Discrete Element Simulation of the Behavior of Bulk Granular Material During Truck Braking

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EC 23,1 Discrete element simulation of the behavior of bulk granular material during truck braking 4 Shu-chun Zuo, Yong Xu and Quan-wen Yang Department of Applied Mechanics, China Agricultural University, Beijing, Received July 2004 Revised May 2005 People’s Republic of China, and Accepted June 2005 Y.T. Feng Civil & Computational Engineering Centre, School of Engineering, University of Wales Swansea, Swansea, UK

Abstract Purpose – To simulate the dynamic feature of bulk granular material (such as agricultural products) during sudden braking of a truck by applying discrete element method. Design/methodology/approach – The bulk granular material was modeled by the discrete element approach, in which the spherical elements were used to represent the granular particles; the interaction between two in-adhesive particles was modeled by Hertz for normal interaction, and by Mindlin and Deresiewicz for tangential interaction; the interaction between two particles with adhesion was modeled by the JKR theory for normal interaction, and by Thornton’s theory for the tangential interaction. Different initial conditions (braking speeds/accelerations) were considered. The dynamic system was numerically solved by the central difference based explicit time integration, and the dynamic impact forces were recorded to further analysis. Findings – The computation predicted that the resultant dynamic force acting upon the front wall behaves in four stages, i.e. increasing, plateau, sharp increasing and drop with damped ﬂuctuation. It was observed that, the shorter the breaking time is, the faster the force reaches its peak, and the greater the peak value is. The phenomenon was in good agreement with physical principals and common knowledge. Research limitations/implications – It is an application of the discrete element method and, therefore, no important contribution is made to advance the methodology. Practical implications – The proposed modeling approach may serve as a useful tool for advanced design of trucks. Originality/value – This paper is the ﬁrst to apply the advanced discrete element method to the problem concerned. Keywords Land transport, Linear motion, Mechanical properties of materials Paper type Research paper

1. Introduction Transport of particulate materials with trucks is common in various industries and road freight. Such a granular matter has its special characteristics, e.g. flow-ability and Engineering Computations: International Journal for fluctuation in physical properties. Therefore, its macroscopic responses and behavior Computer-Aided Engineering and may differ from other solid structured products during braking, pitching or turning, Software Vol. 23 No. 1, 2006 pp. 4-15 This work is funded by the Natural Science Foundations of China under Grant No. 10372113. q Emerald Group Publishing Limited 0264-4401 The authors gratefully acknowledge the encouragement from Dr C. Thornton at Birmingham DOI 10.1108/02644400610638943 University. and these responses could significantly affect the dynamic load on the truck frame and Discrete element on the operation by drivers. Consequently it may cause potential strength problems or simulation even accidents. Therefore, the estimation of the workload involving the additional dynamic load is of great importance. In addition, inter-particle interaction of a particle bed is also important if the carrying goods are agricultural products, such as fruits or other vulnerable particle-shaped products, since damage the breakage will result in the loss of commercial profits. 5 To the authors’ best knowledge, up to date the workload value of the particulate materials used by the conventional truck design is still considered as a point mass and how to deal with the dynamic effect is unclear. It seems lack reports on the microscope mechanism of granular matter and the description of the dynamic interactions between the assembly and the truck box due to the difficulties inherent in the conventional theory which assumes the particulate material as a continuum media. The discrete element method (DEM), originated by Cundall and Strack (1979a), is a numerical method for analyzing a discrete system, which consists of an assembly of blocky or particulate elements. Over the past 20 years the method has been significantly developed and successfully applied to solve many practical problems (Couroyer et al., 2000; Owen and Feng, 2001; Han et al., 2002). Thornton (1991, 1993) presented a new model for adhesive spherical particles and modified the TRUBAL program originated by Cundall and Strack (1979b) into a new version known as Aston-TRUBAL or GRANULE. In this work, an assembly of spherical particles within a truck box in motion was simulated under a sudden braking using GRANULE. The effects of inertia dynamic force with different braking speeds and inter-particle interactions were considered.

