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Numerical simulations of impact breakage of a spherical crystalline agglomerate
Powder Technology 109 Ž2000. 113–132 www.elsevier.comrlocaterpowtec
Numerical simulations of impact breakage of a spherical crystalline agglomerate K.D. Kafui ) , C. Thornton School of Engineering and Applied Science, Aston UniÕersity, Birmingham B4 7ET, UK Accepted 21 September 1999
Abstract A numerical study of the micromechanics of impact of a sub-millimetre sized crystalline agglomerate with a target wall has been carried out using granular dynamics Žor discrete element. simulations. The agglomerate, a spherical face-centred cubic array of ca. 8000 autoadhesive elastic primary particles, was assigned different interparticle bond strengths and impacted at different velocities. The effect of impact velocity and bond strength on the evolution of various impact parameters is reported. During loading, a shear-induced pattern of partially fractured planes was created which was dictated by the geometry of the impact area and the orientation of the packing planes. During unloading, fracture patterns were observed to be sub-sets of the preformed shear-induced weakened planes. The fragment size and mass distributions after impact have also been examined and related to the intrinsic properties of the agglomerate and the impact parameters. q 2000 Elsevier Science S.A. All rights reserved. Keywords: Discrete element method; Agglomerates; Fracture; Impact
1. Introduction The storage, transport, handling and processing of particulate materials constitutes a significant part of the operations in most chemical, pharmaceutical and allied industries. These particulate materials are frequently in the form of powders which are themselves agglomerations of much smaller sized primary particles. A common problem inherent in the handling of powders is the degradation resulting from attrition andror fragmentation of the agglomerates as they collide with each other and with the process equipment. Intentional breakage of particles by impact, e.g., jet milling, is a common technique used in comminution to bring about a desired size reduction. Impact breakage has been studied experimentally for glass, Shipway and Hutchings w1x, Salman and Gorham w2x, w3x, Papadopoulos and Ghadiri w4x, polymers, Schonert ¨ limestone spheres, Santurbano and Fairhurst w5x, sand-cement spheres and limestone-cement spheres, Arbiter et al. w6x, aluminium oxide spheres, Salman et al. w7x, salt crys) Corresponding author. Tel.: q44-121-359-3611; fax: q44-121-3333389
tals, Yuregir et al. w8x, sodium carbonate monohydrate crystals, Cleaver et al. w9x, fertiliser pellets, Salman and Gorham w10x and lactose agglomerates, Ning et al. w11x. The above references cover a range of material types resulting in either brittle or semi-brittle fracture, except in the case of lactose agglomerates which disintegrated upon impact without any evidence of fracture planes forming. Due to the short duration of an impact event, information from physical experiments is normally restricted to post-impact examinations of the fragments and debris produced. Explanations tend to rely on inferences which are based on solid mechanics concepts of brittle or semi-brittle fracture. However, it is not clear to what extent such solid mechanics ideas are applicable to particulate systems such as agglomerates. In the context of indentation, Lawn w12x highlights the mechanistic differences between fine-grained and coarse-grained ceramics but the extension to the impact breakage of coarse-grained agglomerates is not obvious. Experimental information about the evolution of the impact breakage processes is lacking although Santurbano and Fairhurst w5x did record the evolution of the force generated on the target wall during impact. Even when high speed digital video recordings are used, Ning et al.
0032-5910r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 0 3 2 - 5 9 1 0 Ž 9 9 . 0 0 2 3 1 - 4
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w11x, they currently fail to capture the loading stage in sufficient detail. Numerical simulations of systems of discrete particles are not restricted by small time or length scales and a recent development has been the application of such techniques to impact fracture problems. Two dimensional numerical simulations of fracture due to impact of solid particles have been reported by Potapov and Cambell w13x and Kun and Hermann w14x. Numerical simulations of agglomerate impact fracture using the discrete element, or granular dynamics, method was initiated by Yin w15x and results of 2D simulations were presented by Thornton et al. w16x. Ning et al. w11x reported impact simulations on spherical agglomerates composed of a polydisperse system of primary particles but observed that fracture did not occur in any of their simulations. In contrast, Thornton et al. w17x present results of 3D simulations of microstructurally disordered agglomerates impacting with a target wall which result in rebound, fracture or shattering, depending on the magnitude of the impact velocity specified. In the paper we report numerical simulations of a crystalline agglomerate impacting orthogonally with a tar-
get wall at different impact velocities and for different interparticle bond strengths. Previous simulated impacts of crystalline agglomerates were reported by Kafui and Thornton w18x but no clear evidence of fracture occurring was noted. Subsequently, it was discovered that, due to a small error in the 3D computer code, spurious rotations of the primary particles occurred which reduced the agglomerate stiffness and, hence, insufficient elastic energy was stored to create fracture. Having corrected the error, further simulations were performed and the results are presented below. Agglomerates occurring in industrial situations normally do not have a crystalline structure and their impact behaviour is intimately related to the microstructure and distribution of internal stored energy prior to impact. Realistic simulations of industrial agglomeration processes leading to sub-millimetre sized agglomerates composed of thousands of primary particles are not yet feasible with current day computer hardware. Even when simpler but physically realistic simulated agglomerate preparation techniques, such as the application of a centripetal gravity field, are used to create microstructurally disordered ag-
Fig. 1. Effect of impact velocity on the evolution of Ža. the force generated at the agglomerate-wall interface, Žb. the kinetic energy of the system of primary particles composing the agglomerate Žnormalised to the initial kinetic energy, agglomerate mass s 87 mg. and Žc. the proportion of initial interparticle bonds broken. The example shown is for an interface energy, G s 2.0 Jmy2 .
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glomerates, Thornton et al. w16,17x, in order to ensure a stable equilibrated configuration, the computations extend over weeks using workstations with clock speeds less than 100 MHz. Thornton et al. w17x also note that elucidation of the detailed evolution of the micromechanical processes leading to breakage is extremely difficult to confirm in three-dimensional polydisperse disordered systems of spheres. Consequently, it was decided to re-examine the impact breakage of an agglomerate composed of equalsized spheres in face-centred cubic arrangement in order to obtain clear 3D visualisations of the evolution of fracture and to identify aspects of the impact behaviour which are insensitive to the microstructural details. The results may have some relevance to real agglomerates which are produced by crystallisation but we also hope that, being a limiting case in terms of structure, the results may help to understand fracture in disordered systems.
2. Numerical methodology and simulation procedures The granular dynamics model used in this study originated as the distinct element method ŽDEM., Cundall and
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Strack w19x, which was extended to 3D applications by the development of the program TRUBAL, Cundall w20x. In granular dynamics simulations, particles are treated as discrete elements which are indestructible but interact with each other and any other elements present such as walls when they make physical contact. The particle interaction rules are based on contact mechanics theories which relate the contact force to the relative approach of the particle centroids and, hence, particles do not overlap but there is an implied local deformation of the particle adjacent to the contact interface. The Aston version of the TRUBAL code, which has been adapted to simulate agglomerates Žand is now called GRANULE., is capable of modelling elastic, frictional, adhesive or non-adhesive spherical primary particles with or without plastic yield at the interparticle contacts. In this study, the adhesive option with no plastic deformation at the contacts is used. Details of the implementation of the interaction laws have been reported by Thornton and Yin w21x, Thornton and Ning w22x. The particle interactions are modelled as a dynamic process, the evolution of which is advanced using an explicit finite difference scheme to obtain the incremental contact forces and then the incremental displacements of
Fig. 2. Effect of interface energy on the evolution of Ža. the wall force, Žb. the kinetic energy and Žc. the proportion of bonds broken. The example shown is for an impact velocity, V s 1.0 msy1 .
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Fig. 3. Effect of impact velocity on the damage ratio for various values of interface energy.
the particles. Each cycle of calculations that takes the system from time t to t q D t involves the application of incremental force-displacement interaction laws at each contact, resulting in new interparticle forces which are resolved to obtain new out-of-balance forces and moments
for each particle. Numerical integration of Newton’s second law of motion yields the linear and rotational velocities for each particle. A second integration yields the incremental particle displacements and using the new particle positions and velocities, both linear and rotational, the
Fig. 4. Relationship between threshold velocity V0 and interface energy.
