Granular Matter (2015) 17:21–31 DOI 10.1007/s10035-015-0547-3
ORIGINAL PAPER
Impact response of granular layers Amnaya Awasthi · Ziyi Wang · Natalie Broadhurst · Philippe Geubelle
Received: 2 May 2014 / Published online: 22 January 2015 © Springer-Verlag Berlin Heidelberg (outside the USA) 2015
Abstract Motivated by the wave tailoring potential of granular media, this study aims at evaluating force transmission through granular layers made of spherical particles. 2D simulations based on Hertzian contact law between adjacent particles are performed on two distinct systems: (1) layers consisting of ordered bimaterial lattices, and (2) single material layers with random packing. For the ordered systems, force transmission properties are found to vary with material mismatch and layer thickness. Transmitted forcedecay in random configurations is substantially higher than those in the ordered systems.
A. Awasthi (B) Department of Aerospace Engineering, University of Illinois Urbana Champaign, 319 E Talbot Laboratory 104 S Wright St, Urbana, IL 61801, USA e-mail:
[email protected];
[email protected] Z. Wang Department of Aerospace Engineering, University of Illinois Urbana Champaign, Urbana, IL, USA e-mail:
[email protected] N. Broadhurst Department of Materials Science and Engineering, University of Illinois Urbana Champaign, Urbana, IL, USA e-mail:
[email protected] P. Geubelle Department of Aerospace Engineering, University of Illinois Urbana Champaign, 306B Talbot Laboratory 104 S Wright St, Urbana, IL 61801, USA e-mail:
[email protected]
Keywords Granular media · Hertz potential · Force transmission · Ordered packing · Random packing · Material mismatch
1 Introduction The impact response of granular media has unique properties due to intrinsic nonlinearity associated with the contact law [1], including the existence of nonlinear solitary waves. Key attributes of wave propagation in 1D granular systems have been extensively studied in the past [2–4] and are being further investigated for wave tailoring and mitigation applications [5,6] including vibration damping [7], energy scattering [8,9], trapping and reflection [10,11]. 1D granular chains have also been investigated as protectors against force transfer. The influence of randomness in granular chains was first investigated by Nesterenko in 1983 [12], and since then, research involving randomness in granular media has diversified substantially. In [13], chains of granular media having variations in size, mass and stiffness were optimized to minimize the force transmitted across the chain. Additional studies [14,15] have shown how to tailor this distribution of energy by varying particle properties and sizes in the chain. This process allows for the disintegration of high amplitude incident pulse into several lower amplitude components, hence reducing the amplitude of force transmitted. Additionally, recent studies on wave propagation in random granular chains [16] have demonstrated two modes of force decay along a granular chain: an exponential decay associated with the propagation of attenuated solitary waves and a power law decay, associated with the accumulation of dispersed force pulses. It was also shown that randomness in sizes of granular particles leads to a higher force decay than randomness in mass or stiffness.
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More limited wave propagation and force transmission studies have also been performed on 2D granular layers. Experiments and Discrete Element Modeling (DEM) simulations have been used [17] to examine the rebound and force transmission characteristics of granular media when impacted by a projectile. It was found that an increase in thickness of granular layer reduces the transmitted force and the rebound velocity of the projectile. Introduction of additional layers of dissimilar materials reduces the force and increases the projectile’s rebound speed. DEM has also been used to model force transmission in 2D random granular layers focusing on linking transmission characteristics to contact interaction law and microstructural orientation [18,19]. In these studies, random configurations of equal size spherical granules were generated through a biasing scheme which produced preferential directionality in particle alignment. Force transmission responses demonstrated that wave propagation along directions of particle alignment occurs with higher speed and low attenuation. Additionally, several studies have focused on utilizing DEM to investigate dynamics of wave propagation and force transmission in soil [20–22]. The present study is motivated by the simulation results summarized in [23], where wave propagation was analyzed for infinite 2D granular media consisting of frictionless elastic spheres arranged in close-packed square lattice containing interstitial intruders. Depending on the stiffness and mass mismatch between intruders and main beads, the 2D granular medium supports different modes of wave propagation. For lower mass ratios and higher stiffness ratios, a 2D solitary wave propagates with various levels of anisotropy, while high mass ratios and low stiffness ratios result in directional propagation with no 2D solitary wave. These findings make the material tunable to wave propagation modes and hence a good contender in wave tailoring and protection applications. Effects of weak disorder on such close-packed square lattices have demonstrated [24] that small randomness in particle sizes causes wide deviation from the response of perfect packing. Additionally, investigation on randomness in material properties of square close-packed systems [25] has shown predominance of backscatter in wave propagation behavior which increases with randomness. In the present work, the impact response of a similar close-packed square lattice of spherical particles is investigated to evaluate the effect of material mismatch and layer thickness on the force transmission response. We also examine the force transmission characteristics of randomly organized granular packings comprising of the same two-particle granular sizes and volume fractions as the perfect lattice system. The paper is organized as follows: in Sect. 2, we describe the impact problem of interest and the numerical model adopted to simulate it. Section 3 summarizes the force transmission results for ordered system, including the effect of stiffness ratio, mass ratio and layer thickness, while Sect. 4
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describes the force transmission properties of random granular media.
