Microscopic mechanical analysis of K₀ of granular soils with particle size distribution and rolling resistance effects

Abstract

The coefficient of lateral earth pressure at rest, K₀, is an essential parameter for analyzing earth pressure distribution and the safe reliability of structures in geotechnical engineering. This paper presents a series of numerical one-dimensional compression tests on granular soils with particle size distribution (PSD) and rolling resistance (RR) effects using a real-particle 3D discrete element model. The corresponding macro–micro behaviors are investigated in a parallel way. Both PSD and RR affect K₀ and the related compression characteristics. A higher coefficient of uniformity (C_u) or rolling resistance coefficient (μ_r) results in a monotonic decrease in the mean coordination number, and too much consideration of RR makes the mean coordination number less realistic in a particle system. The influence of PSD is more sensitive to the local-ordering structure and contact force network than the RR. The inhomogeneity of normal contact forces enhances as C_u increases and slightly reduces as μ_r increases. The strong contacts are much more anisotropic than the weak ones. Specimen with lower C_u or higher μ_r induces higher anisotropy and more strong contacts during compression, in which a lower K₀ is measured. A unique macro–micro relationship exists between K₀ and deviatoric fabric when strong contacts are considered only.

1 Introduction

The coefficient of lateral earth pressure at rest, K₀, is commonly used to quantify the effective horizontal earth pressure, which is relevant to many geotechnical engineering issues, including tunnels, pile foundations, high rockfill dams, and deep shaft walls [1,2,3]. K₀ represents the ratio between the effective horizontal pressure ($\sigma^{\prime}_{h}$) and the effective vertical pressure ($\sigma^{\prime}_{v}$) under the condition of zero horizontal movements. Although the mathematical description is given clearly, there is no fully accepted theoretical calculation of K₀ [2, 4, 5]. In practice, the widely-used K₀ equation proposed by Jaky [6] is adopted to predict the values of K₀, which is simply related to the internal friction angle of granular soils and given as follows

$$ K_{0} = 1 - \sin \phi^{\prime} $$

(1)

where $\phi^{\prime}$ stands for the internal friction angle of granular soils. Numerous researchers have verified the validity of Eq. (1) through experimental studies [1, 2, 7,8,9].

On the whole, most previous experimental methods on the measurement of the stress states of granular soils under at-rest or K₀ conditions fall into four classes: flexible or thin wall oedometer tests [1, 4, 8, 10, 11], rigid wall oedometer tests [5, 12,13,14,15], triaxial compression tests [16,17,18,19], and in-situ shear wave velocity tests [20,21,22,23]. The K₀ condition means zero horizontal strain and movement. Talesnick [5] stressed that the testing methodology must have the capacity to properly maintain at-rest soil conditions and accurately measure soil pressures. However, the flexible or thin wall oedometers and the triaxial cells can hardly control the specimen to be zero horizontal strain as axial strain increases, making the mechanical state inconsistent with the K₀ condition. Unlike the triaxial cells, the main drawback of the rigid wall oedometers is the existence of frictional stress generating on the soil-wall interface, which reduces the vertical pressure and makes the vertical pressure imprecise along the height, especially for the soils under high-pressure loading. The seismic wave method is susceptible to environmental disturbance and is limited to surveying depth.

It is fortunate that the discrete element method (DEM) proposed by Cundall and Strack [24] enables overcoming the limitations in experimental tests and allows a link between macro and micro mechanical behaviors. With DEM, numerous studies have been carried out to investigate the macroscopic factors affecting the microstructure of granular soils and how the microstructure further affects K₀. For example, Gu et al. [25, 26] found that K₀ of a certain soil depends on the coordination number regardless of the void ratio. Lopera Perez et al. [27] reported that K₀ increases with void ratio, and the variation of K₀ is related to the degree of structural anisotropy and normal contact force anisotropy. Khalili et al. [28] prepared both isotropic and anisotropic samples in the initial state and found that K₀ is related to the evolution of force anisotropy. Chen et al. [29] conducted a series of DEM simulations with two kinds of particle shapes and built a relationship between K₀ and anisotropy of fabric measures (i.e., contact normal and contact force).