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calculation cycle is repeated in the next time step. The time step D t used is a fraction of the critical time step D t c determined from the Rayleigh wave speed for the solid particles, Thornton and Randall w23x. Agglomerate preparation begins with the generation of spherical particles in a prescribed region. For the agglomerate used in this study, particles of a given size and material properties were generated at incremental distances in the X, Y and Z coordinate directions so as to form a face-centred cubic array with a very small gap Žca. 0.1% of the particle radius. between the particles. A spherical agglomerate shape was ensured by accepting only particles in the array which were centred inside a specified inner spherical region. A small centripetal gravity field Ž g s 0.05 msy2 . was then used to bring the particles into contact after which adhesive bonding at the contacts was imposed by attributing each particle with an initially low value of surface energy Žg s 0.03 Jmy2 . and then slowly increasing the value to obtain agglomerates with different interparticle bond strengths. The centripetal gravity field was slowly reduced to zero to complete the agglomerate preparation. To initiate impact, a wall was located at a very small distance Žca. 1 to 100 nm, depending on the impact velocity. from the agglomerate and then the desired impact velocity was attributed to all the constituent primary particles. Cyclic calculations were continued until the end of a single impact event which was indicated by an approximately zero wall force, zero rate of bond breakage and a constant kinetic energy of the system.
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The agglomerate was made up of 7912 spheres of diameter d p s 20 mm and had an overall diameter d a s 0.461 mm with an average coordination number Z s 11.1 after preparation Ž Z - 12 due to finite size of the agglomerate.. The solid density r , Young’s modulus E, coefficient of interparticle friction m and Poisson’s ratio n of the primary particles were 2650 kg my3 , 70 GPa, 0.35 and 0.3 respectively. Five values of interface energy G Žs 2g . s 0.2, 0.4, 1.0, 2.0 and 4.0 Jmy2 were used and impact was carried out at velocities in the range 0.05 to 20 msy1 .
3. Evolution of impact parameters Typical time evolutions of the force generated at the agglomerate-wall interface, the kinetic energy of the system of primary particles composing the agglomerate and the proportion of initial interparticle bonds broken during impact are shown in Fig. 1, illustrating the effect of impact velocity. At sufficiently low impact velocities when no fracturerfragmentation occurs Žnot illustrated. the behaviour is similar to that obtained during impact of a solid sphere in that the maximum force on the wall occurs when the kinetic energy is a minimum and, as in the case of an elastoplastic sphere, the final kinetic energy is less than the initial kinetic energy due to some internal bond breakage and irrecoverable deformation of the microstructure adjacent to the agglomerate-wall interface. As the impact velocity is increased, the maximum wall force increases, the duration of the impact decreases and the proportion of
Fig. 5. Relationship between damage ratio, impact velocity and interface energy.
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bonds broken increases. The final kinetic energy increases with impact velocity but, as shown in Fig. 1b, the proportion of the initial kinetic energy remaining at the end of the impact decreases with impact velocity until the velocity is sufficient to produce multiple fracture and then increases with impact velocity due to the increasing kinetic energy of the broken fragments and debris. Fig. 1 shows that, when fracturerfragmentation occurs, the kinetic energy reaches a minimum value during unloading and that the time lapse between the maximum wall
force and the minimum kinetic energy increases with impact velocity. In all the tests simulated it was observed that interparticle bonds were broken at an increasing rate and then at a decreasing rate as the kinetic energy decreased to its minimum value. As the kinetic energy increased the rate of bond breakage increased and then decreased until a constant final value was attained just after the end of the impact. Fig. 2 illustrates how the evolution of the wall force, the kinetic energy and the proportion of bonds broken varies
Fig. 6. Relative orientation of the agglomerate microstructure Ža. impact direction orthogonal to the square-packed planes and Žb. impact direction parallel to the square-packed planes.
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Fig. 7. Evolution of bond breakage during impact orthogonal to the square-packed planes Žinterface energy G s 4.0 Jmy2 , impact velocity V s 2.0 msy1 ..
with interface energy for an impact velocity of 1.0 msy1 . As the interface energy is increased, the maximum wall force increases, the minimum kinetic energy decreases and there is a reduction in the proportion of bonds broken. Interface energy does not have a significant effect on the impact duration and although the minimum kinetic energy occurs during unloading the time lapse does not vary monotonically with interface energy. For G s 4.0 Jmy2 , no debris was produced; the agglomerate rebounded from the wall with a few Ž- 2%. bonds broken internally. When the interface energy was reduced to G s 2.0 Jmy2 a clean fracture was observed which resulted in two approximately hemispherical fragments. Multiple fracture was observed for G s 1.0 Jmy2 with a limited amount of shattering of some small fragments. Further decreases in interface energy resulted in an increased amount of shattering. For impacts with G s 0.2 Jmy2 and 0.4 Jmy2 , approximately 90% of the interparticle bonds were broken resulting in lots of small fragments consisting of no more than 20 primary particles per fragment. Fig. 2 indicates that single fracture leads to an increase in the elapsed times when the maximum wall force and minimum kinetic energy occur but with further fragmentation and shattering due to reducing the bond strength the maximum wall force and minimum kinetic energy occur at smaller elapsed times. The relationship between the evolution of the proportion of broken bonds
with the evolutions of the wall force and kinetic energy is similar to that observed in Fig. 1. The proportion of bonds broken during an impact was defined as the damage ratio D by Thornton et al. w16x who observed, from the results of their 2D simulations, that the relationship between the damage ratio and the impact velocity could be approximated by the expression D s a ln
V
ž / V0
Ž 1.
where V0 is the threshold velocity below which no significant damage occurs. Deviations from Ž1. occurred for the limits D ™ 0 and D ™ 1 because Ža. there will be a range of low velocities at which the agglomerate rebounds but suffers a small amount of internal damage and Žb. even at very high velocities not all contacts will be broken since it is possible that some doublets and triplets will survive. Fig. 3 shows that the data sets obtained from 3D simulations of crystalline agglomerates also satisfy Ž1. with a s 0.35 for the range 0.2 - D - 0.8. Thornton et al. w16x suggested that the threshold velocity V0 , defined by Ž1., increased exponentially with interface energy. However, as shown in Fig. 4, the results presented here are well represented by the following power law. V0 s 0.17G 1.5
Ž 2.
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Fig. 8. Evolution of bond breakage during impact parallel to the square-packed planes Žinterface energy G s 4.0 Jmy2 , impact velocity V s 2.0 msy1 ..
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Fig. 9. Equivalent space lattice viewed from below illustrating the breakage patterns obtained for a strong agglomerate Ž G s 4.0 Jmy2 . impacted at velocities of 2.0, 3.0, 5.0 and 10.0 msy1 .
This implies that the damage ratio D should scale with lnŽ V 2rG 3 . and this is confirmed reasonably well in Fig. 5 except for the results of the weakest agglomerate Ž G s 0.2 Jmy2 .. Dimensional analysis suggests that the damage ratio is a function of the following dimensionless group
r d 3E 2
V
2
G
3
ž /
W
3
C
2
s
s CA3
Ž 3.
in which the Weber number Ws
r dV G
2
Ž 4.
is the ratio of inertia force to bonding force, the Cauchy number Cs
rV 2
Ž 5.
E
is the ratio of inertia force to elastic force and As
W
Ed s
C
G
Ž 6.
is the ratio of elastic force to bonding force and may be termed an elastic adhesion index. However, more simulations are required to clarify whether the parameters r , d and E in Ž3. should be the density, size and elastic
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modulus of the primary particles or the agglomerate bulk properties.
