Influence of particle size on vertical plate penetration into dense cohesionless granular materials (large-scale DEM simulation using real particle size)

Abstract

The influence of the particle size on the vertical plate penetration into dense cohesionless granular materials was numerically investigated. Simulations were performed in quasi-two-dimensional conditions by changing the mean particle diameters d₅₀ but maintaining the plate thickness B from B/d₅₀ = 63–2.6. The initial bulk packing fraction was kept high, irrespective of the particle size. In the smallest particle size case (B/d₅₀ = 63), the size ratio reached almost the same level as that in the laboratory experiments using natural sand particles. The results demonstrated that the mean penetration resistance force acting on the plate tip surface increases with a decrease of B/d₅₀, while the tangential force acting on the side surfaces does not change with B/d₅₀. Tip resistances increase linearly with the penetration depth, while the tangential resistances increase with the square of the depth regardless of B/d₅₀. The behavior of the resistance fluctuations changes qualitatively between B/d₅₀ = 31 and 21. For all cases, we confirmed the formation of a wedge-shaped flow with a high forward velocity in front of the plate tip. The wedge flow width was larger than the plate thickness by almost a mean particle diameter, and was responsible for the increase in the mean resistance depending on the particle size. For the large B/d₅₀ cases only, the resistance exhibited quasi-periodic fluctuations, which was attributable to the intermittent nucleation and disappearance of the shear bands. Moreover, we investigated the dependence of B/d₅₀ on the band evolutions by analyzing the band thickness.

Graphic abstract

The influence of the particle size on the vertical plate penetration into dense cohesionless granular materials was numerically investigated using DEM. Simulations were performed in quasi-two-dimensional conditions by changing the median particle diameters d₅₀ but maintaining the plate thickness B. The initial bulk packing fraction was kept high, irrespective of the particle size. Upper and lower figures show the result of small (B/d₅₀ = 63) and large particle size case (B/d₅₀ = 21), respectively. In the small particle size case (B/d₅₀ = 63), the size ratio reached almost the same level as that in the laboratory and the dynamics of 35.5 million particles was considered. Right and left figures illustrate instantaneous shear strain rate and local packing fraction distributions, respectively. Large qualitative change in the granular behaviors as well as penetration resistance was observed between B/d₅₀ = 31 and 21. The intermittent nucleation and disappearance of the shear bands were clearly observed only for large B/d₅₀ cases.

1 Introduction

The resistance force acting on intruders during penetration into granular materials is of interest not only in granular physics [1,2,3,4,5,6,7,8,9,10,11,12], but also in engineering applications, such as cone penetration tests [13,14,15,16,17,18,19,20,21], pile installations [22,23,24,25,26,27,28], the digging of mining machineries [29], and the locomotion of legged robots over rough terrain [30]. Previous studies have demonstrated that the penetration resistance is affected by the particle size [12, 14, 16, 17, 20, 21, 31], bulk packing fraction [2, 9, 12, 17, 19, 21, 25], penetration depth [3, 4, 7, 8, 12, 24, 25], intruder shape [3, 4, 20, 27, 29], and penetration velocity [9, 18]. Despite numerous efforts in these previous studies, the understanding of the underlying mechanisms of penetration resistance remains insufficient, owing to the inherent complexity of the particle–object interactions.

In this study, we focused on the influence of the particle size on the penetration resistance and behavior of granular materials. Bolton et al. [14] experimentally demonstrated that the penetration resistance acting on a cone penetrometer is influenced by the particle size. These authors used sand particles with different median diameters (d₅₀ = 0.225, 0.40, and 0.90 mm). These sand particles had almost the same mechanical properties, with different particle size distributions and bulk packing fractions. By means of penetration tests with varying cone diameters B from 6.35 to 19.05 mm, they evaluated the relative particle size effect. For cases of fine sand particles with B/d₅₀ ranging from 28 to 85, the resistances were not affected. However, for medium and coarse sand particle cases using the smallest cone (B/d₅₀ = 16 and 7, respectively), the resistances increased. They suggested that cone penetration tests should be performed using cones with diameters 20 times larger than the particle diameter to diminish the dependence on B/d₅₀, but the detailed mechanism of this dependence was not clarified in their study. Experiments by Peng et al. [3] and Kang et al. [7] also demonstrated that the influence of the particle size on the penetration resistance was negligibly small in the regime (B/d₅₀ > 20). Cerato and Lutenegger [32] suggested that the size effect was closely related to mesoscopic structure evolutions, such as the formation of shear bands. They speculated that shear bands induced by penetrations could not propagate fully with a decreasing B/d₅₀. However, they did not observe shear band formations in their study, and their theories have not been confirmed. Hamm et al. [1] and Tapia et al. [2] observed plate penetrations using an image analysis technique under sufficiently larger B/d₅₀ conditions (B/d₅₀ ≈ 60). They demonstrated that shear band formation could be observed at a dense initial bulk packing fraction (φ₀ > 0.6). Moreover, they found that, for the dense case, the shear band evolutions were strongly related to the penetration resistance response. These results indicated that mesoscopic structural evolutions play an important role in object penetrations. However, the mechanism of the particle size effects on the penetration resistance and its influence on granular behaviors are still not effectively understood, as it is inherently difficult to change the particle size solely and observe the dynamics of mesoscopic structures and individual particles in experiments.

In recent years, numerical simulations based on the discrete element method (DEM) have been extensively employed to investigate the micro–macro behaviors of granular materials. The DEM is capable of changing the particle-scale properties as intended. However, in general, there exists a limitation on the total number of particles for obtaining DEM simulation results within a reasonable computation time. Numerous researchers have reduced the total number of particles by increasing the particle size. It is apparent that this approach is not adequate for investigating the particle size effect. As indicated in Table 1, DEM studies have been limited to small B/d₅₀ conditions excepting our previous study [31], and the shear band formation in granular materials has not been clearly observed during object penetration. In our previous study [33], a DEM program code was parallelized based on the dynamic domain decomposition technique, and could handle tens of millions of particles. To investigate the particle size effect, we conducted large-scale DEM simulations of vertical plate penetration into dense cohesionless granular materials in quasi-two-dimensional conditions using 67.9 million particles for the maximum case (d₅₀ = 0.233 mm) [31]. In the simulations, by varying the mean particle size but maintaining the plate width (B = 20 mm), we changed B/d₅₀ from 1.3 to 86. The results demonstrated that the penetration resistance increases with a decreasing B/d₅₀, and were consistent with the findings of Bolton’s cone penetration tests [14]. We demonstrated that for small particles only (B/d₅₀ = 86), discontinuous shear band evolution could be clearly observed in the bed, and penetration resistance appeared with quasi-periodicity. The results exhibited qualitative agreements with the experimental studies by Hamm et al. [1] and Tapia et al. [2]. In contrast, for small B/d₅₀ cases, discontinuous shear band formation could not be clearly confirmed. In the paper, however, a value of B/d₅₀ where the change occurs was not clarified. Moreover, the mechanisms that B/d₅₀ affects penetration resistance and shear band formation have not been revealed.

