Three-dimensional discrete element modelling of

Powder Technology 337 (2018) 119–126

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Three-dimensional discrete element modelling of three point bending tests: The effect of surface energy on the tensile strength Simone Loreti, Chuan-Yu Wu ⁎ Department of Chemical and Process Engineering, University of Surrey, Guildford GU2 7XH, UK

a r t i c l e

i n f o

Article history: Received 2 February 2017 Received in revised form 14 November 2017 Accepted 3 December 2017 Available online 12 January 2018 Keywords: DEM Three point bending Surface energy Tensile strength Agglomerate

a b s t r a c t Understanding the dependence of the strength of agglomerates on material properties, interfacial properties and structure of the agglomerate is critical in many processes involving agglomerates. For example, for manufacturing of pharmaceutical tablets and pellets with dry granulation, understanding the relationship between the ribbon properties and the properties of granules is critical in controlling the granulation behaviour, and the ribbon properties (e.g. tensile strength and density distribution) is determined by the material properties of feed powders, interfacial properties between particles and the process condition, which determine the structure of ribbons. This study aims to investigate the effect of surface energy and porosity on the bending strength of pharmaceutical ribbons, for which three-dimensional discrete element modelling with a cohesive particle model based upon the JKR theory was performed. Simulations were carried out using specimens of various porosities and surface energies. The dependence of the bending strength on the surface energy and the ribbon porosity was examined. It was found that there is a strong correlation between the bending strength with the porosity and the surface energy. In particular, the bending strength is proportional to the surface energy and is an exponential function of the porosity. © 2017 Published by Elsevier B.V.

1. Introduction In pharmaceutical technology, ribbons and flakes are the intermediate products in the manufacturing of tablets and pellets, which are produced through roll compaction of feed powders and then fragmented into granules using mills. The microscopic and mechanical properties of ribbons and flakes are hence of fundamental importance in controlling the granular properties. A common test to characterise the mechanical properties of ribbons is the three-point bending test (3−PBT) that measures the bending strength σT [1]. In recent years the discrete element method (DEM) was employed to simulate the bending test for several purposes: calibration of material properties [2–4], modelling of mechanical behaviour of beams [3–5], and investigation of the fracture process in bending tests [6, 7]. In these studies, the bonded-particle model (BPM) was generally used. For instance, Zhang et al. [4] used the bonded-particle model, proposed by Potyondy and Cundall [8], and modelled the 3−point bending test and the single edge-notch bending test (SENB), in order to calibrate the SiC ceramic material. When the Particle Flow Codes, i.e. PFC2D and PFC3D [8, 9], were used, numerical calibration of material properties is generally required to derive microscopic parameters that were difficult to measure experimentally [10]. In this process, microscopic parameters ⁎ Corresponding author. E-mail address: [email protected] (C.-Y. Wu).

https://doi.org/10.1016/j.powtec.2017.12.023 0032-5910/© 2017 Published by Elsevier B.V.

are first assigned in numerical simulations that mimic small-scale experiments. Then, through an iterative trial-and-error process, these parameters are tuned until the numerical simulations produce the same macroscopic quantities measured in the experiments, such as Young's modulus or Poisson's ratio. Through numerical calibration, Zhang et al. [4] derived the material properties of SiC ceramic to be employed in modelling SiC-C laminates, i.e. the Young's modulus E, the Poisson's ratio ν, the bending strength σT, the fracture toughness K IC, the unconfined compression strength σc, and the Brazilian tensile strength σt. They also simulated three-point bending tests of SiC-C laminates with weak interfaces of graphite, and examined the process of the crack growth under various test conditions. Varying the number of weak interfaces and the bond strength between the particles, they observed an increase of the bending strength with the increasing number of weak interfaces. They referred this effect to as the “toughening mechanism”. In addition, they found that the fracture energy of the SiC-C laminates increased monotonically as the number of weak interfaces increased. However, in their study, the crack propagation was only examined qualitatively from two dimensional DEM simulations. Calibration of poly-crystalline SiC with the bonded particle model (BPM) was also performed by Tan et al. [3], who simulated different tests, such as the unconfined compressive test, the Brazilian test, the 3−point bending test and SENB, using PFC2D. They calibrated the mechanical properties of polycrystalline SiC, such as the Young's modulus, the Poisson's ratio, the bending strength and the fracture toughness, and then