4. Fracture and fragmentation In this section agglomerate breakage is illustrated by computer generated images of the configuration of the primary particles, in which the particles are colour coded to indicate the size of the fragment to which they belong. Fracture planes are more clearly identified by representing the agglomerate by the equivalent space lattice which is formed by connecting the centres of particles in contact by
solid lines. However, in order to visualise the evolution of the bond breakage leading to fracture, it is more appropriate to use solid lines to connect the centres of particles which were initially in contact but which have broken contact during the impact. A face-centred cubic agglomerate possesses a strongly anisotropic structure. Consequently, we might expect that the fracture pattern would be sensitive to the orientation of the microstructure relative to the impact velocity direction. The structural orientation of a face-centred cubic array can be illustrated, as shown in Fig. 6, by nine spheres. There are two sets of four contiguous spheres, each set forming a square-packed plane. The ninth sphere is located in the
Fig. 10. Equivalent space lattice viewed from below illustrating the breakage patterns obtained for a weak agglomerate Ž G s 0.4 Jmy2 . impacted at velocities of 0.15, 0.30, 0.50 and 1.0 msy1 .
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cusp between the two square-packed planes leading to four sets of close-packed Žtriangular. planes inclined at 54.78 to the square-packed planes. Since the central sphere in Fig. 6 is itself part of a square-packed plane, it has a total of twelve contacts. Fig. 6a illustrates the relative orientation of the agglomerate when the square-packed planes are parallel to the wall. An example of fracture obtained for this orientation is shown in Fig. 7. For a strong Ž G s 4.0 Jmy2 . agglomerate impacted at a velocity of 2.0 msy1 , Fig. 7 illustrates the evolution of bond breakage during the impact, as viewed from above. In addition, to the right of the figure, two views are shown of the equivalent space lattice, showing an oblique fracture plane, a long time after the end of the impact when the large residual fragment is rebounding from the wall. As the force generated at the wall increases, the contacts along the horizontal square-packed planes are broken in the region of high vertical compression immediately above the agglomerate-wall interface. Consequently, the forces generated at the wall are transmitted along the inclined dense-
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packed planes producing a deceleration of the particles transmitting the large forces. Particles in adjacent densepacked planes which do not intersect the agglomerate-wall interface are not decelerated and consequently there is a relative shear movement between the loaded and unloaded dense-packed planes which propagates from the perimeter of the contact area. Initially, each particle is in contact with three particles in the adjacent dense-packed plane. As a result of the shear propagation two sets of contacts are broken between the loaded and unloaded planes thereby creating weakened planes propagated from the perimeter of the contact area. During unloading, as the agglomerate begins to rebound, there is a tendency to break the remaining bonds in the weakened planes. For reasons which presumably are associated with small heterogeneities in the contact force distribution and structure, in the example shown in Fig. 7, only one Žnot four. inclined fracture plane resulted from the impact. The equivalent space lattice illustrating the fracture plane in Fig. 7 may be considered to show an example of what has been termed ‘chipping’,
Fig. 11. Configuration of primary particles Žtop. and equivalent space lattice Žbottom., as viewed from below, illustrating fracture patterns resulting from impacting a strong agglomerate Ž G s 2.0 Jmy2 . at three velocities.
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Zhang and Ghadiri w24x. However, it is emphasised that the micromechanics of the fracture process evolution is the same as for the other cases described below. With the above exception, all the results reported in the paper were obtained from impact simulations in which the impact direction was parallel to the square-packed planes, as shown in Fig. 6b. An example of the bond breakage evolution for this orientation is illustrated in Fig. 8 for the same interface energy Ž G s 4.0 Jmy2 . and impact velocity Ž V s 2.0 msy1 . as the example shown in Fig. 7. The development of the bond breakage pattern is illustrated for different elapsed times showing two orthogonal views: one elevation and a view from above. During loading, bonds are broken along four vertical planes which form a diamond shaped pattern, as viewed from above. The four planes coincide with the perimeter of the agglomerate-wall interface and are the consequence of relative shear motion between adjacent load transmitting and load free densepacked vertical planes. The relative shear motion results in the breaking of one set of contacts between the loaded and unloaded dense-packed planes. The bond breaking propa-
gates upwards from the wall until it reaches the top of the agglomerate when the kinetic energy attains its minimum value at t s 1.065 ms. Due to the spherical shape of the agglomerate, the number of bonds broken along the shear propagation front increases and then decreases as the front traverses a circular plane section. This provides the explanation for the rate of evolution of bond breakage illustrated in Figs. 1 and 2. During unloading, as elastic energy is converted into kinetic energy, a second set of contacts are broken, propagating downwards from the top of the agglomerate. With further increase in kinetic energy, some bonds are also broken in the third set of contacts leading to fracture along some of the shear induced weakened planes just prior to the end of the impact. The breakage pattern resulting from impact fracture of the agglomerate, for G s 4.0 Jmy2 and V s 2.0 msy1 , is illustrated by two orthogonal views of the equivalent space lattice at the bottom of Fig. 8. The effect of impact velocity on the breakage patterns obtained is shown in Figs. 9 and 10 for agglomerates with interface energies G s 4.0 Jmy2 and G s 0.4 Jmy2 re-
Fig. 12. Configuration of primary particles Žtop. and equivalent space lattice Žbottom., as viewed from below, illustrating fracture patterns resulting from impacting a weak agglomerate Ž G s 0.2 Jmy2 . at three velocities.
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spectively. For the strong agglomerate, Fig. 9 clearly shows that at an impact velocity of 5.0 msy1 all four of the shear induced weakened planes are fractured plus two short fracture planes which are parallel to the square-packed planes. At lower impact velocities the fracture pattern is a subset of that obtained for V s 5.0 msy1 . If the impact velocity is increased above 5.0 msy1 no extra fracture planes are created but the residual fragments are weakened due to internal bond breakage and this leads to shattering of the agglomerate at high impact velocities. The effect of varying the velocity at which the weak agglomerate was impacted produced similar results but, as seen in Fig. 10, the large fragments suffered significant internal bond breakage even at low impact velocities due to the relatively low interface energy between the primary particles.
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For an agglomerate with interface energy G s 2.0 Jmy2 the fracture patterns produced at different impact velocities were similar to those obtained for G s 4.0 Jmy2 , as can be seen in Fig. 11. Also shown in the figure are images of the corresponding particle configurations, as viewed from below the impact area, which illustrate the increasing amount of fine debris produced around the impact area when the velocity is increased. When impacted at a velocity of 3.0 msy1 , extensive shattering occurred with the largest fragment consisting of only 312 primary particles. Fig. 12 provides similar images obtained for a very weak agglomerate for which G s 0.2 Jmy2 . It may be noted that when impacted at a velocity of 0.2 msy1 fracture occurred between square-packed planes and between planes orthogonal to the square-packed planes. This was the only occur-
Fig. 13. Fragmentation resulting from impacting a strong agglomerate Ž G s 4.0 Jmy2 . at 2.0 msy1 and 4.0 msy1.
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Fig. 14. Fragmentation resulting from impacting a weak agglomerate Ž G s 0.4 Jmy2 . at 0.15 msy1 and 0.30 msy1.
rence of this fracture pattern in all the simulations carried out. Finally in this section, elevation and plan views of fragmentation resulting from impact for G s 4.0 Jmy2 , V s 2.0 msy1 and V s 4.0 msy1 are shown in Fig. 13; and for G s 0.4 Jmy2 , V s 0.15 msy1 and V s 0.30 msy1 in Fig. 14. In both figures the small debris particles are not shown in the case of the higher velocity impact in order to illustrate the shape of the large residual fragments more clearly.
5. Fragment size distribution Although the size distribution of the fragments resulting from an impact is of practical importance in powder
technology, the definition of size for non-spherical fragments is imprecise. In physical experimentation, sieve sizes are used to classify fragments into different size ranges but ambiguities arise since further breakage may occur due to the vibratory nature of the sieving process and the fact that elongated particles may or may not pass through the sieve depending on the orientation of their long axis. Similar uncertainties arise in numerical simulations when considering the size of the clusters representing the fragmentation products of an impact. One option is to define the size of a fragment d f by the diameter of the circumscribing sphere and to normalise this by dividing by the circumscribing sphere diameter d a of the original agglomerate. An alternative is to use the mass of a cluster m divided by the mass of the original agglomerate M, which is equivalent to using a normalised size based on the
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Fig. 15. Effect of impact velocity on the fragment size distribution of a weak agglomerate Ž G s 0.2 Jmy2 ..
size of the equivalent solid spheres. In this section we use the latter definition to illustrate the numerical simulation data but also quote parametric values obtained using the former definition.
