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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2011, Article ID 123695, 9 pages doi:10.1155/2011/123695 Research Article Discrete Element Simulation of Elastoplastic Shock Wave Propagation in Spherical Particles M. Shoaib and L. Kari Marcus Wallenberg Laboratory for Sound and Vibration Research (MWL), Royal Institute of Technology (KTH), 100 44 Stockholm, Sweden Correspondence should be addressed to M. Shoaib, shoaibse@yahoo.com Received 18 October 2010; Accepted 15 June 2011 Academic Editor: Mohammad Tawﬁk Copyright © 2011 M. Shoaib and L. Kari. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Elastoplastic shock wave propagation in a one-dimensional assembly of spherical metal particles is presented by extending well- established quasistatic compaction models. The compaction process is modeled by a discrete element method while using elastic and plastic loading, elastic unloading, and adhesion at contacts with typical dynamic loading parameters. Of particular interest is to study the development of the elastoplastic shock wave, its propagation, and reﬂection during entire loading process. Simulation results yield information on contact behavior, velocity, and deformation of particles during dynamic loading. Eﬀects of shock wave propagation on loading parameters are also discussed. The elastoplastic shock propagation in granular material has many practical applications including the high-velocity compaction of particulate material. 1. Introduction using the micromechanical modeling of contact between particles. These studies focused on equivalent macroelastic The dynamic response of the granular media has become constitutive constants during dynamic loading. Similarly increasingly important in many branches of engineering. It experimental work using dynamic photoelasticity and strain includes material processing involving dynamic compaction gage are performed to investigate contact loads between par- and material processing, as well as acoustics and wave ticles both under static and dynamic loading [12, 13]. Sadd propagation in geomechanics. The granular matter shows et al. [12] perform numerical simulations to investigate the discrete behavior when subjected to static or dynamic eﬀects of the contact laws on wave propagation in granular loading [1–3]. The dynamic wave propagation in granular matter. Similarly Sadd et al. [13] use the discrete element media shows distinct behavior from the wave propagation method (DEM) to simulate wave propagation in granular in continues media [1]. Shukla and Damania [4] discuss the materials. Results of this study show wave propagation speed wave velocity in granular matter and shown experimentally and amplitude attenuation for two-dimensional assembly of that it depends upon elastic properties of the material spherical particles. However, this study is restricted to the and on geometric structure. Similarly, Shukla and Zhu [5] elastic range only while the material stiﬀness and damping investigate explosive loading discs assembly and found that constants used in the model are determined by photoelas- the force propagation through granular media depends on ticity. DEM was initially developed by Cundall and Strack impact duration, arrangement of the discs, and the diameter [14] and this numerical method has been widely used for of discs. Tanaka et al. [6]investigate numericallyand granular material simulations [15–17]. Diﬀerent engineering experimentally the dynamic behavior of a two-dimensional granular matter subjected to the impact of a spherical approaches are discussed in [18, 19] to model the behavior of granular matter using DEM. Dynamic compaction of metal projectile. To investigate dynamic response, many researchers [7– powder is also reported in the literature [20–24] and studies 11] have modeled the granular matter as spherical particles the distribution of stress, strain, and wave propagation. 2 Advances in Acoustics and Vibration die wall during loading-unloading-reloading stages. Here the compact is modeled as a one-dimensional assembly of spherical particles that indent each other. These particles are F F 2R 2R assumed to be materially isotropic and homogeneous while depicting elastoplastic material behavior. Equivalent elastic modulus E [38]isgiven by E = , (1) 2(1 − ν ) Figure 1: Schematic of two particles in quasistatic contact. h is the overlap and I is the distance between the centers of the two where E represents Young’s modulus and ν is Poisson’s ratio particles. of the material. The eﬀective radius R of the two particles in contact, here labeled 1 and 2 is determined by the relation 1 1 1 However, these studies treat the powder as a continuum and = + . (2) R R R 0 1 2 determine the material constants experimentally. Force transmission in spherical particles occurs in a As compaction