Table 1 Review of studies on B/d₅₀ dependence of penetration resistance

In this study, we investigated the B/d₅₀ dependence of the penetration resistance and shear band formation in a dense granular material by changing B/d₅₀ extensively and maintaining the other conditions, such as the particle shape and initial bulk packing fraction. Firstly, we confirmed the validity of the DEM plate penetration model employed in this study by means of comparisons with the experimental results. Secondly, we performed numerical penetration tests by changing B/d₅₀ from 63 to 2.6 in quasi-two-dimensional conditions. For B/d₅₀ = 63, the ratio was equivalent to that in the experiments by Hamm et al. [1] and Tapia et al. [2], which was a considerably large-scale DEM simulation with a maximum of 35.5 million particles used. Thereafter, we illustrated the B/d₅₀ dependence of the penetration resistance and explored a value of B/d₅₀ at which the tendencies of the resistance and granular behaviors changed. Finally, we analyzed the granular flow and shear band formation at the particle scale and discussed the mechanisms whereby B/d₅₀ affects the penetration resistance and shear band formation.

2 Methodology

2.1 DEM

In the DEM, the dynamics of granular materials were obtained by solving the equations of motion for individual particles. The contact forces were modeled by combinations of a linear spring, dashpot, and frictional slider [34]. Although the use of commercially available glass beads with high sphericity was assumed in this study, real glass beads will always have a certain degree of non-sphericity. A rolling resistance model that suppresses the excessive relative rotation of contacting particles owing to the non-sphericity was also introduced. Several rolling resistance models have been proposed [35,36,37], and the rolling spring-dashpot-slider model presented by Jiang et al. [38] was employed in this study. This model has two components in the rolling and twisting directions, respectively. As illustrated in Fig. 1, the rolling axis is defined on the contact plane, which is perpendicular to the vector from the center of particle i to the center of particle j, while the twisting axis is defined in the direction perpendicular to the contact plane. The translational and rotational motion equations of particle i are respectively expressed by

$$m_{i} \frac{{d{\varvec{v}}_{i} }}{dt} = \sum\limits_{j} {\left( {{\varvec{F}}_{N}^{j} + {\varvec{F}}_{S}^{j} } \right) + m_{i} {\varvec{g}}}$$

(1)

$$I_{i} \frac{{d{\varvec{\omega}}_{i} }}{dt} = \sum\limits_{j} {\left( {{\varvec{r}}_{i}^{j} \times {\varvec{F}}_{S}^{j} + {\varvec{M}}_{R}^{j} + {\varvec{M}}_{T}^{j} } \right),}$$

(2)

where m_i, I_i, v_i, and ω_i are the mass, moment of inertia, velocity, and angular velocity of particle i, respectively; F^j_N and F^j_S are the normal and tangential forces acting on particle i by contacting particle j, respectively; r^j_i is the vector from the center of particle i to the contact point with particle j; M^j_R is the rolling moment; M^j_T is the twisting moment owing to the rolling resistance; and g is the gravity acceleration. Frictional sliders were provided in the tangential, rolling, and twisting directions, which limited the forces and moments to certain values, as illustrated in Fig. 2. The tangential force, and the rolling and twisting resistive moments, respectively, are described as follows:

$${\varvec{F}}_{N}^{j} = \left( {k_{N} \delta_{N} + c_{N} v_{N} } \right)\frac{{{\varvec{v}}_{N} }}{{\left| {{\varvec{v}}_{N} } \right|}}$$

(3)

$${\varvec{F}}_{S}^{j} = {{\rm min}}\left( {k_{S} \delta_{S} + c_{S} v_{S} {, }\mu \left| {{\varvec{F}}_{N}^{j} } \right|} \right)\frac{{{\varvec{v}}_{S} }}{{\left| {{\varvec{v}}_{S} } \right|}}$$

(4)

$${\varvec{M}}_{R}^{j} = {{\rm min}}\left( {k_{R} \theta_{R} + c_{R} \omega_{R} {, }0.25\zeta_{C} \beta r\left| {{\varvec{F}}_{N}^{j} } \right|} \right)\frac{{{\varvec{\omega}}_{R} }}{{\left| {{\varvec{\omega}}_{R} } \right|}}$$

(5)

$${\varvec{M}}_{T}^{j} = {{\rm min}}\left( {k_{T} \theta_{T} + c_{T} \omega_{T} {, }0.65\mu \beta r\left| {{\varvec{F}}_{N}^{j} } \right|} \right)\frac{{{\varvec{\omega}}_{T} }}{{\left| {{\varvec{\omega}}_{T} } \right|}},$$

(6)

where k and c are the spring constant and viscous damping, respectively, while the suffixes N, S, R, and T represent the normal, tangential, rolling, and twisting directions, respectively. Moreover, δ_N and δ_S is the normal and tangential displacements, while θ_R and θ_T are the rolling and twisting rotations; v_N, v_S, ω_R, and ω_T are the relative normal, tangential velocity, rolling angular velocity, and twisting angular velocity, respectively; ω_R and ω_T are calculated as ω_R + ω_T = ω_i − ω_j and ω_T = [(ω_i − ω_j)·n_ij]n_ij, respectively, where n_ij is the unit vector from particle j to particle i; r is the common radius of the particles, defined as r = 2r_ir_j/(r_i + r_j); μ is the coefficient of sliding friction; and β and ζ_C are the rolling resistance model parameters. Jiang et al. [38] described the particle shape effect according to the dimensionless shape parameter β. For a perfect spherical particle, β is set to 0. The local resistance owing to asperity crushing is described by the dimensionless local crushing parameter ζ_C. Furthermore, they proposed a simplified model in which ζ_C is equal to 2.1.

