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Particle shape consideration in numerical simulation of assemblies of irregularly shaped particles
Particuology 9 (2011) 387–397
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Particle shape consideration in numerical simulation of assemblies of irregularly shaped particles Saba Abedi, Ali Asghar Mirghasemi ∗ School of Civil Engineering, College of Engineering, University of Tehran, Tehran P.O. Box 11155-4563, Iran
a r t i c l e
i n f o
Article history: Received 30 June 2010 Received in revised form 23 November 2010 Accepted 30 November 2010 Keywords: Discrete element method Granular soil Angularity Confining pressure
a b s t r a c t The mechanical behavior of granular materials depends much on the shape of the constituent particles. Therefore appropriate modeling of particle, or grain, shape is quite important. This study employed the method of direct modeling of grain shape (Matsushima & Saomto, 2002), in which, the real shape of a grain is modeled by combining arbitrary number of overlapping circular elements which are connected to each other in a rigid way. Then, accordingly, a discrete-element program is used to simulate the assembly of grains. In order to measure the effects of grain shape on mechanical properties of assembly of grains, three types of grains—high angular grains, medium angular grains and round grains are considered where several biaxial tests are conducted on assemblies with different grain types. The results show that the angularity of grains greatly affects the behavior of granular soil. © 2011 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.
1. Introduction Published data show that particle shape has considerable influence on the engineering properties of granular soils. Tests conducted by Holubec and D’Appolonia (1973) on medium to fine sands with varying particle shapes indicated that granular materials with the same relative density could have different mechanical behavior due to angularity. They concluded that the variation of engineering properties due to particle shape could be of the same order of magnitude as the variation of properties due to changes in relative density, thus suggesting that particle shape could be considered an index property to correlate the properties of granular materials. Rothenburg and Bathurst (1992) reported the results of numerical simulations of planar assemblies of elliptical particles. Packing simulations of the initial assembly showed that, for ellipses with eccentricities up to about 0.2, the coordination numbers of the generated assemblies increased with increasing eccentricity. The resultant peak angle of internal friction in biaxial shear simulations showed the same trend and correlated well with the initial coordination number. Ting, Khawaja, Meachum, and Rowell (1993) reported similar conclusion from isotropic compression and biaxial shear test simulations on assemblies of two-dimensional elliptical particles.
∗ Corresponding author. Tel.: +98 21 66957783; fax: +98 21 66957787. E-mail address: [email protected] (A.A. Mirghasemi).
By means of numerical simulations of assemblies of polygonshaped particles, Mirghasemi, Rothenburg, and Matyas (2002) concluded that particle angularity had an important effect on the compressibility and shear strength of the granular media. In order to obtain a precise modeling method that would be comparable with experimental results, the specifications of grains such as form, size, elasticity, plasticity should be modeled carefully. Since the shape of the grain affects tremendously the mechanical behavior of granular media such as shear resistance, appropriate modeling of grain shape is considered very important. In the early discrete element method (DEM) scheme presented by Cundall and Strack (1979), the grains were modeled as discs in 2D and as spheres in 3D simulations with the advantage of simplicity in calculation: contacts between grains can be detected with simple algorithms, and in the simulated tests circular or spherical grains can move and rotate easily, though not without problems in conforming reality. The internal friction angle of shearing resistance is much less for circular or spherical grains compared to that of actual grains with irregular shapes. Resistance to rotation for circular or spherical grains is much less than that of actual grains, particularly for grains with inherent tendency to roll. On the other hand, the direction of the normal contact forces is always toward the center; these forces, however, never contribute to the moments acting on the grains, rotation being only affected by tangential contact forces. Because of these problems different shapes for grains were presented for use in DEM simulations in order to improve the results of simulation. In most cases of modeling that have been applied so far, grain form is generally considered elliptical,
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doi:10.1016/j.partic.2010.11.005
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Nomenclature A As AV Fi fiC Fn Fn Fs Fs I Kn Ks ljC m M n s SF x¨ i ¨ mobilized 1, 2 ij εa εv
total area of the assembly area occupied by grains in the assembly void area between grains in the assembly grain unbalanced force contact force vector normal contact force between two particles Incremental normal contact force shear contact force between two particles incremental shear contact force grain moment of inertia grain stiffness in normal direction grain stiffness in shear direction contact length vector grain mass grain resultant moment relative normal displacement at a contact relative shear displacement at a contact shape factor grain translational acceleration grain rotational acceleration mobilized friction angle major principal stresses stress tensor axial strain volumetric strain
ellipsoidal, polygonal, etc., none of which, however, directly models any irregularly shaped grain, hence will not project the expected mechanical behavior, thus pointing to the need of direct modeling of grains with irregular shapes.
2. Methods for modeling grain shape 2.1. Brief review of particle shape modeling Many researchers studied granular soil behavior in terms of ellipse-shaped grains in 2D and ellipsoid-shaped grains in 3D simulations (Bagherzadeh-Khalkhali & Mirghasemi, 2009; Lin & Ng, 1997; Ng, 1994; Rothenburg & Bathurst, 1992; Ting et al., 1993), apparently with the following advantages: grains have unique outward normal so calculations of forces are simple; grains have fewer tendencies to rotate, making simulated mechanical behavior similar to that of real soils. Such modeling, however, does not accurately represent the shape of grains (Fig. 1(a)). Many studies considered granular soil behavior in terms of polygon-shaped grains (Bagherzadeh-Khalkhali, Mirghasemi, & Mohammadi, 2008; Barbosa & Ghaboussi, 1992; Matuttis, Luding, & Herrmann, 2000; Mirghasemi, Rothenburg, & Matyas, 1997; Mirghasemi et al., 2002; Seyedi Hosseininia & Mirghasemi, 2006, 2007), to show rather realistic representation of soil behavior (Fig. 1(b)), though the method is time-consuming. Potapov and Campbell (1998) proposed the elliptical approximation with oval-shaped boundaries determined by four continuously jointed circular arches. This method involves simple calculations because contacts between two circular segments of grains are similar to contacts between circles in the original DEM code. The results of simulated tests using this method show that this method is effective; though the real shape of the grain is not effectively represented (Fig. 1(c)).
In the method presented by Favier, Abbaspour-Fard, Kremmer, and Raji (1999) to model axi-symmetrical grains, particles are modeled by combining multiple overlapping spheres with fixed inner-sphere connections. Simulations with this method showed good agreement with experimental results. This method can model any shape of grains, unless they are highly angular (Fig. 1(d)). Jensen, Bosscher, Plesha, and Edil (1999) presented a clustering method in which the grains are modeled by combining nonoverlapping circular elements in a semi rigid configuration; The contacts between circular elements are linear-elastic with a high stiffness. Although this method has some improvement over earlier methods, the outlines of simulated grain do not resemble those of the actual grains. Also computational time is long (Fig. 1(e)). In all of the above methods, direct modeling of grain shape is not considered. Because of the considerable effects of grain shape on mechanical behavior of soils, it seems worth conducting discrete element method with grains whose shapes are directly modeled.
