DOI: 10.1103/PhysRevE.85.051302

INTRODUCTION

Granular materials are ubiquitous in everyday life, ranging from industrial applications to geophysical processes (e.g., fault core dynamics). If the steady-state granular flow is reasonably well understood, the transition from the jammed state to the flowing state still remains elusive, including stick-slip behavior and precursor events before failure.

颗粒材料在日常生活中无处不在,从工业应用到地球物理过程(如断层核心动力学)都是如此。如果说人们对颗粒物料的稳态流动已经有了一定的了解,那么从堵塞状态到流动状态的转变过程,包括粘滑行为和破坏前的前兆事件,仍然令人难以捉摸。


Many works on numerical simulations in two-dimensional (2D) or three-dimensional (3D) granular media have been performed to examine the evolution of the structure and the mechanical properties of a granular medium at micro- and mesoscales during the jamming transition and/or shear-band formation. Meanwhile, only a few experiments are available, via photoelastic visualization in 2D and x-ray and magnetic resonance imaging (MRI) tomography in 3D granular models, to investigate their microstructure changes and the particle velocity fields during slow flow. However, the application of these experimental methods to real 3D opaque granular materials appears difficult.

许多关于二维(2D)或三维(3D)颗粒介质的数值模拟工作都是为了研究颗粒介质在堵塞转换和/或剪切带形成过程中微观和介观尺度上的结构和机械性能演变。同时,只有少数实验通过二维光弹性可视化和三维颗粒模型中的 X 射线和磁共振成像(MRI)断层扫描来研究其微观结构变化和缓慢流动过程中的颗粒速度场。然而,将这些实验方法应用于真实的三维不透明颗粒材料似乎很难。


Sound waves provide a unique and sensitive probe for investigating the contact force networks in real granular media. Speed measurements of long-wavelength sound waves allow one to determine the nonlinear elasticity and anisotropic effects of the granular material, while short-wavelength multiply scattered sound waves enable one to detect tiny changes of the contact network configuration at the grain level.

声波为研究实际颗粒介质中的接触力网络提供了一种独特而灵敏的探针。通过对长波声波的速度测量,可以确定粒状材料的非线性弹性和各向异性效应,而通过短波多散射声波,可以检测到颗粒级接触网络配置的微小变化。

Sound waves also provide useful information about the nonlinear dynamics of granular materials under slow shear flow and during granular avalanche. However, is there any specific acoustic signature of the shear-band formation obtainable via shear sound reflection or transmission? Probing with the shear wave might allow inferring the specific behavior of the shear modulus near unjamming transition.

声波还能提供有关颗粒材料在缓慢剪切流动和颗粒崩落过程中的非线性动力学的有用信息。然而,剪切带的形成是否有任何特定的声学特征可以通过剪切声的反射或传播获得?用剪切波进行探测可能会推断出剪切模量在解堵过渡附近的特定行为。

In this work, we show that the sound speed measurements in transmission and particularly of the shear wave provide an assessment of changes in the granular structure of a glass bead packing under shear loading. Our data reveal a clear acoustic signature when the shear band is formed inside the medium. Special attention will be paid to the hysteretic behavior of the mechanical responses and the shear wave speed evolutions under cyclic shear loading with ramped strain amplitude. We will also investigate the irreversible rearrangements of the contact networks under shear by multiply scattered sound waves in both dense and loose granular packings.

在这项工作中,我们展示了透射声速测量结果,尤其是剪切波的测量结果,可以评估玻璃珠堆积在剪切载荷作用下颗粒结构的变化。当剪切带在介质内部形成时,我们的数据揭示了一个清晰的声学特征。我们将特别关注机械响应的滞后行为,以及循环剪切加载下的剪切波速度演变。我们还将研究在致密和疏松颗粒堆积中,多散射声波在剪切作用下接触网络的不可逆重排。

APPARATUS

To create the localization zones of shear strain away from the wall boundaries, we utilize here a direct shear apparatus shown in Fig. 1(a). It consists of a metal box of square cross section $40\times 40\text{ mm}^{2}$ separated into two parts and each has a height of $15\text{ mm}$ (with a filling height $H = 30\text{ mm}$). The shear loading is applied horizontally to the lower part of the box at the constant velocity $V_{\text{drive}} = 0.6 \mu\text{m/s}$, while the upper part is kept fixed.