Fig. 15 shows a Gates–Gaudin–Schuhmann doublelogarithmic plot of cumulative mass fraction undersize f against normalised size mrM obtained from impacting a weak agglomerate Ž G s 0.2 Jmy2 . at three different veloc-
Fig. 16. Effect of impact velocity on the fragment size distribution of a strong agglomerate Ž G s 2.0 Jmy2 ..
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ities. A similar plot is shown in Fig. 16 for a strong Ž G s 2.0 Jmy2 . agglomerate. Both figures show that the fragment size distributions exhibit bilinear characteristics which distinguish the residue of large fragments from the complement of small fragments Ždebris.. Similar results were obtained by Arbiter et al. w6x for impact velocities which produced ‘‘semi-brittle’’ fracture. The normalised size which identifies the transition from residue to debris is independent of the bond strength and corresponds to mrM
s 0.1 or d frd a s 0.5 depending on the size definition used. These values may be compared with values obtained from Arbiter et al.’s experiments of 0.21 and 0.27 for sand-cement spheres and limestone-cement spheres respectively. Figs. 15 and 16 show that the exponent for the residue decreases with impact velocity implying that, at sufficiently high impact velocities, there is no residue as a consequence of the extensive shattering that occurs. Except
Fig. 17. Effect of interface energy on the fragment size distribution.
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Fig. 18. Normalised fragment size distribution.
for very low Žno fracture. or very high Žshattering. levels of breakage, the exponent n which defines the size distribution of the debris, f ; Ž mrM . n , is independent of impact velocity but decreases if the bond strength is increased. Strongly bonded agglomerates exhibit clean fractures and fine debris originating from the impact re-
gion and, hence, low values of n. In weak agglomerates larger debris fragments are created along the less well defined fracture planes leading to higher values of n. In the fracturermultifracture regime, for any size mrM, the cumulative mass fraction undersize f increases with impact velocity. Arbiter et al. w6x demonstrated that, for
Fig. 19. Dependence of the complement size distribution exponent n on the interface energy.
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Fig. 20. Dependence of the complement size distribution prefactor l on the interface energy.
their sand-cement spheres, the size distribution of the debris Žcomplement. correlates with the specific impact energy according to the expression f s lV 2 s n
Ž 7.
where V is the impact velocity, s is the normalised size Ž mrM or d frd a . and l is a constant of proportionality
which depends on the material properties. Using values of the exponent n obtained from Figs. 15 and 16 and other similar plots for other interface energy values, the size distribution data obtained from all the simulations performed are plotted in Fig. 17Ža,b. confirming the form of Ž7. and indicating that, for face-centred cubic crystalline agglomerates, the prefactor l depends on the strength of the interparticle bonds. Using values of l obtained from
Fig. 21. Relationship between the normalised mass of the largest fragment and the ratio of impact velocity and interface energy.
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Fig. 17, Fig. 18 demonstrates that all the simulation data collapse reasonably well onto a single curve. In general, however, l is expected to depend also on other properties such as solid fraction and coordination number which define the microstructure; but this is beyond the scope of this investigation. As indicated in Figs. 15–18, both the exponent n and the prefactor l are functions of the interface energy G . The two relationships are illustrated in Figs. 19 and 20 and may be approximated by n s 0.25 Gy0 .2
Ž 8.
and
l s Gy2
Ž 9.
However, further work is required to obtain more data points in order to reliably quantify the two relationships. The size of the largest surviving fragment, in terms of its normalised mass m L , is plotted against the ratio of impact velocity to interface energy in Fig. 21 which shows that the data satisfies a power law scaling given by m L s 0.15
V
y1 .5
ž / G
Ž 10 .
with the condition m L - 1.
6. Conclusions Results have been presented of numerical simulations of a spherical, crystalline Žface centred cubic. agglomerate impacting orthogonally with a target wall. The effects of impact velocity and bond strength on the evolution of the wall force, kinetic energy of the agglomerate and the proportion of bonds broken have been demonstrated. As might be expected, increasing the impact velocity results in a higher maximum wall force, shorter impact event and more broken bonds. Increasing the bond strength increases the maximum force generated at the wall, reduces the number of bonds broken but does not significantly affect the duration of the impact event. The proportion of bonds broken during an impact has been defined by a damage ratio D. In the range 0.2 - D - 0.8, the initial specific energy required to break a given number of bonds scales with G 3. The size distributions of the fragments produced by impact breakage have been examined and shown to exhibit a bilinear distribution on a Gates–Gaudin–Schuhmann plot. Independent of bond strength, the residue consists of fragments of mass greater than 10% of the mass of the original agglomerate, the complement Žor debris. of fragments of mass less than 10% of the agglomerate mass. For the complement, the exponent of the power law relationship decreases if the bond strength is increased and the cumulative mass fraction undersize scales with Ž VrG . 2 . The size of the largest fragment Žin terms of its mass. scales with Ž VrG .y1 .5.
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Agglomerate breakage has been illustrated by computer generated images of the configuration of primary particles and by the equivalent space lattice. The visualisations have shown that for any given bond strength there is an impact velocity which produces a complete set of fracture planes. Subsets of this fracture pattern are produced at lower impact velocities. Higher impact velocities do not produce extra fracture planes but the residual fragments are weakened due to internal bond breakage, which results in shattering at high impact velocities. By examining the evolution of bond breakage, it has been shown that there is a shear-induced weakening of a set of potential fracture planes during deceleration of the agglomerate. The pattern of weakened planes is dictated by the geometry of the agglomeraterwall interface and the orientation of the agglomerate microstructure with respect to impact direction. As the agglomerate recovers kinetic energy, further shearinduced bond breakage occurs along the same set of planes. At the end of the impact, depending on the amount of kinetic energy recovered, complete fracture occurs along some of the weakened planes. The structure of the initial agglomerate is crystalline with no structural flaws although there is a small random distribution of residual stored elastic energy due to the centripetal gravity field used to prepare the agglomerate. From the observations of bond breakage evolution, it is concluded that fracture is shear-induced and, in effect, the agglomerate creates its own flaw population during loading and that pre-existing flaws do not have a significant effect on agglomerate ‘‘strength’’.
Acknowledgements The work reported in the paper was supported by the Engineering and Physical Science Research Council.
References w1x P.H. Shipway, I.M. Hutchings, Powder Technology 76 Ž1993. 23–30. w2x A.D. Salman, D.A. Gorham, in: R.P. Behringer, J.T. Jenkins ŽEds.., Powders and Grains 97, Balkema, 1997, pp. 563–566. w3x K. Schonert, Proc. Harold Heywood Memorial Symp., Loughbor¨ ough Ž1973. 2–16. w4x D.G. Papadopoulos, M. Ghadiri, Adv. Powder Technol. 7 Ž1996. 183–197. w5x R. Santurbano, C. Fairhurst, in: A. Roegiers ŽEd.., Rock Mechanics as a Multidisciplinary Science, Balkema, 1991, pp. 441–449. w6x N. Arbiter, C.C. Harris, G.A. Stamboltzis, Trans. AIME 244 Ž1969. 118–133. w7x A.D. Salman, D.A. Gorham, A. Verba, Wear 186 Ž7. Ž1995. 92–98. w8x K.R. Yuregir, M. Ghadiri, R. Clift, Chem. Eng. Sci. 42 Ž1987. 843–853. w9x J.A.S. Cleaver, M. Ghadiri, N. Rolfe, Powder Technology 76 Ž1993. 15–22. w10x A.D. Salman, D.A. Gorham, Proc. 2nd. Israel Conf. for Conveying and Handling of Particulate Solids, Ž1997. 8.29–8.34.
