J. Cent. South Univ. (2016) 23: 1132-1141

DOI: 10.1007/s11771-016-0363-x

Frictional characteristics of granular system under high pressure

Cao Miao-yan(曹秒艳)¹, Peng Ya-xin(彭雅新)², Zhao Chang-cai(赵长财)²,

Dong Guo-jiang(董国疆)³, Du Bing(杜冰)²

1. National Engineering Research Center for Equipment and Technology of Cold Strip Rolling (Yanshan University), Qinhuangdao 066004, China;

2. Key Laboratory of Advanced Forging & Stamping Technology and Science of Ministry of Education of China(Yanshan University), Qinhuangdao 066004, China;

3. College of Vehicles and Energy, Yanshan University, Qinhuangdao 066004, China

Central South University Press and Springer-Verlag Berlin Heidelberg 2016

Abstract: In order to reveal the force transmission features of the granules in the solid granule medium forming (SGMF) technology, the frictional characteristics of the non-metallic granule medium (NGM) under high pressure were investigated by tests and simulations. And the relevant changing curves of the internal friction coefficient of the granular system under different normal pressures were obtained by self-designed shear test. By the granule volume compression test, the accurate discrete element simulation parameters were obtained, based on this, the discrete element method (DEM) was adopted to reveal the evolution law of the NGM granules movement in the sample shear process from the microscopic view. Based on the DEM, the influence of granule diameter, surface friction coefficient, normal pressure and shear velocity on the internal friction coefficient of the granular system were studied. And the parameters were conducted to be dimensionless by introducing the inertia coefficient. Finally, the expression showing power-law relationship of inertia coefficient, surface friction coefficient and internal friction coefficient is obtained.

Key words: granule; friction coefficient; discrete element method; inertia coefficient

1 Introduction

The granular matter is a complex system composed of large amount of interactional discrete solid granules, which widely exists in nature and has many extraordinary characteristics different from those of solid and fluid. Even in the simplest case of dry and noncohesive granules, there still exist extremely sophisticated phenomenon and behavior [1–3]. The movement of single granule in the system obeys the Newton’s laws of motion, yet when the external force or internal stress condition changes, overall flow will take place and fluid characteristics will be exhibited, thus granular flow eventually forms. Granular media, whose features range from the particle scale to the force-chain scale and even the bulk scale, are usually modeled as either particulate or continuum materials. In contrast with other approaches, the network of force-chains is the natural first choice for the study of microscopic, mesoscopic, and macroscopic features [4-5]. Due to their widespread applications in various industrial fields (such as pharmaceutical production, chemical industry and food), agricultural industry and energy production, the granular flows have become the frontier research focus both in physics and mechanics [6-8].

Solid granule medium forming (SGMF) technology proposed by ZHAO et al [9], is a new forming process for sheets and tubes, wherein the rigid punch (or elastomer, liquid) is replaced by solid granules. Compared with other flexible-die forming processes, it is simple to be manipulated with simply-structured die, and multiple forming processes can be saved. Moreover, traditional hydraulic medium could not endure high temperature, although a concern in the past, has been significantly reduced with SGMF technology by using special granules with heat-resistant up to 600 °C. SGMF technology is fit for manufacturing all kinds of thin-walled components of light alloys such as titanium, magnesium and aluminum and some work-pieces shown in Figs. 1 and 2. This process has a broad application prospect [10-12].

During the SGMF process, the granules under compression state will be squeezed with each other and then deform. Among the dense granule flows, those with larger contact deformations will connect together to form quasi-lines and relatively stable strong force chains, which will constitute the force chain network to transmit most of the external load, while those with smaller deformations will form weak force chains. It is just through these force chains that the granule medium unevenly transmits the external load to the surfaces of the tubes or sheets to be formed, leading to their deformation gradually and sticking to the inner wall of the die, thus the forming process will be eventually completed.

