Philip J.
Tuckman
^{a},
Kyle
VanderWerf
^{a},
Ye
Yuan
^{bc},
Shiyun
Zhang
^{cd},
Jerry
Zhang
^{c},
Mark D.
Shattuck
^{e} and
Corey S.
O’Hern
*^{acfg}
^{a}Department of Physics, Yale University, New Haven, Connecticut 06520, USA
^{b}Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, China. E-mail: yuanyepeking@pku.edu.cn
^{c}Department of Mechanical Engineering and Materials Science, Yale University, New Haven, Connecticut 06520, USA
^{d}Department of Physics, University of Science and Technology of China, Hefei, Anhui 230026, China. E-mail: zsy12@mail.ustc.edu.cn
^{e}Benjamin Levich Institute and Physics Department, The City College of New York, New York, New York 10031, USA
^{f}Department of Applied Physics, Yale University, New Haven, Connecticut 06520, USA. E-mail: corey.ohern@yale.edu
^{g}Graduate Program in Computational Biology and Bioinformatics, Yale University, New Haven, Connecticut 06520, USA

Received
21st June 2020
, Accepted 1st September 2020

First published on 17th September 2020

We investigate the mechanical response of packings of purely repulsive, frictionless disks to quasistatic deformations. The deformations include simple shear strain at constant packing fraction and at constant pressure, “polydispersity” strain (in which we change the particle size distribution) at constant packing fraction and at constant pressure, and isotropic compression. For each deformation, we show that there are two classes of changes in the interparticle contact networks: jump changes and point changes. Jump changes occur when a contact network becomes mechanically unstable, particles “rearrange”, and the potential energy (when the strain is applied at constant packing fraction) or enthalpy (when the strain is applied at constant pressure) and all derivatives are discontinuous. During point changes, a single contact is either added to or removed from the contact network. For repulsive linear spring interactions, second- and higher-order derivatives of the potential energy/enthalpy are discontinuous at a point change, while for Hertzian interactions, third- and higher-order derivatives of the potential energy/enthalpy are discontinuous. We illustrate the importance of point changes by studying the transition from a hexagonal crystal to a disordered crystal induced by applying polydispersity strain. During this transition, the system only undergoes point changes, with no jump changes. We emphasize that one must understand point changes, as well as jump changes, to predict the mechanical properties of jammed packings.

Numerous theoretical and computational studies have focused on simplified descriptions of dry granular media, where they are modeled as packings of frictionless, purely repulsive spherical grains.^{8,9} These studies have provided significant insights into the jamming transition in packings of frictionless, spherical particles. Disordered packings of frictionless spherical particles are typically isostatic at jamming onset,^{10}i.e. they possess the same number of interparticle contacts N_{c} as the number of non-trivial degrees of freedom: N_{c} = N^{iso}_{c}, where N^{iso}_{c} = dN − d + 1 (for systems with periodic boundary conditions), N is the number of (non-rattler^{11}) grains, and d = 2, 3 is the spatial dimension. Ordered or compressed jammed packings can be hyperstatic with N_{c} ≥ N^{iso}_{c}.^{12} Each jammed packing exists in a local energy minimum in configuration space, and therefore possesses a percolating network of non-zero interparticle forces and nonzero bulk and shear moduli. In contrast, packings with fewer contacts than the isostatic value, N_{c} < N^{iso}_{c}, are unjammed and all interparticle forces are zero.^{13} Several studies have shown that isostatic jammed packings possess unique structural and mechanical properties, such as an excess number of low-frequency vibrational modes above the Debye prediction for the density of states^{14,15} and the power-law scaling of the shear modulus with increasing pressure.^{16}

In prior studies, we considered jammed packings of frictionless, spherical particles undergoing quasistatic deformation (i.e. steps of applied simple or pure shear strain with each step followed by energy minimization).^{17} During quasistatic deformation, grains in the packings undergo continuous motions along “geometric families”, in which the network of interparticle contacts does not change.^{18,19} The continuous geometric families are punctuated by particle rearrangements, which cause the contact networks to change. Such rearrangements determine the structural and mechanical properties of jammed packings. For example, particle rearrangements control the power-law scaling of the ensemble-averaged shear modulus as a function of pressure during isotropic compression.^{20} Prior studies of sheared particulate materials have shown that there are two types of changes in the contact networks.^{21} We refer to these contact network changes as (1) jump changes and (2) point changes. These previous studies also found that the relative frequency of jump and point changes is roughly constant with increasing system size.