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w11x Z. Ning, R. Boerefijn, M. Ghadiri, C. Thornton, Adv. Powder Technol. 8 Ž1997. 15–37. w12x B. Lawn, in: M.J. Adams, S.K. Biswas, B.J. Briscoe ŽEds.., Solid– Solid Interactions, Imperial College Press, 1996, pp. 29–40. w13x A.V. Potapov, C.S. Campbell, Powder Technology 81 Ž1994. 207– 216. w14x F. Kun, H.J. Hermann, Comput. Meth. Appl. Mech. Engng. 138 Ž1996. 3–18. w15x K.K. Yin, PhD thesis, Aston University, 1992. w16x C. Thornton, K.K. Yin, M.J. Adams, J. Phys. D: Appl. Phys. 29 Ž1996. 424–435. w17x C. Thornton, M.T. Ciomocos, M.J. Adams, Proc. 9th Euro. Symp. Comminution Ž1998. 109–118.
w18x K.D. Kafui, C. Thornton, in: C. Thornton ŽEd.., Powders and Grains 93, Balkema, 1993, pp. 401–406. w19x P.A. Cundall, O.D.L. Strack, Geotechnique 29 Ž1979. 47–65. ´ w20x P.A. Cundall, in: M. Satake, J.T. Jenkins ŽEds.., Micromechanics of Granular Materials, Elsevier, 1988, pp. 113–123. w21x C. Thornton, K.K. Yin, Powder Technology 65 Ž1991. 153–166. w22x C. Thornton, Z. Ning, Powder Technology 99 Ž1998. 154–162. w23x C. Thornton, C.W. Randall, in: M. Satake, J.T. Jenkins ŽEds.., Micromechanics of Granular Materials, Elsevier, 1988, pp. 245–252. w24x Z. Zhang, M. Ghadiri, Proc. 1st. Int. Particle Technology Forum, Ž1994. 247–252.
Numerical simulations of impact breakage of a spherical crystalline agglomerate K.D. Kafui ) , C. Thornton School of Engineering and Applied Science, Aston UniÕersity, Birmingham B4 7ET, UK Accepted 21 September 1999
Abstract A numerical study of the micromechanics of impact of a sub-millimetre sized crystalline agglomerate with a target wall has been carried out using granular dynamics Žor discrete element. simulations. The agglomerate, a spherical face-centred cubic array of ca. 8000 autoadhesive elastic primary particles, was assigned different interparticle bond strengths and impacted at different velocities. The effect of impact velocity and bond strength on the evolution of various impact parameters is reported. During loading, a shear-induced pattern of partially fractured planes was created which was dictated by the geometry of the impact area and the orientation of the packing planes. During unloading, fracture patterns were observed to be sub-sets of the preformed shear-induced weakened planes. The fragment size and mass distributions after impact have also been examined and related to the intrinsic properties of the agglomerate and the impact parameters. q 2000 Elsevier Science S.A. All rights reserved. Keywords: Discrete element method; Agglomerates; Fracture; Impact
1. Introduction The storage, transport, handling and processing of particulate materials constitutes a significant part of the operations in most chemical, pharmaceutical and allied industries. These particulate materials are frequently in the form of powders which are themselves agglomerations of much smaller sized primary particles. A common problem inherent in the handling of powders is the degradation resulting from attrition andror fragmentation of the agglomerates as they collide with each other and with the process equipment. Intentional breakage of particles by impact, e.g., jet milling, is a common technique used in comminution to bring about a desired size reduction. Impact breakage has been studied experimentally for glass, Shipway and Hutchings w1x, Salman and Gorham w2x, w3x, Papadopoulos and Ghadiri w4x, polymers, Schonert ¨ limestone spheres, Santurbano and Fairhurst w5x, sand-cement spheres and limestone-cement spheres, Arbiter et al. w6x, aluminium oxide spheres, Salman et al. w7x, salt crys) Corresponding author. Tel.: q44-121-359-3611; fax: q44-121-3333389
tals, Yuregir et al. w8x, sodium carbonate monohydrate crystals, Cleaver et al. w9x, fertiliser pellets, Salman and Gorham w10x and lactose agglomerates, Ning et al. w11x. The above references cover a range of material types resulting in either brittle or semi-brittle fracture, except in the case of lactose agglomerates which disintegrated upon impact without any evidence of fracture planes forming. Due to the short duration of an impact event, information from physical experiments is normally restricted to post-impact examinations of the fragments and debris produced. Explanations tend to rely on inferences which are based on solid mechanics concepts of brittle or semi-brittle fracture. However, it is not clear to what extent such solid mechanics ideas are applicable to particulate systems such as agglomerates. In the context of indentation, Lawn w12x highlights the mechanistic differences between fine-grained and coarse-grained ceramics but the extension to the impact breakage of coarse-grained agglomerates is not obvious. Experimental information about the evolution of the impact breakage processes is lacking although Santurbano and Fairhurst w5x did record the evolution of the force generated on the target wall during impact. Even when high speed digital video recordings are used, Ning et al.
0032-5910r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 0 3 2 - 5 9 1 0 Ž 9 9 . 0 0 2 3 1 - 4
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w11x, they currently fail to capture the loading stage in sufficient detail. Numerical simulations of systems of discrete particles are not restricted by small time or length scales and a recent development has been the application of such techniques to impact fracture problems. Two dimensional numerical simulations of fracture due to impact of solid particles have been reported by Potapov and Cambell w13x and Kun and Hermann w14x. Numerical simulations of agglomerate impact fracture using the discrete element, or granular dynamics, method was initiated by Yin w15x and results of 2D simulations were presented by Thornton et al. w16x. Ning et al. w11x reported impact simulations on spherical agglomerates composed of a polydisperse system of primary particles but observed that fracture did not occur in any of their simulations. In contrast, Thornton et al. w17x present results of 3D simulations of microstructurally disordered agglomerates impacting with a target wall which result in rebound, fracture or shattering, depending on the magnitude of the impact velocity specified. In the paper we report numerical simulations of a crystalline agglomerate impacting orthogonally with a tar-
get wall at different impact velocities and for different interparticle bond strengths. Previous simulated impacts of crystalline agglomerates were reported by Kafui and Thornton w18x but no clear evidence of fracture occurring was noted. Subsequently, it was discovered that, due to a small error in the 3D computer code, spurious rotations of the primary particles occurred which reduced the agglomerate stiffness and, hence, insufficient elastic energy was stored to create fracture. Having corrected the error, further simulations were performed and the results are presented below. Agglomerates occurring in industrial situations normally do not have a crystalline structure and their impact behaviour is intimately related to the microstructure and distribution of internal stored energy prior to impact. Realistic simulations of industrial agglomeration processes leading to sub-millimetre sized agglomerates composed of thousands of primary particles are not yet feasible with current day computer hardware. Even when simpler but physically realistic simulated agglomerate preparation techniques, such as the application of a centripetal gravity field, are used to create microstructurally disordered ag-
Fig. 1. Effect of impact velocity on the evolution of Ža. the force generated at the agglomerate-wall interface, Žb. the kinetic energy of the system of primary particles composing the agglomerate Žnormalised to the initial kinetic energy, agglomerate mass s 87 mg. and Žc. the proportion of initial interparticle bonds broken. The example shown is for an interface energy, G s 2.0 Jmy2 .
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glomerates, Thornton et al. w16,17x, in order to ensure a stable equilibrated configuration, the computations extend over weeks using workstations with clock speeds less than 100 MHz. Thornton et al. w17x also note that elucidation of the detailed evolution of the micromechanical processes leading to breakage is extremely difficult to confirm in three-dimensional polydisperse disordered systems of spheres. Consequently, it was decided to re-examine the impact breakage of an agglomerate composed of equalsized spheres in face-centred cubic arrangement in order to obtain clear 3D visualisations of the evolution of fracture and to identify aspects of the impact behaviour which are insensitive to the microstructural details. The results may have some relevance to real agglomerates which are produced by crystallisation but we also hope that, being a limiting case in terms of structure, the results may help to understand fracture in disordered systems.
