Energy dissipation from two-glass-bead chains under impact

Accepted Manuscript

Energy dissipation from two-glass-bead chains under impact Sheng Jiang , Luming Shen , Franc¸ois Guillard , Itai Einav PII: DOI: Reference:

S0734-743X(17)30767-4 10.1016/j.ijimpeng.2018.01.002 IE 3053

To appear in:

International Journal of Impact Engineering

Received date: Revised date: Accepted date:

8 September 2017 3 January 2018 3 January 2018

Please cite this article as: Sheng Jiang , Luming Shen , Franc¸ois Guillard , Itai Einav , Energy dissipation from two-glass-bead chains under impact, International Journal of Impact Engineering (2018), doi: 10.1016/j.ijimpeng.2018.01.002

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Highlights

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Impact tests on two-glass-bead chains accompanied by micro-CT scan of crushed fragments are performed; Intermediate and large fragments are the main contributor of the newly created fragment surface area; Fragmentation of each glass bead satisfies a surface area fractal condition; Surface areas of undetected micro-cracks could considerably increase the energy conversion ratio.

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Energy dissipation from two-glass-bead chains under impact Sheng Jiang, Luming Shen*, François Guillard and Itai Einav School of Civil Engineering, The University of Sydney, NSW 2006, Australia *

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Corresponding author: [email protected]

Abstract

The mechanisms of energy dissipation during the dynamic fracture and fragmentation of granular materials play a significant role in controlling the evolution of geological structures, enhancing the efficiency of industrial comminution and blasting, and reducing the geoenvironmental hazards. In this

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research, impact tests on two-glass-bead chains accompanied by micro-CT of crushed fragments are performed. The results reveal that only a small portion of the input stress wave energy is dissipated through the fragmentation of the two-bead-chain system and that the energy dissipation efficiency increases with increasing input energy. It is found that generally the glass bead at the back of the chain always experiences more damage regardless of the input stress wave energy level. The micro-CT results

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show that the fracture energy only accounts for a small portion (0.3%~1.4%) of the chain system absorbed energy. Such a low conversion rate is attributed to the underestimated surface area of the microcracks, caused by the finite resolution of the micro-CT. By pursuing a fractal dimension analysis,

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designed to account for the undetected micro-crack surface area, it is estimated that the energy conversion rate could be increased to 0.7%~5.1%. It appears that friction dissipation appears to always be

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predominant, in particular during severe fragmentation. Keywords: Granular material, Impact, Fracture energy, Hopkinson bar, X-ray micro-computed

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tomography

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1. Introduction

The study of stress waves travelling through crushable granular materials has been a fruitful field of research [1-7]. The research outcomes in this field can benefit geological studies and have implications in areas such as seismic hazards and meteoritic impacts, as well as in many industries, including mineral processing, mining, pharmaceutics and petroleum production [8-12]. When assemblies of brittle particles are impacted, the stress wave energy tends to dissipate due to local interplays at contacts that include inelastic deformation, fracture and friction [13, 14]. When immersed in an interstitial fluid (eg., water or air), further complexity including the role of adhesive, capillary and viscous forces, will affect the energy

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dissipation [15, 16]. The stress wave energy propagates and disperses, with more and more particles being involved in the shock process [17, 18]. Grain crushing may occur after the passing of strong stress waves. Irreversible alterations to the porosity, stiffness and permeability of the crushable granular materials after impact may crucially affect subsequent stress wave propagations [19]. The failure pattern of a single particle has been investigated in terms of impact velocity, particle size and

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particle structure [20-23]. By testing the impact of individual glass particles against rigid targets, Andrews and Kim [24] suggested that a critical threshold velocity was required to fracture the glass particle. Below the threshold velocity, the glass particle bounced back after impact, while above it the glass particle experienced fragmentation. In terms of multi-particle systems, Mishra and Thornton [25] studied the effect of contact densities and solid fractions on the various particle failure patterns. Subero et al. [26] found that interfacial energy had a greater effect on the breakage behaviour under low impact velocities.

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Nguyen and Einav [27] highlighted the dissipative contribution arising from the redistribution of lockedin strain energy from surrounding particles; using a simple 1D model (see their appendix) they proved that this mechanism contributes more prominently to the overall dissipation compared to the fracture surface energy, which was later verified by [28] through experimental tests. Following that work in [27], the 1D model was extended for a 3D model by [29], which illustrated the effects of Poisson’s ratio and

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the stress anisotropy on the ratio between dissipation due to redistribution and dissipation due to fracture surface energy. The propagation of stress waves through multi-particle systems is rather intricate due to

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the effects of the interactions among particles and the resulting damage and fragmentation of the individual particles. Hence, the use of chain-like granular material systems is a widely common approach to simplify the problem [30-32]. Most previous studies look at the propagation of stress waves through an

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elastic or elastoplastic chain system, in which the particles remain unbroken after the impact [33-35]. Less attention has been given to chains of brittle particles undertaking extensive fractures. However, particle

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fracture critically determines the energy dissipation and thus controls the dynamic behaviour of granular materials under impact.

