Abstract

Nonlinear phononic materials enable superior wave responses by combining nonlinearity with their inherent periodicity, creating opportunities for the development of novel acoustic devices. However, the field has largely focused on reversible nonlinearities, whereas the role of hysteretic nonlinearity remains unexplored. In this work, we investigate nonlinear shear wave responses arising from the hysteretic nonlinearity of frictional rough contacts, and harness these responses to enable programmable functions. By using a numerical approach, we solve the strongly nonlinear problem of shear wave propagation through a single contact and a periodic array of contacts, accounting for frictional effects. Specifically, the Jenkin friction model with experimentally obtained properties is used to capture the effects of stick–slip transition at the contacts. Results show that friction gives rise to shear-polarized eigenstrains, which are residual static deformations within the system. We then demonstrate how eigenstrain generation in multiple contacts can enable programmable functionalities such as an acoustically controlled mechanical switch, precision position control, and surface reconfigurability. Overall, our findings open new avenues for designing smart materials and devices with advanced functionalities via acoustic waves using the hysteretic nonlinearity of frictional contacts.

1 Introduction

Phononic materials are periodic media designed to manipulate the propagation of mechanical waves in both space and time. While these materials have been predominantly studied in the linear regime [1], recent studies have shown that nonlinearity yields unprecedented wave responses such as solitary waves [2], discrete breathers [3], amplitude-dependent band gaps [4,5], energy transfer between frequencies/modes [5], among others, as detailed in a review by Patil and Matlack [6]. Yet, the focus so far has been on the propagation of longitudinal waves and nonlinear mechanisms, either weakly or strongly, that govern these wave responses. On the other hand, shear waves have received very less attention despite being the fundamental type of wave supported by solid materials. It remains an open question of how shear waves evolve in the presence of nonlinearities in phononic materials. Addressing this question is of foundational significance and could open opportunities for developing novel acoustic devices with advanced functionalities, such as mechanical programmability and surface reconfigurability, simply by employing shear waves.

Furthermore, the types of nonlinearities explored in phononic materials are thus far largely limited to reversible nonlinear laws, where nonlinear forces (or stresses) depend only on the current displacements (or strains). Such memory-independent nonlinearity typically stems from material constitutive laws under elastic limit [5], finite strains [7], and normal contact laws [2]. On the other hand, hysteretic (or irreversible) nonlinearity, where current forces also depend on their history, offers the potential to obtain nonlinear wave responses typically not possible with reversible nonlinearity. Although the effects of hysteresis on wave propagation have been considered, either through viscous damping [811] or elastic-plastic law [12], there still exists a lack of knowledge of how wave interacts when nonlinearity itself is memory dependent. Such knowledge could provide a new platform to control wave-energy propagation in materials by introducing hysteretic nonlinearity and intentionally programming loading history.

One of the physical sources of hysteretic nonlinearity is friction between contacting surfaces. Coulomb’s law, which is postulated to model friction between two surfaces with a coefficient of friction (μ), dictates the existence of two physical states of contact—stick (T < μN, where T and N are tangential and normal forces acting at the contact, respectively) and slip (T = μN). As a result, applied forces trigger hysteresis that physically correspond to surface wearing [13] and heat generation [14] at the contact. Consequently, friction between surfaces is widely researched in tribology [13], nonlinear dynamics [15], structural health monitoring [16], and nondestructive evaluation [1719]. It has been traditionally studied to control its effects and increase the component life [15], but it is now being explored as a tool to enable soft robotic devices that latch onto objects [20] and design mechanical metamaterials that absorb energy under quasi-static loading [21,22]. However, it is yet to be explored how friction can be utilized to control wave propagation and enable smart functionalities.