为了创建远离壁边界的剪切应变局部区域,我们在此使用了图 1(a) 所示的直接剪切设备。它由一个横截面为 $40\times 40\text{ mm}^{2}$ 的正方形金属盒组成,分为两部分,每部分的高度为 $15\text{ mm}$(填充高度为 $H = 30\text{ mm}$)。剪切载荷以恒定速度 $V_{\text{drive}} = 0.6\mu\text{m/s}$ 水平施加到盒子的下部,而上部保持固定。

The shear force $T$ required to maintain the upper part immobile is measured by a load cell of stiffness $k = 10^{7}\text{ N/m}$. The constant normal load $N = 330\text{ N}$ is applied to a piston on the top of the granular sample, corresponding to a normal stress of $P = 206\text{ kPa}$. A position sensor records the relative horizontal displacement between the two parts of the shear box. A second one measures the vertical displacement of the piston to determine the dilatancy of the material induced by shear.

保持上部不动所需的剪切力 $T$ 由刚度为 $k = 10^{7}\text{ N/m}$ 的称重传感器测量。恒定法向载荷 $N = 330\text{ N}$ 施加在颗粒样本顶部的活塞上,对应的法向应力为 $P = 206\text{ kPa}$。位置传感器记录剪切箱两部分之间的相对水平位移。第二个传感器测量活塞的垂直位移,以确定剪切引起的材料膨胀。


Spherical glass beads of diameter $d = 700\pm 20 \mu\text{m}$ are used. The material constants of the glass beads are $\rho = 2500\text{ kg/m}$ for the bulk density, $G = 25\text{ GPa}$ for the shear modulus, and $\nu = 0.2$ for the Poisson ratio. To investigate the influence of the packing density on the unjamming transition in sheared granular materials, two different packing protocols are performed.

使用直径为 $d = 700\pm 20\mu\text{m}$ 的球形玻璃珠。玻璃珠的材料常数为:体积密度 $\rho = 2500\text{ kg/m}^{3}$ ,剪切模量 $G = 25\text{ GPa}$ ,泊松比 $\nu = 0.2$。为了研究堆积密度对剪切颗粒材料解堵塞过渡的影响,我们采用了两种不同的堆积方案。

For the “rainfall” packing preparation, the beads are poured into the box through two spaced grids. This packing protocol creates the dense packing with a solid volume fraction of $\phi = 0.641 \pm 0.002$. For the decompaction packing protocol, we slowly remove an inner horizontal grid through the packing. This preparation method gives a loose packing with $\phi = 0.600 \pm 0.002$. Note that these two preparation methods also produce different granular fabrics.

在 “雨落法” 堆积制备中,珠子通过两个间隔的网格倒入箱中。这种堆积方案产生的致密堆积的固体体积分数为 $\phi = 0.641 \pm 0.002$。在减压堆积方案中,我们通过堆积缓慢移除内层水平网格。这种制备方法得到的松散堆积的体积分数为 $\phi = 0.600 \pm 0.002$。请注意,这两种制备方法也会产生不同的颗粒结构。


Combined ultrasonic measurements are conducted simultaneously with the shear experiments. One is the speed measurement of coherent shear waves and the other is the detection of multiply scattered waves. As shown in Fig. 1(a), a large shear transducer of diameter $30\text{ mm}$ is placed on the lower part of the shear box. For the speed measurement, this source transducer of broadband, centered at $250\text{ kHz}$, is excited by a short pulse of $4 \mu\text{s}$ and coherent shear pulses with central frequencies around $40\text{ kHz}$ are measured by the same large transducer placed in the upper piston at various shear loading [Fig. 1(b)]. The low-frequency spectra of these coherent waves are basically due to scattering attenuation; no dispersion of sound speed is observed here.

综合超声波测量与剪切实验同时进行。一个是相干剪切波的速度测量,另一个是多重散射波的检测。如图 1(a)所示,剪切箱下部放置了一个直径为 $30\text{ mm}$ 的大型剪切换能器。为了进行速度测量,用一个 $4 \mu\text{s}$ 的短脉冲激励这个中心频率为 $250\text{ kHz}$ 的宽带源换能器,并用放置在活塞上部的同一个大型换能器在不同剪切载荷下测量中心频率在 $40\text{ kHz}$ 左右的相干剪切脉冲[图 1(b)]。这些相干波的低频频谱基本上是由于散射衰减造成的;在这里没有观察到声速的频散。

To avoid the interference between the compressional signals and the dominant shear pulses caused by the small size of the shear box, we measure the shear wave speed $V_{S}$ via the time of flight of the pulses maxima (the real value of $V_{S}$ is thus underestimated). The relative change of the shear wave speed $\Delta V_{S}/V_{S}$ due to shearing shown in Fig. 1(b) is about $22\%$, $24\%$, or $27\%$ if one follows the evolution of the maxima, first, or second minima, respectively; these results are consistent showing thus fairly accurate measurements of $\Delta V_{S}/V_{S}$. For monitoring the local rearrangement by the acoustic speckles, the multiply scattered waves are generated by a ten-cycle burst centered at $250\text{ kHz}$ and are detected by a small transducer of $2\text{ mm}$ placed at the top.

