Physica A 419 (2015) 457–463

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Physica A

journal homepage: www.elsevier.com/locate/physa

Bottom stresses of static packing of granular chainsPingPing Wen a, Guan Wang a, Degan Hao a, Ning Zheng a,∗, Liangsheng Li b,Qingfan Shi aa School of Physics, Beijing Institute of Technology, Beijing 100081, Chinab Science and Technology on Electromagnetic Scattering Laboratory, Beijing 100854, China

h i g h l i g h t s

• Bottom stress of a granular chains column is measured precisely and reproducibly.• Scaling behavior of the stress saturation curves agrees with the Janssen model.• Saturation mass is shown as a nonmonotonic function of the chain length.• A transition of the saturation mass is found at the persistence length of chains.• Entanglements as a possible explanation for nonmonotonic behavior are presented.

a r t i c l e i n f o

Article history:Received 4 May 2014Received in revised form 18 September2014Available online 18 October 2014

Keywords:StressJanssen modelGranular chainsStatic packingInter-chains entanglement

a b s t r a c t

We experimentally measure the static stress at the bottom of a granular chains columnwith a precise and reproducible method. The relation, between the filling mass and theapparent mass converted from the bottom stress, is investigated on various chain lengths.Our measurements reconfirm that the scaling behavior of the stress saturation curves is inaccord with the theoretical expectation of the Janssen model. Additionally, the saturationmass is displayed as a nonmonotonic function of the chain length, where a distinguishingtransition of the saturation mass is found at the persistence length of the granular chain.We repeat themeasurementwith anothermeasuringmethodology and a silowith differentsize, respectively, the position of the peak maintains robust. In order to understand thetransition of the saturationmass, the friction coefficient and the volume fraction of granularchains are also measured, from which Janssen parameter can be calculated. Finally, wepreliminarily measure the bottom stress for two distinct packing structures of long chains,find the effect of the entanglements on the bottom stress, and argue that the entanglementsmight be responsible for the transition of the saturation mass.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

The static packing of granular materials has been one of long-lasting issues due to its practical importance in many fieldsincluding civil engineering, soil science, and storage and processing of rawmaterials, while the unity of basic physics remainsunsolved [1]. Among the studies attempting to address the puzzle, generally most researches deal with spherical grains[2–4], and several studies have focused on the aspherical grains, such as ellipsoids [5,6] or rods [7,8], which have inducedqualitatively new features of the volume fraction of granular piles [9,10], the orientation order and coordination number

∗ Corresponding author.E-mail addresses: ningzheng@bit.edu.cn (N. Zheng), liliangsheng@gmail.com (L. Li), qfshi123@bit.edu.cn (Q. Shi).

http://dx.doi.org/10.1016/j.physa.2014.10.0620378-4371/© 2014 Elsevier B.V. All rights reserved.

458 P. Wen et al. / Physica A 419 (2015) 457–463

[11,12], stress distribution [13] and propagation [14] in granular piles, and the jamming transition [15,16]. Whatever grainsare used in investigating the packing properties, nearly all researches have focused on individual grains. Until recentlygranular chains began to draw more attention in the researches on the packing. For instance, Zou et al. investigated thestatic packing structure of granular chains in a cylinder, finding the reciprocal of the volume fraction of chains packingis analogous to the glass transition temperature of polymer solutions. The similar behavior between jamming state ofgranular chainswith glass state of polymerswas thus illustrated [17]. Our experiments subsequently revealed that statisticalscaling characteristics of two-dimensional granular chains are in good accord with the theoretical expectations of polymermodels [18–20].

The previous contributions on the static packing of granular chains mainly put emphasis on the packing structures nearthe jamming transition [17,21], or on an analogy between their statistical properties and theoretical predictions in polymerscience [17–20], aiming to provide a possible way to uncover the physics that is experimentally inaccessible for polymers.Although the packing column confined in a container can also act as a simple but valid experimental system to test manytheoretical models for the repartition or the transmission of stresses in granular materials [22–25], themechanical behaviorof the packing of granular chains did not receive enough attention until Brown et al. revealed that a stress response ingranular chains was very distinct from common grains, namely individual grains [26]. They found that strain stiffeningoccurred in the packing of long chains due to the inter-chains entanglements, which are absent in the packing system ofindividual grains or short chains. However, the lateral wall of the silo in their experiment is a soft membrane that can allowfor free radial expansion when applying a compressive stress. The investigation on the mechanical properties of confinedgranular chains with a rigid, lateral boundary appears to be still scarce. Measuring the stresses under various boundaryconditions is very necessary for the establishment of the macroscopic constitutive relations of stress–strain.

