Comparison of shear and compression jammed packings of frictional disks

Fansheng Xiong; Philip Wang; Abram H. Clark; Thibault Bertrand; Nicholas T. Ouellette; Mark D. Shattuck; Corey S. O’Hern

doi:10.1007/s10035-019-0964-9

Granular Matter

November 2019, 21:109 | Cite as

Comparison of shear and compression jammed packings of frictional disks

Authors
Authors and affiliations

Fansheng Xiong
Philip Wang
Abram H. Clark
Thibault Bertrand
Nicholas T. Ouellette
Mark D. Shattuck
Corey S. O’Hern

Original Paper

First Online: 29 October 2019

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In Memoriam of Robert P. Behringer, late Editor in Chief of Granular Matter

Abstract

We compare the structural and mechanical properties of mechanically stable (MS) packings of frictional disks in two spatial dimensions (2D) generated with isotropic compression and simple shear protocols from discrete element modeling (DEM) simulations. We find that the average contact number and packing fraction at jamming onset are similar (with relative deviations $< 0.5\%$) for MS packings generated via compression and shear. In contrast, the average stress anisotropy $\langle {{\hat{\varSigma}}}_{xy} \rangle = 0$ for MS packings generated via isotropic compression, whereas $\langle {{\hat{\varSigma}}}_{xy} \rangle >0$ for MS packings generated via simple shear. To investigate the difference in the stress state of MS packings, we develop packing-generation protocols to first unjam the MS packings, remove the frictional contacts, and then rejam them. Using these protocols, we are able to obtain rejammed packings with nearly identical particle positions and stress anisotropy distributions compared to the original jammed packings. However, we find that when we directly compare the original jammed packings and rejammed ones, there are finite stress anisotropy deviations $\varDelta {{\hat{\varSigma }}}_{xy}$. The deviations are smaller than the stress anisotropy fluctuations obtained by enumerating the force solutions within the null space of the contact networks generated via the DEM simulations. These results emphasize that even though the compression and shear jamming protocols generate packings with the same contact networks, there can be residual differences in the normal and tangential forces at each contact, and thus differences in the stress anisotropy.

Keywords

Granular materials Jamming Friction Shear jamming Force chains

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Notes

Acknowledgements

The authors (P.W., N.T.O. and C.S.O.) thank the Army Research Laboratory for supporting the research carried out in this work. This work also benefited from the facilities and staff of the Yale University Faculty of Arts and Sciences High Performance Computing Center. P.W. and F.X. contributed equally to the paper.

Funding

This research was sponsored by the Army Research Laboratory under Grant No. W911NF-17-1-0164 (P.W., N.T.O., and C.S.O.). Computing resources were provided by the Army Research Laboratory Defense University Research Instrumentation Program Grant No. W911NF-18-1-0252. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein. In addition, we acknowledge support from the China Scholarship Council Grant No. 201806210283 (F. X.).

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Conflict of interest

The authors declare that they have no conflict of interest.

Appendix 1: Normal stress difference in jammed and rejammed packings

In this Appendix, we describe the results for the normal stress difference for jammed packings of frictional disks generated via simple shear and isotropic compression. In Fig. 12, we show the probability distribution $P(\hat{\varSigma }_{N})$ of the normal stress difference for both shear and comrpession jammed packings with $N=64$ at $\mu =0$, 0.1, and 1.0. Simple shear and isotropic compression do not strongly couple to ${{\hat{\varSigma }}_N}$ and thus we find that $P({{\hat{\varSigma }}}_N)$ is a Gaussian distribution centered at zero with a width that depends on $\mu$. We also calculated $\varDelta {{\hat{\varSigma }}}_N$, which is the rms deviation in the normal stress difference between the rejammed and original jammed packings for protocols A and B. In Fig. 13, we show that (as for the stress anisotropy), the rms deviation in the normal stress difference $\varDelta {{\hat{\varSigma }}}_N$ is typically larger between shear jammed packings and compression rejammed packings, compared to that between compression jammed packings and shear rejammed packings.