2. Discrete element method models The GRANULE code for a spherical dry particle system provides mainly two contact models. The interaction between two in-adhesive particles was modeled by Hertz ( Johnson, 1985) for normal interaction, and by Mindlin and Deresiewicz (1953) for tangential interaction. The interaction between two particles with adhesion was modeled by the JKR theory (Johnson et al., 1971) for normal interaction, and by Thornton’s theory (1991, 1993) for the tangential interaction, which is a combination of the theory by Mindlin and Deresiewicz (1953) and the theory by Savkoor and Briggs (1977). In reality the term “adhesion” is normally used to describe the adhesive property of very small particles with surface energy. However, the adhesive interaction model can be used to take into account of the non-spherical and “sticky” effects, since the element is modeled as perfect round sphere. Besides adhesion, the effects of other properties, e.g. friction, were also considered and their significance to the modelling was also investigated. Note that in the context of DEM, each particle is assumed rigid and that its deformability is modeled via penalty springs. Therefore, Young’s modulus of particles in the model is used to determine the penalty value. Also a central difference explicit time integration is employed to numerically solve the dynamic system of equations. The critical time step is selected automatically in the program. EC 3. Problem description 23,1 3.1 Problem and the related data Our study on the behavior of the particle assembly during braking was focused on the interaction between the particle bed and the wall of the truck box and the movement of the assembly. Here a truck box with 4.0 m in length, 2.0 m in width, and 1.0 m in height was considered. The discrete element model contains 102,000 mono-size, soft spherical 6 particles with a radius of 4.5 cm. The term “soft” refers to a kind of agricultural products with Young’s modulus of around 70 MPa, which is 1/1000 of the elastic property of the wall. Previous work by Xu et al. (2002) indicates that the use of a smaller material modulus in the model has insignificant effect on the physical behaviour of the system but could substantially reduce the simulation time due to the increase of the critical time step. The values of the selected parameters used in the modelling are listed in Table I. The interaction between a sphere and a wall is treated as the interaction between a sphere and another sphere with an infinite radius. In our DEM simulation, a proper workspace is needed to contain all the particles and allow them to move in all possible ways. The workspace is divided into a certain number of cubic cells so as to detect the movement of any particle and identify their contact status. The dimensions of the workspace are the key factor affecting the overall computing efficiency and, therefore, it is impractical to specify a very large workspace to allow the box to move with a normal speed and then to stop in a certain distance. In our approach, an equivalent transformation was employed to allow the truck box being stationary and the relative motion of the particles with respect to the box was analyzed. In the simulation the speed of a truck before braking was chosen as 72 km/h, and four cases of different braking time durations were selected and the corresponding braking distances and de-accelerations were listed in Table II.

Radius (particle) 0.045 m Density (particle) 1,200 kg/m3 Young’s modulus (particle) 70 MPa Poisson’s ratio (particle) 0.25 Friction (particle to particle) 0.35 Number of particles 102,000 Total mass of particles 5,726 kg Table I. Young’s modulus (wall) 70 GPa Properties of particle Poisson’s ratio (wall) 0.3 and wall Friction (wall to particle) 0.35

Case Braking time (s) Braking distance (m) De-acceleration (m/s2)

1 1 10 20 2 0.8 8 25 3 0.667 6.67 30 4 0.5 5 40 Table II. Kinetic properties Note: Motion speed before braking is 72 km/h 3.2 Initial preparation of the particle bed Discrete element The particles were generated randomly within the specified volume. A large number of simulation preliminary iterative cycles were performed under the gravity, during which the positions of the particles and contact links were repeatedly updated until all the particles settle down to form a bed. Then an additional acceleration due to the de-acceleration of the truck was assigned to each particle, the height of the simulation domain was reduced to the original value and the real cyclic computation began. It is 7 noted that a top wall was introduced to avoid the escape of the particles during the simulation. The initial particle bed is shown in Figure 1. During the simulation the “real time” state of the system was continuously fed back to an interface platform in the form of graphic output or data files. The graphic output includes particles profiles, force line graphs and velocity vectors. The walls could be made transparent so that the motion of particles can be observed conveniently. The force line is designed in such a way that the lines are laid along the orientation of force and the width of the line represents the magnitude of the force. Due to the fact that the graph for 3D force lines may be difficult to visualise, the force line graph can be cut into sections. Moreover, in order to trace the movement of some specific particles, their locations and velocities can be output to a data file during the whole course of the simulation for subsequent analysis.

