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Quasi-static response of two-dimensional composite granular layers to a localized force
Powder Technology 261 (2014) 272–278
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Quasi-static response of two-dimensional composite granular layers to a localized force Yahui Yang, Dengming Wang ⁎, Qi Qin Key Laboratory of Mechanics on Environment and Disaster in Western China, The Ministry of Education of China, Lanzhou University, Lanzhou, Gansu 730000, PR China Department of Mechanics and Engineering Sciences, School of Civil Engineering and Mechanics, Lanzhou University, Lanzhou, Gansu 730000, PR China
a r t i c l e
i n f o
Article history: Received 23 December 2013 Received in revised form 3 April 2014 Accepted 5 April 2014 Available online 16 April 2014 Keywords: Composite granular layers External localized force Propagating process of contact forces Thickness ratio of soft layers to rigid layers Discrete element method
a b s t r a c t The quasi-static response of two-dimensional composite layers to a localized external force is investigated based on the discrete element method. The transmitting features of contact forces under various composite modes of granular layers are mainly considered. From simulating results, we find that the contact forces under various composite modes exhibit distinct different features comparing with that inside a monodisperse granular system. The rigid granular layers may commonly promote the propagation of the contact forces to pass through the interface. However, the soft granular layers play roles of disintegration and constrain to contact forces which try to get inside. In addition, for a rigid–soft composite granular system, we find that the thickness ratio of soft layers to total layers has a significant influence on the distributing feature of floor pressure, which implies different propagating processes of contact forces. Finally, a primary phase diagram is proposed to further explicate three propagating modes of contact forces. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Granular material is one of the most prevalent forms of matter in nature and engineering fields. The high degree of discreteness makes it exhibit abundant and complex mechanical properties which are quite different from other common materials and still far from being perfectly understood. In static state, granular material has a solid-like behavior whereas it may exhibit a fluid-like behavior in flow state [1]. Granular material usually exhibits some essential features, which present fundamental difficulties and great challenges in quantitatively modeling and simulating their macroscopic behaviors. Among these problems, the stress propagation in granular material under an external localized load has attracted much attention in recent years. In a granular material, the forces between internal particles, transmitted from one particle to the next via their contacts, may commonly form an inhomogeneous contact network at a mesoscopic scale, which carries most of external loads by way of force chains, while the whole system may exhibit some localized deformation features at a macroscopic scale. Traditionally, engineers have used some phenomenological elastic models to describe such stress propagation inside static or quasi-static granular materials. However, some experiments have revealed the existence of anisotropic force chains, which were interpreted as
⁎ Corresponding author at: Department of Mechanics and Engineering Sciences, School of Civil Engineering and Mechanics, Lanzhou University, Lanzhou, Gansu 730000, PR China. E-mail address: [email protected] (D. Wang).
http://dx.doi.org/10.1016/j.powtec.2014.04.032 0032-5910/© 2014 Elsevier B.V. All rights reserved.
evidence against the applicability of elastic theories of granular material under certain conditions [2]. Some scholars have proposed some stress response functions to describe the stress field in granular materials [3–7]. They supposed that the stress below the plastic yield satisfied elastic partial differential equation (PDE), while the stress corresponding to the plastic deformation satisfied a hyperbolic PDE. Geng et al. have further verified the above conclusions through similar experiments [8]. Goldenberg et al. [9] noted that a two-peak response function can be found in the classical anisotropic spring damping model. Several other experiments [10–12] and simulations [13–15] were also performed and showed that the friction between particles and the magnitude of external force were key factors affecting the response of granular system. However, in actual engineering fields, granular material is usually composed by various components with different physical properties, which belongs to a typical composite material. The stress propagation inside such granular system under external load is very complicated, and so far, the relevant research is quite limited. Prior research has ever proposed a statistical mathematical model of granular composite material to predict some of their mechanical properties [16]. Recently, some investigations focused on the feature of energy transmission in one-dimensional composite granular materials, and found the efficiency of pulse trapping and disintegration in a composite granular protector [17,18]. Sparisoma et al. [19] have proposed a 2-D composite granular model and discussed the elasticity of system using molecular dynamic method. In this paper, we perform a numerical investigation on the transmission of contact forces between internal particles in a 2-D composite
Y. Yang et al. / Powder Technology 261 (2014) 272–278
granular material. The granular assembly consists of two types of particle layers with same density and different rigidity. Based on the discrete element method, we systemically analyze the propagating features of contact forces under an external localized force for different granular composite modes. This paper is organized as follows. In Sec. 2, we provide a description of the model and method. In Sec. 3, the basic features of force transmission in composite granular layers under localized force are analyzed. In Sec. 4, the influence of the thickness ratio of soft layers to rigid layers in composite granular layers on the force response is mainly discussed. A brief conclusion and further interpretations are given in Sec. 5. 2. Model and method We establish a 2D rectangular composite granular model composed of same-sized particles, which contains two types of disks with different stiffness, denoted by different colors in Fig. 1. The initial configuration of non-touching disks is prepared in an enclosure composed of rigid side walls and floor, similar to existing models [9,15]. The system is then relaxed under gravity to form a stationary granular slab. A vertical localized force Fext is applied to the top layer of granular material, positioned on a horizontal surface. Considering the effect of particle gravity, here the localized force adopted in this paper is much larger than per particle’s gravity. In our simulating process, the vertical localized force is gradually applied on the top of the system until it reaches the given value. Finally, a relaxing process is needed until the stable state of whole granular system is achieved. Here the judging criterion of the static equilibrium is that the kinetic energy of per particle is sufficiently small, i.e. Ek ≤ 2 × 10−8mgR [20]. The simulations are carried out using the discrete element method (DEM), which was originally developed by Cundall and Strack [21]. In this method, the position of each particle in the system is obtained by integrating twice with respect to time in Newton's second law of motion. The discrete element method is in fact a time-driven softparticle method that allows two particles’ interpenetrating so as to mimic particles’ deformations. The governing equations of motion for particle i are i d! vi ! X ! ! ¼ mi g þ F n;ij þ F s;ij dt j¼1 k
mi
i dωi X ¼ T ij þ Mij dt j¼1
ð1Þ
k
Ii
ð2Þ
where mi and Ii are the particle mass and moment inertia, ! v i , ωi and Ii are the velocity vector, angular velocity and the moment of inertia of particle ! F n;ij and F s;ij are the normal and tangential force particle j to i respectively; ! * F s is the tangential contact moment and Mij is the particle i, T ij ¼ R i ! * rotation frictional moment; R i is the vector directed from the center of particle i to the contact point. The interaction between two particles can be described using the relative distance between centers of the two particles. The contact
Fig. 1. A two-dimensional composite granular system.