These published results have clearly shown that K₀ is related to many factors, including void ratio [8, 16, 18, 25, 27], friction angle [1, 2, 5, 6], initial preparation method [13, 15, 26, 28, 30], stress history [1, 2, 5, 7, 13], particle shape [8, 29, 31, 32], and particle size distribution (PSD) [15]. Of these, it is well recognized that particle shape and PSD significantly influence the mechanical responses of granular soils. For example, Zhu et al. [15] found that K₀ of gravel decreases with increasing maximum particle size under the same effective vertical stress. Still, studies of the effect of PSD on K₀ for traditional sands are reported rarely. Guo and Stolle [33] found that the relation between K₀ and particle shape is not unique because the variation of particle shape may change particle connectivity. Lee et al. [8] showed that the correlation of K₀ to $\phi^{\prime}$ is effective for uniformly round particles, while some errors exist in angular ones due to interlocking effects. Based on the mobilized strength and inter-particle resistance between particles, Lee et al. [8] further proposed an inter-particle strength-based relationship for describing K₀, which takes the interlocking effect into account. Nevertheless, the effect of particle shape on K₀ remains unclear due to the differences in testing methods and diversities in particle shapes. Particle shape quantification based on shape parameters, such as sphericity, aspect ratio, convexity, roundness, roughness, and overall regularity [34,35,36,37,38,39,40,41,42,43,44,45,46], is not only complicated but also difficult to evaluate the microstructure at the particle level. To take the effect of particle shape into account for simplicity, a common method is to incorporate a torque acting on each particle to counteract the rolling motion, i.e., rolling resistance (RR) [47,48,49,50,51,52]. However, the effect of RR on K₀ of granular soils has not been thoroughly analyzed.

The paper aims to examine the effects of PSD and RR on K₀ of granular soils using 3D DEM with non-spherical particles. The non-spherical particles enable a more realistic simulation and a better understanding of the macro- and micro-mechanical responses of granular soils during K₀ conditions. Numerical results are analyzed in detail from macroscopic and microscopic points of view, e.g., evolutions of K₀, coordination number, contact force distribution, and fabric anisotropy.

2 DEM model description

A series of one-dimensional compression tests were numerically conducted using 3D DEM to study the effects of PSD and RR on K₀ of granular soils. The RR model employed here is based on the linear model, to which a RR mechanism is added [53,54,55,56,57], as shown in Fig. 1. The interaction response between particles includes the normal, tangential, and rotational forces. The contact forces satisfy the following equations:

$$ \left\{ {\begin{array}{*{20}l} {\left| {{\varvec{f}}_{n} } \right| = k_{n} \left| {\varvec{\delta}} \right|} \hfill \\ {\left| {{\varvec{f}}_{s} } \right| = \min \left\{ {\left| {{\varvec{f}}_{s}^{\prime } + k_{s} \Delta {\varvec{u}}} \right|,\mu \left| {{\varvec{f}}_{n} } \right|} \right\}} \hfill \\ \end{array} } \right. $$

(2)

where k_n and k_s are the normal and shear stiffness constants, δ is the penetration depth of two particles at contact, Δu is the relative displacement at each time step, f_n and f_s are the normal and shear contact forces, ${\varvec{f}}_{s}^{\prime }$ is the previous shear contact force, and μ is the interparticle friction coefficient. Given a particle, its motion satisfies the following Newton–Euler equations:

$$ m_{i} \frac{{{{d}}v_{i} }}{{{{d}}t}} = \sum\limits_{c = 1}^{{n_{c} }} {\left( {f_{i}^{l} + f_{i}^{d} } \right)} + m_{i} g $$

(3)

$$ I_{i} \frac{{{{d}}\omega_{i} }}{{{{d}}t}} - \left( {I_{j} - I_{k} } \right)\omega_{j} \omega_{k} = \sum\limits_{c = 1}^{{n_{c} }} {\left( {M_{i}^{s} + M_{i}^{r} } \right)} $$

(4)

where i, j, k are subsequent indexes, m_i is the particle mass, v_i is the translational velocity, n_c is the number of contacts, f_i^l is the elastic force at contact c, f_i^d is the viscous damping force at contact c, ω_i is the angular velocity, I_i is the principal moment of inertia, and M_i^s and M_i^r are the moments caused by the shear force and RR at contact c. The RR moment M ^r is given by

$$ M^{r} = \left\{ {\begin{array}{*{20}c} {M^{r} \begin{array}{*{20}c} , & {\left\| {\left. {M^{r} } \right\| \le \mu_{r} \overline{R} f_{n} } \right.} \\ \end{array} } \\ {\mu_{r} \overline{R} f_{n} \begin{array}{*{20}c} , & {{\text{otherwise}}} \\ \end{array} } \\ \end{array} } \right. $$