Fig. 1

Rolling and twisting moments of contact between particles

Fig. 2

Tangential, rolling and twisting contact response system

2.2 Calculation conditions

2.2.1 DEM parameter setup

The properties of the glass beads used in this study are shown in Table 2. Glass beads #1 and #2 had different median diameters, while the other properties such as the shape and density were similar. For all of the plate penetration tests discussed in the following section, the properties of glass beads #1 (d₅₀ = 0.32 mm), such as the particle size distribution, particle density ρ, coefficient of sliding friction μ (the coefficients of the particle–particle, particle–acrylic wall, and particle–stainless plate frictions were 0.31, 0.20, and 0.27, respectively), and the coefficient of restitution e (0.9) were used. Glass beads #2 were used to obtain the shape parameter β and validate our DEM model. Figure 3 illustrates the size distributions of the glass beads and DEM particles. In the DEM simulations, different particle diameter sets of d = 0.31, 0.32, 0.33, and 0.34 mm and d = 1.8, 1.9, and 2.0 mm were respectively used so as the median diameter d₅₀ and uniformity coefficient d₆₀/d₁₀ are equivalent to glass beads #1 and #2. In the linear spring model, the spring constant cannot be directly related to the material properties of the particles. The normal spring constant k_N was determined such that the particle stress based on the amount of overlap in the normal direction becomes equivalent regardless of d₅₀ [39, 40]. When the spring constant becomes excessively small, the amount of particle overlaps becomes large, and the overall behaviors of the granular materials may be altered [41]. For glass beads #2, the normal spring constant k_N was chosen as 8.3 × 10⁴ N/m according to preliminary computations in which k_N was varied within the range from 6.0 × 10³ to 1.6 × 10⁶ N/m. This value was used because the values in the range from 8.3 × 10⁴ to 1.6 × 10⁶ N/m did not substantially affect the penetration force working on the plate. Furthermore, in the case k_N ≥ 8.3 × 10⁴ N/m, particle overlaps in the normal direction were always less than 1% of the particle diameters. The normal viscous damping c_N was determined from the coefficient of restitution of the particles [42]:

$$c_{N} = - \frac{2\ln e}{{\sqrt {\pi^{2} + \left( {\ln e} \right)^{2} } }}\sqrt {m_{i} k_{N} } .$$

(7)

Table 2 Properties of glass beads #1 and #2

Fig. 3

Size distributions of glass beads #1 (red line) and #2 (blue line). Uniformity coefficients d₆₀/d₁₀ were 1.1 for both types of beads. In the calibration and validation tests, the distributions for the DEM model were given based on glass beads #2. After Sect. 2.2.3, the distributions for the DEM model were determined based on glass beads #1 (color figure online)

The tangential spring constant k_S and viscous damping c_S were calculated by ν = k_S/k_N and ν² = c_S/c_N, respectively, where ν is the Poisson’s ratio of glass (0.25). The rolling spring constant k_R, twisting spring constant k_T, rolling viscous damping c_R, and twisting viscous damping c_T were calculated by k_R = 0.25(βr)² k_N, k_T = 0.5(βr)² k_S, c_R = 0.25(βr)² c_N, and c_T = 0.5(βr)² c_S, respectively [38]. The shape parameter β was calibrated by the comparison between the experimental and numerical results of the cone penetration tests, because it is not trivial to measure the parameter. The calibration study was conducted using glass beads #2 (d₅₀ = 1.86 mm). A stainless cone was penetrated vertically into the cohesionless granular materials with a constant speed of 10 mm/s in a rectangular container with a 218 mm width, 210 mm depth, and 150 mm height. As illustrated in Fig. 4, the cone was equipped with a tip of 30° and base of 28.6 mm in diameter. The cone penetration calculations were performed by varying the shape parameter β. When β = 0.04, the cone penetration force exhibited good agreements with the experimental results, as indicated in Fig. 4.

Fig. 4

Comparison of penetration force in simulation (lines) and experiment (triangles) in calibration tests for shape factor β. Triangles and their error bars represent the means and standard deviations of the two experimental trials, respectively

2.2.2 Validation testing of the DEM model

To validate the presented DEM model, we also performed plate penetration tests experimentally, and the results were compared. As illustrated in Fig. 5, glass beads #2 were randomly packed in a rectangular container with a 1000 mm width (W), 100 mm depth (D), and 400 mm height. The initial bulk packing fraction φ₀ was set to φ₀ ≈ 0.63 by tapping the container. The initial bed height was set to 300 mm. The container side walls were made of acrylic to enable visual observations during penetration. A stainless flat plate with a 96 mm depth, 20 mm (B) thickness, and 250 mm height was used in the experiments. The plate had a similar surface roughness to the cone used in the parameter calibrations. The plate was penetrated vertically into the bed at a constant speed of V_P = 100 mm/s until the front tip reached a depth of 150 mm from the initial bed surface. We confirmed that the penetration resistance does not depend on the penetration speed in this regime. Further discussions will be developed in Sect. 3.1. During the penetrations, the penetration force F_z acting on the plate was measured by a load cell attached to the plate base. The container width was sufficiently large relative to the plate thickness, and the influence of the sidewalls at both ends was negligible. The parameters used in the DEM calculations are displayed in Table 3. The particles with d = 1.8, 1.9, and 2.0 mm were used such that the number ratio became 3:4:3. The initial random bed was obtained by falling particles with random initial velocities under gravity. The initial bulk packing fraction φ₀ was controlled by varying the coefficient of sliding friction and shape factor during the packing process; φ₀ was set to 0.63, corresponding to the experiments. Following packing, the coefficient of sliding friction and the shape factor were set to the predetermined values displayed in Table 3. After the change of coefficients, the beds were relaxed until they reached a steady state. We confirmed that the packing fraction of the beds was the same before and after the relaxation.

Fig. 5

Plate penetration conditions for the validation test of the DEM model

Table 3 DEM parameters for glass beads #2 for validation test

Figure 6 illustrates the penetration resistance force, while Fig. 7 illustrates the particle bed deformation when the plate reached a depth of 150 mm. The particles are colored according to their initial vertical positions (10 mm intervals) to make the deformations visible. Good agreements were observed between the presented DEM and experiment for both the force and bed deformation.