273
force between particles i and j, which can be decomposed into the normal contact force and the tangential contact force, is given by ! Fn
! ¼ −kn ! δ n −mr γ n v n
! Fs
¼
! δt
j! δ tj
! ! min½kt j! δ t j−γ t mr j v t j; μ j F n j:
ð3Þ
ð4Þ
! ! δ n denotes overlap of two particles. v n and v t are normal Here ! δ t is relative velocity and tangential relative velocity respectively. ! tangential deformation vector of particles. mr is equivalent mass = m−1 + m−1 (m−1 r A B , mA and mB are equivalent mass of A and B). We define kn and kt as normal stiffness coefficient and tangential stiffness coefficient respectively, and here we set kt = 0.8kn. Correspondingly, γn and γt are defined as the normal and tangential damping coefficients, respectively. In our composite granular model of Fig. 1, we assume the particles, respectively belonging to its upper and lower parts, have different stiffness. For the purpose of description, we define k1n and k1t as the normal stiffness and tangential contact stiffness between particles belonging to the upper part of granular slab, donated by light gray particles in Fig. 1, and correspondingly define k2n and k2t as the normal stiffness and tangential stiffness between particles located at the lower part of the system, denoted by light red particles. The radius and the density of particles are R = 0.004 m and ρ = 1150 kg/m3 respectively. The normal and tangential damping coefficients are γn = γt = 8.3 × 10−2N s/m. The friction coefficient between particles μ equals to 0.2. For the particle= 2k1n,kwall = 2k1t ,and μwall = 0.2. wall interaction,we choose kwall n t In this paper we mainly focus on two types of composite modes of granular layers as follows: (a) rigid–soft composite granular system, k1n = 107kg/s2,k2n = 105kg/s2; (b) soft–rigid composite granular system, k1n = 107kg/s2,k2n = 109kg/s2. For the purpose of comparison, a corresponding monodisperse granular system,k1n = k2n = 107kg/s2, is also considered here. We systematically investigate varying features of force transmission in composite granular systems comparing with monodisperse system, and further consider the effects of different composite modes on its force transmission.
3. Propagating characters of contact forces Existing studies have shown that the stress in granular solid propagates in a manner described by hyperbolic equations, which is not consistent with that intuitively predicted by the elliptic equation of static elasticity [3,4,9]. The later numerical simulation of a 2D granular slab to an external load reveals that the different response features may be induced by the size of system [9,13], which is also influenced by the magnitude of external force and the static friction between particles. In order to evaluate the effectiveness of our numerical model, we have
Fig. 2. Contact forces in a monodisperse granular system under a local vertical force Fext = 5 N. The linewidths and lengths are proportional to the magnitudes of contact forces.
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0.6
z=0.866 z=2.598 z=4.330 z=6.052 z=7.794
0.5
Fn/Fext
0.4 0.3 0.2 0.1 0.0 -60
-40
-20
0
20
40
60
x/R Fig. 3. Normalized force magnitudes vs the horizontal distance at various depths in a monodisperse granular system, a local vertical force Fext = 5 N.
firstly performed some simulations on the response of monodisperse model (60 or 61 particles wide and 15 deep), in which a local vertical force Fext = 5 N is applied to the particle at the center of top row of particles. The contact forces inside system are shown in Fig. 2. It can be found two prominent force chains along the lattice directions occur in the monodisperse model, which is also observed in a similar experiment [9]. In order to further verify our numerical model, we also present the magnitudes of contact forces between internal particles vs horizontal distance at various depths, as shown in Fig. 3. Here the depth z is measured from the application point of external force and normalized by particle radius R. The result clearly shows that two peaks emerge from the central source and broaden with the depth, whereas the magnitudes of two peaks decrease with increase of depth. Such result is also consistent with existing simulating results [9,15]. Existing studies have also shown that a solitary wave may be disintegrated and reflected at the interface of different types of particles [22,23]. In a one-dimensional granular chain containing two sets of particles, another interesting phenomenon is observed that the energy is always trapped in the middle light section [17,18]. However, for a composite granular material, how do the contact forces between internal particles propagate inside the whole system, especially pass through the interface from one type of granular layers to another? Here we mainly focus on the propagating characters of contact forces inside granular systems under two types of composite modes. Fig. 4(a) shows the configuration of contact chains in a rigid–soft composite granular system, where the contact forces pass through the interface from upper rigid granular layers into lower soft granular layers. From the figure, it can be found that although part of contact forces continues
to propagate along the initial direction, most of them seem to be disintegrated to a symmetric direction. In fact, the soft particles are easily to deform and only can bear a smaller external load before they move, so it is obvious that these soft particles along the directions of initial strong force chains cannot balance the contact forces propagated from the upper rigid particle layers. So from the loading point, we find that each strong force chain has been gradually decomposed into two directions layer by layer: one is along the horizontal direction, another is along the symmetric direction of strong force chain. It is noteworthy that such decompositions mainly occur inside upper rigid granular layers because the contact forces prefer to propagate between rigid particles. As a result, each strong force chain can induce a series of parallel weak force chains along its symmetric direction. We are surprised to find that these weak force chains may easily propagate through the interface of rigid and soft granular layers. This may be due to that such small contact forces are more easily to be balanced from below soft granular layers. Therefore, two sets of parallel weak force chains from two strong force chains may intersect together and form a force network below two strong force chains, where the force response at the interface between two force chains is nearly uniform and exhibits a plateau distribution comparing with the monodisperse granular layers, as shown in Fig. 5(a). From Fig. 4, we also find that these new generating parallel weak force chains may decay rapidly inside soft granular layers as lacking of intersecting constrains from another corresponding set of parallel weak force chains. Therefore, for this rigid–soft composite granular system with nearly same thickness of rigid and soft layers, the majority of contact forces seem to be trapped into a diamond region below the position of applied force. Thus the contact forces between particles inside soft granular layers may gradually converge toward the central region below the interface, and eventually the response on the floor exhibits a single-peaked feature, as shown in Fig. 5(b), which is obviously different from a two-peaked response in a monodisperse granular system. However, for the soft–rigid composite granular system, we find that the propagating directions of strong contact force chains are nearly unchanged when they transmit from soft layers to rigid layers, as shown in Fig. 4(b). In fact, the large contact force can generate between rigid particles even if their contact deformation is quite small. Thus, under the identical external load, the lower rigid granular layers can easily balance the external load propagated from the upper soft particles layers before they move. Inside such granular system, the final result is the contact forces seem to prefer to pass through the interface and continue to propagate along the directions of initial main force chains inside rigid part of the whole system. As a result, the responses at the interface and on the floor all exhibit two peaks, as shown in Fig. 5, which are very similar to those in monodisperse granular system. The different aspect is that the peak values of responses at the interface and on the floor for soft–rigid composite granular layers are all larger than those values at same positions in a corresponding monodisperse granular system.
Fig. 4. Contact forces obtained in composite granular systems under a local vertical force Fext = 5 N. (a) for a rigid–soft composite mode; (b) for a soft–rigid composite mode.
Y. Yang et al. / Powder Technology 261 (2014) 272–278
0.20
0.14
Monodisperse model Rigid-soft model Soft-rigid model
(a)
0.12
Monodisperse model Rigid-soft model Soft-rigid model
(b)
0.10
Fy/Fext
Fn/Fext
0.16
275
0.12 0.08
0.08 0.06 0.04
0.04 0.02 0.00 -60
-40
-20
0
20
40
60
x/R
0.00 -60
-40
-20
0
20
40
60
x/R
Fig. 5. (a) Distribution of normal force at interface; (b) vertical force profile on the floor.