(5)

$$ M^{r*} = M^{r} - k_{r} \Delta \theta_{r} $$

(6)

where μ_r is the RR coefficient, k_r is the RR stiffness, Δθ_r is the incremental rotational angle in the rolling direction, and $\stackrel{\mathrm{-}}{{R}}$ is the effective contact radius. The normal stiffness k_n is given by

$$ k_{n} = \frac{{\pi r^{2} E^{*} }}{{r_{1} + r_{2} }} $$

(7)

where E^* is the effective modulus, r₁ and r₂ are the equivalent radii of particles 1 and 2, and r equals min (r₁, r₂). The shear stiffness k_s is calculated via $k_{s} = k_{n} /\kappa^{*}$, where κ^* is the normal-to-shear stiffness ratio. The RR stiffness k_r is calculated via ${{k}}_{{r}} \, {=}{ \, {{k}}}_{{s}}\stackrel{\mathrm{-}}{{R^{2}}}$.

Fig. 1

The rolling resistance linear model in DEM: a behavior and rheological components of the model; b shear force–displacement law; c rotational moment–angle law

The specimen used in the 3D DEM was represented in a cylinder (Φ 10 mm × 20 mm) by several strain or stress-controlled rigid and frictionless walls. The specimen is kept small to improve computing efficiency and has sufficient size to capture the mechanical behavior while reducing the size effect as the specimen diameter is larger than 8 × mean particle size d₅₀ [57]. The input parameters for DEM simulations are listed in Table 1. Figure 2a shows the PSDs of sands modeled in this study. The properties of the PSDs can be characterized by different parameters, such as coefficient of uniformity (C_u), and mean particle size (d₅₀). In this study, d₅₀ is fixed as 1.2 mm for all specimens, and C_u varies from 1.0 to 2.7. The non-spherical particle used in the model is convex, as shown in Fig. 2b, and its corresponding sphericity and aspect ratio are 0.864 and 0.738, respectively. Sphericity [58] is defined as the surface ratio of a sphere having the same volume as the particle to the surface of the particle itself, i.e. $S = {{\sqrt[3]{{36\pi V^{2} }}} \mathord{\left/ {\vphantom {{\sqrt[3]{{36\pi V^{2} }}} {{\text{SA}}}}} \right. \kern-0pt} {{\text{SA}}}}$, where V = the volume of a particle, and SA = the surface area of a particle. Aspect ratio describing the anisotropy of the form of a particle is the mean of the elongation index (EI = b/a) and flatness index (FI = c/b) (i.e., a, b, and c refer to the major, intermediate, and minor principal dimensions respectively). Table 2 lists the simulation plan in this study.

Table 1 Parameters used in the DEM

Fig. 2

a Particle size distributions and b specimens with non-spherical particles modeled in the DEM simulations

Table 2 Details of the tested specimens

The specimen was randomly distributed and then rearranged without overlap between particles in the cylinder wall. The interparticle friction coefficient μ was temporarily set to zero during this rearrangement. The initial void ratio of all specimens was 0.562, and the corresponding particle numbers were 1111, 1400, 2108, and 4076, respectively. Then the specimens were compressed in the one-dimensional condition by moving the top and bottom walls towards each other with the constant rate of 0.01 m/s until the vertical stress σ_v reached 10 MPa. The one-dimensional compression process is performed with the constant interparticle friction coefficient (μ = 0.5) and different RR coefficients varying from 0.0 to 0.4.

3 Results and discussion

3.1 Typical macroscale behaviors

In Fig. 3, the one-dimensional compression responses of specimens with different PSDs are presented. A-0.1 means that a specimen of grading A was compressed with an μ_r of 0.1. The K₀ values of the specimen with a larger C_u run above those with lower C_u, as shown in Fig. 3a. The lower values of K₀ from the specimen with a lower C_u can be attributed to the strong force chain along the vertical direction due to more significant interlocking, simultaneously resulting in a lower degree of stress transfer in the horizontal direction. From the e-lgσ_v curves shown in Fig. 3b, it is also observed that the specimen with lower C_u is harder to compress, further indicating that more strong forces are along the vertical direction and form a more solid skeleton.

