Abstract

We investigate the change of the static friction threshold of weakly adhesive amorphous interfaces in the presence of the shear ultrasonic oscillation. Prior to sliding, a softening of the shear interfacial stiffness is observed under either static or high-amplitude oscillatory shear. We find that the nonlinear shear ultrasound, regardless of its polarization, triggers the macroscopic sliding at these interfaces far below the static threshold.

Such unjamming transition is due to the vibration-induced decrease of the apparent coefficient of static friction, which provides a mechanism for understanding the reduction of the yielding threshold of granular media by the acoustic fluidization.

我们研究了在超声波剪切振荡作用下，弱粘合无定形界面静摩擦阈值的变化。在滑动之前，无论是静态还是高振幅振荡剪切，都能观察到剪切界面刚度的软化。我们发现，非线性剪切超声，无论其极化如何，都能在这些界面上触发远低于静态阈值的宏观滑动。

这种解堵塞转变是由于振动引起的表观静摩擦系数的降低，这为理解声学流化降低颗粒介质屈服阈值提供了一种机制。

Introduction

The stick-slip motion between solids is associated with the transition of the amorphous interfacial layer from a static solid state to a sliding fluid state under shear load.

This transition plays an essential role in a wide range of natural processes and technological applications involving amorphous systems such as molecular thin films, glasses, granular materials, and seismic fault gouges.

The threshold rheology at the wide range of length scales could be described with the jamming phase diagram and rationalized by various rate- and state-dependent constitutive laws.

固体之间的粘-滑运动与无定形界面层在剪切载荷作用下从静态固体状态向滑动流体状态的转变有关。

这种转变在涉及无定形系统（如分子薄膜、玻璃、颗粒材料和地震断层沟）的各种自然过程和技术应用中起着至关重要的作用。

在大长度尺度范围内，阈值流变学可以用堵塞相图来描述，并通过各种速率和状态相关的本构定律加以合理化。

For solid friction, the elastic dissipation occurs within the amorphous thin films on the nanometer scale via mechanical instabilities. These flips primarily affect a small cluster of a few molecular units, termed shear transformation zones (STZs), and are activated by a thermal noise displaying a logarithmic rate dependence.

Moreover, a dynamical noise generated by the flip of a STZ, i.e., acoustic emission random in time and space, may act in parallel with the thermal one on the other STZs and trigger avalanches of correlated flips. These vibrational effects can be modeled in terms of an effective temperature.

对于固体摩擦，弹性耗散是通过机械不稳定性发生在纳米尺度的无定形薄膜内。这些翻转主要影响由几个分子单元组成的小群，即剪切转变区域（STZ），并由显示对数速率依赖性的热噪声激活。

此外，一个 STZ 的翻转所产生的动态噪音（即在时间和空间上随机的声发射）可能会与其他 STZ 上的热噪声平行作用，并引发相关翻转的雪崩。这些振动效应可以用等效温度来模拟。

In athermal amorphous systems such as granular media, the experiments have clearly shown that externally applied vibrations significantly reduce the yielding stress $\sigma_{s}$ and the angle of avalanche $\theta_{s}$. However, it still remains unclear whether these effects are due to a collective effect, a reduction of the normal stress by the acoustic pressure so that slip of granular media can occur at low shear stress, or a vibration-induced decrease of the static friction coefficient between solid grains.

在无定形热系统（如颗粒介质）中，实验清楚地表明，外部施加的振动会显著降低屈服应力 $\sigma_{s}$ 和雪崩角 $\theta_{s}$。然而，目前仍不清楚这些效应是由于集体效应，还是由于声压降低了法向应力，从而使颗粒介质在低剪切应力下发生滑移，抑或是由于振动导致固体颗粒之间的静摩擦系数降低。

The frictional rheology is controlled by confined thin films, e.g., obtained by the surface grafting of nanometric organic films. Oscillatory rheological measurements performed on various thin films showed a linear viscoelas- tic response for $\sigma < \sigma_{s}$; however, for $\sigma > \sigma_{s}$ the elastic modulus suddenly drops below the viscous one due to the onset of sliding.

摩擦流变受限薄膜控制，例如通过纳米有机薄膜的表面接枝获得的薄膜。对各种薄膜进行的振荡流变测量显示，当$\sigma < \sigma_{s}$时，薄膜具有线性粘弹性响应；然而，当$\sigma > \sigma_{s}$时，由于开始滑动，弹性模量会突然下降到粘性模量以下。

To study precisely the precursor behaviour before the macroscopic sliding, Bureau, Baumberger, and Caroli have measured the displacement response of a slider at a multicontact interface submitted to a biased oscillating shear force of low frequency $f < 1\text{ kHz}$ as compared to the eigenfrequency $f_{0}$ of the slider interface. In this quasistatic regime, the oscillating force $F_{\text{ac}}$ is added to the static shear $F_{\text{dc}}$.

为了精确研究宏观滑动之前的前兆行为，Bureau、Baumberger 和 Caroli 测量了多接触界面上滑块的位移响应，与滑块界面的本征频率 $f_{0}$ 相比，该界面受到的偏置振荡剪切力的频率 $f < 1\text{ kHz}$ 很低。在这种准静态条件下，振荡力 $F_{\text{ac}}$ 是加在静态剪切力 $F_{\text{dc}}$ 上的。

When the maximum shear force $F = F_{\text{dc}} + F_{\text{ac}}$ is close below the static threshold $F_{s}$, the slider undergoes incipient creep. If $F_{\text{ac}}$ is such that $F \approx F_{s}$, the slider undergoes abrupt accelerating motion, i.e., triggered sliding.