proceeds, the particles overlap each other and chain of contacts, which is usually referred to as the force elastic normal force follows the Hertzian law chain. Force chains in granular matter have been widely ∗ 3/2 investigated, experimentally [25, 26] and in simulations [27, F = E R h , (3) e 0 28]. However these studies do not consider wave propagation velocity during loading. Liu and Nagel [29]and Jiaetal. where h denotes the indentation or overlap between particles, [30] found experimentally that sound propagates in granular as shown in Figure 1. In the plastic regime, as described by media along strong force chains. Somfai et al. [31]investigate Storak ˚ ers et al. [40, 41] for two spherical particles undergoing the sound waves propagation in a conﬁned granular system. plastic deformation, the strain hardening relationship is Recently Abd-Elhady et al. [32, 33] studied contact time and given as force transferred due to an incident particle impact while using the Thornton and Ning approach [34]and founda σ = σ ε , i = 1, 2, (4) good agreement between DEM simulation and experimental measurements. However, these studies are restricted to the where σ is a material constant, M is the strain hardening particle collision and are not modeling the shock wave exponent, and σ and ε are stress and strain in the uniaxial propagation during dynamic loading of particles. case. Normal contact force F is given by the relation [42] In the present work, DEM is used to simulate dynamic (2+M)/2 loading of a one-dimensional chain of spherical particles. F = ηh , (5) The contact between particles is modeled using elastic where and plastic loading, elastic unloading, and adhesion at contacts. Recently many researchers [35–39] used these 1−(M/2) 1−(M/2) 1−M 2+M (6) η = 2 3 πc σ R . contact models, however only to investigate diﬀerent aspects of static compaction of particulate matter. In the current Here σ is a material parameter and, for ideally plastic investigation, typical dynamic loading parameters are used, material behavior, invariant c = 1.43. By considering the which are commonly found in high velocity compaction material as perfectly plastic M = 0 while the particles having process. The 1D chain of spherical particles is chosen as a identical yielding stress preliminary step towards the understanding of elastoplastic shock wave propagation and its eﬀects during the entire σ = σ = σ , (7) 1 2 y loading process. Computer simulations reveal generation, transmission, and reﬂection of the elastoplastic shock wave normal contact force F can be written as [39] through the particles. The shock wave eﬀects on contact between particles, particle velocity, and its deformation are F = 6πc σ R h. (8) p y 0 investigated. Eﬀects of shock wave propagation on loading The contact radius a is deﬁned as [40] parameters are also investigated. In addition to trans- ducer design, earthquake engineering, and soil mechanics, 2 2 a = 2c R h, (9) elastoplastic shock propagation in particulate materials has many other practical applications including the high-velocity while contact stiﬀness compaction of powder material. k = 6πc σ R . (10) y 0 2. Basic Contact Equations The parameters F , h ,and a denote normal contact 0 0 0 This section summarizes the theories that describe the force, overlap, and contact radius, respectively, at the end of contact behavior between particles and between particles and plastic compaction process, before the load is removed. The Advances in Acoustics and Vibration 3 Dynamic load limit between a fully elastic unloading and a partially plastic unloading, as shown by Mesarovic and Johnson [42], is Hammer Tool Rigid wall π wE χ = , (11) 2π − 4 p a 0 0 Impact end where p is the uniform pressure at contact p = 3σ and w 0 0 y Hydraulic is the work of adhesion. In the present case, it is set to w = Container pressure 0.5J/m . During the unloading stage, the distance I between the Figure 2: One-dimensional dynamic particles compaction model. centers of the two particles which are pressed together is I = R + R − h + h , (12) 12 1 2 0 u mass of the dynamic load. Hydraulic pressure is maintained and the overlap during the complete compaction process. The dynamic compaction process is simulated by the h = h − h , (13) 0 u discrete element method. This numerical method is used by Martin and Bouvard [37] and other authors to simulate where indentation recovered h is given by [38, 43] static compaction. In the present work, DEM is used to extend fully developed contact models to simulate shock 2p a a 0 0 (14) wave propagation in a chain of spherical particles. Here, h = 1 − . E a each particle is modeled independently and interaction between neighboring particles is governed by contact laws During unloading, the contact radius a is determined as described in the previous section. This contact response from (14) and normal