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DEM analysis of compression breakage of 3D printed agglomerates with different structures
Powder Technology 356 (2019) 1045e1058
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Powder Technology journal homepage: www.elsevier.com/locate/powtec
DEM analysis of compression breakage of 3D printed agglomerates with different structures Ruihuan Ge a, b, Lige Wang c, Zongyan Zhou a, * a
Department of Chemical Engineering, Monash University, Australia Birmingham Centre for Energy Storage (BCES), School of Chemical Engineering, University of Birmingham, UK c School of Engineering, University of Sheffield, UK b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 1 June 2019 Received in revised form 30 August 2019 Accepted 31 August 2019 Available online 5 September 2019
Understanding agglomerate breakage characteristics is of great importance in industries of particulate manufacturing. In this work, Discrete Element Method (DEM) combined with Timoshenko Beam Bond Model (TBBM) is employed to simulate the agglomerate breakage with different structures subject to compressive loads. The agglomerates are made of 4 mm diameter rigid primary particles connected by cylindrical inter-particle bonds. Using 3D printing technology, test agglomerates with tuneable properties are produced and experimentally crushed. The simulation results of agglomerate crushing show similar qualitative trends with experimental results under identical conditions. Different agglomerate types with well-defined bond properties are considered in numerical simulations, e.g. cube/spherical shaped agglomerates with regular packed structures. Detailed damage ratios, particle velocities and bond stress distributions are obtained and analysed by DEM simulations. The results demonstrate that the breakage is strongly influenced by the agglomerate internal bond network orientations and external shapes. © 2019 Elsevier B.V. All rights reserved.
Keywords: 3D printing Discrete element method (DEM) Agglomerates Breakage Bond model
1. Introduction Agglomeration process is widely used in pharmaceutical, food and chemical industries. During processing, the agglomerate can experience different stressing conditions which may lead to their deformation, attrition and/or fragmentation. To ensuring processing efficiency and high product quality, a detailed understanding of agglomerate breakage mechanisms is essential. The agglomerate mechanical properties are strongly dependent on its morphology, internal particle arrangements and interparticle bond properties [1,2]. Rumpf [3] and Kendall [4] have proposed theoretical models to describe the relationships between macroscopic agglomerate strength and its microscopic bonding forces. However, these two strength models cannot take the complex compositions and irregular shapes of realistic agglomerates into account. There have also been several established experimental approaches for measuring single agglomerate breakage. Using molded agglomerates with glass ballotini bonded by epoxy resin binders, Subero et al. [5] investigated the impact breakage of agglomerates with macro-voids, they concluded that the
* Corresponding author. E-mail address: [email protected] (Z. Zhou). https://doi.org/10.1016/j.powtec.2019.08.113 0032-5910/© 2019 Elsevier B.V. All rights reserved.
agglomerate breakage patterns change from localised damage to fragmentation, to multiple fragmentations with increased void fractions and impact velocities. Antonyuk et al. [6] studied the effect of binder contents on the agglomerate strength. Using cylindrical agglomerates made of g-Al2O3 primary particles, it is demonstrated that the agglomerate stiffness and strength vary proportionally with both the binder content and agglomerate size. Recently, we have proposed a new method for producing model agglomerates by 3D printing technology, enabling fully reproducible structures and inter-particle bond properties [7,8]. Apart from theoretical and experimental research, Discrete Element Method (DEM) simulation has been widely used for investigating agglomerate breakage. The main advantage of using DEM is that it can generate detailed force and velocity evolution information at microscopic particle scales. Substantial agglomerate breakage studies have been performed using Johnson-KendallRoberts (JKR) contact model which describes adhesive forces between fine powders [9e12]. Thornton et al. [13,14] studied the impact and compressive breakage of spherical agglomerates with random assembled internal structures. The results mainly show that the breakage pattern changes from disintegration to fragmentation with increased solid fractions or inter-particle contact densities. Furthermore, Thornton et al. [15,16] investigated the effect of agglomerate shapes, including spherical, cuboidal and
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cylindrical agglomerates on breakage characteristics and found that the contact regions adjacent to the impact site play important roles in determining agglomerate breakage characteristics. Recently, Deng et al. [17] studied the influences of fractal dimensions on agglomerate breakage. The results reveal that the damage ratio decreases with increased fractal dimensions ranging from 2.0 to 2.6. The Linear Parallel Bond (LPB) model [18e20] is considered for agglomerates bound by binders, in which primary particles are connected by idealised elastic solid bonds. Using LPB model, Spettl et al. [21] studied the influences of primary particle size distributions and stochastic microstructures on the agglomerate breakage. Eckhard et al. [22] investigated the spray dried agglomerates with defined internal porosities and shell thicknesses. Their results mainly revealed the agglomerate strength is influenced by the variability of microstructures and bond sizes. Nowadays X-ray micro-tomography has been applied to characterizing the particle position, binder and porosity distributions within realistic agglomerates [23]. Dosta et al. [24] used the LPB model to predict agglomerate breakage with their microstructures and inter-particle bond contents obtained from corresponding realistic granules by Xray micro-tomography. The comparison with the experimental breakage results revealed that the simulated breakage strengths showed a similar qualitative increase trend with increasing binder contents. As neighbour particles inside agglomerates are physically attached to each other by a cylindrical geometry which can be treated as a beam element, a newly developed Timoshenko Beam Bond Model (TBBM) bond model is applied to investigating agglomerate breakage in this research [25]. Previous work [1] has demonstrated that JKR model is not applicable for describing the properties of agglomerates with solid bonds, in which the bonded particles can undergo irreversible breakage with large deformation of the inter-particle bonds. Although the LPB model was applied for modelling bonded agglomerates, this model only resists the axial and shear forces. Compared with the above mentioned contact models, the TBBM model is desirable for transmitting normal, shear, bending and torsion movements at two bond ends, and is suitable to describe the deformation and failure behaviours of the stubby beams. In this study, we compared compressive breakage simulation results with corresponding experiments of 3D printed agglomerates. After that, using DEM simulations, we systematically investigated the deformation and breakage behaviours of agglomerates with different internal crystalline structures. The effects of external shapes, internal microstructures and bond properties on the evolutions of breakage, velocity and bond stresses were analysed in detail. The purpose of this work is to numerically investigate how the packing structure and well-defined inter-particle bond properties affect the agglomerate mechanical properties including deformation and breakage characteristics. 2. Methodology 2.1. Experimental protocol As illustrated in Fig. 1, two different agglomerate structures, i.e. cubic tetrahedral structured agglomerate and spherical random structured agglomerate, were designed and applied in experiments. The corresponding doublet dimension is shown in Fig. 1 (a). Fig. 1 (b) shows a cube shaped agglomerate with a tetrahedral internal structure. The cubic tetrahedral structured agglomerate has 91 primary particles. Each particle is 4 mm diameter (Dp ¼ 4 mm), and is connected by 2.6 mm diameter cylindrical inter-particle bonds (Db ¼ 2.6 mm). The inter-particle distance (centre to
centre) is L ¼ 4.25 mm. Fig. 1 (c) shows a spherical agglomerate with random internal structure. The spherical random structured agglomerate has 120 primary particles, all 4 mm in diameter (Dp ¼ 4 mm). This random structure was generated in a 25 mm diameter spherical space using particle factory function in EDEM software. During the generation process, a gravity field was assigned and changed in different directions for ensuring the particles filled the whole space. The generated random structure has a 44% packing void fraction. After generating the structure, particles within 4.3 mm of each other (centre to centre length) are considered to be joined by a 2 mm diameter bond (Db ¼ 2 mm). As the random structured agglomerate is densely packed, the average inter-particle distance (centre to centre) is L ¼ 4 mm. These two structures have also been used in our previous work, for representing two different classes of agglomerates, i.e. Regular structured agglomerate and spherical random structured agglomerate [7,8]. Different bond diameters of the two structures were chosen is mainly because of a limitation of the 3D printing technique. We need to ensure that the inter-particle bond size is big enough compared to the 3D printing resolution, and also is suitable to facilitate the support removal process after 3D printing. The agglomerates were produced using PolyJet 3D printing technology (Stratasys Objet 500). The primary particles were printed using a rigid polymer (VeroWhitePlus™) while the interparticle bonds were printed using a rubber-like material (DM 9895). Quasi-static compressive tests were performed on these 3D printed agglomerates. As shown in Fig. 1, agglomerates were crushed at a constant cross-head speed of 0.02 mm/s. The breakage process was filmed using a Nikon D7000 camera. To ensure the reliability and repeatability, all the tested samples were crushed with the printing layers perpendicular to the loading direction. 2.2. Discrete element method (DEM) simulation 2.2.1. Agglomerate structures DEM simulations of the agglomerate crushing were used to produce the two identical structures used in experiments. In addition, simulations were also performed on agglomerates with regular internal structures. As illustrated in Fig. 2, two different internal crystalline structures, tetrahedron structure and simple cubic structure were studied. The particle size is Dp ¼ 4 mm, and the bond geometry is L ¼ 4.25 mm, Db ¼ 2.6 mm. As the orientations of internal microstructures with respect to the loading direction may affect their deformation and breakage characteristics, two orientations were considered for each structure, namely “Orientation I” and “Orientation II” (see Fig. 2). In Fig. 2, the relative position of the contiguous particle sets inside the structures is depicted on the left side of each design to illustrate different orientations. The particle positions of different types were generated using MATLAB code and imported into the DEM simulation software. By implementing the TBBM contact model, a DEM simulation of agglomerate crushing was created. The crushing directions of these structures are also illustrated in Fig. 2. In this work, the orientation effect of agglomerate structures was only considered and investigated by DEM simulations. 2.2.2. The TBBM contact model DEM models the particle motions and rotations based on Newton’s second law. The interaction forces between primary particles are determined by the contact model. In this study, TBBM contact model [25] is used to describe the bond deformation between primary particles, in which a Timoshenko beam element is used to connect the centres of two primary particles. The bond ends share same degrees of motion freedom with corresponding
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Fig. 1. Schematic of breakage experiment setup (a) doublet inside agglomerate, and (b) cubic agglomerate with a tetrahedral internal structure (L ¼ 4.25 mm, Db ¼ 2.6 mm and Dp ¼ 4 mm), (c) spherical agglomerate with a random structure (L ¼ 4 mm, Db ¼ 2 mm and Dp ¼ 4 mm).