2.2. Direct modeling of grain shape In this method, the real shape of a grain is modeled by combining an arbitrary number of overlapping circular elements which are connected to each other in a rigid way. Fig. 2 shows a simple algorithm to find the optimized number of circular elements to model the real shape of a grain (Matsushima & Saomto, 2002; Matsushima, Hidetaka, Matsumoto, Toda, & Yamada, 2003), the resulting outline closely coinciding with that of the real grain (Fig. 3). Fig. 3(a) shows the outline of an arbitrary grain that is shown in Fig. 3(b). By combining overlapping circles, closely real shape of the grain is generated (Fig. 3(c)), showing that the created shape with circular elements (Fig. 3(d)) closely resembles the real shape of the grain shown in Fig. 3(b). The actual shape of a grain can be precisely modeled with a maximum of 10–15 circular elements. The number of circular elements for modeling a grain shape depends on the degree of nonuniformity and angularity of the actual grain shape, the desired level of accuracy for the grain shape and the limitation of computation time. This direct modeling method can be used to realize realistic results for mechanical behavior of granular soils. The results of simulated assemblies of grains have been compared quantitatively with experimental results of actual assemblies (Matsushima & Saomto, 2002). The calculations for simulated assemblies are similar to assemblies with circular grains, for which no complicated algorithm is required. The program DISC (Bathurst, 1985), which is a modified version of BALL (Cundall, 1978) is adopted and modified to simulate assembly of grains with real shape. This program, calls for the Newton’s second law to compute displacements for each grain from the current resultant forces and the moments acting on the grain, as follows:
Fi = m¨xi , ¨ M = I ,
i = 1, 2
(1)
where Fi is the unbalanced force on each grain, M is the resultant moment on each grain, m is the mass of the grain, I is the moment of inertia of the grain, and x¨ i and ¨ are the grain accelerations. For each grain, accelerations are integrated over small time-steps to give velocities and displacements. The time-step is chosen small enough that the velocities and accelerations can be assumed constant over it. The calculated displacements are used in contact law or force–displacement law through which the con-
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Fig. 1. Several modeling methods in DEM simulations with: (a) elliptical and ellipsoidal elements, (b) polygonal elements, (c) oval elements, (d) axisymmetrical elements and (e) overlapping rigid clusters.
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Fig. 2. Optimized number of circular elements to model real shape of a grain (after).
tact forces between two circular grains (Fig. 4(a)) are computed as follows: (Fn )new = (Fn )old + Fn = (Fn )old + Kn n, (Fs )new = (Fs )old + Fs = (Fs )old + Ks s,
(2)
where Fn and Fs are the normal and shear contact forces, Fn and Fs are the incremental normal and shear contact forces, Kn and Ks are the grain stiffness in normal and shear directions, and n and s are the normal and shear relative displacements at a contact, which are obtained by integrating the relative velocity component with respect to time. The relative velocity components are calculated by projecting relative velocities at the contact point in the normal and tangential directions to the contact plane. Calculated forces and resultant moment of these forces are used in Newton’s second law and calculations are repeated till the unbalanced forces on the grains are very small or close to zero. In the modified program, the contact considerations and calculations of contact forces are conducted for each circular element of the grain that is in contact with the circular element of another grain (Fig. 4(b)), and the equation of motion is solved for each grain. With this procedure, rigid connections between circular elements of each grain are obtained without definition of stiffness between these elements.
Fig. 3. (a) Outline of actual particle, (b) actual particle, (c) modeling with circular elements and (d) modeled grain.
3. Simulated tests The method illustrated above is an appropriate tool to investigate the effects of different shapes of grains on their mechanical behavior. Three series of grains with different angularities are considered: high angular grains, medium angular grains and round grains, as shown in Fig. 5. Then a series of simulated biaxial tests are conducted on the assemblies created by each series of grains, as described below.
Fig. 4. (a) Two circular grains in contact with each other and (b) two circular elements of two grains in contact with each other.
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Fig. 5. (a) High angular grain, (b) medium angular grain and (c) round grain.
3.1. Selection process of grain shape As mentioned above, three series of grains with different angularities are considered. In the first step, 16 types of high angular grains were chosen with equivalent radius ranging from 5 mm to 20 mm. Then, medium angular grains and round grains are generated through reduction of angularity in high angular grains. The general shape and size are similar for three series as shown in Fig. 5. In order to have a quantitative comparison between the shapes of the three series of grains, a shape factor (SF) developed by Sukumaran and Ashmawy (2001) is used to measure the deviation of a grain outline from that of a circle. The polygon shape of the grain is first sliced into n sections. The orientations of the lines connecting points on the perimeter of the polygon are compared with the orientations on lines connecting the polygon created by slicing a circle. The angle which constitutes the difference in orientations, ˛, and the number of sampling points N are used to define the shape factor as: n ˛i grain
SF =
i=1
N × 45◦
× 100%.
(3)
For all practical grain shapes, values of shape factor will lie between zero and one, with zero to correspond to a circle, and one to correspond to a flat plate. In this research, the range and average values of shape factors for three series of grains are summarized in Table 1. Research showed that grain elongation has considerable effects on mechanical behavior. Elongation is defined as the ratio of the length of a particle to its width. In order to consider only the effects of angularity on behavior, the effects of elongation are minimized by selecting grains with equal length and width (elongation equal to 1). In the next step, the grains are modeled by combining circular elements as described in Section 2.2. 3.2. Assembly preparation In this study, circular assemblies consisting of 1000 grains each are used with grain size distribution shown typically in Fig. 6, in which equivalent diameter refers to that of an equivalent circle having the same area as the grain. Circular assembly is first generated for high angular grains. By trial and error, the radius of this circular assembly is chosen to be the lowest possible value that could contain 1000 grains. In other words, a densest initial assembly is generated to minimize computation time for compacting the assembly. Then, the location of the center of gravity of each grain in the assembly is determined and Table 1 Range and average values of shape factors SF for three series of grains. Group
SF range (%)
SF average (%)
High angular grains Medium angular grains Round grains
38–47 27–40 8–15
42 37 12
Fig. 6. Grain size distribution in assemblies created by each series of grains.
registered in an output file. For generating assemblies with medium angular and round grains, this output file is used and the location of each grain is selected coincident with the location of similar high angular grain. This procedure provides better comparison between results of assemblies generated with each series of grains. Fig. 7 shows the generated assemblies with each series of grains. 3.3. Biaxial simulation Each simulated biaxial test includes four stages, as shown in Fig. 8. The generated assembly is first subjected to a defined strain rate to compact the initial loose assembly. In the next stage, zero strain rate is applied in order to bring the assembly to equilibrium. In Stage 3, the assembly is subjected to a defined isotropic confining pressure. In Stage 4, biaxial test is carried out on the assembly. In the simulated biaxial test, the horizontal stress is held constant and the vertical stress is increased by applying constant deviatoric strain rate. In a biaxial test, the mobilized internal friction angle of a cohesionless material can be determined by the following equation as a function of major principal stresses, 1 and 2 : Sin mobilized =
2 − 1 . 2 + 1
(4)
Major principal stresses are determined on the basis of the average stress tensor within an assembly. The stress tensor of an assembly can be calculated by the following equation (Rothenburg, 1980): ij =
1 C C fi lj , A
i, j = 1, 2
(5)
C ∈A
in which ij is stress tensor, A is the area of the assembly, fiC is contact force and ljC is contact vector between two circular elements of two grains that are in contact with each other. To consider the effect of different factors on mechanical behavior, three series of simulated biaxial tests were conducted on assemblies with each series of grains: tests with different confining pressures, tests with different friction coefficients and tests with different void ratios. This section describes the details of these three series of tests. 3.3.1. Tests with different confining pressures Table 2 lists the parameters used in these tests for the discrete element method (DEM). Normal stiffness and tangential stiffness are used in force–displacement equations for computation of con-
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Fig. 7. Assemblies generated by 1000 (a) high angular grains, (b) medium angular grains and (c) round grains.