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The focus of this study is on the energy dissipation mechanisms in a chain of brittle particles during high speed fracture. In such configuration the dissipative contribution from energy redistribution to nearby particles [27] does not play a role, leaving the creation of new surface area via fracture and fragmentation as the primary energy dissipation mechanism [36, 37]. Subsequently, this may lead for further fragment kinetic energy dissipation, acoustic emission, heat and friction dissipation. The newly created surface area is related to the changes in fragment size distribution, as well as the growing cracks within the particles. However, experimental results show that the newly created surface area only accounts for less than 0.1% of the total dissipated energy, far smaller than expected [38, 39]. Hogan and Spray [39] also detected

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micro-cracks along fracture surfaces, but could not quantify their surface area. Also, some of the detached fragments eject from particle surface, transferring the wave energy into fragment kinetic energy. Nakamura and Fujiwara [40] estimated that roughly 1% of the energy was turned into fragment kinetic energy. The high speed fracture also produces acoustic emission, which is treated as a comparably minor contribution to the energy dissipation [41]. In terms of frictional dissipation, no quantitative analysis was made owing to the difficulty in calculation. However, Hogan and Spray [39] discovered the formation of - and

- spheroids on the fracture surface of ceramic under impact, which indicated that

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temperature in excess of 3500K was reached during fracture, suggesting a significant creation of thermal energy that could be attributed to either friction or surface creation. Huang et al. [42] discovered that the frictional dissipation was much higher under impact loading compared to the ones under quasi-static loading for the same breakage extent. Although all the dissipation mechanisms are clear, the contribution

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of each dissipation mechanism remains unknown, particularly the fracture surface energy and friction dissipation.

Motivated by the abovementioned discussion, this paper focuses on assessing accurately the energy dissipated through fracture during the impact of brittle particle. We will tackle this problem experimentally through the impact of a two-glass-bead chain system. In this study, the impact loading is

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provided by the split Hopkinson pressure bar device. A specially designed tube and a separator are used for separating the bead fragments during fragmentation. Micro-CT scanning of the bead fragments is

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carried out after the impact test. The fracture energy is then calculated by evaluating the newly created surface area, which provides the further quantitative analysis of the fracture mechanics. Finally a simple fractal dimension analysis based method is proposed for estimating the surface area of the undetected

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micro-fragments and micro-cracks in order to more accurately measure the dissipated energy through

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surface creation via fracturing and fragmentation.

2. Methodology

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2.1 Split Hopkinson bar Dynamic compression experiments at high strain rate are conducted on a modified split Hopkinson bar (SHPB) device. A schematic diagram of the SHPB setup is shown in Figure 1. The SHPB consists of solid high strength steel bars, a strain data acquisition system and a momentum trap system. The incident and transmitted bar are 1.5m long and the striker bar is 0.12m long, all with a diameter of 15mm.

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Figure 1 Schematic of the split Hopkinson bar

During the experiment, the striker bar launched by the gas gun impacts the incident bar to generate a compression wave (referred as incident wave), which travels along the incident bar to load the specimen

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(here, the bead chain). When the incident wave arrives at the interface between the specimen and the incident bar, the wave is partially reflected back into the incident bar (referred as reflected wave) and partially transmitted through the specimen into the transmitted bar (referred as transmitted wave). The strain histories are measured using two pairs of strain gauges attached to the middle of the incident and transmitted bars, and acquired by the digital oscilloscope, representing the incident, transmitted and

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reflected signals. The typical strain histories for the incident and transmitted bars are shown in Figure 2. According to the one-dimensional stress wave theory, when the stress wave propagates in the long rod,

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the mechanical energy of the stress wave takes the form of the elastic strain energy of the bar deformation and the kinetic energy through bar motion [43]. The elastic strain energy ( ) carried by the incident wave

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can be calculated through the incident strain ( ), ∫

is the deformed volume in the incident bar. It is noted that, at any moment, only a portion of the

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where

(1)

incident bar is involved in the elastic deformation by the incident pulse when the stress waves propagate

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in the bar. The deformed volume ( ) in an incident bar depends on bar cross-sectional area ( ), elastic wave speed in the bar material ( ) and loading duration ( ), which can be expressed as pulse shaper is used in the experiment. Hence is determined by the striker bar length ( ), linearly elastic bar,

, where

. Here, no . For a

is the Young’s modulus for the bar material.

Thus, equation (1) can be rewritten as (2)

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The kinetic energy ( ) of the incident bar after the incident wave passes can be expressed as (3) where

and

are mass and particle velocity of the deformed portion in the incident bar, respectively.

The deformed portion mass is

, where

is the bar material density. The conservation of mass

stress wave theory, expressed as

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yields the relationship between the particle velocity ( ) and the strain ( ) according to one-dimensional .

Equation (3) is thus rewritten as

from the incident bar is the sum of the elastic strain energy ( ) and

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Hence, the total input energy

(4)

kinetic energy ( ),

(5)

The reflected and transmitted energies, associated with the reflected wave and transmitted waves, respectively, can be similarly expressed. Detailed calculations can be found in [43]. If the energy loss in

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the form of sound and heat is ignored, the difference between the incident wave energy and the sum of the transmitted and reflected wave energy is regarded as the energy absorbed

by the specimen. The two-

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glass bead chains are subjected to impacts under six different input energy levels (2.2 ~16.5

).

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corresponding to different striker bar impact velocities of 3.3

Figure 2. The typical strain histories from the incident and transmitted bars.

53.8

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2.2 Specially designed tube and separator Multiple impacts on the specimen might occur during each test if the incident bar is not immediately stopped after the first impact. Since we are only interested in the fragmentation of glass beads and the associated energy dissipation during the first impact, a special specimen holding tube is designed to prevent multiple impacts in each test. The specimen holding tube contains two parts, which can close and

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be fixed by four screws. Half of the tube and the separator are shown in Figure 3. Before the test, two glass beads of diameter 7 mm, namely the front bead (close to the incident bar), the back bead (close to the transmitted bar), and a separator are sandwiched by the incident and transmitted bars in the internal chamber. The separator contains a hollow-cylinder plastic frame ( inner and outer diameters of the frame, respectively, and

and

are the

is the frame thickness) and a 0.1mm thickness

copper shim, which is used for separating the fragments from each bead for future analysis. A groove

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(0.5 mm width) on the frame side is designed for inserting the copper shim. The rigid plastic frame ensures the copper shim remains orthogonal to the impact direction. It is assumed that the copper shim is sufficiently thin so that it will not cause noticeable stress wave reflection at the glass bead and copper shim interface.