2 Numerical model and problem setup The model of granular media in the present work is similar to that studied in [23] and consists of elastic spherical beads with radii ri , masses m i , Young’s moduli E i and Poisson’s ratio νi interacting in the absence of any dissipative mechanisms (plasticity and/or friction). The inter-particle contact force Fi j is given by the Hertz law as [26] Ei E j ri r j 3/2 Fi j = Δi j 2 2 r + r E i (1 − ν j ) + E j (1 − νi ) i j 3/2
= ki j Δi j (no sum),
(1)
where Δi j is the relative displacement between particles i and j (assumed positive in compression). The present study focuses on the impact response of two classes of granular layers. The first model (Fig. 1a) consists of a perfectly ordered square packing of identical larger spherical particles (referred to hereafter as ‘main beads’ with the identifier ‘1’) with the interstitial vacancies occupied by smaller size particles (referred to hereafter as ‘intruder beads’ with the identifier ‘2’). The size of the intruder beads is fixed such that they exactly fit into the interstitial void, leading to√ an intruder to main bead radius ratio (r2 /r1 ) equal to 2 − 1 as shown in Fig. 1a. In the actual simulations, these granular layers are constructed using main beads of radius 5 mm and intruder beads of radius equal to 2.07 mm. The time scale associated with force transmission at elastic granular contacts in the present system is given by τ = πρ R ∗ /4E ∗ [27] (where 1/R ∗ = 1/r1 + 1/r2 and 1/E ∗ = (1 − ν12 )/E 1 + (1 − ν22 )/E 2 ), and all results can be normalized in time by τ and space by R ∗ to represent a wide range of responses. Note that the numbers of main and intruder beads are equal in this model. Features of wave propagation in extended 2D layers of the type shown in Fig. 1a have been presented in detail in [23], highlighting that wave propagation in these systems does occur either as 2D solitary waves or in a directional fashion, with, in both cases, attributes tunable by varying the material mismatch between main and intruder beads as shown in Fig. 2. In the present study of layered granular media, the analysis of wave propagation is performed for the entire map of the mass-stiffness parametric space shown in Fig. 2. In that figure, the parametric space is represented by the mass ratio m 2 /m 1 together with the stiffness counterpart, defined as the spring constant ratio given as a variant of stiffness ratio, is given by [23] 2
√ √ 2E 2 /E 1 k12 = 2( 2 − 1), k11 1 + E 2 /E 1
(2)
Impact response of granular layers
23
a
b E1 E2
1 r1 2 r2
2 r2
2r1
r1 2r1 V0
H
2D Granular layer L
Fig. 1 Schematic of granular assemblies investigated in present work. The 2D granular layer (represented by shaded region) is composed of spherical beads of two distinct sizes arranged in a perfect square lattice of larger size (main beads) and smaller size particles (intruders), which occupy interstitial voids, and b random packing containing equal distribution of main and intruder particles. The thickness of the granular
layer is H and the width L H . The system is driven by an initial mechanical disturbance V0 imparted at the top of the layer, as shown. The force transmission response is studied by monitoring the transmitted force acting on the chain of larger fixed particles located along the bottom edge of the granular layer
Fig. 2 Different modes of wave propagation mapped over the mass ratio stiffness ratio space observed in infinite (square packed) ordered 2D granular media [23]. The ordered layer systems analyzed in the present work (Fig. 1a) cover the complete map of mass and stiffness ratios while multiple realizations of the random layers (Fig. 1b) are evaluated for five cases denoted with circles. The dashed curve repre-
sents the theoretical estimate of critical mass and stiffness ratios that support solitary wave propagation in ordered systems [23]. The regions enclosed by solid curves represent common material combinations, with ‘1’ and ‘2’ representing the main and intruder beads, respectively. Special material combinations labeled A through G in the table also marked
with added simplification of equal Poisson ratios for main and intruder beads (ν1 = ν2 ). The spring constant ratio k12 /k11 represents the ratio of stiffness between main and intruder
beads and that between two main beads. The limit of k12 /k11 as E 2 /E 1 tends to infinity is about 1.53 while that limit is zero when E 2 /E 1 tends to zero. The reference system, relative to