In this work, we further investigate jump and point changes in the contact network and show that these two types of contact network changes occur during a wide range of quasistatic deformations in model granular materials. We carry out discrete element method simulations of purely repulsive, frictionless disks in 2D, focusing on several types of quasistatic deformations: simple shear strain, changes in the size polydispersity of the grains, and isotropic compression. For jump changes, jammed packings become mechanically unstable during quasistatic deformation,^{22} the particles rearrange, and as a result, the total energy, pressure, shear stress, and other thermodynamic quantities are discontinuous at the strain where the particle rearrangement occurs.^{23} At a point change, a contact is added or removed from the interparticle contact network at a given strain, but the particles do not move significantly. The positions of the particles are continuous with strain, but the derivatives of the particle positions with respect to strain are discontinuous. As a result, for point changes, the potential energy (in the case of strain applied at fixed packing fraction) or enthalpy (in the case of strain applied at fixed pressure) and their first derivatives are continuous as a function of strain.^{24} For repulsive linear spring interactions, second- and higher-order derivatives of the potential energy/enthalpy are discontinuous at a point change, while for Hertzian spring interactions, third- and higher-order derivatives of the potential energy/enthalpy are discontinuous. We illustrate the importance of point changes by starting with a perfectly ordered jammed disk packing, adding small increments of size polydisperity to the system, and minimizing the potential energy (at fixed packing fraction) or enthalpy (at fixed pressure). This system undergoes a series of point changes as it proceeds from a hyperstatic toward an isostatic state.^{25,26}

The remainder of the article is organized as follows. In Section 2, we describe the numerical methods that we use to generate disk packings at jamming onset and that we use to deform the jammed packings. In Section 3, we show results for the coordination number (z = 2N_{c}/N), total potential energy, shear stress, pressure, and other thermodynamic properties of jammed packings as a function of strain for each type of deformation, which allows us to illustrate point and jump changes. These studies are performed for both ordered packings of monodisperse disks and disordered packings of polydisperse disks. In Section 4, we summarize the conclusions and provide several possible future research directions including determining how point and jump changes separately contribute to the power-law scaling of the shear modulus with pressure during isotropic compression and investigating the effects of point changes in disk packings that interact via repulsive Hertzian spring interactions^{27} and in jammed systems containing frictional and non-spherical particles.

The disks interact via the following purely repulsive pair potential:

(1) |

Note that the Hertzian theory for the force between two contacting elastic spherical particles depends on the spatial dimension. The theory gives an exponent of α = 5/2 for the interaction energy between two elastic spheres in 3D and an exponent of α = 2 for the interaction between two parallel cylinders,^{29} which can mimic interactions between elastic disks in 2D. Thus, formally, “Hertzian” interactions between elastic disks should consider α = 2 in 2D, not α = 5/2. However, our goal was to investigate the effect of variations of the power-law exponent in eqn (1) on contact changes. Thus, we study both α = 2 and 5/2 for disk packings in 2D, and refer to the 5/2 exponent as the “Hertzian” value since this is value of the exponent in 3D.^{8}