4. Results and discussion 4.1 Static pressure on the front wall Once the particle bed is formed before braking, the static interactive contact forces between the particles and the box walls can be obtained. These data can be treated in a statistically averaging manner to obtain the distribution of the pressure or normal stress acting on the walls. It was found, as expected, that the pressure distributions

Figure 1. Initial particle bed before braking EC along the front wall, back wall and the both sidewalls are almost the same. The 23,1 pressure distributions along the front and side walls are shown in Figure 2, from which it can be seen that the pressure varies linearly. This obviously different pressure distribution comparing to the result of a deep silo is due to the small vertical depth with a limited number of particles. The static pressure proﬁle is useful for the purpose of comparison with the dynamic data. 8 4.2 Dynamic pressure distribution on the front wall The dynamic pressure on the front wall of the truck at certain time instances can be obtained in the same way as the static pressure. The evolution of the dynamic pressure distribution on the front wall at different time instances with an interval of 0.8 s is depicted in Figure 3, in which the corresponding time of each individual time label is summarized in Table III, and the label t0 corresponds to the initial static state.

Figure 2. Static pressure proﬁles on the front and side-walls

Figure 3. Dynamic pressure distributions on frontal wall for different braking times

Label Table III. t0 t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 Labels for different braking times Time (s) 0 0.076 0.152 0.227 0.265 0.303 0.379 0.455 0.531 0.607 0.682 0.724 It is notable from the ﬁgure that, as the braking time increases, the dynamic pressure is Discrete element not smoothly distributed along the wall surface but shows zigzag lines with different simulation magnitude. The similarity of these lines suggests that, using the DEM method with a relatively limited number of particles, local arching can be easily formed. Particularly, strong arching is observed on the corner of the lowest location of the front wall where the pressure drops signiﬁcantly due to the arching and the friction between the particle and the baseboard. This leads to a rise of the acting point of the resultant 9 force.

4.3 Dynamic force on the front wall The more important feature comes from the resultant dynamic force, denoted as Fd,on the front wall due to the inertia of the particles during braking. Such a force can be calculated by the summation of all the contact forces acting along the front wall at a certain time, and apparently provides the evidence of the dynamic response for the estimation of the strength of the truck. In order to describe the inertia effects more universally a dimensionless form of the dynamic force, fd, is introduced:

f d ¼ Fd=Mg where g is the gravitational acceleration and M is the total mass of the particle bed. The dynamic force Fd and the mass M can be calculated by: Xn Xn Fd ¼ N i; M ¼ mi i¼1 i¼1 in which Ni is the normal contact force between the front wall and the ith particle with mass mi,andn is the total number of the particles. The peak value of the dimensionless dynamic force kd ¼ f dmax is also termed the impact factor. The evolution of the dynamic force in terms of the dimensionless dynamic force fd vs braking time for the four acceleration cases is shown in Figure 4, from which it is clear that as the time increases, the variation of the dimensionless dynamic force fd for all the cases can be classiﬁed into four stages:

Figure 4. Dimensionless dynamic force fd with different de-accelerations EC (1) Monotonic increasing. At this stage the force increases monotonically to a certain 23,1 value. It reflects the squeezing between the front wall and the adjacent particles. (2) Plateau stage. After the first stage, there is a short period of time during which the force delays to increase and, therefore, a flat segment is formed. The shorter the braking time is, the greater the plateau value is, and the shorter the segment is. 10 (3) Sharp increasing. At this stage fd increases rapidly and reaches its peak value kd. The magnitude of the peak value varies depending on the braking time. The shorter the breaking time is, the faster the peak arrives, and the greater the peak value is. (4) Fluctuation and damping. At this stage, the factor quickly reduces from its peak value and begins to fluctuate with damping until a stable state reaches.