Despite extensive studies on phononic materials with contacts (or interfaces), the effects of friction have been studied only in a limited capacity. Granular crystals, which are ordered arrangements of spherical particles, have majorly focused on normal contact interaction only, i.e., Hertzian law [23]. A few of those studies analyzed torsional (or rotational) and transverse waves but modeled the tangential interactions through the Hertz–Mindlin theory, which assumes no sliding at the contacts [2426]. Only a few studies have incorporated friction to study dissipation [8,27] and hysteretic torsional coupling [28] in granular crystals. Recently, an array of frictional disks [29] and an acoustic metamaterial with internal frictional contacts [30] have been studied to report the transition from static to dynamic friction [29] and low-frequency attenuation bands [30], respectively. These limited studies demonstrate the potential of the hysteretic nonlinearity of friction in enriching the wave dynamics of phononic materials.

Previously, we studied one-dimensional (1D) phononic materials with periodic rough contacts, which only involved normal interaction [3134]. Our studies revealed a plethora of interesting nonlinear responses of longitudinal waves. In the weakly nonlinear regime (externally precompressed), the material supported nonlinear frequency conversion, wave self-demodulation, acoustic dilation, and wave mixing [31,32]. In the strongly nonlinear regime (uncompressed or lightly compressed), we reported energy localization through acoustic resonance, propagation of solitary waves of stegoton nature, and disintegration of input waves into compression pulses, rarefaction fronts, and oscillatory tails [33,34]. These findings highlighted the potential of locally nonlinear rough contacts in designing materials with global nonlinear responses. This article shifts focus to the tangential interaction of these contacts and aims to (1) illustrate nonlinear shear wave responses arising from hysteretic nonlinearity of frictional contact and (2) harness these responses to enable advanced acoustic/mechanical functionalities with programmable characteristics.

To this end, we study shear wave propagation through rough contacts with single and array configurations, which are externally precompressed. The frictional instability of the contacts gives rise to strong nonlinearity within the system. To solve this problem, an interactive numerical framework that couples a finite element model solving 1D wave motion and an analytical model with contact friction law is employed. Specifically, we consider the Jenkins friction model with experimentally obtained properties at the contacts. We first reveal how contact instability gives rise to amplitude-dependent eigenstrains and then illustrate programmable functions by extending the analysis to complex configurations with multiple contacts.

This article is organized as follows: Sec. 2 presents the numerical framework employed in this study, including the reduced-order model of rough contacts, experimentally obtained frictional properties, and finite element wave propagation model. By using the model, Sec. 3 discusses eigenstrains generated from wave-induced friction at a single contact. Then, Sec. 4 includes the analysis of configurations with multiple contacts and numerical demonstrations of their programmability. Finally, we conclude the study and discuss open questions and potential research directions in Sec. 5.

2 Modeling Wave Propagation Through Frictional Contacts

2.1 Reduced-Order Model of Rough Contacts.

Rough contacts are interfaces between surfaces that have microscale asperities (Fig. 1(a)). These asperities exhibit a nonlinear response due to sliding when subjected to tangential loads beyond the friction limit. Here, we study shear wave interactions with these contacts at wavelength, λ, much larger than the asperity size, δ, focusing on the effective nonlinear response of a collection of asperities. To study such mesoscale interactions, we employ a reduced-order model of the contacts, which involves treating each contact as a nonlinear spring with experimentally obtained effective contact properties (Fig. 1). We neglect the interfacial mass at the contact since the effects of gap (or cavity) resonances are negligible at these wavelength–asperity ratios [35].

Fig. 1
Experimentally informed reduced-order model of rough contacts: (a) modeling of rough contact as a nonlinear tangential spring defined through (b) Jenkins friction law. Experimentally obtained (c) absolute contact stiffness, Kt, and (d) coefficient of friction, μ, as contacting surfaces evolve. Insets in (d) highlight wear during surface evolution and the actual area of contact (shaded black patch) during the steady-state response.
Fig. 1
Experimentally informed reduced-order model of rough contacts: (a) modeling of rough contact as a nonlinear tangential spring defined through (b) Jenkins friction law. Experimentally obtained (c) absolute contact stiffness, Kt, and (d) coefficient of friction, μ, as contacting surfaces evolve. Insets in (d) highlight wear during surface evolution and the actual area of contact (shaded black patch) during the steady-state response.
Close modal

