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Effect of local non-convexity on the critical shear strength of granular materials determined via the discrete element method
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Effect of local non-convexity on the critical shear strength of granular materials determined via the discrete element method Zhihong Nie a , Shunkai Liu a , Wei Hu b , Jian Gong c,∗ a
School of Civil Engineering, Central South University, Changsha, Hunan 410075, China Hunan Province Key Laboratory of Geotechnical Engineering Stability Control and Health Monitoring, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China c School of Hydraulic Engineering, Changsha University of Science and Technology, Changsha 410114, China b
a r t i c l e
i n f o
Article history: Received 22 May 2019 Received in revised form 16 October 2019 Accepted 19 December 2019 Available online xxx Keywords: Multi-sphere clump Local non-convexity Discrete element method Biaxial shear test Contact type Critical shear strength
a b s t r a c t Multi-sphere clumps are commonly used to simulate non-spherical particles in discrete element method simulations. It is of interest whether the degree of local non-convexity affects the mechanical behaviour of granular materials with the same non-convexity . A series of discrete-element-method biaxial shear tests are conducted on rough particle packings with = 0.075 and different values (ranging from 0.134 to 0.770). The microscale results show that the contact type changes with an increase in . However, the critical strength is independent of . The evaluation of the contributions of different contact types to the critical shear strength and a detailed analysis of the anisotropies help clarify the microscopic mechanisms that result in the independence of the critical shear strength from . © 2020 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.
Introduction There are three important scales regarding the particle shape of granular materials: sphericity (eccentricity or plainness), roundness (angularity), and roughness (smoothness or non-convexity) (Cho, Dodds, & Santamarina, 2006). Various previous experimental and numerical studies have reported that sphericity and roundness affect the mechanical behaviours of granular materials (Azéma & Radjai, 2010; Azéma, Radjai, & Dubois, 2013; Brown et al., 2011; Feng, Zhao, Kato, & Zhou, 2017; Jensen, Bosscher, Plesha, & Edil, 1999; Meng, Li, Lu, Li, & Jin, 2012). However, only a few studies have focused on the effect of roughness, which also affects the properties of granular materials. In terms of experimental studies, Narayan and Hancock (2003) noted that the indentation hardness, elastic modulus, and brittle fracture index of particles gradually decrease with increasing roughness. Additionally, Santamarina and Cascante (1998) and Otsubo, O’sullivan, Sim, and Ibraim (2015)) reported that rough spheres have a smaller shear stiffness than smooth spheres. Furthermore, Anthony and Marone (2005) found that an increase in particle roughness increases the frictional
∗ Corresponding author. E-mail address: gj [email protected] (J. Gong).
strength and affects the distribution of stress within sheared layers. Alternatively, the discrete element method (DEM) can be adopted for the efficient numerical modelling of the effects of roughness on the macroscale and microscale properties of granular materials (Otsubo, O’Sullivan, Hanley, & Sim, 2017; Rémond, Gallias, & Mizrahi, 2008; de Graaf, van Roij, & Dijkstra, 2011; Yang, Cheng, & Sun, 2017). Using clumps composed of sub-spheres is the most common method of simulating non-spherical particles with the DEM. The effect of roughness can then be studied by changing the properties (e.g., radius and position) of the sub-spheres of the clumps. Taking this approach, Ludewig and Vandewalle (2012) found that when particles are modified to have a high roughness, particle interlocking and multiple contact points develop in the packing; nonconvex particle packings have a lower density and are more stable than spherical packings. Saint-Cyr et al. (SaintCyr, Delenne, Voivret, Radjai, & Sornay, 2011; Saint-Cyr, Radjai, Delenne, & Sornay, 2013) reported that both the internal angle of friction and the maximum cohesion increase linearly as the roughness increases, but the former approaches a certain value. Furthermore, Azéma, Radjai, Dubois et al. (2013) observed that the critical shear strength is an increasing function of roughness. In the above-mentioned DEM simulations, roughness was quantified as the concavity of the entire particle surface, which is composed of various local non-convexities. Note that non-convexity is of concern in practical engineering, particularly in the storage,
https://doi.org/10.1016/j.partic.2019.12.008 1674-2001/© 2020 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.
Please cite this article in press as: Nie, Z., et al. Effect of local non-convexity on the critical shear strength of granular materials determined via the discrete element method. Particuology (2020), https://doi.org/10.1016/j.partic.2019.12.008
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Table 1 Input parameters in the DEM simulation. Parameter
Value
Particle density, (kg/m3 ) Number of particles Coefficient of inter-particle friction, b Coefficient of wall–particle friction, w Wall stiffness (N/m) Contact effective modulus, Ec (Pa) kn /ks Damping constant, ˇ
2600 5000 0.5 0.0 1 × 109 1 × 108 4/3 0.7
Fig. 1. Different local non-convexities of particles with the same roughness.
handling, and recycling of crushed hollow industrial by-products and sintered powders (Azéma, Radjai, & Saussine, 2009; Rémond et al., 2008). For a given overall roughness, the form of local nonconvexity may not be the same (see Fig. 1). Indeed, the variation in local non-convexity may be a noise factor when using clumps in DEM simulations. The critical shear strength reflects the intrinsic shear strength of the granular material and is independent of the initial state. Thus, it is of interest whether different local nonconvexities affect the critical shear strength of granular materials with identical roughness. This issue is the motivation of the present work, wherein we conduct grain-scale modelling using the DEM, which has previously been demonstrated to reproduce certain key features of granular materials (Azéma, Radjai, Peyroux, & Saussine, 2007; Brown et al., 2011; Gong, Wang, Li, & Nie, 2019; Nie, Zhu, Wang, & Gong, 2019). In this paper, the effect of local non-convexity on the shear behaviour of particle packings is explored through a series of drained biaxial compression tests using the DEM. The rest of the paper is organized as follows. First, a brief introduction of DEM modelling is given. Then, contact types and microscale simulation results of each contact type, including the coordination number and proportions related to the contact type, are studied. Next, the stress–strain characteristics and the contributions of certain contact types to the critical shear strength are analysed. Subsequently, the effect of the local non-convexity on fabric anisotropy is evaluated. Finally, main conclusions are presented. DEM modelling Biaxial compression tests were conducted using the wellrecognized DEM program PFC2D (Itasca, 2014), which was originally developed by Cundall and Strack (1979). Simulations were carried out using the linear elastic contact model, as used in several previous studies (Gong, Nie, Zhu, Liang, & Wang, 2019; Gu, Huang, & Qian, 2014; Minh & Cheng, 2013). The microscale parameters used in the present study were selected by referring to many previous experimental and numerical studies. Specifically, Rowe (1962) reported that the inter-particle friction b of quartz particles is about 0.445–0.601. Considering values of b adopted in many previous DEM simulations (e.g., Li, Yu, & Li, 2009; Abbireddy & Clayton, 2010; Abedi & Mirghasemi, 2011; Yang & Dai, 2011; Saint-Cyr et al., 2011, 2013; Yang, Cheng, & Wang, 2016; Gong & Liu, 2017a; Gong, Nie et al., 2019), the value of b = 0.5 was used in the present study. The contact effective modulus Ec = 1 × 108 Pa
Fig. 2. Geometry of regular aggregates.