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The diameter of the internal chamber is 7.05 mm, with a 13.5 mm internal length. The whole glass bead chain system is 14.1 mm long (two 7 mm-diameter glass beads and 0.1mm-thickness copper shim). As a result, a small part of the chain system is exposed outside the internal chamber thanks to the 0.6 mm gap

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between the internal chamber and incident bar. This design ensures a maximum 0.6mm compressive deformation of the glass bead chain, as further impact load would be carried by the chamber wall instead

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of the glass beads. Based on our calculation, the 0.6 mm gap can guarantee the single impact condition and that sufficient strain will be applied for both beads to crush [44]. Although the level of adsorbed energy will be affected if the gap is too small, it is not our purpose to control the input and/or absorbed

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energy and we will accept any measured input/absorbed energy as long as there is only a single impact on

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the specimen.

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Figure 3. Schematic of half of the specimen tube and the separator

2.3 Micro-CT scanning

All the fragments are scanned at the Australian Centre for Microscopy and Microanalysis (ACMM) at the

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University of Sydney using the SkyScan 1172 high-resolution micro-CT at a spatial resolution of 17.66 µm. During scanning, the fragments of each bead are put in a specimen container. The container is located on the platform and scanned by rotating the platform

around its central axis between the X-

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ray tube and a detector. A series of the absorption radiographs of the sample are acquired during the stepwise rotation. The CT images can be visualized as a stack of images, each of which is a cross section of

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the scanned sample with the thickness of one voxel size (17.66 µm here). The reconstructed three-dimensional (3D) CT images are analysed through a series of image-processing

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and analysis procedures to obtain a quantitative description of the fragments. The image processing is made while reducing the image noise and enhancing the contrast between the air and the solid, with the objective of identifying the individual fragment. The reconstruction software used in this study is the

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NRecon software (

). All the parameters are kept unchanged for all the scans in order to keep the

analysis systematic. Table 1 provides the complete list of the scanning and reconstruction parameters. The next step is to analyse the reconstructed images to measure the morphology parameters of the fragments.

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Table 1. X-ray micro-CT scanning and reconstruction parameters Scanning

Reconstruction SkyScan 1172

Program

NRecon

Camera

Hamamatsu 10Mp

Version

1.6.9.18

Spatial resolution

17.66 µm

Smoothing

2

Source voltage

100 kV

Ring artefact correction

7

Source current

100 µA

Beam hardening correction

20%

Number of rows × columns

1048×3968

Image format

16-bit TIFF

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Scanner

Rotation step

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3. Results and discussions 3.1 Damage pattern

For the purpose of qualitative failure analysis of the impacted glass beads, three representative failure patterns are identified corresponding to increasing beads alteration upon impact, namely damage, splitting and crushing, as shown in Figure 4. The input energy

and absorbed energy

of each test are

energy

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summarized in Table 2. Dissipation ratio (R) is calculated as the ratio of absorbed energy . Striker bar velocity (V) is back-calculated from the input energy

to input

. Due to random

microstructures of the material and different loading conditions, fracture tests of brittle particles often

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exhibit scattered results [45]. In this study, we carry out a total of 25 impact tests under different impact velocities and the results are quite repeatable. It is undeniable that in some cases (2 out of 25) the front

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beads crush and back beads are only damaged slightly. However, the rest of the cases show consistent failure pattern where the back beads experience a more severe damage state than the front beads. Only six

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tests are scanned by the micro-CT due to the very long scanning time and the corresponding analysis time.

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(c)

Figure 4. Representative patterns of the failed glass beads: (a) Damage, (b) Splitting, (c) Crushing

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Table 2. Input and absorbed energy for each test

(J)

V (m/s)

(J)

Damage pattern Front bead

Back bead

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Test No.

2.2

0.05

2.3

3.3

Damage

Damage

2

6.5

0.8

12.3

5.7

Damage

Crushing

3

11.9

1.7

14.3

7.8

Crushing

Crushing

4

18.1

2.7

14.9

9.6

Splitting

Crushing

5

23.1

4.4

19.0

10.8

Crushing

Crushing

6

53.8

10.8

20.1

16.5

Crushing

Crushing

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1

In Test 1, the front bead is slightly damaged with a typical Hertzian ring cracks [20, 46] at both contact points. The back bead has some annular chips spalled from the surface but still retains its spherical shape. As the input energy increases in Test 2, the front bead begins to develop chips, while the back bead

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crushes into small fragments. With a further increase to the input energy a ‘transition input energy regime’ (roughly between 12 J and 23 J) develops, where the damage patterns are inconsistent between tests.

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Specifically, at the relatively low input energy regime both glass beads crush (e.g., in Test 3: crushingcrushing), while under the higher input energy only the back bead crushes while the front bead is only splitting (e.g., in Test 4: splitting-crushing). This is likely due to the heterogeneity and statistics of the

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beads’ microstructure and chemical composition. Similar experimental findings were reported during the impact breakage of rocks, whose fracture patterns were random under intermediate impact velocities [47].

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Then, with a further increase of the input energy in Test 6, this unstable phenomenon vanish, and both the front and back beads crush into small debris.