为了避免由于剪切盒尺寸较小而造成的压缩信号与主要剪切脉冲之间的干扰,我们通过脉冲最大值的飞行时间来测量剪切波速度 $V_{S}$(因此会低估 $V_{S}$ 的真实值)。图 1(b)中显示的剪切力导致的剪切波速度 $\Delta V_{S}/V_{S}$ 的相对变化约为 $22%$、$24%$ 或 $27%$,如果按照最大值、第一个或第二个最小值的演变来计算的话;这些结果是一致的,因此显示了对 $\Delta V_{S}/V_{S}$ 的相当精确的测量。为了监测声学斑点的局部重排,倍频散射波是由中心频率为 $250 \text{ kHz}$ 的十周期脉冲串产生的,并由放置在顶部的 $2 \text{ mm}$ 小型传感器检测到。

1

(a) Direct shear apparatus combined with acoustic measurements where $E$ and $R$ are the acoustic source and receiver. (b) Typical transmitted ultrasonic signals obtained before shear loading (dashed line) and at the peak shear force (solid line). (c) Evolution under shear of force ratio $T/N$ and (d) dilatancy in the loose ($\square$), dense ($\circ$), and stratified packings ($\Delta$, reloading in the same direction; ♦, reloading in the reverse direction).

(a) 结合声学测量的直接剪切仪器,其中 $E$ 和 $R$ 为声源和接收器。(b) 剪切加载前(虚线)和剪切力峰值时(实线)获得的典型透射超声波信号。(c) 力比 $T/N$ 和 (d) 松散($\square$)、致密($\circ$)和层状堆积($\Delta$,同方向加载;♦ ,反方向加载)在剪切作用下的膨胀率变化。

MECHANICAL RESPONSES

We first examine the mechanical response of a granular medium under shear. Figures 1(c) and 1(d) show typical evolutions of the force ratio $T/N$ and the dilatancy as a function of the relative horizontal displacement $U$ between the two parts of the shear box. We observe the classical behavior of sheared granular media, undergoing the transition from the jammed state to the flowing state.

我们首先研究颗粒介质在剪切力作用下的机械响应。图 1(c) 和 1(d) 显示了力比 $T/N$ 和膨胀率随剪切箱两部分之间相对水平位移 $U$ 的典型变化。我们观察到剪切颗粒介质的经典行为,即从堵塞状态过渡到流动状态。

For the densely packed sample, the shear force $T$ rises rapidly with the shear displacement in the early stage before reaching a peak value and then decreases and tends to a stationary value when the flow is fully developed. For the loosely packed sample, the behavior is similar except that there is no peak force. In both cases, there are small compressions of the media at the initial stage preceding the dilatations [Fig. 1(d)].

对于致密堆积的样品,剪切力 $T$ 在达到峰值之前的早期阶段随剪切位移迅速上升,然后下降,并在流动完全发展后趋于静止。对于松散包装的样品,除了没有峰值力之外,其他行为类似。在这两种情况下,介质在扩张之前的初始阶段都有小的压缩[图 1(d)]。


Note that the steady-state flowing is characterized by a constant value of $T/N$, referred to as the residual strength; it is independent of the initial sample density and the fabric anisotropy. It has been shown that when a granular medium is sheared to the steady state in a direct shear box, the shear strain is essentially localized in a narrow zone located at the midheight of the box where a shear band is formed. Such a shear localization zone exhibits distinct features compared to the rest of the medium, including extremely large voids and the presence of a highly anisotropic network of force chains.

请注意,稳态流动的特征是恒定值 $T/N$,称为残余强度;它与初始样品密度和结构各向异性无关。研究表明,当颗粒介质在直接剪切箱中被剪切到稳定状态时,剪切应变基本上会局部集中在位于箱体中部高度的狭窄区域,在该区域会形成剪切带。与介质的其他部分相比,这种剪切定位区具有明显的特征,包括空隙极大以及存在高度各向异性的力链网络。


We then wonder what may be the mechanical behavior of a granular medium comprising a preexisting shear band when applying a shear force. Figure 1(c) (insets) displays the shear force displacement response of such a sample, sheared either in the same or the reverse direction as the first shearing. In such stratified medium possessing a shear band at the middle, the peak force disappears and the medium reaches closely the same steady flowing state. When the reloading is performed in the same direction as the prior shear direction, we observe a similar mechanical behavior to that in a loosely packed sample. As mentioned previously, when all the samples are driven to the steady-state flow, the shear strain is localized in the shear band. The same stationary value of the ratio $T/N$ implies that a fully developed shear band has the same properties, independent of the sample history.