In order to explain the above phenomenon Figure 5(a-d), extracted from Figure 4, respectively, depict the four particle profiles corresponding to four time instances (0.02275, 0.11373, 0.32755 and 0.55346 s) at four different stages for the case of a de-acceleration of 25 m/s2. Note that the front wall is on the left side in the figures. At the first stage (Figure 5(a)), the pressure increases because the particles near the front wall hit the wall immediately resulting in a monotonic pressure increase. At the second stage (Figure 5(b)), the “front particles” began to rebound from the front wall, while the particles next to the “front particles” keep moving forward and impact with the “front particles”, which reduces the increase of the pressure and forms a plateau curve segment. At the third stage (Figure 5(c)), almost all the particles move forward that contributes to a continuous increase of the contact forces until their peak values are reached. At the final stage (Figure 5(d)), many particles in the box impact and rebound from each other to form a periodically oscillating contact force wave, resulting in the fluctuation and damping in the pressure profile until a steady state is reached. The most important feature in this four-stage response is the peak value of the dynamic force, which is represented by the impact factor. The values of the impact factor for the four cases with a braking speed of 72 km/h are listed in Table IV. It is illustrated that, when the braking duration time is 1 s with a de-acceleration of 20 m/s2, the impact factor at 0.297 s after braking is 2.658 times of the bed weight. While when the braking duration is 0.5 s, the impact factor at 0.22 s is 5.96 times of the weight. The relationship between the dynamic force and the braking time is further investigated by simulating the problem with more braking durations. The peak dynamic force and the corresponding time instance for different braking times are shown in Figure 6(a) and (b), respectively. It can be seen that the impact force increases almost linearly with the increase of the de-acceleration value, while the corresponding peak time instance initially decreases rapidly but appears to gradually approach to a constant value when the de-acceleration becomes large. This suggests that as the braking time decreases, there is a significant increase in the impact force but the peak time instance decreases not significantly when the braking duration is short.

4.4 Effects of particle size distribution on the dynamic force The particles used for the simulation in all the four cases are identical. In order to examine the effect of different particle sizes on the dynamic force, a comparative numerical study on a mono-size system and a multi-disperse one was carried out. The mono-disperse system consisted of 6,000 particles with a diameter of 0.115 m, while the Discrete element simulation

Figure 5. Particle proﬁles for the four stages with the braking acceleration of 25 m/s2

2 Case Braking duration (s) De-acceleration (m/s ) Time at peak force (s) kd 1 1 20 0.2957 2.6583 Table IV. 2 0.8 25 0.2654 3.5708 Inertia impact factor 3 0.667 30 0.2426 4.3751 (with initial speed 4 0.5 40 0.2199 5.9554 72 km/h) multi-disperse bed had three different sizes: 1,500 particles with a diameter of 0.120 m, 3,000 particles with a diameter of 0.115 m and 1,500 particles with 0.110 m in diameter. In this way the total mass in the both systems were maintained approximately the same value. Other parameters were taken exactly the same value as in the previous simulation. The simulated impact force distributions are shown in Figure 7. EC 23,1

Figure 6. Peak force and the correspondent time vs de-acceleration

From the plots it is evident that both systems exhibit the same dynamic characteristics as identiﬁed earlier and almost the same peak value of the dynamic force. A very small difference appears in the evolution of the dynamic force where the curve for the multi-disperse system is slightly delayed after the plateau stage. This is simply because the small particles within the main contact network of the large particles block the massive movement due to the inertia. These results suggest that on the whole the size distribution of the particles has little effect on the dynamic force and, therefore, the use of a mono-disperse system instead of multi-disperse systems is valid.

4.5 Effects of adhesion and friction on the dynamic force Friction and adhesion are two important physical properties affecting the dynamic force. In order to estimate the effects on the behavior of granule bed when braking, four numerical test cases (A – rough wall and in-adhesive, B – rough wall and adhesive, C – smooth wall and adhesive, D – smooth wall and non-adhesive) were carried out, and the coefﬁcients of friction and adhesive used are described in Table V. The variations of the dynamic factor for the four tests are shown in Figure 8. From the plots Discrete element simulation

Figure 7. Effects of size distributions on the dynamic force

Test case Wall friction Adhesion

A 0.35 0.0 B 0.35 200.0 J/m2 Table V. C 0.0 200.0 J/m2 Test for friction and D 0.0 0.0 adhesion

Figure 8. Effects of friction and adhesion on the dynamic force it can be seen that, for both the in-adhesive cases (A and D), when the wall is very smooth, the dynamic force is much greater than that of a rough wall and the oscillation in the box is also more signiﬁcant. On the other hand, for both the adhesive cases (B and C), the very large adhesion sticks the particles together, and, therefore, EC no notable plateau stage appears as in tests A and D. In addition the friction can 23,1 signiﬁcantly reduce the magnitude of the force and eliminate the oscillation due to the smooth wall. This means that for a very sticky and large frictional granular material, the effect on the dynamic force may be neglected.