Fig. 6

Comparison of penetration forces in simulation (red line) and experiment (triangles) in the validation test for the DEM model. Triangles and their error bars represent the means and standard deviations of the five experimental trials, respectively (color figure online)

Fig. 7

Comparison of granular bed behaviors in a experiment and b simulation in the validation test for the DEM model, showing snapshots at the penetration depth z_P = 150 mm. Dashed lines indicate the initial free surfaces of the bed

2.2.3 Plate penetration model setup

As illustrated in Fig. 8, all simulations were performed in quasi-two-dimensional computational domains, in which the periodic boundary condition was used in the depth (x) direction. A flat plate assuming periodicity in the x direction was pushed vertically into the cohesionless granular beds with a constant speed of V_P = 100 mm/s. Cases A–F were compared, with different mean particle diameters, as listed in Table 4. For all cases, the plate thickness B was set to 20 mm. Hence, the ratio of the plate thickness to the median particle diameter B/d₅₀ was changed from 2.6 to 63. The smallest particles were used in Case A (d₅₀ = 0.32 mm), where the relative particle size (B/d₅₀ = 63) reached almost the same level as that in previous laboratory experiments [1, 2]. This was a considerably large-scale DEM simulation, because the total number of particles in the bed was 35.5 million (Case A: d₅₀ = 0.32 mm).

Fig. 8

Plate penetration DEM simulation in quasi-two-dimensional computational domain. Periodic boundary conditions are used in the depth (x) direction

Table 4 Summary of quasi-two-dimensional penetration tests using DEM

The computational domain size in the periodic direction was at least 20 times larger than the maximum particle diameter in each case. The computational domain width W in the y direction was set to be large as the domain size effect became negligible. To examine the influence of the domain size effect and boundary conditions on the penetration force, we performed simulations using different conditions, and the results are presented in Fig. 9. With the exception of the domain width W in the y direction and boundary conditions in the x direction, the other calculation parameters were the same as in the validation simulation in the former section. We found that in the case of W = 500 mm, the penetration force was almost the same as in the case of W = 1000 mm, until the penetration depth reached 150 mm. This result indicates that the value was sufficient to prevent the domain size effect if W ≥ 500 mm. Thus, for the cases from d₅₀ = 0.32–3.84 mm (Case A–E), W was set to 500 mm, while for d₅₀ = 7.68 mm (Case F), W was set to 1000 mm. In terms of the boundary conditions, the penetration forces in the frictional and frictionless fixed-wall boundary condition were larger than that in the periodic boundary condition because the fixed-walls prevent particle motion in the x direction.

Fig. 9

Influence of boundary conditions in the x direction and domain width W in the y direction on penetration forces using DEM. The green dotted line, blue solid line, and red dashed line represent the cases of domain width W of 250, 500, and 1000 mm, respectively. The yellow dashed line and black solid line represent cases of frictionless wall and periodic boundary conditions with W = 500 mm (color figure online)

The particle size distribution and particle density in DEM calculations were determined according to the glass beads #1 in Fig. 3 and Table 2. For Case A, particles with d = 0.31, 0.32, 0.33 and 0.34 mm were randomly packed such that the number ratio became 1:2:2:1. For Cases B–F, the relative particle size relation was kept but it was adjusted to realize the targeted median diameter d₅₀. For example, d = 0.62, 0.64, 0.66, and 0.68 mm were adopted for Case B. The particle density ρ was set to 2610 kg/m³. For all particle size cases, the initial packing fraction φ₀ of the beds was set to 0.63. Simulations were performed until the penetration depth z_p, which is the distance from the plate front tip to the initial bed free surface, reached 150 mm. All calculations were performed in three dimensions.

3 Results

3.1 Penetration resistance

The penetration resistance force F_z can be decomposed into the normal force working on the front surface F_Tip, and the tangential forces on the left surface F_Left and right surface F_Right. Figure 10 illustrates the penetration depth dependence of F_Tip and the sum of F_Left and F_Right. The vertical axis represents the penetration resistance, while the horizontal axis represents the penetration depth z_P. To enable comparisons, the penetration resistances were normalized by S_P (depth of periodic domain D × plate thickness B). In Fig. 10a, the results of penetration speed V_p = 50 mm/s are also shown for Case A (B/d₅₀ = 63). It was confirmed that the resistances including fluctuations are not sensitive to the penetration speed in this regime. For all cases, both F_Tip and the sum of F_Left and F_Right increased with the penetration depth and F_Tip is the major contribution to the penetration resistance. Figure 11 shows the transitions of F_Tip with respect to penetration depth using F_Tip/(S_P × z_P). F_Tip increased immediately after the penetration started and increased almost linearly with the penetration depth after z_P ≈ 20 mm. Hereafter, we refer to the depth with z_P ≥ 20 mm as the linear regime. In Fig. 10, linear fitting lines p(z_P) for F_Tip/S_P of z_P ≥ 20 mm are also plotted. F_Tip is a hydrostatic-like force that depends on the penetration depth. Figure 12 shows the depth dependence of F_Left and F_Right of all the cases. The sum of F_Left and F_Right increased super-linearly with the penetration depth irrespective of B/d₅₀. It is expected that friction between particles and plates is the major contribution to the tangential forces working on the side surfaces and can be given by

$$F_{Left} , F_{Right} = \mu_{particle - plate} D\mathop \int \nolimits_{0}^{{z_{B} }} P_{y} \left( z \right)dz,$$

(8)

where P_y (z) is the earth pressure working normal to the side surfaces of the plate, μ_{particle–plate} is the coefficient of friction between particles and the plate, and z_B is the depth from the free surface of the bed to the plate tip. Assuming that the earth pressure increases linearly with depth P_y (z) ~ z and z_B is equal to z_P, F_Left and F_Right are proportional to the square of z_P; F_Left and F_Right ~ z_P². A fitting curve q(z_P) as a function of the square of penetration depth follows the data quite closely, as shown in Fig. 12. Superposing the linearity of the normal force working on the front surface and the non-linearity of tangential forces on the side surfaces, a total force F_z shows a slight non-linearity as demonstrated in Fig. 10d.

