ENAbstract. Recent simulations have predicted that near jamming for collections of spherical particles, there will be a discontinuous increase in the mean contact number Z at a critical volume fraction φ_c①. Above φ_c, Z and the pressure P are predicted to increase as power laws in φ − φ_c. In experiments using photoelastic disks we corroborate a rapid increase in Z at φ_c and power-law behavior above φ_c for Z and P. Specifically we find a power-law increase as a function of φ − φ_c for Z − Z_c with an exponent β around 0.5, and for P with an exponent around 1.1②. These exponents are in good agreement with simulations. We also find reasonable agreement with a recent mean-field theory for frictionless particles.

中译摘要。近期模拟预测，球形颗粒体系临近堵塞时，平均接触数 Z 会在临界堆积分数 φ_c 处不连续跃增①；φ > φ_c 后，Z 与压强 P 随 φ − φ_c 呈幂律增长。光弹圆盘实验支持 Z 在 φ_c 附近快速上升，并支持其上方 Z 与 P 的幂律标度。具体而言，Z − Z_c 的指数β 约为 0.5，而 P 的指数 ψ 约为 1.1②。这些指数与模拟结果相容，数据也与近期针对无摩擦颗粒的平均场理论大致吻合。

ENA solid, in contrast to a fluid, is characterized by mechanical stability that implies a finite resistance to shear and isotropic deformation. While such stability can originate from long-range crystalline order, there is no general agreement on how mechanical stability arises for disordered systems, such as molecular and colloidal glasses, gels, foams, and granular packings [1]. For a granular system, in particular, a key question concerns how stability occurs when the packing fraction φ increases from below to above a critical value φ_c for which there are just enough contacts per particle Z to satisfy the conditions of mechanical stability③. In recent simulations on frictionless systems it was found that Z exhibits a discontinuity at φ_c followed by a power-law increase for φ > φ_c [2–5]. The pressure is also predicted to increase as a power law above φ_c.

中译固体与流体的区别在于力学稳定性——即对剪切形变和各向同性形变具有有限的抵抗力。虽然这种稳定性可以源自长程晶体序，但对于无序系统（如分子与胶体玻璃、凝胶、泡沫以及颗粒堆积 [1]），力学稳定性如何产生尚无普遍共识。特别是对颗粒系统，一个关键问题是：当堆积分数 φ 从下方增大并越过临界值 φ_c（此时每个颗粒恰好拥有满足力学稳定条件所需的接触数 Z③）时，稳定性是如何出现的。在近期的无摩擦系统模拟中发现，Z 在 φ_c 处表现出不连续性，随后在 φ > φ_c 时呈幂律增长 [2–5]；压强 P 也被预测在 φ_c 之上按幂律增长。

ENA number of recent theoretical studies address jamming, and we note work that may be relevant to granular systems. Silbert, O’Hern et al. have shown in computer simulations of frictionless particles [2–4] that (a) for increasing φ, Z increases discontinuously at the transition point from zero to a finite number, Z_c, corresponding to the isostatic value needed for mechanical stability; (b) for both two- and three-dimensional systems, Z is expected to continue increasing as (φ − φ_c)^β above φ_c, where β ≈ 0.5; and (c) pressure P is expected to grow as (φ − φ_c)^ψ, where ψ = α_f − 1 ≈ 1 in the simulations④, and α_f is the exponent of the interparticle potential. More recent simulations by Donev et al. for hard spheres in three dimensions found β = 0.6 in maximally random jammed packings [5]. It is interesting to note that a model for foam exhibits quite similar behavior for Z [6]⑤. Henkes and Chakraborty [7] constructed a mean-field theory of the jamming transition in 2D based on entropy arguments. They predict power-law scaling for P and Z in terms of a variable α, the pressure derivative of entropy. Eliminating α yields an algebraic relation between P and Z − Z_c, which we compare with the data below.