Fig. 3

Particle size distribution effect on the macroscale behaviors: a K₀; b e

Figure 4 shows the one-dimensional compression responses of specimens with different RR coefficients. The K₀ values of the specimen with a lower μ_r run above those with a higher μ_r, as shown in Fig. 4a. Similar to the above, the lower values of K₀ from the specimen with a higher μ_r can be attributed to the strong force chain along the vertical direction due to more intense friction between particles, resulting in a lower degree of stress transfer in the horizontal direction. Figure 4b reveals that the specimen with a higher μ_r forms a more solid skeleton. The effect of RR on the K₀ values can be identified by comparing the results from Lee et al. [8], who found that the K₀ values for irregular sands are lower than those for glass beads due to the higher degree of friction and interlocking between particles.

Fig. 4

Rolling resistance effect on the macroscale behaviors: a K₀; b e

3.2 Coordination number

One advantage of DEM modeling is that the evolution of microscale response can be observed and analyzed to reveal the underlying mechanism. The coordination number (CN) quantifies the contact number of each particle and reflects the microstructural evolution. The mean CN defined by Thornton and Antony [59] is given by

$$ Z = \frac{{2N_{c} - N_{p}^{1} }}{{N_{p} - N_{p}^{0} - N_{p}^{1} }} $$

(8)

where N_c = the total contact number, N_p = the total particle number, ${{N}}_{{p}}^{0}$ and ${{N}}_{{p}}^{1}$ = numbers of particles with zero and one contact, respectively. The reason for this definition is that particles with no contact or one contact miss the contribution to stress transmission.

Figure 5a shows the evolutions of mean CNs under the PSD effect. It can be seen that Z increases rapidly with increasing vertical stress at the early stage and then gradually stabilizes. The higher C_u, the lower Z is. It means that a wide particle grading range increases the number of floating particles with contact numbers less than two, as shown in Figs. 6 and 7. Figure 6 also shows the evolution of the percentage of particles with more than one contact ${{N}}_{{p}}^{2}$; it is observed that the percentage of ${{N}}_{{p}}^{2}$ decreases with increasing C_u. The mean CNs for A-0.1, B-0.1, C-0.1, and D-0.1 at 10 MPa are nearly 6.89, 6.01, 5.62, and 4.84, respectively, and the relative mean CNs (R_CN, ratios of A-0.1, B-0.1, C-0.1, and D-0.1 to A-0.1) are 1.000, 0.872, 0.816, and 0.702, which decrease with increasing relative C_u (R_Cu). Figure 8 shows the relationship between R_CN and R_Cu as the vertical stress ranges from 0.5 to 10 MPa, and the result indicates that R_CN decreases linearly with increasing R_Cu regardless of the influence of vertical stress.

Fig. 5

Evolutions of coordination number during one-dimensional compressions: a PSD effect; b RR effect

Fig. 6

Evolutions of particle numbers for particles with no contact, one contact, and more than one contact: a A-0.1; b B-0.1; c C-0.1; d D-0.1

Fig. 7

Illustrations of particles with no contact (burgundy), one contact (bright blue), and more than one contact (gray)

Fig. 8

Relationship between R_CN and R_Cu in a wider range of vertical stress

Figure 5b shows the evolutions of CNs under the RR effect. It can be seen that increasing μ_r causes a monotonic decrease in the mean CN, where specimen D-0.4 has the lowest mean CN (3.5–4.5) throughout the simulation. Previous simulation studies of frictional spheres compressed in a gravity-free environment have shown that the mean CN is significantly larger than 4.5 [60]. Existed CT scanning tests of silica sands have reported that the mean CN is larger than 6 as the vertical stress reaches 10 MPa [38, 61]. Obviously, too much consideration of RR makes the mean CN less realistic in a particle system [52].