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simulated the scratching test and the cutting process of ceramics using the calibrated parameters. In order to calibrate the material properties of agglomerates of bonded particles, Hare et al. [2] performed the three-point bending test experimentally and numerically. In their study, the numerical simulations were performed using the software EDEM, in which an inter-particle bonded contact model described in Brown et al. [11] was employed. They considered two ribbon formulations: one with density 4,000 kg/m3 and a second one with a density 7,000 kg/m3. Both formulations had a Poisson ratio of 0.25 and a shear modulus of 0.1 GPa. The bond strength and the Young's modulus in DEM were calibrated by comparing the numerical results of the three-point bending simulations with the corresponding experimental measurements. Ribbons with the calibrated bond strength were then employed to explore the size reduction process in milling dominated by impact and shearing. Wolff et al. [5] modelled the mechanical behaviour of ceramic beams in 3− and 4−point bending tests using DEM. They prepared cuboid beams of randomly distributed ceramic particles bonded together using the BPM model [11]. Beams with an average packing density of 63% were considered. The Hertz-Mindlin-Tsuji contact law was used for contact modelling, while a linear-elastic solid bond was introduced to model the polymeric binder between the particles. They investigated how the loading speed, the loading scheme and the bond stiffness affected the modulus of elasticity and observed that the modulus of elasticity varied almost linearly with the bond stiffness, but did not depend on the loading speed nor the position of the supports. Tarokh and Fakhimi [7] investigated the development of fractures in quasi-brittle materials with specific localised micro-cracks. They examined the effect of particle size on the width of the fracture process zone that was the region of damage around the crack tip from the two-dimensional 3−point bending simulations. In their study, the BPM was also used and a tension softening contact bond model was introduced to mimic the behaviour of the fracture process zone in quasibrittle materials like rocks (sedimentary rocks as sandstone) or concrete. The grains were represented as rigid circular particles interacting through normal and shear springs. The “tension softening” parameter is represented by the ratio K n/Knp, where Kn is the normal stiffness and Knp is the slope of the softening line. Tarokh and Fakhimi [7] analysed different materials ranging from perfectly brittle (K n/Knp = 0) to less brittle (K n/Knp = 100) and found that the width of the process zone was a function of both the specimen and the particle size. They showed that DEM with the tension softening contact bond model could be used to explore the crack propagation (in loading condition) and the effect of particle size on the width of the fracture process zone of quasi-brittle materials. Nevertheless, only a two-dimensional discrete element model with circular particles was used. In the above-mentioned investigations, the bonded-particle model (BPM) was generally used, in which artificial bars were introduced to connect a pair of particles and could transmit force and torque between particles. A model calibration is generally required in order to ensure the numerical modelling can represent the mechanical response of a given material. Even so, the importance of the interfacial properties (such as the surface energy) and the structure of the ribbons cannot be explored. In this study, a different inter particle contact model will be used. The particles are connected through attractive forces induced by the interfacial surface energy and the classical JKR theory was used to describe the cohesive interactions. Furthermore, according to Etzler and Pisano [12], the effect of the surface energy on the tensile strength of ribbbons is not understood yet. Therefore, the aim of this study is to examine how the interfacial energy and porosity affect the mechanical properties of ribbons, especially the bending strength.