2.1.1 Tangential Contact Law.

We consider the Jenkins friction model [36] to account for frictional effects and tangential compliance of roughness (Fig. 1(b)). This model captures stick–slip transition at the contacts and further assumes that the compliance at real contacts is governed by the stiffness corresponding to the elastic deformation of the micro-asperities at the contact interfaces [37] and the bulk elastic deformation at the macroscopic contact scale [38]. Accordingly, the tangential interaction at the contacts is expressed as follows:
τ(t)={τ(tΔt)+κtΔb(t),stick,sgn[b˙(t)]τc,slip
(1)
where τ is shear stress at the contact, b is corresponding tangential contact deformation, and b˙ (= db/dt) is tangential velocity. t is time variable and Δt is time-step such that displacement increment, Δb(t) = b(t) − b(t − Δt). κt is tangential contact stiffness per unit area, and τc (= μσ0) is the friction limit at which the contacts start to slide. The contact is under the state of normal pre-stress, σ0. We treat the static and kinematic coefficient of friction as the same, i.e., no abrupt jump in the friction force on the onset of sliding. From Eq. (1), it can be seen that the contact law is history dependent and follows different loading–unloading paths due to contact sliding (Fig. 1(b)). While an intermediate state of micro-slip (or partial slip) exists in real contacts, we assume no such state in our study, i.e., all asperities slide simultaneously once the friction limit is reached.

2.1.2 Experimentally Obtained Friction Properties.

We conducted experiments to inform the contact properties, κt and μ, to be used as input parameters in the tangential contact law of Eq. (1). Specifically, we measured friction hysteresis loops on a high-frequency friction rig (refer to the experimental schematics of Fig. 2(a) and Ref. [39] for more details). The setup consists of an electrodynamic shaker that applies a harmonic sinusoidal excitation in the tangential direction to two contacting specimens, while two laser Doppler vibrometers and a force transducer measure relative motion between the specimens, b, and friction (or tangential) force, T, respectively. Experiments were conducted on two aluminum (AL6061-T6511) specimens, which were roughened to a root-mean-square surface roughness of Rq ∼ 1.1 μm using P1200 grit sandpaper. The loading conditions for hysteresis measurements were as follows: N = 60 N and b = 15 μm at an excitation frequency of 100 Hz at room temperature, which is commonly used for reliable measurements on the friction rig.

Fig. 2
Modeling nonlinear shear wave propagation through rough contacts. (a) Coupled finite element-analytical model informed by experimental contact properties. (b) Physical model of an externally precompressed single and array of rough contacts and corresponding (c) finite element setup. Black vertical lines in (c) are contacts modeled through nonlinear spring-equivalent boundary conditions. (d) A typical wave excitation in the form of a wave packet (solid black) with a certain number of cycles, which is achieved by imposing a rectangular function (red) on a harmonic wave (dashed back). (Color version online.)
Fig. 2
Modeling nonlinear shear wave propagation through rough contacts. (a) Coupled finite element-analytical model informed by experimental contact properties. (b) Physical model of an externally precompressed single and array of rough contacts and corresponding (c) finite element setup. Black vertical lines in (c) are contacts modeled through nonlinear spring-equivalent boundary conditions. (d) A typical wave excitation in the form of a wave packet (solid black) with a certain number of cycles, which is achieved by imposing a rectangular function (red) on a harmonic wave (dashed back). (Color version online.)
Close modal

We recorded the hysteresis loops (Tb relation) and evaluated absolute contact stiffness, Kt, as the slope of the stick portion of the loop and coefficient of friction as, μ = Tc/N (schematics of Fig. 2(a)). These values are evaluated while the surface texture evolved from wear until a steady-state response was reached (Figs. 1(c) and 1(d)). The steady-state properties are listed in Table 1. Microscopic measurements were also used to determine the actual contact area (shaded black patch in the inset of Fig. 1(d)), from which the contact stiffness per unit area (κt = Kt /a, where a is the actual contact area) was obtained and used as input for the friction law of Eq. (1) along with the coefficient of friction.