3.3 Radial distribution function

The radial distribution function (RDF), used to explore the local-ordering structure of a granular assembly, is the probability of finding the center of a particle within a spherical shell at a certain distance from a reference particle [62]. The RDF is defined as follows:

$$ n\left( {r_{2} } \right) - n\left( {r_{1} } \right) = \int_{{r_{1} }}^{{r_{2} }} {g\left( r \right)} 4\pi r^{2} {{d}}r $$

(9)

where n(r) is the number of particles within a spherical shell of radius r. Figure 9 shows the normalized radial distribution of particle numbers in a spherical shell as a function of the dimensionless distance ${{r}}/\langle {{d}}\rangle$, where $\langle {{d}}\rangle$ is the mean particle diameter. For the monodisperse particles in Fig. 9a, a clear first peak of g(r) can be seen at ${{r}}/\langle {{d}}\rangle$ less than 1. This position of the first peak is consistent with the result from Conzelmann et al. [63] and is lower than the position for spherical particles (${{r}}/\langle {{d}}\rangle \, {=} \, {1}$) [62, 64, 65]. Then, g(r) decreases to minimum at r = 1.6 $\langle {{d}}\rangle$ indicating a minimum probability of finding particles in contact. g(r) continues to increase to another peak at r = 2.35 $\langle {{d}}\rangle$. RDF of the specimen with higher C_u shows different behaviors; the first peak (${{r}}/\langle {{d}}\rangle$ < 1) shifts to a lower value, and the peak is more prominent, representing the higher coordination of the polydisperse specimen compared with the monodisperse one and also indicating an increasing organization in the packing structure [65,66,67].

Fig. 9

Radial distribution functions for specimens with a different PSDs and b different RR coefficients

Figure 9b shows the RDFs of the four specimens with different RR coefficients. The first peak of an RDF appears at the same position r = 0.5 $\langle {{d}}\rangle$, regardless of RR. Additionally, the number of peaks and the corresponding amplitudes is almost identical regardless of RR. Similar results have been found by Zhao et al. [67] that the position of the first peak is independent of particle shape, and Kramar et al. [68] found that the RDF is regardless of the friction coefficient.

3.4 Contact force

The microstructure of granular materials can be described in terms of force chain characteristics. Figure 10 presents the closeup views of the contact force network in A-0.1, B-0.1, C-0.1, and D-0.1 as the vertical stress equals 10 MPa. As C_u increases, the distribution of forces broadens, which reflects in the increase in the maximum contact force.

Fig. 10

Closeup views of the contact force network under the vertical stress of 10 MPa

The probability distribution function (PDF) of the contact force is commonly used to quantify the contact force network. For the specimen with monodisperse particles (C_u = 1), the PDF for normal contact force f_n less than the average〈f_n〉fits well with the Gaussian distribution (see Fig. 11a) defined as

$$ {\text{PDF}}\left( {f_{n} } \right) = a + \frac{b}{{c\sqrt {{\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-0pt} 2}} }}e^{{{{ - 2\left( {{{f_{n} } \mathord{\left/ {\vphantom {{f_{n} } {\left\langle {f_{n} } \right\rangle }}} \right. \kern-0pt} {\left\langle {f_{n} } \right\rangle }} - d} \right)^{2} } \mathord{\left/ {\vphantom {{ - 2\left( {{{f_{n} } \mathord{\left/ {\vphantom {{f_{n} } {\left\langle {f_{n} } \right\rangle }}} \right. \kern-0pt} {\left\langle {f_{n} } \right\rangle }} - d} \right)^{2} } {c^{2} }}} \right. \kern-0pt} {c^{2} }}}} $$

(10)

where a, b, c, and d are fitting parameters of the Gaussian function. As C_u increases, PDF(f_n) has an upturn at very small forces and PDF(f_n) fits well with the power law (see Fig. 11b–d)

$$ {\text{PDF}}\left( {f_{n} } \right) = \beta_{2} \left( {{{f_{n} } \mathord{\left/ {\vphantom {{f_{n} } {\left\langle {f_{n} } \right\rangle }}} \right. \kern-0pt} {\left\langle {f_{n} } \right\rangle }}} \right)^{{\beta_{1} }} $$

(11)

where β₁ and β₂ are fitting parameters of the power function. As usually observed, PDF(f_n) above〈f_n〉for each specimen is characterized by an exponential distribution

$$ {\text{PDF}}\left( {f_{n} } \right) = \alpha_{2} e^{{ - \alpha_{1} \left( {{{f_{n} } \mathord{\left/ {\vphantom {{f_{n} } {\left\langle {f_{n} } \right\rangle }}} \right. \kern-0pt} {\left\langle {f_{n} } \right\rangle }}} \right)}} $$