Fig. 12
Probability distribution of the normal stress difference $P({{\hat{\varSigma }}}_{N})$ for jammed packings generated via isotropic compression (open symbols) and simple shear (closed symbols) for $N=64$ and friction coefficients $\mu =0$ (triangles), 0.1 (circles), and 1.0 (squares). The distributions were obtained from more than $10^3$ independently generated jammed packings. The solid lines are Gaussian distributions with zero mean and standard deviations $\varDelta \approx 0.091$, 0.093, and 0.114 for $\mu =0$, 0.1, and 1.0, respectively

Fig. 13
The frequency distribution $p(\varDelta {\hat{\varSigma }}_{N})$ of the root-mean-square deviation in the normal stress difference between shear jammed packings and those generated using protocol A (triangles) and between compression jammed packings and those generated using protocol B (circles)

Appendix 2: Parabolic geometrical families

Geometrical families are collections of jammed packings that share the same interparticle contact network at different values of the packing fraction at jamming onset $\phi _J$ and either pure or simple shear strain $\gamma$. In this Appendix, we present a derivation of the relation between $\phi _J$ and $\gamma$ for a simple example of jammed packings of two hard disks (one small disk with diameter $\sigma _s$ and one large disk with diameter $\sigma _l$) undergoing pure shear strain $\gamma =\ln \left( \frac{L_{x}}{L_{y}}\right)$ in a box with side walls with lengths $L_x$ and $L_y$ in the x- and y-directions. We first express the box lengths $L_{x}=\sigma _{ls}(1+\cos \theta )$ and $L_{y}=\sigma _{ls}(1+\sin \theta )$ in terms of the angle $\theta$ between the horizontal axis of the box and ${\vec {r}}_{ij}$ connecting the centers of the disks. See Fig. 14a. Thus, the pure shear strain satisfies

$$\begin{aligned} \gamma =\ln \left( \frac{1+\cos \theta }{1+\sin \theta }\right) . \end{aligned}$$

(B1)

In addition, we can write the jammed packing fraction as

$$\begin{aligned} \phi _J=\frac{\phi _1}{(1+\cos \theta )(1+\sin \theta )}, \end{aligned}$$

(B2)

where

$$\begin{aligned} \phi _{1}=\frac{\pi (\sigma _s^{2}+\sigma _l^{2})}{(\sigma _s+\sigma _l)^{2}}. \end{aligned}$$

(B3)

Using Eqs. B1 and B2, we can eliminate the dependence on $\theta$ and write $\phi _J$ in terms of $\gamma$:

$$\begin{aligned} \phi _J=\phi _{1} \left[ \sqrt{2}-2\cosh \left( \frac{\gamma }{2}\right) \right] ^{2}. \end{aligned}$$

(B4)

Equation B4 is similar to a parabola, which can be seen by expanding it in powers of $\gamma -\gamma _0$ about the minimum at $\gamma _0=0$ and retaining terms up to $(\gamma -\gamma _0)^{2}$:

$$\begin{aligned} \phi _J=\phi _{0} + A (\gamma -\gamma _0)^{2}, \end{aligned}$$

(B5)

where $\phi _0 = (6-4\sqrt{2})\phi _1$ and $A=\left( 1 - \frac{\sqrt{2}}{2}\right) \phi _1$. We carried out discrete element method simulations to generate jammed packings at each value of the pure shear strain $\gamma$ for two bidisperse hard disks. The DEM results for $\phi _J(\gamma )$ agree with the analytical result as shown in Fig. 14. For small systems, a single geometrical family can exist over a wide range of strain. Also, for pure shear in small systems, the parabolic geometrical family is centered on $\gamma =0$. In contrast, for simple shear of two hard bidisperse disks (in fixed wall boundary conditions), the parabolic geometrical family is not centered on $\gamma =0$ as shown in Fig. 15.

Fig. 14
a Illustration of a jammed packing of two bidisperse disks i and j in a simulation cell with side lengths $L_x$ and $L_y$ in the x- and y-directions and periodic boundary conditions. $\theta$ gives the angle between the center-to-center separation vector ${\vec {r}}_{ij}$ and the x-axis. b The packing fraction at jamming onset $\phi _J$ versus the pure shear strain $\gamma$ for packings of two bidisperse disks obtained from Eq. B4 (solid blue line) and DEM simulations (open triangles). The jammed packing fraction $\phi _0$ and pure shear strain $\gamma _0$ at the minimum and the curvature A of the parabola are given in the main text (color figure online)

