- Email: [email protected]
Force fluctuations in granular media
ELSEVIER
PhysicaD 107 (1997) 183-185
Force fluctuations in granular media S.N. C o p p e r s m i t h 1 The James Franck Institute, The University of Chicago, Chicago, IL 60637, USA
Abstract
The forces in packed granular materials are distributed very differently than those in "normal" solids. Forces much larger than the mean occur, but are exponentially rare. These force inhomogeneities can be understood using an exactly solvable statistical model, in which the fluctuations in the forces arise because of variations in the contact angles together with the constraints imposed by the force balance on each grain. The details of this work are published elsewhere, see Liu et al. (1995) and Coppersmith et al. (1996). Keywords: Granular materials; Arching; Force fluctuations; Statistical mechanics
Granular materials are of interest not only because of their technological relevance but also because their complex dynamical properties pose a difficult scientific challenge [1-5]. Understanding them is difficult because thermal effects are negligible, so the system is extremely far from thermal equilibrium. Granular materials exhibit some solid-like and some liquid-like aspects; an open container of sand can support weight like a solid, and yet the sand can be poured, like a liquid. Large periodic vibrations can cause the appearance of convection rolls similar to those observed in liquids heated from below [6], yet there are important differences between convection in liquids and convection in sand [7-16]. Understanding how forces are distributed in stationary granular media is important not only as a first step towards elucidating dynamics, but also because of the insight it yields into failure mechanisms for materials such as composites. It is also important for other fields where yield stresses are crucial, for example in I E-mail: [email protected].
the problem of earthquake initiation. We have used experiments, simulations, and theory to characterize the large force inhomogeneities that are observed in stationary packs of spherical particles [17,18]. Fig. 1 is a photograph of a three-dimensional pack of beads which was subjected to a large external load, illuminated with polarized light, and viewed through crossed polarizers. The spheres were immersed in an indexmatched medium to avoid scattering from the bead surfaces. The stressed beads can be seen as the bright regions due to their stress-induced birefringence. We introduced and solved exactly a model which reproduces many aspects of these experiments and of computer simulations. In this model the inhomogeneities in the force distribution arise because of variations in the contact angles and the constraints imposed by force balance on each bead in the pack. The distribution of forces at a given depth converges to a "fixed function" as the depth is increased. The insight that the normalized weight distribution at a given depth is the crucial quantity allows quantitative comparison
0167-2789/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved PII S0167-2789(97)00085-7
S.N. Coppersmith/Physica D 107 (1997) 183-185
184
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Fig. 1. Optical image of force three-dimensional bead pack [17].
inhomogeneities in a
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,
I 0.5
,
I 1.0
,
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,
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Fig. 2. Comparison of force distribution at the bottom of simulation of sphere packing (points) and for simplified statistical model (solid line) [17].
between theory and experiment, since one can measure quantitatively the force distribution at the bottom of a bead pack. This distribution has a width of the same order of magnitude as the mean and decays exponentially at large forces (see Fig. 2). The force inhomogeneities in bead packs are intermediate between "usual" ~ fluctuations (where the size of the fluctuations relative to the mean decreases as the system size is increased) and "critical" fluctuations typical of, e.g., percolation networks (where the probability of obtaining a large force decays as
a power law rather than an exponential). A crucial simplification of the model is the assumption that the geometrical complexity of the granular material can be characterized statistically. This type of description leads to enormous analytical simplifications, so that the force fluctuations predicted by the model of a static system in a uniaxial geometry with a certain probability distribution of contact forces can be obtained exactly. As Fig. 2 demonstrates, this theory yields a force distribution function which agrees quantitatively with that obtained in a simulation of spheres interacting via Hertzian contacts. Neither our simulations nor the q-model captures all features of real bead packs. In our simulations, we have included only central forces and have ignored friction; the q-model ignores the vector nature of the forces, assuming that only the component along the direction of gravity plays a vital role. The qualitative consistency between the results obtained using the different methods provides some indication that the effects that we have neglected do not determine the main qualitative features of the force distribution at large v. The physical picture of the force fluctuations presented here is different from that proposed by Pavlovitch [19] who uses a picture of a percolation network of force chains with a buckling mechanism. In our theory, large fluctuations arise because of the variations in the contact angles combined with the constraints imposed by force balance on each bead. Still, the fluctuations (relative to the mean) are limited: they do not grow with the system size as critical fluctuations do. Thus, the force-chain patterns seen in Fig. 1 should not get increasingly tenuous with increasing system size. Nonetheless, these fluctuations are n o t self-averaging; they scale with depth in the same way as the mean rather than as its square root. They should have important effects not only on the static properties but also on the dynamical and flow characteristics of these materials.