tact forces. Friction coefficient indicates the roughness of grain surface and damping coefficients are defined to dissipate kinetic energy so that static equilibrium condition for assembly of grains can be reached. For better comparison of results, equal parameters were considered for the three assemblies. Table 2 DEM parameters used in simulated tests. Parameter
Value
Normal contact stiffness (N/m) Tangential contact stiffness (N/m) Inter-particle friction coefficient Inter-particle cohesion Density of particles (kg/m3 ) Damping coefficients
245 × 107 245 × 107 0.5 0.0 2000 5 and 0.01
The loose generated assemblies were subjected to a strain rate equal to 0.005 in 500,000 cycles to become compact. After relaxing assemblies by applying zero strain rate to boundary particles, the assemblies were subjected to different isotropic confining pressures (0.1, 0.5, 1, 2 and 4 MPa). Table 3 shows the ratio of the area occupied by grains to the area of the entire assembly, which is called density, for three series of assemblies when they reached equilibrium under different confining pressures. According to the above table, density increases slightly with increasing confining pressure and decreasing angularity, the differences being by no means substantial. In the last stage, biaxial tests were carried out on the assemblies. In the simulated biaxial test the deviatoric strain rate was chosen to be 0.0005.
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Fig. 8. (a) Generated assembly of high angular grains; (b) Stage 1: assembly is subjected to strain rate in 1-1 and 2-2 directions; (c) Stage 2: assembly is subjected to zero strain rate in 1-1 and 2-2 directions; (d) Stage 3: assembly is subjected to isotropic confining pressure; and (e) Stage 4: sheared assembly (biaxial simulation).
3.3.2. Tests with different friction coefficients DEM parameters and their values used in these simulated tests are similar to those in Table 2 except for the friction coefficient which was chosen as a variable. The first and two stages of these
tests were similar to what was described in Section 3.3.1. In Stage 3, assemblies were subjected to isotropic confining pressure of 1 MPa for the different friction coefficients (0, 0.25, 0.5 and 0.75). Table 4 shows the densities for the three series of assemblies when they
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Table 3 Density for assemblies under different confining pressures. Confining pressure (MPa)
0.1 0.5 1 2 4
Density
High angular grains
Medium angular grains
Round grains
0.858 0.867 0.867 0.869 0.870
0.876 0.878 0.880 0.882 0.885
0.880 0.884 0.884 0.880 0.886
Table 4 Density for assemblies for different friction coefficients. Friction coefficient ()
0 0.25 0.5 0.75
Density
High angular grains
Medium angular grains
Round grains
0.889 0.872 0.867 0.869
0.903 0.888 0.880 0.880
0.918 0.890 0.884 0.883
reached equilibrium at the end of this stage, indicating that the density of the assemblies all increased with decreasing friction coefficient and angularity. For considering the effect of different densities on mechanical behavior of assemblies, tests were simulated and described in the next section. In the last stage biaxial tests were simulated with different friction coefficients and deviatoric strain rate equal to 0.0005. 3.3.3. Tests with different void ratios As mentioned earlier, when assemblies were subjected to isotropic confining pressure of 1 MPa for different friction coefficients, slight difference between the densities of assemblies was noted. If we consider the void ratio as the ratio of the void area between grains to the area occupied by grains, the relation between void ratio and density can be expressed as follows: Av A − As 1 Void ratio = = = − 1. As As Density
(6)
Using above equation and the values for density of assemblies in Table 4, Table 5 lists the values of void ratios for assemblies when they reached equilibrium after Stage 3 of the simulated biaxial tests. In order to consider only the effect of initial void ratios on mechanical behavior of assemblies, the last stage of simulated biaxial tests was conducted on assemblies with constant friction coefficient. So the simulated tests were similar to Section 3.3.2 except for the last stage for which the constant friction coefficient and constant strain rate were set to 0.5 and 0.0005, respectively.
0 0.25 0.5 0.75
4. Results The results of simulated tests are presented in two forms of charts in this paper: • Sin of the mobilized friction angle (sin mobilized ) versus axial strain (εa ). • Volumetric strain (εv ) versus axial strain (εa ). 4.1. Tests results with different confining pressures Fig. 9 shows the results of biaxial tests for assemblies with high angular grains under different confining pressures, and Fig. 10 compares results of biaxial tests for three series of assemblies under confining pressure of 0.5 MPa. Results for all simulations are summarized in Tables 6 and 7. Table 6 (mobilized )max values for three series of assemblies under different confining pressures.
Table 5 Void ratios for assemblies for different friction coefficients. Friction coefficient ()
Fig. 9. (a) Sin mobilized and (b) εv versus εa for assemblies with high angular grains under different confining pressures.
Confining pressure (MPa)
Void ratio
High angular grains
Medium angular grains
Round grains
0.125 0.147 0.153 0.151
0.107 0.126 0.137 0.136
0.089 0.123 0.131 0.133
0.1 0.5 1 2 4
(mobilized )max (◦ ) High angular grains
Medium angular grains
Round grains
41.8 42.1 41.5 41 40.5
39.7 38 36.8 35.8 36.4
30.3 30.8 30.4 28.4 27.1
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Fig. 10. (a) Sin mobilized and (b) εv versus εa for three series of assemblies under confining pressure of 0.5 MPa.
395
Fig. 11. (a) Sin mobilized and (b) εv versus εa for assemblies with high angular grains with different inter-particle surface friction coefficients.