in the present study was consistent with values reported by Yang, Cheng et al. (2016) and Gong and Liu (2017a). Alternatively, the shear stiffness ratio kn /ks (where kn and ks respectively denote the normal and shear contact stiffness of particles) suggested by Goldenberg and Goldhirsch (2005) for realistic granular materials is in the range of 1.0 < kn /ks < 1.5, which correlates well with the Cattaneo–Mindlin model (Johnson, 1987) for elastic spherical contacts. Hence, kn /ks = 4.0/3 was adopted in the present study. Local damping with damping constant ˇ = 0.7 as suggested in Itasca (2014) has been used for effectively dissipating kinetic energy, but it is expected to have little effect on the simulation results. All basic parameters used in our simulations are listed in Table 1. The model particles comprised clumps, which were rigid sets of n overlapping disks of the same radius r, with n-fold rotational symmetry (see Fig. 2). Following previous studies (e.g., Saint-Cyr et al., 2011; 2013), the surface roughness was described by the nonconvexity (), which characterizes the degree of distortion from a perfectly circular shape and is defined as
=R/R = 1– (/tan˛ +
1– 2 )/(1+/sin˛)
(1)
where R = R – R´, with R and R´respectively being the radii of the circumscribing circle and the inscribed circle; ˛ = /n; and = l/2r, with l being the distance between disk centres. In this paper, the local non-convexity is defined as cos(/2), where is the average of the angles formed by the tangent lines on either side of a concave point, as shown in Fig. 2. According to their definitions, and have the geometric relationship = cos(/2). The local non-convexity can therefore be represented by . When the radius r of the disk is fixed, can be varied by changing the number of disks owing to the rotationally symmetric distribution of the sub-disks. Fig. 3 displays the non-convexity as a function of for different values of n. Clearly, Fig. 3 shows that can be unified by adjusting and becomes monotonically greater as n increases for a
Please cite this article in press as: Nie, Z., et al. Effect of local non-convexity on the critical shear strength of granular materials determined via the discrete element method. Particuology (2020), https://doi.org/10.1016/j.partic.2019.12.008
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cal compression while ensuring that the shear process remained in a quasi-static state. This process reduces the inertia parameter I, ˙ /p´ (Da Cruz, Emam, Prochnow, Roux, which is defined as I = εd ´ the mean pressure, ε˙ is the axial strain & Chevoir, 2005), where pis rate, and d is the particle diameter. The shear process can be considered quasi-static when I is below 2.5 × 10−3 (Lopera Perez, Kwok, O’Sullivan, Huang, & Hanley, 2016), and I is less than 10−4 in this paper. In a 2D drained biaxial compression test, the effective mean (p)´ and effective deviatoric (q)´ stresses are defined as p´= ( 1 + 2 )/2
(2)
q´ = ( 1 − 2 )/2
(3)
where 1 and 2 respectively denote the axial stress and lateral stress. The axial strain ε1 and volumetric strain εv can be estimated from the boundary displacements: Fig. 3. Change in with the number of disks for a non-convexity parameter = 0.075.
given . In this study, the non-convexity = 0.075 was selected, as shown in Fig. 3. Accordingly, the values corresponding to n = 3, 4, 5, 6, 7, 8, 11, 15, 17, and 19 were 0.134, 0.186, 0.236, 0.285, 0.332, 0.377, 0.505, 0.652, 0.715, and 0.770, respectively. Fig. 4 illustrates 10 non-convex particles having different local non-convexity . Each sample contained 5000 particles, which is consistent with previous numerical simulations on non-convex particles (Saint-Cyr et al., 2011; 2013). To avoid long-range ordering, a small variation in size was introduced by varying R from 0.01 to 0.02 m, with a uniform distribution. Particles were initially generated with random orientations and zero contacts within a large rectangular region modelled by four rigid walls. These particles were then subjected to isotropic compression under a small strain rate. To avoid force gradients and obtain dense isotropic packings, the gravity g and coefficients b and w of friction between the particles and the walls were set to zero during the compression. Finally, the sample was considered to have reached equilibrium when the ratio of the average static unbalanced force to the average contact force was less than 10−5 , and the tolerance of the stress from the wall was less than 0.5% of the required confining stress. Biaxial shear was then applied to all two-dimensional (2D) numerical samples under drained conditions. Before the shearing process, the interparticle frictional coefficient was adjusted to 0.5. The frictional coefficient of the particle wall remained unchanged at zero. While a constant confining pressure was applied to the lateral walls by servo control, the top wall moved downward at a constant velocity sufficiently low to subject the sample to verti-
ε1 =( h0 – h )/h0
(4)
εv = ( v0 – v)/v0
(5)
where h0 and h are respectively the initial and current heights of the sample; v0 is the initial volume of the sample; and v is the current volume at the same h. Volumetric compression is considered to be positive in this study. The internal angle of friction, ϕ, which reflects the shear strength of granular materials, is defined as sinϕ = ( 1 − 2 )/( 1 + 2 ) = q´/p´
(6)
Results and discussion The present paper focuses on critical state behaviour. All samples were sheared to axial strain of ε1 = 30%. Note that the char´ volume, are acteristic critical conditions, such as a constant q´/pand basically satisfied at deformation of ε1 ≥ 10% for all samples. This paper defines axial strain ε1 being in the range [10%, 30%] as the critical state, and the values of each parameter in the critical state are the average values within this interval. Previous numerical studies investigated the effect of the particle number on the critical shear strength. As an example, Huang, Hanley, O’Sullivan, and Kwok (2014) used three cylindrical samples (having 6783, 16,073, and 31,392 particles) to observe the material response. They found that the particle number has a limited effect on the critical shear strength, as reported for other DEM studies (e.g., Wang & Gutierrez, 2010; Gong, Nie et al., 2019; Gong, Wang, et al., 2019). Alternatively, previous numerical (Cantor, Azéma, Sornay, & Radjai, 2018) and experimental (Yang & Luo, 2018) studies investigated the effect of the particle size distribution on the critical shear strength of granular materials. They reported that
Fig. 4. Particles with different local non-convexities.
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Fig. 5. Stress ratio–axial strain responses for different particle numbers and s values: (a) different particle numbers and (b) different s values.
Fig. 6. Five different contact types: (A) simple contact (s), (B) double contact (d), (C) double-simple contact (ds), (D) triple contact (t), and (E) quadruple contact (q).
the particle size distribution had an insignificant effect on the critical shear strength. To verify the reliability of these conclusions, different particle numbers and particle size distribution are studied in the present paper. The particle size distribution is generally described as the size span, which can be defined by s = (dmax − dmin ) / (dmax + dmin ), where dmin and dmax are respectively the smallest and largest diameters of the particles (Voivret, Radjai, Delenne, & El Youssoufi, 2009). Note that a greater s indicates a wider size span. The numbers of particles in the validation tests were 5000, 10,000, 15,000, and 20,000 while the values of s were 0.5 (dmin =0.02 m and dmax =0.06 m), 0.67 (dmin =0.01 m and dmax =0.05 m), and 0.82 (dmin ´ =0.0075 m and dmax =0.075 m). Fig. 5(a) and (b) presents q´/pversus the axial strain ε1 for samples with different particle numbers and s. The results show that the peak and critical shear strength are nearly unchanged with increasing s and particle number, despite fluctuations in the critical state. The effects of the particle size distribution and particle number on the critical shear strength can therefore be excluded in this paper. The effect of non-convexity is mainly reflected in the variety of types of contact between particles (Saint-Cyr et al., 2011). The present study considered five different types of contact between two particles, as shown in Fig. 6: simple contact (s), double contact (d), double-simple contact (ds), triple contact (t), and quadruple contact (q). Note that quadruple contact can only occur when the overlap of particles is allowed, as when calculating the normal contact force for the soft contact model in DEM simulations. The coordination number is an indispensable index for measuring the internal structural characteristics of particle systems. There are two methods of calculating the coordination number for multi-disk clump simulations (Markauskas, Kaˇcianauskas, Dˇziugys, & Navakas, 2010). In one method, Zn = 2Ncp /Np , where Ncp denotes the contacts between particles (and not between sub-disks) and Np is the particle number. The other method assumes that Zc = 2Ncs /Np ,
Fig. 7. Coordination numbers as a function of in the initial and critical states.