Fig. 2. Agglomerates with crystalline structures in terms of different orientations (the crushing direction is illustrated by down arrows).
particles. At each time step, the relationship between the bond displacement Δui and resulting forces ΔFi is related with the stiffness matrix [K]:
fDFg ¼ DFax ; DFay ;DFaz ;DMax ;DMay ;DMaz ;DFbx ;DFby ;DFbz DMbx ; DMby ;DMbz T (2)
fDFg ¼ ½K,fDug
(1)
where {DF} is the incremental force vector that contains six force (ΔF) and six moment (ΔM) increments at the two bond ends:
fDug ¼ Ddax ; Dday ;Ddaz ;Dqax ;Dqay ;Dqaz ;Ddbx ;Ddby ;Ddbz ;Dqbx ;Dqby ;Dqbz T
where the subscript a or b denotes the two bond ends, and x, y or z denotes the force direction in the local coordinate system. The superscript T in this equation denotes the transpose operation of this row vector. {Δu} is the incremental displacement vector that contains six displacements (d) and six rotation (q) increments at the two bond ends a and b:
(3)
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The stiffness matrix [K] is a 12 12 matrix containing six different stiffness constants to describe the axial, shear and bending forces of the beam. The stiffness constants are determined by the bond mechanical and geometry parameters including bond Young’s modulus (Eb), Possion’s ratio (nb), bond length (Lb) and diameter (db), which are given in the following section. Detailed formulations of the stiffness matrix is in [26,27]. The total internal forces Fi and moments Mi at the bond ends are calculated by summing the incremental forces ΔFi and moments ΔMi at each time step. The inter-particle bond behaves in an elasticbrittle way and it breaks once its maximum stress exceeds the given strength. Three failure criteria are considered in the TBBM model: compressive (sC), tensile (sT) and shear strength (t). The maximum stresses are determined according to beam theory and summarized as follows:
sCi
3. Results and analysis
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0 2 2 Fbx rb Miy þ M iz A i ¼ a; b ¼@ b Ib A
(4)
sCmax ¼ minðsC a ; sC b Þ 0
sTi ¼ @
Fbx Ab
þ
rb
(5)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 M 2iy þ M2iz A i ¼ a; b Ib
(6)
sTmax ¼ maxðsT a sT b Þ tmax
jMax jrb 4 ¼ þ 2Ib
(7)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F 2ay þ F 2az
(8)
3Ab
Eq. (8) is used to determine the maximum shear stress tmax at the bond ends. As the twisting moments and shear forces in the equation are equal and opposite at two bond ends, the maximum shear stress tmax is calculated at the a end. 2.2.3. DEM simulation parameters The simulations were carried out using EDEM 2.7 software. Input parameters for the DEM simulation setup including particle and inter-particle bond properties are listed in Table 1. The particle and bond parameters are taken from PolyJet digital 3D printing materials [28], which are in accordance with our previous research
Table 1 Parameters used in DEM simulations.
Particle
Parameter
Description
Value
Dp
Diameter (m) Inter-particle static friction Inter-particle restitution Density (kg/m3) Poisson’s ratio Young’s modulus (GPa) Length (m) Diameter (m) Young’s modulus (MPa) Poisson’s ratio Compressive strength (MPa) Tensile strength (MPa) Shear strength (MPa) Steel-particle static friction Steel-particle restitution Density (kg/m3) Poisson ratio Shear modulus (GPa)
0.004 0.3 0.9 1200 0.3 1.0 0.00425/0.004 0.0026/0.002 50e200 0.4 50 10 10 0.3 0.9 7800 0.3 70
mpp epp
rp np Bond
Steel platen
Ep Lb Db Eb
nb sC sT t msp esp
rs ns
Gs
[7,8]. The primary particle is a rigid material (VeroWhite polymer), while the inter-particle bond represents a relatively soft material (DM 9895). The bond strength parameters represent the properties of soft bond material DM 9895 that are obtained from the datasheet offered by the vendor [28]. The bond Young’s modulus Eb is 100 MPa in breakage simulations. In addition, the effect of bond Young’s modulus values Eb ranging from 50 to 200 MPa on the compressive modulus of the agglomerate structures Eagg is also addressed. The mechanical properties of steel plates are referred from the material library of EDEM software. The time-step of this simulation is 1 107 s. The compressive breakage simulation setup is the same as experiments. The sample was loaded by moving the upper plate with a constant 2 mm/s loading rate. The compressive loads of the upper plate were recorded, and detailed force and velocity evolutions at particle scales were obtained and analysed.
3.1. Comparison of simulation and experimental results The experimental compressive curves of cubic tetrahedral structured agglomerate and spherical random structured agglomerate are shown in Fig. 3. The simulated compressive breakage patterns of these two structures are compared with the corresponding experimental results, as illustrated in Figs. 4 and 5. The simulations indicate the damage ratio (i.e. the fraction of broken bonds to the initial bond contacts for a single particle) by changing its colour from blue to red, where blue means that no contacts have been broken for this particle and red means that all the original particle contacts are broken. The average damage ratio for the whole agglomerate (i.e. the fraction of the total broken bonds to the initial total bond amount) during the compression process is illustrated in Fig. 6. Fig. 4 refers to the breakage process of the cube shaped agglomerate with tetrahedral structure (the “Orientation I” case). The experiments and simulations show similar breakage patterns. At the beginning of the compression, i.e. displacement <3 mm (13% strain), the agglomerate mainly shows elastic deformation without damage. After this elastic deformation stage, local damage appears and the agglomerate deforms along a 45o plane. Subsequently, with the increased displacement, the breakage expands from the local region to the whole particle assembly. The experimental compressive curve in Fig. 3 (a) and damage ratio evolution in Fig. 6 clearly show the two stages during the compression breakage progress. Initially the contacts break in the local region at 3 mm displacement (13% strain). After that, extensive slip takes place with a small number of bonds broken. This is followed by a further contact breakage near 8 mm displacement (30% strain), which stands for the whole assembly has failed along slip planes. Fig. 5 illustrates the breakage process of the spherical agglomerate with random structure. The snapshots of the simulation case show that minor damage appears first near the contact region at around 2 mm of displacement (8% strain). At approximately 3 mm of displacement (12% strain), the agglomerate fractures into two large fragments. Similar breakage behaviours can be observed for the experimental camera stills, in which the agglomerate breaks into two hemispheres. The fracture is accompanied by a load drop in the load-displacement curve shown in Fig. 3 (b). From Fig. 6 (b), the damage ratio experiences a sudden increase near 3 mm displacement (12% strain), which corresponds to the occurrence of fracture. This fracture pattern has been reported in previous research on the breakage of densely packed agglomerates [14,16]. The qualitative agreement between the simulation and experimental results confirms the validity of TBBM model for investigating agglomerate breakage characteristics. In the following,
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Fig. 3. Experimental compressive curves (a) Cubic tetrahedral structured agglomerate (Orientation I case) (b) Spherical agglomerate with random internal structure.
Fig. 4. Simulated and experimental breakage results for a cube shaped agglomerate with ordered, tetrahedral internal structure as a function of the displacement of the upper test platen.
detailed evolutions of damage ratios, particle velocities and bond stresses are investigated using DEM combined with the TBBM contact model.
3.2. Deformation and breakage of cubic shaped structures The deformation and breakage characteristics of cubic shaped agglomerates with different crystalline structures are analysed and shown in Figs. 7-10. The relative orientations of the microstructure with respect to the loading directions are illustrated in each figure, by using a single unit cell inside the particle assemblies. At the initial loading stage, the structures show elastic deformations. The Young’s modulus of agglomerate structures Eagg is defined as:
Eagg ¼
kagg ,Lagg Aagg
(9)
where kagg the agglomerate structure stiffness obtained from the compressive force curves. Lagg and Aagg are the height and crosssectional area of the cubic sample, respectively. The Young’s modulus Eagg, as calculated from DEM simulations, are plotted for particle numbers of N in Fig. 7. The number of particles N was varied by keeping the same aspect ratio of the structure, i.e. the cubic structure size increases equal proportionally along the x, y and z axis in a Cartesian coordinate system. In Fig. 7, the Young’s modulus increases with particle number for the tetrahedron structure and reduces for the simple cubic structure. According to the definition in Eq. (9), as the structure stiffness kagg increases with increased particle number, the different variation
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Fig. 5. Simulated and experimental breakage results for the spherical agglomerate with random internal structure as a function of the displacement of the upper test platen.