2. Numerical methodology and simulation procedures The granular dynamics model used in this study originated as the distinct element method ŽDEM., Cundall and
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Strack w19x, which was extended to 3D applications by the development of the program TRUBAL, Cundall w20x. In granular dynamics simulations, particles are treated as discrete elements which are indestructible but interact with each other and any other elements present such as walls when they make physical contact. The particle interaction rules are based on contact mechanics theories which relate the contact force to the relative approach of the particle centroids and, hence, particles do not overlap but there is an implied local deformation of the particle adjacent to the contact interface. The Aston version of the TRUBAL code, which has been adapted to simulate agglomerates Žand is now called GRANULE., is capable of modelling elastic, frictional, adhesive or non-adhesive spherical primary particles with or without plastic yield at the interparticle contacts. In this study, the adhesive option with no plastic deformation at the contacts is used. Details of the implementation of the interaction laws have been reported by Thornton and Yin w21x, Thornton and Ning w22x. The particle interactions are modelled as a dynamic process, the evolution of which is advanced using an explicit finite difference scheme to obtain the incremental contact forces and then the incremental displacements of
Fig. 2. Effect of interface energy on the evolution of Ža. the wall force, Žb. the kinetic energy and Žc. the proportion of bonds broken. The example shown is for an impact velocity, V s 1.0 msy1 .
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Fig. 3. Effect of impact velocity on the damage ratio for various values of interface energy.
the particles. Each cycle of calculations that takes the system from time t to t q D t involves the application of incremental force-displacement interaction laws at each contact, resulting in new interparticle forces which are resolved to obtain new out-of-balance forces and moments
for each particle. Numerical integration of Newton’s second law of motion yields the linear and rotational velocities for each particle. A second integration yields the incremental particle displacements and using the new particle positions and velocities, both linear and rotational, the
Fig. 4. Relationship between threshold velocity V0 and interface energy.
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calculation cycle is repeated in the next time step. The time step D t used is a fraction of the critical time step D t c determined from the Rayleigh wave speed for the solid particles, Thornton and Randall w23x. Agglomerate preparation begins with the generation of spherical particles in a prescribed region. For the agglomerate used in this study, particles of a given size and material properties were generated at incremental distances in the X, Y and Z coordinate directions so as to form a face-centred cubic array with a very small gap Žca. 0.1% of the particle radius. between the particles. A spherical agglomerate shape was ensured by accepting only particles in the array which were centred inside a specified inner spherical region. A small centripetal gravity field Ž g s 0.05 msy2 . was then used to bring the particles into contact after which adhesive bonding at the contacts was imposed by attributing each particle with an initially low value of surface energy Žg s 0.03 Jmy2 . and then slowly increasing the value to obtain agglomerates with different interparticle bond strengths. The centripetal gravity field was slowly reduced to zero to complete the agglomerate preparation. To initiate impact, a wall was located at a very small distance Žca. 1 to 100 nm, depending on the impact velocity. from the agglomerate and then the desired impact velocity was attributed to all the constituent primary particles. Cyclic calculations were continued until the end of a single impact event which was indicated by an approximately zero wall force, zero rate of bond breakage and a constant kinetic energy of the system.
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The agglomerate was made up of 7912 spheres of diameter d p s 20 mm and had an overall diameter d a s 0.461 mm with an average coordination number Z s 11.1 after preparation Ž Z - 12 due to finite size of the agglomerate.. The solid density r , Young’s modulus E, coefficient of interparticle friction m and Poisson’s ratio n of the primary particles were 2650 kg my3 , 70 GPa, 0.35 and 0.3 respectively. Five values of interface energy G Žs 2g . s 0.2, 0.4, 1.0, 2.0 and 4.0 Jmy2 were used and impact was carried out at velocities in the range 0.05 to 20 msy1 .
3. Evolution of impact parameters Typical time evolutions of the force generated at the agglomerate-wall interface, the kinetic energy of the system of primary particles composing the agglomerate and the proportion of initial interparticle bonds broken during impact are shown in Fig. 1, illustrating the effect of impact velocity. At sufficiently low impact velocities when no fracturerfragmentation occurs Žnot illustrated. the behaviour is similar to that obtained during impact of a solid sphere in that the maximum force on the wall occurs when the kinetic energy is a minimum and, as in the case of an elastoplastic sphere, the final kinetic energy is less than the initial kinetic energy due to some internal bond breakage and irrecoverable deformation of the microstructure adjacent to the agglomerate-wall interface. As the impact velocity is increased, the maximum wall force increases, the duration of the impact decreases and the proportion of
Fig. 5. Relationship between damage ratio, impact velocity and interface energy.
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bonds broken increases. The final kinetic energy increases with impact velocity but, as shown in Fig. 1b, the proportion of the initial kinetic energy remaining at the end of the impact decreases with impact velocity until the velocity is sufficient to produce multiple fracture and then increases with impact velocity due to the increasing kinetic energy of the broken fragments and debris. Fig. 1 shows that, when fracturerfragmentation occurs, the kinetic energy reaches a minimum value during unloading and that the time lapse between the maximum wall
force and the minimum kinetic energy increases with impact velocity. In all the tests simulated it was observed that interparticle bonds were broken at an increasing rate and then at a decreasing rate as the kinetic energy decreased to its minimum value. As the kinetic energy increased the rate of bond breakage increased and then decreased until a constant final value was attained just after the end of the impact. Fig. 2 illustrates how the evolution of the wall force, the kinetic energy and the proportion of bonds broken varies
Fig. 6. Relative orientation of the agglomerate microstructure Ža. impact direction orthogonal to the square-packed planes and Žb. impact direction parallel to the square-packed planes.
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Fig. 7. Evolution of bond breakage during impact orthogonal to the square-packed planes Žinterface energy G s 4.0 Jmy2 , impact velocity V s 2.0 msy1 ..
with interface energy for an impact velocity of 1.0 msy1 . As the interface energy is increased, the maximum wall force increases, the minimum kinetic energy decreases and there is a reduction in the proportion of bonds broken. Interface energy does not have a significant effect on the impact duration and although the minimum kinetic energy occurs during unloading the time lapse does not vary monotonically with interface energy. For G s 4.0 Jmy2 , no debris was produced; the agglomerate rebounded from the wall with a few Ž- 2%. bonds broken internally. When the interface energy was reduced to G s 2.0 Jmy2 a clean fracture was observed which resulted in two approximately hemispherical fragments. Multiple fracture was observed for G s 1.0 Jmy2 with a limited amount of shattering of some small fragments. Further decreases in interface energy resulted in an increased amount of shattering. For impacts with G s 0.2 Jmy2 and 0.4 Jmy2 , approximately 90% of the interparticle bonds were broken resulting in lots of small fragments consisting of no more than 20 primary particles per fragment. Fig. 2 indicates that single fracture leads to an increase in the elapsed times when the maximum wall force and minimum kinetic energy occur but with further fragmentation and shattering due to reducing the bond strength the maximum wall force and minimum kinetic energy occur at smaller elapsed times. The relationship between the evolution of the proportion of broken bonds
with the evolutions of the wall force and kinetic energy is similar to that observed in Fig. 1. The proportion of bonds broken during an impact was defined as the damage ratio D by Thornton et al. w16x who observed, from the results of their 2D simulations, that the relationship between the damage ratio and the impact velocity could be approximated by the expression D s a ln
V
ž / V0
Ž 1.
where V0 is the threshold velocity below which no significant damage occurs. Deviations from Ž1. occurred for the limits D ™ 0 and D ™ 1 because Ža. there will be a range of low velocities at which the agglomerate rebounds but suffers a small amount of internal damage and Žb. even at very high velocities not all contacts will be broken since it is possible that some doublets and triplets will survive. Fig. 3 shows that the data sets obtained from 3D simulations of crystalline agglomerates also satisfy Ž1. with a s 0.35 for the range 0.2 - D - 0.8. Thornton et al. w16x suggested that the threshold velocity V0 , defined by Ž1., increased exponentially with interface energy. However, as shown in Fig. 4, the results presented here are well represented by the following power law. V0 s 0.17G 1.5
Ž 2.