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which mass and stiffness ratios are varied, is composed of steel beads with density ρ = 7,840 kg/m3 , Young’s modulus E = 200 GPa and Poisson’s ratio ν = 0.3. Using the analogy between the equations describing the dynamic response of the granular system and that for a set of atoms, we analyze this problem using a specially adapted molecular dynamics (MD) solver, Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [28] with the inter-particle potential under compression corresponding to the Hertzian contact law (1). To capture the absence of sphere-to-sphere interaction when the beads are separated, we adopt in the MD framework a cut-off distance equal to the sum of the radii (r1 + r2 ) of the two adjacent beads. Motivated by the inevitable randomness that arises in this type of packing, we also investigate a second model of granular layer consisting of an equal number of main and intruder beads (also of sizes 5 and 2.07 mm) arranged randomly as illustrated in Fig. 1b. Even though the number density for both ordered and random sets is the same (equal number of main and intruder beads), they differ in arrangement of particles, and thus, the random samples are also different in the distribution of porosity and average mass density. The random packings are created by utilizing an in-house random packing generator also based on LAMMPS (see more details in Sect. 4). For the present study of layers made of randomly distributed beads, we have examined specific cases of mass-stiffness mismatch, with ratios (E 2 /E 1 , m 2 /m 1 ) equal to (1.0, 0.07107), (0.01, 0.05), (0.01, 50) (10, 50) and (10, 0.05). These ratios are depicted as circles in Fig. 2. The time step, Δt for investigating wave propagation is taken such that Δt/τ ≈ 2.9 × 10−2 . Force transmission studies are performed by constructing a wide random layer and evaluating its average transmission characteristics. This is obtained by impacting the layer at several different locations along the top boundary and averaging the force transmission of the bottom edge. This methodology produces force transmission averages similar to the setup of constructing smaller sized multiple random realizations and studying them individually. For both ordered and random layer problems, we construct layers with width L sufficiently larger than the thickness H to ensure that wave propagation across the thickness is unaffected by the lateral boundaries. The lower edge of the granular layer is kept fixed. The granular layer is impacted on its top surface by imparting an initial velocity of V0 to a large particle as shown in Fig. 1. The disturbance hence generated propagates through the granular medium, eventually hits the fixed bottom edge and reflects back towards the top. We evaluate the force transmission response of the granular layer by monitoring the compressive force defined as the norm of the force vector experienced by every bead lying along the bottom edge. The characteristics of force transmission are obtained by performing parametric studies
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to evaluate the effectiveness of the granular layer as tailored medium for transmitting or mitigating disturbances.
3 Force transmission in ordered granular layers We first investigate force transmission and wave propagation of the ordered square packed granular media shown in Fig. 1a. The presence of intruder beads in the square packed layer provides medium for the disturbance to spread laterally. As observed both numerically [23] as well as experimentally [24] in configurations where intruder beads are absent, loads applied to the boundary bead only travel along the principal directions and never propagate diagonally. In the following three sections, we successively analyze the dependence of the impact force transmission on the intruder-to-main-bead stiffness and mass ratios, and on the layer thickness. 3.1 Effect of stiffness ratio As described earlier, impact force transmission across the layer is quantified by monitoring the total compressive force along the bottom edge. Figure 3 shows the evolution of the compressive force along the bottom edge with the origin of the spatial coordinate x (normalized by the diameter 2r1 of the large beads) located just below the point of impact for three different stiffness ratios, E 2 /E 1 = 0.05, 1.0 and 25.0. The mass ratio m 2 /m 1 is kept fixed at 0.07107, which corresponds to a density ratio (ρ2 /ρ1 ) equal to 1. Unless otherwise indicated, the impact velocity V0 for all the simulations in the present work is fixed at 1 m/s. The choice of impact velocity is arbitrary, because forces in 2D elastic 6/5 granular systems scale as F ∝ V0 [23]. The total compressive force is depicted by the color palette shown as inset in Fig. 3a. In each case, we indicate the finite time interval (ta ) needed for the disturbance to reach the lower edge. We also note that the location of the