Fig. 6. Damage ratio evolutions for each agglomerate design (a) cubic tetrahedral structured agglomerate (Orientation I case) (b) spherical agglomerate with random internal structure.
tendency for these two structures can be explained by the dimension differences induced by different internal structures. In addition, with increased particle numbers N, it can be seen that for both structures the predicted modulus approaches an asymptotic value and are independent of N. As illustrated in Fig. 7, the asymptotic values of Young’s modulus for the tetrahedron structure and simple cubic structure are 2.75 MPa and 35 MPa, correspondingly. In the following Figs. 8-10, we adopt particle number N > 2500 for the tetrahedron structure, and N > 1000 for the simple cubic structure. Fig. 8 shows the effects of different bond Young’s modulus Eb and internal structure orientations on the Young’s modulus Eagg of each cubic shaped structure. In both cases, the structure Young’s modulus increases with increased inter-particle bond modulus. Furthermore, for both structures, at any given bond Young’s modulus, the “Orientation II” structure provides a higher modulus
value than “Orientation I”. This difference can be explained by the unit cell orientations as illustrated in Fig. 8. The structure carries the loads by compression, bend or shear deformations of the beam elements inside the structures. Under uniaxial compression direction, the axially orientated beams parallel to the loading directions can give a stiffer response of the whole structure. This effect is more pronounced for the simple cubic structures. As illustrated in Fig. 8 (b), the “Orientation I” case carries the loads by bending and shear deformations of the beam elements. While when loaded in “Orientation II”, as this arrangement shows a pure stretchdominated deformation [29,30], it has a relatively higher modulus value. Figs. 9 and 10 show the stress-strain curves and corresponding breakage patterns of different structures and loading orientations. The total bond stresses including compressive, shear and tensile component with respect to the nominal strain are calculated and
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Fig. 7. Dependence of the compressive modulus Eagg of cube shaped structures on the number of particles (N), the bond modulus Eb ¼ 100 MPa.
Fig. 8. The compressive modulus Eagg of cubic shaped structures considering different bond Young’s modulus Eb and internal structure orientations.
presented. The nominal strain is obtained from the corresponding compressive displacement divided by the sample height (Lagg). In these two figures, the stress-strain curves show a sudden stress change stage during compression which corresponds to the breakage point of the structures. Typical breakage patterns of each structure after the breakage point are depicted on the right side of each figure. The particle colors change from blue to red, representing the increased damage ratio of bond contacts. Fig. 9 (a) shows that the “Orientation I” tetrahedron structure breaks after a nominal strain of 10%. The fracture initially happens at the contact region to the test platen, and then it propagates to the whole particle assembly along a 45o diagonal plane. The “Orientation II” tetrahedron structure breaks at 6% nominal strain with different breakage patterns. As depicted in Fig. 9 (b), after the initial local breakage near the contact region, the broken bonds gradually extend to the middle plane of the structure accompanied by a slight increase of the total bond stress from 8% to 14% nominal strain. Compared with “Orientation I”, the “Orientation II” structure breaks at a lower strain (6% nominal strain) and stress level. As
shown in Fig. 8, the compressive modulus of “Orientation II” structure (Eagg ¼ 5.3 MPa) is higher than that of “Orientation I” (Eagg ¼ 2.8 MPa), indicating the stress level of the inter-particle bonds is higher at the same strain. As the inter-particle bonds of these two cases have same bond strength values (listed in Table 1), breakage event happens at a lower strain for “Orientation II” structure. Fig. 10 shows breakage characteristics of cubic shaped simple cubic structures. For the “Orientation I” simple cubic structure, the breakage happens along shear planes at around 7% nominal strain. A breakage pattern of two fracture planes can be observed along the diagonal line (see Fig. 10 (a)). For the “Orientation II” case, the whole particle assembly is under normal stress at the initial elastic deformation stage. As illustrated in Fig. 10 (b), with the increased nominal strain, the structure shows buckling behaviours and breaks along a horizontal plane. From the stress-strain curve, it can be seen that the bond stress inside this simple cubic structure changes from pure compression to more complex distributions after the breakage point at around 6.5% nominal strain. Especially,
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Fig. 9. Stress-strain curves and breakage patterns of cubic shaped tetrahedron structures, the colour change from blue to red represents the increasing proportion of broken bond contacts for a given particle. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
during deformation stage, the tensile stresses are negative as the bonds is under pure compressive stress. After the breakage event, the bond tensile stresses become positive, indicating the occurrence of tensile deformation and failure inside the structure. 3.3. Deformation and breakage of spherical shaped structures As illustrated in Figs. 11-16, compression breakage tests were also simulated for spherical shaped agglomerates. The spherical shaped structures were prepared by generating the particle assemblies inside a certain spherical region. The primary particle numbers N of the spherical tetrahedron and simple cubic structure are 1000 and 800, respectively. The orientations of the microstructure on the breakage were considered and depicted in each figure. Simulation parameters and setups are kept the same as used for the cubic shaped structures reported above. Fig. 11 presents the load-strain curves and breakage patterns of spherical shaped tetrahedron structures. From the load-strain curves, it can be observed that the first breakage event occurs at an applied load of 100 N for “Orientation I” case, and 150 N for “Orientation II” case. After then, for both cases, the compressive loads increase with nominal strains and reach an almost identical magnitude of 300 N after 15% nominal strain. The fluctuations of compressive loads indicate that local breakage happens during compression. It can be seen from Fig. 11 that for both cases, breakage happens progressively near the contact region adjacent to the test platen, and there is no clear fracture plane observed with compression continued to 40% nominal strain. Fig. 12 demonstrates the velocity evolution profiles of spherical shaped tetrahedron structures during compression. The particle velocities are represented by vectors with directions. As shown in Fig. 12, the particles
move downwards, and some disordered vectors appear near the contact region, indicating the rearrangement of primary particles and occurrence of breakage events. Fig. 13 refers to the compression test curves and breakage patterns of spherical shaped simple cubic structures. The load-strain curves show that compressive loads of “Orientation I” are much less than that of “Orientation II”. As shown in Fig. 13, for “Orientation I” case, the first breakage happens at 7% nominal strain with a wedge-shaped broken bonds region formed adjacent to the test platen. With the increased nominal strain, the load slightly increases to around 400 N at about 17% nominal strain. This is followed by a crushing of the whole particle assembly after 20% nominal strain with slip fracture planes observed from the snapshots (See Fig. 13 (a)). In contrast, when the structure is crushed according to “Orientation II” loading direction, the compression loads quickly rise to around 1400 N, reaching the first breakage point of around 6% nominal strain. After the first breakage event, additional breakages can be observed from the load-strain curves (see Fig. 13). The snapshots in Fig. 13 (b) illustrate that the whole structure falls apart along a vertical fracture plane following on the initial local breakage near the contact region. The particle velocity evolutions during compression is also recorded in the simulation and shown in Fig. 14. In this figure, significant velocity discontinuities can be observed as the particles move vertically downwards, indicating the formation and movement of fracture planes, i.e. inclined shear planes for “Orientation I” while a cleavage-like failure plane for “Orientation II”. Similar breakage patterns of this spherical simple cubic structure have also been observed in our previous experimental research [7]. Compared with the velocity profiles of spherical shaped tetrahedron structures in Fig. 12, velocity discontinuities and induced
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Fig. 10. Stress-strain curves and breakage patterns of cubic shaped simple cubic structures, the colour change from blue to red represents the increasing proportion of broken bond contacts for a given particle. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
fracture planes are easily formed in the simple cubic structured agglomerate. The velocity vectors indicate the relative movement of fragments. By contrast, in Fig. 12, the velocity discontinuity cannot propagate through the tetrahedron structures, and the breakage events mainly happen near the contact region without clear fracture planes. The macroscopic breakage behaviours of different structures are closely related to their microscopic bond stress distributions during compression. The bond stress distributions at different nominal strains including compressive, shear and tensile components are calculated by Eqs. (4)e(8) and presented in Figs. 15 and 16. For all cases shown in Figs. 15 and 16, the peak stress rises and has a concentrated distribution until breakage event occurs. Once the breakage happens, the bond stress inside the structures relaxes, and the intact bond stress increases again by a subsequent compression. Therefore after breakage the bond stress values have more complex distributions for each structure. Specifically, for the spherical shaped tetrahedron structure, the stress distributions become wider. This tendency is more obvious for the “Orientation I” case from 16% to 40% strain (see Fig. 15 (a)). For the spherical shaped simple cubic structures shown in Fig. 16, the stress values have more concentrated distributions during the crushing process. These different stress distribution characteristics can be explained by different breakage patterns shown in Figs. 11 and 13. For the spherical shaped tetrahedron structures, the whole particle assembly breaks progressively from the contact area. The disordered particle movements and rearrangements (see Figs. 11 and 12) lead to a large scatter of bond stress distributions. By contrast, for the spherical shaped simple cubic structures, clear fracture planes are
induced and formed during compression. The contact bonds break in an oriented way, resulting in relatively concentrated bond stress distributions. The fore mentioned illustrations demonstrate that for both structures, when compressed in two separate orientations, they exhibit different deformation and breakage characteristics emanating from the relative positions of the bond networks. In addition, the breakage is influenced by the external shapes of agglomerates. For the cubic shaped structures and spherical shaped simple cubic structure, large cracks are easily formed and propagate into the whole structure. This is accompanied by significant load drops and velocity discontinuities along the fracture plane. For the spherical shaped tetrahedron structure, breakage is restricted to the contact area. The structure is crushed progressively with fluctuated platen forces and complex bond stress distributions. 4. Discussion and future work Agglomerates have complex breakage characteristics that are influenced by their structures and inter-particle bond properties. Using DEM simulations with TBBM contact model, detailed information about the bond stresses and velocity evolutions at particle scales was obtained for analysing the macroscopic deformation and breakage characteristics of agglomerates. The simulation results show that the bond orientations affect the agglomerate mechanical properties. Especially, for both tetrahedron and simple cubic structures, the “Orientation II” structures have higher modulus than samples tested under “Orientation I”. This is because the axially orientated bonds inside the “Orientation II” structures
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Fig. 11. Load-strain curves and breakage patterns of spherical shaped tetrahedron structures, the colour change from blue to red represents the increasing proportion of broken bond contacts for a given particle. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 12. Velocity profiles of spherical shaped tetrahedron structures.