Fig. 15
The packing fraction at jamming onset $\phi _J$ versus the simple shear strain $\gamma$ for packings of two bidisperse disks obtained from DEM simulations (open triangles). We also show a fit to a parabolic form, $\phi _J = \phi _0 + A(\gamma -\gamma _0)^2$, where $\phi _0 \approx 0.49$, $A \approx 0.34$, and $\gamma _0 \approx 1.03$ (solid blue line) (color figure online)

Appendix 3: Sensitivity of results on numerical parameters of packing-generation protocols

In this Appendix, we investigate how the ability to generate the original jammed packing from the rejamming protocols A and B depends on parameters associated with the packing-generation protocols. When determining the packing fraction at jamming onset, we seek particle configurations for which the total potential per particle U/N is nonzero, but small, i.e. $U_{\mathrm{tol}}< U/N <2 U_{\mathrm{tol}}$ and $U_{\mathrm{tol}} = 10^{-16}$ for the results provided in the main text. To better understand the sensitivity of our results on $U_\mathrm{tol}$, we calculate the frequency distribution for rms deviations in the positions between shear jammed packings and compression rejammed packings as a function of $U_{\mathrm{tol}}$. For $U_{\mathrm{tol}}=10^{-16}$, $p(\varDelta r)$ is narrow with a peak near $\varDelta r \approx 10^{-12}$ as shown in Fig. 16. However, for $U_{\mathrm{tol}} =10^{-14}$ and $10^{-12}$, $p(\varDelta r)$ broadens dramatically, with non-zero probability between $\varDelta r = 10^{-11}$ and $10^{-7}$. These results emphasize that it is more difficult to recover the original jammed packing for packings that are over-compressed because over-compressed packings are further from the unjammed state, increasing the likelihood that the system can find a pathway to another jammed configuration during the re-jamming process.

Does the stress anisotropy for shear jammed and compression re-jammed packings (or for compression jammed and shear re-jammed packings) differ for systems at $\mu =0$? In general, the stress anisotropy distributions are similar for jammed and re-jammed packings, but the precise values of the stress anisotropy can differ for each original jammed packing and its rejammed counterpart. We find that the average value of the stress anisotropy difference $\langle \varDelta {{\hat{\varSigma }}}_{xy} \rangle \approx 10^{-5}$ (between shear jammed and compression re-jammed packings) for $\mu =0.1$, but $\langle \varDelta {{\hat{\varSigma }}}_{xy} \rangle \approx 10^{-7.5}$ is much lower for $\mu =0$. (See Fig. 17.) In contrast to the results for $\mu >0$, $\langle \varDelta {{\hat{\varSigma }}}_{xy} \rangle$ for $\mu =0$ scales to zero with the degree to which the simulations can maintain force balance.

Fig. 16
Frequency distribution $p(\varDelta r)$ of the root-mean-square deviations in the positions of the disks between shear jammed packings and those generated using protocol A for $U_{\mathrm{tol}}$ = $10^{-16}$ (black triangles), $10^{-14}$ (blue circles), and $10^{-12}$ (red squares), N=16 and $\mu$=0.1 (color figure online)

Fig. 17
The frequency distribution $p(\varDelta {\hat{\varSigma }}_{xy})$ of the root-mean-square deviations in the stress anisotropy between shear jammed packings and those generated using protocol A for $N = 16$ and $\mu = 0$

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Authors and Affiliations

Fansheng Xiong
- 1
- 2
Philip Wang
- 2
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Abram H. Clark
- 3
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Thibault Bertrand
- 4
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Nicholas T. Ouellette
- 5
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Mark D. Shattuck
- 6
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Corey S. O’Hern
- 2
- 7
- 8
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1.Zhou Pei-Yuan Center for Applied MathematicsTsinghua UniversityBeijingChina
2.Department of Mechanical Engineering and Materials ScienceYale UniversityNew HavenUSA
3.Department of PhysicsNaval Postgraduate SchoolMontereyUSA
4.Department of MathematicsImperial College LondonLondonUK
5.Department of Civil and Environmental EngineeringStanford UniversityStanfordUSA
6.Benjamin Levich Institute and Physics DepartmentThe City College of the City University of New YorkNew YorkUSA
7.Department of PhysicsYale UniversityNew HavenUSA
8.Department of Applied PhysicsYale UniversityNew HavenUSA

Comparison of shear and compression jammed packings of frictional disks

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