中译多项近期理论研究讨论了堵塞问题。Silbert、O’Hern 等人对无摩擦颗粒的模拟 [2–4] 表明：(a) 随 φ 增大，Z 在转变点由零不连续跃增到与等静定条件相应的有限值 Z_c；(b) 在二维和三维中，φ > φ_c 时 Z 继续按 (φ − φ_c)^β 增长，β ≈ 0.5；(c) 压强按 (φ − φ_c)^ψ 增长，模拟中ψ = α_f − 1 ≈ 1④，其中 α_f 是颗粒间相互作用势的幂指数。Donev 等人对三维硬球最大随机堵塞堆积得到 β = 0.6 [5]。泡沫模型中的 Z 也表现出相近行为 [6]⑤。Henkes 与 Chakraborty [7] 基于熵建立了二维堵塞转变的平均场理论，以熵对压强的导数 α 表示 P 与 Z 的幂律标度；消去 α 后可得到 P 与 Z − Z_c 的代数关系，作者将在下文与实验数据比较。

ENWhile the simulations agree among themselves at least qualitatively, so far, these novel features have not been identified in experiments⑥. Hence, it is crucial to test these predictions experimentally. In the following, we present experimental data for Z and P vs φ, based on a method that yields accurate determination of the contacts and identifies power laws in Z and P for a two-dimensional experimental system of photoelastic disks. By measuring both P and Z, we can also obtain a sharper value for the critical packing fraction φ_c, for the onset of jamming, and we can test the model of Henkes and Chakraborty.

中译尽管这些模拟彼此之间至少在定性上一致，但迄今为止，这些新颖特征尚未在实验中被辨识出来⑥。因此，对这些预测进行实验检验至关重要。下文中，我们基于一种能够精确判定接触的方法，给出二维光弹圆盘实验系统中 Z 与 P 随 φ 变化的数据，并识别出 Z 与 P 中的幂律。通过同时测量 P 与 Z，我们还能更精确地确定堵塞起始的临界堆积分数 φ_c，并对 Henkes 与 Chakraborty 的模型进行检验。

ENThe relevant simulations have been carried out predominantly for frictionless particles. For real frictional particles there will clearly be some differences. For instance, in the isostatic limit, Z equals 4 for frictionless disks, whereas for frictional disks, Z is around 3, depending on the system details [8]. Other predictions such as specific critical exponents may also need modification. However, one might hope that the observed experimental behavior, in particular, critical exponents, might be similar to that for frictionless particles if the frictional forces are typically small relative to the normal forces. Indeed, in recent experiments, the typical intergrain frictional forces in a physical granular system were found to be only about 10% of the normal forces [9]⑦.

中译相关模拟主要是针对无摩擦颗粒进行的，而真实的摩擦颗粒显然会带来一些差异。例如，在等静定极限下，无摩擦圆盘的 Z 等于 4，而摩擦圆盘的 Z 约为 3（取决于系统的具体细节）[8]；其他预测（如具体的临界指数）也可能需要修正。不过，如果摩擦力通常远小于法向力，则可以期望实验观测到的行为——尤其是临界指数——与无摩擦颗粒的情形相近。事实上，近期实验发现，真实颗粒系统中典型的颗粒间摩擦力仅约为法向力的 10% [9]⑦。

ENFigure 1(a) shows a schematic of the apparatus. We use a bidisperse mixture (80% small and 20% large particles)⑧ of approximately 3000 polymer (PSM-4) photoelastic (birefringent under stress) disks⑨ with diameter 0.74 or 0.86 cm. This ratio preserves a disordered system. The disks have Young’s modulus of 4 MPa, and a static coefficient of friction of 0.85. The model granular system is confined in a biaxial test cell (42 cm × 42 cm with two movable walls) which rests on a smooth Plexiglas sheet. The displacements of the walls can be set very precisely with stepper motors. The linear displacement step size used in this experiment is 40 μm, which is approximately 0.005D, where D is the average diameter of the disks. The deformation δ per particle is less than 1% in the compressed state. The setup is horizontal and placed between crossed circular polarizers. It is imaged from above with an 8 MP CCD color camera which captures roughly 1200 disks in the center of the cell, enabling us to visualize the stress field within each disk (Fig. 1). We then obtain good measurements of the vector contact forces (normal and tangential — frictional — components) [9].