Fig. 10

Penetration resistance acting on the plate as function of penetration depth z_P. Front surface F_Tip (black solid lines), and the sum of tangential forces on the left surface F_Left and right surface F_Right (red solid lines). Linear fitting lines p(z_P) for F_Tip/S_P (dashed lines) in the depth z_P ≥ 20 mm. In a, the green dotted lines show the resistances for the slow penetration case V_P = 50 mm/s, the yellow dashed line in d shows the total resistance F_z/S_P (color figure online)

Fig. 11

Transitions of F_Tip with respect to penetration depth using F_Tip/(S_P × z_P). The inset shows a magnified view of the transition depth (z_P ~ 20 mm) from the initial regime to the linear regime

Fig. 12

B/d₅₀ dependence of the lateral penetration resistance. The curve q(z_P) (black dashed line) is a quadratic regression fit to the all the cases

To investigate the influence of the particle size, F_Tip/S_P and the linear fitting lines p(z_P) for all the cases are plotted in Fig. 13. Clearly, as B/d₅₀ decreased, the mean tip penetration resistance increased, which is consistent with the experiments by Bolton et al. [14]. In Fig. 14, fluctuations in F_Tip can be observed for all the cases. Figure 14 illustrates the resistance fluctuations ΔP = F_Tip/S_P − p(z_P) for Case A (B/d₅₀ = 63), Case B (B/d₅₀ = 31), Case C (B/d₅₀ = 21), and Case D (B/d₅₀ = 10). The fluctuations became quasi-periodic when B/d₅₀ ≥ 31. For B/d₅₀ = 63, ΔP exhibited a quasi-periodic cycle at almost 20 mm intervals along with the plate advancement. For B/d₅₀ = 31, ΔP exhibited a quasi-periodic cycle at almost 25 mm, until z_P reached almost 100 mm. The quasi-periodic fluctuations in the resistance were similar to the results reported by Hamm et al. [1] and Tapia et al. [2]. Meanwhile, for B/d₅₀ = 21 and 10, ΔP did not exhibit a clear trend. We found that the tendency of the resistance fluctuations changed between B/d₅₀ = 31 and 21.

Fig. 13

B/d₅₀ dependence of the penetration resistance, showing a raw data F_Tip/S_P and b fitting lines p(z_P)

Fig. 14

Fluctuations of tip resistance ΔP = F_Tip/S_P − p(z_P). a B/d₅₀ = 63, b B/d₅₀ = 31, c B/d₅₀ = 21, and d B/d₅₀ = 10

3.2 Deformation in granular materials

Figure 15 illustrates the deformation responses of the granular beds when z_P reached 150 mm. Focusing on the deformation patterns of the layer, for B/d₅₀ ≥ 31, we observed that large deformation was localized as indicated by the arrow in the enlarged views of Fig. 15a, b. For B/d₅₀ < 31, although bed deformation occurred, localized deformation could not be clearly observed (see online resource 1). The results demonstrated that changes in the granular behaviors as well as penetration resistance occurred between B/d₅₀ = 31 and 21. According to the above results, we develop discussions mainly based on the comparisons between the large B/d₅₀ case (Case A: B/d₅₀ = 63) and small B/d₅₀ case (Case C: B/d₅₀ = 21) hereafter.

Fig. 15

Deformation of beds at penetration depth z_P = 150 mm. The animation of these figures is shown in online resource 1. Insets in a and b display magnified views of localized large deformations

To observe the deformation in more detail, we investigated the bed dilatancy characteristics. Tapia et al. [2] experimentally demonstrated that the bulk volume increased with the penetration depth in densely packed beds. Using DEM simulation, the local compaction and dilation in beds could easily be evaluated. The local packing fraction φ_local distributions were obtained by calculating the total particle volume inside a spherical sampling volume with a radius of R_c = 4d₅₀ located at the center of each particle. Figure 16a, b illustrate the results of Case A (B/d₅₀ = 63) and Case C (B/d₅₀ = 21) corresponding to Fig. 15a, c, respectively. For B/d₅₀ = 63, narrow dilated regions with a low φ_local (≈ 0.57–0.60) emerged diagonally from the plate tip to the free surface. It is clearly confirmable that the narrow dilated regions correspond to the large deformation regions indicated in Fig. 16a. Moreover, φ_local remained unchanged and a dense packing state was maintained in the other regions. From the results of the numerical simulations of plate drag in dense dry glass beads, Kobayakawa et al. [43] demonstrated that the low φ_local was localized near the shear bands, and the values were approximately 0.580–0.590. For B/d₅₀ = 21, the dilated regions with a low φ_local (≈ 0.57–0.60) could be observed only in front of the plate tip. It appears that the slightly dilated regions were widely distributed from the plate tip to the free surface (see online resource 2).

Fig. 16

Distributions of local packing fraction φ_local in beds at z_P = 150 mm. The animation of these figures is shown in online resource 2

3.3 Granular flows

The contours in Fig. 17 indicate the magnitudes of the individual particle velocity |v_i| normalized by the plate penetration speed V_P in the vertical cross-section for Case A (B/d₅₀ = 63) and Case C (B/d₅₀ = 21) when the tip resistance F_Tip was in the linear regime in z_P = 47–67 mm. Moreover, the directions of the mean velocity fields u at z_P = 57 are plotted in Fig. 17c, h, where the vectors have the same length regardless of their magnitude. For all cases, the granular flows were confined to a region formed from below the plate tip up to the free surface. The materials outside this region were undisturbed and almost at rest. The boundaries between the disturbed flow region and undisturbed region (dashed lines in Fig. 17c, h) resemble the outlines (arcs a–d and a–e) of the failure surfaces introduced in the theory of Meyerhof [24], as indicated in Fig. 18. Focusing on the flow region, for all cases, we observed a wedge-shaped flow formed in front of the plate tip, moving with approximately plate speed (|v_i|/V_P ≈ 1). It is observable that the size of the wedge-shaped flow region varies in the z direction as the plate penetration proceeds for the B/d₅₀ = 63 case, while it remains approximately constant for B/d₅₀ = 21 (see online resource 3). This characteristic flow below the plate tip was called the dead zone or stagnant zone in Murthy et al. [5], Viswanathan et al. [6], Kang et al. [7], Feng et al. [12], and Aguilar et al. [30]. In plastic failure theories [22,23,24,25], it is assumed that an elastic wedge zone in front of the plate behaves as part of the plate. As illustrated in Fig. 18, in the case of cohesionless granular materials, the resistance force working on the plate tip is balanced with the sum of the vertical component of the plunging force working on the wedge and the wedge weight. Hence, the wedge size is an important factor to consider for the resistance. In these theories, it is assumed that the wedge width in the y direction is the same as the plate thickness, and the wedge height in the z direction is determined from the internal frictional angle of the granular material and plate roughness, regardless of the particle size. Moreover, because the theories were constructed under static equilibrium conditions, the wedge zone dynamics during the penetration process have not been considered at all. Kang et al. [7] and Feng et al. [12] have proposed a model considering the dynamics of the wedge zone. Feng et al. explained that the initial rapid increase of resistance was caused by an initial compression of the material immediately ahead of the intruder. The transition from the nonlinear to the linear regime was due to the growth of a wedge zone in the penetration direction, and the resistance increased linearly after the wedge zone reached a steady size. As shown in Fig. 17, on the contrary, our results showed that for the B/d₅₀ = 63 case, the wedge shape did not keep a steady-state even after the tip resistance had already reached the linear regime (z_P ≥ 20 mm). Feng et al. conducted their penetrations in loose materials (φ₀ = 0.575), while we focused on dense materials (φ₀ = 0.63). In our previous study, we investigated the influence of the initial packing fraction on the plate drag [43], and found that a stable granular flow could be observed in front of the plate for a loose material (φ₀ = 0.586), whereas an unsteady granular flow was observed for the dense material (φ₀ = 0.638). It is expected that this qualitative difference in behavior can be attributed to the difference in the initial packing fraction.