These results show that when the contact forces inside a composite granular material transmit from its rigid layers into soft layers, its propagating features undergo an obvious variation. The soft granular layers can disintegrate the initial main force chains propagating from external granular systems and induce to generate some new assistant weak force chains inside it. These new generating weak force chains are trapped in a relatively small and concentrated region closer to the loading boundary to form a stable contact force network to sustain the external loading, which is similar to energy trapping and shock disintegration in a onedimensional composite granular chain [17,18]. To gain a better understanding to the propagating feature of contact forces inside a composite granular material, we present another composite model as shown in Fig. 6, in which the external vertical force is parallel to the interface between the soft and rigid granular layers. The model is also composed of two different types of particles. k1n is normal stiffness and tangential stiffness of particles placed on the left part, and k2n is normal stiffness of particles at the right side of the system. In this model, a local vertical force Fext = 5 N is also applied to the top of this rectangular layer of a composite granular material, positioned on a horizontal surface. Fig. 7(a) presents force network of composite model with two sets of particles, the left part of granular system composed by relative rigid particles with a normal stiffness k1n = 107kg/s2and the right part of system composed by soft particles with a normal stiffness k2n = 105kg/s2. If the vertical force is applied on the top of rigid granular layers, we find that the contact forces prefer to transmit inside the rigid granular layers, and an obvious strong force chain occurs along the lattice direction inside the left part of system, as shown in Fig. 7(a). Although the vertical force is very closer to the interface, we can hardly observe an obvious force chain in right part of granular system. It shows that few contact forces can pass through the interface and enter the soft granular layers. However, for another composite granular material, where the left part of granular system are composed by relative soft particles with a normal stiffness k1n = 107kg/s2 and the right part of system are composed by rigid particles with a normal stiffness k2n = 109kg/s2, the result is completely different. We find that although a strong force chain also occurs inside the left part of system, most of contact forces prefer to
pass through the interface and transmit into rigid particle layers, as shown in Fig. 7(b). In this composite granular system, the vertical force generates another strong force chain passing through the interface, which seems obviously to be stronger than the left one. To further investigate the process of force transmission in a composite granular material mentioned in Fig. 7, we also present the transmitting details of contact forces at several different instantaneous moments during the initial stage of loading, as shown in Fig. 8. From the figure, we find that the transmitting feature of contact forces inside a composite granular material is almost identical to that in a monodisperse system before the contact forces pass through the interface. When the contact forces continue to propagate and arrive at the interface from rigid granular layers to soft granular layers, its transmission along the right lattice direction is clearly constrained due to the existence of soft granular layers, as shown in Fig. 8(a). Therefore, the contact forces along the left lattice direction may gradually form a dominant force chain inside the left rigid granular layers whereas its corresponding one inside right soft granular layers becomes more and more weak during the initial loading processes. However, when the contact forces transmit from soft granular layers into rigid layers, most of them prefer to pass through the interface and converge into a stronger force chain inside rigid granular layers, as shown in Fig. 8(b). From the above results, we can find that under a localized loading, the contact forces inside a composite granular material exhibit an interesting transmitting feature, which is distinctly different from that inside a monodisperse granular system. When the contact forces transmit from soft granular layers to rigid granular layers, they prefer to pass through the interface and converge into a stronger force chain inside rigid part of the whole system. However, under a case where the contact forces transmit from rigid granular layers to soft granular layers, such variation of transmitting path of contact forces becomes more obvious. For a composite granular material where the external local force is perpendicular to the interface and the contact forces inevitably pass through the interface, the strong force chains in rigid layers may be disintegrated into a series of parallel weak force chains along its symmetric direction. Then the soft granular layers can trap the majority of these contact forces into a relatively small and concentrated region closer to the loading boundary, and form a stable contact force network to sustain the external loading. For another typical composite granular material where the external local force is parallel to the interface as shown in Fig. 7(a), the soft granular layers may directly constrain the contact forces to pass through the interface and induce them to propagate inside the rigid layers along the symmetric lattice direction. 4. The effect of thickness ratio of soft layers to rigid layers on response
Fig. 6. A left–right composite granular model composed by two sets of particles with different stiffnesses.
Above results demonstrate that the contact forces inside different types of composite granular layers exhibit different transmitting features under a localized force, especially inside a rigid–soft composite
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Fig. 7. The configurations of force chains in different composite granular systems under external force. (a) k1n = 107kg/s2, k2n = 105kg/s2. (b) k1n = 107kg/s2, k2n = 109kg/s2.
granular system, where soft granular layers have a significant effect on its force transmission within the whole system. Next we mainly focus on the influence of thickness ratio of soft layers to rigid layers on force transmission in composite granular materials mentioned above in Fig. 4. At first, we present the vertical force profiles on the floor for different thickness ratios of soft layers to rigid layers for two typical composite granular systems under a vertical external force, as shown in Fig. 9. Here m denotes the thickness ratio of soft layers to total layers in a rigid–soft composite granular system and m = 0 represents a monodisperse granular system, which is also presented in Fig. 9 as a comparison with other results. Fig. 9(a) presents a vertical force profile on the floor for different m inside a rigid–soft composite granular system. From the figure, we can observe three different types of force responses on the floor. For an identical external load, when m is small, the response exhibits a plateau floor pressure distribution, as shown in Fig. 9(a). With the increasing of m, the width of plateau response reduces gradually. When m approaches 0.5, which represents the thickness of soft granular layers is close to that of rigid layers, the floor pressure evolves into a one-peaked distribution. When m excesses 0.5 and continues to increase, we find that the floor pressure distribution undergoes a crossover from one-peaked to two-peaked features, and the width between two peaks also increases with m. So it is obvious that the thickness ratio of soft layers to rigid layers has a significant effect on its force transmission within a rigid–soft composite granular material. As a comparison, we also investigate the influence of the thickness ratio on the floor force distribution in soft–rigid composite granular systems, as shown in Fig. 9(b). Here n denotes the thickness ratio of
rigid layers to total layers in a soft–rigid granular system, and n = 0 also represents a monodisperse granular system. From the figure, we find that the floor pressure for different n all exhibits a two-peaked distribution, although there are some subtle changes quantitatively. This result demonstrates that the thickness ratio of soft layers to rigid layers almost does not affect the distributing feature of floor pressure inside soft–rigid composite granular systems. For a rigid–soft composite granular system, the thickness ratio of soft layers to total layers has a significant influence on the distributing feature of floor pressure, which is obviously due to the variation of propagating path of contact forces inside the whole system. In order to further explore such propagating feature, we have performed a serial of numerical experiments, and find that different propagating processes of contact forces lead to three different distributing features of floor pressure inside a rigid–soft composite granular system. Here we use several schematic diagrams to further explicate such propagating processes of contact forces for different m under identical external loads, as shown in Fig. 10. When the thickness of soft granular layers is smaller than that of rigid layers(i.e. m b 0.5), we find that each strong force chain can induce a serial of parallel weak force chains along its symmetric lattice directions inside upper rigid granular layers and these weak force chains can propagate through the interface. Therefore, two sets of parallel weak force chains, generated from two strong force chains respectively, may intersect together and form a contact force network below the loading point. A more interesting aspect is that the pressure distribution at certain height almost is almost uniform along the horizontal direction, which is also valid for floor pressure distribution. On these cases, the floor pressure exhibits a plateau distribution, as shown in Fig. 10(a).
(a)
3.5 10 5 s
4.5 10 5 s
5.5 10 5 s
6.5 10 5 s
3.5 10 5 s
4.5 10 5 s
5.5 10 5 s
6.5 10 5 s
(b)
Fig. 8. The transmission of force chains at different time instantaneous moments for different composite granular systems. (a) k1n = 107kg/s2, k2n = 105kg/s2. (b) k1n = 107kg/s2, k2n = 109kg/s2.