因此,我们不禁要问,当施加剪切力时,包含一个预先存在的剪切带的颗粒介质会产生怎样的机械行为。图 1(c)(插图)显示了这样一种样品的 对剪切力的位移响应,剪切方向与第一次剪切相同或相反。在这种中间具有剪切带的层状介质中,峰值力消失,介质达到了几乎相同的稳定流动状态。当重新加载的方向与之前的剪切方向相同时,我们观察到的机械行为与松散堆积样品的机械行为相似。如前所述,当所有样品都被驱动至稳态流动时,剪切应变会集中在剪切带中。相同的静态比值 $T/N$ 意味着完全发展的剪切带具有相同的特性,与样品历史无关。

ACOUSTIC RESPONSES

We now examine the evolution of shear wave speed $V_{S}$ through the granular samples during the first shear loading before the strain localization. As shown in Fig. 2, significant decreases of shear wave speed $V_{S}$ up to $20\%$ are observed in both densely and loosely packed samples, displaying a comparable overall acoustic response under shear. For large shear displacement, the steady-state flow is characterized by a constant wave speed. As a comparison, we note that the decrease of the compression wave speed $V_{P}$ during shear [inset of Fig. 2(a)], measured with longitudinal transducers, is about two times less than that of $V_{S}$.

现在我们来研究在应变局部化之前的第一次剪切加载过程中通过颗粒样品的剪切波速 $V_{S}$ 的演变。如图 2 所示,在致密和松散样品中都观察到了剪切波速 $V_{S}$ 的显著下降,最高可达 $20\%$,在剪切作用下显示出相似的整体声学响应。对于大的剪切位移,稳态流动的特点是波速恒定。作为比较,我们注意到在剪切过程中,用纵波传感器测量到的压缩波速度 $V_{P}$ 的减小量[图 2(a) 插图]大约是 $V_{S}$ 的两倍。


To account for our speed measurements during the transition from the jammed state to the flowing one, we may compare qualitatively the experiments with the predictions by the effective medium theory (EMT). In the limit of long wavelength as in our experiments, i.e., $\lambda\sim 10\text{ mm}$ much larger than the bead size $d\sim 0.7\text{ mm}$, the speeds of compression and shear waves $V_{P}$ and $V_{S}$ are related to the effective bulk and shear modulus $K$ and $G$ via $V_{P} = [(K + 2G/\rho)]^{1/2}$ and $V_{S} = (G/\rho)^{1/2}$, where $\rho = \rho_{0}\phi$ and $\rho_{0}$ is the glass bead density. For isotropic compression $P$ , the scaling expression of elastic moduli $K$ and $G$ yields

$$ V_{P,S} \propto Z^{1/3}\phi^{-1/6}P^{1/6}\tag{1}\label{eq1} $$

with $Z$ the coordination number, i.e., the mean number of contact per grain. This simplified expression illustrates a general relationship between the microstructural parameters and the macroscopic acoustic properties. The overall speed decreases of compression and shear waves observed under shear is thus likely related to the change of the geometric structure (i.e., texture) with a decrease of the mean coordination number $Z$. The inset of Fig. 2(b) shows 3D numerical simulations of direct shear test in which the decrease of the coordination number induced by shearing is clearly evidenced, supporting thus our interpretation.

为了解释我们在从堵塞状态过渡到流动状态时的速度测量结果,我们可以将实验结果与等效介质理论(EMT)的预测结果进行定性比较。在我们实验中的长波极限,即 $\lambda\sim 10\text{ mm}$ 远大于珠子尺寸 $d\sim 0.7\text{ mm}$,压缩波和剪切波的速度 $V_{P}$ 和 $V_{S}$ 通过 $V_{P} = [(K + 2G/\rho)]^{1/2}$ 和 $V_{S} = (G/\rho)^{1/2}$ 与有效体积模量和剪切模量 $K$ 和 $G$ 相关, 其中 $\rho = \rho_{0}\phi$, 而 $\rho_{0}$ 是玻璃珠密度。对于各向同性压缩 $P$,弹性模量 $K$ 和 $G$ 的比例表达式为

$$ V_{P,S} \propto Z^{1/3}\phi^{-1/6}P^{1/6} $$

$Z$ 为配位数,即每个晶粒的平均接触数。这个简化表达式说明了微观结构参数与宏观声学特性之间的一般关系。因此,在剪切作用下观察到的压缩波和剪切波的整体速度下降很可能与几何结构(即纹理)的变化有关,而几何结构的变化又与平均配位数 $Z$ 的减少有关。图 2(b) 的插图显示了直接剪切试验的三维数值模拟,其中清楚地显示了剪切引起的配位数下降,从而支持了我们的解释。

2

Evolutions of the shear wave speed under shear in the dense samples (a) and loose samples (b). $\square$ and $\circ$, first loading; $\Delta$, reloading in the same direction; ♦, reloading in the reverse direction. Inset in (a): decrease of the compression wave speed during shear. Inset in (b): illustration of the evolution of the coordination number obtained in 3D discrete element model simulation in [8] with the polar distribution evolution of contact normal force inspired from [9].