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Fig. 13. Load-strain curves and breakage patterns of spherical shaped simple cubic structures, the colour change from blue to red represents the increasing proportion of broken bond contacts for a given particle. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 14. Velocity profiles of spherical shaped simple cubic structures.
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Fig. 15. Bond stress distributions of spherical shaped tetrahedron structures.
Fig. 16. Bond stress distributions of spherical shaped simple cubic structures.
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provide a stiffer response under compressive loadings. The breakage characteristics of agglomerates are affected by external shapes and internal microstructures. The cubic shaped agglomerates show various fracture planes depending on the bond orientations: shear planes for the “Orientation I” structure, and horizontal planes for the “Orientation II” structure. As for the spherical shaped agglomerates, the bond stress distributions are significantly different for different bond orientations. For the spherical simple cubic structured agglomerates, clear fracture planes are induced and formed, while the spherical tetrahedron structured agglomerates breaks progressively without fracture planes. Compared with the above mentioned regular structured agglomerates, real granules introduce more complexities, i.e. variations in particle size, shape, inter-particle bond distributions and internal cavities. The breakage characteristics of random structured agglomerates have been revealed by our previous experimental work [8]. As the TBBM model has a representative description of the bond mechanics, it provides a way to investigate the breakage characteristics of real granules. With well characterised morphology and internal structures by X-ray micro-tomography [31], it is possible to analyse the breakage strength of real granules by DEM simulations. In addition, the optimum granule mechanical properties can be achieved by tuning the internal bond networks. Agglomerates with stretch-dominated microstructures have enhanced stiffness effects, which has been presented in this work. By tailoring the microstructures, the maximum granule strength can also be obtained. In a recently proposed research, architected materials demonstrate damage-tolerant characteristics by mimicking the strengthening mechanisms of crystalline structures [32]. This design concept can be adopted for designing the agglomerate structures with maximum strength. Using 3D printing technology, experimental breakage results under identical conditions were obtained and compared with those of DEM simulation results. The comparison results between simulation and experiments demonstrate that for both structures i.e. cubic tetrahedral structure and spherical random structure, experimental and simulation results show similar macroscopic breakage patterns. However, as the current TBBM model can only describe the linear elastic and brittle manner of materials, more sophisticated model considering plasticity and ductile failure need to be developed to enable quantitative comparisons of breakage. Furthermore, although 3D printing has been used to fabric agglomerate structures with tuneable properties [7], for validating these numerical results at microscopic particle scales, new experimental techniques for visualising the bond stress distribution and particle velocity field inside the granule structure are still required in future. 5. Conclusions In this work, DEM simulations with TBBM bond model were applied to investigate agglomerate breakage subject to uniaxial compressive conditions. The effects of agglomerate structures and shapes on breakage characteristics are addressed. Conclusions are drawn from the work and given below. 3D printed agglomerates with tuneable properties were tested under identical conditions and compared with corresponding DEM simulations. The comparison results show that, for both agglomerate structures, the simulation and experiment have similar breakage patterns. The simulation predictions show that the deformations of agglomerates are influenced by their internal microstructure orientations and bond properties. For both tetrahedron structures and simple cubic structures investigated in this research, they have higher Young’s modulus values when tested under the “Orientation II” loading direction. Detailed forces, velocities and
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bond stress evolutions show the dependence of breakage patterns on agglomerate orientations and shapes. For the cubic shaped structures and spherical shaped simple cubic structures, the whole assemblies are crushed with fracture planes. By contrast, localised breakage was found in the spherical tetrahedron structures. Future work can be extended to investigate the breakage of realistic granules with heterogeneous structures. In addition, combining with the 3D printing technology used in our research, it offers a possibility of tailoring mechanical properties of agglomerates. List of symbols A D d E e F I K k L M N r u
q m n r s t
Area (m2) Diameter (m) Translational displacement (m) Young’s modulus (Pa) Coefficient of restitution Force (N) Second moment of area (m4) Stiffness (N m1) Stiffness constant (N m1) Length (m) Moment (N.m) Primary particle number Radius (m) Displacement vector (m) Rotational angle Static friction coefficient Poisson’s ratio Density (kg/m3) Compressive/Tensile strength (MPa) Shear strength (MPa)
Indices
a, b Agg b C min max p S T x, y, z
Ends of a single beam Agglomerate Bond Compressive stress Minimum Maximum Particle Shear stress Tensile stress Cartesian coordinates
Acknowledgements Ruihuan Ge’s PhD scholarship was supported by the China Scholarship Council (CSC). The support of DEM Solutions for providing EDEM licences for the purpose of this work is greatly acknowledged. References [1] D.J. Golchert, Application of X-Ray Microtomography to Discrete Element Method Simulations of Agglomerate Breakage, PhD dissertation, University of Queensland, 2003. [2] J. Subero, Impact Breakage of Agglomerates, PhD dissertation, University of Surrey, 2001. [3] H. Rumpf, The Strength of Granules and Agglomerate, Agglomeration, Interscience, New York, 1962, pp. 379e418. [4] K. Kendall, Agglomerate strength, Powder Metall. 31 (1988) 28e31. [5] J. Subero, M. Ghadiri, Breakage patterns of agglomerates, Powder Technol. 120 (2001) 232e243. [6] S. Antonyuk, S. Palis, S. Heinrich, Breakage behaviour of agglomerates and crystals by static loading and impact, Powder Technol. 206 (2011) 88e98. [7] R. Ge, M. Ghadiri, T. Bonakdar, K. Hapgood, 3D printed agglomerates for
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granule breakage tests, Powder Technol. 306 (2017) 103e112. [8] R. Ge, M. Ghadiri, T. Bonakdar, Z. Zhou, I. Larson, K. Hapgood, Experimental study of the deformation and breakage of 3D printed agglomerates: effects of packing density and inter-particle bond strength, Powder Technol. 340 (2018) 299e310. [9] K. Johnson, K. Kendall, A. Roberts, Surface energy and the contact of elastic solids, in: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, The Royal Society, 1971, pp. 301e313. [10] C. Thornton, Z. Ning, A theoretical model for the stick/bounce behaviour of adhesive, elastic-plastic spheres, Powder Technol. 99 (1998) 154e162. [11] C. Thornton, K. Yin, Impact of elastic spheres with and without adhesion, Powder Technol. 65 (1991) 153e166. [12] Z. Tong, R. Yang, A. Yu, S. Adi, H. Chan, Numerical modelling of the breakage of loose agglomerates of fine particles, Powder Technol. 196 (2009) 213e221. [13] C. Thornton, M. Ciomocos, M. Adams, Numerical simulations of agglomerate impact breakage, Powder Technol. 105 (1999) 74e82. [14] C. Thornton, M.T. Ciomocos, M.J. Adams, Numerical simulations of diametrical compression tests on agglomerates, Powder Technol. 140 (2004) 258e267. [15] L. Liu, K.D. Kafui, C. Thornton, Impact breakage of spherical, cuboidal and cylindrical agglomerates, Powder Technol. 199 (2010) 189e196. [16] C. Thornton, L. Liu, How do agglomerates break? Powder Technol. 143-144 (2004) 110e116. , Breakage of fractal agglomerates, Chem. Eng. Sci. 161 [17] X. Deng, R.N. Dave (2017) 117e126. [18] D.O. Potyondy, P.A. Cundall, A bonded-particle model for rock, Int. J. Rock Mech. Min. Sci. 41 (2004) 1329e1364. [19] Y. He, T. Evans, A. Yu, R. Yang, DEM investigation of the role of friction in mechanical response of powder compact, Powder Technol. 319 (2017) 183e190. [20] Y. He, Z. Wang, T. Evans, A. Yu, R. Yang, DEM study of the mechanical strength of iron ore compacts, Int. J. Miner. Process. 142 (2015) 73e81.