当最大剪切力 $F = F_{\text{dc}}+ F_{\text{ac}}$ 接近静态阈值 $F_{s}$时，滑块就会发生初期蠕变。如果 $F_{\text{ac}}$ 使得 $F \approx F_{s}$，则滑块会发生突然加速运动，即诱发滑动。

These observations raise another important question. Indeed, the transition from the static state to the sliding state occurs over a typical slip distance ~ $\mu\text{m}$; for low velocity sliding (~ $\text{mm}/\text{s}$), this corresponds to a characteristic time ~$\text{ms}$.

What would be the role, if any, of a high-frequency oscillation $f \gg f_{0}$ in the triggering of macroscopic sliding? Addressing these issues may be of fundamental interest for understanding the rheology of jammed granular media under vibration, but also the fault gouge weakening by the acoustic fluidization.

这些观察结果提出了另一个重要问题。事实上，从静态到滑动状态的转变发生在一个典型的滑动距离 ~ $\mu\text{m}$ 上；对于低速滑动（~ $\text{mm}/\text{s}$），这相当于一个特征时间 ~$\text{ms}$。

如果有的话，高频振荡 $f \gg f_{0}$ 在触发宏观滑动中的作用是什么？解决这些问题对于理解振动作用下颗粒介质的流变学以及声学流化对断层破碎带的劣化都具有重要意义。

In this Letter, we monitor the change of the static friction threshold of weakly adhered interfacial films, in the presence of the high-frequency shear oscillation. The shear ultrasound is used here both as a nondestructive probe and a controlled pump.

We observe that the shear stiffness is weakened by either static or high-amplitude oscillatory shear before sliding. Our main finding is the onset of sliding, triggered well below the static threshold by nonlinear shear ultrasound, whatever the direction of polarization.

Beyond the nonlinear behavior shown in [23], this result points to the important role of an effective temperature played by the high-frequency shear oscillations in a jamming transition diagram.

在这封快讯中，我们监测了在高频剪切振荡作用下，弱粘附界面薄膜静摩擦阈值的变化。在这里，剪切超声既是一种无损探头，也是一种可控泵浦(源)。

我们观察到，在滑动之前，剪切刚度会因静态或高振幅振荡剪切而减弱。我们的主要发现是，无论极化方向如何，非线性剪切超声都能在远低于静态阈值的情况下触发滑动。

除了 [23] 中显示的非线性行为之外，这一结果还表明了高频剪切振荡在堵塞转变图中的有效温度所起的重要作用。

Experiments

The experimental setup is shown Fig. 1(a), which combines static shear and ultrasonic measurements of interfacial films. The sphere-plane contact geometry [Fig. 1(b)] is composed of three equidistant steel beads of radius $1\text{ mm}$ clamped in a thin steel disk, referred to as probe, and a shear quartz resonator coated with a self assembled monolayer of undecanethiol or mercaptoundecanoic acid, providing, respectively, the low and high adhesion to the steel beads.

实验装置如图 1(a) 所示，它结合了界面薄膜的静态剪切和超声波测量。球体-平面接触几何体[图 1(b)]由夹在薄钢盘（称为探头）中的三个半径为 $1\text{ mm}$ 的等距钢珠和涂有自组装的 十一硫醇 或 巯基十酸 单层的剪切石英谐振器组成，这两种物质分别提供了钢珠的低附着力和高附着力。

Before each measurement, the clamped beads are slightly polished with diamond paste, followed by water rinsing and air drying, and let sit at ambient condition ($17^{\circ}C$, $50\%\text{ RH}$) for $15\text{ min}$. The quartz is cleaned with a “piranha” solution for $5\text{ min}$ followed by thorough rinsing with water, then immersed for $24\text{ h}$ in a $1\text{ mmol}$ solution of thiol. The samples are rinsed with milli-Q water, dried with a flow of nitrogen, and stored under vacuum until use.

在每次测量之前，先用金刚石膏对夹紧的珠子进行轻微抛光，然后用水冲洗和风干，并在环境条件下（$17^{\circ}C$, $50\%\text{RH}$）放置 $15\text{ min}$。用 “食人鱼” 溶液清洗石英 $5\text{ min}$，然后用水彻底冲洗，再在 $1\text{ mmol}$ 的硫醇溶液中浸泡 $24\text{ h}$。样品用毫升-Q 水冲洗，用氮气吹干，真空保存，直至使用。

For making the shear experiment, the homemade cell is mounted on an inclined plane [Fig. 1(a)] whose angle $\theta$ is controlled by means of a rack-and-pinion at $\pm 0.5^{\circ}$ with a protractor. The probe is gently placed on the quartz (at $\theta = 0^{\circ}$) such that its axis coincides with the centre of the quartz.

To obtain the reproducible measurements, a high-amplitude ultrasonic oscillation is applied to shear the interface for $10\text{ s}$; the contact is left further ageing for $10\text{ min}$.

为了进行剪切实验，自制电池被安装在一个倾斜平面上[图 1(a)]，其角度 $\theta$ 是通过一个带量角器的齿条和小齿轮控制在 $\pm 0.5^{\circ}$ 处。将探针轻轻放在石英上（角度为 $\theta = 0^{\circ}$），使其轴线与石英中心重合。

为了获得可重复的测量结果，使用高振幅超声波振荡对界面进行剪切，持续时间为 $10\text{s}$；让接触面进一步老化，持续时间为 $10\text{min}$。

We determine the static friction coefficient $\mu_{s}$ by measuring the angle of sliding $\theta_{s}$ and the threshold force $F_{s} = W\sin{\theta_{s}}$ for various weights of the probe $W = 2.3-21\text{ mN}$. The results are correctly fitted by a modified Coulomb’s law $F_{s} = F_{0} + \mu_{s}W\cos{\theta_{s}}$ where $F_{0}$ is the zero-load threshold.