致密样品(a)和松散样品(b)在剪切作用下的剪切波速度变化。$\square$ 和 $\circ$,第一次加载;$\Delta$,同方向重载;♦,反方向重载。(a) 中的插图:剪切过程中压缩波速度的下降。(b)插图:[8]中三维离散元模型模拟获得的配位数演变图,以及[9]中启发的接触法向力极性分布演变图。


Nevertheless, the anisotropy of the stress field induced by shearing may also affect our measured wave speeds. As shown previously, the scaling relationship in Eq. $\eqref{eq1}$ holds qualitatively for anisotropic loading if $P$ corresponds to the stress component along the direction of wave propagation. Applying a shear load here induces the principal axis rotation of the stress and the modulus tensor, which reduces effectively the elastic moduli $K$ and $G$ and consequently, the sound speeds $V_{P}$ and $V_{S}$ along the vertical direction.

然而,剪切所引起的应力场的各向异性也可能影响我们测量到的波速。如前文所示,如果 $P$ 对应的是沿波传播方向的应力分量,则公式 $\eqref{eq1}$ 中的比例关系在各向异性加载时定性成立。在此施加剪切载荷会引起应力和模量张量的主轴旋转,从而等效于降低弹性模量 $K$ 和 $G$,进而降低沿垂直方向的声速 $V_{P}$ 和 $V_{S}$。

This anisotropic effect could also explain the evolution of wave speed in the densely packed sample, which passes through a minimum before reaching the stationary value. Indeed, when the dense packing reaches the pronounced peak force [Fig. 1(c)], the principal stress direction is deviated from the vertical direction significantly reducing the sound speeds.

这种各向异性效应也可以解释密集堆积样品中波速的演变,波速在达到稳定值之前会经过一个最小值。事实上,当致密堆积达到明显的力峰值时[图 1(c)],主应力方向偏离垂直方向,大大降低了声速。

When the medium attains the flowing state, the principal axis of stress rotates back to the vertical direction. This behavior has been confirmed by numerical simulation on the evolution of either the deviator fabric in 3D simulations or the distribution of normal contact forces in 2D simulations.

当介质达到流动状态时,应力主轴会旋转回垂直方向。这种行为已通过三维模拟中偏差结构的演变或二维模拟中法向接触力分布的数值模拟得到证实。


The main finding of this work lies in the drastically different acoustic response in a granular medium where the shear band is fully developed, e.g., by the first loading. Under further shearing, whatever the direction of reloading versus that of the prior shearing, the shear wave speed in such a granular medium remains nearly constant (Fig. 2). This observation provides a clear acoustic signature of the irreversible modification inside the granular structure with the formation of a shear band. We understand this result as follows. During the first loading, a shear band is formed which becomes the weakest zone of the medium.

这项工作的主要发现在于,在剪切带充分发展的颗粒介质中,例如在第一次加载时,声学响应截然不同。在进一步剪切的情况下,无论重新加载的方向与之前剪切的方向如何,这种颗粒介质中的剪切波速度几乎保持不变(图 2)。这一观察结果提供了一个清晰的声学特征,即随着剪切带的形成,颗粒结构内部发生了不可逆的改变。我们对这一结果的理解如下。在首次加载过程中,剪切带形成,成为介质的最薄弱区。

Upon reloading, this weak zone localizes the strain and yields immediately, driving the medium to the flowing state. However, the structure away from this narrow shear band (a few grain size) in the rest of the sample, is little deformed; hence, the coordination number and the anisotropy of the medium remains globally unchanged. The different stationary values of VS between the dense and loose packings reveal a dependence on the initial packing condition. Note that there is no detectable acoustic reflection by the shear band in these experiments probably due to the weak impedance contrast.

重新加载时,这个薄弱区域将应变局部化,并立即屈服,使介质进入流动状态。然而,在样品的其他部分,远离这一狭窄剪切带(几个晶粒大小)的结构变形很小;因此,介质的配位数和各向异性总体上保持不变。致密堆积和疏松堆积之间不同的 VS 固定值揭示了对初始堆积条件的依赖。需要注意的是,在这些实验中,可能由于阻抗对比较弱,剪切带并没有产生可检测到的声波反射。


Let us now compare quantitatively the shear waves speed decrease measured during the first shear loading with the prediction by the heuristic EMT model. Using together with a decrease of $Z$ by $10\%$ at a global shear strain $\varepsilon = U/H\sim 10%$ [inset of Fig. 2(b)] and a decrease of a normal stress by $30\%$ (due to the rotation of the principal stress axis ~$45^{\circ}$ and the inhomogeneous repartition of the force network), Eq. $\eqref{eq1}$ predicts a sound speed decrease by $8\%$. This estimation is in agreement with the decrease of $V_{P}$ (~$10\%$), but only half the variation of $V_{S}$ observed in experiments. Two possible reasons might be responsible for such discrepancy.