arrival of the disturbance is vertically below the impacted bead as expected. Thereafter, the lower edge experiences a symmetrical spread of the disturbance while the disturbance is reflected into the granular medium. The lateral spread is less prominent for system (a), which corresponds to very low stiffness ratio, due to the more directional nature of wave propagation. Defining the average speed of wave propagation C = H/ta , we note that the disturbance travels faster for system (a) compared to cases (b) and (c), for which the average speeds of propagation are almost equal. It can also be observed that systems (b) and (c) consist of wave propagation via 2D solitary waves, while system (a) yields a directional wave propagation. This observation is consistent with [23] where the transition between directional and solitary wave propagation takes place at E 2 /E 1 ∼ 0.2 for m 2 /m 1 = 0.07107. For very
Impact response of granular layers
25
1.6
1.4
0 80ms
a
C
-50
0 95ms
50
ta
C
-50
0 x 2r1
50
C
-50
539m s 0 x 2r1
50
0.6
is the average speed of the disturbance. Time axis graduations are common for each plot. Time scale at granular contacts, τ , for each case is approximately 1.1, 0.3 and 0.2 µs respectively
25
25
20
20
15
15
Fp
50 40
F max , N
c
527m s
Fig. 3 Evolution of the compressive force along the bottom edge of the granular layer: a E 2 /E 1 = 0.05, b E 2 /E 1 = 1.0 and c E 2 /E 1 = 25.0. In all cases, ρ2 /ρ1 = 1 (i.e., m 2 /m 1 = 0.07107), H˜ = H/2r1 = 50, ta is the time of arrival of the disturbance at the bottom edge and C = H/ta
60
0 93ms 0.8
b
629m s 0 x 2r1
1
ta
50
0
30 10
10
5
5
x max 2r1
ta
1.2
t ms
F, N
60 50 40 30 20 10 0
20 10
-50
a
b
c
0 0.6 0.8
0 0.6 0.8
0 0.6 0.8
1
1.2 1.4 1.6
t ms
1
1.2 1.4 1.6
t ms
1
1.2 1.4 1.6
t ms
Fig. 4 Time evolution of the amplitude F max and location x max of the maximum compressive force along the bottom edge of the layer for E 2 /E 1 equal to a 0.05, b 1.0 and c 25.0. In all cases, ρ2 /ρ1 = 1 and
H˜ = 50. F p is the peak compressive force experienced by the bottom edge. In the cases shown, it occurs right below the impacted bead
low stiffness ratios (0 < E 2 /E 1 1), the disturbance travels as a 1D solitary wave along a vertical chain of beads as mentioned before. In Fig. 4, we present the evolution of the maximum force (F max ) and its location (x max ) along the bottom edge for the three cases shown in Fig. 3. In each case, we observe that F max is zero until the disturbance hits the bottom edge. Thereafter, it rises quickly and reaches a peak at time t p with magnitude F p , which in each case also represents the global maxima of F max . Note that we will explore later in this manuscript cases for which F p is not the first local maximum. In the cases shown in Fig. 4, when the disturbance reaches
the bottom edge, the maximum F p is experienced vertically below the impact point on the bottom edge. Beyond time t p , the amplitude of the maximum force F max falls more rapidly for low stiffness ratio (a) than higher stiffness ratios (b) and (c). As the disturbance reflects back into the medium and simultaneously spreads laterally over the bottom edge, the location of force maxima shifts symmetrically to either side of the center bead. This symmetrical spread is more prominent for higher stiffness ratio (c) than lower stiffness ratio (a), as also apparent in Fig. 3. It can be observed from Fig. 4 that the peak force F p is three times higher for E 2 /E 1 = 0.05 than for E 2 /E 1 = 1 and E 2 /E 1 = 25.
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A. Awasthi et al. E 2 /E 1 0.5
1
2
0.9 0.8 0.7
5
10 25
V0 =1m/s V0 =2m/s V0 =5m/s
Directional 2D Solitary Wave
1
0.2
1
0.9 1/5
C/ V0
F p / V0
6/5
0.6 0.8
0.5 0.4 0.3
0.7 0.2 0.1 0
0
0.2
0.4
0.6
0.8
1
1.2
0.6
1.4
k12 /k11
Fig. 5 Variation of the peak compressive force F p and average wave 6/5 1/5 speed C (respectively scaled by V0 and V0 ) obtained along the bottom edge with respect to the spring constraint ratio (lower axis) and the material stiffness ratio (upper axis), obtained for three load levels (defined by the initials velocity V0 ). F p and C are normalized by their respective values obtained in the absence of intruders. The vertical line indicates a transition between the directional wave propagation regime, characterized by a rapid decay of the transmitted force, and the solitary wave regime, for which the transmitted force and average propagation speed are almost independent of the stiffness ratio
As shown in [23], the force and wave velocity scale as 6/5 1/5 F ∝ V0 and C ∝ V0 . Based on this observation, we present in Fig. 5, the variation of the scaled peak force F p and wave speed C on the spring constant ratio k12 /k11 (Eq. 2). As expected, the peak transmitted compressive force and the average speed are maximum for k12 /k11 = 0 (i.e., for