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To summarise the findings, in general the back bead is more susceptible to crushing, and even when both beads crush, it always shows a more severe crushing than the front bead. The experimental results are in agreement with the results from [6], where a ten-glass-bead chain was impacted and the second glass bead (close to the incident bar) always fractured more severely than the first glass bead. According to their detailed simulation results, the second glass bead was the first to reach the tensile strength of the glass and thus was the first to fail. The high maximum principal stress in the second glass bead mainly resulted from the tensile stress generated as the reflected waves of the compressive waves at the free surface. On

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the other hand, these tensile stresses in the first glass bead were somehow cancelled by the continuously incoming compressive stress waves induced by the striker bar. Relationship between the experimental input energy

and dissipation ratio (R) is shown in Figure 5,

which also includes an additional fitting curve as a guide for the eye. The intercept of the fitting curve on the x-axis (input energy) is around 2 J. Below this critical value of input energy the chain system is

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expected to undergo only elastic deformation, with most of the input energy being transferred to the transmitted bar while the rest generates negligible acoustic dissipation and friction along the boundaries. In the high input energy regime, the dissipation ratio increases slightly. It seems that the dissipation ratio has an upper limit of around 20%, as shown in Figure 5. It is believed that this upper limit is dependent on the mechanical properties, geometry and boundary conditions of the investigated system.

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As can be seen from Figure 5, for the two-bead chain system, a higher input energy corresponds to a higher dissipation ratio. The fracture and breakage of the glass bead affects the stress wave energy dissipation. Once the glass bead is fractured, free fracture surfaces are created. The input compression stress wave will then be reflected when it reaches the newly created free surface. More free surfaces are created when the glass bead is broken into many fragments. As a result, more reflections of the stress wave will happen at the newly created free surfaces. This process greatly precludes the propagation of the

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stress wave and thus increases the energy dissipation efficiency. A sharp increase in the dissipation ratio is observed when the failure pattern of the glass bead chain changes from damage-damage to damage-

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crushing, indicating that more input energy is dissipated when severe breakage has occurred.

Figure 5. Relationship between the input energy and the dissipation ratio.

3.2 Fragment analysis The damaged specimens are scanned using the micro-CT, with the fragments being analysed using the corresponding 3D dataset, in term of their surface area and volume.

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3.2.1 Analysis of individual beads from the chain The cumulative frequency distributions (based on the fragment number) for three tests corresponding to the representative failure patterns are plotted in Figure 6. The generalized extreme value distribution function is adopted to fit the data [48], as follows:

where

(

is the local parameter,

)) ]

(6)

is the scale parameter,

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[ (

is the shape parameter, and

logarithm of the fragment volume.

is the value of the

As shown in Figure 6, the three-parameter generalized extreme value distribution makes a good fitting for the fragments of each bead generated from the impact test. An obvious shift of the curves towards the

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smaller volume is shown from the front bead to the back bead in all the different tests (simply because the

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back bead contains more fine particles and has a smaller average volume than the front bead).

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(c)

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(a)

Figure 6. Fragment volume cumulative frequency distribution of single bead in three damage patterns: (a)

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Damage-Crushing, (b) Crushing-Crushing and (c) Splitting-Crushing 3.2.2 Full chain analysis

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For the two-bead chain system, the fragments of each test are grouped into eight bins based on their volumes. The data of each bin is then normalized over the total fragment number, total fragment surface area and total fragment volume to get the contribution percentages, as shown in Figure 7. In the following, if the fragment volume is smaller than whose volume is greater than

, it will be treated as fine powder, while a fragment

will be called large one. A fragment with a volume between these

two values will be considered as an intermediate fragment.

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It can be seen from Figure 7(a) that the fragment number contributions of different tests follow a similar trend, and in terms of fragment number, fragment sizes between majority in all the tests. Fine powders (volume smaller than

and

make the

) contribute nearly 90% number of

the fragments in each test and are the main products during the high-speed fragmentation process. However, they only contribute a tiny portion of the total scanned surface area and volume due to their extremely small size. It is clear from Figure 7(b) and Figure 7(c) that the percentages of the fine powder

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surface area and volume contributions keep decreasing with the reduction of the particle size. In fact, the intermediate and large fragments are the major contributors of the scanned fragment surface area and

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volume.

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(c)

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(a)

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Figure 7. The percentage contribution of (a) fragment number; (b) surface area; (c) volume for each test. 3.2.3 Fragment morphology analysis

The shape of the fragment makes another parameter that one can probe in order to study the particle

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fracture process. The sphericity is usually used to characterize the morphology of the fragment and describe the similarity between the fragment and a sphere. Since the fragment volume varies in several

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∑

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orders of magnitude, we add the volume weight term here to calculate the sphericity.

where

(7)

is the surface area of a sphere having the same volume of the ith fragment,

surface area and volume of the ith fragment, and

and

are the

is the total volume of all fragments.

Table 3 shows the volume-weighted sphericity parameter of each damaged bead for each test. As can be seen from Table 3, the sphericity value is greatly affected by the damage pattern. The crushed beads have a much smaller sphericity value (0.28-0.52) than the damaged ones (0.90-0.95). Thus, the fragments from the crushed beads are more irregular and angular than those from the slightly damaged ones. The bead

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chain sphericity value is also highly related to the chain damage pattern. The sphericity value of the damage-damage case (0.95) is larger than that in the damage-crushing case (0.62) and splitting-crushing case (0.57). In the three crushing-crushing cases, although the input energy varies a lot, the values of the chain sphericity (combining both bead fragments) are very similar (0.39, 0.40 and 0.41). This is mainly resulted from the similar fracture mechanisms at this stage of severe fragmentation, no matter how large the input energy is, the glass bead fracture mechanism remains unchanged, which also validates the

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abovementioned dissipation ratio limitation as demonstrated in Figure 5.