现在让我们将第一次剪切加载时测得的剪切波速度下降与启发式 EMT 模型的预测进行定量比较。在全局剪切应变 $\varepsilon = U/H\sim 10\%$ [图 2(b) 插图]和法向应力减少 $30\%$(由于主应力轴 ~$45^{\circ}$ 的旋转以及力网络的不均匀再分配)的情况下,公式 $\eqref{eq1}$ 预测声速将减少 $8\%$。这一估计与 $V_{P}$ 的下降(~$10\%$ )一致,但只有实验中观察到的 $V_{S}$ 变化的一半。造成这种差异的原因可能有两个。

One stems from the heuristic model which does not account for quantitatively either the inhomogeneous field of contact forces, inherent of the direct shear box or the important fabric variation; note that the shear wave speed $V_{S}$ was shown to be particularly sensitive to the fabric anisotropy.

其中一个原因是启发式模型没有定量考虑直接剪切箱固有的不均匀接触力场或重要的结构变化;请注意,剪切波速度 $V_{S}$ 对结构各向异性特别敏感。

The other, more fundamentally, originates from the breakdown of the EMT in sheared media near unjamming transition (i.e., failure) where the elastic weakening due to relaxation or slippage of grains especially at low confining pressure might induce a supplementary decrease of $V_{S}$ by $20\%–40\%$.

另一个更基本的原因是,在剪切介质中,EMT 在接近解堵塞转变(即失效)时被破坏,在这种情况下,由于松弛或颗粒滑动而导致的弹性减弱,尤其是在低约束压力下,可能会导致 $V_{S}$ 以 $20\%-40\%$ 的形式补充减小。

Indeed, if one considers the distinct properties of the shear band formed after the failure (with a height $h = 5–10d$) compared to the rest of the medium assumed weakly modified, an increase of $20\%$ of the time of flight in this stratified medium $t = (H − h)/V_{0} + h/V_{\text{SH}}$ (at $\varepsilon = 10\%$) relative to that under zero shear $t_{0} = H /V_{0}$, might reveal a decrease of shear wave speed $V_{\text{SH}}$ inside the shear band up to $45\%–70\%$, not predicted by the EMT model.

事实上,如果考虑到破坏后形成的剪切带(高度为 $h = 5-10d$)与介质的其余部分(假定为轻微改变)相比所具有的不同特性,则与零剪切力下的时间 $t_{0} = H /V_{0}$ 相比,该分层介质中的飞行时间 $t = (H - h)/V_{0} + h/V_{\text{SH}}$(在 $\varepsilon = 10\%$ 时)增加了 $20\%$、可能会发现剪切带内的剪切波速度 $V_{\text{SH}}$ 下降到 $45\%-70\%$,这是 EMT 模型无法预测的。

A. Cyclic shear

To better understand the structural changes which lead to the strain localization, we investigate the response of the medium subjected to cyclic shear with increasing shear amplitude. In the following, we will focus our attention on the densely packed samples. At a given shear displacement before reaching the peak force, we obtain the typical hysteretic loop between normalized shear force ($T/N$) and displacement [Fig. 3(a)]. For this small shear displacement, a slight strengthening with the number of cycling is observed.

为了更好地理解导致应变局部化的结构变化,我们研究了介质在剪切振幅不断增大的循环剪切作用下的响应。下面,我们将重点关注致密样品。在达到峰值剪切力之前的给定剪切位移下,我们会得到归一化剪切力($T/N$)和位移之间的典型滞回[图 3(a)]。对于这种小的剪切位移,可以观察到随着循环次数的增加,剪切力略有增强。

3

Evolutions of force ratio (a) and shear wave speed (b) under cyclic shear. (c) Evolutions of the shear wave speed during the eighth cyclic loading for different displacement amplitudes. The left inset illustrates the evolution of force ratio during the eighth cycle and the right inset illustrates the evolution of the wave speed variation within the eighth cycle.

