DOI: arXiv:2209.02146

Abstract

We report the simultaneous softening and compaction of a confined dense granular pack in acoustic resonance experiments. Elastic softening is manifested by a reduction of the shear-wave speed, as the wave amplitude increases beyond some threshold.

No macroscopic rearrangement of grains or dilatancy is observed; instead, elastic softening is accompanied by a tiny amount of compaction on the scale of grain asperities. We explain these apparent contradictory observations using a theoretical model, based on shear transformation zones (STZs), of soft spots and slipping contacts.

我们报告了在声共振实验中压缩致密颗粒堆积同时发生软化和压实的情况。弹性软化表现为剪切波速度的降低，而波幅的增加超过了一定的阈值。

没有观察到宏观的颗粒重新排列或扩张现象；相反，弹性软化伴随着微量的颗粒表面压实。我们用一个基于剪切转变区域（STZs）、软点和滑动接触的理论模型来解释这些明显矛盾的观察结果。

It predicts a linear shear stress-strain response with negligible macro-plastic deformation due to the small-amplitude acoustic oscillation. However, these waves reduce the interparticle friction and contact stiffness through the acoustic lubrication of grain contacts, resulting in an increase in the structural disorder or compactivity and softening of dynamic modulus. The compaction associated with this microscopic friction decrease is consistent with the prediction by an Ising-like correlation between STZs in the subyield regime.

它预测了线性剪应力-应变响应，由于小振幅声波振荡，宏观塑性变形可以忽略不计。然而，这些声波通过对颗粒接触点的声学润滑，降低了晶粒间的摩擦力和接触刚度，从而导致结构无序性/致密性增加以及动态模量软化。与这种微观摩擦力降低相关的致密性与亚屈服机制中 STZ 之间的 Ising 类相关性预测一致。

Introduction

Granular matter are ubiquitous in nature, and can demonstrate liquidlike or solidlike properties depending on the packing fraction and external load. The viscoelastic and elastoplastic properties of dense granular media can be probed by bulk acoustic waves in a noninvasive fashion.

Moreover, waves of sufficiently large amplitude are known to unjam the granular material via acoustic fluidization and reduce the aggregate elastic moduli, manifested by a decrease of sound speed through the granular pack.

颗粒物质在自然界中无处不在，根据堆积(体积)分数和外加负载的不同，它们可以表现出类似液体或固体的特性。致密颗粒介质的粘弹性和弹塑性特性可以通过体声波进行非侵入式探测。

此外，众所周知，足够大振幅的声波可通过声学流化作用解除颗粒材料的堵塞，并降低集合弹性模量，具体表现为声速通过颗粒堆积的速度降低。

Ref. [15] reported precise measurements of elastic softening and compaction induced by traveling compression waves and the subsequent healing, and proposed a microscopic model based on the Hertz-Mindlin theory of contacts for their observations within the mean-field framework.

That study revealed that acoustic fluidization operates mainly through wave-induced shear contact stiffness weakening (via microslips), termed henceforth acoustic lubrication, rather than the acoustic pressure fluctuation (via contact opening) initially proposed and akin to the vibration fluidization.

参考文献 [15] 报告了对传播的压缩波引起的弹性软化和压实以及随后愈合的精确测量，并提出了一个基于 Hertz-Mindlin 接触理论的微观模型，用于平均场框架内的观测。

该研究揭示了声学流化主要是通过波诱导的剪切接触刚度减弱（通过微滑动）（以下称为声学润滑）而非最初提出的声压波动（通过接触开口）来实现的，与振动流化类似。

Nevertheless, numerical simulations show that large-amplitude sound waves in a dense pack of elastic beads may change locally the contact network, on the mescoscopic scale, by a decrease of the coordination number, and create soft spots whose population increases with acoustic amplitude.

尽管如此，数值模拟显示，在致密弹性球珠中，大振幅声波可能会局域性地改变接触网络，在微观尺度上降低配位数，并产生软点，其数量随声波振幅而增加。

In this work, we propose a model, based on shear transformation zones (STZs), to interpret qualitatively the observation of elastic softening and simultaneous compaction in confined granular glass bead packs, induced by shear waves which cause much more pronounced irreversible sound-matter interactions than by compressional waves.

This theoretical concept, originally developed for deformation in glassy, amorphous materials, provides new insights into the mesoscopic physics of non-affine displacement and slippage of grains and the associated change in the force chain network at dynamic STZ heterogeneities.