Table 3. Volume weighted sphericity of each impacted bead and combined result in each test Sphericity

1 2 3 4 5 6

Front bead Back bead 0.95 0.90 0.50 0.76 0.52 0.46

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3.3 Fractal dimension

0.90 0.35 0.28 0.37 0.30 0.32

Combined in bead chain 0.95 0.62 0.40 0.57 0.41 0.39

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Test No.

Fragmentation of particles shows a statistically scale-invariant topology concerning the fragment size

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distribution [49], which may thus be described as a fractal process [50]. Here, a new analysis of the surface area fractal dimension is performed and examined to find out if the fragments satisfy such a

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fractal condition.

The fractal dimension of the fragment surface area can be defined by the relationship between the number of the particles and the surface area of the particles. Different from the previous definition between the

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fragment size and number, the definition here directly uses the full 3D dataset from the micro-CT scanning results. Previous studies usually convert an irregular shape particle into an equivalent sphere

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with the same volume. The equivalent sphere omits the irregular particle morphology and will not accurately reflect the real particle surface area. The surface area frequency distribution of the fragments is given by

, where

is the cumulative number of the fragments with surface area equal

to or greater than a certain value , and D is the surface area fractal dimension. Usually, the value of D represents the negative slope in a log

vs. log

plot [49]. Because the log

hide systematic errors in the data, a new plot of [3 log ( ) + log slope of each line (tan ) is

] vs. log

[51, 52]. Figure 8 shows the [3 log ( ) + log

vs log

plot tends to

is used here, where the ] vs. log

plot for

the fragments in each test. The plot for the front bead of Test 1 is not shown in the figure because this

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bead remains intact after the test. As can be seen from Figure 8, fragmentation at the high speed fracture level satisfies the surface area fractal condition. The damage and splitting damage patterns have the same fractal dimension value of 0.5, while those of the crushing patterns are 0.6 and 0.7. It appears that the fragment surface area fractal dimension reflects the damage state to a certain degree with a larger D value

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corresponding to a more severe fragmentation state.

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Figure 8. Fractal dimension of the fragment surface area for each test. Another interesting phenomenon is that the micro-CT detects ‘voids’ inside the fragments of splitting and

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crushing cases. No ‘void’ is discovered in the damage cases, which likely means that these ‘voids’ do not initially exist in the beads and are newly formed during the impact. In fact, ‘voids’ should be part of the newly created internal cracks inside the fragments. A typical fragment containing the internal cracks is

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shown in Figure 9(a). Main cracks propagate through the particle bulk and split the fragments. These

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internal cracks are the by-products of main cracks during the high-speed fracture [53].

(a)

(b)

Figure 9 (a) Grey-scale slice image of a fragment (b) the corresponding binary image

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Image analysis of these internal cracks needs segmentation, which involves converting the raw grey-scale images (Figure 9(a)) to binary images of ‘voids’ (black) and ‘solid’ (white), as shown in Figure 9(b). In this study, the segmentation is achieved using a simple thresholding method. Each pixel in an image is replaced by a black pixel if the image intensity is less than the thresholding value, or given a white pixel if the image intensity is greater than that value. Owing to the partial volume effect, the internal crack boundary is impossible to identify with confidence during the thresholding process. Nevertheless, for

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each detected internal crack, part or all of it is treated as ‘solid’, thus the ‘voids’ shown do indeed represent the real internal cracks. Furthermore, previous studies showed that not only the fragments satisfied a fractal distribution, but also the cracks looked the same at all scales and thus could be described by fractals [54]. Inspired by the fractal dimension of the fragment surface area, the fractal dimension of the crack surface area is plotted in Figure 10. In addition to the damage cases, the result for

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the front bead of Test 4 (splitting) is also excluded in the figure because only one void is detected. It is

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very clear from the figure that all the crushing cases have the same crack fractal dimension value of 1.4.

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Figure 10. Fractal dimension of the internal crack surface area

3.4 Energy dissipation 3.4.1 Fracture energy The scanned fragments and internal crack surface areas from all the tests are summarized in Table 4. and

are the newly created fragment surface areas for the front bead and the back bead, respectively. and

represent the scanned internal crack surface areas of the front bead and the back bead,

respectively.

is the total newly created surface area, which is the sum of

,

,

and

.

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The standard fracture energy the total fracture energy

(energy used to create a unit surface area) is introduced here to estimate

(energy used to create total new surface area). The standard surface energy is

treated as a material constant. Hence, the fracture energy can be calculated by (8) is between 3.5 and 5.3

[36]. In the following

=5.3

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For glass, the standard fracture energy

will be used to get an upper bound or the energy dissipated by surface creation. It is found that the energy conversion rate

(the ratio of fracture energy to absorbed energy) is higher than the previous

experimental results of CR<0.1% reported in [39]. This could result from the more accurate 3D measurement of the fragment surface area thanks to a comparatively higher resolution. It needs to be noted that the actual conversion rate is higher than the values listed in Table 4 because the absorbed

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energy also includes the plastic work of the copper shims, which is estimated to be no more than 1% of the absorbed energy. However, the overall conversion rate is still quite small.

Test No.