where Ncs is the number of contacts between sub-disks. Throughout this paper, the superscripts i and c respectively indicate initial and critical states. Fig. 7 shows the coordination number in the initial and critical states as a function of . Clearly, Zin and Zic are greater than Zcn and Zcc , respectively. This is because the dilatancy effect of the dense samples used in this study leads to the samples becoming looser. As increases, Zin and Zcn decrease but Zic and Zcc increase. The decreased Zn and increased Zc indicate that with an increase in , the number of contacts between sub-disks gradually increases but the number of contacts between particles gradually decreases. That is to say, the contact type changes with increasing .
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Fig. 8. Proportions of different contact types as a function of : (a) initial state and (b) critical state.
Fig. 9. Stress ratio–axial strain responses for all samples.
Fig. 8 shows the proportions of different contact types (that is, ks , kd , kds , kt , and kq ) as a function of . Here, ks , kd , kds , kt , and kq respectively denote the proportions of the numbers of simple, double, double-simple, triple, and quadruple contacts in the total number of contacts between sub-disks. A consistent trend is observed between initial and critical states. The proportions of simple and double contacts over all contacts (that is, ks and kd ) indicate that these contacts are the major contact types (ks + kd ≥ 75%). ks and kd have a decreasing and increasing trend, respectively, with increasing . The kds , kt , and kq values are small and remain nearly unchanged as increases, despite some fluctuations. As increases, the decrease in ks and increase in kd are consistent with the observation that the number of contacts between sub-disks gradually increases while the number of contacts between particles gradually decreases, as shown in Fig. 7. The change in contact type may affect the contact network, which is the backbone of stress transmission in quasi-static equilibrium and links the relevant microstructural variables in a granular material (Radjai, Troadec, & Roux, 2004). Local non-convexity may therefore affect the critical shear strength of granular materials. ´ Fig. 9 presents q´/pversus the axial strain ε1 for samples with dif´ ferent . All the curves have the same trend; that is, q ´/pincreases rapidly to a peak and then decreases to a state of slight fluctuation. All samples roughly reach the critical state after ε1 = 10%. The inset of Fig. 9 illustrates the peak (qp /p)´ and critical (qc /p)´ stress ratios as functions of . The qp /p´ratio has a unimodal pattern with ´ ; that is, qp /pfirst increases, reaching a maximum at = 0.332, and then decreases monotonically as increases. However, qc /p´remains
nearly unchanged and fluctuates only in the range of [0.307, 0.334] as increases. This indicates that the critical shear strength is unaffected by the local non-convexity. This finding is contrary to intuition because the contact types change. Note that rotationally symmetric particles were used in DEM simulations in this study, as shown in Fig. 4. However, for real granular materials, most particles have uneven roughness. To study whether the critical shear strength is also independent of for uneven particles, additional tests using uneven particles with different were conducted, as shown in Fig. 10. Stress ratio–axial strain curves are presented in Fig. 11. For the uneven particles, the critical shear strength is almost the same and the peak shear strength declines slightly; these results are consistent with the observations in Fig. 9. This finding indicates that the critical shear strength is also independent of for the uneven particles. Fig. 12 illustrates the contribution of each contact type to the critical shear strength as a function of . Following Gong and Liu (2017b), qcm /p´ is defined as the shear strength of the correspondc – c )/2p´, where m denotes either ing contact type, qcm /p´ = ( 1m 2m s, d, ds, t, or q. As increases, qcs /p´ and qcd /p´ respectively decrease and increase while qcds /p´, qct /p´, and qcq /p´are small and remain constant. The contributions of the different contact types to the critical shear strength are related to the proportions, as shown in Fig. 8(b). Furthermore, the decreasing contribution of simple contacts to the critical shear strength is compensated for by the increase in the contribution of double contacts, and the critical strength thus basically remains constant. Previous DEM studies considered that, from the perspective of micromechanics, force and fabric anisotropies are the sources of the shear resistance of granular materials (Guo & Zhao, 2013; Ouadfel & Rothenburg, 2001; Zhao, Evans, & Zhou, 2018). For cohesionless granular materials, this correlation can be expressed as q´/p´= (ac + an + at + adn + adt )/2 ,
(7)
where ac is the anisotropic coefficient of the contacts; an and adn are respectively the anisotropic coefficients of the normal contact forces and normal branch vectors; and at and adt are respectively the anisotropic coefficients of the tangential contact forces and tangential branch vectors. Notably, all the anisotropic coefficients can be calculated from the generalized fabric and force tensors given in the Appendix; these tensors were confirmed in previous studies (Markauskas et al., 2010; Ouadfel & Rothenburg, 2001). Variations in the anisotropic coefficients in the critical state with are plotted in Fig. 13. In the figure, the results of Eqs. (6) and (7) are well matched; thus, the method of conversion between shear strength and fabric anisotropy for the entire contact network is reliable. Clearly, ac and an dominate the critical shear strength. They are followed by at , while adn and adt may be ignored. As increases, ac fluctuates slightly, while an and at remain roughly the same.
Please cite this article in press as: Nie, Z., et al. Effect of local non-convexity on the critical shear strength of granular materials determined via the discrete element method. Particuology (2020), https://doi.org/10.1016/j.partic.2019.12.008
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Fig. 10. Deviation in particle shape under the same non-convexity.
Fig. 13. Variations in the anisotropic coefficients in the critical state as a function of .
Fig. 11. Stress ratio–axial strain responses for uneven samples.
Fig. 12. Contributions of different contact types to the critical shear strength.