Fig. 1 Schematic diagram of SGMF experimental device and work-pieces of metal tubes:

Fig. 2 Schematic diagram of SGMF experimental device and work-pieces of sheet metals:

During the forming process, the granule medium continuously changes its original shape along with the deformation of the work-piece, and shear movement of the granules will happen under high pressure. Due to the existence of the granular surface friction, the force chains of the granular system are able to resist certain shear deformation, thus obstruct the tangential movements of the granules, which affects the fittability and forming quality of work-piece finally. So the study on the granular frictional characteristics is important to obtain the force-transfer performance of granules and control the work-piece precision forming.

Some scholars have studied the friction characteristics of granular system by direct shear test (DST) under the pressure within 1 MPa [13-14], yet there is a certain gap between granules and the shear dilatancy is obvious. In SGMF technology, granule medium is compacted under normal pressure of 30- 200 MPa, which is different from that of former works. In this work, friction coefficient tests were carried out, and discrete element simulations of the tests were performed to reveal the mechanical properties of granular system under high pressure, since the DEM can provide mesoscopic information that is difficult to obtain by test. The coordination number, volume fraction and force chains etc. were compared and the friction mechanism of granular system was further studied.

2 Experimental study on friction coefficient of granular system

In order to study the internal friction coefficient of the granule medium under high pressure, the shear test device was designed, composing of punch, square block, upper box, shear box, push rod, etc., as shown in Fig. 3. During the test, the granules were filled into the square cavity constituted by upper box and the shear box, and certain pressure was exerted on the granules through the square press block by the punch, and then through uniform twisting of the bolt. transverse displacement of the sliding box was generated, thereby the granular system was sheared. Through the pressure sensor and the displacement sensor, the data were simultaneously output, then collected by the collector and finally input into the computer for processing. This test was conducted on WEW-1000 universal testing machine, also YD-28A dynamic resistance strain gauge, 5CB-10C precise digital displacement meter, INV306G (LF) intelligent signal collecting, processing and analyzing system were used.

In the test, the approximately uniform spherical non-metallic granule medium (NGM) of certain diameter was studied, the major ingredient of which is SiO₂ and the other associated material parameters are listed in Table 1.

It can be seen from Fig. 3, the granule sample exhibits ‘_┗’ shape before shear, which gradually becomes ‘_┻’ shape with the shear proceeding, and turns into the ‘_┛’shape when the shear process completes. As a result, this test device can effectively reduce the errors caused by the gradual shrinkage of the shear surface of the sample during the shear process of DST. During the whole shear process of this device, the shear area was always kept consistent with the bottom area of the square block.

Fig. 3 Schematic diagram and device for internal friction coefficient test of granule medium system

Table 1 Material parameters of NGM

In the test, a constant force N was applied on the granular system through the punch, and then the load F was applied on the shear box through the horizontal loading device, causing the uniform transverse shear movement of the granular system, wherein both N and F could be obtained by the collection system. Due to the friction coefficient (μ_s) between the sliding block and the baseplate, the friction coefficient of test system μ_a which could be gotten by μ_a=F/N, is the sum of μ_p and μ_s. Without granules filled, μ_s could be gained by the same way as μ_a. Thus, μ_p is eventually obtained as

In the test, the sizes of the upper box and the shear box are 30 mm×30 mm×20 mm and 40 mm×30 mm× 20 mm, respectively. For each test, the sample was prepared with new granules poured into the shear box at a constant speed, and then naturally deposited relying on the gravity, after the shear box was fully filled, the normal pressure was applied to the upper surface of the granules by punch, subsequently, the shear test began at a speed of 0.1 mm/s.

The variation curves of internal friction coefficient μ_p with the shear displacement (s) of NGM under different normal pressures are obtained, as shown in Fig. 4(a). As can be seen, when a certain normal pressure is applied on the granule sample, with the shear proceeding, the internal friction coefficient gradually increases from zero, and the variation curve of μ_p with s is approximately linear within 1 mm of the shear displacement, which indicates that elastic deformation mainly occurs after each extrusion of granules in this stage, thus μ_p gradually rises, and rare granules are sheared on the shear surface.

When s reaches 4 mm, μ_p no longer rises basically, and it slightly fluctuates around a certain value. Index μ_pm represents the peak value of μ_p, and μ_pm=0.34 when normal pressure p=11 MPa. The relation curves of μ_pm–p is shown in Fig. 4(b). As can be seen, with increasing of p, μ_pm gradually decreases, but it keeps stable when p≥ 78 MPa.