中译图 1(a) 为实验装置示意图。我们采用约 3000 个聚合物（PSM-4）光弹圆盘（受应力时产生双折射）⑨的双分散混合物（80% 小颗粒、20% 大颗粒）⑧，直径分别为 0.74 cm 与 0.86 cm；这一比例可使系统保持无序。圆盘的杨氏模量为 4 MPa，静摩擦系数为 0.85。该模型颗粒系统被约束在一个双轴试验腔室中（42 cm × 42 cm，两面壁可移动），腔室置于一块光滑的有机玻璃（Plexiglas）底板上。壁的位移可由步进电机非常精确地设定；本实验所用线性位移步长为 40 μm，约为 0.005D（D 为圆盘平均直径）。压缩态下每个颗粒的形变 δ 小于 1%。装置水平放置于一对正交圆偏振片之间，从上方用 800 万像素 CCD 彩色相机成像，可拍摄到腔室中心约 1200 个圆盘，使我们能够观察每个圆盘内部的应力场（图 1）。由此，我们可以很好地测量矢量接触力（法向分量与切向即摩擦分量）[9]。

ENWe also use the particle photoelasticity to accurately determine the presence or absence of contacts between particles. In numerical studies one can use a simple overlap criterion to determine contacts: a contact occurs if the distance between particle centers is smaller than the sum of the particle radii. However, in experimental systems, a criterion based solely on the particle centers is susceptible to relatively large errors which include false positives [Fig. 1(b) (squares)] as well as false negatives (circles)⑩. As seen in Fig. 1(b), the contacts through which there is force transmission appear as source points for the stress pattern. Further details are given in Sec. I of the supplementary material [10].

中译我们还利用颗粒的光弹性来精确判定颗粒之间是否存在接触。在数值研究中，可以用简单的重叠判据来确定接触：若两颗粒中心间距小于两颗粒半径之和，即发生接触。然而在实验系统中，仅依据颗粒中心的判据容易产生较大的误差，包括假阳性 [图 1(b) 中的方块] 与假阴性（圆圈）⑩。如图 1(b) 所见，传递力的接触表现为应力图样的“源点”。更多细节见补充材料第 I 节 [10]。

ENWe use two protocols to produce different packing fractions: we either compress the system from an initially stress-free state, or decompress the system until the end state is essentially a stress-free state. The results for both protocols are the same within error bars above φ_c; below jamming, the data for Z obtained by compression are a few percent below those for decompression. Below, we will present decompression data. Figures 1(d) and 1(e) show the initial highly stressed state and the end state after decompression, respectively. After each decompression step, we apply tapping to relax stress in the system⑪. This could be seen as roughly analogous to the annealing process invoked in some simulations. Two images are captured at each state: one without polarizers to determine the disk centers and one with polarizers to record the stress.

中译我们采用两种实验规程来产生不同的堆积分数：要么从初始无应力态压缩系统，要么让系统卸压直至末态基本处于无应力态。在 φ_c 之上，两种规程的结果在误差范围内相同；在堵塞以下，压缩方式得到的 Z 数据比卸压方式低几个百分点。下文将给出卸压数据。图 1(d) 与 1(e) 分别显示初始的高应力态和卸压后的末态。每一步卸压之后，我们对系统施加轻敲（tapping）以弛豫其中的应力⑪；这大致类似于某些模拟中引入的退火过程。在每个状态采集两幅图像：一幅不带偏振片，用于确定圆盘中心；另一幅带偏振片，用于记录应力。

ENThe average Z can be computed either by counting only the force bearing disks or by counting all the disks including rattlers which do not contribute to the mechanical stability of the system. We consider as rattlers, all the disks which have less than two contacts⑫. For the number of rattlers beyond the transition point we find an exponential decrease with φ − φ_c; hence, a divergence in the number of rattlers at φ_c is not indicated by the data.