References [1] H.M. Jaeger, J.B. Knight, C.-H. Liu and S.R. Nagel, Mat. Res. Soc. Bull. 19 (1994) 25. [2] H.M. Jaeger and S.R. Nagel, Science 255 (1992) 1523.
S.N. Coppersrnith/Physica D 107 (1997) 183-185 [3] D. Bideau and A. Hansen, Disorder and Granular Media (North-Holland, San Francisco, 1993). [4] C. Thornton, ed., Powder and Grains 93 (Balkema, Rotterdam, 1993). I511 A. Mehta, Granular Matter (Springer, Berlin, 1993). 161 M. Faraday, Philos. Trans. Roy. Soc. London 52 (1831) 299. [711 S.B. Savage, J. Fluid Mech. 194 (1988) 457. [8] P. Evesque and J. Rajchenbach, Phys. Rev. Lett. 62 (1989) 44. [9] E. Clement, J. Durand and J. Rajchenbach, Phys. Rev. Lett. 69 (1992) 1189. [10] P. Evesque, E. Szmatula and J.-P. Denis, Europhys. Lett. 12 (1990) 623. [11] J.B. Knight, H.M. Jaeger and S.R. Nagel, Phys. Rev. Lett. 70 (1993) 3728.
185
[12] Y.H. Taguchi, Phys. Rev. Lett. 69 (1992) 1367. [13] A. Gallas, H.J. Herrman and S. Sokolowski, Phys. Rev. Lett. 69 (1992) 1371. [14] H.K. Pak and R.P. Behringer, Phys. Rev. Lett. 71 (1993) 1832. [15] E.E. Ehrichs, H.M. Jaeger, G.S. Karczmar, J.B. Knight, V. Yu. Kuperman and S. R. Nagel, Science 267 (1995) 1632.
[16] M. Bourzutschky and J. Miller, Phys. Rev. Lett. 74 (1995) 2216. [17] C.-H. Liu, S.R. Nagel, D.A. Schecter, S.N. Coppersmith, S. Majumdar, O. Narayan and T.A. Witten, Science 269 (1995) 513. [18] S.N. Coppersmith, C.-H. Liu, S. Majumdar, O. Narayan and T.A. Witten, Phys. Rev. E 53 (1996) 4673. [19] A. Pavlovitch, unpublished.
PhysicaD 107 (1997) 183-185
Force fluctuations in granular media S.N. C o p p e r s m i t h 1 The James Franck Institute, The University of Chicago, Chicago, IL 60637, USA
Abstract
The forces in packed granular materials are distributed very differently than those in "normal" solids. Forces much larger than the mean occur, but are exponentially rare. These force inhomogeneities can be understood using an exactly solvable statistical model, in which the fluctuations in the forces arise because of variations in the contact angles together with the constraints imposed by the force balance on each grain. The details of this work are published elsewhere, see Liu et al. (1995) and Coppersmith et al. (1996). Keywords: Granular materials; Arching; Force fluctuations; Statistical mechanics
Granular materials are of interest not only because of their technological relevance but also because their complex dynamical properties pose a difficult scientific challenge [1-5]. Understanding them is difficult because thermal effects are negligible, so the system is extremely far from thermal equilibrium. Granular materials exhibit some solid-like and some liquid-like aspects; an open container of sand can support weight like a solid, and yet the sand can be poured, like a liquid. Large periodic vibrations can cause the appearance of convection rolls similar to those observed in liquids heated from below [6], yet there are important differences between convection in liquids and convection in sand [7-16]. Understanding how forces are distributed in stationary granular media is important not only as a first step towards elucidating dynamics, but also because of the insight it yields into failure mechanisms for materials such as composites. It is also important for other fields where yield stresses are crucial, for example in I E-mail: [email protected].
the problem of earthquake initiation. We have used experiments, simulations, and theory to characterize the large force inhomogeneities that are observed in stationary packs of spherical particles [17,18]. Fig. 1 is a photograph of a three-dimensional pack of beads which was subjected to a large external load, illuminated with polarized light, and viewed through crossed polarizers. The spheres were immersed in an indexmatched medium to avoid scattering from the bead surfaces. The stressed beads can be seen as the bright regions due to their stress-induced birefringence. We introduced and solved exactly a model which reproduces many aspects of these experiments and of computer simulations. In this model the inhomogeneities in the force distribution arise because of variations in the contact angles and the constraints imposed by force balance on each bead in the pack. The distribution of forces at a given depth converges to a "fixed function" as the depth is increased. The insight that the normalized weight distribution at a given depth is the crucial quantity allows quantitative comparison
0167-2789/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved PII S0167-2789(97)00085-7
S.N. Coppersmith/Physica D 107 (1997) 183-185
184
: ~ -,
>.