Fig. 17

Distributions of individual particle velocities |v_i|/V_P from z_P = 47–67 mm for a–e B/d₅₀ = 63 and f–j B/d₅₀ = 21. The animation of these figures is shown in online resource 3. In c and h, the direction of the mean velocity vector u is illustrated when |u| ≥ 0.9

Fig. 18

For cohesionless granular materials, resistance force working on the plate tip predicted by Meyerhof’s theory [24]

To investigate the dynamics of the wedge-shaped flow in the height (z) direction, we plotted the temporal change in the normalized vertical particle velocity u_z/V_P distributions along the vertical centerline of the plate for B/d₅₀ = 63 and 21 in Fig. 19, where u_z is the vertical component of the mean velocity field u. In both cases, firstly, at the initial moment the plate penetrates the material, a flow region with the middle velocity (u_z/V_P ≈ 0.5) propagates deeply into the material, which induces the initial rapid increase of the tip resistance; secondly, the growth of the flow region corresponding to the wedge zone was observed by the early stage (z_p < 20–30 mm). In the linear regime, for B/d₅₀ = 63, a quasi-periodic change occurred in the flow region and the onset of rapid forward expansions of the flow region could be observed at certain points, such as point a (z_P = 47 mm) and point e (z_P = 67 mm). In contrast, for B/d₅₀ = 21, a quasi-periodic change in the flow region was not clearly observed. To examine the flow region in more detail, Fig. 20 presents the u_z/V_P distributions for B/d₅₀ = 63 and 21 at five different penetration depths corresponding to points a–e and f–j in Fig. 19, respectively. The figure indicates that u_z/V_P was almost in unity with the plate tip, and decreased with an increasing distance from the plate tip. To investigate the expansion processes of B/d₅₀ = 63 (Fig. 20a), the u_z/V_P distributions at point a (z_P = 47 mm) and point b (z_P = 52 mm) are compared. Particles existing at a 70 mm distance from the free surface exhibited forward velocity when the plate tip reached point a, while they were almost at rest at point b. At the onset of the flow expansions (point a), the flow region was instantaneously propagated far forward from the plate tip. Focusing on the length of the flow region with a high velocity (u_z/V_P > 0.95), it increased to 9 mm from the plate tip at point b, compared to 3 mm at point a. This indicates that a finite time scale exists for the development of a high-velocity flow region in front of the plate tip. At point e (z_P = 67 mm), similarly to point a, instantaneous propagation of the low-velocity flow region occurred again, while expansion of the high-velocity flow region still did not occur. These results reveal that the flow regions were not in a steady state at all; moreover, deeper rapid expansions of the flow region occurred first, with the high-velocity flow region subsequently formed near the plate tip, and these processes were repeated (see online resource 3). In contrast, for B/d₅₀ = 21 (Fig. 20b), the u_z/V_P distributions in the flow region remained almost unchanged along with the plate movement. A jump in the height of the high-velocity flow region could not be confirmed.

Fig. 19

Change in normalized vertical velocity u_z/V_P distributions along center of plate in penetration (z) direction

Fig. 20

u_z/V_P at the five different penetration depth which correspond to points a–j in Fig. 19

Figure 21 illustrates the temporal changes in the normalized velocity u_z/V_P distributions in the width (y) direction at the position of one mean particle radius ahead of the plate tip for B/d₅₀ = 63 (Case A), 5.2 (Case E), and 2.6 (Case F). The horizontal axis represents the distance from the plate center. It can be observed that the fast downward flow regions were confined beneath the plate. For each B/d₅₀ case, the width of the flow region remained approximately constant and independent of the penetration depth. Meanwhile, it is clearly observable that the width of the flow region changed corresponding to the particle size d₅₀. This result indicates that the wedge width has a dependence on the particle size as well as plate width. We assumed that the wedge width was related to the size of the region in which particles can make direct contact with the plate tip. As illustrated in Fig. 22, this direct contact region is wider, by one particle diameter, than the path of the plate. Here, we introduced an effective plate thickness of B_Eff = B + d₅₀. The concept is similar to the effective penetration depth and effective vane diameter proposed by Soller and Koehler [44]. They performed experiments with rotating vanes in glass beads and investigated the influence of the vane and bead diameters. They demonstrated that the measured rotational drag and lift forces working on the vane increased with the bead size, and followed a scaling behavior that depended on the effective penetration depth and effective vane diameter, expressed as a function including particle size. Figure 23 illustrates the tip penetration resistance F_Tip/S_Eff, where S_Eff is defined by S_Eff = (B + d₅₀) × D. It was found that the mean tip penetration resistances using the effective plate thickness almost agreed with one another. This result indicates that, as the particle size became larger, the effective plate thickness increased and, as a result, the mean tip penetration resistance increased with the particle size.