中译平均接触数 Z 既可只统计承力颗粒，也可统计包括游离颗粒（rattlers，即不属于维持整体稳定的承力骨架者）在内的全部圆盘。本文把接触数少于 2 的圆盘定义为 rattler⑫。转变点以上，rattler 数量随 φ − φ_c 指数衰减；数据没有显示其数量在 φ_c 处发散。

ENWe next compute the Cauchy stress tensor for each disk, σ_ij = (1/2A) Σ_c(F_ix_j + F_jx_i), with P = tr(σ). Here F_i and x_j are components of the contact force and branch vector, respectively; A is the Voronoi area of the disk⑬, and the sum runs over that disk’s contacts. We then average the particle pressure over the observed disks. For the data presented below, we performed two sets of experiments: a broad scan over 0.8390 ≤ φ ≤ 0.8650 with step size δφ = 0.016⑭, followed by a finer scan over 0.840745 ≤ φ ≤ 0.853312 with δφ = 0.000324 after the jamming region had been identified. The inset in Fig. 2 shows Z over a broad range of φ, with rattlers included (stars) or excluded (squares). These data show a significant rise in Z at the jamming transition.

中译接下来计算每个圆盘的柯西应力张量：σ_ij = (1/2A) Σ_c(F_ix_j + F_jx_i)，并按 P = tr(σ) 定义压强。其中 F_i 与 x_j 分别为接触力和支矢量的分量；A 是该圆盘的 Voronoi 胞元面积⑬，求和遍及该圆盘的所有接触。随后在观测区域的颗粒集合上对颗粒压强取平均。作者进行了两组实验：先在 0.8390 ≤ φ ≤ 0.8650 范围内以 δφ = 0.016 作粗扫描⑭；确定堵塞区间后，再在 0.840745 ≤ φ ≤ 0.853312 内以 δφ = 0.000324 作精细扫描。图 2 插图给出较宽 φ 范围内的 Z，并分别统计包含 rattlers（星形）和排除 rattlers（方块）的结果。

ENWhile this rise is not sharply discontinuous, it occurs over a very small range in φ. At higher φ, the variations of the curves are similar with and without rattlers. At lower φ, their behavior differs: The values of Z drop lower for the case with rattlers. The pressure P(φ − φ_c) in Fig. 2 shows a flat background below jamming, and then a sharp positive change in slope at a well defined φ. The pressure is not identically zero below jamming for similar reasons that the jump in Z is not perfectly sharp, as discussed below.

中译虽然这一上升并非尖锐的不连续跃变，但它发生在非常小的 φ 范围内。在较高 φ 处，含与不含 rattler 的曲线变化相似；在较低 φ 处，二者行为不同：含 rattler 时 Z 的数值降得更低。图 2 中的压强 P(φ − φ_c) 在堵塞以下呈现平坦的背景，随后在一个明确限定的 φ 处斜率发生急剧的正向变化。堵塞以下压强并非严格为零，其原因与 Z 的跃变不完全尖锐的原因相似，将在下文讨论。