force F is given by [38, 42]as plays an important role in the use of DEM to simulate shock ⎡ ⎤ wave propagation through the particles. During calculations a a a ⎣ ⎦ F = 2p a arcsin − 1 − at time t + Δt,where t is previous time and Δt is the u 0 a a a 0 0 0 (15) time step, contact force between particles is calculated which determines the net force or compaction force F acting on 3/2 − 2 2πwE a , each particle. By using Newton’s second law, these resultant forces enable new acceleration, velocity, and position of each which can be written in terms of χ as [42] particle. At time t = 0, force, velocity, and position of each ⎡ ⎤ particle are known because it is the moment of the ﬁrst hit. F 2 a a a (t+Δt) ⎣ ⎦ = arcsin − 1 − The velocity v of a particle i at a time t +Δt is determined F π a a a 0 0 0 0 by adopting a central diﬀerence scheme as (16) 1/2 3/2 1 2 a F Δt − 4 1 − χ . (t+Δt) (t+Δt/2) i v = v + , (17) π π a i i m 2 In (15)and (16), the second term on the right hand side where the position x is given by gives the contact force due to adhesion traction. During the present work it has been realized that the second term must (t+Δt) (t+Δt/2) (18) x = x + v Δt. i i i be included in elastic and plastic loading equations when adhesion traction is considered. Both cases with and without During iterative calculations, the size of time step Δt adhesion traction are treated in the present study. plays an important role to ensure numerical stability. For problems of a similar nature, Cundall and Strack [14]have 3. Discrete Element Method proposed a relationship to calculate the time step which is further developed by O’Sullivan and Bray [44] for the central The system under study is the one-dimensional dynamic diﬀerence time integration scheme as compaction model shown in Figure 2. There is a chain of micron-sized identical, spherical particles aligned in a max (Δt) = f , (19) container with one end open and the other blocked. At the open end, these particles are in contact with a compaction tool which has the same diameter as those of the particles. where correction factor f = 0.01 for the present case, m is Friction between the particles and the container walls is the mass of the lightest particle, and k is the approximate not considered. To start the compaction process, hydraulic contact stiﬀness given by expression (10). This value of the pressure is used to accelerate the hammer which strikes the time step is shown to be suﬃcient to ensure numerical compact at a certain impact velocity. The hammer along with stability during the calculations. the compaction tool form the dynamic load. Compaction During the compaction process, particle contact energy is mainly determined by the impact velocity and goes through several loading, unloading, and reloading 4 Advances in Acoustics and Vibration −3 ×10 0.06 0.04 0.02 8 One cycle −0.02 −0.04 0 1 2 2.9 2.95 3 −4 −5 ×10 ×10 Time (s) Time (s) Particle 1 Particle 60 Particle 10 Particle 13 Particle 20 Particle 80 Particle 11 Particle 14 Particle 40 Particle 100 Particle 12 Particle 15 (a) (b) Figure 3: Elastoplastic shock wave during dynamic compaction. (a) Propagation and reﬂection during various cycles. (b) Shock front is shape preserving while amplitude decreases during propagation. sequences. In the beginning of compaction, contact force stress, therefore, during loading, these elements deform only is initially elastic for small values of contact radius a and elastically. In the simulation, particle numbering starts from it is given by (3). The contact force follows the same curve the compaction end. As a convention, resultant force, veloc- during unloading and reloading in the elastic regime. At ity,and displacement are taken positive from compaction larger contact radius, the contact becomes plastic and to dead end, that is, along the positive x-axis, otherwise contact force follows (8). The term relating the adhesion negative. Dynamic eﬀects during particle compaction like 3/2 −2 2πwE a is added in elastic and plastic equations elastoplastic shock wave propagation, particle contact behav- when considering adhesion traction. If the contact is ior, and particle velocity along with loading parameters are unloaded, normal force follows elastic unloading (15). investigated in this section. When contact is reloaded during unloading then it follows the same equation up to the value of the contact radius on which it was unloaded. Beyond this point, plasticity is 4.1. Elastoplastic Wave Propagation. The dynamic load trans- reactivated and (8) applies. ferred in particles is described using elastoplastic shock wave propagation variables like shock wave front velocity. The shock wave front is interpreted as the maximum absolute 4. Results and Discussion compaction force at a particular time while shock wave For simulation, one hundred aluminum particles of diameter velocity