E 2 /E 1 = 0), where the disturbance travels along a vertical chain of beads. With increase in stiffness ratio, both the transmitted peak force and the average speed fall rapidly before reaching a plateau. The vertical line in Fig. 5 indicates the theoretical critical value of the stiffness ratio at which the wave propagation transitions from the directional regime to the solitary wave one. For the system of interest, this transition occurs for k12 /k11 ∼ 0.2 or E 2 /E 1 ∼ 0.15. Once in the solitary wave regime, further increase in stiffness ratio makes the leading wavefront circular and further increase in stiffness ratio does not affect the spatial dispersion of the wave. This spatial restriction limits the transmitted forces as shown in Fig. 5 and they reach saturation with increase in stiffness ratio [23]. 3.2 Effect of mass ratio We now evaluate the effect of mass ratio m 2 /m 1 , keeping the stiffness ratio E 2 /E 1 fixed. As shown in [23], low mass ratios favor the formation of 2D solitary waves in the square packed granular system, while higher mass ratios offer greater resistance to propagation due to inertia, thereby reducing the speed of propagation. Figures 6 and 7 present the total compressive force response of the bottom edge for three different mass ratios: (a) m 2 /m 1 = 0.1, (b) m 2 /m 1 = 1 and (c) m 2 /m 1 = 10 with the stiffness ratio E 2 /E 1 fixed at 1. As apparent in Fig. 6, an increase in mass ratio leads to a reduction of the wave speed. Additionally, the shape of the disturbance obtained along the lower edge of the layer undergoes a transition, consistent with the observations obtained for the unbounded domains studied in [23] and alluded to in 3
1.5
5 4.5
1.4
4
2.5
1.3
20 15 F, N
1.2 2
5
ta
0 98ms
a -50
C
509m s
0 x 2r1
50
0.9 0.8
1.5 ta
1 90ms
b -50
C
344m s
0 x 2r1
50
c 1
-50
C
2.5 2 1.5
1 45ms
Fig. 6 Evolution of the compressive force along bottom edge of the granular layer, along with times of arrival and average wave speeds for ordered granular layers with mass ratios, m 2 /m 1 , equal to a 0.1, b 1.0
123
0
ta
3
10
1.1 1
3.5
t ms
0.1
251m s
0 x 2r1
50
1 0.5
and c 10.0. For each case, E 2 /E 1 = 1 and H˜ = 50. Time scales at granular contact, τ , for the three cases are approximately 0.3, 4 and 0.8 µs respectively
Impact response of granular layers
27
25
F max , N
20
Fp
12 Fp
10
15
10
8
8
6
6
4
4
2
2
50
Fp
0
10
5
a
b
0 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
t ms
0
1
x max 2r1
12
-50
c 1.5
2
2.5
3
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
t ms
t ms
Fig. 7 Time evolution of peak force F max and location x max for m 2 /m 1 equal to a 0.1, b 1.0 and c 10.0. In each case, E 2 /E 1 = 1 and H˜ = 50
m2 m1
1 , k 11 2−5/4 +1 k12
(3)
which, for E 2 /E 1 = 1, yields m 2 /m 1 0.756. This theoretical boundary of transition between 2D solitary wave behavior and directional propagation is shown as the vertical solid line in Fig. 8, in which the peak force F p and the average speed C are normalized by their counterpart value corresponding to m 2 /m 1 = 0, F pm0 and Cm0 . Due to the finite value of the stiffness ratio (E 2 /E 1 = 1), the peak force and the average speed drop very rapidly when the mass ratio
1 0.8
0.4
Solitary wave propagation
0.1
C Cm0
F p F pm0
Fig 2. This transition is also clearly illustrated by the evolution of the amplitude (F max ) and location (x max ) of the maximum transmitted force shown in Fig. 7. Note that the peak force experienced by the bottom edge is about two times larger for the case with smaller mass ratio in Fig. 7a than higher mass ratios of Fig. 7b, c. Also apparent in Fig. 7, the peak total compressive force F p corresponding to low mass ratios is associated with the primary wave and occurs vertically below the impact, while the secondary disturbances have smaller magnitudes and decay rapidly. With increase in mass ratios, the peak force F p becomes linked to secondary disturbances and its location deviates from the point on the lower edge immediately below the point of impact. The mass ratio might therefore be exploited to devise protection layers that direct the impact load away from point of impact. The complete picture of the effect of mass ratios (with stiffness ratio fixed at 1) is shown in Fig. 8, which presents the effect of the mass ratio on the peak force F p and average velocity C normalized by the corresponding values obtained in the absence of intruders. The approximate transition from directional waves to solitary waves in square packed systems corresponds to the following critical mass ratio [23]:
Directional propagation
0.2
0.01 1e-05
F p F pm0 C Cm0
0.0001
0.001
0.01