3 Mesoscopic structural analysis of granule medium shear process

The granular system is a multi-scale structural system, which can be divided into three scales according to the structure level: the microscale single granule, the mesoscale force chains and the macroscale granular system, wherein the formation and evolution laws of the force chains are the core of the multi-scale mechanics research on the granule medium [15-16]. Therefore, the mesoscopic simulating using DEM is proved to be aneffective method in revealing the internal deformation characteristics and evolution laws of the granular system under high pressure [17-19].

Fig. 4 Measured curves of internal friction coefficient μ_p of granular system:

DEM for granules was first proposed as a numerical method by CUNDALL and STRACK in 1979 [20], when it was applied on the basis of a single granule to research on the physical structure and motion law of granular material. The granular material description of this method differs from that of traditional continuous medium theory, and it is established on the basis of Newton’s second law. For its solution, two equations are employed alternatively and cyclically, one is the motion equation, and the other is the contact force equation.

In this work, particle flow code in 2-dimension (PFC2D) [21] is employed with the contact model which is a linear contact stiffness one, as shown in Fig. 5. As seen, each contact is constituted by two linear springs, i.e. the normal and the tangential ones, with the corresponding stiffness coefficients of k_n and k_s, respectively. A slip exists in the tangential direction with a friction coefficient of μ_m. The granules are allowed to overlap, and the overlap amount can be used to calculate the contact force through combination with the spring stiffness. The normal and tangential forces F_n can be given by the following two equations:

where ΔU_n and ΔU_s are the overlap increments in the normal and tangential directions, respectively. DEM has earned a more and more widespread application in the research on the microstructure of granular system.

Fig. 5 DEM contact model

3.1 Establishment of basic DEM mechanical parameters

There exist many unmanageable factors between the computing elements in DEM and the real materials, wherein the nonlinear factors are obvious and influence mutually. As a consequence, it is difficult to match the macroscopic mechanical parameters with the mesoscopic structural ones of the granular physical model. In order to seek the relationship between them and obtain the proper mesoscopic discrete element parameters, DEM is adopted to simulate the volume compression test and the associated parameters are adjusted to get the simulation values consistent with the measured ones, then they are applied to the shear test simulation. The volume compression test device of granule medium is designed, as shown in Fig. 6, and NGM is selected as the medium, with the barrel diameter of 80 mm and the granular assembly height of 170 mm. The variation curves of theaxial strain ε with the normal pressure p of the punch are eventually obtained, as shown in Fig. 7, and the relationship of ε-p could be accurately fitted by power function.

Fig. 6 Schematic diagram of volume compression test

Fig. 7 Relation curves between ε and p of NGM granules with fitting formula

In this discrete element simulations, the physical system can be completely described by a series of independent parameters of granules and the shear velocities, wherein the former includes the granule diameter d, the material density ρ, the granular surface friction coefficient μ_m, the friction coefficient μ_w between the granules and the wall, the elastic modulus E of granular material, the normal and tangential stiffness coefficients k_n and k_s, respectively, wherein k_s is of the same order of magnitude as k_n [22]. As can be obtained from Ref. [23], the tangential stiffness exhibits a minor influence on the simulation result. Consequently, it is set as 0.4k_n in this work, and the detailed parameters are listed in Table 2. The numerical model of the granular system is consisted of spherical granules with diameters of 0.12, 0.13 and 0.14 mm at a ratio of 2:5:3. The granules are deposited under the gravity effect firstly, and then compressed by punch. The simulation values obtained by DEM keep consistent with the measured ones in Fig. 7, which demonstrates the reliability of DEM eventually.

Table 2 DEM simulation parameters

3.2 Shear test simulation under high pressure

3.2.1 Numerical model establishment of shear process

In order to study the friction characteristics of the granular system under high pressure from mesoscopic view, based on the calculation parameters in Table 2, two-dimensional DEM simulation is performed on the shear test, and the shear simulation process is divided into four steps, as shown in Fig. 8.