is deﬁned as the velocity of the wave front. The 2R = 100 μm are used. Dynamic compression load is movement of the shock wave from compaction to dead applied on the particles by supplying a hydraulic pressure end and then back to the compaction end is described of 13.5 MPa which gives the hammer an impact velocity of as one compaction cycle. As the hammer moves forward 10 m/s. This impact velocity along with the diﬀerent choices to compact the material, particles overlap each other and of loading mass results in a compaction energy of 1 J/g to thus contact forces are developed as a result of material 6.5 J/g. These loading parameters correspond to a typical stiﬀness and damping characteristics. The diﬀerence between high-velocity compaction process. The time step used is contact forces results in a net force on the particle. This net Δt = 2.3 ns which is estimated as explained in the previous force increases and the particle starts to move approximately section. The material properties of the aluminum particles with piston velocity after which this net force decreases and are density 2700 kg/m , yielding stress 146 MPa, Young’s eventually becomes zero. During this period, the shock is modulus 70 GPa, and Poissons’s ratio 0.30. The material also transferred continuously to the next particles. In one properties of the loading elements are density 7800 kg/m , compaction cycle, the shock travels from the ﬁrst particle to Young’s modulus 210 GPa, and Poissons’s ratio 0.35. The the last particle and then it is reﬂected back from the dead loading elements are made of steel with a high yielding end towards the compaction end. During the backward part Net force (N) Net force (N) Advances in Acoustics and Vibration 5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 1 2 3 4 5 0 1 2 −6 −4 ×10 ×10 Overlap (m) Time (s) Contact between particles 2-3 Contact between particles 2-3 Contact between particles 40-41 Contact between particles 40-41 Contact between particles 80-81 Contact between particles 80-81 (a) (b) Figure 4: Contact force between particles during loading-unloading process. (a) During compaction, contact is mainly in the plastic range except for elastic unloading at few points. (b) Contact force increases until the hammer stops. Oscillations during unloading are due to the adhesion between particles. 10 10 8 8 6 6 4 4 2 2 0 0 −2 −2 2 4 6 8 10 2 4 6 8 10 −5 −5 ×10 ×10 Time (s) Time (s) Particle 1 Particle 60 Particle 1 Particle 60 Particle 20 Particle 80 Particle 20 Particle 80 Particle 40 Particle 100 Particle 40 Particle 100 (a) (b) Figure 5: Velocity of the particles during the compaction process. (a) No adhesion. (b) With adhesion. of the cycle, net contact force on a particle becomes negative ﬁrst half cycle. Figure 3(b) shows the enlarged view of as shown in Figure 3(a). neighboring particles. It can be seen that wave front is The shock wave velocity is approximately 750 m/s (with approximately shape preserving as it propagates through adhesion) for the ﬁrst cycle and it decreases approximately the particles. However, wave amplitude decreases slightly 5% from one cycle to the next as in Figure 3(a).Periodin while shock wave passes from one particle to the next. It which shock passes through individual particle also changes is mainly due to energy loss in the plastic deformation. from one cycle to the next and it is about 1.5 μs for the In this particular case of a single chain of particles, shock Contact force (N) Particle velocity (m/s) Particle velocity (m/s) Contact force (N) 6 Advances in Acoustics and Vibration −5 −4 ×10 ×10 1.5 0.5 One cycle 0 1 2 0 1 2 −4 −4 ×10 ×10 Time (s) Time (s) Particle 2 Particle 60 Particle 2 Particle 60 Particle 20 Particle 80 Particle 20 Particle 80 Particle 40 Particle 99 Particle 40 Particle 99 (a) (b) Figure 6: (a) Deformation of all the particles approximately remains the same. (b) Displacement covered by the particles depends upon their position from compaction end. −5 −5 ×10 ×10 2.5 1.5 0.5 0 0 2 4 6 8 1 2 3 4 5 6 7 −5 −5 ×10 ×10 Time (s) Time (s) Piston Hammer velocity = 5m/s Hammer velocity = 15 m/s All particles Hammer velocity = 10 m/s Hammer velocity = 20 m/s (a) (b) Figure 7: Kinetic energy varies as shock wave propagates through the particles. (a) The hammer kinetic energy and collective kinetic energy of all the particles. (b) Variation in collective kinetic energy of particles with hammer velocity. wave velocity and wave front mainly depend upon material Contact history for diﬀerent particles is shown in Figure 4(a). properties and they are only slightly aﬀected by changing the Contact response between neighboring particles plays an loading mass or initial impact velocity. important role in transfer of mechanical energy through particles. The compaction process is initially elastic which 4.2. Particles Behavior during Compaction. This section