(

)

(

)

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Table 4. The scanning results for each test

*

68.4

2

35.3

2117.1

3

1597.7

4

)

(

)

(

)

(J)

*

*

68.4

0.00036

0.7%

*

2.5

2154.8

0.011

1.4%

2298.8

1.1

6.3

3903.9

0.021

1.2%

154.9

2740.5

0.01

17.1

2912.5

0.015

0.6%

5

1977.2

2682.4

2.6

17.4

4679.6

0.024

0.5%

6

2584.3

3086.1

3.1

17.6

5691.1

0.030

0.3%

PT

ED

1

(

CE

There are two main reasons for such a low conversion rate. The first reason is the existence of other energy dissipation mechanisms during the process of fragmentation. In addition to the particle fracture

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dissipation, frictional dissipation between the fragments should also be considered as a main dissipation mechanism during the high speed dynamic fracture. The relations between the input energy and fragment number as well as the conversion rate are plotted in Figure 11. Since the input energy is highly related to the chain damage pattern, a higher input energy corresponds to a larger number of fragments. On the other hand, the conversion rate peaks at 1.4% and starts to drop in the following tests. This dropping in the conversion rate can be related to the fragment number in the severe damage cases, as well as the input energy. The reason is that the frictional dissipation in the severe damage is much larger than that in the slightly damaged case. More fragments with more irregular shape are created in the crushing process. The

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frequency of contact between fragments increases, leading to a higher friction dissipation level in the severe damage cases. The decreasing trend of the conversion rate also illustrates the breakage efficiency is much lower when more severe fragmentation occurs, which mainly results from the continually rising

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CR IP T

friction dissipation component.

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Figure 11. Relations between the input energy and fragment number as well as energy conversion rate

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(the ratio of fracture energy to absorbed energy).

The second possible reason is the underestimated fracture energy resulting from undetected surface areas

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of smaller fragments and cracks. Limited by the micro-CT scanning resolution, the minimum detected surface area is

( ) for both fragment and internal cracks. Parab et al. [55] estimated that

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the flaw size in the glass was around to

. Hence, there is an undetected surface area range (from

). In the following, these undetected fragments and cracks will be

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called micro-fragments and micro-cracks. To illustrate the influence of the undetected micro-fragments and micro-cracks, the undetected surface area is estimated from the surface area fractal distribution. Since the surface area

and the cumulative number

satisfy the fractal dimension (

),

(the number of the micro-fragments and micro-cracks for a certain surface area ) is the differential gain of

and can be computed as follows (the absolute value is used in order to make sure |

|

): (9)

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Then, the undetected surface area ̃ can be estimated as follows: ̃

∫

∫

(10)

The estimated surface area for micro-fragments and micro-cracks are summarized in Table 5. ̃

and ̃

are the estimated micro-fragment surface areas for the front and back beads, respectively, while ̃ ̃

and

is the total

newly created surface area, including the detected and undetected fragments and cracks.

represents

CR IP T

are the estimated micro-crack surface areas for the front and back beads, respectively.

the new conversion rate after considering the contribution of the undetected micro-fragments and microcracks.

As can be seen from Table 5 the estimated surface area of the undetected micro-fragments accounts for a

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small part of the total surface area. The estimated results are consistent with what shown in Figure 6(b), where the fine powders only contribute a small part to the total fragment surface area. On the other hand, the estimated surface area of the undetected micro-cracks is much larger. They can affect the total surface area more significantly than the micro-fragments. If the surface area of the undetected micro-cracks is considered, the new conversion rate is about 3~4 times higher than the original value, with a maximum

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value of 5.1%. However, the fracture energy still only contributes a small portion of the absorbed energy. This is consistent with what suggested by Nguyen and Einav [27] that the frictional dissipation should prevail over the fracture energy. Although the proposed fractal analysis of the undetected micro-cracks

ED

may explain the underestimated fracture energy to some extent, a more accurate estimation of the undetected micro-crack surface area may warrant further work.

(

)

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Test No.

PT

Table 5. Estimated surface area of the micro-fragments and micro-cracks ̃ (

)

̃ (

)

̃ (

)

̃ (

)

(

)

68.4

0.7%

*

0.1

*

*

68.5

0.7%

2

2154.8

1.4%

0.1

0.2

*

5516

7671.1

5.1%

3

3903.9

1.2%

0.2

0.4

5516

5516

14936.5

4.7%

4

2912.5

0.6%

0.1

0.4

*

5516

8429.0

1.7%

5

4679.5

0.5%

0.2

0.4

5516

5516

15712.1

1.9%

6

5691.0

0.3%

0.2

0.4

5516

5516

16723.6

0.8%

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1

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Conclusions Energy dissipation from a two-glass-bead chain under impact is investigated experimentally using the split Hopkinson bar and micro-CT. In general, the back bead crushes more easily than the front bead and damages more severely. Under lower input energies, the energy dissipation efficiency of the chain system greatly increases when severe fragmentation occurs. Under higher input energies, the dissipation ratio increases slowly and tends to saturate at an upper limit of about 20% (whose exact value likely depends

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on the mechanical properties, geometry and boundary conditions of the investigated system).

The generalized extreme value distribution is used to characterize the fragment size distribution. Fragmentation results highlight that the fine powder is the main product during the high speed fracture process. Considering the relatively small order of magnitude of the powder surface area and volume, the total powder surface area and volume are concluded to have a negligible effect; instead, the intermediate

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and large fragments are the main contributor of the newly created fragment surface area. Morphology analysis show that these fragments created during severe fragmentation are more irregular and angular than those in the slightly damaged cases. The chain sphericity values in the crushing-crushing cases are very similar due to the similar fracture mechanisms of the glass bead during the severe fragmentation process. Fragmentation of each glass bead satisfies a surface area fractal condition. Energy dissipation during the fragmentation is also examined. The fracture energy used to create new

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surface areas only occupies a small portion of the absorbed energy under current scanning resolution. Two reasons are discussed to explain this phenomenon. Firstly, the breakage efficiency becomes lower

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when the input energy increases. The increasing irregularity of the fragments created in the higher dynamic breakage process increases the chances for contacts, and thus elevates the role of frictional dissipation. The decreasing trend of the conversion rate indicates that the frictional dissipation component

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keeps rising with input energy. Secondly, the fracture energy is underestimated owing to the undetected micro-cracks. These micro-crack surface area, estimated from the fractal analysis, greatly increases the

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total fractured surface area. The new conversion rate is increased by about 3~4 times after considering the contribution of undetected micro-cracks. Under current scanning resolution and image processing

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software, it is still very difficult to accurately calculate the micro-crack surface area, which warrants further research. However our study takes into account the contribution of fragments below the experimental resolution using fractal dimension, and therefore strongly suggests that the surface creation only accounts for a small portion of the dissipated energy during fragmentation under impact.