The following results can be deduced: • Both mobilized friction angle (sin mobilized ) and dilation decrease with increasing confining pressure, attributable to particles breakage (Fumagalli, Mosconi, & Rossi, 1970), though in this study particles breakage is not modeled. High confining pressure tends to compress the particle assembly, not letting it to dilate, that is, it does not allow particles to move against each other, thus reducing the mobilized friction angle. In general, the more the assembly dilates, the larger the mobilized friction angle becomes, as happens in experimental test results (Charles & Watts, 1980). • For any specified confining pressure, both mobilized friction angle and dilation increase significantly with increase in angularity. Shear resistance arises from friction and interlocking between grains. For the same friction coefficients for the three series of assemblies, higher shear resistance is due to higher interlocking between grains. For highly angular grains significant interlocking Table 7 Dilation values at εa = 18% for three series of assemblies under different confining pressures. Confining pressure (MPa)
0.1 0.5 1 2 4
Dilation value (%) at εa = 18% High angular grains
Medium angular grains
Round grains
7.4 7.1 6.2 6.1 5.4
6.6 6.8 5.5 5.1 3.6
3.4 4.1 3.9 2.5 2.4
between grains leads to high shear resistance and dilation of the assembly during biaxial test. • Fig. 10(a) shows that maximum mobilized friction angle for high angular and medium angular grains occurs at higher axial strain as compared to round grains. Also, the residual mobilized friction angle increased for assemblies with high angular and medium angular grains as compared to assemblies with circular grains. High angular and medium angular grains show approximately the same residual mobilized friction angle, implying that the effect of angularity on residual shear resistance decreases for higher angularity. • Table 6, on maximum mobilized friction angles for three series of assemblies with different confining pressures, shows that maximum mobilized friction angle for high angular grains under a specified confining pressure is approximately 3◦ to 5◦ greater than that for equivalent medium angular grains, and 11◦ to 14◦ greater than that for equivalent round grains. Table 7 shows dilation values for three series of assemblies under different confining pressures in axial strain of 18%, indicating the value of dilation for high angular grains under any specified confining pressure, is approximately 0.5–2% and 2–4% greater than its value for equivalent medium angular grains and round grains, respectively. 4.2. Tests results with different friction coefficients Fig. 11 shows results of simulated biaxial tests for assemblies with high angular grains and different friction coefficients, and Fig. 12 compares results of biaxial tests for three series of assem-
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0 0.25 0.5 0.75
Table 8 (mobilized )max values for three series of assemblies for different friction coefficients. (mobilized )max (◦ )
0 0.25 0.5 0.75
High angular grains
Medium angular grains
Round grains
18.3 34.6 41.5 44
14.3 30.2 36.8 38.7
12.9 25.2 30.4 33.2
High angular grains
Medium angular grains
Round grains
1 3.3 6.2 8.8
0.5 2 5.5 6.3
0.6 0.7 3.9 5.3
decreases at higher values of friction coefficient. With variation of friction coefficient from zero to 0.25, shear strength increases significantly, demonstrating that frictional strength constitutes a considerable portion of shear strength. High angular and medium angular grains show higher increase in shear strength as compared to round grains while friction coefficient varies from zero to 0.25. Also, residual shear strength increases with increasing friction coefficient. Assemblies with higher friction coefficient show lower decrease in volume and higher dilation during biaxial test. • For specified friction coefficient, shear strength and dilation increase with increasing angularity. Also, residual shear strength increases with increasing angularity (Fig. 12(a)). These results are compatible with the results presented in earlier sections. In the case for zero friction coefficient, strength is due to interlocking between grains. In assemblies of highly angular grains, higher interlocking between grains leads to higher shear strength. Also, increase in shear strength with increasing angularity for friction coefficients greater than zero, is greater than its increase for zero friction coefficient. • Table 8 reports maximum mobilized friction angles for the three series of assemblies for different friction coefficients. When friction coefficient increases from 0 to 0.75, maximum mobilized friction angle increases by 25◦ for high angular and medium angular grains and 20◦ for round grains. Also for friction coefficients greater than zero, maximum mobilized friction angle for high angular grains is approximately 4◦ to 5◦ greater than that for medium angular grains, and 9◦ to 11◦ greater than that for round grains. Table 9 shows dilation values for the three series of assemblies for different friction coefficients at axial strain of 18%, indicating that when friction coefficient increases from zero to 0.75, dilation increases by 5–8%, and, for any specified friction coefficient, dilation for high angular grains is 0.5–2.5% and 0.5–3.5% higher, respectively, as compared to that for medium angular grains and round grains.
Fig. 12. (a) Sin mobilized and (b) εv versus εa for three series of assemblies with interparticle surface friction coefficient of 0.25.
Friction coefficient ()
Dilation value (%) at εa = 18%
blies for friction coefficient of 0.25. The results for other cases are summarized in Tables 8 and 9. The following results can be extracted from these figures and tables:
4.3. Tests results for different void ratios Fig. 13 shows results of simulated biaxial tests for assemblies with high angular grains and different initial void ratios. The results for all test simulations are summarized in Table 10. The following conclusions can be extracted from these charts and table:
• Shear strength and dilation increase with increasing friction coefficient. The difference between maximum shear strengths for friction coefficients of 0.5 and 0.75 is small and it can be deduced that the effect of friction coefficient on maximum shear strength
Table 10 Maximum mobilized friction angle and dilation values at εa = 18% for three series of assemblies with different initial void ratios (e). High angular grains
Medium angular grains ◦
Round grains ◦
e
()max ( )
Dilation (%)
e
()max ( )
Dilation (%)
e
()max (◦ )
Dilation (%)
0.125 0.147 0.151 0.153
44 42.2 42.1 41.5
11.1 8.1 7.7 6.2
0.107 0.126 0.136 0.137
39.9 38.9 37 36.8
8.2 7 5.3 5.5
0.089 0.123 0.131 0.133
37.5 32.9 30.5 31.2
8 5.1 3.9 3.8
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• For grains with the same angularities, shear strength, dilation and residual shear strength increase with increasing inter-particle surface friction coefficient. • Increasing initial void ratio leads to less dilation and lower mobilized friction angle.
References
Fig. 13. (a) Sin mobilized and (b) εv versus εa for assemblies with high angular grains and different initial void ratios.
• The shear strength (sin mobilized ) decreases with increasing void ratio for the three series of assemblies. These results could be attributed to less interlocking and fewer contacts between grains. Also maximum shear strength occurs at higher axial strain and with increase in void ratio. The residual shear strengths for each series of assemblies are approximately equal in spite of different void ratios. • Increase in void ratio leads to less dilation, while on the other hand, assemblies with higher void ratio show greater decrease in volume and less dilation which occur at higher axial strains.
5. Summary and conclusions Three series of grains with different angularities were simulated using a method for direct modeling of grain shape. For each series of grains, assemblies consisting of 1000 particles were generated and sheared with different confining pressures, inter-particle surface friction coefficients and initial void ratios. The results can be summarized as follows: • For grains with equal angularities, mobilized friction angle and dilation decrease with increasing confining pressure. • Under any specified confining pressure, shear strength (or mobilized friction angle), dilation and residual shear strength increase considerably with increasing angularity of grains.