(12)

where α₁ and α₂ are fitting parameters of the exponential function. Notably, the differences in the PDF(f_n) for a certain specimen are almost negligible. That is, PDF(f_n) maintains a nearly constant distribution regardless of the influence of vertical stress. The main reason for this phenomenon is that the stress field or K₀ in a one-dimensional state is less varied. The result is in accordance with the findings in previous one-dimensional tests [37], and isotropic compression tests [69, 70]. Additionally, Fig. 11 shows that the distribution of normal forces varies in the other two ways as C_u increases. One is that the distribution becomes broader with the average of α₁ decreasing from 1.20 to 0.71, as shown in Fig. 12, and maximum forces get to be as large as sixteen times the average, implying that the inhomogeneity of normal forces enhances as C_u increases. Another is that the average proportion of weak contacts increases from 59.29% for C_u = 1 to 68.77% for C_u = 2.7. Similar observations were made in 2D DEM simulation investigated by Estrada and Oquendo [71], and 3D simulations investigated by An et al. [72], Cantor et al. [73], and Mutabaruka et al. [74].

Fig. 11

Probability distribution function of normal contact forces f_n normalized by the average〈f_n〉in log-linear scale: a A-0.1; b B-0.1; c C-0.1; d D-0.1

Fig. 12

Variation trend of α₁

Figure 13 shows the PDF(f_n) of normal forces normalized by the average under the effect of RR. It can be seen that the PDF(f_n) below〈f_n〉for each specimen fits well with the power law, and the PDF(f_n) above〈f_n〉is characterized by an exponential distribution, implying that the function type of PDF(f_n) is independent of RR. The average proportion of weak contacts decreases slightly from 69.36% for μ_r = 0.0 to 67.96% for μ_r = 0.4. The average of α₁ increases from 0.66 for μ_r = 0.0 to 0.77 for μ_r = 0.4 in a narrow range, indicating that the homogeneity of normal forces slightly enhances as μ_r increases.

Fig. 13

Probability distribution function of normal contact forces f_n normalized by the average〈f_n〉in log-linear scale: a D-0.0; b D-0.1; c D-0.2; d D-0.4

The second-order fabric tensor introduced by Satake [75] is frequently used to quantify the fabric anisotropy, which characterizes the distribution in contact normal orientations.

$$ \phi_{ij} = \frac{1}{{N_{c} }}\sum\limits_{k = 1}^{{N_{c} }} {n_{i}^{k} n_{j}^{k} } \left( {i,j = x,y,z} \right) $$

(13)

where ${n}_{i}^{k}$ = the contact normal vector of the contact k in the ith direction. The principal values of $\phi_{ij}$, ordered by decreasing magnitude, are $\phi_{1}$, $\phi_{2}$, $\phi_{3}$. To quantify the fabric anisotropy, a deviatoric fabric $\delta_{d}$ proposed by Barreto et al. [76] is adopted as follows

$$ \delta_{d} = \sqrt {\frac{{\left( {\phi_{1} - \phi_{2} } \right)^{2} + \left( {\phi_{2} - \phi_{3} } \right)^{2} + \left( {\phi_{1} - \phi_{3} } \right)^{2} }}{2}} $$

(14)

Radjai et al. [77] proposed that the average normal force〈f_n〉is a characteristic force separating the interparticle contacts into two complementary groups: the “weak” contacts bearing forces smaller than the average and the “strong” contacts bearing forces larger than the average. Numerous numerical studies have shown that the distribution of weak contact forces is nearly isotropic, indicating that the weak forces only contribute to the isotropic stress or have little contribution to the deviatoric stress [64, 77,78,79,80]. Take specimen D-0.1 for example, the value of $\delta_{d}$ for strong contacts ($\delta_{d}^{s}$) is higher than the weak contacts, indicating that the strong contacts are much more anisotropic and much more similar to that of the K₀ versus σ_v curve, as shown in Fig. 14a. Furthermore, the shape of the contact normal distribution for weak contacts is close to a sphere because the distribution of weak contacts is approximately isotropic, and the shape for strong contacts, by contrast, is thin in the middle, as shown in Fig. 14b and c.