In this paper, we accurately measure the average mass at the bottom of a granular chain column confined in a rigidcylinder, with the measurement method proposed by Vanel et al. [27]. In the measurement, eleven different chains,consisting of the same material but the chain length N spanning three orders of magnitude from N = 2 to 2048, areused to measure the average mass at the bottom of the chains column as a function of the granular chains filling mass,respectively. The relation between them is in a good agreementwith the description of the Janssenmodel.We thus apply theJanssen model to fit the experimental curves, and then extract the saturation masses for all chains. The saturation mass as anonmonotonic function of the chain length is also shown,where a transition of the saturationmass is found at the persistencelength of the granular chain. Moreover, these relevant parameters to determine the saturation mass are measured, and theJanssen parameter is calculated for all chains as well. Finally, qualitative arguments for the nonmonotonic dependence ofthe saturation mass are presented.

2. Experimental setup

A diagrammatic sketch of the experimental setup is illustrated in Fig. 1(a). The semi-rigid granular chains are confinedin an acrylic cylinder with an inner diameter of 45 mm and a height of 700 mm. The cylindrical silo is vertically mounted toa heavy stand, and its bottom is closed by a movable piston connected to a force sensor which records the average pressureon the piston. The diameter of the piston is slightly smaller than the inner diameter of the silo in order that the piston nevertouches the inner wall, and the gap between the piston and the wall is small enough to avoid the leakage or the blockageof filled materials. The linear chains with free ends used in our experiment are composed of hollow, steel beads of diameter3.0 ± 0.1 mm and steel rods, as the link connecting beads. The beads are uniform in size and they are also free to rotateabout the rods. The rod between two neighboring beads is not fixed, but rather retractile. The maximum length of the rodconnecting two beads can reach 1.3± 0.1mm, as shown in Fig. 1(b) and (c). The chain lengthN is represented by the numberof the beads on a chain. These ball-chains are not arbitrarily flexible: the number of the beads required to form a smallestring, which is generally equivalent to the persistence length of the chain, ξ = 8 is used to characterize the stiffness of thegranular chain, see Fig. 1(d). Due to hollow structure, the measured apparent density of the granular chain is not equal toits solid density (material density), but 3.56 ± 0.12 g/cm3.

A filling massMfill of granular chains is poured into the silo through a hopper. We also scrunch a long chain up and throwinto the silo, finding little influence on the bottomstressmeasurement provided the rest of themeasuring procedure remainsidentical except for filling methods. After the filling procedure is completed, the steppingmotor controls the piston tomovedownward at a constant speedV = 0.2mm/s. The pistonwould not stop towait 90 s until it goes through a 2mmdownwarddisplacement. The downward motion allows the granular column to slide down so that the friction along the inner wall issupposed to be fully mobilized. During the waiting time, we make use of a measuring protocol proposed by Vanel et al.,which can precisely and reproducibly obtain the mean pressure at the bottom of a granular column [27]. Then, the pistoncontinues its downward movement and the bottom stress measurement repeats. When a total displacement 20 mm ends,a 10-steps stress measurement is fulfilled. Fig. 2(a) shows a typical time evolution of the apparent mass that is convertedfrom the bottom stress. It is clearly observed that when the displacement of the piston starts, the apparent mass abruptlydecreases. During the slow downward motion, the apparent mass fluctuates violently due to unstable friction mobilization.Once the stepping motor stops, the apparent mass sharply jumps up to a point, in which a slow relaxation of the apparentmass occurs and a stable value can be reached at the end of the waiting time. The final values of the apparent mass, notedMa, is our desired measurement as indicated in Fig. 2(a). Then the silo is entirely emptied and refilled to repeat the 10-stepsmeasurement about 5–7 times. Thus, every data point is at least averaged by 50 times.