de- remains for a very short time. Then particles are in plastic scribes particle contact behavior, particle velocity, and com- deformation where the overlap between particles increases linearly with contact force. However, contact force decreases paction during the dynamic loading process. These param- eters are mainly inﬂuenced by the shock wave propagation. at few points which depicts elastic unloading during Particle deformation (m) Kinetic energy (J) Kinetic energy (J) Particle displacement (m) Advances in Acoustics and Vibration 7 −4 ×10 0.4 0.3 0.2 0.1 Mh = 1.3 times the total mass H = 17.5 MPa of all the particles −0.1 0 1 2 3 4 0 0.5 1 1.5 2 2.5 −4 −4 ×10 ×10 Time (s) Time (s) Hydraulic pressure = H Hydraulic pressure = 4H Hammer mass = Mh Hammer mass = 0.25 Mh Hydraulic pressure = 2H Hydraulic pressure = 12 H Hammer mass = 4Mh Hammer mass = 16 Mh (a) (b) Figure 8: (a) Compaction of the particles with the same hammer kinetic energy. (b) Hydraulic pressure counters the elastic unloading energy and also prevents the particles from instantaneous separation. compaction. It mainly occurs when shock wave travels Furthermore, shock wave propagation also plays an back from the dead end and passes through the particle. important role in particles deformation as shown in Contact force increases as shock wave passes through the Figure 6(a). Particles are deformed plastically as shock passes during both parts of the cycle despite particles movement. particle during both parts of the cycle, which can be seen However, displacement covered by the particles depends in Figure 4(b). When hammer stops and there is no applied upon their position from the hammer as in Figure 6(b).All hydraulic pressure, contact force decreases and eventually particles are compacted approximately to the same amount becomes zero. During this elastic unloading, the small at shock wave propagation. amount of overlap is recovered. Oscillations after unloading are due to adhesion traction between particles. In case of no adhesion between particles, those oscillations disappear. 4.3. Eﬀects of Changing Loading Parameters. Like particles During compaction, velocity of the particles does not contact force and velocity, hammer kinetic energy (KE) remain uniform and constant as illustrated in Figure 5.As is inﬂuenced by shock wave propagation. Piston KE and the shock wave passes during forward part of the cycle, collective KE of all the particles are shown in Figure 7(a). it results in the particle motion. Velocity of individual There are three shock cycles for this compaction period. particle is increased from zero to approximately hammer Hammer KE has the same pattern during a particular cycle velocity during this period. All particles start to move by the and it changes when shock hits back the hammer at the end end of forward cycle after which shock hits the dead end of the cycle. Particles KE increase as shock moves forward and is reﬂected back. Now, as disturbance passes through from compaction to the dead end. It reaches maximum value the individual particle, its velocity decreases and eventually which is about 7% of hammer KE when shock hits the dead becomes zero. Time for motion of a particle is determined by end. On the return cycle, particles KE decrease and become the period between shock wave passes through the particle zero when shock hits the compaction end. At this point, during forward and backward parts of the cycle. The hammer all energy is converted to plastic deformation and elastic compresses particles until its velocity becomes zero. Up potential energy. For various choices of hammer KE, particles till this point, both cases of adhesion and no adhesion KE have diﬀerent values but they all have the same pattern depict almost similar behavior. After compaction, net force as illustrated in Figure 7(b). In all the cases, maximum and becomes zero and particles expand and push back the minimum value occurs when shock hits the dead end and hammer due to the elastic energy. In case of no adhesion, as compaction end, respectively. in Figure 5(a), particles are moving with diﬀerent velocities It is obvious that particles compaction energy is mainly which indicates that particles are separated. In adhesion case, directly proportional to hammer kinetic energy which as illustrated in Figure 5(b), particles oscillations can be seen depends on its mass and impact velocity. Particles com- which are caused by adhesion between particles. It indicates paction for the the same hammer KE, but with diﬀerent that particles are not fully separated. mass and velocity combinations is shown in Figure 8(a). Piston displacement (m) Contact force (N) 8 Advances in Acoustics and Vibration Compaction is the same in all the cases. 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Published: Aug 9, 2011
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