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Fig. 8. Evolution of bond breakage during impact parallel to the square-packed planes Žinterface energy G s 4.0 Jmy2 , impact velocity V s 2.0 msy1 ..
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Fig. 9. Equivalent space lattice viewed from below illustrating the breakage patterns obtained for a strong agglomerate Ž G s 4.0 Jmy2 . impacted at velocities of 2.0, 3.0, 5.0 and 10.0 msy1 .
This implies that the damage ratio D should scale with lnŽ V 2rG 3 . and this is confirmed reasonably well in Fig. 5 except for the results of the weakest agglomerate Ž G s 0.2 Jmy2 .. Dimensional analysis suggests that the damage ratio is a function of the following dimensionless group
r d 3E 2
V
2
G
3
ž /
W
3
C
2
s
s CA3
Ž 3.
in which the Weber number Ws
r dV G
2
Ž 4.
is the ratio of inertia force to bonding force, the Cauchy number Cs
rV 2
Ž 5.
E
is the ratio of inertia force to elastic force and As
W
Ed s
C
G
Ž 6.
is the ratio of elastic force to bonding force and may be termed an elastic adhesion index. However, more simulations are required to clarify whether the parameters r , d and E in Ž3. should be the density, size and elastic
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modulus of the primary particles or the agglomerate bulk properties.
4. Fracture and fragmentation In this section agglomerate breakage is illustrated by computer generated images of the configuration of the primary particles, in which the particles are colour coded to indicate the size of the fragment to which they belong. Fracture planes are more clearly identified by representing the agglomerate by the equivalent space lattice which is formed by connecting the centres of particles in contact by
solid lines. However, in order to visualise the evolution of the bond breakage leading to fracture, it is more appropriate to use solid lines to connect the centres of particles which were initially in contact but which have broken contact during the impact. A face-centred cubic agglomerate possesses a strongly anisotropic structure. Consequently, we might expect that the fracture pattern would be sensitive to the orientation of the microstructure relative to the impact velocity direction. The structural orientation of a face-centred cubic array can be illustrated, as shown in Fig. 6, by nine spheres. There are two sets of four contiguous spheres, each set forming a square-packed plane. The ninth sphere is located in the
Fig. 10. Equivalent space lattice viewed from below illustrating the breakage patterns obtained for a weak agglomerate Ž G s 0.4 Jmy2 . impacted at velocities of 0.15, 0.30, 0.50 and 1.0 msy1 .
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cusp between the two square-packed planes leading to four sets of close-packed Žtriangular. planes inclined at 54.78 to the square-packed planes. Since the central sphere in Fig. 6 is itself part of a square-packed plane, it has a total of twelve contacts. Fig. 6a illustrates the relative orientation of the agglomerate when the square-packed planes are parallel to the wall. An example of fracture obtained for this orientation is shown in Fig. 7. For a strong Ž G s 4.0 Jmy2 . agglomerate impacted at a velocity of 2.0 msy1 , Fig. 7 illustrates the evolution of bond breakage during the impact, as viewed from above. In addition, to the right of the figure, two views are shown of the equivalent space lattice, showing an oblique fracture plane, a long time after the end of the impact when the large residual fragment is rebounding from the wall. As the force generated at the wall increases, the contacts along the horizontal square-packed planes are broken in the region of high vertical compression immediately above the agglomerate-wall interface. Consequently, the forces generated at the wall are transmitted along the inclined dense-
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packed planes producing a deceleration of the particles transmitting the large forces. Particles in adjacent densepacked planes which do not intersect the agglomerate-wall interface are not decelerated and consequently there is a relative shear movement between the loaded and unloaded dense-packed planes which propagates from the perimeter of the contact area. Initially, each particle is in contact with three particles in the adjacent dense-packed plane. As a result of the shear propagation two sets of contacts are broken between the loaded and unloaded planes thereby creating weakened planes propagated from the perimeter of the contact area. During unloading, as the agglomerate begins to rebound, there is a tendency to break the remaining bonds in the weakened planes. For reasons which presumably are associated with small heterogeneities in the contact force distribution and structure, in the example shown in Fig. 7, only one Žnot four. inclined fracture plane resulted from the impact. The equivalent space lattice illustrating the fracture plane in Fig. 7 may be considered to show an example of what has been termed ‘chipping’,
Fig. 11. Configuration of primary particles Žtop. and equivalent space lattice Žbottom., as viewed from below, illustrating fracture patterns resulting from impacting a strong agglomerate Ž G s 2.0 Jmy2 . at three velocities.
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Zhang and Ghadiri w24x. However, it is emphasised that the micromechanics of the fracture process evolution is the same as for the other cases described below. With the above exception, all the results reported in the paper were obtained from impact simulations in which the impact direction was parallel to the square-packed planes, as shown in Fig. 6b. An example of the bond breakage evolution for this orientation is illustrated in Fig. 8 for the same interface energy Ž G s 4.0 Jmy2 . and impact velocity Ž V s 2.0 msy1 . as the example shown in Fig. 7. The development of the bond breakage pattern is illustrated for different elapsed times showing two orthogonal views: one elevation and a view from above. During loading, bonds are broken along four vertical planes which form a diamond shaped pattern, as viewed from above. The four planes coincide with the perimeter of the agglomerate-wall interface and are the consequence of relative shear motion between adjacent load transmitting and load free densepacked vertical planes. The relative shear motion results in the breaking of one set of contacts between the loaded and unloaded dense-packed planes. The bond breaking propa-
gates upwards from the wall until it reaches the top of the agglomerate when the kinetic energy attains its minimum value at t s 1.065 ms. Due to the spherical shape of the agglomerate, the number of bonds broken along the shear propagation front increases and then decreases as the front traverses a circular plane section. This provides the explanation for the rate of evolution of bond breakage illustrated in Figs. 1 and 2. During unloading, as elastic energy is converted into kinetic energy, a second set of contacts are broken, propagating downwards from the top of the agglomerate. With further increase in kinetic energy, some bonds are also broken in the third set of contacts leading to fracture along some of the shear induced weakened planes just prior to the end of the impact. The breakage pattern resulting from impact fracture of the agglomerate, for G s 4.0 Jmy2 and V s 2.0 msy1 , is illustrated by two orthogonal views of the equivalent space lattice at the bottom of Fig. 8. The effect of impact velocity on the breakage patterns obtained is shown in Figs. 9 and 10 for agglomerates with interface energies G s 4.0 Jmy2 and G s 0.4 Jmy2 re-
Fig. 12. Configuration of primary particles Žtop. and equivalent space lattice Žbottom., as viewed from below, illustrating fracture patterns resulting from impacting a weak agglomerate Ž G s 0.2 Jmy2 . at three velocities.
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spectively. For the strong agglomerate, Fig. 9 clearly shows that at an impact velocity of 5.0 msy1 all four of the shear induced weakened planes are fractured plus two short fracture planes which are parallel to the square-packed planes. At lower impact velocities the fracture pattern is a subset of that obtained for V s 5.0 msy1 . If the impact velocity is increased above 5.0 msy1 no extra fracture planes are created but the residual fragments are weakened due to internal bond breakage and this leads to shattering of the agglomerate at high impact velocities. The effect of varying the velocity at which the weak agglomerate was impacted produced similar results but, as seen in Fig. 10, the large fragments suffered significant internal bond breakage even at low impact velocities due to the relatively low interface energy between the primary particles.
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For an agglomerate with interface energy G s 2.0 Jmy2 the fracture patterns produced at different impact velocities were similar to those obtained for G s 4.0 Jmy2 , as can be seen in Fig. 11. Also shown in the figure are images of the corresponding particle configurations, as viewed from below the impact area, which illustrate the increasing amount of fine debris produced around the impact area when the velocity is increased. When impacted at a velocity of 3.0 msy1 , extensive shattering occurred with the largest fragment consisting of only 312 primary particles. Fig. 12 provides similar images obtained for a very weak agglomerate for which G s 0.2 Jmy2 . It may be noted that when impacted at a velocity of 0.2 msy1 fracture occurred between square-packed planes and between planes orthogonal to the square-packed planes. This was the only occur-
Fig. 13. Fragmentation resulting from impacting a strong agglomerate Ž G s 4.0 Jmy2 . at 2.0 msy1 and 4.0 msy1.