[21] A. Spettl, M. Dosta, S. Antonyuk, S. Heinrich, V. Schmidt, Statistical investigation of agglomerate breakage based on combined stochastic microstructure modeling and DEM simulations, Adv. Powder Technol. 26 (2015) 1021e1030. [22] S. Eckhard, M. Fries, S. Antonyuk, S. Heinrich, Dependencies between internal structure and mechanical properties of spray dried granuleseexperimental study and DEM simulation, Adv. Powder Technol. 28 (2017) 185e196. [23] S. Dale, C. Wassgren, J. Litster, Measuring granule phase volume distributions using X-ray microtomography, Powder Technol. 264 (2014) 550e560. [24] M. Dosta, S. Dale, S. Antonyuk, C. Wassgren, S. Heinrich, J.D. Litster, Numerical and experimental analysis of influence of granule microstructure on its compression breakage, Powder Technol. 299 (2016) 87e97. [25] N.J. Brown, J.-F. Chen, J.Y. Ooi, A bond model for DEM simulation of cementitious materials and deformable structures, Granul. Matter 16 (2014) 299e311. [26] N.J. Brown, Discrete Element Modelling of Cementitious Materials, PhD dissertation, University of Edinburgh, 2013. [27] J.S. Przemieniecki, Theory of Matrix Structural Analysis, Courier Corporation, 1985. [28] PolyJet Materials, available online at, http://www.stratasys.com/materials/ polyjet. [29] L. Wang, J. Lau, E.L. Thomas, M.C. Boyce, Co-continuous composite materials for stiffness, strength, and energy dissipation, Adv. Mater. 23 (2011) 1524e1529. [30] X. Zheng, H. Lee, T.H. Weisgraber, M. Shusteff, J. DeOtte, E.B. Duoss, J.D. Kuntz, M.M. Biener, Q. Ge, J.A. Jackson, Ultralight, ultrastiff mechanical metamaterials, Science 344 (2014) 1373e1377. [31] M. Dadkhah, E. Tsotsas, Influence of process variables on internal particle structure in spray fluidized bed agglomeration, Powder Technol. 258 (2014) 165e173. [32] M.-S. Pham, C. Liu, I. Todd, J. Lertthanasarn, Damage-tolerant architected materials inspired by crystal microstructure, Nature 565 (2019) 305.
Contents lists available at ScienceDirect
Powder Technology journal homepage: www.elsevier.com/locate/powtec
DEM analysis of compression breakage of 3D printed agglomerates with different structures Ruihuan Ge a, b, Lige Wang c, Zongyan Zhou a, * a
Department of Chemical Engineering, Monash University, Australia Birmingham Centre for Energy Storage (BCES), School of Chemical Engineering, University of Birmingham, UK c School of Engineering, University of Sheffield, UK b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 1 June 2019 Received in revised form 30 August 2019 Accepted 31 August 2019 Available online 5 September 2019
Understanding agglomerate breakage characteristics is of great importance in industries of particulate manufacturing. In this work, Discrete Element Method (DEM) combined with Timoshenko Beam Bond Model (TBBM) is employed to simulate the agglomerate breakage with different structures subject to compressive loads. The agglomerates are made of 4 mm diameter rigid primary particles connected by cylindrical inter-particle bonds. Using 3D printing technology, test agglomerates with tuneable properties are produced and experimentally crushed. The simulation results of agglomerate crushing show similar qualitative trends with experimental results under identical conditions. Different agglomerate types with well-defined bond properties are considered in numerical simulations, e.g. cube/spherical shaped agglomerates with regular packed structures. Detailed damage ratios, particle velocities and bond stress distributions are obtained and analysed by DEM simulations. The results demonstrate that the breakage is strongly influenced by the agglomerate internal bond network orientations and external shapes. © 2019 Elsevier B.V. All rights reserved.
Keywords: 3D printing Discrete element method (DEM) Agglomerates Breakage Bond model
1. Introduction Agglomeration process is widely used in pharmaceutical, food and chemical industries. During processing, the agglomerate can experience different stressing conditions which may lead to their deformation, attrition and/or fragmentation. To ensuring processing efficiency and high product quality, a detailed understanding of agglomerate breakage mechanisms is essential. The agglomerate mechanical properties are strongly dependent on its morphology, internal particle arrangements and interparticle bond properties [1,2]. Rumpf [3] and Kendall [4] have proposed theoretical models to describe the relationships between macroscopic agglomerate strength and its microscopic bonding forces. However, these two strength models cannot take the complex compositions and irregular shapes of realistic agglomerates into account. There have also been several established experimental approaches for measuring single agglomerate breakage. Using molded agglomerates with glass ballotini bonded by epoxy resin binders, Subero et al. [5] investigated the impact breakage of agglomerates with macro-voids, they concluded that the
* Corresponding author. E-mail address: [email protected] (Z. Zhou). https://doi.org/10.1016/j.powtec.2019.08.113 0032-5910/© 2019 Elsevier B.V. All rights reserved.
agglomerate breakage patterns change from localised damage to fragmentation, to multiple fragmentations with increased void fractions and impact velocities. Antonyuk et al. [6] studied the effect of binder contents on the agglomerate strength. Using cylindrical agglomerates made of g-Al2O3 primary particles, it is demonstrated that the agglomerate stiffness and strength vary proportionally with both the binder content and agglomerate size. Recently, we have proposed a new method for producing model agglomerates by 3D printing technology, enabling fully reproducible structures and inter-particle bond properties [7,8]. Apart from theoretical and experimental research, Discrete Element Method (DEM) simulation has been widely used for investigating agglomerate breakage. The main advantage of using DEM is that it can generate detailed force and velocity evolution information at microscopic particle scales. Substantial agglomerate breakage studies have been performed using Johnson-KendallRoberts (JKR) contact model which describes adhesive forces between fine powders [9e12]. Thornton et al. [13,14] studied the impact and compressive breakage of spherical agglomerates with random assembled internal structures. The results mainly show that the breakage pattern changes from disintegration to fragmentation with increased solid fractions or inter-particle contact densities. Furthermore, Thornton et al. [15,16] investigated the effect of agglomerate shapes, including spherical, cuboidal and
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cylindrical agglomerates on breakage characteristics and found that the contact regions adjacent to the impact site play important roles in determining agglomerate breakage characteristics. Recently, Deng et al. [17] studied the influences of fractal dimensions on agglomerate breakage. The results reveal that the damage ratio decreases with increased fractal dimensions ranging from 2.0 to 2.6. The Linear Parallel Bond (LPB) model [18e20] is considered for agglomerates bound by binders, in which primary particles are connected by idealised elastic solid bonds. Using LPB model, Spettl et al. [21] studied the influences of primary particle size distributions and stochastic microstructures on the agglomerate breakage. Eckhard et al. [22] investigated the spray dried agglomerates with defined internal porosities and shell thicknesses. Their results mainly revealed the agglomerate strength is influenced by the variability of microstructures and bond sizes. Nowadays X-ray micro-tomography has been applied to characterizing the particle position, binder and porosity distributions within realistic agglomerates [23]. Dosta et al. [24] used the LPB model to predict agglomerate breakage with their microstructures and inter-particle bond contents obtained from corresponding realistic granules by Xray micro-tomography. The comparison with the experimental breakage results revealed that the simulated breakage strengths showed a similar qualitative increase trend with increasing binder contents. As neighbour particles inside agglomerates are physically attached to each other by a cylindrical geometry which can be treated as a beam element, a newly developed Timoshenko Beam Bond Model (TBBM) bond model is applied to investigating agglomerate breakage in this research [25]. Previous work [1] has demonstrated that JKR model is not applicable for describing the properties of agglomerates with solid bonds, in which the bonded particles can undergo irreversible breakage with large deformation of the inter-particle bonds. Although the LPB model was applied for modelling bonded agglomerates, this model only resists the axial and shear forces. Compared with the above mentioned contact models, the TBBM model is desirable for transmitting normal, shear, bending and torsion movements at two bond ends, and is suitable to describe the deformation and failure behaviours of the stubby beams. In this study, we compared compressive breakage simulation results with corresponding experiments of 3D printed agglomerates. After that, using DEM simulations, we systematically investigated the deformation and breakage behaviours of agglomerates with different internal crystalline structures. The effects of external shapes, internal microstructures and bond properties on the evolutions of breakage, velocity and bond stresses were analysed in detail. The purpose of this work is to numerically investigate how the packing structure and well-defined inter-particle bond properties affect the agglomerate mechanical properties including deformation and breakage characteristics. 2. Methodology 2.1. Experimental protocol As illustrated in Fig. 1, two different agglomerate structures, i.e. cubic tetrahedral structured agglomerate and spherical random structured agglomerate, were designed and applied in experiments. The corresponding doublet dimension is shown in Fig. 1 (a). Fig. 1 (b) shows a cube shaped agglomerate with a tetrahedral internal structure. The cubic tetrahedral structured agglomerate has 91 primary particles. Each particle is 4 mm diameter (Dp ¼ 4 mm), and is connected by 2.6 mm diameter cylindrical inter-particle bonds (Db ¼ 2.6 mm). The inter-particle distance (centre to