frictional-elastic particles [13, 14] the JKR-Thornton models for the autoadhesive particles [15,16] were used. The detailed description of the implementation of these models into DEM can be found in Thornton and Barnes [17]; Thornton and Yin [16]. Only adhesive contacts without plastic deformation were considered. An agglomerate made of 1460 polydispersed spheres was created with material properties of Mannitol Pearlitol SD 200 used in Mirtič and Reynolds [18] (see Table 1); the spheres are of sizes between 77 μm and 312 μm mimicing the particle size distribution (PSD) measured in the experiments and showed in Fig. 1. This PSD was divided into five segments accordingly to the empirical 68–95–99.7 rule, also called the three sigma rule [20], to create a polydisperse system. For each segment, the weighted arithmetic mean diameter was taken as the representative diameter and used in the DEM modelling. Table 2 lists the fi ve representative diameters and the corresponding number of spheres in each segment used in the DEM modelling. The parallelepiped agglomerate was constructed using confining walls as follows: i) consolidation and ii) relaxation. During consolidation, spherical particles were generated in a region limited by walls (see Table 1) gravitational forces were then applied, making particles settle to the bottom of the region. Once the particles deposited, a piston moved downwards to compress the spheres to form the agglomerate. During consolidation, no adhesive bonding at the contacts was introduced, i.e. the surface energy was zero, while the particle-particle and particle-wall friction were considered. To complete the agglomeration preparation, the particle deformation was recovered during relaxation, where two of the four vertical lateral walls containing the agglomerate were released, while the other two lateral walls and the horizontal ones were fi xed. The surface energy was introduced at the beginning of the second step to make the particles into adhesive contact, with a high initial value γ = 300 J/m2. The surface energy was gradually reduced to γ = 0.01 J/m2 during relaxation. The entire process of agglomeration production, i.e. consolidation and relaxation, was performed with 11 piston displacements to obtain agglomerates with different porosities. The piston height was varied from h = 1.193 mm to h = 0.623 mm, leading to an increase of the pressure on the agglomerate and a decrease of porosity from ϵ = 0.369 to ϵ = 0.088. For clarity, the process of relaxation can b e illustra ted using th e temporal evolution of the assembly porosity as shown in Fig. 2. The symbols represent the val ues of po rosity at each comp utational iter ation, with the iteration = 0 representing the m oment when two of t he four vertical lateral wal ls were released and the relaxation commenced. The different colours and symbols in Fig. 2 indicate different cases of conso lidation, whic h can be define d as the pressure P exerted from the piston on the assembly at the iteration = 0 (see the legend of Fig. 2). For all cases considered, the evolution of porosity during the phase of relaxation was similar. When the lateral vertical walls were removed, the assembly expanded rapidly, leading to a dramatic increase in the porosity (about 10 % − 15% for each case) before a peak was reached. Then, the porosity gradually decreased and stabilised after around 1.0 × 10 7 iterations. As mentioned previously, the surface energy was introduced at t he iteration = 0 and gradually decreased to obta in ribbons with different values of surface energy ( see th e top x-axis in Fig. 2). The variation of porosity during the relaxation can be explained with the balance between the elastic resistance, which tends to keep the particles apart under

Table 1 Material Properties for Mannitol-200SD and walls used in DEM. The Young's modulus for the mannitol comes from Bassam et al. [19].

2. DEM model

Material type

Poisson's ratio

Density (kg/m3)

Friction coefficient

In this study, to model a 3-point bending test, a cuboid shaped agglomerate was created using spherical particles in DEM, in which contact laws based on the Hertz-Mindlin-Deresiewicz models for

Young's modulus (GPa)

Mannitol-200SD Walls (Steel)

12.2 210

0.30 0.30

1470 7850

0.30 0.30


Fig. 2. Evolution of the porosity during the relaxation. The different cases of consolidation are indicated by the piston pressure P at iteration = 0.

Fig. 1. The particle size distribution for Mannitol-200SD.

loading, and the surface energy, which holds them together. Immediately after the release of the lateral walls, the elastic resistance was dominant over the surface energy, leading to the assembly expansion and a rapid increase of the porosity. The porosity increase continued until the elastic force was balanced by the adhesive force between the particles, i.e. a steady state (i.e. an almost constant porosity) was reached. For each case of consolidation, the representative values of porosity used in the present study were calculated as the ave rage among the po rosities in the range between 1.0 × 107 and 3.0 × 107 iterations. The simulated assembly dimensions were proportional to the one experi mentally used for the three-point bending test in Mirtič and Reynolds [18], but scaled down by a factor fs = 5, i.e. L = 3 mm, b = 1.2 mm and h = 1.193~0.623 mm. Therefore, the ribbon's dimensions used in the experiments were Lexp = 15 mm , bexp = 6 mm and h exp = 5 mm. Once the agglomerate was prepared, 3− PBT simulations were performed by removing the confining walls and introducing two supports and a loading beam, as illustrated in Fig. 3. The distance between two supports was set as l = 2.4 mm. The bending tests were simulated by lowering gradually a loading beam, to compress the specimen until the agglomerate breaks. The tensile strength σT was calculated from the maximum normal force of the loading beam FP, as follows [18]:

σT

3 FP l 2bh

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value of porosity, the 3−PBT was repeated 13 times by varying the surface energy.