$F_{0}\approx 1\text{ mN}$ is not significantly affected by the nature of the monolayer, while a larger value of the friction coefficient is obtained for the adhesive substrate (COOH) $\mu_{s}\approx 0.30$, compared to $0.15$ for the less adhesive one (CH3). Our data are consistent with those obtained by others.

我们通过测量不同重量探针的滑动角度 $\theta_{s}$ 和临界力 $F_{s} = W\sin{\theta_{s}}$ 来确定静摩擦系数 $\mu_{s}$。结果被修正的 Coulomb 定律正确拟合 $F_{s} = F_{0} + \mu_{s}W\cos{\theta_{s}}$ 其中 $F_{0}$ 是零负载阈值。

$F_{0}\approx 1\text{ mN}$ 并没有受到单层性质的显著影响，而粘附性较强的基底（COOH）的摩擦系数值为 $\mu_{s}\approx 0.30$，而粘附性较弱的基底（CH3）的摩擦系数值为 $0.15$。我们的数据与其他人获得的数据一致。

1 $\tag{FIG.1}\label{FIG1}$

(a) The cell for shear stiffness measurements is mounted on an inclined plane. (b) Side view of the interfacial film between the probe and the shear quartz resonator covered with electrode.

(a) 用于测量剪切刚度的单元安装在倾斜平面上。 (b) 探针与覆盖有电极的剪切石英谐振器之间的界面薄膜侧视图。

The ultrasonic measurement is realized by bringing the probe in contact with the adsorbed surface of the quartz resonator, which shifts the resonance peak towards higher frequency.

The increased frequency shift $\Delta f$ is related to the shear stiffness of the contact by $k_{T}(=F_{\text{ac}}/U_{\text{ac}}) = \xi\Delta f$ where $\xi = 4\pi(MK)^{1/2}$, $M = 3.5\times 10^{-5}\text{ kg}$ and $K= 3 \times 10^{10}\text{ N/m}$ are the effective mass and stiffness of the quartz, $F_{\text{ac}}$ and $U_{\text{ac}}$ are its oscillating shear force and displacement.

Figure $\eqref{FIG.2}$(a) shows the typical elastic response of the adhesive interface (COOH) as a function of the oscillatory amplitude $U_{\text{ac}}$ at an inclination well below the angle of sliding $\theta_{s}\sim 34.5^{\circ}$.

超声波测量是通过使探头与石英谐振器的吸附表面接触来实现的，这使得谐振峰向更高的频率移动。

频率增移 $\Delta f$ 与接触的剪切刚度的关系为 $k_{T}(=F_{\text{ac}}/U_{\text{ac}}) = \xi\Delta f$, 其中 $\xi = 4\pi(MK)^{1/2}$, $M = 3.5 \times 10^{-5}\text{ kg}$ 和 $K= 3 \times 10^{10}\text{ N/m}$ 是石英的等效质量和刚度，$F_{\text{ac}}$ 和 $U_{\text{ac}}$ 是其振荡剪切力和位移。

图$eqref{FIG.2}$(a)显示了粘合剂界面（COOH）的典型弹性响应，它是振动振幅 $U_{\text{ac}}$ 在远小于滑动角 $\theta_{s}\sim 34.5^{\circ}$ 时的函数。

Two distinct regimes can be identified before the macroscopic sliding: a linear visco-elastic response at low amplitude ($U_{\text{ac}} < 1\text{ nm}$) and a nonlinear frictional regime at high-amplitude oscillation ($U_{\text{ac}} > 2\text{ nm}$), accompanied with an important decrease of the interfacial stiffness.

在宏观滑动之前，可以确定两种截然不同的状态：低振幅的线性粘弹响应（$U_{\text{ac}} < 1\text{ nm}$）和高振幅振荡的非线性摩擦状态（$U_{\text{ac}} > 2\text{ nm}$），同时伴随着界面刚度的显著下降。

In the linear regime, when increasing the inclination angle $\theta$ or shear load, we observe a significant softening of the stiffness of about $5\%$ [Fig. 2(a)], both with COOH and CH3 interfaces before sliding [Fig. 3(a)]. Furthermore, when the macroscopic sliding occurs at the angle of sliding $\theta_{s}$ the resonance peak exhibits abrupt erratic shifts, providing a supplementary sensitive measurement of $\theta_{s}$.

在线性机制中，当增大倾斜角 $\theta$ 或剪切载荷时，我们观察到滑动前 COOH 和 CH3 界面刚度都有明显的软化，软化幅度约为 $5\%$[图 2(a)][图 3(a)]。此外，当宏观滑动发生在滑坡角度为 $\theta_{s}$ 时，共振峰会出现突然的不规则移动，从而为 $\theta_{s}$ 的灵敏测量提供了补充。

2 $\tag{FIG.2}\label{FIG.2}$

(a) Decrease of the shear stiffness for the COOH interface under various angles $\theta$ and $W = 6.7\text{ mN}$. Inset: experimental protocol for measuring $\Delta f$ at $\theta = 19^{\circ}$. Each color (ten measurements) corresponds to a given oscillatory $U_{\text{ac}}$ where the last $5$ points are averaged for a data point shown in the main panel. At $U_{\text{max}} \approx 5\text{ nm}$ the delayed slip occurs and the probe slides. (b) Schematic illustration of a Hertzian contact of radius $a_{H}$ and a microslip annulus of width $a_{H} - c$ induced by shear.