P. Wen et al. / Physica A 419 (2015) 457–463 459

Fig. 1. (a) Sketch of the experimental setup (not to scale). An acrylic silo clamped vertically is filled with granular chains. The bottom of the silo is closedby a piston attached to a force sensor. The diameter of the piston is slightly smaller than that of the silo such that the design can avoid the friction betweenthe piston and the inner wall, as well as the leakage of granular chains. The enlarged portions of the schematic show a static packing of the chains with thechain length N = 2048 (upper right). (b) The diameter of the bead on the chain is 3.0 ± 0.1 mm, and the rod between two neighboring beads is retractile.(c) The maximum length of the rod connecting two beads can reach 1.3 ± 0.1 mm. (d) The persistence length, ξ = 8, namely the number of beads requiredto form aminimum loopwhich is highlighted by a red dashed circle. (For interpretation of the references to color in this figure legend, the reader is referredto the web version of this article.)

Fig. 2. (Color online) (a) Typical evolution of the apparent massMa with time. Circles indicate themeasured values of the apparent mass at the end of timerelaxation. (b) Apparent mass Ma as a function of filling mass Mfill is plotted for short chains with different chain lengths N = 2 (circle), N = 4 (square),and N = 8 (diamond). (c) Apparent mass as a function of filling mass is plotted for long chains with different chain lengths N = 16 (up triangle), N = 32(down triangle), N = 64 (square), N = 128 (diamond), N = 256 (Hexagon), N = 512 (cross), N = 1024 (circle), and N = 2048 (star). All data points in (b)and (c) have been averaged about 50 times from 7 experiment trails. (d) Apparent mass Ma vs. filling mass Mfill , rescaled by the saturation mass Msat , forall chain lengths. All data points cluster together to form a universal scaling curve that agrees with the prediction of the Janssen model (solid line).

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Following a complete stressmeasurement, a thin and flat cardboard is used to flatten the top surface of the chains columncarefully and then a well-defined surface is obtained. The volume fraction of the chains column is estimated by monitoringits height H . The volume fraction ν is given by

ν =M

HπR2

ρa (1)

where M is the filling mass, H is the height of the chains column, R is the radius of the silo, and ρa =mV is the apparent

density measured, where m is the mass of the chain, and the V is the volume of the chain. To eliminate other undesirableinfluences which disturb the measurement as much as possible, temperature and humidity are maintained at 23± 1 °C and45%–50%, respectively.

3. Results and discussion

In the bottom stress measurement, the chain length ranges three orders of magnitude from N = 2 to 2048. Accordingto the persistence length ξ = 8, these chains may be categorized into two groups, short chains (N ≤ ξ ) and long chains(N > ξ ). Fig. 2(b) and (c) shows the relationship between the apparent mass, which is converted from the bottom stress,and the filling mass for short and long chains, respectively. In Fig. 2, all curves approach their own saturation values. Forshort chains, the saturation mass increases with the chain length. However, the situation is reverse for long chains.

For the asymptotic saturation of the filling mass in a silo, Janssen proposed a pioneering work to explain the counter-intuitive phenomenon [28]. The model treated a granular packing column as a continuous medium, and assumed that thefriction between the grains and the wall was fully mobilized along the upward direction, that thus the part of the verticalstress is transferred to the horizontal orientation. The apparent mass Ma at the bottom of the silo as a function of Mfill is inthe form of the equation as follows:

Ma = Msat

1 − exp

−

Mfill

Msat

with Msat =

ρνπR3

2Kµs(2)

where ρ is the density of granular materials, µs is the friction coefficient between the silo wall and the granular materials,K is the Janssen parameter, redirecting the part of vertical stresses to horizontal direction via the friction, R is the radius ofthe silo and ν is the volume fraction of the granular column. To verify that the experimental curves can be described by theJanssen model, we rescale the apparent mass and the filling mass with the saturation mass. It is found that although theexperimental curves separate in Fig. 2(b) and (c), all rescaled data cluster together to form a universal scaling curve whichagrees with the one predicted by Janssen model:Ma/Msat = f (Mfill/Msat), with f (x) = 1 − exp(−x), see Fig. 2(d).