Fig. 8 Simulation of shear process:

Step 1: Boundary establishment. Boundary walls 1 -8 are generated according to the shear box sizes in the actual test in Fig. 8(a).

Step 2: Granular system generation. Within the zones constituted by walls, a sample composed of 37800 granules are generated and fully contacted with each other in the multi-layer by undercompaction method [15]. The zone constituted by walls 3, 4, 5, 6 and 8 represents the upper box, while that by walls 1, 2, and 7 represents the shear box. So as to clearly show the deformations of each part of the granular system during the shear process, the sample is divided into multiple strip regions, endowed with different colors respectively, as shown in Fig. 8(b).

Step 3: Application of normal pressure. Wall 5 is removed and replaced by a raft punch that is formed by the cluster composed of 30 balls. And the cluster punch can apply a servo pressure on the top surface of granular system, as shown in Fig. 8(c).

Step 4: Shear the granules. Walls 1, 2 and 7 move at a specified speed along the x direction to shear the granules with constant normal pressure, and the position of the punch is real-time adjusted as the shear proceeding. The shear state of the granules at the end of the shear process is shown in Fig. 8(d), as can be seen, the area with larger deformation of the granular system is located in the shear band.

3.2.2 Velocity fields during shear process

When the normal pressure p=60 MPa, the quasi- static velocity fields of granular system are shown in Fig. 9. At the beginning of the shear process, for the granular system starting to shear with a static state, the whole sample has trend to move along the shear direction, as shown in Fig. 9(a). With the proceeding of the process, obvious ‘vortex cloud’ of velocity appears at the shear band. Every time after the vortex velocity field varies from slight gathering state to a certain strength in Fig. 9(b), it dissipates rapidly to form a new vortex, as shown in Fig. 9(c), whose vortex center moves along the shear plane, and soon the new vortex disappears with a certain strength. As a result, a lager translational velocity field appears in whole of the shear box, as shown in Fig. 9(d).

The phenomenon that the vortex cloud appears in the velocity field of the shear band goes throughout the whole shear process, which is due to the fact that the granules in the shear box need to extrude and get pass the ones ahead of them, and part of the granules rotate under the contract force. Owing to the special structure of the granular system, the rotation can keep for some time with the energy concentration until the break of the force chains, the release of the energy and the disappearance of the vortex cloud. The shear band is formed through the aggravation of the rotation and the concentration [23].

During the shear process, the internal structure of the granules system changes continuously, wherein the equilibrium state among the granules is broken and restructured constantly, and that cycle repeats.

Fig. 9 Quasi-static velocity fields:

3.2.3 Organizational structure and force chain network

In order to reveal the structure variation laws of the granular system during the shear process, the force chains across the entire test under p=78 MPa is displayed in Fig. 10.

The granules persistently keep close contact with each other to transmit the internal momentum in the shear process. The elastic wave caused by the contact spreads at the sound velocity v_s along the force chains in the granular system, and v_s is associated with the elastic modulus E and Poisson ratio ν of the granules. In the dense granule flow, the transmission time of the energy and force between the granules along the force chains is far shorter than the duration one of the force chains. Consequently, the force chains can be regarded as a stable structure, the evolution process of which can be treated as a quasi-static one [24].

Fig. 10 Force chains evolution diagram of solid granules during shear process

The original state of the granular sample is shown in Fig. 10(a), the contact force chain network is relatively uniform. While the granules begin to bear normal pressure as shown in Fig. 10(b), and the dendritic force chains stretch into the far distance from punch, and its line thickness is proportional to the contact force. In Figs. 10(c) and (d), the evolution process of shearing and the end state are shown respectively, at this moment, the granular system is in a high-pressure state, the force chains near the shear plane approximately presents an angle of 30° to the horizontal plane, while the force chain structure outside the shear band exhibits unobvious variation. As can be seen from Fig. 10, the force chain structure will be changed and follow the principle of begin to generate → gradually strengthen → shear fracture → regenerate new strong chain during the entire process.