To generate jammed packings, we first randomly place N disks in the simulation cell at small packing fraction ϕ_{0} ≈ 0.1. We set the particle diameters to be σ_{i} = 〈σ〉 + ηδ_{i}, where −0.5 ≤ δ_{i}/〈σ〉 ≤ 0.5 is uniformly distributed, 〈δ_{i}〉 = 0, is the standard deviation of the disk diameters, and defines the average diameter. For disordered packings, we employ a square box, whereas for crystalline packings, we employ a rectangular box with aspect ratio , which allows a hexagonal packing of contacting disks to fit in the simulation cell without any defects. We isotropically compress the system in small packing fraction steps, Δϕ, until the system develops a small nonzero pressure, . After each compression step, the total potential energy is minimized using the FIRE algorithm^{30} until the magnitude of the total net force on the disks, . We study the coordination number, total potential energy, pressure, shear stress, and elastic moduli in jammed packings as a function of the packing fraction and strain. We measure energy, stress, and force in units of ε, ε/〈σ〉^{2}, and ε/〈σ〉, respectively.

To understand the effects of jump and point changes in the interparticle contact networks, we consider jammed disk packings undergoing several types of quasistatic deformations: (1) simple shear at constant packing fraction, (2) simple shear at constant pressure, (3) increments of increasing size polydispersity at constant packing fraction, (4) increments of increasing size polydispersity at constant pressure, and (5) isotropic compression.

During the simple shear strain deformation, we calculate several quantities as a function of γ including the shear stress,

(2) |

(3) |

(4) |

The shear modulus can be decomposed into the affine and nonaffine contributions,^{32}G_{γ} = G^{a}_{γ} + G^{na}_{γ}, respectively. To calculate G^{a}_{γ}, we assume that all particles move according to the affine deformation, (x_{i}′,y_{i}′) = (x^{0}_{i} + γL_{x}y^{0}_{i}/L_{y},y^{0}_{i}). G^{na}_{γ} includes the nonaffine particle motion in response to potential energy minimization at fixed packing fraction and boundary strain. For repulsive linear spring interactions (α = 2 in eqn (1)), the affine contribution to the shear modulus can be calculated analytically,

(5) |

(6) |

(7) |

(8) |

As discussed for applied simple shear strain, G_{η} can also be decomposed into the affine and nonaffine contributions: G_{η} = G^{a}_{η} + G^{na}_{η}. For repulsive linear spring interactions, the affine contribution can be calculated analytically, which becomes

(9) |

(10) |

(11) |

(12) |

(13) |

At small η, the isostatic network in Fig. 2(a) has the lowest enthalpy of the three contact networks. At 1.190 < η_{1}* < 1.191, H of the configuration in (b) becomes less than that of the configuration in (a), and the system becomes hyperstatic with an additional interparticle contact. At a higher strain 1.191 < η_{2}* < 1.192, H for the configuration in (c) becomes less than that of the configuration in (b), and the system transitions to a different isostatic contact network. Most importantly, the particle positions do not change discontinuously during each point change. In other words, the contact change happens between two energy minimized configurations. In contrast, for jump changes, as shown in Fig. 1(d), the contact change occurs between a non-minimized configuration (point b_{1}) and a minimized configuration (point b_{2}).

The changes of the particle trajectories in Fig. 2(a)–(c) demonstrate the importance of point changes. If contact 2 did not form in panel (b), the two particles that form that contact would continue to move towards each other as they do in panel (a). These particle trajectories would cause a dramatic increase in enthalpy, as shown by H(η) for the first isostatic contact network in panel (d). However, due to the formation of the new contact, the particle trajectories are altered following the point change as shown in panel (c). Despite the continuous particle motion that occurs during point changes, the particle trajectories are significantly altered with further strain.

Fig. 3 displays the values of the polydispersity strain η_{1}* (η_{2}*) at which several example polydisperse N = 8 packings transition from an isostatic packing to a hyperstatic packing (and from the same hyperstatic packing to an isostatic packing) as a function of the target pressure p_{t}. For each packing, we find that both η_{1}* and η_{2}* are linear in p_{t} with vertical intercept η_{0} = η_{1,2}*(p_{t} = 0). In Fig. 3, we show that the values of η_{1,2}*, corresponding to when the packing either gains a contact or loses a contact, possess the same η_{0}. Thus, the width of the strain region over which the system is hyperstatic between the two successive point changes (first from an isostatic packing to a hyperstatic packing and then from the same hyperstatic packing to another isostatic packing) tends to zero in the zero-pressure limit. We find similar behavior for disk packings undergoing simple shear strain, as well as for larger system sizes.