Fig. 14

Deviatoric fabric and contact normal distribution in D-0.1: a $\delta_{d}$; b 4 MPa; c 10 MPa

In terms of the link between the macroscopic behavior and the strong force network, Essayah et al. [81] found that the deviatoric stress in the triaxial test is carried by strong contacts, and Mahmud Sazzad et al. [82] and Mahmud Sazzad [83] found that the tendency of $\delta_{d}$ for strong contacts coincides with the stress–strain curve of granular material during cyclic loading and true triaxial loading. To emphasize the main ideas and to allow for concise analytical discussion, the contribution of the strong contacts to the stress tensor is considered here only.

Figure 15 shows the evolution of $\delta_{d}^{s}$ under the effect of PSD. The $\delta_{d}^{s}$ increases initially with increasing vertical stress σ_v and then gradually levels off. The value of ${\delta }_{{{d}}}^{{{s}}}$ and strong contact proportion depends on the PSD. The specimen with lower C_u induces higher anisotropy and more strong contacts during compression, in which a lower value of K₀ is measured.

Fig. 15

Evolution and percentage of ${\delta}_{\text{d}}^{\text{s}}$ under the effect of PSD

Figure 16 shows the evolution of $\delta_{d}^{s}$ under the effect of RR. The value of $\delta_{d}^{s}$ and strong contact proportion also depends on the RR coefficient, which has less influence than the PSD. The specimen with higher μ_r induces higher anisotropy and more strong contacts during compression, in which a lower value of K₀ is measured. These findings imply that a possible correlation exists between $\delta_{d}^{s}$ and K₀ in one-dimensional tests.

Fig. 16

Evolution and percentage of ${\delta}_{\text{d}}^{\text{s}}$ under the effect of RR

3.5 Relationship between K₀ and fabric anisotropy

Figure 17 shows the relationship between K₀ and fabric anisotropy of strong contacts $\delta_{d}^{s}$. It is worth noting that a good linear relationship between K₀ and $\delta_{d}^{s}$ is established as the strong contacts are used to quantify the fabric tensors. This linear relationship demonstrates unequivocally that K₀ measured through the rigid walls on the macro-level is directly connected with the fabric anisotropy of strong contacts on the micro-level.

Fig. 17

Relationship between K₀ and ${\delta}_{\text{d}}^{\text{s}}$ for all specimens

4 Conclusion

DEM simulations of one-dimensional compression tests were carried out to investigate the effects of PSD and RR on K₀ and the corresponding microscopic behaviors of sands. A non-spherical particle was introduced in the DEM model. A macro–micro relationship between K₀ and fabric anisotropy of strong contacts ${\delta}_{\text{d}}^{\text{s}}$ is established. Some interesting findings are summarized below.

• (1) After the vertical stress reaches a certain value, K₀ gradually decreases and approaches a comparatively stable value as vertical stress reaches 10 MPa. K₀ of the specimen with a larger C_u runs above those with a lower C_u. This is attributed to the strong force chain along the vertical direction due to more significant interlocking, resulting in a lower degree of stress transfer in the horizontal direction.

• (2) K₀ of the specimen with a lower μ_r runs above those with a higher μ_r. Similar to the above, lower K₀ from the specimen with a higher μ_r can be attributed to the strong force chain along the vertical direction due to more intense friction between particles.

• (3) PSD and RR significantly affect the evolution of the coordination number. The higher C_u, the lower the mean CN is. The relative mean CN decreases linearly with increasing relative C_u regardless of the influence of vertical stress. Additionally, increasing μ_r causes a monotonic decrease in the mean CN, and too much consideration of RR makes the mean CN less realistic in a particle system.

• (4) RDF of the specimen with higher C_u shows that the polydisperse specimen is more ordered than the monodisperse one. However, the effect of RR on the RDF is negligible. The PDF(f_n) maintains a nearly constant distribution regardless of the influence of vertical stress. For the specimen with monodisperse particles, the PDF(f_n) for normal contact force f_n less than the average〈f_n〉fits well with the Gaussian distribution, while PDF(f_n) fits well with the power law as C_u increases. PDF(f_n) above〈f_n〉for each specimen is characterized by an exponential distribution, from which the inhomogeneity of normal forces enhances as C_u increases and slightly reduces as μ_r increases.

• (5) Strong contacts are much more anisotropic than weak ones. A unique macro–micro relationship exists between K₀ and deviatoric fabric when strong contacts are considered only.