ENTo compare these experimental results with predictions above φ_c, we perform least-squares fits of Z − Z_c and P against φ − φ_c. These fits depend on the choice of φ_c, which is ambiguous because of transition rounding; the data allow a range from about 0.840 to 0.843⑮. φ_c can be estimated from the point where Z reaches 3, where P rises above background, and other criteria (cf. Ref. [10]). Figure 2 shows power-law fits (Z − Z_c) ∝ (φ − φ_c)^β. The fitted β depends on φ_c: 0.494 ≤ β ≤ 0.564 without rattlers and 0.363 ≤ β ≤ 0.525 with rattlers. Details are given in Sec. II of Ref. [10]. Figure 2 uses φ_c = 0.84220, where P rises above background, and gives consistent fits for P and Z. The point where Z reaches 3 without rattlers agrees within δφ_c = 0.0005, with similar exponents. Our β ≈ 0.55 without rattlers is above the 0.5 reported for frictionless simulations [2,3] but below the 0.6 obtained by Donev et al. in 3D [5]. Aharonov and Sparks [11] obtained 0.36 for frictional disks under shear, but that system is not directly comparable with isotropic jamming here.

中译为与 φ_c 之上的理论预测比较，作者对 Z − Z_c 和 P 关于 φ − φ_c 作最小二乘拟合。由于转变展宽（rounding），φ_c 的选取存在不确定性，数据允许的范围约为 0.840–0.843⑮。φ_c 可由 Z 达到 3、P 超过背景等不同判据估计（参见文献 [10]）。图 2 对 (Z − Z_c) ∝ (φ − φ_c)^β 作拟合；β 随 φ_c 改变，不含 rattlers 时为 0.494–0.564，含 rattlers 时为 0.363–0.525。图 2 采用 P 刚超过背景处的 φ_c = 0.84220，可同时拟合 P 和 Z；不含 rattlers 时 Z 达到 3 的判据与其相差 δφ_c = 0.0005，所得指数相近。实验中不含 rattlers 的 β≈0.55，高于无摩擦模拟 [2,3] 的 0.5、低于 Donev 等人三维结果 [5] 的 0.6。Aharonov 与 Sparks [11] 对受剪摩擦圆盘得到 0.36，但该体系不能与本文各向同性堵塞直接比较。

ENThe lower panel of Fig. 2 shows P versus φ and indicates a clear transition at φ_c = 0.8422 ± 0.0005. For this φ_c, P ∝ (φ − φ_c)^ψ with ψ = 1.1 ± 0.05 above φ_c. This value is fitted over the full φ − φ_c range of Fig. 2; restricting the fit to points very close to φ_c would produce a larger exponent⑯. The value is close to ψ = 1.0 found for a linear force law [2,3], shown as a dashed line. A linear force law with a logarithmic correction is expected for ideal disks, but direct mechanical calibration of the cylinders gives F(δ) closer to δ^3/2⑰ (see Ref. [10]). The larger exponent is attributed to small asperities that affect the law at small deformations. However, the photoelastic response is detectable only for δ > 150 μm, where the force law is locally close to linear in δ.

中译图 2 下图给出 P 随 φ 的变化，并在 φ_c = 0.8422 ± 0.0005 处显示清晰转变。采用该 φ_c，φ > φ_c 时 P ∝ (φ − φ_c)^ψ，ψ = 1.1 ± 0.05。该值来自图 2 全部 φ − φ_c 区间的拟合；若只拟合极接近 φ_c 的数据，会得到更大的指数⑯。ψ 接近文献 [2,3] 在线性力律下得到的 1.0，图中虚线表示线性律。理想圆盘预计遵循带对数修正的线性力律，但对本文圆柱的直接机械标定得到更接近 F(δ)∝δ^3/2 的接触力律⑰（见文献 [10]）；较高指数被归因于小形变下表面微凸起的影响。光弹响应仅在 δ > 150 μm 时可探测，而该区间内力律在局域上接近线性。

ENFrom P versus φ we can derive the bulk modulus B = −A∂P/∂A, where A is the area enclosed by the boundaries. Since φ = A_p/A and A_p is the presumably fixed area occupied by the disks, B = φ∂P/∂φ. Thus B ∝ (φ − φ_c)^ψ−1⑱, implying only weak variation of B above φ_c. Jia’s acoustic experiments showed anomalous bulk-modulus behavior, discussed by Makse et al. [12], in which B near φ_c varied faster with P than previously expected because Z changed.