Fig. 3 For three sample polydisperse N = 8 packings, we measure the polydispersity strain values at which the system transitions from an isostatic to a hyperstatic packing (η_{1}*) and from the same hyperstatic packing to another isostatic packing (η_{2}*) as shown in Fig. 2, at 10 target pressures p_{t}. We plot η_{1,2}*–η_{0}, where η_{0} = η_{1,2}*(p_{t} = 0), versus p_{t} for each contact change in each packing. The strain at which the packings transition from isostatic to hyperstatic, (i.e. between Fig. 2(a) and (b)), are represented by blue diamonds, red rightward triangles, and green downward triangles. The strains at which the packings transition from hyperstatic to isostatic, (i.e. between Fig. 2(b) and (c)) are represented by blue squares, red leftward triangles, and green upward triangles. Since all of the lines meet at η_{1,2}* = η_{0}, the width of the hyperstatic strain region tends to zero in the p_{t} = 0 limit. |

As an example, in Fig. 4, we show the enthalpy H as a function of simple shear strain γ − γ* for an N = 8 packing (with repulsive linear spring interactions) undergoing simple shear at fixed pressure. For the jump change at γ*, H is discontinuous. For the point change at γ*, H and dH/dγ (in the inset) are both continuous, but d^{2}H/dγ^{2} is discontinuous. The fact that the second derivative of the enthalpy, G_{γ} + p_{t}(d^{2}V/dγ^{2}), is discontinuous at a point change can be illustrated by analyzing the affine contribution of the shear modulus, G^{a}_{γ} in eqn (5), when contacts with zero overlap, r_{ij} → σ_{ij}, are added to or removed from the contact network. For the same reason, point changes give rise to discontinuities in the second derivatives with respect to strain of the potential energy/enthalpy for disk packings with repulsive linear spring interactions undergoing other applied strains.

In Fig. 5, we visualize polydisperse N = 8 disk packings in the packing fraction ϕ and simple shear strain γ plane. The color of a region indicates the type of contact network: regions that are red indicate isostatic contact networks and regions that are green indicate hyperstatic contact networks. Regions with different hues of red and green correspond to different contact networks. The white regions represent unjammed states. The lines provide contours of constant shear stress Σ_{γ}. Σ_{γ} is discontinuous at jump changes, whereas it is continuous at point changes.

The ϕ–γ landscape in Fig. 5 has two lines of point changes, which can be traversed by compressing or decompressing the packing at fixed γ, by applying simple shear strain at fixed ϕ, or by a combination of changes in ϕ and γ. The packing undergoes a point change when a contact is added (i.e. transitioning from an isostatic packing to a hyperstatic packing) or a contact is removed (i.e. transitioning from a hyperstatic packing to an isostatic packing). As discussed in Section 3.2, the two lines of point changes merge into a single point near (0.04, 0.81) in the zero-pressure limit. Traversing a point change in the forward direction leads to the same behavior as traversing it in the reverse direction.

Lines of jump changes in Fig. 5 occur when moving from an isostatic jammed region to an unjammed region. As we found for point changes, jump changes can be induced by compressing the packing at fixed γ, by applying simple shear strain at fixed ϕ, or by a combination of changes in ϕ and γ. When undergoing a jump change to an unjammed state, the total potential energy and shear stress drop discontinuously from a finite value to zero. In Fig. 5, there is also a line of jump changes between two different isostatic packings near (0.015, 0.81).