Y. Yang et al. / Powder Technology 261 (2014) 272–278
0.14
0.10
Fn/Fext
0.090
m=0.00 m=0.20 m=0.33 m=0.47 m=0.60 m=0.73
(a)
0.08 0.06
0.045 0.030
0.02
0.015
0.00 -40
-20
0
20
40
n=0.00 n=0.20 n=0.33 n=0.47 n=0.60 n=0.73
0.060
0.04
-60
(b)
0.075
Fy/Fext
0.12
277
60
0.000 -60
-40
x/R
-20
0
20
40
60
x/R
Fig. 9. (a) Vertical force profiles on the floor for different ratio, m is the thickness ratio of softer layers to total layers in a rigid–soft composite granular system. (b) Vertical force profiles on the floor for different ratio; n is the ratio of rigid layers to total layers in a soft–rigid composite granular system.
We also find that once these new generating parallel weak force chains pass through the interface, they may decay rapidly without intersecting constrains from another corresponding set of parallel weak force chains inside soft granular layers. So it can be obviously observed that the distributing width of floor pressure is reducing with the increasing of m. When the thickness of soft granular layers is close to that of rigid layers (i.e. m = 0.5), the distributing width of floor pressure may tend to a point and the floor pressure may evolve into a one-peaked distribution from a plateau distribution, as shown in Fig. 10(b). With the increasing of m (i.e. m N 0.5), two sets of parallel weak force chains may intersect and converge into a vertical force located at a certain depth below the interface, approximately equaling to the thickness of upper rigid granular layers. In this situation, the propagation of contact forces inside soft granular layers below this vertical force is very similar to that inside a monodisperse system, and then the floor pressure exhibits a twopeaked distribution, as shown in Fig. 10(c). Above results show that the thickness ratio of soft layers to rigid layers is a very important factor for propagating features of contact forces inside a rigid–soft composite granular system. On the other hand, the existing investigation shows that the size of granular system is also a decisive factor affecting the propagation of contact forces inside a monodisperse granular system [15]. Here we present a phase diagram to identify the propagating features of contact forces inside a rigid–soft composite granular material, and here the thickness ratio of soft to rigid layers and the size of the whole system are two main factors for consideration. In our all simulations, we have employed a series of rigid–soft composite granular models with total layers from 5 to 35 since the floor response of deeper models becomes very weak under an identical local force Fext = 5 N. We find that with the increasing of soft granular layers, the floor response always exhibits three different distributing forms: plateau, one-peaked and two-peaked distribution, regardless of the number of total granular layers, as shown in Fig. 11. It indicates that indeed there are three different propagating modes of contact forces inside rigid–soft composite granular systems. In this phase diagram, we present three distributing regions just corresponding to
these three propagating modes. It can be found that the distribution of floor pressure is one-peaked when the thickness ratio of soft layers to rigid layers is close to 1, also as shown in the inset of Fig. 11, and it may evolve into a plateau and two-peaked distributions when this thickness ratio deviates from 1 to lower and to upper respectively.
5. Conclusions The transmitting feature of contact forces in a composite granular material is investigated based on the discrete element method, and the influences of the thickness ratio of soft layers to rigid layers are mainly considered. From the simulating results, we find that the transmission of contact forces under various composite modes of granular layers exhibit distinct different features comparing with that inside a monodisperse granular system. For a soft–rigid composite granular system, the propagating directions of strong contact force chains are nearly unchanged when they transmit from softer layers to harder layers. However, for a rigid–soft composite granular system, if the external local force is perpendicular to the interface, the strong force chains in rigid layers may be disintegrated into a series of parallel weak force chains along its symmetric direction. Then the soft granular layers can trap the majority of these contact forces into a relatively small and concentrated region closer to the loading boundary, and form a stable contact force network to sustain the external loading. In addition, for a rigid–soft composite granular system, the thickness ratio of soft layers to total layers has a significant influence on the distributing feature of floor pressure. We have performed a serial of numerical experiments, and find that different propagating processes of contact forces lead to these three different distributing features of floor pressure. Therefore, we have presented three schematic diagrams to further explicate such propagating processes of contact forces. Finally, a primary phase diagram is proposed for a rigid–soft composite granular system, in which three distributing regions just correspond to these three propagating modes of contact forces.
Fig. 10. Schematic diagrams of propagating processes of contact forces inside a rigid–soft composite granular material for different m.
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Two peaks
0.6 0.5
Single peak
0.4
20 18 16
0.3
Plateau
0.2
soft layers
Ratio of soft to total layers
0.7
14
two peaks
12
6
plateau
4
0.1
signal peak
10 8
2 0
5
10
15 20 25 30 total granular layers
35
0.0 5
10
15
20
25
30
35
Total granular layers Fig. 11. The phase diagram describing the propagating features of contact forces inside a rigid–soft composite granular material. The horizontal axis is the total layers of the system. The vertical axis represents the thickness ratio of soft layers to total layers. The phase diagram is also shown in the inset for which the vertical axis represents the number of soft layers.
Acknowledgments This work was supported by the Science Fund for Creative Research Groups of the National Natural Science Foundation of China (No.: 11121202), the National Natural Science Foundation of China (Grant No.: 11002064), and the Postdoctoral Science Foundation of China (2013M530434). References [1] Heinrich M. Jaeger, Sidney R. Nagel, Robert P. Behringer, Granular solids, liquids, and gases, Rev. Mod. Phys. 68 (4) (1996) 1259–1273. [2] A. Drescher, G. de Josselin de Jong, Photoelastic verification of a mechanical model for the flow of a granular material, J. Mech. Phys. Solids 20 (1972) 337–340.