中译由 P(φ) 还可推导体积模量 B = −A∂P/∂A，其中 A 为边界所围面积。由于 φ = A_p/A，且 A_p 是圆盘占据的、假定不变的面积，所以 B = φ∂P/∂φ。因此 B ∝ (φ − φ_c)^ψ−1⑱，意味着 φ_c 上方 B 变化很弱。Jia 的声学实验曾观察到异常的体积模量行为；Makse 等人 [12] 将其与 Z 的变化联系起来，指出 φ_c 附近 B 随 P 的变化快于早先预期。

ENSince P in Fig. 2 corresponds closely to expectations for a linear force law, we performed a computer simulation for a polydisperse system of 1950 particles with a linear force law (k_n = 10⁵ N/m) without friction; details can be found in [13]. In Fig. 3 the results are shown for a larger range in density than done in earlier studies. All the data in Fig. 3 can be fitted with a single value for the transition density of φ_c = 0.84005⑲. While the average overlap per particle (equivalent of the deformation δ for physical particles) is clearly linear in φ, the pressure P is not: P increases faster than linear with an exponent close to the one found in the experiment. Z is also consistent with a power-law exponent close to 0.5. With the rattlers included, Z at φ_c, Z_c = 3.94, is slightly below the isostatic value of 4⑳ for a frictionless system of disks.

中译由于图 2 中的 P 与线性力律的预期符合得很好，我们对一个 1950 颗粒的多分散系统做了线性力律（k_n = 10⁵ N/m）、无摩擦的计算机模拟；细节见文献 [13]。图 3 给出了比以往研究的密度范围更大的结果。图 3 中的所有数据都可以用单一的转变密度值 φ_c = 0.84005 来拟合⑲。虽然每颗粒的平均重叠（相当于物理颗粒的形变 δ）随 φ 明显呈线性，压强 P 却不然：P 的增长快于线性，其指数接近实验值；Z 也与接近 0.5 的幂律指数相一致。含 rattler 时，φ_c 处的 Z 值（Z_c = 3.94）略低于无摩擦圆盘系统的等静定值 4⑳。

ENTo connect with the predictions of Henkes and Chakraborty [7], we consider P − P_c versus Z. Their Eq. (10) is equivalent to (P − P_c)/P_c = u − [(4u² + 1)^1/2 − 1]/2, where u = C(Z − Z_c) and C = ε/α_c is a system-dependent constant. Here ε measures grain elasticity, with ε = 0 corresponding to perfectly rigid grains, and α_c is the critical value of α. In fitting this form, P_c, Z_c within reason, and C may be adjusted. Figure 4 shows reasonable but imperfect agreement above φ_c, and gives Z_c = 3.04, close to the frictional isostatic value Z_c = 3㉑.

中译为与 Henkes 和 Chakraborty [7] 的预测比较，作者考察 P − P_c 随 Z 的变化。其式 (10) 等价于 (P − P_c)/P_c = u − [(4u² + 1)^1/2 − 1]/2，其中 u = C(Z − Z_c)，C = ε/α_c 为系统相关常数。ε 衡量颗粒弹性，ε = 0 对应完全刚性颗粒；α_c 是 α 的临界值。拟合时可调整 P_c、合理范围内的 Z_c 和 C。图 4 显示 φ_c 之上的数据与该预测在合理程度上吻合，但并不完美；拟合得到 Z_c = 3.04，接近摩擦圆盘的等静定值 3㉑。

ENWe now turn to the rounding of Z near the transition and the background pressure near φ_c. One possible explanation is friction between the disks and the Plexiglas base, which could help freeze in contact forces and contacts. However, an upper-bound estimate shows that base friction cannot account for a significant pressure background. Taking the maximum base-friction force per grain as F_f = μ_bamg = 2.8 × 10⁻³ N, with μ_ba < 1, and assuming Z interparticle contacts plus one particle-base contact per grain, the pressure perturbation is bounded by δP ≈ (ZF_fR)/(πR²) ≈ 0.22Z N/m. For Z ≈ 3, this is almost two orders of magnitude below the observed background pressure㉒. A separate issue is apparatus-induced anisotropy during compression or expansion. It is difficult to eliminate near φ_c, even in simulation, although it remains small. In Fig. 1(e), a weak array of force chains tends to slant from lower left to upper right㉓. Such anisotropy may be induced by friction at the lateral confining walls of the biaxial apparatus.