centre) is L ¼ 4.25 mm. Fig. 1 (c) shows a spherical agglomerate with random internal structure. The spherical random structured agglomerate has 120 primary particles, all 4 mm in diameter (Dp ¼ 4 mm). This random structure was generated in a 25 mm diameter spherical space using particle factory function in EDEM software. During the generation process, a gravity field was assigned and changed in different directions for ensuring the particles filled the whole space. The generated random structure has a 44% packing void fraction. After generating the structure, particles within 4.3 mm of each other (centre to centre length) are considered to be joined by a 2 mm diameter bond (Db ¼ 2 mm). As the random structured agglomerate is densely packed, the average inter-particle distance (centre to centre) is L ¼ 4 mm. These two structures have also been used in our previous work, for representing two different classes of agglomerates, i.e. Regular structured agglomerate and spherical random structured agglomerate [7,8]. Different bond diameters of the two structures were chosen is mainly because of a limitation of the 3D printing technique. We need to ensure that the inter-particle bond size is big enough compared to the 3D printing resolution, and also is suitable to facilitate the support removal process after 3D printing. The agglomerates were produced using PolyJet 3D printing technology (Stratasys Objet 500). The primary particles were printed using a rigid polymer (VeroWhitePlus™) while the interparticle bonds were printed using a rubber-like material (DM 9895). Quasi-static compressive tests were performed on these 3D printed agglomerates. As shown in Fig. 1, agglomerates were crushed at a constant cross-head speed of 0.02 mm/s. The breakage process was filmed using a Nikon D7000 camera. To ensure the reliability and repeatability, all the tested samples were crushed with the printing layers perpendicular to the loading direction. 2.2. Discrete element method (DEM) simulation 2.2.1. Agglomerate structures DEM simulations of the agglomerate crushing were used to produce the two identical structures used in experiments. In addition, simulations were also performed on agglomerates with regular internal structures. As illustrated in Fig. 2, two different internal crystalline structures, tetrahedron structure and simple cubic structure were studied. The particle size is Dp ¼ 4 mm, and the bond geometry is L ¼ 4.25 mm, Db ¼ 2.6 mm. As the orientations of internal microstructures with respect to the loading direction may affect their deformation and breakage characteristics, two orientations were considered for each structure, namely “Orientation I” and “Orientation II” (see Fig. 2). In Fig. 2, the relative position of the contiguous particle sets inside the structures is depicted on the left side of each design to illustrate different orientations. The particle positions of different types were generated using MATLAB code and imported into the DEM simulation software. By implementing the TBBM contact model, a DEM simulation of agglomerate crushing was created. The crushing directions of these structures are also illustrated in Fig. 2. In this work, the orientation effect of agglomerate structures was only considered and investigated by DEM simulations. 2.2.2. The TBBM contact model DEM models the particle motions and rotations based on Newton’s second law. The interaction forces between primary particles are determined by the contact model. In this study, TBBM contact model [25] is used to describe the bond deformation between primary particles, in which a Timoshenko beam element is used to connect the centres of two primary particles. The bond ends share same degrees of motion freedom with corresponding
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Fig. 1. Schematic of breakage experiment setup (a) doublet inside agglomerate, and (b) cubic agglomerate with a tetrahedral internal structure (L ¼ 4.25 mm, Db ¼ 2.6 mm and Dp ¼ 4 mm), (c) spherical agglomerate with a random structure (L ¼ 4 mm, Db ¼ 2 mm and Dp ¼ 4 mm).
Fig. 2. Agglomerates with crystalline structures in terms of different orientations (the crushing direction is illustrated by down arrows).
particles. At each time step, the relationship between the bond displacement Δui and resulting forces ΔFi is related with the stiffness matrix [K]:
fDFg ¼ DFax ; DFay ;DFaz ;DMax ;DMay ;DMaz ;DFbx ;DFby ;DFbz DMbx ; DMby ;DMbz T (2)
fDFg ¼ ½K,fDug
(1)
where {DF} is the incremental force vector that contains six force (ΔF) and six moment (ΔM) increments at the two bond ends:
fDug ¼ Ddax ; Dday ;Ddaz ;Dqax ;Dqay ;Dqaz ;Ddbx ;Ddby ;Ddbz ;Dqbx ;Dqby ;Dqbz T
where the subscript a or b denotes the two bond ends, and x, y or z denotes the force direction in the local coordinate system. The superscript T in this equation denotes the transpose operation of this row vector. {Δu} is the incremental displacement vector that contains six displacements (d) and six rotation (q) increments at the two bond ends a and b:
(3)
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The stiffness matrix [K] is a 12 12 matrix containing six different stiffness constants to describe the axial, shear and bending forces of the beam. The stiffness constants are determined by the bond mechanical and geometry parameters including bond Young’s modulus (Eb), Possion’s ratio (nb), bond length (Lb) and diameter (db), which are given in the following section. Detailed formulations of the stiffness matrix is in [26,27]. The total internal forces Fi and moments Mi at the bond ends are calculated by summing the incremental forces ΔFi and moments ΔMi at each time step. The inter-particle bond behaves in an elasticbrittle way and it breaks once its maximum stress exceeds the given strength. Three failure criteria are considered in the TBBM model: compressive (sC), tensile (sT) and shear strength (t). The maximum stresses are determined according to beam theory and summarized as follows:
sCi
3. Results and analysis
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0 2 2 Fbx rb Miy þ M iz A i ¼ a; b ¼@ b Ib A
(4)
sCmax ¼ minðsC a ; sC b Þ 0
sTi ¼ @
Fbx Ab
þ
rb
(5)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 M 2iy þ M2iz A i ¼ a; b Ib
(6)
sTmax ¼ maxðsT a sT b Þ tmax
jMax jrb 4 ¼ þ 2Ib
(7)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F 2ay þ F 2az
(8)
3Ab
Eq. (8) is used to determine the maximum shear stress tmax at the bond ends. As the twisting moments and shear forces in the equation are equal and opposite at two bond ends, the maximum shear stress tmax is calculated at the a end. 2.2.3. DEM simulation parameters The simulations were carried out using EDEM 2.7 software. Input parameters for the DEM simulation setup including particle and inter-particle bond properties are listed in Table 1. The particle and bond parameters are taken from PolyJet digital 3D printing materials [28], which are in accordance with our previous research
Table 1 Parameters used in DEM simulations.
Particle
Parameter
Description
Value
Dp
Diameter (m) Inter-particle static friction Inter-particle restitution Density (kg/m3) Poisson’s ratio Young’s modulus (GPa) Length (m) Diameter (m) Young’s modulus (MPa) Poisson’s ratio Compressive strength (MPa) Tensile strength (MPa) Shear strength (MPa) Steel-particle static friction Steel-particle restitution Density (kg/m3) Poisson ratio Shear modulus (GPa)
0.004 0.3 0.9 1200 0.3 1.0 0.00425/0.004 0.0026/0.002 50e200 0.4 50 10 10 0.3 0.9 7800 0.3 70
mpp epp
rp np Bond
Steel platen
Ep Lb Db Eb
nb sC sT t msp esp
rs ns
Gs
[7,8]. The primary particle is a rigid material (VeroWhite polymer), while the inter-particle bond represents a relatively soft material (DM 9895). The bond strength parameters represent the properties of soft bond material DM 9895 that are obtained from the datasheet offered by the vendor [28]. The bond Young’s modulus Eb is 100 MPa in breakage simulations. In addition, the effect of bond Young’s modulus values Eb ranging from 50 to 200 MPa on the compressive modulus of the agglomerate structures Eagg is also addressed. The mechanical properties of steel plates are referred from the material library of EDEM software. The time-step of this simulation is 1 107 s. The compressive breakage simulation setup is the same as experiments. The sample was loaded by moving the upper plate with a constant 2 mm/s loading rate. The compressive loads of the upper plate were recorded, and detailed force and velocity evolutions at particle scales were obtained and analysed.
3.1. Comparison of simulation and experimental results The experimental compressive curves of cubic tetrahedral structured agglomerate and spherical random structured agglomerate are shown in Fig. 3. The simulated compressive breakage patterns of these two structures are compared with the corresponding experimental results, as illustrated in Figs. 4 and 5. The simulations indicate the damage ratio (i.e. the fraction of broken bonds to the initial bond contacts for a single particle) by changing its colour from blue to red, where blue means that no contacts have been broken for this particle and red means that all the original particle contacts are broken. The average damage ratio for the whole agglomerate (i.e. the fraction of the total broken bonds to the initial total bond amount) during the compression process is illustrated in Fig. 6. Fig. 4 refers to the breakage process of the cube shaped agglomerate with tetrahedral structure (the “Orientation I” case). The experiments and simulations show similar breakage patterns. At the beginning of the compression, i.e. displacement <3 mm (13% strain), the agglomerate mainly shows elastic deformation without damage. After this elastic deformation stage, local damage appears and the agglomerate deforms along a 45o plane. Subsequently, with the increased displacement, the breakage expands from the local region to the whole particle assembly. The experimental compressive curve in Fig. 3 (a) and damage ratio evolution in Fig. 6 clearly show the two stages during the compression breakage progress. Initially the contacts break in the local region at 3 mm displacement (13% strain). After that, extensive slip takes place with a small number of bonds broken. This is followed by a further contact breakage near 8 mm displacement (30% strain), which stands for the whole assembly has failed along slip planes. Fig. 5 illustrates the breakage process of the spherical agglomerate with random structure. The snapshots of the simulation case show that minor damage appears first near the contact region at around 2 mm of displacement (8% strain). At approximately 3 mm of displacement (12% strain), the agglomerate fractures into two large fragments. Similar breakage behaviours can be observed for the experimental camera stills, in which the agglomerate breaks into two hemispheres. The fracture is accompanied by a load drop in the load-displacement curve shown in Fig. 3 (b). From Fig. 6 (b), the damage ratio experiences a sudden increase near 3 mm displacement (12% strain), which corresponds to the occurrence of fracture. This fracture pattern has been reported in previous research on the breakage of densely packed agglomerates [14,16]. The qualitative agreement between the simulation and experimental results confirms the validity of TBBM model for investigating agglomerate breakage characteristics. In the following,
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Fig. 3. Experimental compressive curves (a) Cubic tetrahedral structured agglomerate (Orientation I case) (b) Spherical agglomerate with random internal structure.