0.1
1
10
0.1 100
m2 m1 Fig. 8 Dependence of normalized peak compressive force (F p ) and wave speed (C) on mass m 2 /m 1 for E 2 /E 1 = 1. The normalization factors F pm0 and Cm0 are respectively the peak force and average speed of disturbance for m 2 /m 1 = 0. The vertical line at m 2 /m 1 = 0.756 corresponds to the theoretical mass ratio for stiffness ratio E 2 /E 1 = 1 at which wave propagation transitions from 2D solitary wave to directional propagation
assumes a non-zero (yet very small). Then these two quantities remain almost independent of m 2 /m 1 until the transition from solitary to directional propagation. Within the solitary propagation regime, the primary disturbance contains the peak force. With further increase in mass ratio beyond the transition from solitary to directional propagation, the peak force attains a minimum around m 2 /m 1 = 2.5 and rises in magnitude with further increase in mass ratio. The minimum represents the transition at which the primary disturbance becomes very small compared to the peak force, the majority of the load is distributed sideways, and the transmitted force load immediately below the impact point becomes
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A. Awasthi et al. 0.9
0.35
Simulation Exp. fit
0.8
Table 1 Coefficients of exponential and power law dependence of peak force with respect to layer thickness for different stiffness ratios
Simulation Power fit
0.3
0.7
E 2 /E 1
Functional form
a
b
0.001
˜ F˜p = ae H b
F p F p0
0.25 0.6 0.2
0.5
Fp
0.4
0.15
0.3 0.1 0.2 0.05
0.1 0
a 0
E2 E1 20
b
0 05 40
60
80
H
0
0
E2 E1 20
40
1 60
80
H
Fig. 9 Effect of layer thickness H˜ on the decay of normalized peak transmitted force F p obtained for E 2 /E 1 equal to a 0.05 and b 1. The exponential and power law fits of the numerical results point to the existence of two different decay regimes associated with the type (directional versus solitary wave) of wave propagation in the granular medium
1.0028
−0.001
0.003
1.0058
−0.004
0.005
1.0130
−0.006
0.008
1.0154
−0.009
0.01
1.0210
−0.011
0.05
1.0468
−0.038
0.08
0.9931
−0.048
0.10
0.8997
−0.049
F˜p = a H˜ b
14.027
−1.385
0.50
3.7258
−1.057
1.00
2.1065
−0.927
10
0.8842
−0.707
100
0.8469
−0.694
0.15
0.4
0.35
3.3 Effect of layer thickness To study the effect of layer thickness, we prepare samples of different thickness H keeping the initial impact conditions the same (V0 = 1 m/s). The width W is kept sufficiently large as before to accommodate lateral propagation of the disturbance and keep it free from edge effects. The influence of thickness is evaluated for two categories of problems: in the first type, the stiffness ratio is varied while the density ratio is kept fixed at 1 (mass ratio 0.07017). In the second type, the mass ratio is varied while the stiffness ratio is fixed at 1. We describe the results in that order below. Granular layers are constructed with thirteen stiffness ratios ranging from 0.001 to 100 and the analysis is performed for each stiffness ratio with five different thicknesses H˜ = H/2r1 = 10, 25, 40, 55 and 70. For each case, the peak transmitted force, F p , is monitored. The results, such as those shown in Fig. 9, demonstrate the existence of two distinct regimes of decay of F p with respect to H˜ : exponential decay for E 2 /E 1 < 0.15 and power law decay for E 2 /E 1 > 0.15. The values of fitting parameters are provided in Table 1. We now extend the investigation of the impact of layer thickness on transmitted forces over different mass ratios. Here, we keep the stiffness ratio fixed at E 2 /E 1 = 1 and systems with fourteen different mass ratios are analyzed, varying m 2 /m 1 from 10−5 to 100. For each system, six different real-
123
0.3
F p F pm0
Simulation Exp. fit
0.35 0.3
0.25
Fp
vanishingly small. Further increase in mass ratio causes the average speed to stay almost at a constant value while the peak force continues to increase. In this regime, the qualitative nature of wave propagation is same as in Figs. 6c and 7c.
Simulation Power fit
a
0.2
m2 m1
0 75
0.25
b
m2 m1
10 0
40
60
0.2 0.15 0.15 0.1
0.1
0.05 0
0.05 0
20
40
H
60
0
0
20
H
Fig. 10 Effect of layer thickness H˜ on normalized peak compressive force F˜p for m 2 /m 1 equal to a 0.75 with power decay and b 10 with exponential decay. F pm0 is the peak compressive force for m 2 /m 1 = 0
izations are set up with H˜ equal to 5, 10, 20, 30, 50 and 60. We observe that smaller mass ratios are associated with a power law decay of the transmitted forces while higher mass ratios yield an exponential decay with the transition occurring around m 2 /m 1 = 5. Figure 10 shows two representative cases for these two decay modes. Fitting parameters for the two modes for this case are given in Table 2.