循环剪切下的力比(a)和剪切波速(b)的变化。(c) 不同位移振幅下第八次循环加载时剪切波速度的演变。左侧插图说明了第八个周期内力比的演变,右侧插图说明了第八个周期内波速变化的演变。

In contrast, for larger amplitude we obtain an elastic weakening (not shown). The evolution of the shear wave speed during this cyclic shear loading is depicted in Fig. 3(b). The remarkable speed decrease between the beginning and the end of the first cycle of shear loading corresponds to some plastic rearrangement and probably small adaptation of the initial packing. Under subsequent unloading-reloading cycles, we observe a repeated hysteretic response of the shear wave speed or elastic modulus as a function of shear displacement; this is likely associated with the evolution of the microstructure when the system explores different metastable configurations via flips events.

相反,当振幅较大时,我们会发现弹性减弱(未展示)。图 3(b) 描述了循环剪切加载过程中剪切波速度的变化。在剪切加载的第一个周期开始和结束时,速度明显下降,这与一些塑性重排以及初始堆积的微小适应性有关。在随后的卸载-再加载循环中,我们观察到剪切波速度或弹性模量作为剪切位移函数的重复滞后响应;这可能与微观结构的演变有关,当系统通过 翻转 事件探索不同的亚稳态构型时。

Such elastic hysteretic behavior is consistent with the evolution of fabric anisotropy obtained in 2D simulation during cyclic tilting of a granular pile below the avalanche angle. Indeed, the wave speed varies when the anisotropy and the mean coordination number change; the agreement between our wave speed measurements and numerical simulation confirms the ability of sound waves for probing the structural evolution under shear.

这种弹性滞后行为与在二维模拟中获得的低于雪崩角的颗粒堆积周期性倾斜过程中结构各向异性的演变相一致。事实上,当各向异性和平均配位数发生变化时,波速也会发生变化;我们的波速测量结果与数值模拟结果之间的一致性证实了声波探测剪切作用下结构演变的能力。


Let us now investigate the amplitude effect of the shear displacement. We present in Fig. 3(c) the variations of the shear wave speed during the eighth cyclic loading with different displacement amplitudes [left inset of Fig. 3(c)]. As the amplitude increases, we observe that the maximum variation of the wave speed $\Delta V_{S}$ within the hysteretic loop decreases. As shown in [9], for small displacement, there is no strain localization and the structural change entails the whole packing.

现在我们来研究剪切位移的振幅效应。我们在图 3(c) 中展示了不同位移振幅下第八次循环加载时剪切波速度的变化 [图 3(c) 左侧插图]。随着振幅的增大,我们观察到滞回线内波速 $\Delta V_{S}$ 的最大变化减小。如文献[9]所示,对于小位移,不存在应变局部化,结构变化涉及整个堆积。

Hence, in this regime, the larger the shear displacement, the larger the wave speed variation. Beyond a certain threshold of the shear displacement, the variation of the wave speed reduces importantly due to the onset of the strain localization. In that regime, as was stated above, the packing away from the narrow shear zone does not respond significantly to the shear loading. $V_{S}$ measured from the time of flight across the entire packing varies little and becomes almost constant with the full development of the shear band.

因此,在这种情况下,剪切位移越大,波速变化越大。当剪切位移超过某一临界值时,由于应变局部化的出现,波速的变化会明显减小。在这种情况下,如上所述,远离狭窄剪切区的堆积对剪切荷载的反应并不明显。从整个堆积的飞行时间开始测量的 $V_{S}$ 变化很小,并且随着剪切带的充分发展变得几乎恒定。

B. Intermittent dynamics before failure

If the residual strength of a granular medium is independent of the initial state of the granular packing, the peak force and the prepeak behavior such as creep and plastic deformation depend strongly on the packing density and the fabric. Several works have shown that significant precursor events or irreversible rearrangement of grains occur before the peak force, i.e., failure.

如果颗粒介质的残余强度与颗粒堆积的初始状态无关,那么力峰值以及蠕变和塑性变形等峰值前行为则与堆积密度和结构密切相关。一些研究表明,在力峰值(即破坏)出现之前,会发生显著的前兆事件或颗粒的不可逆重新排列。

To investigate the precursor behavior, we use the configuration-specific specklelike scattered waves to monitor the granular packings under shear, from the jammed state to the flowing state. More specifically, we define the degree resemblance $\Gamma_{i,j}$ between two successive acoustic speckles $S_{i}(t)$ and $S_{j}(t)$ (inset of Fig. 4):

$$ \Gamma_{i,j}(\tau = 0) = \frac{C_{ij}(\tau = 0)}{\sqrt{C_{ii}(0)C_{jj}(0)}}\tag{2}\label{eq2} $$

where $C_{ij}(\tau)$ is the cross-correlation function between $S_{i}(t)$ and $S_{j}(t)$ and $\tau$ is the time lag. Figure 4 illustrates $\Gamma_{i,i+1}$ deduced from two successive scattered acoustic signals recorded at a rate of $1\text{ Hz}$ in dense and loose bead packings, respectively.