(a) 在不同角度 $\theta$ 和 $W = 6.7\text{ mN}$ 下 COOH 界面剪切刚度的下降。插图：在 $\theta = 19^{\circ}$ 时测量 $\Delta f$ 的实验方案。每种颜色（$10$ 次测量）对应一个给定的振荡 $U_{\text{ac}}$，其中最后 $5$ 个点是主面板中显示的数据点的平均值。在 $U_{\text{max}} \approx 5\text{ nm}$ 时，延迟滑移发生，探针滑动。 (b) 半径为 $a_{H}$ 的 Hertzian 接触和宽度为 $a_{H} - c$ 的微滑环的示意图。

The main finding is obtained in the nonlinear regime: at $\theta = 19^{\circ}$ far below the static threshold $\theta_{s}$, a high-amplitude ultrasonic oscillation ($U > 5\text{ nm}$) provokes a discontinuous shift in $\Delta f$ and triggers the macroscopic sliding of the probe. Before sliding, a creeplike softening is observed [inset of $\eqref{FIG.2}$(a)], pointing to a delayed slip induced by shear ultrasound.

主要的发现是在非线性机制中获得的：在远低于静态阈值 $\theta = 19^{\circ}$ 时，高振幅超声波振荡（$U > 5\text{ nm}$）引起了 $\Delta f$ 的不连续位移，并引发了探针的宏观滑动。在滑动之前，可以观察到蠕变状软化[$\eqref{FIG.2}$(a)的插图]，这表明剪切超声诱发了延迟滑动。

3 $\tag{FIG.3}\label{FIG.3}$

(a) Softening of the shear stiffness for COOH (red) and CH3 interfaces (blue) as a function of $\theta$, extracted from measurements similar to $\eqref{FIG.2}$(a). The average angle of sliding is indicated by the vertical lines. Various symbols correspond to different experiments. (b) Schematic illustration of a noncohesive (Hertzian) contact and cohesive (JKR) contact.

(a) COOH（红色）和 CH3（蓝色）界面的剪切刚度软化与 $\theta$ 的函数关系，提取自类似于 $\eqref{FIG.2}$(a) 的测量值。垂直线表示平均滑动角度。不同的符号对应不同的实验。

Discussion

We seek to understand the combined shear and ultrasonic measurements before sliding [$\eqref{FIG.2}$(a)]. At low shear displacement $U < U_{c}\sim 1\text{ nm}$, the interfacial layer is pinned and responds to shear force elastically as $F(U) = k_{T0}U$ with $k_{T0}$ a constant stiffness.

Beyond a certain threshold $U > U_{c}$, the contact zone flows plastically due to the structure change within the nanometric amorphous film.

我们试图理解滑动前的剪切力和超声波综合测量结果[$\eqref{FIG.2}$(a)]。在低剪切位移 $U < U_{c}\sim 1\text{ nm}$时，界面层被钉住，对剪切力的弹性响应为 $F(U) = k_{T0}U$，其中 $k_{T0}$ 为恒定刚度。

超过一定的临界值 $U > U_{c}$ 后，接触区会因纳米无定形薄膜内的结构变化而发生塑性流动。

In the sphere-plane geometry, this plastic flow may initiate by a fracture at the edge of the adhesive contact [$\eqref{FIG.3}$(b)], along with the growth of a microslip annulus towards the center [$\eqref{FIG.3}$(b)]. To account for the interplay between friction and adhesion, we use here a modified friction model $F = F_{0}(U) + F_{M}(U)$ where the Mindlin friction force is $F_{M}(U) = \mu_{s}W\cos{\theta}\{1 - [1 - 16G^{*}a_{H}U/(3\mu_{s}W\cos{\theta})]^{3/2}\}$ with $G^{*}$ (~ $12\text{ GPa}$) the reduced shear modulus and $a_{H}$ the radius of the Hertzian contact area.

在球体-平面几何中，这种塑性流动可能起始于 粘附接触 边缘的断裂 [$\eqref{FIG.3}$(b)] 以及向中心增长的微滑动环。为了解释摩擦力和附着力之间的相互作用，我们在此使用了修正的摩擦力模型 $F = F_{0}(U) + F_{M}(U)$，其中 $\color{red}{M}$indlin 摩擦力为 $F_{\color{red}{M}}(U) = \mu_{s}W\cos{\theta}\{1 - [1-16G^{*}a_{H}U/(3\mu_{s}W\cos{\theta})]^{3/2}\}$ 其中 $G^{*}$ (~ $12\text{ GPa}$) 是约化剪切模量，$a_{H}$ 是 Hertzian 接触区半径。

$F_{0}(U)\sim k_{T0}S(U)U$ is an effective force associated with the pinned sites or adhesive area; $S(U)$ ( ~ $1$ for $U < U_{c}$) is a parameter depending on the adhesive area and decreases with increasing $U$ (see below). At the macroscopic sliding $U_{\text{max}}$, $F_{M} = \mu_{s}W\cos{\theta}$ leads to the Coulomb-like law $F_{s} = F_{0}(U_{\text{max}}) + \mu_{s}W\cos{\theta}$.

$F_{0}(U)\sim k_{T0}S(U)U$ 是与钉固部位或粘附面积相关的有效作用力；$S(U)$（$U < U_{c}$ 时为 ~ $1$）是一个取决于粘附面积的参数，并随着 $U$ 的增大而减小（见下文）。在宏观上滑动 $U_{\text{max}}$，$F_{M} = \mu_{s}W\cos{\theta}$ 导致类似 Coulomb 定律的 $F_{s} = F_{0}(U_{\text{max}}) + \mu_{s}W\cos{\theta}$。

Let us examine the measured shear stiffness $k_{T}$ [$\eqref{FIG.2}$(a)] in terms of the contact area. Two types of approaches appear available for these weakly adhered interfaces.

让我们用接触面积来检验测量到的剪切刚度 $k_{T}$ [$\eqref{FIG.2}$(a)]。对于这些弱粘附界面，有两种方法可供选择。

(i) One relates $k_{T}$ to a bonded contact of $k_{T}\approx \pi a_{E}^{2}G/h$, where $h$ (~ $1\text{ nm}$) and $G$ (~ $10\text{ MPa}$) are the thickness and elastic modulus of the interfacial film. $a_{E}\approx a_{\text{JKR}}$ is the radius of the adhesive area closely predicted by the Johnson-Kendall-Roberts (JKR) model [$\eqref{FIG.3}$(b)] and can be reduced by either static or high-amplitude oscillatory shear.