3. Results and discussion 3.1. Breakage patterns Fig. 4 shows typical force evolution curves for specimens of the same porosity (0.271) but different interfacial energies during the 3-point bending tests. All the curves with a surface energy from γ = 20 J/m2 to γ = 200 J/m2 show a peak, after a steep increase of the loading force, while the variations for the curves γ = 0.1 J/m2 and γ = 1 J/m2 are too small to be visible in Fig. 4a, therefore they are plotted in Fig. 4b. Once the agglomerate breaks, the force drops down to zero. The peaks represent the maximum loads required to break the specimen, they are higher and sharper for specimen with a higher surface energy and they are used to calculate the tensile strength using Eq. (1). Fig. 6a and b show the front and top view of the specimen (ϵ = 0.271) before the loading, respectively. Fig. 6c shows the front view of the agglomerate when the maximum loading force is reached, while Fig. 6d illustrates the front view of a typical breakage pattern. Two failure planes are developed in proximity to the supports, resulting in three large daughter fragments.

1

2

In this study, the virtual 3−point bending test was performed with specimens produced using 11 porosities and 13 values of surface energy, i.e. a total of 143 cases were simulated. It means that for each

Table 2 Representative particle diameters Φi and number of particles Ni for each segment i used in the polydisperse system in DEM. 3

Segment i

Φi (μm)

Vi (μm )

Qi (%)

Ni

1 2 3 4 5

76.882 126.509 189.095 261.635 311.589

2.379e + 05 1.060e + 06 3.540e + 06 9.377e + 06 1.584e + 07

0.037 0.167 0.641 0.113 0.042

443 454 521 35 7

Fig. 3. The three-point bending system used in the DEM simulations. L is length, h is height and b is thickness of the agglomerate.

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Fig. 5. Fracture energy evaluation for different values of surface energy, but the same ribbon porosity, = 0.271.

specimen with t he same porosit y, the tensil e strengt h increases with the increase of surface energy. A re-plot of the data shown in Fig. 6 in a semi-logarithmic scale is shown in Fig. 8. It can be seen that the tensile strength varies approximately four orders of magnitude between σT = 10−2 N/mm2 and σ T = 102 N/mm2 for the surface energies considered. It is interesting to note that the data shown in Fig. 8 can be approximated with straight lines. In order to infer the relationship between the tensile strength σT and the porosity ϵ for each value of surface energy, a non-linear regression was applied. The best fitting functions for all datasets are superimposed in Fig. 8 using black solid lines. These functions have the same exponential form as the RyshkewitchDuckworth equation [23]: σT Fig. 4. Typical time evoluti on of the loading beam force for a ribbon with a porosity of = 0.271.

3.2. Fracture energy The fracture energy G f for the three point bending test was calculated according to Santos et al. [22] and Javier Malvar and Warren [22]: Gf

U A

2

where U is the area under the loading-displacement curve and A is called the ligament area. In a three-point bending test with a single notch, the ligament area can be written as A = b(h − a0), where a 0 is the notch depth, while b and h are the thickness and the height of the ribbon, respectively. Since the notch was not reproduced in this study, the ligament area was calculated simply as A = bh (a0 = 0). The area under the loading-displacement curve was calculated with the trapezoidal numerical integration (trapz function in MATLAB) for specimens with different values of surface energy, but the same porosity (ϵ = 0.271). Fig. 5 shows the fracture energy G f calculated using Eq. (2), for several values of surface energy (but the same porosity ϵ = 0.271), indicating a linear relationship between the fracture energy and the surface energy. 3.3. Relat ionship between the tensile strength, the porosity and the surface energy The variation of the tensile strength with the porosity is shown in Fig. 7 for various surface energies. For the simulations with the same value of surface energy, the tensile strength decreases with the increase of porosity. For example, when the surface energy γ = 200 J/m2 , the tensile strength decreases from σ T = 43 N/mm2 at the porosity ϵ = 0.088 to σ T = 10 N/mm2 at the porosity ϵ = 0.38. While for the