[3] J.-P. Bouchaud, M.E. Cates, P. Claudin, Stress distribution in granular media and nonlinear wave equation, J. Phys. I 5 (1995) 639–656 (France). [4] J.P. Wittmer, P. Claudin, M.E. Cates, Stress propagation and arching in static sandpiles, J. Phys. I 7 (1997) 39–80 (France). [5] Guillaume Reydellet, Eric Clément, Green's function probe of a static granular piling, Phys. Rev. Lett. 86 (2001) 3308–3311. [6] J.-P. Bouchaud, P. Claudin, D. Levine, M. Otto, Force chain splitting in granular materials: a mechanism for large-scale pseudo-elastic behavior, Eur. Phys. J. E 4 (2001) 451–457. [7] Jean-Philippe Bouchaud, Philippe Claudin, Eric Clément, Matthias Otto, Guillaume Reydellet, The stress response function in granular materials, C. R. Phys. 3 (2002) 141–151. [8] Junfei Geng, D. Howell, E. Longhi, R.P. Behringer, G. Reydellet, L. Vanel, E. Clément, Footprints in sand: the response of a granular material to local perturbations, Phys. Rev. Lett. 87 (2001) 035506. [9] C. Goldenberg, I. Goldhirsch, Force chains, microelasticity, and macroelasticity, Phys. Rev. Lett. 89 (2002) 084302. [10] Junfei Geng, G. Reydellet, E. Clément, R.P. Behringer, Green's function measurements of force transmission in 2D granular materials, Phys. D 182 (2003) 274–303. [11] Nathan W. Mueggenburg, Heinrich M. Jaeger, Sidney R. Nagel, Stress transmission through three-dimensional ordered granular arrays, Phys. Rev. E. 66 (2002) 031304. [12] Srdjan Ostojic, Debabrata Panja, Elasticity from the force network ensemble in granular media, Phys. Rev. Lett. 97 (2006) 208001. [13] C. Goldenberg, I. Goldhirsch, Friction enhances elasticity in granular solids, Nature 435 (2005) 188–191. [14] N. Gland, P. Wang, H.A. Makse, Numerical study of the stress response of dense granular packings, Eur. Phys. J. E 20 (2006) 179–184. [15] C. Goldenberg, I. Goldhirsch, Effects of friction and disorder on the quasistatic response of granular solids to a localized force, Phys. Rev. E. 77 (2008) 041303. [16] R.I. Budeshtskii, Mathematical model of granular composite materials, Strength of Materials, vol. 3, 1971, pp. 912–916. [17] C. Daraio, V.F. Nesterenko, E.B. Herbold, S. Jin, Energy trapping and shock disintegration in a composite granular medium, Phys. Rev. E. 96 (2006) 058002. [18] P.J. Wang, J.H. Xia, Y.D. Li, C.S. Liu, Crossover in the power-law behavior of confined energy in a composite granular chain, Phys. Rev. E. 76 (2007) 041305. [19] Sparisoma Viridi, Widayani, Siti Nurul Khotimah, 2-D granular model of composite elasticity using molecular dynamics simulation, International Conference on Physics and its Applications: (ICPAP 2011), AIP Conference Proceedings, vol. 1454, 2012, pp. 219–222. [20] James W. Landry, Gary S. Grest, Granular packings with moving side walls, Phys. Rev. E. 76 (2004) 031303. [21] Cundall, Strack, A discrete numerical model for granular assemblies, Geotechnique 29 (1979) 47–65. [22] V.F. Nesterenko, Dynamics of Heterogeneous Materials, Springer, New York, 2001. (Chap. 1). [23] Jongbae Hong, Universal power-law decay of the impulse energy in granular protectors, Phys. Rev. Lett. 94 (2005) 108001.
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Powder Technology journal homepage: www.elsevier.com/locate/powtec
Quasi-static response of two-dimensional composite granular layers to a localized force Yahui Yang, Dengming Wang ⁎, Qi Qin Key Laboratory of Mechanics on Environment and Disaster in Western China, The Ministry of Education of China, Lanzhou University, Lanzhou, Gansu 730000, PR China Department of Mechanics and Engineering Sciences, School of Civil Engineering and Mechanics, Lanzhou University, Lanzhou, Gansu 730000, PR China
a r t i c l e
i n f o
Article history: Received 23 December 2013 Received in revised form 3 April 2014 Accepted 5 April 2014 Available online 16 April 2014 Keywords: Composite granular layers External localized force Propagating process of contact forces Thickness ratio of soft layers to rigid layers Discrete element method
a b s t r a c t The quasi-static response of two-dimensional composite layers to a localized external force is investigated based on the discrete element method. The transmitting features of contact forces under various composite modes of granular layers are mainly considered. From simulating results, we find that the contact forces under various composite modes exhibit distinct different features comparing with that inside a monodisperse granular system. The rigid granular layers may commonly promote the propagation of the contact forces to pass through the interface. However, the soft granular layers play roles of disintegration and constrain to contact forces which try to get inside. In addition, for a rigid–soft composite granular system, we find that the thickness ratio of soft layers to total layers has a significant influence on the distributing feature of floor pressure, which implies different propagating processes of contact forces. Finally, a primary phase diagram is proposed to further explicate three propagating modes of contact forces. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Granular material is one of the most prevalent forms of matter in nature and engineering fields. The high degree of discreteness makes it exhibit abundant and complex mechanical properties which are quite different from other common materials and still far from being perfectly understood. In static state, granular material has a solid-like behavior whereas it may exhibit a fluid-like behavior in flow state [1]. Granular material usually exhibits some essential features, which present fundamental difficulties and great challenges in quantitatively modeling and simulating their macroscopic behaviors. Among these problems, the stress propagation in granular material under an external localized load has attracted much attention in recent years. In a granular material, the forces between internal particles, transmitted from one particle to the next via their contacts, may commonly form an inhomogeneous contact network at a mesoscopic scale, which carries most of external loads by way of force chains, while the whole system may exhibit some localized deformation features at a macroscopic scale. Traditionally, engineers have used some phenomenological elastic models to describe such stress propagation inside static or quasi-static granular materials. However, some experiments have revealed the existence of anisotropic force chains, which were interpreted as
⁎ Corresponding author at: Department of Mechanics and Engineering Sciences, School of Civil Engineering and Mechanics, Lanzhou University, Lanzhou, Gansu 730000, PR China. E-mail address: [email protected] (D. Wang).
http://dx.doi.org/10.1016/j.powtec.2014.04.032 0032-5910/© 2014 Elsevier B.V. All rights reserved.
evidence against the applicability of elastic theories of granular material under certain conditions [2]. Some scholars have proposed some stress response functions to describe the stress field in granular materials [3–7]. They supposed that the stress below the plastic yield satisfied elastic partial differential equation (PDE), while the stress corresponding to the plastic deformation satisfied a hyperbolic PDE. Geng et al. have further verified the above conclusions through similar experiments [8]. Goldenberg et al. [9] noted that a two-peak response function can be found in the classical anisotropic spring damping model. Several other experiments [10–12] and simulations [13–15] were also performed and showed that the friction between particles and the magnitude of external force were key factors affecting the response of granular system. However, in actual engineering fields, granular material is usually composed by various components with different physical properties, which belongs to a typical composite material. The stress propagation inside such granular system under external load is very complicated, and so far, the relevant research is quite limited. Prior research has ever proposed a statistical mathematical model of granular composite material to predict some of their mechanical properties [16]. Recently, some investigations focused on the feature of energy transmission in one-dimensional composite granular materials, and found the efficiency of pulse trapping and disintegration in a composite granular protector [17,18]. Sparisoma et al. [19] have proposed a 2-D composite granular model and discussed the elasticity of system using molecular dynamic method. In this paper, we perform a numerical investigation on the transmission of contact forces between internal particles in a 2-D composite
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granular material. The granular assembly consists of two types of particle layers with same density and different rigidity. Based on the discrete element method, we systemically analyze the propagating features of contact forces under an external localized force for different granular composite modes. This paper is organized as follows. In Sec. 2, we provide a description of the model and method. In Sec. 3, the basic features of force transmission in composite granular layers under localized force are analyzed. In Sec. 4, the influence of the thickness ratio of soft layers to rigid layers in composite granular layers on the force response is mainly discussed. A brief conclusion and further interpretations are given in Sec. 5. 2. Model and method We establish a 2D rectangular composite granular model composed of same-sized particles, which contains two types of disks with different stiffness, denoted by different colors in Fig. 1. The initial configuration of non-touching disks is prepared in an enclosure composed of rigid side walls and floor, similar to existing models [9,15]. The system is then relaxed under gravity to form a stationary granular slab. A vertical localized force Fext is applied to the top layer of granular material, positioned on a horizontal surface. Considering the effect of particle gravity, here the localized force adopted in this paper is much larger than per particle’s gravity. In our simulating process, the vertical localized force is gradually applied on the top of the system until it reaches the given value. Finally, a relaxing process is needed until the stable state of whole granular system is achieved. Here the judging criterion of the static equilibrium is that the kinetic energy of per particle is sufficiently small, i.e. Ek ≤ 2 × 10−8mgR [20]. The simulations are carried out using the discrete element method (DEM), which was originally developed by Cundall and Strack [21]. In this method, the position of each particle in the system is obtained by integrating twice with respect to time in Newton's second law of motion. The discrete element method is in fact a time-driven softparticle method that allows two particles’ interpenetrating so as to mimic particles’ deformations. The governing equations of motion for particle i are i d! vi ! X ! ! ¼ mi g þ F n;ij þ F s;ij dt j¼1 k
mi
i dωi X ¼ T ij þ Mij dt j¼1
ð1Þ
k
Ii
ð2Þ
where mi and Ii are the particle mass and moment inertia, ! v i , ωi and Ii are the velocity vector, angular velocity and the moment of inertia of particle ! F n;ij and F s;ij are the normal and tangential force particle j to i respectively; ! * F s is the tangential contact moment and Mij is the particle i, T ij ¼ R i ! * rotation frictional moment; R i is the vector directed from the center of particle i to the contact point. The interaction between two particles can be described using the relative distance between centers of the two particles. The contact
Fig. 1. A two-dimensional composite granular system.