在这项工作中，我们提出了一个基于剪切转换区域（STZs）的模型，用于定性解释在约束颗粒玻璃球堆积中观察到的弹性软化和伴随的压实现象，这种现象是由剪切波引起的，剪切波引起的不可逆声物质相互作用要比压缩波明显得多。

这一理论概念最初是针对玻璃状无定形材料的变形而提出的，它为了解颗粒的 非仿射位移和滑动 以及 动态 STZ 异质性条件下 力链网络的协同变化的介观物理学提供了新的视角。

Unlike the large-amplitude oscillatory shear measurements, the STZ model gives rise to hysteresis curves whose encircled areas are narrow enough to resemble straight lines in the stress-strain phase space (i.e., almost linear stress-strain response), for shear waves of relatively small amplitude considered here. The elastic modulus can be inferred by the mean value over one oscillation cycle (see the Supplementary Material).

与大振幅振荡剪切测量不同，STZ 模型产生的滞后曲线的包围区域足够窄，类似于应力-应变相空间中的直线（即几乎是线性的应力-应变响应），适用于此处考虑的振幅相对较小的剪切波。弹性模量可通过一个振荡周期的平均值推断（见 Supplementary Material）。

Our experimental observation thus raises an important question: Is it possible to reconcile a linear STZ dynamics and a nonlinear acoustic response with a softening of shear wave velocity (dynamic modulus), as well as plastic deformation with a tiny amount of compaction on the scale of grain asperities?

This situation is particularly relevant in dynamic earthquake triggering where the fault core may be impacted by various seismic waves from afar.

因此，我们实验中的观察提出了一个重要问题：是否可能通过剪切波速软化（动态模量）以及塑性变形与在颗粒(表面)微凸尺度的微量压实来协调 线性 STZ 动力学 和 非线性声学响应?

这种情况与动态地震触发尤其相关，因为断层核心可能会受到来自远处的各种地震波的影响。

We will attribute these seemingly contradictory observations to an increase in disorder, manifested by an increase of an internal state variable or configurational temperature termed the compactivity due to the decrease of interparticle friction and subsequent unlocking between neighboring grains through acoustic lubrication. Such a mechanism may also allow the system to transition between metastable configurations, causing granular compaction.

我们将这些看似矛盾的观察结果归因于无序性的增加，表现为内部状态变量或结构温度的增加，即由于粒子间摩擦的减小以及随后相邻晶粒间通过声学润滑的解锁而产生的致密性。这种机制还可使系统在可变构型之间转换，从而导致颗粒压实。

Experiment

Our granular materials consist of dry glass beads of diameter $a = 0.6 − 0.8\text{ mm}$ and density $\rho_{G} = 2.4 \times 10^{3}\text{kg m}^{−3}$, confined in a oedometer cell of diameter $D\approx 30\text{ mm}$ and filled by rain deposition to a height $L ≈ 18.5\text{ mm}$. A normal load is applied on the bead pack across the top piston (piezoelectric transducer) using an electromechanical servo press, keeping the load constant within an error of $2\%$, corresponding to effective axial pressure $p$ ranging from $70\text{ kPa}$ to $2.8\text{ MPa}$.

我们的颗粒材料由直径为 $a = 0.6 - 0.8\text{ mm}$、密度为 $\rho_{G} = 2.4 \times 10^{3}\text{kg m}^{-3}$ 的干玻璃珠组成，它们被限制在直径为 $D\approx 30\text{ mm}$、通过落雨法填充到高度为 $L ≈ 18.5\text{ mm}$ 的测径池中。使用一个机电伺服压力机，通过顶部活塞（压电传感器）对球珠堆积施加法向载荷，保持载荷恒定，误差在 $2\%$ 以内，对应的等效轴向压力 $p$ 从 $70\text{ kPa}$ 到 $2.8\text{ MPa}$ 不等。

Before acoustic measurements, one cycle of loading and unloading up to $p$ is performed on the granular packing in order to consolidate the sample and ensure repeatability of material preparation with a packing density $\phi = 0.63 \pm 0.01$. To optimize the propagation of coherent shear waves, a large shear transducer is used as a plane-wave source transmitting a continuous wave (Fig. 1(a)).