Fig. 4. Simulated and experimental breakage results for a cube shaped agglomerate with ordered, tetrahedral internal structure as a function of the displacement of the upper test platen.
detailed evolutions of damage ratios, particle velocities and bond stresses are investigated using DEM combined with the TBBM contact model.
3.2. Deformation and breakage of cubic shaped structures The deformation and breakage characteristics of cubic shaped agglomerates with different crystalline structures are analysed and shown in Figs. 7-10. The relative orientations of the microstructure with respect to the loading directions are illustrated in each figure, by using a single unit cell inside the particle assemblies. At the initial loading stage, the structures show elastic deformations. The Young’s modulus of agglomerate structures Eagg is defined as:
Eagg ¼
kagg ,Lagg Aagg
(9)
where kagg the agglomerate structure stiffness obtained from the compressive force curves. Lagg and Aagg are the height and crosssectional area of the cubic sample, respectively. The Young’s modulus Eagg, as calculated from DEM simulations, are plotted for particle numbers of N in Fig. 7. The number of particles N was varied by keeping the same aspect ratio of the structure, i.e. the cubic structure size increases equal proportionally along the x, y and z axis in a Cartesian coordinate system. In Fig. 7, the Young’s modulus increases with particle number for the tetrahedron structure and reduces for the simple cubic structure. According to the definition in Eq. (9), as the structure stiffness kagg increases with increased particle number, the different variation
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Fig. 5. Simulated and experimental breakage results for the spherical agglomerate with random internal structure as a function of the displacement of the upper test platen.
Fig. 6. Damage ratio evolutions for each agglomerate design (a) cubic tetrahedral structured agglomerate (Orientation I case) (b) spherical agglomerate with random internal structure.
tendency for these two structures can be explained by the dimension differences induced by different internal structures. In addition, with increased particle numbers N, it can be seen that for both structures the predicted modulus approaches an asymptotic value and are independent of N. As illustrated in Fig. 7, the asymptotic values of Young’s modulus for the tetrahedron structure and simple cubic structure are 2.75 MPa and 35 MPa, correspondingly. In the following Figs. 8-10, we adopt particle number N > 2500 for the tetrahedron structure, and N > 1000 for the simple cubic structure. Fig. 8 shows the effects of different bond Young’s modulus Eb and internal structure orientations on the Young’s modulus Eagg of each cubic shaped structure. In both cases, the structure Young’s modulus increases with increased inter-particle bond modulus. Furthermore, for both structures, at any given bond Young’s modulus, the “Orientation II” structure provides a higher modulus
value than “Orientation I”. This difference can be explained by the unit cell orientations as illustrated in Fig. 8. The structure carries the loads by compression, bend or shear deformations of the beam elements inside the structures. Under uniaxial compression direction, the axially orientated beams parallel to the loading directions can give a stiffer response of the whole structure. This effect is more pronounced for the simple cubic structures. As illustrated in Fig. 8 (b), the “Orientation I” case carries the loads by bending and shear deformations of the beam elements. While when loaded in “Orientation II”, as this arrangement shows a pure stretchdominated deformation [29,30], it has a relatively higher modulus value. Figs. 9 and 10 show the stress-strain curves and corresponding breakage patterns of different structures and loading orientations. The total bond stresses including compressive, shear and tensile component with respect to the nominal strain are calculated and
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Fig. 7. Dependence of the compressive modulus Eagg of cube shaped structures on the number of particles (N), the bond modulus Eb ¼ 100 MPa.
Fig. 8. The compressive modulus Eagg of cubic shaped structures considering different bond Young’s modulus Eb and internal structure orientations.
presented. The nominal strain is obtained from the corresponding compressive displacement divided by the sample height (Lagg). In these two figures, the stress-strain curves show a sudden stress change stage during compression which corresponds to the breakage point of the structures. Typical breakage patterns of each structure after the breakage point are depicted on the right side of each figure. The particle colors change from blue to red, representing the increased damage ratio of bond contacts. Fig. 9 (a) shows that the “Orientation I” tetrahedron structure breaks after a nominal strain of 10%. The fracture initially happens at the contact region to the test platen, and then it propagates to the whole particle assembly along a 45o diagonal plane. The “Orientation II” tetrahedron structure breaks at 6% nominal strain with different breakage patterns. As depicted in Fig. 9 (b), after the initial local breakage near the contact region, the broken bonds gradually extend to the middle plane of the structure accompanied by a slight increase of the total bond stress from 8% to 14% nominal strain. Compared with “Orientation I”, the “Orientation II” structure breaks at a lower strain (6% nominal strain) and stress level. As
shown in Fig. 8, the compressive modulus of “Orientation II” structure (Eagg ¼ 5.3 MPa) is higher than that of “Orientation I” (Eagg ¼ 2.8 MPa), indicating the stress level of the inter-particle bonds is higher at the same strain. As the inter-particle bonds of these two cases have same bond strength values (listed in Table 1), breakage event happens at a lower strain for “Orientation II” structure. Fig. 10 shows breakage characteristics of cubic shaped simple cubic structures. For the “Orientation I” simple cubic structure, the breakage happens along shear planes at around 7% nominal strain. A breakage pattern of two fracture planes can be observed along the diagonal line (see Fig. 10 (a)). For the “Orientation II” case, the whole particle assembly is under normal stress at the initial elastic deformation stage. As illustrated in Fig. 10 (b), with the increased nominal strain, the structure shows buckling behaviours and breaks along a horizontal plane. From the stress-strain curve, it can be seen that the bond stress inside this simple cubic structure changes from pure compression to more complex distributions after the breakage point at around 6.5% nominal strain. Especially,
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Fig. 9. Stress-strain curves and breakage patterns of cubic shaped tetrahedron structures, the colour change from blue to red represents the increasing proportion of broken bond contacts for a given particle. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
during deformation stage, the tensile stresses are negative as the bonds is under pure compressive stress. After the breakage event, the bond tensile stresses become positive, indicating the occurrence of tensile deformation and failure inside the structure. 3.3. Deformation and breakage of spherical shaped structures As illustrated in Figs. 11-16, compression breakage tests were also simulated for spherical shaped agglomerates. The spherical shaped structures were prepared by generating the particle assemblies inside a certain spherical region. The primary particle numbers N of the spherical tetrahedron and simple cubic structure are 1000 and 800, respectively. The orientations of the microstructure on the breakage were considered and depicted in each figure. Simulation parameters and setups are kept the same as used for the cubic shaped structures reported above. Fig. 11 presents the load-strain curves and breakage patterns of spherical shaped tetrahedron structures. From the load-strain curves, it can be observed that the first breakage event occurs at an applied load of 100 N for “Orientation I” case, and 150 N for “Orientation II” case. After then, for both cases, the compressive loads increase with nominal strains and reach an almost identical magnitude of 300 N after 15% nominal strain. The fluctuations of compressive loads indicate that local breakage happens during compression. It can be seen from Fig. 11 that for both cases, breakage happens progressively near the contact region adjacent to the test platen, and there is no clear fracture plane observed with compression continued to 40% nominal strain. Fig. 12 demonstrates the velocity evolution profiles of spherical shaped tetrahedron structures during compression. The particle velocities are represented by vectors with directions. As shown in Fig. 12, the particles
move downwards, and some disordered vectors appear near the contact region, indicating the rearrangement of primary particles and occurrence of breakage events. Fig. 13 refers to the compression test curves and breakage patterns of spherical shaped simple cubic structures. The load-strain curves show that compressive loads of “Orientation I” are much less than that of “Orientation II”. As shown in Fig. 13, for “Orientation I” case, the first breakage happens at 7% nominal strain with a wedge-shaped broken bonds region formed adjacent to the test platen. With the increased nominal strain, the load slightly increases to around 400 N at about 17% nominal strain. This is followed by a crushing of the whole particle assembly after 20% nominal strain with slip fracture planes observed from the snapshots (See Fig. 13 (a)). In contrast, when the structure is crushed according to “Orientation II” loading direction, the compression loads quickly rise to around 1400 N, reaching the first breakage point of around 6% nominal strain. After the first breakage event, additional breakages can be observed from the load-strain curves (see Fig. 13). The snapshots in Fig. 13 (b) illustrate that the whole structure falls apart along a vertical fracture plane following on the initial local breakage near the contact region. The particle velocity evolutions during compression is also recorded in the simulation and shown in Fig. 14. In this figure, significant velocity discontinuities can be observed as the particles move vertically downwards, indicating the formation and movement of fracture planes, i.e. inclined shear planes for “Orientation I” while a cleavage-like failure plane for “Orientation II”. Similar breakage patterns of this spherical simple cubic structure have also been observed in our previous experimental research [7]. Compared with the velocity profiles of spherical shaped tetrahedron structures in Fig. 12, velocity discontinuities and induced