273
force between particles i and j, which can be decomposed into the normal contact force and the tangential contact force, is given by ! Fn
! ¼ −kn ! δ n −mr γ n v n
! Fs
¼
! δt
j! δ tj
! ! min½kt j! δ t j−γ t mr j v t j; μ j F n j:
ð3Þ
ð4Þ
! ! δ n denotes overlap of two particles. v n and v t are normal Here ! δ t is relative velocity and tangential relative velocity respectively. ! tangential deformation vector of particles. mr is equivalent mass = m−1 + m−1 (m−1 r A B , mA and mB are equivalent mass of A and B). We define kn and kt as normal stiffness coefficient and tangential stiffness coefficient respectively, and here we set kt = 0.8kn. Correspondingly, γn and γt are defined as the normal and tangential damping coefficients, respectively. In our composite granular model of Fig. 1, we assume the particles, respectively belonging to its upper and lower parts, have different stiffness. For the purpose of description, we define k1n and k1t as the normal stiffness and tangential contact stiffness between particles belonging to the upper part of granular slab, donated by light gray particles in Fig. 1, and correspondingly define k2n and k2t as the normal stiffness and tangential stiffness between particles located at the lower part of the system, denoted by light red particles. The radius and the density of particles are R = 0.004 m and ρ = 1150 kg/m3 respectively. The normal and tangential damping coefficients are γn = γt = 8.3 × 10−2N s/m. The friction coefficient between particles μ equals to 0.2. For the particle= 2k1n,kwall = 2k1t ,and μwall = 0.2. wall interaction,we choose kwall n t In this paper we mainly focus on two types of composite modes of granular layers as follows: (a) rigid–soft composite granular system, k1n = 107kg/s2,k2n = 105kg/s2; (b) soft–rigid composite granular system, k1n = 107kg/s2,k2n = 109kg/s2. For the purpose of comparison, a corresponding monodisperse granular system,k1n = k2n = 107kg/s2, is also considered here. We systematically investigate varying features of force transmission in composite granular systems comparing with monodisperse system, and further consider the effects of different composite modes on its force transmission.
3. Propagating characters of contact forces Existing studies have shown that the stress in granular solid propagates in a manner described by hyperbolic equations, which is not consistent with that intuitively predicted by the elliptic equation of static elasticity [3,4,9]. The later numerical simulation of a 2D granular slab to an external load reveals that the different response features may be induced by the size of system [9,13], which is also influenced by the magnitude of external force and the static friction between particles. In order to evaluate the effectiveness of our numerical model, we have
Fig. 2. Contact forces in a monodisperse granular system under a local vertical force Fext = 5 N. The linewidths and lengths are proportional to the magnitudes of contact forces.
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0.6
z=0.866 z=2.598 z=4.330 z=6.052 z=7.794
0.5
Fn/Fext
0.4 0.3 0.2 0.1 0.0 -60
-40
-20
0
20
40
60
x/R Fig. 3. Normalized force magnitudes vs the horizontal distance at various depths in a monodisperse granular system, a local vertical force Fext = 5 N.
firstly performed some simulations on the response of monodisperse model (60 or 61 particles wide and 15 deep), in which a local vertical force Fext = 5 N is applied to the particle at the center of top row of particles. The contact forces inside system are shown in Fig. 2. It can be found two prominent force chains along the lattice directions occur in the monodisperse model, which is also observed in a similar experiment [9]. In order to further verify our numerical model, we also present the magnitudes of contact forces between internal particles vs horizontal distance at various depths, as shown in Fig. 3. Here the depth z is measured from the application point of external force and normalized by particle radius R. The result clearly shows that two peaks emerge from the central source and broaden with the depth, whereas the magnitudes of two peaks decrease with increase of depth. Such result is also consistent with existing simulating results [9,15]. Existing studies have also shown that a solitary wave may be disintegrated and reflected at the interface of different types of particles [22,23]. In a one-dimensional granular chain containing two sets of particles, another interesting phenomenon is observed that the energy is always trapped in the middle light section [17,18]. However, for a composite granular material, how do the contact forces between internal particles propagate inside the whole system, especially pass through the interface from one type of granular layers to another? Here we mainly focus on the propagating characters of contact forces inside granular systems under two types of composite modes. Fig. 4(a) shows the configuration of contact chains in a rigid–soft composite granular system, where the contact forces pass through the interface from upper rigid granular layers into lower soft granular layers. From the figure, it can be found that although part of contact forces continues
to propagate along the initial direction, most of them seem to be disintegrated to a symmetric direction. In fact, the soft particles are easily to deform and only can bear a smaller external load before they move, so it is obvious that these soft particles along the directions of initial strong force chains cannot balance the contact forces propagated from the upper rigid particle layers. So from the loading point, we find that each strong force chain has been gradually decomposed into two directions layer by layer: one is along the horizontal direction, another is along the symmetric direction of strong force chain. It is noteworthy that such decompositions mainly occur inside upper rigid granular layers because the contact forces prefer to propagate between rigid particles. As a result, each strong force chain can induce a series of parallel weak force chains along its symmetric direction. We are surprised to find that these weak force chains may easily propagate through the interface of rigid and soft granular layers. This may be due to that such small contact forces are more easily to be balanced from below soft granular layers. Therefore, two sets of parallel weak force chains from two strong force chains may intersect together and form a force network below two strong force chains, where the force response at the interface between two force chains is nearly uniform and exhibits a plateau distribution comparing with the monodisperse granular layers, as shown in Fig. 5(a). From Fig. 4, we also find that these new generating parallel weak force chains may decay rapidly inside soft granular layers as lacking of intersecting constrains from another corresponding set of parallel weak force chains. Therefore, for this rigid–soft composite granular system with nearly same thickness of rigid and soft layers, the majority of contact forces seem to be trapped into a diamond region below the position of applied force. Thus the contact forces between particles inside soft granular layers may gradually converge toward the central region below the interface, and eventually the response on the floor exhibits a single-peaked feature, as shown in Fig. 5(b), which is obviously different from a two-peaked response in a monodisperse granular system. However, for the soft–rigid composite granular system, we find that the propagating directions of strong contact force chains are nearly unchanged when they transmit from soft layers to rigid layers, as shown in Fig. 4(b). In fact, the large contact force can generate between rigid particles even if their contact deformation is quite small. Thus, under the identical external load, the lower rigid granular layers can easily balance the external load propagated from the upper soft particles layers before they move. Inside such granular system, the final result is the contact forces seem to prefer to pass through the interface and continue to propagate along the directions of initial main force chains inside rigid part of the whole system. As a result, the responses at the interface and on the floor all exhibit two peaks, as shown in Fig. 5, which are very similar to those in monodisperse granular system. The different aspect is that the peak values of responses at the interface and on the floor for soft–rigid composite granular layers are all larger than those values at same positions in a corresponding monodisperse granular system.