Note that in Fig. 5, the system can transition from a jammed packing to unjammed packing through isotropic compression. Indeed, in recent computational studies, we showed that “compression unjamming” occurs frequently near jamming onset. We also showed that the probability for compression unjamming (averaged over a finite range of strain) approaches a finite value in the large-system limit, and thus compression unjamming occurs in the large-system limit.^{20}

In Fig. 6, we show a portion of the packing fraction and polydispersity strain landscape for N = 8 disk packings. The lines provide contours of constant polydispersity stress Σ_{η}. Σ_{η} is discontinuous at jump changes, whereas it is continuous at point changes. In Fig. 6, there are two lines of point changes, which can be traversed by compressing or decompressing the packing at fixed η, by applying polydispersity strain at fixed ϕ, or by a combination of changes in ϕ and η. Again, the two lines of point changes merge into a single point near (0.135, 0.815) in the zero-pressure limit. We find one line of jump changes in Fig. 6 that can cause a transition between two isostatic packings, between a hyperstatic and an isostatic packing, and between two hyperstatic packings.

(14) |

In Fig. 8, we plot the ensemble-averaged excess coordination number 〈z − z_{iso}〉, where z_{iso} = 2N^{iso}_{c}/N, as a function of polydispersity strain η at fixed p_{t}. 〈z − z_{iso}〉 ≈ 2 at small η, and then begins to decrease toward zero at a characteristic η_{c}. As shown in the inset to Fig. 8, η_{c} ∼ p_{t} since 〈z − z_{iso}〉 collapses when plotted versus η/p_{t}. Thus, in the zero-pressure limit, the hexagonal crystal at ϕ = ϕ_{x} becomes isostatic with z = z_{iso} in the limit of zero applied strain.

We find similar behavior for the transition from a hexagonal crystal to a disordered crystal when we apply polydispersity strain at fixed packing fraction. In Fig. 9, we plot the total potential energy U and elastic modulus G_{η}versus η at fixed ϕ for an N = 16 packing initialized in a hexagonal crystal. We show that at each change in the contact network U is continuous, but G_{η} is discontinuous, which signals that the changes in the contact network are point changes. In Fig. 10, we show the ϕ–η landscape for an N = 16 packing initialized in a hexagonal crystal. There are many contact networks near the hexagaonal crystal, which are separated by point changes since there are no discontinuities in the polydispersity stress Σ_{η}. In the zero-pressure limit, all of the point changes coincide and the system transitions from a hexagonal network to an isostatic network at zero strain.

In principle, one can also use particle displacements (i.e. nonaffine particle motion) to identify changes in the contact networks.^{35} For example, one could apply polydispersity strain from η_{1} to η_{2} yielding particle positions (η_{1}) and (η_{2}), and then reverse the strain from η_{2} to η_{1} to measure the new particle positions ′(η_{1}). The particle displacements Δr = |(η_{1}) − ′(η_{1})| from this process will be large when there is a jump change between η_{1} and η_{2}, whereas Δr → 0 (in the small strain limit) for strain intervals where there is no jump change. Thus, measuring non-affine particle motions cannot be used to identify point changes. For this reason, we recommend measurements of ΔG and ΔU to identify point and jump changes in particulate media.

The fact that point changes cause discontinuities with respect to strain in the second derivative of the potential energy/enthalpy (for disk packings with repulsive linear spring interactions) stems from the shape of the interparticle potential energy (eqn (1)). The purely repulsive linear spring potential has a discontinuity in d^{2}U/dr_{ij}^{2} across a point change, and thus the elastic moduli, G_{γ}, G_{η}, and B, are discontinuous across a point change. For the purely repulsive Hertzian spring potential with α = 5/2 in eqn (1), d^{3}U/dr_{ij}^{3} is discontinuous across a point change, and thus the derivatives of the elastic moduli with respect to strain (not the moduli themselves) are discontinuous. The discontinuities caused by point changes will occur in higher-order derivatives of the potential energy (when the strain is applied at constant packing fraction) if higher-order derivatives of the interparticle potential are continuous. Similar results are found for the derivatives of the enthalpy when the strain is applied at fixed pressure.