Considering the force and fabric anisotropies supported by the s, d, ds, t, and q contacts in the particle packing, the anisotropy of each contact type is quantified by partitioning the anisotropies ( m ac-m , m an-m , and m at-m ), where m is the weight coefficient and m denotes the s, d, ds, t, or q contacts. In this case, assuming that the principal directions of the branch vectors and contact forces of each contact type are the same as the principal directions of the global branch and contact forces, we have ac ≈ s ac-s + d ac-d + ds ac-ds + t ac-t + q ac-q ,
(8)
an ≈ s an-s + d an-d + ds an-ds + t an-t + q an-q
(9)
at ≈ s at-s + d at-d + ds at-ds + t at-t + q at-q
(10)
Note that for some cases of granular systems, the assumption of the principal directions of the branch vectors and contact forces of subsets of the contact network is not fulfilled. As an example, a common partitioning approach is to consider subnets of strong and weak contact forces (Radjai, Wolf, Jean, & Moreau, 1998), where the delineation is made using the mean normal interparticle contact force. The weak contact forces are generally lateral (perpendicular to the principal direction) and provide stability against forces propagating through strong contacts (Radjai et al., 1998). Therefore, this assumption is not fulfilled for the subsets of strong and weak contact networks. In this context, the assumption needs verification for the subsets of contact networks in this study. Fig. 13 also illustrates the anisotropies (that is, 0.5ac , 0.5an , and 0.5at ) derived from the anisotropy of each contact type. It is clear that the values of 0.5ac , 0.5an , and 0.5at compare well with Eqs. (8), (9) and (10), respectively. The assumption is thus valid for the subnets of the contact network in this study. Fig. 14(a)–(c) respectively present the evolutions of m ac-m , m an-m , and m at-m as functions of . Clearly, s contacts and d contacts dominate the three anisotropic coefficients, and the contribution of q contacts is nearly zero. This result is consistent with results presented in Figs. 8(b) and 12 . The proportions of ds and t contacts are relatively small in Fig. 9; however, as shown in Fig. 14(a), they play an important role in determining ac . Specifically, the increases in ds ac-ds and t ac-t are offset by the decrease in s ac-s , ensuring the equilibrium of ac , while d ac-d remains constant with some fluctuation. In Fig. 14(b), the decrease in s an-s compensates for the increase in d an-d as increases, which maintains the stability of an , while ds an-ds and t an-t are small and stable. Fig. 14(c) and (b) shows the same trends for changes in s, d, and ds contacts; that is, s at-s decreases, d at-d increases and ds at-ds fluctuates with an increase in , whereas t at-t increases slightly. Therefore, the stability of at against is attributed to the increases in d at-d and t at-t compensating for the decrease in s at-s .
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butions of the different contact types to the critical shear strength and the fabric anisotropic coefficients were determined. The dominant contribution was related to the combined effect of simple and multipoint contacts; that is, an increase in multipoint (especially double) contacts is compensated by a decrease in simple contacts. Declaration of interest statement The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments The study is financially supported by the National Natural Science Foundation of China, China (No. 51809292, 51478481 and 51508141), Postdoctoral Fund of Central South University, China (No. 205455) and Beijing Municipal Science and Technology Project: Research and Application of Design and Construction Technology of Railway Engineering Traveling the Rift Valley, China (No. Z181100003918005). The authors express their appreciation for the financial assistance. Appendix A. Fabric and force tensors The anisotropies can be conveniently estimated from the following fabric and force tensors: ac−m = 4ϕijm’ , ϕijm = ij
1 Nc ˛ ˛ ni nj , Nm ˛=1 c
4
Nmc
m
an-m = ij
Fn-m’ , Fn-m = ij ij −fn-m 0
adn−m = ij
=
= Fig. 14. Evolution of m ac- m , m an- m , and m at- m as a function of : (a) ac- m , (b) an- m , and (c) at- m .
Conclusions This study investigated the effects of local non-convexity on the critical shear strength of granular materials using the DEM. A series of drained biaxial tests were conducted on rough particle packings with a uniform non-convexity of 0.075 and different values of local non-convexity ranging from 0.134 to 0.770. The results show that the contact type changes with an increase in but the critical shear strength is not affected by . Through analysis of the contact types and their proportions and coordination numbers, the contri-
m c-m n˛ n˛ ) ˛=1 Nc (1+˛ k l kl
˛ ˛ d˛ dn ni nj
= −ddn-m 0 m c-m n˛ n˛ ) ˛=1 Nc (1+˛ k l kl 4
f¯0t−m
adt−m = ij
˛ ˛ f˛ n ni nj
= Fn-m , −fn-m 0 ii A2)
4 Dijdn−m’ , Ddn-m ij d¯ 0dn−m
Nmc
= at−m ij
(A1)
Fijt−m’ , Ft-m ij =
Nmc
Ddn-m , ii ˛ ˛ f˛ t ni nj
−ft-m 0 m c-m n˛ n˛ ) ˛=1 Nc (1+˛ k l kl
(A3)
= Ft-m ii ,A4)
4 Dijdt−m’ , Ddt-m ij d¯ 0dt−m
Nmc
˛ ˛ d˛ dt ni nj
−ddt-m = 0 m c-m n˛ n˛ ) ˛=1 Nc (1+˛ k l kl
Ddt-m , ii
(A5)
dn-m’ where ϕijm’ , Fn-m’ , Ft-m’ , and Ddt-m’ are the deviatoric tenij ij , Dij ij t-m dn-m sors corresponding to ϕijm , Fn-m , and Ddt-m , respectively, ij , Fij , Dij ij given by Guo and Zhao (2013). ´ The effective mean (p) and deviatoric (q´) stresses can be given
by p´= ii /2 and q´ =
ij’ ij’ /2, where ij’ is the deviation corre-
sponding to the stress tensor ij . The fabric is anisotropic, and it is convenient to describe the degree of anisotropy of the fabric by using a scalar a*-m , which is attained from the invariant of every anisotropy tensor, i.e., ac-m , an-m , at-m , adn-m , and adt-m , as follows: ij ij ij ij ij
a*-m = sign(Sr )
a*-m a*-m /2 ij ij
(A6)
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where a*-m indicates the anisotropic coefficients ac-m , ad-m , at-m , adn-m and adt-m , corresponding to one of the four cases of fabric anisotropy mentioned above. Sr is a normalized quantity of the douand ij’ , given by (Guo & Zhao, 2013). Then, ble contraction of a*-m ij ´ the stress-force-fabric relationship between q/pand am (neglecting * the cross product between ac-m and an-m ) can be modified as q´/p´=
m
m
ac-m + ad-m + at-m + adn-m + adt-m . 2
(A7)