Bagherzadeh-Khalkhali, A., & Mirghasemi, A. A. (2009). Numerical and experimental direct shear tests for coarse-grained soils. Particuology, 7(1), 83–91. Bagherzadeh-Khalkhali, A., Mirghasemi, A. A., & Mohammadi, S. (2008). Micromechanics of breakage in sharp-edge particles using combined DEM and FEM. Particuology, 6(5), 347–361. Barbosa, R., & Ghaboussi, J. (1992). Discrete finite element method. Engineering Computations, 9, 253–266. Bathurst, R. J. (1985). A study of stress and anisotropy in idealized granular assemblies. Unpublished doctoral dissertation. Department of Civil Engineering, Queen’s University, Kingston, Ontario. Charles, J. A., & Watts, K. S. (1980). The influence of confining pressure on the shear strength of compacted rockfill. Geotechnique, 30(4), 353–367. Cundall, P. A. (1978). Ball: A computer program to model granular media using the distinct element method. Technical note TN-LN-13, Advance Technology group. London: Dames and Moore. Cundall, P. A., & Strack, O. D. L. (1979). A discrete numerical-model for granular assemblies. Geotechnique, 29, 47–65. Favier, J. F., Abbaspour-Fard, M. H., Kremmer, M., & Raji, A. O. (1999). Shape representation of axi-symmetrical, non-spherical particles in discrete element simulation using multi-element model particles. Engineering Computations, 6, 467–480. Fumagalli, E., Mosconi, B., & Rossi, P. P. (1970). Laboratory tests on materials and static models for rockfill dams. In 10th international congress on large dams Montreal, (pp. 531–551). Holubec, I., & D’Appolonia, E. (1973). Effect of particle shape on the engineering properties of granular soils. In Proceedings symposium on evaluation of relative density Los Angeles, ASTM STP 523, (pp. 304–318). Philadelphia: American Society for Testing and Materials. Jensen, R. P., Bosscher, P. J., Plesha, M. E., & Edil, T. B. (1999). DEM simulation of granular media-structure interface: Effects of surface roughness and particle shape. International Journal for Numerical and Analytical Methods in Geomechanics, 23, 531–547. Lin, X., & Ng, T. T. (1997). A three-dimensional discrete element model using arrays of ellipsoids. Geotechnique, 47(2), 319–329. Matsushima, T., & Saomto, H. (2002). Discrete element modeling for irregularlyshaped sand grains. In P. Mestat (Ed.), Proceedings of NUMGE2002: Numerical methods in geotechnical engineering (pp. 239–246). Matsushima, T., Hidetaka, S., Matsumoto, M., Toda, K., & Yamada, Y. (2003). Discrete element simulation of an assembly of irregularly shaped grains: Quantitative comparison with experiments. In ASCE 16th engineering mechanics conference University of Washington, Seattle. Matuttis, H. G., Luding, S., & Herrmann, H. J. (2000). Discrete element simulations of dense packings and heaps made of spherical and non-spherical particles. Powder Technology, 109(1), 278–292. Mirghasemi, A. A., Rothenburg, L., & Matyas, E. L. (1997). Numerical simulations of assemblies of two-dimensional polygon-shaped particles and effects of confining pressure on shear strength. Soils & Foundations, 37(3), 43–52. Mirghasemi, A. A., Rothenburg, L., & Matyas, E. L. (2002). Influence of particle shape on engineering properties of assemblies of two-dimensional polygon-shaped particles. Geotechnique, 52(3), 209–217. Ng, T. T. (1994). Numerical simulation of granular soil using elliptical particles. Computers and Geotechnics, 16(2), 153–169. Potapov, A. V., & Campbell, C. S. (1998). A fast model for the simulation of non-round particles. Granular Matter, 1, 9–14. Rothenburg, L. (1980). Micromechanics of idealized granular system. Unpublished doctoral dissertation. Carleton University, Ottawa, Ontario, Canada. Rothenburg, L., & Bathurst, R. J. (1992). Micromechanical features of granular assemblies with planar elliptical particles. Geotechnique, 42(1), 79–95. Sukumaran, B., & Ashmawy, A. K. (2001). Quantitative characterization of the geometry of discrete particles. Geotechnique, 51(7), 619–627. Seyedi Hosseininia, E., & Mirghasemi, A. A. (2006). Numerical simulation of breakage of two-dimensional polygon-shaped particles using discrete element method. Powder Technology, 166, 100–112. Seyedi Hosseininia, E., & Mirghasemi, A. A. (2007). Effect of particle breakage on the behavior of simulated angular particle assemblies. Particuology, 5, 328–336. Ting, J. M., Khawaja, M., Meachum, L. R., & Rowell, J. D. (1993). An ellipse-based discrete element model for granular materials. International Journal for Numerical and Analytical Methods in Geomechanics, 17, 603–623.
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Particle shape consideration in numerical simulation of assemblies of irregularly shaped particles Saba Abedi, Ali Asghar Mirghasemi ∗ School of Civil Engineering, College of Engineering, University of Tehran, Tehran P.O. Box 11155-4563, Iran
a r t i c l e
i n f o
Article history: Received 30 June 2010 Received in revised form 23 November 2010 Accepted 30 November 2010 Keywords: Discrete element method Granular soil Angularity Confining pressure
a b s t r a c t The mechanical behavior of granular materials depends much on the shape of the constituent particles. Therefore appropriate modeling of particle, or grain, shape is quite important. This study employed the method of direct modeling of grain shape (Matsushima & Saomto, 2002), in which, the real shape of a grain is modeled by combining arbitrary number of overlapping circular elements which are connected to each other in a rigid way. Then, accordingly, a discrete-element program is used to simulate the assembly of grains. In order to measure the effects of grain shape on mechanical properties of assembly of grains, three types of grains—high angular grains, medium angular grains and round grains are considered where several biaxial tests are conducted on assemblies with different grain types. The results show that the angularity of grains greatly affects the behavior of granular soil. © 2011 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.
1. Introduction Published data show that particle shape has considerable influence on the engineering properties of granular soils. Tests conducted by Holubec and D’Appolonia (1973) on medium to fine sands with varying particle shapes indicated that granular materials with the same relative density could have different mechanical behavior due to angularity. They concluded that the variation of engineering properties due to particle shape could be of the same order of magnitude as the variation of properties due to changes in relative density, thus suggesting that particle shape could be considered an index property to correlate the properties of granular materials. Rothenburg and Bathurst (1992) reported the results of numerical simulations of planar assemblies of elliptical particles. Packing simulations of the initial assembly showed that, for ellipses with eccentricities up to about 0.2, the coordination numbers of the generated assemblies increased with increasing eccentricity. The resultant peak angle of internal friction in biaxial shear simulations showed the same trend and correlated well with the initial coordination number. Ting, Khawaja, Meachum, and Rowell (1993) reported similar conclusion from isotropic compression and biaxial shear test simulations on assemblies of two-dimensional elliptical particles.
∗ Corresponding author. Tel.: +98 21 66957783; fax: +98 21 66957787. E-mail address: [email protected] (A.A. Mirghasemi).
By means of numerical simulations of assemblies of polygonshaped particles, Mirghasemi, Rothenburg, and Matyas (2002) concluded that particle angularity had an important effect on the compressibility and shear strength of the granular media. In order to obtain a precise modeling method that would be comparable with experimental results, the specifications of grains such as form, size, elasticity, plasticity should be modeled carefully. Since the shape of the grain affects tremendously the mechanical behavior of granular media such as shear resistance, appropriate modeling of grain shape is considered very important. In the early discrete element method (DEM) scheme presented by Cundall and Strack (1979), the grains were modeled as discs in 2D and as spheres in 3D simulations with the advantage of simplicity in calculation: contacts between grains can be detected with simple algorithms, and in the simulated tests circular or spherical grains can move and rotate easily, though not without problems in conforming reality. The internal friction angle of shearing resistance is much less for circular or spherical grains compared to that of actual grains with irregular shapes. Resistance to rotation for circular or spherical grains is much less than that of actual grains, particularly for grains with inherent tendency to roll. On the other hand, the direction of the normal contact forces is always toward the center; these forces, however, never contribute to the moments acting on the grains, rotation being only affected by tangential contact forces. Because of these problems different shapes for grains were presented for use in DEM simulations in order to improve the results of simulation. In most cases of modeling that have been applied so far, grain form is generally considered elliptical,
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doi:10.1016/j.partic.2010.11.005
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Nomenclature A As AV Fi fiC Fn Fn Fs Fs I Kn Ks ljC m M n s SF x¨ i ¨ mobilized 1, 2 ij εa εv
total area of the assembly area occupied by grains in the assembly void area between grains in the assembly grain unbalanced force contact force vector normal contact force between two particles Incremental normal contact force shear contact force between two particles incremental shear contact force grain moment of inertia grain stiffness in normal direction grain stiffness in shear direction contact length vector grain mass grain resultant moment relative normal displacement at a contact relative shear displacement at a contact shape factor grain translational acceleration grain rotational acceleration mobilized friction angle major principal stresses stress tensor axial strain volumetric strain
ellipsoidal, polygonal, etc., none of which, however, directly models any irregularly shaped grain, hence will not project the expected mechanical behavior, thus pointing to the need of direct modeling of grains with irregular shapes.