In Fig. 2, the saturation masses for different chains appear to change with the chain length. Thus, the saturation mass isplotted as a function of the chain length, shown in Fig. 3(a). The relation between the saturation mass with the chain lengthis a nonmonotonic function, showing a peak at the persistence length. Within the range of short chains, the saturation massincreases with the chain length, finally reaching a maximum at N = ξ . But at the range of long chains, the saturation massis a monotonically decreasing function of the chain length. To examine the validity of the measuring methodology (methodI) developed by Vanel, another methodology (method II) suggested by Ovarlez et al. [29] has been adopted to remeasure therelation of the saturation masses and the chain length. Although it is found the saturation mass measured by the method IIis smaller than that measured by themethod I, the feature of the curve still remains robust. Additionally, to clarify the effectof the silo size on the occurrence of the peak at the persistent length, another silo with different diameter (inner diameter20 mm) is adopted to repeat the experiment with method I. From Fig. 3(b), the basic shape of the curve is similar to that inFig. 3(a), and the peak still occurs at persistence length.

In the Janssen model, the saturation mass can be expressed byMsat =ρνπR3

2Kµs. To understand the nonmonotonic behavior

of the saturation mass, the chain length dependences of the packing volume ν and the friction coefficient µs are measured.We firstmeasure the volume fraction before and after the stressmeasurement, and the difference of both packing fractions ishard to distinguish, as shown in Fig. 4(a). It is clearly observed that the volume fraction ν, which reflects the average structureproperties of the packing column, declines monotonically with the chain length N , from ν ≈ 0.62 for N = 1 to a saturationvalue ν ≈ 0.41 at largeN on the characteristic curve. The curves in Fig. 3 exhibit a nonmonotonic characteristic, but the chainlength dependence of the volume fraction is amonotonic function. Consequently, even if the saturationmass depends on thevolume fraction, it is not the only parameter to determine the nonmonotonic behavior. From the expression of the saturationmass, the saturationmass also relates to the friction coefficientµs and the Janssen parameter K , besides the volume fractionν. Then we measure the friction coefficients of different chains using the sliding angle of a chains sledge. Fig. 4(b) showsthat the friction coefficient is almost independent on the chain length, basically a constant. From the relationship of thesaturation mass, the friction coefficient and the volume fraction, the Janssen parameter K can be calculated. It is found inFig. 4(c) that in short chains case K reduces with the chain length N , but in long chains case the situation is opposite. A dipappears at the persistence length as well. However, Janssen parameter K is only an effective parameter derived from theexperimental data, which cannot be acquired by direct measurements. Thus the underlying physics of the nonmonotonicrelation still remains ambiguous.

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Fig. 3. (a) Chain length dependence of the saturation mass that is extracted from Janssen model is plotted using a semilogarithmic scale. The solid lineis plotted for eye guide. Method I: data represented by the red circle (solid line) are measured by the Vanel methodology; Method II: data represented bythe blue triangle (dashed line) are measured by the Ovarlez methodology. The peak corresponds to the persistence length ξ = 8 at x axis for both curves.(b) Repeat the experiment by method I with using another silo with a diameter 20 mm. The peak still occurs at ξ = 8. (For interpretation of the referencesto color in this figure legend, the reader is referred to the web version of this article.)

Recently strain stiffening was observed in the granular chains column as the chains packings were sheared in uniaxialcompression [26]. Specifically, the distinct mechanical responses are divided by the persistence length. While the chainlengthwas shorter than the persistence length, all measuring results exhibited typical granular strain softening. As the chainlength was longer than the persistence length, the chains column showed strong strain stiffening that further enhancedwith the increase of the chain length. The reason that strain stiffening only occurred for the longer chains system could beattributed to the inter-chains entanglements.