Acknowledgments The work is supported in part by the Australian Research Council through Discovery Projects (DP130101291 and DP140100945) and by the National Natural Science Foundation of China (Grant No.

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11232003). The authors acknowledge the facilities and the scientific and technical assistance of the Australian Microscopy & Microanalysis Research Facility at the Australian Centre for Microscopy & Microanalysis at the University of Sydney.

References

7.

8. 9. 10. 11. 12. 13.

14. 15.

CR IP T

AC

16.

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6.

M

5.

ED

4.

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3.

H. Kolsky, Stress waves in solids. Vol. 1098. 1963: Courier Corporation. J. Subero and M. Ghadiri, Breakage patterns of agglomerates. Powder Technology, 2001. 120(3): p. 232-243. D. Verkoeijen, G.M. Meesters, P.H. Vercoulen, and B. Scarlett, Determining granule strength as a function of moisture content. Powder technology, 2002. 124(3): p. 195-200. A. Salman, G. Reynolds, J. Fu, Y. Cheong, C. Biggs, M. Adams, et al., Descriptive classification of the impact failure modes of spherical particles. Powder Technology, 2004. 143: p. 19-30. F. Guillard, P. Golshan, L. Shen, J.R. Valdes, and I. Einav, Dynamic patterns of compaction in brittle porous media. Nature Physics, 2015. 11(10): p. 835-838. D. Liu, L. Shen, F. Guillard, and I. Einav, Transition failure stress in a chain of brittle elastic beads under impact. International Journal of Impact Engineering, 2016. 93: p. 222-230. S. Wang, L. Shen, F. Maggi, A. El-Zein, and G.D. Nguyen, Uniaxial compressive behavior of partially saturated granular media under high strain rates. International Journal of Impact Engineering, 2017. 102: p. 156-168. E.N. Hiestand, Mechanical properties of compacts and particles that control tableting success. Journal of pharmaceutical sciences, 1997. 86(9): p. 985-990. Y. Ben-Zion and C.G. Sammis, Characterization of fault zones, in Seismic Motion, Lithospheric Structures, Earthquake and Volcanic Sources: The Keiiti Aki Volume. 2003, Springer. p. 677-715. J. Uehara, M. Ambroso, R. Ojha, and D. Durian, Low-speed impact craters in loose granular media. Physical Review Letters, 2003. 90(19): p. 194301. H. Sutar, S.C. Mishra, S.K. Sahoo, and H. Maharana, Progress of red mud utilization: an overview. 2014. I. Vardoulakis, M. Stavropoulou, and P. Papanastasiou, Hydro-mechanical aspects of the sand production problem. Transport in porous media, 1996. 22(2): p. 225-244. Y. Zhang, G. Buscarnera, and I. Einav, Grain size dependence of yielding in granular soils interpreted using fracture mechanics, breakage mechanics and Weibull statistics. Géotechnique, 2016. 66(2): p. 149-160. Y. Zhang and G. Buscarnera, A rate-dependent breakage model based on the kinetics of crack growth at the grain scale. Géotechnique, 2017: p. 1-15. I.S. Aranson and L.S. Tsimring, Patterns and collective behavior in granular media: Theoretical concepts. Reviews of modern physics, 2006. 78(2): p. 641. G. Buscarnera and I. Einav, The yielding of brittle unsaturated granular soils. Géotechnique, 2012. 62(2): p. 147. J. Crassous, D. Beladjine, and A. Valance, Impact of a projectile on a granular medium described by a collision model. Physical Review Letters, 2007. 99(24): p. 248001. J.R. Valdes, F., Guillard, and I. Einav, Evidence that strain-rate softening is not necessary for material instability patterns. Physical Review Letters, 2017: p. In press. J.R. Valdes, F.L. Fernandes, and I. Einav, Periodic propagation of localized compaction in a brittle granular material. Granular Matter, 2012. 14(1): p. 71-76. A. Salman and D. Gorham, The fracture of glass spheres. Powder technology, 2000. 107(1): p. 179-185.

CE

1. 2.

17. 18. 19. 20.

ACCEPTED MANUSCRIPT

27. 28.

29. 30. 31. 32. 33. 34.

35.

36. 37.

AC

38.

CR IP T

26.

AN US

25.

M

24.

ED

23.

PT

22.