Fig. 4. Contact forces obtained in composite granular systems under a local vertical force Fext = 5 N. (a) for a rigid–soft composite mode; (b) for a soft–rigid composite mode.
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0.20
0.14
Monodisperse model Rigid-soft model Soft-rigid model
(a)
0.12
Monodisperse model Rigid-soft model Soft-rigid model
(b)
0.10
Fy/Fext
Fn/Fext
0.16
275
0.12 0.08
0.08 0.06 0.04
0.04 0.02 0.00 -60
-40
-20
0
20
40
60
x/R
0.00 -60
-40
-20
0
20
40
60
x/R
Fig. 5. (a) Distribution of normal force at interface; (b) vertical force profile on the floor.
These results show that when the contact forces inside a composite granular material transmit from its rigid layers into soft layers, its propagating features undergo an obvious variation. The soft granular layers can disintegrate the initial main force chains propagating from external granular systems and induce to generate some new assistant weak force chains inside it. These new generating weak force chains are trapped in a relatively small and concentrated region closer to the loading boundary to form a stable contact force network to sustain the external loading, which is similar to energy trapping and shock disintegration in a onedimensional composite granular chain [17,18]. To gain a better understanding to the propagating feature of contact forces inside a composite granular material, we present another composite model as shown in Fig. 6, in which the external vertical force is parallel to the interface between the soft and rigid granular layers. The model is also composed of two different types of particles. k1n is normal stiffness and tangential stiffness of particles placed on the left part, and k2n is normal stiffness of particles at the right side of the system. In this model, a local vertical force Fext = 5 N is also applied to the top of this rectangular layer of a composite granular material, positioned on a horizontal surface. Fig. 7(a) presents force network of composite model with two sets of particles, the left part of granular system composed by relative rigid particles with a normal stiffness k1n = 107kg/s2and the right part of system composed by soft particles with a normal stiffness k2n = 105kg/s2. If the vertical force is applied on the top of rigid granular layers, we find that the contact forces prefer to transmit inside the rigid granular layers, and an obvious strong force chain occurs along the lattice direction inside the left part of system, as shown in Fig. 7(a). Although the vertical force is very closer to the interface, we can hardly observe an obvious force chain in right part of granular system. It shows that few contact forces can pass through the interface and enter the soft granular layers. However, for another composite granular material, where the left part of granular system are composed by relative soft particles with a normal stiffness k1n = 107kg/s2 and the right part of system are composed by rigid particles with a normal stiffness k2n = 109kg/s2, the result is completely different. We find that although a strong force chain also occurs inside the left part of system, most of contact forces prefer to
pass through the interface and transmit into rigid particle layers, as shown in Fig. 7(b). In this composite granular system, the vertical force generates another strong force chain passing through the interface, which seems obviously to be stronger than the left one. To further investigate the process of force transmission in a composite granular material mentioned in Fig. 7, we also present the transmitting details of contact forces at several different instantaneous moments during the initial stage of loading, as shown in Fig. 8. From the figure, we find that the transmitting feature of contact forces inside a composite granular material is almost identical to that in a monodisperse system before the contact forces pass through the interface. When the contact forces continue to propagate and arrive at the interface from rigid granular layers to soft granular layers, its transmission along the right lattice direction is clearly constrained due to the existence of soft granular layers, as shown in Fig. 8(a). Therefore, the contact forces along the left lattice direction may gradually form a dominant force chain inside the left rigid granular layers whereas its corresponding one inside right soft granular layers becomes more and more weak during the initial loading processes. However, when the contact forces transmit from soft granular layers into rigid layers, most of them prefer to pass through the interface and converge into a stronger force chain inside rigid granular layers, as shown in Fig. 8(b). From the above results, we can find that under a localized loading, the contact forces inside a composite granular material exhibit an interesting transmitting feature, which is distinctly different from that inside a monodisperse granular system. When the contact forces transmit from soft granular layers to rigid granular layers, they prefer to pass through the interface and converge into a stronger force chain inside rigid part of the whole system. However, under a case where the contact forces transmit from rigid granular layers to soft granular layers, such variation of transmitting path of contact forces becomes more obvious. For a composite granular material where the external local force is perpendicular to the interface and the contact forces inevitably pass through the interface, the strong force chains in rigid layers may be disintegrated into a series of parallel weak force chains along its symmetric direction. Then the soft granular layers can trap the majority of these contact forces into a relatively small and concentrated region closer to the loading boundary, and form a stable contact force network to sustain the external loading. For another typical composite granular material where the external local force is parallel to the interface as shown in Fig. 7(a), the soft granular layers may directly constrain the contact forces to pass through the interface and induce them to propagate inside the rigid layers along the symmetric lattice direction. 4. The effect of thickness ratio of soft layers to rigid layers on response
Fig. 6. A left–right composite granular model composed by two sets of particles with different stiffnesses.
Above results demonstrate that the contact forces inside different types of composite granular layers exhibit different transmitting features under a localized force, especially inside a rigid–soft composite
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Fig. 7. The configurations of force chains in different composite granular systems under external force. (a) k1n = 107kg/s2, k2n = 105kg/s2. (b) k1n = 107kg/s2, k2n = 109kg/s2.