(i) 将 $k_{T}$ 与 $k_{T}\approx \pi a_{E}^{2}G/h$ 的粘合接触相关联，其中 $h$ (~ $1\text{ nm}$) 和 $G$ (~ $10\text{ MPa}$) 分别是界面薄膜的厚度和弹性模量。$a_{E}\approx a_{\text{JKR}}$ 是 Johnson-Kendall-Roberts（JKR）模型[$\eqref{FIG.3}$(b)]预测良好的粘合区域半径，且可以通过静态或高振幅振荡剪切来减小。

(ii) Alternatively, $k_{T}$ can be derived from a noncohesive sphere-plane contact using the Hertz-Mindlin model via the static friction $\mu_{s}$ [27] $k_{T}\approx k_{M}[1 - F_{\text{ac}}/(6\mu_{s}W)]$ (at $\theta = 0^{\circ}$), where $k_{M} = 8G^{*}a_{H}(\approx k_{T0})$ is the linear shear stiffness at vanishing $F_{\text{ac}}$.

(ii) 另外，$k_{T}$ 可以通过静摩擦力 $\mu_{s}$ [27] $k_{T}\approx k_{M}[1 - F_{\text{ac}}/(6\mu_{s}W)]$（在 $\theta = 0^{\circ}$ 时）从使用 Hertz-Mindlin 模型的非粘性球面-平面接触中推导出来, 其中 $k_{M} = 8G^{*}a_{H}(\approx k_{T0})$ 是在 $F_{\text{ac}}$ 消失时的线性剪切刚度。

This model does not account for the linear oscillatory response originating from the adhesion, but describes conveniently the decrease of $k_{T}$ in the nonlinear regime, relating $\Delta k_{T}/k_{T}\sim F_{\text{ac}}/\mu_{s}W$ to the growth of the microslip annulus of radius $c = a_{H}[1 - F_{\text{ac}}/(\mu_{s}W)]^{1/3}$ [$\eqref{FIG.2}$(b)], indicated by $\Delta k_{T}/k_{T}\sim (a_{H} - c)/a_{H}$.

These two apparently disconnected approaches predict a similar decrease of $k_{T}$ induced by the high-amplitude $F_{\text{ac}}$ if the reduction of the adhesive area $a_{E}^{2}$ is the same as that of the nonslip area inside $a_{H}^{2}$.

这个模型并没有解释由粘附引起的线性振动响应，但却很方便地描述了非线性机制中 $k_{T}$ 的下降, 并将 $\Delta k_{T}/k_{T}\sim F_{\text{ac}}/\mu_{s}W$ 与半径为 $c = a_{H}[1 - F_{\text{ac}}/(\mu_{s}W)]^{1/3}$ 的微滑环的增长联系起来 [$\eqref{FIG.2}$(b)], 表示为 $\Delta k_{T}/k_{T}\sim (a_{H} - c)/a_{H}$.

如果粘合面积 $a_{E}^{2}$ 的减小与 $a_{H}^{2}$ 内部 非滑动 面积的减小相同，那么这两种看似互不相关的方法就会预测高振幅 $F_{\text{ac}}$ 引发的 $k_{T}$ 下降情况相似。

A softening of $k_{T}$ of about $5\%$ induced by static shear $F_{\text{dc}} = W\sin{\theta}$ is observed before sliding using the low-amplitude ultrasound [$\eqref{FIG.3}$(a)].

It likely arises from the reduction of $a_{E}^{2}$ initiated at the edge via the opening crack. According to the adhesion models, $a_{E}^{2}$ may be reduced under shear from $a_{\text{JKR}}^{2}$ to $a_{H}^{2}$ [$\eqref{FIG.3}$(b)] prior to failure, by developing a fractured zone similar to the microslip during the incipient stage of sliding friction.

在使用低振幅超声波[$\eqref{FIG.3}$(a)]进行滑动之前，可以观察到静态剪切力 $F_{\text{dc}} = W\sin{\theta}$ 引起的 $k_{T}$ 软化，软化程度约为 $5\%$。

这很可能是由于在边缘处通过开口裂纹引起的 $a_{E}^{2}$ 的减少。根据粘附模型，通过在滑动摩擦萌芽阶段形成类似微滑动的断裂带，$a_{E}^{2}$ 在剪切作用下可能会在破坏前从 $a_{\text{JKR}}^{2}$ 降低到 $a_{H}^{2}$ [$\eqref{图3}$(b)]。

This would define an upper limit for the decrease of $S(U_{\text{max}})\sim a_{H}^{2}/a_{\text{JKR}}^{2}\sim 40\%$ and for the softening $\Delta k_{T}/k_{T}\approx 2\Delta a_{E}/a_{E}\sim 50\%$ where we use $a_{H}\sim 3.2\text{ }\mu\text{m}$ for a normal load $W/3\sim 2.2\text{ mN}$ and $a_{\text{JKR}}\sim 5\text{ }\mu\text{m}$ estimated from the interfacial energy of the adhesive layers.

Such decrease of $k_{T}$ larger than $50\%$ is observable at these weakly adhered brittle interfaces ($a_{\text{JKR}}\geq a_{H}$) before failure [$\eqref{FIG.2}$(a)] if applying precisely ramped high-amplitude oscillatory shear, supporting thus the above picture.