Clearly, for m = 1 in Eq. (A7) for the overall contact network, the stress-force-fabric relationship can be reduced to Eq. (7). References Abbireddy, C. O. R., & Clayton, C. R. I. (2010). Varying initial void ratios for DEM simulations. Géotechnique, 60(6), 497–502. Abedi, S., & Mirghasemi, A. A. (2011). Particle shape consideration in numerical simulation of assemblies of irregularly shaped particles. Particuology, 9(4), 387–397. Anthony, J. L., & Marone, C. (2005). Influence of particle characteristics on granular friction. Journal of Geophysical Research, 110, B08409. Azéma, E., & Radjai, F. (2010). Stress-strain behavior and geometrical properties of packings of elongated particles. Physical Review E, 81(5), 051304. Azéma, E., Radjai, F., & Saussine, G. (2009). Quasistatic rheology, force transmission and fabric properties of a packing of irregular polyhedral particles. Mechanics of Materials, 41(6), 729–741. Azéma, E., Radjai, F., Peyroux, R., & Saussine, G. (2007). Force transmission in a packing of pentagonal particles. Physical Review E, 76(1), 011301. Azéma, E., Radjai, F., Saint-Cyr, B., Delenne, J. Y., & Sornay, P. (2013). Rheology of three-dimensional packings of aggregates: Microstructure and effects of nonconvexity. Physical Review E, 87(5), 052205. Azéma, E., Radjai, F., & Dubois, F. (2013). Packings of irregular polyhedral particles: Strength, structure, and effects of angularity. Physical Review E, 87(6), 062203. Brown, E., Zhang, H., Forman, N. A., Maynor, B. W., Betts, D. E., DeSimone, J. M., et al. (2011). Shear thickening and jamming in densely packed suspensions of different particle shapes. Physical Review E, 84(3), 031408. Cantor, D., Azéma, E., Sornay, P., & Radjai, F. (2018). Rheology and structure of polydisperse three-dimensional packings of spheres. Physical Review E, 98(5), 052910. Cho, G. C., Dodds, J., & Santamarina, J. C. (2006). Particle shape effects on packing density, stiffness, and strength: Natural and crushed sands. Journal of Geotechnical and Geoenvironmental Engineering, 132(5), 591–602. Cundall, P. A., & Strack, O. D. L. (1979). A discrete numerical model for granular assemblies. Géotechnique, 29(1), 47–65. Da Cruz, F., Emam, S., Prochnow, M., Roux, J. N., & Chevoir, F. (2005). Rheophysics of dense granular materials: Discrete simulation of plane shear flows. Physical Review E, 72(2), 021309. de Graaf, J., van Roij, R., & Dijkstra, M. (2011). Dense regular packings of irregular nonconvex particles. Physical Review Letters, 107(15), 155501. Feng, Y. T., Zhao, T., Kato, J., & Zhou, W. (2017). Towards stochastic discrete element modelling of spherical particles with surface roughness: A normal interaction law. Computer Methods in Applied Mechanics and Engineering, 315, 247–272. Goldenberg, C., & Goldhirsch, I. (2005). Friction enhances elasticity in granular solids. Nature, 435(7039), 188–191. Gong, J., & Liu, J. (2017a). Effect of aspect ratio on triaxial compression of multi-sphere ellipsoid assemblies simulated using a discrete element method. Particuology, 32, 49–62. Gong, J., & Liu, J. (2017b). Mechanical transitional behavior of binary mixtures via DEM: Effect of differences in contact-type friction coefficients. Computers and Geotechnics, 85, 1–14. Gong, J., Nie, Z., Zhu, Y., Liang, Z., & Wang, X. (2019). Exploring the effects of particle shape and content of fines on the shear behavior of sand-fines mixtures via the DEM. Computers and Geotechnics, 106, 161–176. Gong, J., Wang, X., Li, L., & Nie, Z. (2019). DEM study of the effect of fines content on the small-strain stiffness of gap-graded soils. Computers and Geotechnics, 112, 35–40. Gu, X., Huang, M., & Qian, J. (2014). DEM investigation on the evolution of microstructure in granular soils under shearing. Granular Matter, 16(1), 91–106. Guo, N., & Zhao, J. (2013). The signature of shear-induced anisotropy in granular media. Computers and Geotechnics, 47, 1–15.
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Effect of local non-convexity on the critical shear strength of granular materials determined via the discrete element method Zhihong Nie a , Shunkai Liu a , Wei Hu b , Jian Gong c,∗ a
School of Civil Engineering, Central South University, Changsha, Hunan 410075, China Hunan Province Key Laboratory of Geotechnical Engineering Stability Control and Health Monitoring, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China c School of Hydraulic Engineering, Changsha University of Science and Technology, Changsha 410114, China b
a r t i c l e
i n f o
Article history: Received 22 May 2019 Received in revised form 16 October 2019 Accepted 19 December 2019 Available online xxx Keywords: Multi-sphere clump Local non-convexity Discrete element method Biaxial shear test Contact type Critical shear strength
a b s t r a c t Multi-sphere clumps are commonly used to simulate non-spherical particles in discrete element method simulations. It is of interest whether the degree of local non-convexity affects the mechanical behaviour of granular materials with the same non-convexity . A series of discrete-element-method biaxial shear tests are conducted on rough particle packings with = 0.075 and different values (ranging from 0.134 to 0.770). The microscale results show that the contact type changes with an increase in . However, the critical strength is independent of . The evaluation of the contributions of different contact types to the critical shear strength and a detailed analysis of the anisotropies help clarify the microscopic mechanisms that result in the independence of the critical shear strength from . © 2020 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.
Introduction There are three important scales regarding the particle shape of granular materials: sphericity (eccentricity or plainness), roundness (angularity), and roughness (smoothness or non-convexity) (Cho, Dodds, & Santamarina, 2006). Various previous experimental and numerical studies have reported that sphericity and roundness affect the mechanical behaviours of granular materials (Azéma & Radjai, 2010; Azéma, Radjai, & Dubois, 2013; Brown et al., 2011; Feng, Zhao, Kato, & Zhou, 2017; Jensen, Bosscher, Plesha, & Edil, 1999; Meng, Li, Lu, Li, & Jin, 2012). However, only a few studies have focused on the effect of roughness, which also affects the properties of granular materials. In terms of experimental studies, Narayan and Hancock (2003) noted that the indentation hardness, elastic modulus, and brittle fracture index of particles gradually decrease with increasing roughness. Additionally, Santamarina and Cascante (1998) and Otsubo, O’sullivan, Sim, and Ibraim (2015)) reported that rough spheres have a smaller shear stiffness than smooth spheres. Furthermore, Anthony and Marone (2005) found that an increase in particle roughness increases the frictional
∗ Corresponding author. E-mail address: gj [email protected] (J. Gong).
strength and affects the distribution of stress within sheared layers. Alternatively, the discrete element method (DEM) can be adopted for the efficient numerical modelling of the effects of roughness on the macroscale and microscale properties of granular materials (Otsubo, O’Sullivan, Hanley, & Sim, 2017; Rémond, Gallias, & Mizrahi, 2008; de Graaf, van Roij, & Dijkstra, 2011; Yang, Cheng, & Sun, 2017). Using clumps composed of sub-spheres is the most common method of simulating non-spherical particles with the DEM. The effect of roughness can then be studied by changing the properties (e.g., radius and position) of the sub-spheres of the clumps. Taking this approach, Ludewig and Vandewalle (2012) found that when particles are modified to have a high roughness, particle interlocking and multiple contact points develop in the packing; nonconvex particle packings have a lower density and are more stable than spherical packings. Saint-Cyr et al. (SaintCyr, Delenne, Voivret, Radjai, & Sornay, 2011; Saint-Cyr, Radjai, Delenne, & Sornay, 2013) reported that both the internal angle of friction and the maximum cohesion increase linearly as the roughness increases, but the former approaches a certain value. Furthermore, Azéma, Radjai, Dubois et al. (2013) observed that the critical shear strength is an increasing function of roughness. In the above-mentioned DEM simulations, roughness was quantified as the concavity of the entire particle surface, which is composed of various local non-convexities. Note that non-convexity is of concern in practical engineering, particularly in the storage,
https://doi.org/10.1016/j.partic.2019.12.008 1674-2001/© 2020 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.