Coordination number Z and volume fraction Γ are two important parameters of granular physical properties, which has significant effect on its geometrical structure. It can be found from the simulation that the response of the granular matter under compression is mainly resulted from the dynamic variation of the force chain structure to resist the external load, not just the surface friction between granules. When p=33 MPa, Z is 4.8 near the shear band, and the contact force between granules is small, the force chains are relatively few and mainly weak ones. While the shear force is mainly induced by the edge angle structures formed by the granules in the shear band against the external load, therefore, μ_p is correspondingly larger. Yet for the force chain structure is not mature, μ_p fluctuates dramatically. With the increase of normal pressure, the average coordination number increases rapidly. When p=78 MPa, Z=5.5 near the shear band, the force chain network gradually grows mature, and the entire system tends to be a stable structure. The friction of granular system isn’t mainly provided by the original edge angle structures but the granular surface friction, which contributes to the reasons why μ_p first rises then falls with p increasing in the range of 10-100 MPa. When p≥67 MPa, μ_p keeps stable and almost change no more，as shown in Fig. 11(a), which has the same trend with the test one (Fig. 4). The simulation result maintains consistent with the test one in Fig. 11(b), thus the reliability of the discrete element simulation is demonstrated again.

Fig. 11 Internal friction coefficient μ_p plotted against shear displacement s:

4 Influence of mesoscopic parameters on internal friction coefficient of granule medium

In order to further study the effects of mesoscopic parameters on internal friction coefficient of the granule medium, the surface friction coefficient μ_m, the granule diameter d and the granule rigidities (k_n, k_s) are discussed respectively as follows.

4.1 Influence of surface friction coefficient on internal friction coefficient of system

The surface friction coefficient can affect the connection state between the granules, thus affect the macroscopic shear performances. When p=78 MPa, μ_m are 0.1, 0.2 and 0.4, respectively, the simulation curves of μ_p are shown in Fig. 12(a). As can be seen, μ_p increases accordingly with μ_m. Meanwhile, it can be also found that a linear relationship between μ_pm and μ_m of the system is exhibited in Fig. 12(b).

Fig. 12 Influence of surface friction coefficient μ_m on surface friction coefficient μ_p of d=0.13 mm:

SCOTT [25] found the relationship between the granular surface friction coefficient μ_m and the internal friction angle of the granular system through the shear test on the isometric spherical granules.

where is the internal friction angle corresponding to the internal poeak strength of the granular system, and μ_m is the surface friction coefficient of granular material.

ZENG and ZHOU [26] conducted shear test simulations on combined granules with different shapes, and it was generally considered that there was a linear relationship between μ_m and μ_pm of the granular material, when the granules were irregular with smaller μ_m, certain nonlinear relationship was presented between them.

Especially, when the granular surface is absolutely smooth, i.e. μ_m=0, μ_p is not 0, but close to 0.2. When μ_m=0, the force chain structure of granular system is shown in Fig. 13, wherein obvious lattice fission phenomenon is exhibited, and the granular system is divided into several zones, which is a specific phenomenon when the smooth granules with similar diameters are regularly arranged. A vast majority of force chains in the granular system are weak onesdistributing uniformly, while only a few of strong ones distribute in the lattice crack band, which withstands the extrusion forces between adjacent granules.

Fig. 13 Force chain structure when μ_m=0

The surface friction coefficient μ_m has a significant influence on the volume fraction Γ. When p=78 MPa, the change curves of Z–s and Γ–s of the shear band are obtained, as shown in Fig. 14. It can be seen that Z and Γ rise rapidly with s until a stable value. When μ_m=0, Z gradually increases from the original value 5.59 to the stable value 5.78, and Γ gradually increases to 0.976. Figure 15 shows that the coordination number and the volume fraction both are decreased to a steady value with the increase of μ_m.

4.2 Influence of granule diameter on internal friction coefficient

The curves of μ_p–s under d = 0.13, 0.25 and 0.5 mm are shown in Fig. 16, respectively. As can be seen, μ_p increases with increasing d, so does its fluctuation, due to larger granules can bear greater deformation resistance and move more difficult, a larger value of μ_p is exhibited. Similarly, the equilibrium state of the force chain structure can not be broken easily, once broken, a longer time will be needed to establish a new equilibrium state.