σ 0e

−K

3

where σ0 is the tensile strength of the material at zero porosity (i.e., ϵ = 0) and K is a constant representing the “bonding capacity”. When the tensile strength σT is plotted against the surface energy, as shown in Fig. 9, it is clear that it is proportional to the surface energy, i.e., σT

4

∝a

This is in broad agreement with the Rumpf [24 ] and Kendall [25] models. The Rumpf [24] theory of agglomerate strength was based on the van der Walls force H between two identical spherical particles of diameter d. Summing these forces among the particles in the agglomerate, he proposed a relationship between tensile strength, porosity and the van der Walls force: σT

9 1−ϵ H ϵ 8 d2

5

Eq. (5) implies that the tensile strength σT is proportional to the van der Walls force H. Kendall [25] derived a model for the agglomerate strength by summing the total fracture energy, i.e. the total energy required for fracture, instead of the total force used by Rumpf [24]. The Kendall [25] theory was based on fracture mechanics analysis of agglomerates composed of smooth adhesive elastic spheres. His model relates the tensile strength to the fracture energy c, the interfacial energy , the assembly porosity, the diameter of the spheres and the macroscopic flaw in the assembly of length c, from which the fracture starts when σT

15 6 1−ϵ dc

4

c 1 2

5 6 1 6

6

The main difference between these two models is the failure mechanism: Rumpf [24] assumed the failure occurs simultaneously along the

123


2

2

Fig. 6. Typical breakage pattern for ribbon with porosity = 0.271 and surface energy = 1 J/m and = 0.1 J/m .

agglomerate, while in Kendall [25] theory the failure is dominated by crack propagation. Eqs. (5) and (6) can be expressed using different bonding forces FB between particles, like van der Walls adhesion, capillary liquid bridge, JKR adhesion, viscous liquid bridge, and solid bridge [26]. The equation of the Rumpf [24] model can be rewritten to consider the bonding force as follows [26]: σT

9 1−ϵ FB 2 ϵ 8 d

7

while the equation representing the Kendall [25] model can be given as [26]: σT

3 75 1−ϵ

4

FB d

2

8

Since the simulations in the present study were performed with the JKR contact model [15], Eqs. (7) and (8), representing the Rumpf [24]

Fig. 7. Tensile strength as a function of porosity.

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Fig. 8. Tensile strength as a function of porosity in the semi-logarithmic scale. Six values of surface energy are shown, with the correspondent exponential fitting.

and Kendall [25] models, can be calculated with the bond force FB from the JKR adhesion theory: FB

3πdWA 8

9

Therefore the tensile strength σT in Eqs. (7) and (8) is linearly proportional to the surface energy expressed with the work of adhesion WA = 2γ, for particles of the same material. It is clear from Eqs (7)–(9) that both Rumpf [24] and Kendall [25] models predict that the tensile strength σT is proportional to the surface energy γ, as illustrated in Fig. 9. Substituting Eq. (4) into (3), the tensile strength σT can be written as a function of the porosity ϵ and the surface energy γ, i.e. σT

a e −K

10

Since Eq. (3) was proposed by Ryshkewitch [27] and Duckworth [23] as an empirical relationship between the tablet tensile strength and the tablet porosity, some attempts were made to give a theoretical explanation to the equation's parameters σ0 and K [12, 28]. Andersson [28] provided an analytical basis to Eq. (3), showing the exponential constant, i.e. the bonding capacity, has a strong dependency on the pore shape and the orientation of the pores. He also found the exponential constant is independent of the pore size, but the surface energy was not considered in the theoretical expression. Etzler and Pisano [12] derived a model to predict the tensile strength of binary mixtures from the Ryshkewitch-Duckworth parameters, starting from a principle of the adhesion science, i.e. the tensile strength of a material is dependent on the surface energy [29]. However, their model did not show any relationship with the surface energy but, they argued that the surface

Fig. 9. The variation of tensile strength with surface energy.