4 Force transmission in randomly packed granular layers To generate the random packings numerically as described in Fig. 1b, we make a start by placing big and small beads at sparse uniformly distributed random locations in 2D and
Impact response of granular layers
29 2
m 2 /m 1
Functional form
a
b
1.4
10−5
F˜p = a H˜ b
2.3721
−0.937
10−4
2.3720
−0.936
0.8
10−3
2.3686
−0.936
0.6
0.01
2.3535
−0.936
0.4
0.10
2.3292
−0.972
0.25
1.8174
−0.993
0.2 -50
0.50
1.1200
−0.880
0.75
1.1963
−0.944
1
1.1664
−0.971
2.5
1.4530
−1.086
˜ F˜p = ae H b
0.3318
−0.053
10
0.4540
−0.055
50
0.4833
−0.028
100
0.4624
−0.020
5.0
enclosing them by rigid walls. The complete assembly is then allowed to dynamically settle at the bottom of the rigid frame under the influence of an externally applied gravitational force together with a viscous drag to damp the motion. Once the beads settle, the gravitational and viscous forces are gradually eliminated. These steps, performed using LAMMPS are designed to ensure contact between the granules which is critical for studying wave propagation. Moreover, the dynamical time step for creating the random assembly is chosen as approximately 29τ to accurately resolve the movement of the beads. We keep the stiffness ratio as well as density ratio of the beads as 1 in all the random layer simulations except wherever mentioned. We prepare a wide granular layer of width L = 1.84 m (L/2r1 = 184), and thickness H = 0.25 m(H/2r1 = 25) consisting of about 8,500 beads. The top free boundary of this randomly packed granular layer has a staggard profile. To study the average wave propagation characteristics of the randomly packed layer, we choose several main beads located along the top free boundary and perform individual tests by imparting them an initial velocity of V0 = 1 m/s vertically downwards as shown in Fig. 1. The idea behind creating a wide layer and performing analysis on it by impacting it on different points assumes that the structure is representative of creating individual random samples and performing analysis on each individually. We confirmed this aspect by also preparing samples of similar thickness and lesser width to investigate the representative nature of the random layer in terms of force transmission. Force transmission characteristics are investigated on both these systems by monitoring the forces on the bottom edge.
50 0.25
1.8
F max F po
1.2
25
0.2 0.15
0
a -25
0
x 2r1
F max
F F po
1
F
t ms
1.6
0.2 0.15 0.1 0.05 0
25
x 2r1
Table 2 Coefficients of exponential and power law dependence of the peak force F˜p across granular layers on the layer thickness H˜ obtained for different mass ratios
0.1 -25 0.05
b 50
0
0.4 0.8 1.2 1.6
2
-50
t ms
Fig. 11 Typical force transmission in random granular layer. a x − t plot of the bottom edge and b evolution of normalized F max and x max experienced by bottom edge. The normalization factor F po is the peak compressive force in ordered system of same thickness, showing that the peak force in the random layer is much smaller than that in its ordered counterpart
In the random samples, the lack of periodicity in the particle distribution leads to a distribution of particle contacts and porosity. Figure 11 shows a typical response of the random granular layer due to randomness in the packing. Symmetry and solitary wave behavior are absent unlike those observed in ordered systems. In that figure, the response of the random layer is normalized by that of corresponding ordered layer of same thickness and material properties. We observe that the disturbance reaches the bottom edge with considerable attenuation as shown in Fig. 11b. Force transmission occurs through the granular chain by a network of contacts and, due to randomness in the system, these disturbance propagation paths are also randomly oriented, as also observed in previous studies [18]. To characterize randomness in the system all along the width of the layer, we estimate the volume fractions of main beads, intruder beads and voids, along vertical lines connecting the top free beads of the granular chain and the fixed boundary at the bottom as shown in Fig. 12. These volume fractions are estimated by evaluating intersections of each of these vertical lines with beads lying in their way. The volume fractions are defined as fraction of the total length of the line inside the main, intruder bead or void space. The different vertical lines shown in Fig. 12 correspond to different impact locations on the random layer. An approximate estimate of the volume fraction of main beads, intruder and voids is obtained by summing up the respective components in Fig. 12. These numbers are respectively Vm = 0.733, Vi = 0.119 and Vv = 0.147 for the present random system while, for the square packed ordered system, the values are Vmo = 0.785, Vio = 0.134 and Vvo = 0.080, respectively. It is clear that the differences in force transmission attributes arise from differences in distribution of particles and porosities. However no direct correlation is observed between the