These results raise several important questions for future research. First, how do jammed packings behave when the applied strain is reversed^{36–38} after point and jump changes occur in the interparticle contact networks? Point changes are completely reversible, since the particle motions are continuous during a point change. Jump changes, however, are not reversible in this way. As shown in Fig. 1, the packing immediately after the jump change has a lower potential energy (in the case of applied strain at constant packing fraction) than the packing immediately before the jump change. Thus, when the strain is reversed after the jump change, the system will follow a different path in the energy landscape (than the one followed during the forward strain). However, it is possible that the system can undergo a series of point changes or another jump change during the reversed strain and return to the path in the energy landscape that was traversed during the forward strain. This behavior was termed “loop reversibility” in ref. 39 and “limit cycle” behavior in ref. 40, both of which studied systems undergoing cyclic simple shear strain.

In recent studies,^{20} we found that changes in the contact network during isotropic compression of jammed packings give rise to the power-law scaling of the shear modulus with pressure, i.e. G_{γ} ∼ p^{1/2} for repulsive linear spring interactions in d = 2 and 3. Since both point and jump changes cause jumps in the shear modulus, ΔG_{γ}, an interesting question is to determine whether point changes, jump changes, or both contribute significantly to the increase in the shear modulus during isotropic compression. In addition, G_{γ} ∼ p^{2/3} for Hertzian spring interactions undergoing isotropic compression in d = 2 and 3.^{8} In future studies, we will investigate how jump and point changes give rise to this behavior, given that point changes do not cause discontinuities in G_{γ} for Hertzian interactions.

To understand the mechanical response of jammed packings to applied strain, one must be able to predict the potential energy (and other physical quantities that depend on the particle positions) as the system evolves along geometrical families, as well as across point and jump changes. We emphasize that it is still important to study point changes in packings undergoing quasistatic deformation even if the interparticle potential does not possess discontinuities in its derivatives. Even if there are no discontinuities in the interparticle potential, the particle trajectories change directions when the system undergoes each point change, which influences the evolution of the potential energy, stress, and elastic moduli as a function of strain.

Another possible extension of the current studies is to investigate how point changes behave in packings of non-spherical particles. For example, in packings of circulo-lines in 2D, two particles with an “end–end” contact behave differently than two particles with an “end–middle” contact.^{41} It will be interesting to study packings of circulo-lines that transition between these two types of contacts and determine whether this process can be described as a generalized point change, even though the interparticle contact network does not change.

A similar effect can occur in packings of spherical particles with frictional interactions. Numerous studies have shown that in addition to the number of contacts per particle, the ratio of the tangential to the normal force, ζ_{ij}, at each contact between particles i and j, plays an important role in determining the mechanical stability of frictional packings.^{42} Thus, it is possible that effective “point changes” can occur if ζ_{ij} varies significantly during strain even though particles i and j remain in contact.

Nevertheless, in Fig. 13, we show similar data as in Fig. 11, except for packings of N = 64, 128, and 256 disks undergoing simple shear (with step size Δγ = 7 × 10^{−13}) at fixed packing fraction ϕ = 0.88. Again, we observe that there are three clusters of data points: one for jump changes (large |ΔU|/N and large |ΔG_{γ}|), one for point changes (small |ΔU|/N and large |ΔG_{γ}|), and one for the control group (small |ΔU|/N and small |ΔG_{γ}|), for which point and jump changes do not occur. More importantly, we find that the location and spread of each of the three clusters remain the same for the three system sizes.

In Fig. 14, we show the same plot as in Fig. 13 for the three system sizes N = 64, 128, and 256, except using a larger shear strain step size Δγ = 10^{−11}. The data points for |ΔG_{γ}| and |ΔU|/N corresponding to jump changes remain the same for the two shear strain step sizes. For the data points that correspond to point changes, the values of |ΔU|/N change with the shear strain step size, but the values of |ΔG_{γ}| do not. In addition, for the points that do not correspond to changes in the contact network, both |ΔU|/N and |ΔG_{γ}| shift to larger values with the larger shear strain step size. Thus, |ΔU|/N → 0 and |ΔG_{γ}| → 0 in the limit Δγ → 0 for data points that do not correspond to changes in the contact network.

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