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Table 1 Input parameters in the DEM simulation. Parameter
Value
Particle density, (kg/m3 ) Number of particles Coefficient of inter-particle friction, b Coefficient of wall–particle friction, w Wall stiffness (N/m) Contact effective modulus, Ec (Pa) kn /ks Damping constant, ˇ
2600 5000 0.5 0.0 1 × 109 1 × 108 4/3 0.7
Fig. 1. Different local non-convexities of particles with the same roughness.
handling, and recycling of crushed hollow industrial by-products and sintered powders (Azéma, Radjai, & Saussine, 2009; Rémond et al., 2008). For a given overall roughness, the form of local nonconvexity may not be the same (see Fig. 1). Indeed, the variation in local non-convexity may be a noise factor when using clumps in DEM simulations. The critical shear strength reflects the intrinsic shear strength of the granular material and is independent of the initial state. Thus, it is of interest whether different local nonconvexities affect the critical shear strength of granular materials with identical roughness. This issue is the motivation of the present work, wherein we conduct grain-scale modelling using the DEM, which has previously been demonstrated to reproduce certain key features of granular materials (Azéma, Radjai, Peyroux, & Saussine, 2007; Brown et al., 2011; Gong, Wang, Li, & Nie, 2019; Nie, Zhu, Wang, & Gong, 2019). In this paper, the effect of local non-convexity on the shear behaviour of particle packings is explored through a series of drained biaxial compression tests using the DEM. The rest of the paper is organized as follows. First, a brief introduction of DEM modelling is given. Then, contact types and microscale simulation results of each contact type, including the coordination number and proportions related to the contact type, are studied. Next, the stress–strain characteristics and the contributions of certain contact types to the critical shear strength are analysed. Subsequently, the effect of the local non-convexity on fabric anisotropy is evaluated. Finally, main conclusions are presented. DEM modelling Biaxial compression tests were conducted using the wellrecognized DEM program PFC2D (Itasca, 2014), which was originally developed by Cundall and Strack (1979). Simulations were carried out using the linear elastic contact model, as used in several previous studies (Gong, Nie, Zhu, Liang, & Wang, 2019; Gu, Huang, & Qian, 2014; Minh & Cheng, 2013). The microscale parameters used in the present study were selected by referring to many previous experimental and numerical studies. Specifically, Rowe (1962) reported that the inter-particle friction b of quartz particles is about 0.445–0.601. Considering values of b adopted in many previous DEM simulations (e.g., Li, Yu, & Li, 2009; Abbireddy & Clayton, 2010; Abedi & Mirghasemi, 2011; Yang & Dai, 2011; Saint-Cyr et al., 2011, 2013; Yang, Cheng, & Wang, 2016; Gong & Liu, 2017a; Gong, Nie et al., 2019), the value of b = 0.5 was used in the present study. The contact effective modulus Ec = 1 × 108 Pa
Fig. 2. Geometry of regular aggregates.
in the present study was consistent with values reported by Yang, Cheng et al. (2016) and Gong and Liu (2017a). Alternatively, the shear stiffness ratio kn /ks (where kn and ks respectively denote the normal and shear contact stiffness of particles) suggested by Goldenberg and Goldhirsch (2005) for realistic granular materials is in the range of 1.0 < kn /ks < 1.5, which correlates well with the Cattaneo–Mindlin model (Johnson, 1987) for elastic spherical contacts. Hence, kn /ks = 4.0/3 was adopted in the present study. Local damping with damping constant ˇ = 0.7 as suggested in Itasca (2014) has been used for effectively dissipating kinetic energy, but it is expected to have little effect on the simulation results. All basic parameters used in our simulations are listed in Table 1. The model particles comprised clumps, which were rigid sets of n overlapping disks of the same radius r, with n-fold rotational symmetry (see Fig. 2). Following previous studies (e.g., Saint-Cyr et al., 2011; 2013), the surface roughness was described by the nonconvexity (), which characterizes the degree of distortion from a perfectly circular shape and is defined as
=R/R = 1– (/tan˛ +
1– 2 )/(1+/sin˛)
(1)
where R = R – R´, with R and R´respectively being the radii of the circumscribing circle and the inscribed circle; ˛ = /n; and = l/2r, with l being the distance between disk centres. In this paper, the local non-convexity is defined as cos(/2), where is the average of the angles formed by the tangent lines on either side of a concave point, as shown in Fig. 2. According to their definitions, and have the geometric relationship = cos(/2). The local non-convexity can therefore be represented by . When the radius r of the disk is fixed, can be varied by changing the number of disks owing to the rotationally symmetric distribution of the sub-disks. Fig. 3 displays the non-convexity as a function of for different values of n. Clearly, Fig. 3 shows that can be unified by adjusting and becomes monotonically greater as n increases for a
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cal compression while ensuring that the shear process remained in a quasi-static state. This process reduces the inertia parameter I, ˙ /p´ (Da Cruz, Emam, Prochnow, Roux, which is defined as I = εd ´ the mean pressure, ε˙ is the axial strain & Chevoir, 2005), where pis rate, and d is the particle diameter. The shear process can be considered quasi-static when I is below 2.5 × 10−3 (Lopera Perez, Kwok, O’Sullivan, Huang, & Hanley, 2016), and I is less than 10−4 in this paper. In a 2D drained biaxial compression test, the effective mean (p)´ and effective deviatoric (q)´ stresses are defined as p´= ( 1 + 2 )/2
(2)
q´ = ( 1 − 2 )/2
(3)
where 1 and 2 respectively denote the axial stress and lateral stress. The axial strain ε1 and volumetric strain εv can be estimated from the boundary displacements: Fig. 3. Change in with the number of disks for a non-convexity parameter = 0.075.
given . In this study, the non-convexity = 0.075 was selected, as shown in Fig. 3. Accordingly, the values corresponding to n = 3, 4, 5, 6, 7, 8, 11, 15, 17, and 19 were 0.134, 0.186, 0.236, 0.285, 0.332, 0.377, 0.505, 0.652, 0.715, and 0.770, respectively. Fig. 4 illustrates 10 non-convex particles having different local non-convexity . Each sample contained 5000 particles, which is consistent with previous numerical simulations on non-convex particles (Saint-Cyr et al., 2011; 2013). To avoid long-range ordering, a small variation in size was introduced by varying R from 0.01 to 0.02 m, with a uniform distribution. Particles were initially generated with random orientations and zero contacts within a large rectangular region modelled by four rigid walls. These particles were then subjected to isotropic compression under a small strain rate. To avoid force gradients and obtain dense isotropic packings, the gravity g and coefficients b and w of friction between the particles and the walls were set to zero during the compression. Finally, the sample was considered to have reached equilibrium when the ratio of the average static unbalanced force to the average contact force was less than 10−5 , and the tolerance of the stress from the wall was less than 0.5% of the required confining stress. Biaxial shear was then applied to all two-dimensional (2D) numerical samples under drained conditions. Before the shearing process, the interparticle frictional coefficient was adjusted to 0.5. The frictional coefficient of the particle wall remained unchanged at zero. While a constant confining pressure was applied to the lateral walls by servo control, the top wall moved downward at a constant velocity sufficiently low to subject the sample to verti-
ε1 =( h0 – h )/h0
(4)
εv = ( v0 – v)/v0
(5)
where h0 and h are respectively the initial and current heights of the sample; v0 is the initial volume of the sample; and v is the current volume at the same h. Volumetric compression is considered to be positive in this study. The internal angle of friction, ϕ, which reflects the shear strength of granular materials, is defined as sinϕ = ( 1 − 2 )/( 1 + 2 ) = q´/p´
(6)
Results and discussion The present paper focuses on critical state behaviour. All samples were sheared to axial strain of ε1 = 30%. Note that the char´ volume, are acteristic critical conditions, such as a constant q´/pand basically satisfied at deformation of ε1 ≥ 10% for all samples. This paper defines axial strain ε1 being in the range [10%, 30%] as the critical state, and the values of each parameter in the critical state are the average values within this interval. Previous numerical studies investigated the effect of the particle number on the critical shear strength. As an example, Huang, Hanley, O’Sullivan, and Kwok (2014) used three cylindrical samples (having 6783, 16,073, and 31,392 particles) to observe the material response. They found that the particle number has a limited effect on the critical shear strength, as reported for other DEM studies (e.g., Wang & Gutierrez, 2010; Gong, Nie et al., 2019; Gong, Wang, et al., 2019). Alternatively, previous numerical (Cantor, Azéma, Sornay, & Radjai, 2018) and experimental (Yang & Luo, 2018) studies investigated the effect of the particle size distribution on the critical shear strength of granular materials. They reported that
Fig. 4. Particles with different local non-convexities.