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Fig. 12 Volume fraction for the main (Vm ) and intruder beads (Vi ) estimated at different vertical locations (spaced by x/2r1 = 1) along the width of the randomly packed granular layer. The vertical lines represent 25 x-locations of the main beads that are impacted to obtain average force transmission characteristics. Vmo and Vio are respectively the volume fractions of main and intruder beads in the ordered system and the over-bar denotes average
Vm
Vm Vi
Vm
Vm Vi
Vmo
Vmo Vio
1
Volume fraction
0.8
0.6
0.4
0.2
0
-80
-60
-40
-20
0
20
40
60
80
Distance along width of layer, x 2r1
0.3
F p F po
0.29
0.2
Moving average, F p
F max
F max F po
0.25
0.15 0.1 0.05
a 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
t ms
Sample 1 Sample 2
0.28
0.26
E2 E1
m2 m1
0.25
1.00
0.07107
0.2349
0.24
0.01
0.05
0.0093
0.23
0.01
50.00
0.0007
0.22
10.00
50.00
0.1731
10.00
0.05
0.3253
0.27
0.21 0.2
b 0
F p = F p /F po
5 10 15 20 25 30 35
Experiment number
Fig. 13 Force transmission in multiply-impacted granular layers. a Evolution of F max for different impact cases and b running average of peak force F p evaluated for two sets of random samples. F p0 is the peak force in ordered system for same layer thickness
lengthwise volume fractions and the characteristics of force transmission due to the complexity of the phenomenon. As apparent in Fig. 13a, which presents the evolution of the transmitted maximum force obtained for various impact events, the response is qualitatively similar for the different loading cases and consists of a rising part, a peak force and a decay segment. The peak forces are represented as running average with respect to number of samples (which corresponds to the number of different impacts at the top layer) in Fig. 13b where it can be observed that 15–20 samples are sufficient to capture the average response demonstrating. We further investigate four additional combinations of mass and stiffness ratios of intruder and main beads pertaining to random granular aggregates (circles in Fig. 2). Their
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Table 3 Normalized peak force transmission in random granular layer for five different combination of stiffness and mass ratios
results are summarized in Table 3 showing that low stiffness ratio reduces the average normalized peak force transmitted across the granular layer and together with a high mass ratio, the cumulative effect further reduces the normalized peak force. In contrast, high stiffness ratio increases the average normalized peak force and, if mass ratio is reduced, it leads to a further increase in the normalized peak force. We can also infer that granular layers can be utilized to tailor force transmission spanning a wide range of attenuation.
5 Summary and conclusions A numerical study was performed to evaluate force transmission characteristics of granular layers composed of elastic spherical particles interacting through the Hertz potential. The analysis was focused on two configurations: ordered square packing of larger granules with interstitially occupied small granules, and random packing of granules of same sizes as in ordered pack. These granular layers were point impacted
Impact response of granular layers
on the free boundary and force transmission features were analyzed along the opposite boundary. For ordered systems, the effect of material mismatch and layer thickness on the force transmission response was evaluated, while for random systems, average response was studied for a few chosen material combinations of the beads. Results obtained with the ordered granular systems have shown that force transmission magnitudes and disturbance propagation speeds reach saturation levels after a critical stiffness ratio of 0.148. This value is associated with a transition from directional behavior of propagation to solitary wave propagation in 2D granular media. A parametric study of the impact of the mass ratio has demonstrated that force transmission can be tailored to minimize the forces right below the point of impact and instead offset them away. Variation of thickness showed that peak force variation follows exponential decay with increasing thickness for directional propagation and power law decay for solitary wave propagation. When randomly packed, the granular layers demonstrate high attenuation in force amplitudes than corresponding ordered systems and showed wide ranges (four orders of magnitude) of average force transmission. The present study showed that granular layers have the potential of force transmission tailoring. Organizing them in ordered packing, controlling stiffness, mass ratio and thickness can provide tailored directional or dispersed responses, while random assemblies can result in substantially reducing peak force amplitudes. Acknowledgments Authors acknowledge the support by the United States Army Research Office through the Multi University Research Initiative project number W911NF0910436 (Program Manager Dr. David Stepp). Natalie Broadhurst was partially supported and sponsored by the Illinois Space Grant Consortium, as part of the Undergraduate Research Opportunity Program held every summer in the Department of Aerospace Engineering at the University of Illinois. Conflict of interest The data presented in this work have not been published elsewhere. No other disclosures are associated with the present work.
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