4.6 Effects of density on the dynamic force 14 In order to study the effects of the mass density on the dynamic force we compared the current results with the new results obtained by doubling the density. The comparison for these two sets of the results is shown in Figure 9. It illustrates that the curves of the dynamic force have a similar shape, but the bed with a higher density has a greater peak force (about 1.2 times as much as the lower density) and a shorter but identical plateau segment (about 2 times as the plateau force value), therefore, the peak value responded quicker when the density is higher.

5. Conclusions The current paper has presented a discrete element approach for the numerical simulation of the dynamic response of a particle bed within a truck box during a sudden braking. It has been found that during the braking process the dynamic force on the front wall experiences four different periods, in which the transmission of contact forces play an important part. Particularly, the relationship between the peak value of the dynamic force and the braking time duration has been investigated in detail and the phenomena observed have been explained from a viewpoint at the micro-mechanics level. The results obtained may provide useful information for the examination of the strength of the truck frame and also for a proper design of this type of trucks. It should be pointed out that although the current discrete element model is a simpliﬁed representation of the actual problem considered, it provides a viable approach to the assessment of the dynamic force in transportation trucks and can also achieve an accuracy to a reasonable degree. Further analysis of the behavior of the

Figure 9. Effects of particle mass density on the dynamic force particle bed under vibrating is currently in progress, which might be another step Discrete element forward for achieving a better understanding of the dynamic process involved. simulation In addition, the methodology developed in the current work can be applied to other practical problems, such as railway transportation, and may be further extended to effectively solve the problems of similar nature.

References 15 Couroyer, C., Ning, Z. and Ghadiri, M. (2000), “Distinct element analysis of bulk crushing: effect of particle properties and loading rate”, Powder Technology, Vol. 109 Nos 1/3, pp. 241-54. Cundall, P.A. and Strack, O.D.L. (1979a), “A discrete numerical model for granular assembles”, Geotechnique, Vol. 29 No. 1, pp. 47-65. Cundall, P.A. and Strack, O.D.L. (1979b), “The distinct element method as a tool for research in granular media, Part II”, Report to the National Science Foundation, University of Minnesota. Han, K., Owen, D.R.J. and Peric, D. (2002), “Combined finite/discrete element and explicit/implicit simulations of peen forming processes”, Engng. Comp., Vol. 19 No. 1, pp. 92-118. Johnson, K.L. (1985), Contact Mechanics, Cambridge University Press, Cambridge, MA, pp. 84-104. Johnson, K.L., Kendall, K. and Roberts, A.D. (1971), “Surface energy and the contact of elastic solids”, Proc. R. Soc. Lond. A., Vol. 324, pp. 301-13. Mindlin, R.D. and Deresiewicz, H. (1953), “Elastic spheres in contact under varying oblique forces”, J. Appl. Mech., Vol. 20 No. 3, pp. 327-44. Owen, D.R.J. and Feng, Y.T. (2001), “Parallelised finite/discrete element simulation of multi-fracture solids and discrete systems”, Engng. Comp., Vol. 18 Nos 3/4, pp. 557-76. Savkoor, A.R. and Briggs, G.A.D. (1977), “The effect of tangential force on the contact of elastic solids in adhesion”, Proc. R. Soc. Lond. A., Vol. 356, pp. 103-14. Thornton, C. (1991), “Interparticle sliding in the presence of adhesion”, J. Phys. D: Appl. Phys., Vol. 24, pp. 1942-6. Thornton, C. (1993), “On the relationship between the modulus of particulate media and surface energy of the constituent particles”, J. Phys. D: Appl. Phys., Vol. 26, pp. 1587-91. Xu, Y., Kafui, K.D., Thornton, C. and Guoping Lian (2002), “Effects of material properties on granular flow in silo using DEM simulation”, Particulate Science and Technology, Vol. 20 No. 2, pp. 109-24.

Further reading Jones, C.S., Holt, J.E. and Schoor, I.D. (1991), “A model to predict damage to horticultural produce during transport”, J. Agric. Engng. Res., Vol. 50 No. 4, pp. 259-72. Peleg, K. and Hinga, S. (1986), “Simulation of vibration damage in produce transportation”, Trans. ASAE, Vol. 29 No. 2, pp. 633-41.

Corresponding author Yong Xu can be contacted at: [email protected]

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