Fig. 21

Change of the normalized vertical velocity u_z/V_P distributions in the width (y) direction at the position one particle radius ahead of the plate tip

Fig. 22

Effective plate thickness B_Eff = B + d₅₀

Fig. 23

Tip penetration resistance using effective plate width as function of penetration depth z_P

3.4 Shear band

Localizations of large deformations, known as shear bands, are induced by intruders in granular beds. Vesic [25] experimentally demonstrated that for a densely packed bed, shear band formation emerges from the edges of an intruder tip and extends downwards and laterally outwards in the shallow penetration regime. The propagation of shear bands has been observed by the particle-tracking technique [5] and numerical particle simulation [6]. To identify the shear bands in the granular materials, the shear strain rate in the beds was computed. It was demonstrated that each shear band firstly started from the edges of an intruder tip, and the bands propagated downwards and intersected below the plate tip. It could be observed that two shear bands formed a wedge zone below the plate tip. These authors only focused on very shallow penetration depths (z_P < B) and did not discuss the behaviors of shear bands after they were formed in detail. Hamm et al. [1] and Tapia et al. [2] experimentally demonstrated that shear bands are nucleated and disappear repeatedly as the penetration depth increases further. Moreover, they reported that the evolution of shear bands is largely dependent on the initial bulk packing fraction. Because they were not using plates with a flat-tip, they did not report on the existence of a wedge zone.

To identify the shear band in the materials, we computed the maximal shear strain rate $\dot{\gamma }$ from the DEM results by

$$\dot{\gamma } = \sqrt {\left( {\frac{{\dot{e}_{yy} - \dot{e}_{zz} }}{2}} \right)^{2} + \dot{e}_{yz}^{2} } ,$$

(9)

where $\dot{e}_{yy}$ and $\dot{e}_{zz}$ are the normal strain rates, and $\dot{e}_{yz}$ is the shear strain rate, given by

$$\dot{e}_{yy} = \frac{{\partial u_{y} }}{\partial y},$$

(10)

$$\dot{e}_{zz} = \frac{{\partial u_{z} }}{\partial z},$$

(11)

$$\dot{e}_{yz} = \frac{1}{2}\left( {\frac{{\partial u_{y} }}{\partial z} + \frac{{\partial u_{z} }}{\partial y}} \right).$$

(12)

Figure 24 illustrates the maximal shear strain rate fields for Case A (B/d₅₀ = 63) and Case C (B/d₅₀ = 21). These snapshots correspond to the distributions of particle velocities in Fig. 17. In both cases, it can be observed that the shear bands in front of the tip bordered a wedge-shaped region (ΔABC in Fig. 24c, h), corresponding to the flow region with a high velocity. For large B/d₅₀ cases, it can clearly be observed that the shear bands were developed from the plate edges to the free surface; in contrast, this is not as clear for small B/d₅₀ cases (see online resource 4). These results are consistent with those of our previous study [31]. In the present study, we investigated the B/d₅₀ dependence on the dynamics of the shear band formation in further detail. To develop further discussions, the shear bands are distinguished as (I) and (II), as indicated in Fig. 24c, h. Shear bands (I) are indicated by solid lines, and these were developed respectively from the plate edges B and C to intersection point A, and were bordering a wedge zone (ΔABC). Shear bands (II) are indicated by dashed lines, and developed from near intersection point A to the free surface (points D and E).

Fig. 24

Shear strain rate fields from z_P = 47–67 mm for a–e B/d₅₀ = 63 and f–j B/d₅₀ = 21. The animation of these figures is shown in online resource 4. As shown in c B/d₅₀ = 63 and h B/d₅₀ = 21 at z_P = 57 mm, shear band (I) formed from plate edges B and C to intersection point A (indicated by solid lines), and shear band (II) formed from near intersection point A to the free surface (points D and E). In the inset of c, ψ_R and ψ_L indicate the angles of shear bands (I) $\angle{\text{ABC}}$ and $\angle{\text{BCA}}$

Shear bands (I) In the theories based on static equilibrium, the wedge zone is assumed to be a triangle with angles π/2 + ϕ/2, and it does not change according to time. In contrast, for B/d₅₀ = 63, the simulation results demonstrate that the wedge angles changed during the penetration. Figure 25a illustrates the temporal change in the angles of shear bands (I) ψ_R and ψ_L, and the resistance fluctuation ΔP for Case A (B/d₅₀ = 63), where ψ_R and ψ_L indicate the angles $\angle{\text{ABC}}$ and $\angle{\text{BCA}}$ in Fig. 24c, respectively. From the figure, we found that for B/d₅₀ = 63, ψ_R and ψ_L exhibited quasi-periodic fluctuations, and sudden jumps from between 51° and 71° to between 19° to 41° occurred on both sides. At the timing of the jump events, new shear bands were formed (solid lines in inset of Fig. 24e), while the former shear bands (dotted lines in inset of Fig. 24e) could still be observed. It is noted that the angle of the new shear bands is plotted in Fig. 25 for the case where former and new bands could be observed simultaneously. Figure 25 illustrates that the onset of new band formation sometimes occurred synchronously on both sides. A short while after the new band was formed, the former bands disappeared. The resistance force reached the local maximum when the former bands disappeared. However, for B/d₅₀ = 21, the magnitude of the fluctuations in ψ_R and ψ_L was smaller (34°–67°), and no clear periodicity was exhibited (Fig. 25b). According to Figs. 24f–j and 25b, it appears that the bands continued to form in front of the tip without disappearance, and there was no clear correlation between the angles and resistance. For both cases, the mean values of the angles gradually increased with the penetration depth. The mean angles between z_P = 110 and 150 mm were 55° for both cases. Shear bands (II) Shear bands (II) could be observed clearly for B/d₅₀ = 63, but not for B/d₅₀ = 21. For B/d₅₀ = 63, the evolution of bands (II) was related to the evolution of bands (I). At the timing of the inception of bands (I), several new bands (II) emerged from near intersection point A (dashed-dotted lines in Fig. 24e). Thereafter, it appears that these newly formed bands (II) could not sustain themselves, and only one band with large shear strain remained on each side as a result (Fig. 24b–d). In contrast, for B/d₅₀ = 21, it appears that the inception and disappearance of new shear bands (II) were repeated frequently during the penetration, and clear local concentration of shear strains was not observed.