granular system, where soft granular layers have a significant effect on its force transmission within the whole system. Next we mainly focus on the influence of thickness ratio of soft layers to rigid layers on force transmission in composite granular materials mentioned above in Fig. 4. At first, we present the vertical force profiles on the floor for different thickness ratios of soft layers to rigid layers for two typical composite granular systems under a vertical external force, as shown in Fig. 9. Here m denotes the thickness ratio of soft layers to total layers in a rigid–soft composite granular system and m = 0 represents a monodisperse granular system, which is also presented in Fig. 9 as a comparison with other results. Fig. 9(a) presents a vertical force profile on the floor for different m inside a rigid–soft composite granular system. From the figure, we can observe three different types of force responses on the floor. For an identical external load, when m is small, the response exhibits a plateau floor pressure distribution, as shown in Fig. 9(a). With the increasing of m, the width of plateau response reduces gradually. When m approaches 0.5, which represents the thickness of soft granular layers is close to that of rigid layers, the floor pressure evolves into a one-peaked distribution. When m excesses 0.5 and continues to increase, we find that the floor pressure distribution undergoes a crossover from one-peaked to two-peaked features, and the width between two peaks also increases with m. So it is obvious that the thickness ratio of soft layers to rigid layers has a significant effect on its force transmission within a rigid–soft composite granular material. As a comparison, we also investigate the influence of the thickness ratio on the floor force distribution in soft–rigid composite granular systems, as shown in Fig. 9(b). Here n denotes the thickness ratio of
rigid layers to total layers in a soft–rigid granular system, and n = 0 also represents a monodisperse granular system. From the figure, we find that the floor pressure for different n all exhibits a two-peaked distribution, although there are some subtle changes quantitatively. This result demonstrates that the thickness ratio of soft layers to rigid layers almost does not affect the distributing feature of floor pressure inside soft–rigid composite granular systems. For a rigid–soft composite granular system, the thickness ratio of soft layers to total layers has a significant influence on the distributing feature of floor pressure, which is obviously due to the variation of propagating path of contact forces inside the whole system. In order to further explore such propagating feature, we have performed a serial of numerical experiments, and find that different propagating processes of contact forces lead to three different distributing features of floor pressure inside a rigid–soft composite granular system. Here we use several schematic diagrams to further explicate such propagating processes of contact forces for different m under identical external loads, as shown in Fig. 10. When the thickness of soft granular layers is smaller than that of rigid layers(i.e. m b 0.5), we find that each strong force chain can induce a serial of parallel weak force chains along its symmetric lattice directions inside upper rigid granular layers and these weak force chains can propagate through the interface. Therefore, two sets of parallel weak force chains, generated from two strong force chains respectively, may intersect together and form a contact force network below the loading point. A more interesting aspect is that the pressure distribution at certain height almost is almost uniform along the horizontal direction, which is also valid for floor pressure distribution. On these cases, the floor pressure exhibits a plateau distribution, as shown in Fig. 10(a).
(a)
3.5 10 5 s
4.5 10 5 s
5.5 10 5 s
6.5 10 5 s
3.5 10 5 s
4.5 10 5 s
5.5 10 5 s
6.5 10 5 s
(b)
Fig. 8. The transmission of force chains at different time instantaneous moments for different composite granular systems. (a) k1n = 107kg/s2, k2n = 105kg/s2. (b) k1n = 107kg/s2, k2n = 109kg/s2.
Y. Yang et al. / Powder Technology 261 (2014) 272–278
0.14
0.10
Fn/Fext
0.090
m=0.00 m=0.20 m=0.33 m=0.47 m=0.60 m=0.73
(a)
0.08 0.06
0.045 0.030
0.02
0.015
0.00 -40
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20
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n=0.00 n=0.20 n=0.33 n=0.47 n=0.60 n=0.73
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0.075
Fy/Fext
0.12
277
60
0.000 -60
-40
x/R
-20
0
20
40
60
x/R
Fig. 9. (a) Vertical force profiles on the floor for different ratio, m is the thickness ratio of softer layers to total layers in a rigid–soft composite granular system. (b) Vertical force profiles on the floor for different ratio; n is the ratio of rigid layers to total layers in a soft–rigid composite granular system.
We also find that once these new generating parallel weak force chains pass through the interface, they may decay rapidly without intersecting constrains from another corresponding set of parallel weak force chains inside soft granular layers. So it can be obviously observed that the distributing width of floor pressure is reducing with the increasing of m. When the thickness of soft granular layers is close to that of rigid layers (i.e. m = 0.5), the distributing width of floor pressure may tend to a point and the floor pressure may evolve into a one-peaked distribution from a plateau distribution, as shown in Fig. 10(b). With the increasing of m (i.e. m N 0.5), two sets of parallel weak force chains may intersect and converge into a vertical force located at a certain depth below the interface, approximately equaling to the thickness of upper rigid granular layers. In this situation, the propagation of contact forces inside soft granular layers below this vertical force is very similar to that inside a monodisperse system, and then the floor pressure exhibits a twopeaked distribution, as shown in Fig. 10(c). Above results show that the thickness ratio of soft layers to rigid layers is a very important factor for propagating features of contact forces inside a rigid–soft composite granular system. On the other hand, the existing investigation shows that the size of granular system is also a decisive factor affecting the propagation of contact forces inside a monodisperse granular system [15]. Here we present a phase diagram to identify the propagating features of contact forces inside a rigid–soft composite granular material, and here the thickness ratio of soft to rigid layers and the size of the whole system are two main factors for consideration. In our all simulations, we have employed a series of rigid–soft composite granular models with total layers from 5 to 35 since the floor response of deeper models becomes very weak under an identical local force Fext = 5 N. We find that with the increasing of soft granular layers, the floor response always exhibits three different distributing forms: plateau, one-peaked and two-peaked distribution, regardless of the number of total granular layers, as shown in Fig. 11. It indicates that indeed there are three different propagating modes of contact forces inside rigid–soft composite granular systems. In this phase diagram, we present three distributing regions just corresponding to
these three propagating modes. It can be found that the distribution of floor pressure is one-peaked when the thickness ratio of soft layers to rigid layers is close to 1, also as shown in the inset of Fig. 11, and it may evolve into a plateau and two-peaked distributions when this thickness ratio deviates from 1 to lower and to upper respectively.
5. Conclusions The transmitting feature of contact forces in a composite granular material is investigated based on the discrete element method, and the influences of the thickness ratio of soft layers to rigid layers are mainly considered. From the simulating results, we find that the transmission of contact forces under various composite modes of granular layers exhibit distinct different features comparing with that inside a monodisperse granular system. For a soft–rigid composite granular system, the propagating directions of strong contact force chains are nearly unchanged when they transmit from softer layers to harder layers. However, for a rigid–soft composite granular system, if the external local force is perpendicular to the interface, the strong force chains in rigid layers may be disintegrated into a series of parallel weak force chains along its symmetric direction. Then the soft granular layers can trap the majority of these contact forces into a relatively small and concentrated region closer to the loading boundary, and form a stable contact force network to sustain the external loading. In addition, for a rigid–soft composite granular system, the thickness ratio of soft layers to total layers has a significant influence on the distributing feature of floor pressure. We have performed a serial of numerical experiments, and find that different propagating processes of contact forces lead to these three different distributing features of floor pressure. Therefore, we have presented three schematic diagrams to further explicate such propagating processes of contact forces. Finally, a primary phase diagram is proposed for a rigid–soft composite granular system, in which three distributing regions just correspond to these three propagating modes of contact forces.
Fig. 10. Schematic diagrams of propagating processes of contact forces inside a rigid–soft composite granular material for different m.
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Two peaks
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Total granular layers Fig. 11. The phase diagram describing the propagating features of contact forces inside a rigid–soft composite granular material. The horizontal axis is the total layers of the system. The vertical axis represents the thickness ratio of soft layers to total layers. The phase diagram is also shown in the inset for which the vertical axis represents the number of soft layers.
Acknowledgments This work was supported by the Science Fund for Creative Research Groups of the National Natural Science Foundation of China (No.: 11121202), the National Natural Science Foundation of China (Grant No.: 11002064), and the Postdoctoral Science Foundation of China (2013M530434). References [1] Heinrich M. Jaeger, Sidney R. Nagel, Robert P. Behringer, Granular solids, liquids, and gases, Rev. Mod. Phys. 68 (4) (1996) 1259–1273. [2] A. Drescher, G. de Josselin de Jong, Photoelastic verification of a mechanical model for the flow of a granular material, J. Mech. Phys. Solids 20 (1972) 337–340.
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