2. Methods for modeling grain shape 2.1. Brief review of particle shape modeling Many researchers studied granular soil behavior in terms of ellipse-shaped grains in 2D and ellipsoid-shaped grains in 3D simulations (Bagherzadeh-Khalkhali & Mirghasemi, 2009; Lin & Ng, 1997; Ng, 1994; Rothenburg & Bathurst, 1992; Ting et al., 1993), apparently with the following advantages: grains have unique outward normal so calculations of forces are simple; grains have fewer tendencies to rotate, making simulated mechanical behavior similar to that of real soils. Such modeling, however, does not accurately represent the shape of grains (Fig. 1(a)). Many studies considered granular soil behavior in terms of polygon-shaped grains (Bagherzadeh-Khalkhali, Mirghasemi, & Mohammadi, 2008; Barbosa & Ghaboussi, 1992; Matuttis, Luding, & Herrmann, 2000; Mirghasemi, Rothenburg, & Matyas, 1997; Mirghasemi et al., 2002; Seyedi Hosseininia & Mirghasemi, 2006, 2007), to show rather realistic representation of soil behavior (Fig. 1(b)), though the method is time-consuming. Potapov and Campbell (1998) proposed the elliptical approximation with oval-shaped boundaries determined by four continuously jointed circular arches. This method involves simple calculations because contacts between two circular segments of grains are similar to contacts between circles in the original DEM code. The results of simulated tests using this method show that this method is effective; though the real shape of the grain is not effectively represented (Fig. 1(c)).
In the method presented by Favier, Abbaspour-Fard, Kremmer, and Raji (1999) to model axi-symmetrical grains, particles are modeled by combining multiple overlapping spheres with fixed inner-sphere connections. Simulations with this method showed good agreement with experimental results. This method can model any shape of grains, unless they are highly angular (Fig. 1(d)). Jensen, Bosscher, Plesha, and Edil (1999) presented a clustering method in which the grains are modeled by combining nonoverlapping circular elements in a semi rigid configuration; The contacts between circular elements are linear-elastic with a high stiffness. Although this method has some improvement over earlier methods, the outlines of simulated grain do not resemble those of the actual grains. Also computational time is long (Fig. 1(e)). In all of the above methods, direct modeling of grain shape is not considered. Because of the considerable effects of grain shape on mechanical behavior of soils, it seems worth conducting discrete element method with grains whose shapes are directly modeled.
2.2. Direct modeling of grain shape In this method, the real shape of a grain is modeled by combining an arbitrary number of overlapping circular elements which are connected to each other in a rigid way. Fig. 2 shows a simple algorithm to find the optimized number of circular elements to model the real shape of a grain (Matsushima & Saomto, 2002; Matsushima, Hidetaka, Matsumoto, Toda, & Yamada, 2003), the resulting outline closely coinciding with that of the real grain (Fig. 3). Fig. 3(a) shows the outline of an arbitrary grain that is shown in Fig. 3(b). By combining overlapping circles, closely real shape of the grain is generated (Fig. 3(c)), showing that the created shape with circular elements (Fig. 3(d)) closely resembles the real shape of the grain shown in Fig. 3(b). The actual shape of a grain can be precisely modeled with a maximum of 10–15 circular elements. The number of circular elements for modeling a grain shape depends on the degree of nonuniformity and angularity of the actual grain shape, the desired level of accuracy for the grain shape and the limitation of computation time. This direct modeling method can be used to realize realistic results for mechanical behavior of granular soils. The results of simulated assemblies of grains have been compared quantitatively with experimental results of actual assemblies (Matsushima & Saomto, 2002). The calculations for simulated assemblies are similar to assemblies with circular grains, for which no complicated algorithm is required. The program DISC (Bathurst, 1985), which is a modified version of BALL (Cundall, 1978) is adopted and modified to simulate assembly of grains with real shape. This program, calls for the Newton’s second law to compute displacements for each grain from the current resultant forces and the moments acting on the grain, as follows:
Fi = m¨xi , ¨ M = I ,
i = 1, 2
(1)
where Fi is the unbalanced force on each grain, M is the resultant moment on each grain, m is the mass of the grain, I is the moment of inertia of the grain, and x¨ i and ¨ are the grain accelerations. For each grain, accelerations are integrated over small time-steps to give velocities and displacements. The time-step is chosen small enough that the velocities and accelerations can be assumed constant over it. The calculated displacements are used in contact law or force–displacement law through which the con-
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Fig. 1. Several modeling methods in DEM simulations with: (a) elliptical and ellipsoidal elements, (b) polygonal elements, (c) oval elements, (d) axisymmetrical elements and (e) overlapping rigid clusters.
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Fig. 2. Optimized number of circular elements to model real shape of a grain (after).
tact forces between two circular grains (Fig. 4(a)) are computed as follows: (Fn )new = (Fn )old + Fn = (Fn )old + Kn n, (Fs )new = (Fs )old + Fs = (Fs )old + Ks s,
(2)
where Fn and Fs are the normal and shear contact forces, Fn and Fs are the incremental normal and shear contact forces, Kn and Ks are the grain stiffness in normal and shear directions, and n and s are the normal and shear relative displacements at a contact, which are obtained by integrating the relative velocity component with respect to time. The relative velocity components are calculated by projecting relative velocities at the contact point in the normal and tangential directions to the contact plane. Calculated forces and resultant moment of these forces are used in Newton’s second law and calculations are repeated till the unbalanced forces on the grains are very small or close to zero. In the modified program, the contact considerations and calculations of contact forces are conducted for each circular element of the grain that is in contact with the circular element of another grain (Fig. 4(b)), and the equation of motion is solved for each grain. With this procedure, rigid connections between circular elements of each grain are obtained without definition of stiffness between these elements.
Fig. 3. (a) Outline of actual particle, (b) actual particle, (c) modeling with circular elements and (d) modeled grain.
3. Simulated tests The method illustrated above is an appropriate tool to investigate the effects of different shapes of grains on their mechanical behavior. Three series of grains with different angularities are considered: high angular grains, medium angular grains and round grains, as shown in Fig. 5. Then a series of simulated biaxial tests are conducted on the assemblies created by each series of grains, as described below.
Fig. 4. (a) Two circular grains in contact with each other and (b) two circular elements of two grains in contact with each other.
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Fig. 5. (a) High angular grain, (b) medium angular grain and (c) round grain.
3.1. Selection process of grain shape As mentioned above, three series of grains with different angularities are considered. In the first step, 16 types of high angular grains were chosen with equivalent radius ranging from 5 mm to 20 mm. Then, medium angular grains and round grains are generated through reduction of angularity in high angular grains. The general shape and size are similar for three series as shown in Fig. 5. In order to have a quantitative comparison between the shapes of the three series of grains, a shape factor (SF) developed by Sukumaran and Ashmawy (2001) is used to measure the deviation of a grain outline from that of a circle. The polygon shape of the grain is first sliced into n sections. The orientations of the lines connecting points on the perimeter of the polygon are compared with the orientations on lines connecting the polygon created by slicing a circle. The angle which constitutes the difference in orientations, ˛, and the number of sampling points N are used to define the shape factor as: n ˛i grain
SF =
i=1
N × 45◦
× 100%.