Fig. 25

Dynamics of the wedge region below the plate tip for a B/d₅₀ = 63 and b B/d₅₀ = 21

To investigate the mechanism of the B/d₅₀ dependence on the shear band formation, we focused on the relationship between the shear band formation and microscopic particle motion. It has been reported that the rotations of individual particles become significant inside shear bands [45,46,47]. Figure 26 illustrates the magnitude of the normalized angular velocity of the individual particle $\bar{\omega }_{i}$ for Case A (B/d₅₀ = 63) and Case C (B/d₅₀ = 21), where $\bar{\omega }_{i}$ is defined by

$$\bar{\omega }_{i} = \frac{{d_{p} \left| {{\varvec{\omega}}_{i} } \right|}}{{2V_{P} }}.$$

(13)

In the case where there is no slip between a particle and a plate moving toward the direction parallel to the plate surface, the particle rotation induced by the contact is given by $\bar{\omega }_{i}$ = 1. These snapshots represent the results at z_P = 57 mm and corresponding to Fig. 24c, h. A clear correlation is confirmed between the high shear region (Fig. 24c, h) and high angular velocity region (Fig. 26), as expected. Figure 27 illustrates the $\bar{\omega }_{i}$ distributions along the dashed lines passing through plate edge C in Fig. 26. The solid lines represent the values averaged over each maximum particle diameter interval. For B/d₅₀ = 63, $\bar{\omega }_{i}$ near the plate tip and at a distance of approximately 17 mm from the tip were clearly higher, corresponding to shear bands (I) and (II), respectively. Following Bardet and Proubet [46], we estimated the shear band thickness by measuring the distance between two points on the line crossing band (II), at which the magnitude of the particle rotation became sufficiently small. At z_P = 57 mm, the distance along the dashed line was 21d₅₀ (6.8 mm). Because the angle between the dashed line and band was approximated as 64°, the shear band thickness was estimated as 19d₅₀. For B/d₅₀ = 21, $\bar{\omega }_{i}$ near the plate tip was large, and decreased as the distance from the plate tip increased. We could not confirm a local peak corresponding to shear band (II). Various researchers [32, 48,49,50,51,52] have reported that the shear band thickness is strongly affected by the absolute size of the particles. Nemat-Nasser and Okada [50] reported that when the mean particle diameters d₅₀ were 0.22 and 0.48 mm, the band thicknesses were 11d₅₀ and 15d₅₀, respectively. Viggiani et al. [51] also reported that, for d₅₀ = 0.34 and 0.90 mm, the band thicknesses were between 10d₅₀ and 15d₅₀ and 8d₅₀, respectively. These results indicate that the ratio of the band thickness to the mean particle diameter was almost constant, even though the particle size changed. In contrast, Vangla and Latha [52] observed that the band thickness decreased from between 21d₅₀ and 24d₅₀ to between 10d₅₀ and 12d₅₀ when the mean particle size increased from 0.22 to 0.87 mm. Owing to the particle size dependence of the shear band thickness, with a decreasing B/d₅₀, the thickness became relatively larger with respect to the plate thickness. When B/d₅₀ became approximately 20 or less, the band thickness became almost the same or larger than the plate width. For all cases, shear bands (I) interfered with shear bands (II), and evolutions of the wedge zone formed by bands (I) and the strain concentration in bands (II) did not occur in any cases.

Fig. 26

Distribution of normalized angular velocity of particles at z_P = 57 mm for a B/d₅₀ = 63 and b B/d₅₀ = 21. Figure 27 shows the normalized angular velocity distributions along the dashed lines passing through plate edge C

Fig. 27

Angular velocity distribution along left plane of the plate for a B/d₅₀ = 63 and b B/d₅₀ = 21. Error bars indicate the standard deviations within each sampling space

4 Conclusions

We investigated the influence of particle size on vertical plate penetration into granular materials by large-scale DEM simulations. Particle size d₅₀ was varied from 0.32 to 7.68 mm while keeping the plate thickness B to 20 mm for all the cases. Hence, B/d₅₀ was varied from 2.6 to 63. For B/d₅₀ = 63, the size ratio reaches almost the same level with laboratory experiments in the literatures [1, 2]. As parameters that give major effects on the penetration resistance, initial packing fraction of the beds was kept high and penetration depth was kept in a shallow regime.

We confirmed that the mean penetration resistance acting on the plate tip surface increases with a decrease of B/d₅₀, which is consistent with the experiments by Bolton et al. [14]. From the observations of granular flow and shear band formation, a wedge-shaped granular flow with high forward velocity was observed in front of the plate tip for all the cases and it was similar to the wedge zone assumed in classical plastic failure theories. Only for the large B/d₅₀ cases (B/d₅₀ = 63 and 31), quasi-periodic fluctuations of the resistance similar to those reported by Hamm et al. [1] and Tapia et al. [2] could be observed. We have investigated the influence of particle size effect on the processes of granular flow and shear band propagation at the particle-scale and the following findings were revealed:

• 1. The width of wedged-flow formed in front of the plate tip was larger than the plate thickness almost by a mean particle diameter. The effective plate thickness increased as the particle size became larger and the mean tip resistance increased with particle size as a result. Moreover, the tangential force acting on the side surfaces does not change with B/d₅₀.

• 2. Tip resistance increases linearly with penetration depth, while the tangential resistances increase with the depth squared, regardless of B/d₅₀.

• 3. The behavior of tip resistance fluctuations changes depending on B/d₅₀. In particular, the behavior changed qualitatively between B/d₅₀ = 31 and 21. For the large B/d₅₀ cases, tip resistance fluctuations became quasi-periodic; conversely, for the small B/d₅₀ cases, the fluctuations did not show a clear periodicity.

• 4. For the large B/d₅₀ case, the shear bands nucleated and disappeared quasi-periodically during penetrations. The tip resistance fluctuations were attributed to the band evolutions. Due to particle size dependence of shear band thickness, as decreasing B/d₅₀, the thickness is relatively larger with respect to the plate thickness. When B/d₅₀ becomes about 20 or less, the band thickness is almost same or larger than the plate thickness. As a result, for the small B/d₅₀ case, the evolutions of the bands formed the wedge zone could not be observed. The periodic resistance fluctuations did not occur.

• 5. The behavior of wedged-flow in the vertical direction changes depending on B/d₅₀. For the large B/d₅₀ case, the height of the wedged-flow oscillated quasi-periodically. The evolutions of the wedged-flow in the vertical direction were corresponded to the evolutions of shear bands. For the small B/d₅₀ case, neither the wedged-flow evolutions nor evolutions of shear bands occur.

This study shows that for plate penetration into dense cohesionless granular materials in a shallow penetration regime, the value of B/d₅₀ affects not only the penetration resistance but also granular flow and shear band formation in the beds. We recommend that B/d₅₀ should be more than 30 to neglect particle size effect.