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Fig. 5. Stress ratio–axial strain responses for different particle numbers and s values: (a) different particle numbers and (b) different s values.
Fig. 6. Five different contact types: (A) simple contact (s), (B) double contact (d), (C) double-simple contact (ds), (D) triple contact (t), and (E) quadruple contact (q).
the particle size distribution had an insignificant effect on the critical shear strength. To verify the reliability of these conclusions, different particle numbers and particle size distribution are studied in the present paper. The particle size distribution is generally described as the size span, which can be defined by s = (dmax − dmin ) / (dmax + dmin ), where dmin and dmax are respectively the smallest and largest diameters of the particles (Voivret, Radjai, Delenne, & El Youssoufi, 2009). Note that a greater s indicates a wider size span. The numbers of particles in the validation tests were 5000, 10,000, 15,000, and 20,000 while the values of s were 0.5 (dmin =0.02 m and dmax =0.06 m), 0.67 (dmin =0.01 m and dmax =0.05 m), and 0.82 (dmin ´ =0.0075 m and dmax =0.075 m). Fig. 5(a) and (b) presents q´/pversus the axial strain ε1 for samples with different particle numbers and s. The results show that the peak and critical shear strength are nearly unchanged with increasing s and particle number, despite fluctuations in the critical state. The effects of the particle size distribution and particle number on the critical shear strength can therefore be excluded in this paper. The effect of non-convexity is mainly reflected in the variety of types of contact between particles (Saint-Cyr et al., 2011). The present study considered five different types of contact between two particles, as shown in Fig. 6: simple contact (s), double contact (d), double-simple contact (ds), triple contact (t), and quadruple contact (q). Note that quadruple contact can only occur when the overlap of particles is allowed, as when calculating the normal contact force for the soft contact model in DEM simulations. The coordination number is an indispensable index for measuring the internal structural characteristics of particle systems. There are two methods of calculating the coordination number for multi-disk clump simulations (Markauskas, Kaˇcianauskas, Dˇziugys, & Navakas, 2010). In one method, Zn = 2Ncp /Np , where Ncp denotes the contacts between particles (and not between sub-disks) and Np is the particle number. The other method assumes that Zc = 2Ncs /Np ,
Fig. 7. Coordination numbers as a function of in the initial and critical states.
where Ncs is the number of contacts between sub-disks. Throughout this paper, the superscripts i and c respectively indicate initial and critical states. Fig. 7 shows the coordination number in the initial and critical states as a function of . Clearly, Zin and Zic are greater than Zcn and Zcc , respectively. This is because the dilatancy effect of the dense samples used in this study leads to the samples becoming looser. As increases, Zin and Zcn decrease but Zic and Zcc increase. The decreased Zn and increased Zc indicate that with an increase in , the number of contacts between sub-disks gradually increases but the number of contacts between particles gradually decreases. That is to say, the contact type changes with increasing .
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Fig. 8. Proportions of different contact types as a function of : (a) initial state and (b) critical state.
Fig. 9. Stress ratio–axial strain responses for all samples.
Fig. 8 shows the proportions of different contact types (that is, ks , kd , kds , kt , and kq ) as a function of . Here, ks , kd , kds , kt , and kq respectively denote the proportions of the numbers of simple, double, double-simple, triple, and quadruple contacts in the total number of contacts between sub-disks. A consistent trend is observed between initial and critical states. The proportions of simple and double contacts over all contacts (that is, ks and kd ) indicate that these contacts are the major contact types (ks + kd ≥ 75%). ks and kd have a decreasing and increasing trend, respectively, with increasing . The kds , kt , and kq values are small and remain nearly unchanged as increases, despite some fluctuations. As increases, the decrease in ks and increase in kd are consistent with the observation that the number of contacts between sub-disks gradually increases while the number of contacts between particles gradually decreases, as shown in Fig. 7. The change in contact type may affect the contact network, which is the backbone of stress transmission in quasi-static equilibrium and links the relevant microstructural variables in a granular material (Radjai, Troadec, & Roux, 2004). Local non-convexity may therefore affect the critical shear strength of granular materials. ´ Fig. 9 presents q´/pversus the axial strain ε1 for samples with dif´ ferent . All the curves have the same trend; that is, q ´/pincreases rapidly to a peak and then decreases to a state of slight fluctuation. All samples roughly reach the critical state after ε1 = 10%. The inset of Fig. 9 illustrates the peak (qp /p)´ and critical (qc /p)´ stress ratios as functions of . The qp /p´ratio has a unimodal pattern with ´ ; that is, qp /pfirst increases, reaching a maximum at = 0.332, and then decreases monotonically as increases. However, qc /p´remains
nearly unchanged and fluctuates only in the range of [0.307, 0.334] as increases. This indicates that the critical shear strength is unaffected by the local non-convexity. This finding is contrary to intuition because the contact types change. Note that rotationally symmetric particles were used in DEM simulations in this study, as shown in Fig. 4. However, for real granular materials, most particles have uneven roughness. To study whether the critical shear strength is also independent of for uneven particles, additional tests using uneven particles with different were conducted, as shown in Fig. 10. Stress ratio–axial strain curves are presented in Fig. 11. For the uneven particles, the critical shear strength is almost the same and the peak shear strength declines slightly; these results are consistent with the observations in Fig. 9. This finding indicates that the critical shear strength is also independent of for the uneven particles. Fig. 12 illustrates the contribution of each contact type to the critical shear strength as a function of . Following Gong and Liu (2017b), qcm /p´ is defined as the shear strength of the correspondc – c )/2p´, where m denotes either ing contact type, qcm /p´ = ( 1m 2m s, d, ds, t, or q. As increases, qcs /p´ and qcd /p´ respectively decrease and increase while qcds /p´, qct /p´, and qcq /p´are small and remain constant. The contributions of the different contact types to the critical shear strength are related to the proportions, as shown in Fig. 8(b). Furthermore, the decreasing contribution of simple contacts to the critical shear strength is compensated for by the increase in the contribution of double contacts, and the critical strength thus basically remains constant. Previous DEM studies considered that, from the perspective of micromechanics, force and fabric anisotropies are the sources of the shear resistance of granular materials (Guo & Zhao, 2013; Ouadfel & Rothenburg, 2001; Zhao, Evans, & Zhou, 2018). For cohesionless granular materials, this correlation can be expressed as q´/p´= (ac + an + at + adn + adt )/2 ,
(7)
where ac is the anisotropic coefficient of the contacts; an and adn are respectively the anisotropic coefficients of the normal contact forces and normal branch vectors; and at and adt are respectively the anisotropic coefficients of the tangential contact forces and tangential branch vectors. Notably, all the anisotropic coefficients can be calculated from the generalized fabric and force tensors given in the Appendix; these tensors were confirmed in previous studies (Markauskas et al., 2010; Ouadfel & Rothenburg, 2001). Variations in the anisotropic coefficients in the critical state with are plotted in Fig. 13. In the figure, the results of Eqs. (6) and (7) are well matched; thus, the method of conversion between shear strength and fabric anisotropy for the entire contact network is reliable. Clearly, ac and an dominate the critical shear strength. They are followed by at , while adn and adt may be ignored. As increases, ac fluctuates slightly, while an and at remain roughly the same.