:
:
"
::N:
Fig. 1. Optical image of force three-dimensional bead pack [17].
inhomogeneities in a
10 0
10 "1
> O.. lO "~
10 "l 0.0
,
I 0.5
,
I 1.0
,
t 1.5
I 2.0
,
I 2.5
,
I 3.0
,
3.5
v
Fig. 2. Comparison of force distribution at the bottom of simulation of sphere packing (points) and for simplified statistical model (solid line) [17].
between theory and experiment, since one can measure quantitatively the force distribution at the bottom of a bead pack. This distribution has a width of the same order of magnitude as the mean and decays exponentially at large forces (see Fig. 2). The force inhomogeneities in bead packs are intermediate between "usual" ~ fluctuations (where the size of the fluctuations relative to the mean decreases as the system size is increased) and "critical" fluctuations typical of, e.g., percolation networks (where the probability of obtaining a large force decays as
a power law rather than an exponential). A crucial simplification of the model is the assumption that the geometrical complexity of the granular material can be characterized statistically. This type of description leads to enormous analytical simplifications, so that the force fluctuations predicted by the model of a static system in a uniaxial geometry with a certain probability distribution of contact forces can be obtained exactly. As Fig. 2 demonstrates, this theory yields a force distribution function which agrees quantitatively with that obtained in a simulation of spheres interacting via Hertzian contacts. Neither our simulations nor the q-model captures all features of real bead packs. In our simulations, we have included only central forces and have ignored friction; the q-model ignores the vector nature of the forces, assuming that only the component along the direction of gravity plays a vital role. The qualitative consistency between the results obtained using the different methods provides some indication that the effects that we have neglected do not determine the main qualitative features of the force distribution at large v. The physical picture of the force fluctuations presented here is different from that proposed by Pavlovitch [19] who uses a picture of a percolation network of force chains with a buckling mechanism. In our theory, large fluctuations arise because of the variations in the contact angles combined with the constraints imposed by force balance on each bead. Still, the fluctuations (relative to the mean) are limited: they do not grow with the system size as critical fluctuations do. Thus, the force-chain patterns seen in Fig. 1 should not get increasingly tenuous with increasing system size. Nonetheless, these fluctuations are n o t self-averaging; they scale with depth in the same way as the mean rather than as its square root. They should have important effects not only on the static properties but also on the dynamical and flow characteristics of these materials.
References [1] H.M. Jaeger, J.B. Knight, C.-H. Liu and S.R. Nagel, Mat. Res. Soc. Bull. 19 (1994) 25. [2] H.M. Jaeger and S.R. Nagel, Science 255 (1992) 1523.
S.N. Coppersrnith/Physica D 107 (1997) 183-185 [3] D. Bideau and A. Hansen, Disorder and Granular Media (North-Holland, San Francisco, 1993). [4] C. Thornton, ed., Powder and Grains 93 (Balkema, Rotterdam, 1993). I511 A. Mehta, Granular Matter (Springer, Berlin, 1993). 161 M. Faraday, Philos. Trans. Roy. Soc. London 52 (1831) 299. [711 S.B. Savage, J. Fluid Mech. 194 (1988) 457. [8] P. Evesque and J. Rajchenbach, Phys. Rev. Lett. 62 (1989) 44. [9] E. Clement, J. Durand and J. Rajchenbach, Phys. Rev. Lett. 69 (1992) 1189. [10] P. Evesque, E. Szmatula and J.-P. Denis, Europhys. Lett. 12 (1990) 623. [11] J.B. Knight, H.M. Jaeger and S.R. Nagel, Phys. Rev. Lett. 70 (1993) 3728.
185
[12] Y.H. Taguchi, Phys. Rev. Lett. 69 (1992) 1367. [13] A. Gallas, H.J. Herrman and S. Sokolowski, Phys. Rev. Lett. 69 (1992) 1371. [14] H.K. Pak and R.P. Behringer, Phys. Rev. Lett. 71 (1993) 1832. [15] E.E. Ehrichs, H.M. Jaeger, G.S. Karczmar, J.B. Knight, V. Yu. Kuperman and S. R. Nagel, Science 267 (1995) 1632.
[16] M. Bourzutschky and J. Miller, Phys. Rev. Lett. 74 (1995) 2216. [17] C.-H. Liu, S.R. Nagel, D.A. Schecter, S.N. Coppersmith, S. Majumdar, O. Narayan and T.A. Witten, Science 269 (1995) 513. [18] S.N. Coppersmith, C.-H. Liu, S. Majumdar, O. Narayan and T.A. Witten, Phys. Rev. E 53 (1996) 4673. [19] A. Pavlovitch, unpublished.