Comparedwith Brown et al. results [26], we also find a distinguishing transition of the saturationmass at the persistencelength. For the packings of the long chains (N > 8), the inter-chains entanglements become increasingly common andsignificantly influence the packing structure, leading to change of themechanical response of the packing system. In contrast,the short chains (N ≤ 8) preferentially behave more like individual flexible rods with different aspect ratios, due to theabsence of inter-chains entanglements. Even if the chain N = 8 can be bent into a smallest ring ideally, it has a very smallprobability to form a loop in the packings, as observed in the experiment. Here we preliminarilymeasured the bottom stressfor two distinct packing structures of long chains (N = 256) with the method I. One of the packing structures was randompacking obtained by the method described in the experimental setup, with a volume fraction 0.41 ± 0.01. Another packingstructure was artificial packing, where the chains were folded and put into the container orderly to avoid the entanglementsas much as possible, like stacking chopstick neatly into a box. Obviously, the entanglement in the chopstick-in-the-boxstructure ismuch less than the randompacking, thus resulting in a volume fraction 0.52± 0.01much larger than the randompacking. We find that the bottommass in the chopstick-in-the-box structure is 300 ± 12 g, and that in the random packingis 192 ± 8 g. The result suggests that the entanglement is responsible for the bottom stress. Qualitatively, the entanglementreduces the bottom stress, which agrees with the result in Fig. 3 in which the mass decreases with the increase of the chainlength for long chains (N > 8). With the increase of the chain length, the entanglements become increasingly prevalent.Correspondingly, it is plausible to suspect the inter-chain entanglements might be responsible for the transition of thesaturation mass. The findings of ours and Brown et al. imply that the chains packings solely with different chain lengthsare supposed to be seen as the packings of various materials, at least on some mechanical aspects, although all mechanicalproperties of these chains such as density and friction coefficient are identical. However, the questions whether theinter-chains entanglements are indeed responsible for the transition of the saturation mass, and how to explain theunderlying physics, remain open for future work.

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Fig. 4. (a) Volume fraction ν vs. the chain length N , using a semilogarithmic scale. The red solid line indicates the volume fraction before the massmeasurement, and the blue dashed line represents the volume fraction after the mass measurement. (b) Friction coefficient µs , between the granularchains and the inner wall of the silo, as a function of the chain length N is shown using a semilogarithmic scale. (c) Janssen parameter K as a function of thechain length N . The minimum on the curve corresponds to the persistence length ξ = 8. (For interpretation of the references to color in this figure legend,the reader is referred to the web version of this article.)

4. Summary

In conclusion, we fill a rigid silo with granular chains, and measure the average stress at the bottom by using a preciseand reproducible method. The apparent mass, that is, measured mass at the bottom increases with the filling mass untilreaching a plateau noted saturation mass. We use eleven kinds of granular chains with different chain lengths to test therelation between the apparent mass with the filling mass. When being scaled with the saturation mass, the measured datafrom different chains collapse together to form a universal master curve which agrees with the prediction of the Janssenmodel.We employ the Janssenmodel to fit the experimental curves, and then the saturationmasses are extracted. The chainlength dependence of the saturation mass appears to be nonmonotonic. As the chain length is shorter than the persistencelength (N < 8), the saturation mass increases with the chain length. However, if the chain length is longer (N > 8), thesaturationmass is inversely proportional to the chain length. We repeat themeasurement with another measuringmethod,and the saturation mass still reaches a peak at N = 8. We also find the peak does not move when another silo with differentsize is adopted. The Janssen parameter K is calculated by using the relationship of the saturation mass, the volume fractionand the friction coefficient. The relation between the Janssen parameter K and the chain length N is nonmonotonic as well.

P. Wen et al. / Physica A 419 (2015) 457–463 463

For the nonmonotonic behavior, we speculate that the short chains can be seen as anisotropic, elastic rods due to the absenceof inter-chains entanglements, showing the mechanical response similar to the common grains. But for the long chains, theamount of the entanglements is proportional to the chain length, and the existence of numerous inter-chains entanglementsmight introduce a qualitative difference of packing structures from the short chains system. However, the question whetherinter-chains entanglements are indeed responsible for the nonmonotonic behavior of the saturation mass remains open.

Acknowledgments

The work was supported by the National Natural Science Foundation of China (Grant No. 11104013) and the NationalInnovative Experimental Projects for University Students (Grant No. 201210007044).

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