Please cite this article in press as: Nie, Z., et al. Effect of local non-convexity on the critical shear strength of granular materials determined via the discrete element method. Particuology (2020), https://doi.org/10.1016/j.partic.2019.12.008
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Fig. 10. Deviation in particle shape under the same non-convexity.
Fig. 13. Variations in the anisotropic coefficients in the critical state as a function of .
Fig. 11. Stress ratio–axial strain responses for uneven samples.
Fig. 12. Contributions of different contact types to the critical shear strength.
Considering the force and fabric anisotropies supported by the s, d, ds, t, and q contacts in the particle packing, the anisotropy of each contact type is quantified by partitioning the anisotropies ( m ac-m , m an-m , and m at-m ), where m is the weight coefficient and m denotes the s, d, ds, t, or q contacts. In this case, assuming that the principal directions of the branch vectors and contact forces of each contact type are the same as the principal directions of the global branch and contact forces, we have ac ≈ s ac-s + d ac-d + ds ac-ds + t ac-t + q ac-q ,
(8)
an ≈ s an-s + d an-d + ds an-ds + t an-t + q an-q
(9)
at ≈ s at-s + d at-d + ds at-ds + t at-t + q at-q
(10)
Note that for some cases of granular systems, the assumption of the principal directions of the branch vectors and contact forces of subsets of the contact network is not fulfilled. As an example, a common partitioning approach is to consider subnets of strong and weak contact forces (Radjai, Wolf, Jean, & Moreau, 1998), where the delineation is made using the mean normal interparticle contact force. The weak contact forces are generally lateral (perpendicular to the principal direction) and provide stability against forces propagating through strong contacts (Radjai et al., 1998). Therefore, this assumption is not fulfilled for the subsets of strong and weak contact networks. In this context, the assumption needs verification for the subsets of contact networks in this study. Fig. 13 also illustrates the anisotropies (that is, 0.5ac , 0.5an , and 0.5at ) derived from the anisotropy of each contact type. It is clear that the values of 0.5ac , 0.5an , and 0.5at compare well with Eqs. (8), (9) and (10), respectively. The assumption is thus valid for the subnets of the contact network in this study. Fig. 14(a)–(c) respectively present the evolutions of m ac-m , m an-m , and m at-m as functions of . Clearly, s contacts and d contacts dominate the three anisotropic coefficients, and the contribution of q contacts is nearly zero. This result is consistent with results presented in Figs. 8(b) and 12 . The proportions of ds and t contacts are relatively small in Fig. 9; however, as shown in Fig. 14(a), they play an important role in determining ac . Specifically, the increases in ds ac-ds and t ac-t are offset by the decrease in s ac-s , ensuring the equilibrium of ac , while d ac-d remains constant with some fluctuation. In Fig. 14(b), the decrease in s an-s compensates for the increase in d an-d as increases, which maintains the stability of an , while ds an-ds and t an-t are small and stable. Fig. 14(c) and (b) shows the same trends for changes in s, d, and ds contacts; that is, s at-s decreases, d at-d increases and ds at-ds fluctuates with an increase in , whereas t at-t increases slightly. Therefore, the stability of at against is attributed to the increases in d at-d and t at-t compensating for the decrease in s at-s .
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butions of the different contact types to the critical shear strength and the fabric anisotropic coefficients were determined. The dominant contribution was related to the combined effect of simple and multipoint contacts; that is, an increase in multipoint (especially double) contacts is compensated by a decrease in simple contacts. Declaration of interest statement The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments The study is financially supported by the National Natural Science Foundation of China, China (No. 51809292, 51478481 and 51508141), Postdoctoral Fund of Central South University, China (No. 205455) and Beijing Municipal Science and Technology Project: Research and Application of Design and Construction Technology of Railway Engineering Traveling the Rift Valley, China (No. Z181100003918005). The authors express their appreciation for the financial assistance. Appendix A. Fabric and force tensors The anisotropies can be conveniently estimated from the following fabric and force tensors: ac−m = 4ϕijm’ , ϕijm = ij
1 Nc ˛ ˛ ni nj , Nm ˛=1 c
4
Nmc
m
an-m = ij
Fn-m’ , Fn-m = ij ij −fn-m 0
adn−m = ij
=
= Fig. 14. Evolution of m ac- m , m an- m , and m at- m as a function of : (a) ac- m , (b) an- m , and (c) at- m .
Conclusions This study investigated the effects of local non-convexity on the critical shear strength of granular materials using the DEM. A series of drained biaxial tests were conducted on rough particle packings with a uniform non-convexity of 0.075 and different values of local non-convexity ranging from 0.134 to 0.770. The results show that the contact type changes with an increase in but the critical shear strength is not affected by . Through analysis of the contact types and their proportions and coordination numbers, the contri-
m c-m n˛ n˛ ) ˛=1 Nc (1+˛ k l kl
˛ ˛ d˛ dn ni nj
= −ddn-m 0 m c-m n˛ n˛ ) ˛=1 Nc (1+˛ k l kl 4
f¯0t−m
adt−m = ij
˛ ˛ f˛ n ni nj
= Fn-m , −fn-m 0 ii A2)
4 Dijdn−m’ , Ddn-m ij d¯ 0dn−m
Nmc
= at−m ij
(A1)
Fijt−m’ , Ft-m ij =
Nmc
Ddn-m , ii ˛ ˛ f˛ t ni nj
−ft-m 0 m c-m n˛ n˛ ) ˛=1 Nc (1+˛ k l kl
(A3)
= Ft-m ii ,A4)
4 Dijdt−m’ , Ddt-m ij d¯ 0dt−m
Nmc
˛ ˛ d˛ dt ni nj
−ddt-m = 0 m c-m n˛ n˛ ) ˛=1 Nc (1+˛ k l kl
Ddt-m , ii
(A5)
dn-m’ where ϕijm’ , Fn-m’ , Ft-m’ , and Ddt-m’ are the deviatoric tenij ij , Dij ij t-m dn-m sors corresponding to ϕijm , Fn-m , and Ddt-m , respectively, ij , Fij , Dij ij given by Guo and Zhao (2013). ´ The effective mean (p) and deviatoric (q´) stresses can be given
by p´= ii /2 and q´ =
ij’ ij’ /2, where ij’ is the deviation corre-
sponding to the stress tensor ij . The fabric is anisotropic, and it is convenient to describe the degree of anisotropy of the fabric by using a scalar a*-m , which is attained from the invariant of every anisotropy tensor, i.e., ac-m , an-m , at-m , adn-m , and adt-m , as follows: ij ij ij ij ij
a*-m = sign(Sr )
a*-m a*-m /2 ij ij
(A6)
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where a*-m indicates the anisotropic coefficients ac-m , ad-m , at-m , adn-m and adt-m , corresponding to one of the four cases of fabric anisotropy mentioned above. Sr is a normalized quantity of the douand ij’ , given by (Guo & Zhao, 2013). Then, ble contraction of a*-m ij ´ the stress-force-fabric relationship between q/pand am (neglecting * the cross product between ac-m and an-m ) can be modified as q´/p´=
m
m
ac-m + ad-m + at-m + adn-m + adt-m . 2
(A7)
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