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Fig. 10. Stress-strain curves and breakage patterns of cubic shaped simple cubic structures, the colour change from blue to red represents the increasing proportion of broken bond contacts for a given particle. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
fracture planes are easily formed in the simple cubic structured agglomerate. The velocity vectors indicate the relative movement of fragments. By contrast, in Fig. 12, the velocity discontinuity cannot propagate through the tetrahedron structures, and the breakage events mainly happen near the contact region without clear fracture planes. The macroscopic breakage behaviours of different structures are closely related to their microscopic bond stress distributions during compression. The bond stress distributions at different nominal strains including compressive, shear and tensile components are calculated by Eqs. (4)e(8) and presented in Figs. 15 and 16. For all cases shown in Figs. 15 and 16, the peak stress rises and has a concentrated distribution until breakage event occurs. Once the breakage happens, the bond stress inside the structures relaxes, and the intact bond stress increases again by a subsequent compression. Therefore after breakage the bond stress values have more complex distributions for each structure. Specifically, for the spherical shaped tetrahedron structure, the stress distributions become wider. This tendency is more obvious for the “Orientation I” case from 16% to 40% strain (see Fig. 15 (a)). For the spherical shaped simple cubic structures shown in Fig. 16, the stress values have more concentrated distributions during the crushing process. These different stress distribution characteristics can be explained by different breakage patterns shown in Figs. 11 and 13. For the spherical shaped tetrahedron structures, the whole particle assembly breaks progressively from the contact area. The disordered particle movements and rearrangements (see Figs. 11 and 12) lead to a large scatter of bond stress distributions. By contrast, for the spherical shaped simple cubic structures, clear fracture planes are
induced and formed during compression. The contact bonds break in an oriented way, resulting in relatively concentrated bond stress distributions. The fore mentioned illustrations demonstrate that for both structures, when compressed in two separate orientations, they exhibit different deformation and breakage characteristics emanating from the relative positions of the bond networks. In addition, the breakage is influenced by the external shapes of agglomerates. For the cubic shaped structures and spherical shaped simple cubic structure, large cracks are easily formed and propagate into the whole structure. This is accompanied by significant load drops and velocity discontinuities along the fracture plane. For the spherical shaped tetrahedron structure, breakage is restricted to the contact area. The structure is crushed progressively with fluctuated platen forces and complex bond stress distributions. 4. Discussion and future work Agglomerates have complex breakage characteristics that are influenced by their structures and inter-particle bond properties. Using DEM simulations with TBBM contact model, detailed information about the bond stresses and velocity evolutions at particle scales was obtained for analysing the macroscopic deformation and breakage characteristics of agglomerates. The simulation results show that the bond orientations affect the agglomerate mechanical properties. Especially, for both tetrahedron and simple cubic structures, the “Orientation II” structures have higher modulus than samples tested under “Orientation I”. This is because the axially orientated bonds inside the “Orientation II” structures
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Fig. 11. Load-strain curves and breakage patterns of spherical shaped tetrahedron structures, the colour change from blue to red represents the increasing proportion of broken bond contacts for a given particle. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 12. Velocity profiles of spherical shaped tetrahedron structures.
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Fig. 13. Load-strain curves and breakage patterns of spherical shaped simple cubic structures, the colour change from blue to red represents the increasing proportion of broken bond contacts for a given particle. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 14. Velocity profiles of spherical shaped simple cubic structures.
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Fig. 15. Bond stress distributions of spherical shaped tetrahedron structures.
Fig. 16. Bond stress distributions of spherical shaped simple cubic structures.
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provide a stiffer response under compressive loadings. The breakage characteristics of agglomerates are affected by external shapes and internal microstructures. The cubic shaped agglomerates show various fracture planes depending on the bond orientations: shear planes for the “Orientation I” structure, and horizontal planes for the “Orientation II” structure. As for the spherical shaped agglomerates, the bond stress distributions are significantly different for different bond orientations. For the spherical simple cubic structured agglomerates, clear fracture planes are induced and formed, while the spherical tetrahedron structured agglomerates breaks progressively without fracture planes. Compared with the above mentioned regular structured agglomerates, real granules introduce more complexities, i.e. variations in particle size, shape, inter-particle bond distributions and internal cavities. The breakage characteristics of random structured agglomerates have been revealed by our previous experimental work [8]. As the TBBM model has a representative description of the bond mechanics, it provides a way to investigate the breakage characteristics of real granules. With well characterised morphology and internal structures by X-ray micro-tomography [31], it is possible to analyse the breakage strength of real granules by DEM simulations. In addition, the optimum granule mechanical properties can be achieved by tuning the internal bond networks. Agglomerates with stretch-dominated microstructures have enhanced stiffness effects, which has been presented in this work. By tailoring the microstructures, the maximum granule strength can also be obtained. In a recently proposed research, architected materials demonstrate damage-tolerant characteristics by mimicking the strengthening mechanisms of crystalline structures [32]. This design concept can be adopted for designing the agglomerate structures with maximum strength. Using 3D printing technology, experimental breakage results under identical conditions were obtained and compared with those of DEM simulation results. The comparison results between simulation and experiments demonstrate that for both structures i.e. cubic tetrahedral structure and spherical random structure, experimental and simulation results show similar macroscopic breakage patterns. However, as the current TBBM model can only describe the linear elastic and brittle manner of materials, more sophisticated model considering plasticity and ductile failure need to be developed to enable quantitative comparisons of breakage. Furthermore, although 3D printing has been used to fabric agglomerate structures with tuneable properties [7], for validating these numerical results at microscopic particle scales, new experimental techniques for visualising the bond stress distribution and particle velocity field inside the granule structure are still required in future. 5. Conclusions In this work, DEM simulations with TBBM bond model were applied to investigate agglomerate breakage subject to uniaxial compressive conditions. The effects of agglomerate structures and shapes on breakage characteristics are addressed. Conclusions are drawn from the work and given below. 3D printed agglomerates with tuneable properties were tested under identical conditions and compared with corresponding DEM simulations. The comparison results show that, for both agglomerate structures, the simulation and experiment have similar breakage patterns. The simulation predictions show that the deformations of agglomerates are influenced by their internal microstructure orientations and bond properties. For both tetrahedron structures and simple cubic structures investigated in this research, they have higher Young’s modulus values when tested under the “Orientation II” loading direction. Detailed forces, velocities and
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bond stress evolutions show the dependence of breakage patterns on agglomerate orientations and shapes. For the cubic shaped structures and spherical shaped simple cubic structures, the whole assemblies are crushed with fracture planes. By contrast, localised breakage was found in the spherical tetrahedron structures. Future work can be extended to investigate the breakage of realistic granules with heterogeneous structures. In addition, combining with the 3D printing technology used in our research, it offers a possibility of tailoring mechanical properties of agglomerates. List of symbols A D d E e F I K k L M N r u
q m n r s t
Area (m2) Diameter (m) Translational displacement (m) Young’s modulus (Pa) Coefficient of restitution Force (N) Second moment of area (m4) Stiffness (N m1) Stiffness constant (N m1) Length (m) Moment (N.m) Primary particle number Radius (m) Displacement vector (m) Rotational angle Static friction coefficient Poisson’s ratio Density (kg/m3) Compressive/Tensile strength (MPa) Shear strength (MPa)
Indices
a, b Agg b C min max p S T x, y, z
Ends of a single beam Agglomerate Bond Compressive stress Minimum Maximum Particle Shear stress Tensile stress Cartesian coordinates
Acknowledgements Ruihuan Ge’s PhD scholarship was supported by the China Scholarship Council (CSC). The support of DEM Solutions for providing EDEM licences for the purpose of this work is greatly acknowledged. References [1] D.J. Golchert, Application of X-Ray Microtomography to Discrete Element Method Simulations of Agglomerate Breakage, PhD dissertation, University of Queensland, 2003. [2] J. Subero, Impact Breakage of Agglomerates, PhD dissertation, University of Surrey, 2001. [3] H. Rumpf, The Strength of Granules and Agglomerate, Agglomeration, Interscience, New York, 1962, pp. 379e418. [4] K. Kendall, Agglomerate strength, Powder Metall. 31 (1988) 28e31. [5] J. Subero, M. Ghadiri, Breakage patterns of agglomerates, Powder Technol. 120 (2001) 232e243. [6] S. Antonyuk, S. Palis, S. Heinrich, Breakage behaviour of agglomerates and crystals by static loading and impact, Powder Technol. 206 (2011) 88e98. [7] R. Ge, M. Ghadiri, T. Bonakdar, K. Hapgood, 3D printed agglomerates for
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