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Fig. 14. Fragmentation resulting from impacting a weak agglomerate Ž G s 0.4 Jmy2 . at 0.15 msy1 and 0.30 msy1.
rence of this fracture pattern in all the simulations carried out. Finally in this section, elevation and plan views of fragmentation resulting from impact for G s 4.0 Jmy2 , V s 2.0 msy1 and V s 4.0 msy1 are shown in Fig. 13; and for G s 0.4 Jmy2 , V s 0.15 msy1 and V s 0.30 msy1 in Fig. 14. In both figures the small debris particles are not shown in the case of the higher velocity impact in order to illustrate the shape of the large residual fragments more clearly.
5. Fragment size distribution Although the size distribution of the fragments resulting from an impact is of practical importance in powder
technology, the definition of size for non-spherical fragments is imprecise. In physical experimentation, sieve sizes are used to classify fragments into different size ranges but ambiguities arise since further breakage may occur due to the vibratory nature of the sieving process and the fact that elongated particles may or may not pass through the sieve depending on the orientation of their long axis. Similar uncertainties arise in numerical simulations when considering the size of the clusters representing the fragmentation products of an impact. One option is to define the size of a fragment d f by the diameter of the circumscribing sphere and to normalise this by dividing by the circumscribing sphere diameter d a of the original agglomerate. An alternative is to use the mass of a cluster m divided by the mass of the original agglomerate M, which is equivalent to using a normalised size based on the
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Fig. 15. Effect of impact velocity on the fragment size distribution of a weak agglomerate Ž G s 0.2 Jmy2 ..
size of the equivalent solid spheres. In this section we use the latter definition to illustrate the numerical simulation data but also quote parametric values obtained using the former definition.
Fig. 15 shows a Gates–Gaudin–Schuhmann doublelogarithmic plot of cumulative mass fraction undersize f against normalised size mrM obtained from impacting a weak agglomerate Ž G s 0.2 Jmy2 . at three different veloc-
Fig. 16. Effect of impact velocity on the fragment size distribution of a strong agglomerate Ž G s 2.0 Jmy2 ..
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ities. A similar plot is shown in Fig. 16 for a strong Ž G s 2.0 Jmy2 . agglomerate. Both figures show that the fragment size distributions exhibit bilinear characteristics which distinguish the residue of large fragments from the complement of small fragments Ždebris.. Similar results were obtained by Arbiter et al. w6x for impact velocities which produced ‘‘semi-brittle’’ fracture. The normalised size which identifies the transition from residue to debris is independent of the bond strength and corresponds to mrM
s 0.1 or d frd a s 0.5 depending on the size definition used. These values may be compared with values obtained from Arbiter et al.’s experiments of 0.21 and 0.27 for sand-cement spheres and limestone-cement spheres respectively. Figs. 15 and 16 show that the exponent for the residue decreases with impact velocity implying that, at sufficiently high impact velocities, there is no residue as a consequence of the extensive shattering that occurs. Except
Fig. 17. Effect of interface energy on the fragment size distribution.
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Fig. 18. Normalised fragment size distribution.
for very low Žno fracture. or very high Žshattering. levels of breakage, the exponent n which defines the size distribution of the debris, f ; Ž mrM . n , is independent of impact velocity but decreases if the bond strength is increased. Strongly bonded agglomerates exhibit clean fractures and fine debris originating from the impact re-
gion and, hence, low values of n. In weak agglomerates larger debris fragments are created along the less well defined fracture planes leading to higher values of n. In the fracturermultifracture regime, for any size mrM, the cumulative mass fraction undersize f increases with impact velocity. Arbiter et al. w6x demonstrated that, for
Fig. 19. Dependence of the complement size distribution exponent n on the interface energy.
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Fig. 20. Dependence of the complement size distribution prefactor l on the interface energy.
their sand-cement spheres, the size distribution of the debris Žcomplement. correlates with the specific impact energy according to the expression f s lV 2 s n
Ž 7.
where V is the impact velocity, s is the normalised size Ž mrM or d frd a . and l is a constant of proportionality
which depends on the material properties. Using values of the exponent n obtained from Figs. 15 and 16 and other similar plots for other interface energy values, the size distribution data obtained from all the simulations performed are plotted in Fig. 17Ža,b. confirming the form of Ž7. and indicating that, for face-centred cubic crystalline agglomerates, the prefactor l depends on the strength of the interparticle bonds. Using values of l obtained from
Fig. 21. Relationship between the normalised mass of the largest fragment and the ratio of impact velocity and interface energy.
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Fig. 17, Fig. 18 demonstrates that all the simulation data collapse reasonably well onto a single curve. In general, however, l is expected to depend also on other properties such as solid fraction and coordination number which define the microstructure; but this is beyond the scope of this investigation. As indicated in Figs. 15–18, both the exponent n and the prefactor l are functions of the interface energy G . The two relationships are illustrated in Figs. 19 and 20 and may be approximated by n s 0.25 Gy0 .2
Ž 8.
and
l s Gy2
Ž 9.
However, further work is required to obtain more data points in order to reliably quantify the two relationships. The size of the largest surviving fragment, in terms of its normalised mass m L , is plotted against the ratio of impact velocity to interface energy in Fig. 21 which shows that the data satisfies a power law scaling given by m L s 0.15
V
y1 .5
ž / G
Ž 10 .
with the condition m L - 1.
6. Conclusions Results have been presented of numerical simulations of a spherical, crystalline Žface centred cubic. agglomerate impacting orthogonally with a target wall. The effects of impact velocity and bond strength on the evolution of the wall force, kinetic energy of the agglomerate and the proportion of bonds broken have been demonstrated. As might be expected, increasing the impact velocity results in a higher maximum wall force, shorter impact event and more broken bonds. Increasing the bond strength increases the maximum force generated at the wall, reduces the number of bonds broken but does not significantly affect the duration of the impact event. The proportion of bonds broken during an impact has been defined by a damage ratio D. In the range 0.2 - D - 0.8, the initial specific energy required to break a given number of bonds scales with G 3. The size distributions of the fragments produced by impact breakage have been examined and shown to exhibit a bilinear distribution on a Gates–Gaudin–Schuhmann plot. Independent of bond strength, the residue consists of fragments of mass greater than 10% of the mass of the original agglomerate, the complement Žor debris. of fragments of mass less than 10% of the agglomerate mass. For the complement, the exponent of the power law relationship decreases if the bond strength is increased and the cumulative mass fraction undersize scales with Ž VrG . 2 . The size of the largest fragment Žin terms of its mass. scales with Ž VrG .y1 .5.
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Agglomerate breakage has been illustrated by computer generated images of the configuration of primary particles and by the equivalent space lattice. The visualisations have shown that for any given bond strength there is an impact velocity which produces a complete set of fracture planes. Subsets of this fracture pattern are produced at lower impact velocities. Higher impact velocities do not produce extra fracture planes but the residual fragments are weakened due to internal bond breakage, which results in shattering at high impact velocities. By examining the evolution of bond breakage, it has been shown that there is a shear-induced weakening of a set of potential fracture planes during deceleration of the agglomerate. The pattern of weakened planes is dictated by the geometry of the agglomeraterwall interface and the orientation of the agglomerate microstructure with respect to impact direction. As the agglomerate recovers kinetic energy, further shearinduced bond breakage occurs along the same set of planes. At the end of the impact, depending on the amount of kinetic energy recovered, complete fracture occurs along some of the weakened planes. The structure of the initial agglomerate is crystalline with no structural flaws although there is a small random distribution of residual stored elastic energy due to the centripetal gravity field used to prepare the agglomerate. From the observations of bond breakage evolution, it is concluded that fracture is shear-induced and, in effect, the agglomerate creates its own flaw population during loading and that pre-existing flaws do not have a significant effect on agglomerate ‘‘strength’’.
Acknowledgements The work reported in the paper was supported by the Engineering and Physical Science Research Council.
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