这将确定下降的上限 $S(U_{\text{max}})\sim a_{H}^{2}/a_{\text{JKR}}^{2}\sim 40\%$ 以及软化的上限 $\Delta k_{T}/k_{T}\approx 2\Delta a_{E}/a_{E}\sim 50\%$, 其中我们使用 $a_{H}\sim 3.2\text{ }\mu\text{m}$ 用于法向载荷 $W/3\sim 2.2\text{ mN}$，而 $a_{\text{JKR}}\sim 5\text{ }\mu\text{m}$ 是根据粘合层的界面能估算的。

这些弱粘附的脆性界面（$a_{\text{JKR}}\geq a_{H}$）在破裂之前 [$\eqref{FIG.2}$(a)]，如果应用精确施加的高振幅振荡剪切，$k_{T}$ 的减小大于 $50\%$ 是可观察到的，从而支持上述观点。

Moreover, the zero-load threshold deduced from the ultrasonic measurement $F_{0}(U_{\text{max}})\sim k_{T0}S(U_{\text{max}})U_{\text{max}}\sim 2\text{ mN}$ agrees with those obtained from the sliding experiment $F_{0}\approx 1\text{ mN}$.

The complex interplay between adhesion and friction requires further study to describe more precisely the softening regime.

此外，根据超声波测量值 $F_{0}(U_{\text{max}})\sim k_{T0}S(U_{\text{max}})U_{\text{max}}\sim 2\text{ mN}$ 推导出的零负载阈值与滑动实验值 $F_{0}\approx 1\text{ mN}$ 一致。

附着力和摩擦力之间复杂的相互作用需要进一步研究，以更精确地描述软化机制。

We now interpret the triggering of sliding by the non-linear shear ultrasound, far below the static threshold $F_{s}$ [$\eqref{FIG.2}$ (a)] using the friction approach.

As mentioned above, the onset of sliding has been previously observed for a multicontact interface below the angle of sliding $\theta_{s}$, triggered by a low-frequency ($f/f_{0}\sim 10^{-1}$) oscillating force $F_{\text{ac}}$ parallel to $F_{\text{dc}}$.

现在，我们用摩擦法来解释远低于静态阈值 $F_{s}$ [$\eqref{FIG.2}$ (a)] 的非线性剪切超声引发的滑动。

如上所述，之前已经观察到滑动角 $\theta_{s}$ 以下的多接触界面在与 $F_{\text{dc}}$ 平行的低频（$f/f_{0}\sim 10^{-1}$）振荡力 $F_{\text{ac}}$ 触发下开始滑动。

Here $f_{0} = \sqrt{k_{T}/m}/2\pi\sim 1\text{ kHz}$, $k_{T}$ is the interfacial stiffness and $m$ the mass of the slider.

When $F_{\text{ac}}$ is ramped so that $F = F_{\text{dc}} + F_{\text{ac}}$ precisely approaches the threshold $F_{s}$, a self-accelerated unlimited slip occurs reaching the averaged velocity about $0.1\text{ mm/s}$.

这里 $f_{0} = \sqrt{k_{T}/m}/2\pi\sim 1\text{ kHz}$，$k_{T}$ 是界面刚度，$m$ 是滑块的质量。

当 $F_{\text{ac}}$ 被调整，使得 $F = F_{\text{dc}} + F_{\text{ac}}$ 精确地接近阈值 $F_{s}$ 时，会发生自加速的无限滑动，达到平均速度约为 $0.1\text{ mm/s}$。

We use this range of velocity as a criterion to define the triggering of sliding. The bifurcation between the jamming creep regime ($F\leq F_{s}$) and the sliding regime ($F\approx F_{s}$) is well described by a rate- and state-dependent friction law. Unlike the simple Coulomb failure law, the process of thermal activation included in the Rice-Ruina model implies a creep prior to sliding, as observed in the above experiments.

我们将这一速度范围作为定义滑动触发的标准。堵塞蠕变机制（$F\leq F_{s}$）和滑动机制（$F\approx F_{s}$）之间的分叉可以用速度和状态相关的摩擦定律很好地描述。与简单的 Coulomb 破坏定律不同，Rice-Ruina 模型中包含的 热激活过程 意味着滑动之前的蠕变，正如上述实验中所观察到的那样。

In the ultrasonic measurements, the frequency of the oscillating force $F_{\text{ac}}$ is much higher than the eigenfrequency of the slider interface ($f/f_{0}\sim 10^{2}$).

Unlike the above quasistatic regime, $F_{\text{ac}}$ shall not provoke any macroscopic sliding motion of the slider in the high-frequency limit due to the inertial effect.

在超声波测量中，振荡力 $F_{\text{ac}}$ 的频率远高于滑块界面的本征频率（$f/f_{0}\sim 10^{2}$）。

与上述准静态机制不同，由于惯性效应，$F_{\text{ac}}$ 在高频极限下不会引起滑块的任何宏观滑动运动。

Here we propose a new scenario of the triggering of sliding by the nonlinear shear ultrasound. As stated above, the high-amplitude oscillation $F_{\text{ac}}$ most significantly reduces the nonslip area $\Sigma_{s}$ from the initial $a_{H}^{2}$ to $c^{2}$ via the growth of the microslip annulus.

在此，我们提出了非线性剪切超声触发滑动的新方案。如上所述，高振幅振荡 $F_{\text{ac}}$ 通过 微滑环 的增长，最显著地将非滑动区域 $\Sigma_{s}$ 从初始的 $a_{H}^{2}$ 减少到 $c^{2}$。

Accordingly, such high-frequency oscillation which works as a lubrication decreases the static threshold $F_{s} = \sigma_{s}\Sigma_{s}$ and triggers the macroscopic sliding under a static shear $F_{\text{dc}}$ below the shear threshold $F_{s}$. Figure $\eqref{FIG.4}$(a) displays the reduced static threshold $\sin{\theta_{s}^{*}} = F_{s}^{*}/W$ as a function of the oscillatory shear $F_{\text{ac}}$.