(3)
For all practical grain shapes, values of shape factor will lie between zero and one, with zero to correspond to a circle, and one to correspond to a flat plate. In this research, the range and average values of shape factors for three series of grains are summarized in Table 1. Research showed that grain elongation has considerable effects on mechanical behavior. Elongation is defined as the ratio of the length of a particle to its width. In order to consider only the effects of angularity on behavior, the effects of elongation are minimized by selecting grains with equal length and width (elongation equal to 1). In the next step, the grains are modeled by combining circular elements as described in Section 2.2. 3.2. Assembly preparation In this study, circular assemblies consisting of 1000 grains each are used with grain size distribution shown typically in Fig. 6, in which equivalent diameter refers to that of an equivalent circle having the same area as the grain. Circular assembly is first generated for high angular grains. By trial and error, the radius of this circular assembly is chosen to be the lowest possible value that could contain 1000 grains. In other words, a densest initial assembly is generated to minimize computation time for compacting the assembly. Then, the location of the center of gravity of each grain in the assembly is determined and Table 1 Range and average values of shape factors SF for three series of grains. Group
SF range (%)
SF average (%)
High angular grains Medium angular grains Round grains
38–47 27–40 8–15
42 37 12
Fig. 6. Grain size distribution in assemblies created by each series of grains.
registered in an output file. For generating assemblies with medium angular and round grains, this output file is used and the location of each grain is selected coincident with the location of similar high angular grain. This procedure provides better comparison between results of assemblies generated with each series of grains. Fig. 7 shows the generated assemblies with each series of grains. 3.3. Biaxial simulation Each simulated biaxial test includes four stages, as shown in Fig. 8. The generated assembly is first subjected to a defined strain rate to compact the initial loose assembly. In the next stage, zero strain rate is applied in order to bring the assembly to equilibrium. In Stage 3, the assembly is subjected to a defined isotropic confining pressure. In Stage 4, biaxial test is carried out on the assembly. In the simulated biaxial test, the horizontal stress is held constant and the vertical stress is increased by applying constant deviatoric strain rate. In a biaxial test, the mobilized internal friction angle of a cohesionless material can be determined by the following equation as a function of major principal stresses, 1 and 2 : Sin mobilized =
2 − 1 . 2 + 1
(4)
Major principal stresses are determined on the basis of the average stress tensor within an assembly. The stress tensor of an assembly can be calculated by the following equation (Rothenburg, 1980): ij =
1 C C fi lj , A
i, j = 1, 2
(5)
C ∈A
in which ij is stress tensor, A is the area of the assembly, fiC is contact force and ljC is contact vector between two circular elements of two grains that are in contact with each other. To consider the effect of different factors on mechanical behavior, three series of simulated biaxial tests were conducted on assemblies with each series of grains: tests with different confining pressures, tests with different friction coefficients and tests with different void ratios. This section describes the details of these three series of tests. 3.3.1. Tests with different confining pressures Table 2 lists the parameters used in these tests for the discrete element method (DEM). Normal stiffness and tangential stiffness are used in force–displacement equations for computation of con-
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Fig. 7. Assemblies generated by 1000 (a) high angular grains, (b) medium angular grains and (c) round grains.
tact forces. Friction coefficient indicates the roughness of grain surface and damping coefficients are defined to dissipate kinetic energy so that static equilibrium condition for assembly of grains can be reached. For better comparison of results, equal parameters were considered for the three assemblies. Table 2 DEM parameters used in simulated tests. Parameter
Value
Normal contact stiffness (N/m) Tangential contact stiffness (N/m) Inter-particle friction coefficient Inter-particle cohesion Density of particles (kg/m3 ) Damping coefficients
245 × 107 245 × 107 0.5 0.0 2000 5 and 0.01
The loose generated assemblies were subjected to a strain rate equal to 0.005 in 500,000 cycles to become compact. After relaxing assemblies by applying zero strain rate to boundary particles, the assemblies were subjected to different isotropic confining pressures (0.1, 0.5, 1, 2 and 4 MPa). Table 3 shows the ratio of the area occupied by grains to the area of the entire assembly, which is called density, for three series of assemblies when they reached equilibrium under different confining pressures. According to the above table, density increases slightly with increasing confining pressure and decreasing angularity, the differences being by no means substantial. In the last stage, biaxial tests were carried out on the assemblies. In the simulated biaxial test the deviatoric strain rate was chosen to be 0.0005.
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Fig. 8. (a) Generated assembly of high angular grains; (b) Stage 1: assembly is subjected to strain rate in 1-1 and 2-2 directions; (c) Stage 2: assembly is subjected to zero strain rate in 1-1 and 2-2 directions; (d) Stage 3: assembly is subjected to isotropic confining pressure; and (e) Stage 4: sheared assembly (biaxial simulation).
3.3.2. Tests with different friction coefficients DEM parameters and their values used in these simulated tests are similar to those in Table 2 except for the friction coefficient which was chosen as a variable. The first and two stages of these
tests were similar to what was described in Section 3.3.1. In Stage 3, assemblies were subjected to isotropic confining pressure of 1 MPa for the different friction coefficients (0, 0.25, 0.5 and 0.75). Table 4 shows the densities for the three series of assemblies when they
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Table 3 Density for assemblies under different confining pressures. Confining pressure (MPa)
0.1 0.5 1 2 4
Density
High angular grains
Medium angular grains
Round grains
0.858 0.867 0.867 0.869 0.870
0.876 0.878 0.880 0.882 0.885
0.880 0.884 0.884 0.880 0.886
Table 4 Density for assemblies for different friction coefficients. Friction coefficient ()
0 0.25 0.5 0.75
Density
High angular grains
Medium angular grains
Round grains
0.889 0.872 0.867 0.869
0.903 0.888 0.880 0.880
0.918 0.890 0.884 0.883
reached equilibrium at the end of this stage, indicating that the density of the assemblies all increased with decreasing friction coefficient and angularity. For considering the effect of different densities on mechanical behavior of assemblies, tests were simulated and described in the next section. In the last stage biaxial tests were simulated with different friction coefficients and deviatoric strain rate equal to 0.0005. 3.3.3. Tests with different void ratios As mentioned earlier, when assemblies were subjected to isotropic confining pressure of 1 MPa for different friction coefficients, slight difference between the densities of assemblies was noted. If we consider the void ratio as the ratio of the void area between grains to the area occupied by grains, the relation between void ratio and density can be expressed as follows: Av A − As 1 Void ratio = = = − 1. As As Density
(6)
Using above equation and the values for density of assemblies in Table 4, Table 5 lists the values of void ratios for assemblies when they reached equilibrium after Stage 3 of the simulated biaxial tests. In order to consider only the effect of initial void ratios on mechanical behavior of assemblies, the last stage of simulated biaxial tests was conducted on assemblies with constant friction coefficient. So the simulated tests were similar to Section 3.3.2 except for the last stage for which the constant friction coefficient and constant strain rate were set to 0.5 and 0.0005, respectively.
0 0.25 0.5 0.75
4. Results The results of simulated tests are presented in two forms of charts in this paper: • Sin of the mobilized friction angle (sin mobilized ) versus axial strain (εa ). • Volumetric strain (εv ) versus axial strain (εa ). 4.1. Tests results with different confining pressures Fig. 9 shows the results of biaxial tests for assemblies with high angular grains under different confining pressures, and Fig. 10 compares results of biaxial tests for three series of assemblies under confining pressure of 0.5 MPa. Results for all simulations are summarized in Tables 6 and 7. Table 6 (mobilized )max values for three series of assemblies under different confining pressures.
Table 5 Void ratios for assemblies for different friction coefficients. Friction coefficient ()
Fig. 9. (a) Sin mobilized and (b) εv versus εa for assemblies with high angular grains under different confining pressures.
Confining pressure (MPa)
Void ratio
High angular grains
Medium angular grains
Round grains
0.125 0.147 0.153 0.151
0.107 0.126 0.137 0.136
0.089 0.123 0.131 0.133
0.1 0.5 1 2 4
(mobilized )max (◦ ) High angular grains
Medium angular grains
Round grains
41.8 42.1 41.5 41 40.5
39.7 38 36.8 35.8 36.4
30.3 30.8 30.4 28.4 27.1
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Fig. 10. (a) Sin mobilized and (b) εv versus εa for three series of assemblies under confining pressure of 0.5 MPa.