中译下面讨论转变附近 Z 的展宽及 φ_c 附近的背景压强。一种可能来源是圆盘与有机玻璃底板的摩擦，它可能保留接触力和接触。然而，上限估计表明底板摩擦不足以解释显著的压强背景。取每颗粒最大底板摩擦力 F_f = μ_bamg = 2.8 × 10⁻³ N（μ_ba < 1），并假设每个颗粒有 Z 个颗粒间接触和一个颗粒—底板接触，则压强扰动上限为 δP ≈ (ZF_fR)/(πR²) ≈ 0.22Z N/m。Z ≈ 3 时，该值比观测到的背景压强小近两个数量级㉒。另一个问题是装置在压缩或膨胀时诱发的各向异性；即使在模拟中，它在 φ_c 附近也难以完全消除。图 1(e) 中一组较弱力链倾向于从左下指向右上㉓，这种各向异性可能来自双轴装置侧向约束壁的摩擦。

ENWe conclude that these experiments—the first of which we are aware—demonstrate critical jamming behavior in a real granular material. Photoelasticity enables high-accuracy measurement of Z, which rises rapidly at φ_c = 0.8422. The fine density resolution shows that the transition is not as sharply discontinuous under the present experimental conditions as in the computer simulation㉔. Above φ_c, Z and P follow power laws in φ − φ_c with β = 0.5–0.6 and ψ ≈ 1.1, respectively, consistent with recent frictionless-particle simulations. The data also agree reasonably with a frictionless mean-field model. These results suggest that frictional effects are modest but not necessarily negligible. The narrow yet finite transition range appears to be caused mostly by small residual shear stresses㉕ induced by the confining walls rather than the supporting base. The sensitivity of jamming to weak shear motivates a deeper study of anisotropy.

中译作者总结认为，据其所知，这组实验首次在真实颗粒材料中展示了堵塞的临界特征。光弹性使接触数 Z 得以高精度测量；Z 在 φ_c = 0.8422 附近快速上升。精细的密度分辨率表明，在当前实验条件下，转变不如计算机模拟中那样呈现尖锐不连续㉔。φ > φ_c 时，Z 与 P 分别按 β = 0.5–0.6 和 ψ ≈ 1.1 的幂律随 φ − φ_c 增长，与近期无摩擦颗粒模拟相容；数据也与无摩擦平均场模型大致吻合。结果提示摩擦效应可能较弱，但不可忽略。实验中的转变跨越狭窄却有限的 φ 区间，作者认为其主要可能来源是样品侧壁而非底板诱发的微弱残余剪应力㉕。堵塞对弱剪切的敏感性说明，各向异性应作为独立控制变量深入研究。

ENAcknowledgments. This work was supported by No. NSF-DMR0137119, NSF-DMR0555431, NSF-DMS0244492, the U.S.-Israel Binational Science Foundation No. 2004391, and No. DFG-SP714/3-1. We thank E. Aharonov, B. Chakraborty, D. J. Durian, M. van Hecke, C. S. O’Hern, and S. Torquato for helpful discussions.

中译致谢。本工作受到以下基金资助：NSF-DMR0137119、NSF-DMR0555431、NSF-DMS0244492、美–以双边科学基金 2004391 号以及 DFG-SP714/3-1。感谢 E. Aharonov、B. Chakraborty、D. J. Durian、M. van Hecke、C. S. O’Hern 与 S. Torquato 的有益讨论。