因此，这种起润滑作用的高频振荡会降低静态阈值 $F_{s} = \sigma_{s}\Sigma_{s}$，并在低于剪切阈值 $F_{\text{dc}}$ 的静态剪切下引发宏观滑动。图 $\eqref{FIG.4}$ (a) 显示了减小的静态阈值 $\sin{\theta_{s}^{*}} = F_{s}^{*}/W$ 与振荡剪切力 $F_{\text{ac}}$ 的函数关系。

We additionally notice that the triggering of sliding is independent of the polarization of shear oscillation relative to the sliding direction. Such behavior could be captured by the Mindlin friction model,

$$ \begin{aligned} \sin{\theta_{s}^{*}}/\sin{\theta_{s}} &= F_{s}^{*}/F_{s} \text{~} c^{2}/a_{H}^{2}\\ &\approx 1 - (2/3)F_{\text{ac}}/(\mu_{s}W\cos{\theta_{s}}). \end{aligned}\tag{1}\label{eq1} $$

我们还注意到，滑坡的触发与剪切振荡相对于滑动方向的极化无关。Mindlin 摩擦模型可以捕捉到这种行为:

$$ \begin{aligned} \sin{\theta_{s}^{*}}/\sin{\theta_{s}} &= F_{s}^{*}/F_{s} \text{~} c^{2}/a_{H}^{2}\\ &\approx 1 - (2/3)F_{\text{ac}}/(\mu_{s}W\cos{\theta_{s}}). \end{aligned} $$

4 $\tag{FIG.4}\label{FIG.4}$

(a) Reduced static threshold versus the oscillating force for CH3 (blue symbols) and COOH interfaces (red symbols). Circles and diamonds correspond to two sets of experiments for a given direction of shear oscillation ($\alpha = 0^{\circ}$), stars are related to those where the shear resonator is rotated by $\alpha = 45^{\circ}$ for CH3 and by $90^{\circ}$ for COOH films. The straight lines correspond to best fits with Eq. $\eqref{eq1}$. Inset: angles of sliding of a steel cylinder versus direction of polarization $\alpha$[$\eqref{FIG1}$(a)], measured at low-amplitude (black circles) and high-amplitude shear ultrasound (green points). (b) Normalized vibrational energy necessary for the triggering of sliding (see the text). The solid curve delimits the jammed and flowing states. Inset: jamming phase diagram.

(a) CH3（蓝色符号）和 COOH（红色符号）界面的降低静态阈值与振荡力的关系。圆圈和菱形对应于给定剪切振荡方向（$\alpha = 0^{\circ}$）的两组实验，星形对应于剪切谐振器旋转了 $\alpha = 45^{\circ}$（CH3）和 $\alpha = 90^{\circ}$（COOH）的实验。直线对应于公式 $\eqref{eq1}$ 的最佳拟合。插图：在低振幅（黑圆圈）和高振幅（绿点）剪切超声下测量的钢圆柱体滑动角与偏振方向 $\alpha$[$\eqref{FIG1}$(a)]的关系。 (b) 触发滑动所需的归一化振动能量（见正文）。实心曲线划分了堵塞态和流动态。插图：堵塞相图。

Comparison to the data infers $\mu_{s}\sim 2$ for COOH and $\mu_{s}\sim 1$ for CH3 interfaces, larger than those obtained from the angle of sliding but consistent with the previous data.

Furthermore, we have triggered a similar sliding at a multicontact interface between a steel cylinder of diameter $5\text{ mm}$ and the quartz. As shown in $\eqref{FIG.4}$(a) (inset), we observe the decrease of $\theta_{s}$ at the highest $U_{\text{ac}} > 5\text{ nm}$ polarized in the different azimutal angle $\alpha$.

与数据相比，我们推断出 COOH 和 CH3 界面的滑动角度分别为 $\mu_{s}\sim 2$ 和 $\mu_{s}\sim 1$，比滑动角度得出的结果要大，但与之前的数据一致。

此外，我们还在直径为 $5\text{ mm}$ 的钢圆柱体与石英之间的多接触界面上触发了类似的滑动。如 $\eqref{FIG.4}$(a)（插图）所示，我们观察到在最高 $U_{\text{ac}} > 5\text{ nm}$ 在不同的方位角 $\alpha$ 下极化时，我们观察到 $\theta_{s}$ 的下降。

This result suggests that when the local static threshold is reached, the nonslip contact area of the interface $\Sigma_{s} = N\pi\bar{a}^{2}$ ($N$ is the number of asperities and $\bar{a}$ the average contact radius) would fluidize by progressive sliding of asperities, reducing $F_{s}$.

这一结果表明，当达到局部静态阈值时，界面的非滑动接触面积 $\Sigma_{s} = N\pi\bar{a}^{2}$ （$N$ 为微尖的数量，$\bar{a}$ 为平均接触半径）将通过微尖的逐渐滑动而流化，从而降低 $F_{s}$。

Considering the scalar nature of the fluidization effect insensitive to the oscillation polarization, we examine the decrease of the static threshold versus the vibrational energy $E_{v}\approx(A/2)KU_{\text{ac}}^{2}$.

Here $A$ is the ratio of the vibrating surface area to the area of contact. In $\eqref{FIG.4}$(b), the solid curve described by $E_{v}/B\approx [1 - \sin{\theta_{s}^{*}}/\sin{\theta_{S}}]^{2}$ with $B = 9AK\mu_{s}^{2}W^{2}/8k^{2}\sim 10^{10}\text{ kT}$ (with $kT$ the thermal energy) delimits roughly the jammed and sliding states of the frictional system, as expected in a jamming diagram where $E_{v}$ would play the role of an effective temperature $T_{\text{eff}}$ [inset of $\eqref{FIG.4}$(b)].