为了研究前兆行为,我们采用构型敏感的斑点状散射波来监测颗粒堆积在剪切力作用下从堵塞状态到流动状态的过程。更具体地说,我们定义了两个连续声学斑点 $S_{i}(t)$ 和 $S_{j}(t)$ 之间的相似度 $\Gamma_{i,j}$(图 4 插图):

$$ \Gamma_{i,j}(\tau = 0) = \frac{C_{ij}(\tau = 0)}{\sqrt{C_{ii}(0)C_{jj}(0)}} $$

其中,$C_{ij}(\tau)$ 是 $S_{i}(t)$ 和 $S_{j}(t)$ 之间的交叉相关函数,$\tau$ 是时滞。图 4 展示了分别在致密和松散球珠堆积中以 $1\text{ Hz}$ 的速率记录的两个连续散射声学信号推导出的 $\Gamma_{i,i+1}$。

This correlation function reveals basically a slight global decorrelation lasting over the experimental range, superposed by the very large pulselike components which occur intermittently. These spikes likely correspond to the major irreversible rearrangements in granular packings evidenced by the intermittent events in the force measurement [Fig. 1(b)]. For closer examination, we plot in Fig. 4(c) a zoomed part of Fig. 4(a) (around $U = 2400 \mu\text{m}$) as a function of the experimental time.

该相关函数显示,在整个实验范围内基本上存在轻微的全局不相关性,与之叠加的是间歇出现的非常大的脉冲样成分。这些峰值很可能对应于颗粒堆积中的主要不可逆重排,这在力测量的间歇性事件中得到了证明[图 1(b)]。为了仔细观察,我们在图 4(c) 中绘制了图 4(a) 的放大部分(大约 $U = 2400 \mu\text{m}$)与实验时间的函数关系。

4

Correlation between two successive coda waves before failure in (a) a dense sample; $\circ$, force ratio; (b) a loose sample;$\square$, force ratio; and (c) zoomed part of Fig. 4(a) as a function of the experimental time (see text).

(a)致密样品;$\circ$,力比;(b)松散样品;$\square$,力比;(c)图 4(a)的放大部分与实验时间的函数关系(见正文)。

简单的推算: $2500/0.6\approx 1.15\text{ h}$

Here the abrupt force drops clearly coincide with the important decorrelation of acoustic speckles $\Gamma_{i,i+1}$. We note that the significant decorrelation may also occur due to the force network change but without visible motions of grains. Furthermore, the intermittent events detected in $\Gamma_{i,i+1}$ appears more pronounced in the loose packing than in the dense packing, before the failure (i.e., peak force). In the following state when the strain is localized, $\Gamma_{i,i+1}$ displays the similar intermittent responses in the two packings, showing that the rheological behavior is henceforth dominated by the shear band.

在这里,力的突然下降明显与声学斑点 $\Gamma_{i,i+1}$ 的重要去相关性相吻合。我们注意到,显著的去相关性也可能是由于力网络的变化而发生的,但没有可见的颗粒运动。此外,在破坏(即力峰值)之前,$\Gamma_{i,i+1}$ 中检测到的间歇事件在松散堆积中比在致密堆积中更为明显。在随后应变局部化的状态下,$\Gamma_{i,i+1}$ 在两种堆积中显示出相似的间歇响应,这表明流变行为从此由剪切带主导。

CONCLUSION

The evolutions of the structure and the mechanical properties of glass bead packings under shear are investigated by acoustic measurements, during the transition from the jammed state to the flowing state. Measurements of the shear wave speed clearly evidence the breaking of contacts or decrease of the coordination number in sheared granular media; they also provide the acoustic signature when the shear bands are formed. The hysteretic responses under cyclic shear reveal flips or collective rearrangement events. Moreover, the correlation functions of specklelike scattered sound waves evidence the specific intermittent dynamics of the contact force networks. Our results obtained in the laboratory experiments, both for the shear-band formation and the precursor events, may have important implications on large-scale field experiments for remote probing of the fault core dynamics.

通过声学测量,研究了玻璃珠堆积在剪切力作用下,从堵塞状态过渡到流动状态过程中的结构和机械性能演变。对剪切波速度的测量清楚地证明了剪切颗粒介质中接触的断裂或配位数的减少;它们还提供了剪切带形成时的声学特征。循环剪切下的滞后响应揭示了瞬时或集体重排事件。此外,斑点状散射声波的相关函数证明了接触力网络的间歇动态特性。我们在实验室实验中获得的结果,无论是剪切带的形成还是前兆事件,都可能对远程探测断层核心动态的大规模现场实验产生重要影响。