395
Fig. 11. (a) Sin mobilized and (b) εv versus εa for assemblies with high angular grains with different inter-particle surface friction coefficients.
The following results can be deduced: • Both mobilized friction angle (sin mobilized ) and dilation decrease with increasing confining pressure, attributable to particles breakage (Fumagalli, Mosconi, & Rossi, 1970), though in this study particles breakage is not modeled. High confining pressure tends to compress the particle assembly, not letting it to dilate, that is, it does not allow particles to move against each other, thus reducing the mobilized friction angle. In general, the more the assembly dilates, the larger the mobilized friction angle becomes, as happens in experimental test results (Charles & Watts, 1980). • For any specified confining pressure, both mobilized friction angle and dilation increase significantly with increase in angularity. Shear resistance arises from friction and interlocking between grains. For the same friction coefficients for the three series of assemblies, higher shear resistance is due to higher interlocking between grains. For highly angular grains significant interlocking Table 7 Dilation values at εa = 18% for three series of assemblies under different confining pressures. Confining pressure (MPa)
0.1 0.5 1 2 4
Dilation value (%) at εa = 18% High angular grains
Medium angular grains
Round grains
7.4 7.1 6.2 6.1 5.4
6.6 6.8 5.5 5.1 3.6
3.4 4.1 3.9 2.5 2.4
between grains leads to high shear resistance and dilation of the assembly during biaxial test. • Fig. 10(a) shows that maximum mobilized friction angle for high angular and medium angular grains occurs at higher axial strain as compared to round grains. Also, the residual mobilized friction angle increased for assemblies with high angular and medium angular grains as compared to assemblies with circular grains. High angular and medium angular grains show approximately the same residual mobilized friction angle, implying that the effect of angularity on residual shear resistance decreases for higher angularity. • Table 6, on maximum mobilized friction angles for three series of assemblies with different confining pressures, shows that maximum mobilized friction angle for high angular grains under a specified confining pressure is approximately 3◦ to 5◦ greater than that for equivalent medium angular grains, and 11◦ to 14◦ greater than that for equivalent round grains. Table 7 shows dilation values for three series of assemblies under different confining pressures in axial strain of 18%, indicating the value of dilation for high angular grains under any specified confining pressure, is approximately 0.5–2% and 2–4% greater than its value for equivalent medium angular grains and round grains, respectively. 4.2. Tests results with different friction coefficients Fig. 11 shows results of simulated biaxial tests for assemblies with high angular grains and different friction coefficients, and Fig. 12 compares results of biaxial tests for three series of assem-
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0 0.25 0.5 0.75
Table 8 (mobilized )max values for three series of assemblies for different friction coefficients. (mobilized )max (◦ )
0 0.25 0.5 0.75
High angular grains
Medium angular grains
Round grains
18.3 34.6 41.5 44
14.3 30.2 36.8 38.7
12.9 25.2 30.4 33.2
High angular grains
Medium angular grains
Round grains
1 3.3 6.2 8.8
0.5 2 5.5 6.3
0.6 0.7 3.9 5.3
decreases at higher values of friction coefficient. With variation of friction coefficient from zero to 0.25, shear strength increases significantly, demonstrating that frictional strength constitutes a considerable portion of shear strength. High angular and medium angular grains show higher increase in shear strength as compared to round grains while friction coefficient varies from zero to 0.25. Also, residual shear strength increases with increasing friction coefficient. Assemblies with higher friction coefficient show lower decrease in volume and higher dilation during biaxial test. • For specified friction coefficient, shear strength and dilation increase with increasing angularity. Also, residual shear strength increases with increasing angularity (Fig. 12(a)). These results are compatible with the results presented in earlier sections. In the case for zero friction coefficient, strength is due to interlocking between grains. In assemblies of highly angular grains, higher interlocking between grains leads to higher shear strength. Also, increase in shear strength with increasing angularity for friction coefficients greater than zero, is greater than its increase for zero friction coefficient. • Table 8 reports maximum mobilized friction angles for the three series of assemblies for different friction coefficients. When friction coefficient increases from 0 to 0.75, maximum mobilized friction angle increases by 25◦ for high angular and medium angular grains and 20◦ for round grains. Also for friction coefficients greater than zero, maximum mobilized friction angle for high angular grains is approximately 4◦ to 5◦ greater than that for medium angular grains, and 9◦ to 11◦ greater than that for round grains. Table 9 shows dilation values for the three series of assemblies for different friction coefficients at axial strain of 18%, indicating that when friction coefficient increases from zero to 0.75, dilation increases by 5–8%, and, for any specified friction coefficient, dilation for high angular grains is 0.5–2.5% and 0.5–3.5% higher, respectively, as compared to that for medium angular grains and round grains.
Fig. 12. (a) Sin mobilized and (b) εv versus εa for three series of assemblies with interparticle surface friction coefficient of 0.25.
Friction coefficient ()
Dilation value (%) at εa = 18%
blies for friction coefficient of 0.25. The results for other cases are summarized in Tables 8 and 9. The following results can be extracted from these figures and tables:
4.3. Tests results for different void ratios Fig. 13 shows results of simulated biaxial tests for assemblies with high angular grains and different initial void ratios. The results for all test simulations are summarized in Table 10. The following conclusions can be extracted from these charts and table:
• Shear strength and dilation increase with increasing friction coefficient. The difference between maximum shear strengths for friction coefficients of 0.5 and 0.75 is small and it can be deduced that the effect of friction coefficient on maximum shear strength
Table 10 Maximum mobilized friction angle and dilation values at εa = 18% for three series of assemblies with different initial void ratios (e). High angular grains
Medium angular grains ◦
Round grains ◦
e
()max ( )
Dilation (%)
e
()max ( )
Dilation (%)
e
()max (◦ )
Dilation (%)
0.125 0.147 0.151 0.153
44 42.2 42.1 41.5
11.1 8.1 7.7 6.2
0.107 0.126 0.136 0.137
39.9 38.9 37 36.8
8.2 7 5.3 5.5
0.089 0.123 0.131 0.133
37.5 32.9 30.5 31.2
8 5.1 3.9 3.8
S. Abedi, A.A. Mirghasemi / Particuology 9 (2011) 387–397
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• For grains with the same angularities, shear strength, dilation and residual shear strength increase with increasing inter-particle surface friction coefficient. • Increasing initial void ratio leads to less dilation and lower mobilized friction angle.
References
Fig. 13. (a) Sin mobilized and (b) εv versus εa for assemblies with high angular grains and different initial void ratios.
• The shear strength (sin mobilized ) decreases with increasing void ratio for the three series of assemblies. These results could be attributed to less interlocking and fewer contacts between grains. Also maximum shear strength occurs at higher axial strain and with increase in void ratio. The residual shear strengths for each series of assemblies are approximately equal in spite of different void ratios. • Increase in void ratio leads to less dilation, while on the other hand, assemblies with higher void ratio show greater decrease in volume and less dilation which occur at higher axial strains.
5. Summary and conclusions Three series of grains with different angularities were simulated using a method for direct modeling of grain shape. For each series of grains, assemblies consisting of 1000 particles were generated and sheared with different confining pressures, inter-particle surface friction coefficients and initial void ratios. The results can be summarized as follows: • For grains with equal angularities, mobilized friction angle and dilation decrease with increasing confining pressure. • Under any specified confining pressure, shear strength (or mobilized friction angle), dilation and residual shear strength increase considerably with increasing angularity of grains.
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