Fig. 10. Tablet tensile strength as a function of surface energy for 5 pharmaceutical materials [31].

energy played a role in the tensile strength. A direct link between the tensile strength and the surface energy was indicated by El Gindy and Samaha [30] and Luangtana-Anan and Fell [31]. El Gindy and Samaha [30] explored the correlation between the tensile strength of pharmaceutical compacts and the surface energy of seven powders and demonstrated there was a direct relationship between the tensile strength of the compact and the surface energy: the compact becomes stronger with the increase of surface energy, based upon their experimental data (see Fig. 10). Luangtana-Anan and Fell [31] explored the effect of adhesive forces of seven different materials in the form of tablets, and examined the relationship between the tensile strength of tablets and the Hamaker constant, which is related to the surface energy. They showed that the van der Walls forces contributed significantly to the tensile strength of tablets. The studies of Etzler and Pisano [12], El Gindy and Samaha [30] and Luangtana-Anan and Fell [31] showed a correlation between the tensile strength of tablets and the surface energy. Nevertheless, no mathematical expression was proposed to relate the tensile strength σ T, nor the tablet porosity. Nevertheless, Eq. (10) obtained from the present DEM simulations gives a direct mathematical relationship between the tensile strength and the surface energy as well as the porosity. Furthermore, the bonding capacity in the Ryshkewitch-Duckworth equation can be obtained, and the parameters σ 0 can be related to the surface energy as follows: σ0

a

11

To verify Eq. (10) , the tensile strength shown in Figs. 7–9 was divided by the surface energy and the quantity σT /γ was plotted against

Fig. 11. The variation of tensile strength with porosity. The black lines represent the fitting of simulation data.


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This is in broad agreement with the data reported in the literature. The obtained empirical equation can be used to obtain the surface energy of various materials. However, it shows that this approach significantly overestimated the surface energy value, implies that the van der Waals forces are not the only mechanisms in governing the tensile strength of the agglomerates. Acknowledgments This work was supported by the IPROCOM Marie Curie initial training network, funded through the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme FP7/2007–2013/ under REA grant agreement No. 316555. Special thanks to Alexander Krok and Filippo Loreti for useful discussion. References Fig. 12. Fitting of Eq. (10) of the tensile strength measured experimentally for several materials. Black lines represent the fitting of Eq.(9) for each material.

porosity for different materials in Fig. 11. It is evident that a unified curve can be obtained for most of the surface energies considered, except for γ = 0.01 J/m2, the lowest surface energy considered in this study. The coalescence of all data into a master curve indicates that the parameters a and K in Eq. (10) only depend on material properties. 3.4. Experimental validation Eq. (10) was used to describe various single-component powder systems, i.e. microcrystalline cellulose (MCC), hydroxypropylmethyl cellulose (HPMC) and starch, described in [32], and mannitol (Pearlitol SD 200) employed in Mirtič and Reynolds [18]. The experimental data of tensile strength evaluated by Mirtič and Reynolds [18]; Wu et al. [32] were hence fitted with Eq. (10) (see Fig. 12) using the multivariate fit and the resulting parameters a, K and γ were estimated for each material. The obtained parameters are presented in Table.3, together with the literature values of surface energy for all the materials considered. The values of surface energy reported in literature are: 37.6 mJ/m 2 for MCC Avicel PH101 [33], 48.4 mJ/m2 for HPMC [34, 35], 58.7 mJ/m2 for starch [34, 35] and 72.07 mJ/m2 for mannitol [36]. Using the empirical equation (Eq. (10)), the surface energy appears overestimated significantly. This implies that Van der Waals forces (that directly related to the surface energy) are not the only mechanism governing the bonding strength of agglomerates, as discussed by Rumpf [24]. For the bonding dominated by van der Waals forces, the obtained empirical equation can be used to approximate the correlation between the tensile strength with surface energy and porosity. 4. Conclusions A three-dimensional discrete element model (DEM) for autoadhesive particles was used to model ribbons with a wide range of porosities and to perform three-point bending tests. The three-point bending test was used to obtain the correlation between tensile strength σT , porosity ϵ and surface energy γ of the particles. An empirical equation was obtained, which is in a similar form to the Ryshkewitch-Duckworth equation but considered the effect of both the porosity and the surface energy.

Table 3 Parameters estimated using Eq. (10) for various powders. 2

2

Material type

a

K

γ simul.(J/m )

γliterat. (mJ/m )

MCC HPMC Starch Mannitol

2.4095 3.6433 2.9610 4.8637

6.8347 8.7205 10.5344 8.5721

10.5238 3.6311 3.0119 5.1183

37.6 48.4 58.7 72.07

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Three-dimensional discrete element modelling of

Three-dimensional discrete element modelling of

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