Fig. 14 Curves of Z-s (a) and Г-s (b) under different values of μ_m

Fig. 15 Influence of μ_m Z (a) and Г (b)

Fig. 16 Relation curves of μ_p versus s under different granule diameters d when μ_m=0.2 and p=78 MPa

4.3 Influence of shear velocity on internal friction coefficient of granule medium

The forming rate serves as a key factor in influencing the yield and forming quality of the products in SGMF technology, and μ_p varying with the forming rate results in transmitting pressures different from the granule medium. As shown in Fig. 17, μ_p–s curves under different shear velocities v_w are simulated by DEM. When v_w=0.1, 0.2 and 0.4 mm/s, μ_p increases with v_w, so does its fluctuations, which is consistent with the test one.

Fig. 17 Relation curves of μ_p–s under different v_w

5 Rheological law of granule medium under high pressure

In order to seek the rheological law of the dense flow under high pressure, the dimensionless parameters are introduced referring to Refs. [26-28], where the height of granular system in the shear box is L, and the shear velocity is v_w. In order to guarantee all the parameters to be dimensionless, in the following, = v_w/L is noted as the equivalent shear rate. For there is already one dimensionless quantity μ_m, inertia coefficient I is introduced to conduct dimensionless treatment on the rest dimensional parameters of the granules, such as the diameter d, the density ρ and the equivalent shear rate :

A reasonable interpretation of this parameter in terms of the ratio between two time scales is given in Ref. [28], wherein the microscopic time scale represents the time it takes for a single granule to go through a hole of diameter d under the pressure p, i.e. it provides the typical timescale of granule rearrangements; while the macroscopic scale parameter is associated with the flow of granules. Small values of I correspond to quasi-static flows, wherein the macroscopic deformation of the granular system with microscopic rearrangement is rather slow; on the contrast, large values of I correspond to rapid flows. From the dimension analysis, it can be known that there are two ways to switch from the quasi-static to the inertial flow, one can either increase the shear rate or decrease the pressure.

Compared with the ordinary plane shear [27-30] and the common direct shear [14, 31-33] tests on the granular system, the order of magnitude of the granules inertia coefficient in high-pressure shear process is extremely small. Through the simulation analysis, the volume fraction is greater than 0.85, and the coordination number is close to the limit value 6, both at higher levels, and the granular system of this state belongs to the category of quasi-static flow.

According to the simulation values above, the following expression can be obtained by means of regression analysis

As can be seen, μ_p is not zero actually when μ_m=0, which demonstrates that μ_p is associated with the force chain structure and evolution laws under extrusion or shear, and it increases rapidly with I in the power law.

6 Conclusions

1) The matching parameters for DEM analysis on the NGM are obtained through the volume compression tests, and the shear simulation model with DEM is established based on them. Comparison between simulation results and real shear test results of the granular system verifies the correctness and practicability of the model.

2) The internal friction coefficient of the granular system increases with increasing granule diameter, surface friction coefficient and shear velocity, respectively, while decreases with the rising of normal pressure.

3) To reduce the number of parameters and clearly express the interrelationships between the internal friction coefficient and each parameter, the dimensionless inertia coefficient is introduced. The inertia coefficient order of magnitude of granular material under high pressure is extremely small. The volume fraction is greater than 0.94, and the coordination number is greater than 5; both of them gradually decay to a steady value with the rising of the surface friction coefficient. The granular system of this state belongs to the quasi-static flow. The internal friction coefficient of the granules can be expressed in the power function by two dimensionless parameters, i.e. the inertia coefficient and the surface friction coefficient.

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(Edited by FANG Jing-hua)

Foundation item: Projects (51305385, 51305386) supported by the National Natural Science Foundation of China; Project (QN20131080) supported by the Science Research Youth Foundation of Hebei Provincial Colleges and Universities, China

Received date: 2015-02-27; Accepted date: 2015-05-30

Corresponding author: ZHAO Chang-cai, Professor, PhD; Tel: +86-13633333873; E-mail: zhao1964@ysu.edu.cn