在声学测量之前，要对颗粒堆积进行一次加载-卸载循环，最大加载量为 $p$，以巩固样品并确保材料制备的可复现性，堆积密度(分数)为 $\phi = 0.63 \pm 0.01$。为了优化相干剪切波的传播，使用了一个大型剪切传感器作为发射连续波的平面波源（图 1(a)）。

In general, the shear and compressional wave velocities $v_{s}$ and $v_{p}$ can be measured via traveling waves or the (fundamental) resonance frequency $f_{r}$ by $v_{p,s} = \lambda f_{r}$ with a wavelength $\lambda \approx 2L$ for a resonator with rigid boundary conditions considered here.

To examine the nonlinear response, we vary the input voltage $V_{\text{input}}$ from $5$ to $250\text{ V}$, which correspond to a vibration displacement at the source transducer $u\approx 1$ to $50\text{ nm}$, calibrated by an optical interferometer.

一般来说，对于此处考虑的具有刚性边界条件的谐振器，剪切波和压缩波速度 $v_{s}$ 和 $v_{p}$ 可以通过行波或（基）共振频率 $f_{r}$ 测量，即 $v_{p,s} = \lambda f_{r}$，波长为 $\lambda \approx 2L$。

为了检验非线性响应，我们改变了输入电压 $V_{\text{input}}$ ，范围从 $5$ 到 $250\text{V}$，对应于源换能器上的振动位移 $u\approx 1$ 到 $50\text{ nm}$ (振动位移由光学干涉仪进行校准)。

To construct a resonance curve, we sweep the frequency ranging from $1$ to $10\text{ kHz}$ that contains the fundamental shear modes for roughly $120\text{ s}$, and extract the time-averaged amplitude at each frequency interval. Figure 1(b) shows resonance curves in the glass bead pack under effective pressure $p = 140\text{ kPa}$.

为了构建共振曲线，我们在包含基本剪切模的 $1$ 到 $10 \text{ kHz}$ 的频率范围内扫描了大约 $120\text{ s}$，并提取了每个频率区间的时间平均振幅。图 1(b) 显示了玻璃珠堆积在等效压力 $p = 140\text{ kPa}$ 下的共振曲线。

The graph shows a plot of detected amplitude versus sweeping frequency at progressively increasing drive amplitude. As the excitation amplitude is increased, the resonance frequency $f_{r}$ decreases which corresponds to a decrease in shear wave velocity $v_{s}$ and dynamic modulus $\mu(=\rho_{G}v_{s}^{2})$.

该图显示了在驱动振幅逐渐增大时, 检测到的振幅与扫描频率的关系。随着激励振幅的增加，共振频率 $f_{r}$ 会降低，这与剪切波速度 $v_{s}$ 和动态模量 $\mu(=\rho_{G}v_{s}^{2})$ 的降低相对应。

The broadening of resonance peak with increasing indicates a nonlinear frictional dissipation. Here, the induced modulus reduction in shear resonant modes is about $10\%$ (Fig. 1(c)), corresponding to a decrease in shear modulus of about $20\%$ over a drive strain range ($\epsilon_{a} = \pi u/L$) of $10^{−6}$ to $10^{−5}$. This is about twice the P-wave induced modulus decrease over the same drive strain range.

共振峰随着增大而变宽表明存在非线性摩擦耗散。在此，被诱导的剪切共振模的诱降低约为 $10\%$（图1(c)），对应于在 $10^{-6}$ 至 $10^{-5}$ 的驱动应变范围内（$\epsilon_{a} = \pi u/L$）剪切模量降低约 $20\%$。这大约是同样驱动应变范围内 P 波诱导模量下降的两倍。

To investigate the irreversibility of the sound-matter interaction for increasing wave amplitude, we combine measurements of the resonance frequency downward shift and the packing density change (the sample height). Beyond a certain amplitude of driving depending on the confining pressure, the shear modulus weakening is accompanied by slight plastic deformation corresponding to a compaction of roughly $0.5\text{ }\mu\text{m}$ for one bead layer (Fig. 1(d)). This characteristic scale suggests a microplastic behavior on the contact scale, negligibly small compared to the macro-plastic rearrangement observed in shaking experiments on the grain scale. This finding highlights the sensitivity of the macroscopic elastic weakening on the tiny change of the contact network (via microslips) induced by sound waves or weak vibration without visible macroscopic rearrangement of grains. Interestingly, such microscopic compaction which undergoes a jump upon a step increase of the drive strain (with also a transient stress drop not shown) exhibits a logarithmic-like relaxation under external driving (inset of Fig. 1(d)), reminiscent of those observed in the shaking-induced macroscopic compaction.