考虑到流化效应的标量性质对振荡极化不敏感，我们研究了静态阈值相对于振动能量 $E_{v}\approx(A/2)KU_{\text{ac}}^{2}$ 的下降情况。

这里的 $A$ 是振动表面积与接触面积之比。在 $\eqref{FIG.4}$(b), 由 $E_{v}/B\approx [1 - \sin{\theta_{s}^{*}}/\sin{\theta_{S}}]^{2}$ 描述的实心曲线，其中 $B = 9AK\mu_{s}^{2}W^{2}/8k^{2}\sim 10^{10}\text{ kT}$ （其中 $kT$ 为热能）大致划分了摩擦系统的堵塞和滑动状态. 在堵塞图中，$E_{v}$ 将扮演有效温度 $T_{\text{eff}}$ 的角色[$\eqref{FIG.4}$(b) 插图].

It is important to notice that the oscillation period ~ $0.2\text{ }\mu\text{s}$ used in this work is much smaller than the characteristic relaxation time $>1\text{ ms}$ in the thin interfacial layer. This high-frequency shear (~ $5\text{ MHz}$) allows maintaining the interface in the fluidized state within the microslip annulus (or randomly distributed patches) and prevents from healing, unlike the low-frequency shear ($< 0.1\text{ kHz}$).

值得注意的是，这项工作中使用的振荡周期 ~ $0.2\text{ }\mu\text{s}$ 远远小于薄界面层中的特征弛豫时间 $>1\text{ ms}$。与低频剪切（$< 0.1\text{ kHz}$）不同的是，这种高频剪切（~ $5\text{MHz}$）可以使界面在微滑动环（或随机分布的斑块）内保持流化状态，防止愈合。

Likewise, at a given amplitude, oscillations of normal force $W_{\text{ac}}$ of lower frequency ($< 5\text{ kHz}$) whose effect is further reduced by $W_{\text{ac}} = F_{\text{ac}}/\mu_{s}\sim 5F_{\text{ac}}$, would be less efficient than the present shear oscillations for fluidizing the nonslip contact area. More work is needed to account for high-frequency oscillation or $T_{\text{eff}}$ in the rate and state models.

同样，在给定振幅下，较低频率（$< 5\text{kHz}$）的法向力 $W_{\text{ac}}$ 的振荡（其效果会因 $W_{\text{ac}} = F_{\text{ac}}/\mu_{s}\sim 5F_{\text{ac}}$ 而进一步减弱）对于非滑动接触区域的流化效果会比目前的剪切振荡更差。在速率和状态模型中考虑高频振荡或 $T_{\text{eff}}$ 还需要做更多的工作。

Finally, the proposed scenario for the onset of sliding triggered by nonlinear shear ultrasound is helpful for understanding the reduction of the yielding threshold of granular media by the acoustic fluidization. The applied vibration could generate high-frequency acoustic emission by the rearrangement of grains and the rupture of asperities.

As stated above, the highfrequency oscillatory shear reduces the apparent coefficient of static friction between grains and consequently lowers the static friction coefficient of granular layers which can slip at lower shear stress.

最后，所提出的非线性剪切超声引发滑动的情景有助于理解声学流化降低颗粒介质屈服阈值的原理。外加振动可通过颗粒的重新排列和微尖的破裂产生高频声发射。

如上所述，高频振荡剪切会降低颗粒之间的表观静摩擦系数，从而降低颗粒层的静摩擦系数，使其在较低的剪切应力下发生滑移。

The necessary energy for the rearrangement of grains by sliding is 2 orders of magnitude smaller than the energy barrier by jumping, $E_{j}\sim Wh_{0}\sim 10^{12}\text{ kT}$ where $h_{0}\sim 1\text{ }\mu\text{m}$ is the height of a surface asperity. In terms of acoustic fluidization, the mechanism discussed here offers an alternative to that relying on the balance of the overburden via the acoustic pressure, which needs an unusually high acoustic energy.

通过滑动重新排列颗粒所需的能量比跳跃的能量障碍小两个数量级，即 $E_{j}\sim Wh_{0}\sim 10^{12}\text{ kT}$，其中 $h_{0}\sim 1\text{ }\mu\text{m}$ 是表面微尖的高度。就声波流化而言，这里讨论的机制提供了一种替代方案，即通过声压来平衡覆盖层，这需要异常高的声波能量。

In conclusion, the shear stiffness of weakly adhered interfaces is significantly weakened before sliding, either by static shear or by high-amplitude oscillatory shear, due to the opening crack initiated at the edge and the development of microslip zones. The present measurements would enable us to bridge two distinct approaches for describing the failure, namely, fracture and microslip propagation.

总之，无论是静态剪切还是高振幅振荡剪切，弱粘附界面的剪切刚度在滑动前都会明显减弱，这是由于在边缘处产生了开口裂纹并形成了微滑动区。目前的测量结果将使我们能够用两种不同的方法（即断裂和微滑动传播）来描述失效。

The onset of sliding triggered far below the static threshold by the nonlinear shear ultrasound is due to the fluidization of the nonslip contact area, reducing the apparent coefficient of friction. The intensity rather than the polarisation of the shear ultrasound matters in such unjamming transition from static to sliding friction, suggesting its role as an effective temperature. This work would provide an alternative mechanism to understand the granular fault weakening by the acoustic fluidization via high-frequency scattered shear waves and the effect of the dynamic noise in flowing systems.

非线性剪切超声在远低于静态阈值的情况下触发滑动的起因是非滑动接触区域的流体化，从而降低了表观摩擦系数。在这种从静摩擦到滑动摩擦的解堵塞转变中，剪切超声波的强度, 而不是极化, 极其重要，这表明它起到了有效温度的作用。这项研究提供了另一种机制，可以通过高频散射剪切波的声学流化和流动系统中动态噪声的影响来理解颗粒断层的削弱。