A. Samimi, M. Ghadiri, R. Boerefijn, A. Groot, and R. Kohlus, Effect of structural characteristics on impact breakage of agglomerates. Powder Technology, 2003. 130(1): p. 428435. D. Gorham and A. Salman, The failure of spherical particles under impact. Wear, 2005. 258(1): p. 580-587. S. Jiang, L. Shen, I. Einav, and F. Guillard, Energy dissipation mechanism in single glass bead under impact, in International Conference on Geomechanics, Geo-energy and Geo-resources. 2016: Melbourne, Australia. E. Andrews and K.-S. Kim, Threshold conditions for dynamic fragmentation of glass particles. Mechanics of Materials, 1999. 31(11): p. 689-703. B. Mishra and C. Thornton, Impact breakage of particle agglomerates. International Journal of Mineral Processing, 2001. 61(4): p. 225-239. J. Subero, Z. Ning, M. Ghadiri, and C. Thornton, Effect of interface energy on the impact strength of agglomerates. Powder Technology, 1999. 105(1): p. 66-73. G.D. Nguyen and I. Einav, The energetics of cataclasis based on breakage mechanics. Pure and applied geophysics, 2009. 166(10-11): p. 1693-1724. A. Sufian and A.R. Russell, Microstructural pore changes and energy dissipation in Gosford sandstone during pre-failure loading using X-ray CT. International Journal of Rock Mechanics and Mining Sciences, 2013. 57: p. 119-131. A.R. Russell and I. Einav, Energy dissipation from particulate systems undergoing a single particle crushing event. Granular Matter, 2013. 15(3): p. 299-314. L. Slepyan and L. Troyankina, Fracture wave in a chain structure. Journal of Applied Mechanics and Technical Physics, 1984. 25(6): p. 921-927. C. Coste, E. Falcon, and S. Fauve, Solitary waves in a chain of beads under Hertz contact. Physical review E, 1997. 56(5): p. 6104. J.E. Socolar, D.G. Schaeffer, and P. Claudin, Directed force chain networks and stress response in static granular materials. The European Physical Journal E, 2002. 7(4): p. 353-370. C.Y. Wu, L.Y. Li, and C. Thornton, Energy dissipation during normal impact of elastic and elastic–plastic spheres. International Journal of Impact Engineering, 2005. 32(1): p. 593-604. R. Boettcher, A. Russell, and P. Mueller, Energy dissipation during impacts of spheres on plates: Investigation of developing elastic flexural waves. International Journal of Solids and Structures, 2017. 106: p. 229-239. L.D. Liu, C. Thornton, and S.J. Shaw. Energy Dissipation During Impact of an Agglomerate Composed of Autoadhesive Elastic-Plastic Particles. in Proceedings of the 7th International Conference on Discrete Element Methods. 2017. Springer. S.M. Wiederhorn, Fracture surface energy of glass. Journal of the American Ceramic Society, 1969. 52(2): p. 99-105. G. McDowell, M. Bolton, and D. Robertson, The fractal crushing of granular materials. Journal of the Mechanics and Physics of Solids, 1996. 44(12): p. 2079-2101. R. Woodward, W. Gooch, R. O'donnell, W. Perciballi, B. Baxter, and S. Pattie, A study of fragmentation in the ballistic impact of ceramics. International Journal of Impact Engineering, 1994. 15(5): p. 605-618. J.D. Hogan, J.G. Spray, R.J. Rogers, S. Boonsue, G. Vincent, and M. Schneider, Micro-scale energy dissipation mechanisms during dynamic fracture in natural polyphase ceramic blocks. International Journal of Impact Engineering, 2011. 38(12): p. 931-939. A. Nakamura and A. Fujiwara, Velocity distribution of fragments formed in a simulated collisional disruption. Icarus, 1991. 92(1): p. 132-146. A. Evans and M. Linzer, Acoustic emission in brittle materials. Annual Review of Materials Science, 1977. 7(1): p. 179-208. J. Huang, S. Xu, and S. Hu, Influence of particle breakage on the dynamic compression responses of brittle granular materials. Mechanics of Materials, 2014. 68: p. 15-28.

CE

21.

39.

40. 41. 42.

ACCEPTED MANUSCRIPT

46. 47.

48.

49. 50. 51. 52. 53. 54.

AC

CE

PT

ED

55.

CR IP T

45.

AN US

44.

B. Song and W. Chen, Energy for specimen deformation in a split Hopkinson pressure bar experiment. Experimental Mechanics, 2006. 46(3): p. 407-410. N. Xu and C. Wayne, Dynamic behavior of soda lime glass under biaxial loading. Proceedings of the 2006 SEM Annual Conference and Exposition on Experimental and Applied Mechanics 2006 2006. 4: p. 2266 - 2267. J. Huang, S. Xu, H. Yi, and S. Hu, Size effect on the compression breakage strengths of glass particles. Powder Technology, 2014. 268: p. 86-94. A. Salman and D. Gorham, The fracture of glass spheres under impact loading. Powders & Grains, 1997. 97. L. Bbosa, M. Powell, and T. Cloete, An investigation of impact breakage of rocks using the split Hopkinson pressure bar. Journal of the South African Institute of Mining and Metallurgy, 2006. 106(4): p. 291-296. T.-x. Hou, Q. Xu, and J.-w. Zhou, Size distribution, morphology and fractal characteristics of brittle rock fragmentations by the impact loading effect. Acta Mechanica, 2015. 226(11): p. 36233637. D. Turcotte, Fractals and fragmentation. Journal of Geophysical Research: Solid Earth, 1986. 91(B2): p. 1921-1926. B.B. Mandelbrot and R. Pignoni, The fractal geometry of nature. Vol. 173. 1983: WH freeman New York. B. Zhao, J. Wang, M. Coop, G. Viggiani, and M. Jiang, An investigation of single sand particle fracture using X-ray micro-tomography. Géotechnique, 2015. 65(8): p. 625-641. A. Thompson, Fractals in rock physics. Annual Review of Earth and Planetary Sciences, 1991. 19(1): p. 237-262. E. Sharon, S.P. Gross, and J. Fineberg, Energy dissipation in dynamic fracture. Physical review letters, 1996. 76(12): p. 2117. W.K. Brown and K.H. Wohletz, Derivation of the Weibull distribution based on physical principles and its connection to the Rosin–Rammler and lognormal distributions. Journal of Applied Physics, 1995. 78(4): p. 2758-2763. N.D. Parab, Z. Guo, M.C. Hudspeth, B.J. Claus, K. Fezzaa, T. Sun, and W.W. Chen, Fracture mechanisms of glass particles under dynamic compression. International Journal of Impact Engineering, 2017. 106: p. 146-154.

M

43.