising method in improving the efficiency of mechanical rock breakage.Additionally, the fragmentation largely relies on both irradiation duration and loading rate.For the purpose of obtaining the fragments with a desired size distribution, the pretreatment and loading conditions should be chosen with care.

6.Conclusions

In this study, a series of tests was carried out to investigate the influence of microwave irradiation on the dynamic fragmentation behaviors of FG specimens using dynamic compression tests.The carefully prepared cylindrical FG specimens were exposed to microwave irradiation for different durations up to 4.5 min.The microwave induced damages to the specimens were quantified using both X-ray micro-CT scanning and P-wave velocity.The results indicated that both the CT values and the P-wave velocities decreased with increase of the irradiation duration.

The dynamic compression tests were conducted using a SHPB system,with a momentum-trap to ensure single-pulse loading and valid sieve analysis.The dynamic UCS values of both original and irradiated FG specimens exhibited a dependence on loading rate.Moreover, the dynamic UCS values for FG specimens decreased with increase of microwave irradiation duration.

Three fragmentation types were proposed based on FSD results obtained using the sieve analysis.The correlations among FSD,fragmentation type, and loading rate were established.An average fragment size was used to quantify FSD.Regardless of irradiation duration, the average fragment size decreased with increase of loading rate.A loading rate range characterized by a sharp decrease of the average fragment size was identified,below and above which the average fragment size roughly remained the same.Further, a novel method was also proposed to evaluate the correlation between the degree of fragmentation,microwave irradiation,and loading rate.

Declaration of competing interest

The authors wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Nos.51704211 and 51879184).Prof.Xia-Ting Feng from the Northeastern University was greatly appreciated for providing advisable guidance to the discussion in this work.