Table 1

Friction rig experimental parameters and evaluated steady-state contact frictional properties

ParametersObtained values
Normal load, N60 N
Actual worn-out area, a1.2 mm2
Pre-stress, σ0 = N/a50 MPa
Tangential displacement amplitude, b15 μm
Coefficient of friction, μ0.77
Absolute stiffness, Kt22 N/μm
Stiffness per unit area, κt = Kt /a18.33e12 N/m3
ParametersObtained values
Normal load, N60 N
Actual worn-out area, a1.2 mm2
Pre-stress, σ0 = N/a50 MPa
Tangential displacement amplitude, b15 μm
Coefficient of friction, μ0.77
Absolute stiffness, Kt22 N/μm
Stiffness per unit area, κt = Kt /a18.33e12 N/m3

To ensure that the contact properties evaluated experimentally can be applied in the wave propagation analysis, we employ identical loading conditions and sample material in our numerical model. Additionally, we make the following assumptions:

  1. The contacts are already in a steady-state condition from surface wearing, i.e., frictional properties do not change with time as waves propagate through the contact.

  2. Both the coefficient of friction and tangential stiffness are rate independent, i.e., independent of the frequency of excitation. This is a reasonable assumption since the coefficient of friction obtained from quasi-static and frequency-dependent approaches differ insignificantly [19]. Further, contact stiffnesses obtained through the quasi-spring model are frequency independent for frequencies that correspond to wavelengths much larger than asperity size [35]. Thus, the frictional properties of contacts with a roughness of the orders of microns obtained at 100 Hz through friction rig can be reasonably applied to study wave propagation up to ultrasonic frequencies (i.e., up to a few MHz).

  3. On the basis of parametric measurements of hysteresis loops on friction rig, we assume that both coefficient of friction and tangential stiffness do not (or negligibly) depend upon the tangential displacement amplitude.

2.2 Nonlinear Wave Propagation Model.

We employed a coupled analytical-finite element model, as adopted from Ref. [40], to study 1D strongly nonlinear shear wave propagation through rough contacts (Fig. 2(a)). Specifically, the experimentally informed Jenkins friction law from Sec. 2.1 is externally solved in matlab and linked to finite element analysis of wave propagation in comsol multiphysics. The wave propagation analysis was conducted as a time-dependent study, where the boundary conditions (BCs) at the contact interfaces are determined externally at each time-step through the friction model. To capture the effects of hysteresis, the force history is stored and re-used at each time-step to evaluate the current forces.

The physical system under consideration consists of two semi-infinite waveguides (ΓL and ΓR, where L and R refer to left and right, respectively) with rough contacts at the center, precompressed externally by σ0 (Fig. 2(b)). We study two different configurations of this system: (1) a single rough contact and (2) an array of contacts spaced a distance l apart (referred to as phononic material). The finite element model of this system is based on our previous work on nonlinear longitudinal waves [31,33] but modified for shear wave propagation (Fig. 2(c)). The simulation considers 1D wave propagation in the x-direction, modeled within a two-dimensional (2D) plane strain framework by applying antisymmetry BCs on the top and bottom edges of the model.

The contacts are assumed to be in a state of constant normal stress, σ0, such that σ0 < 0, before the incident shear wave strikes the interface. A shear wave packet of amplitude, U, at frequency, f, is excited at the left boundary (Fig. 2(d)). We conduct simulations with a wave packet consisting of a single cycle and 15 cycles. Single-cycle pulse facilitates sequential excitation of pulses over a short time useful for programmability demonstration. In contrast, a finite number of cycles enables steady-state signals and better signal postprocessing to illustrate wave–contact mechanics. A short smoothing window is also applied at the start and end to suppress transient effects from sudden wave excitation. All the results presented in this article correspond to an excitation frequency of f = 0.5 MHz, as a test case.

For a general understanding of the nonlinear responses, we define two normalized parameters:

  1. ξ = τi/μσ0 is the normalized excitation, where τi is the maximum incident shear stress generated by the excitation wave. Accordingly, ξ increases with the increasing excitation amplitude. Note that ξ < 1 corresponds to contact always being in the stick regime (linear case) as τi < μσ0, whereas ξ > 1 corresponds to the nonlinear case where contacts switch between the stick and slip zones.

  2. Λ = λf /l is the normalized wavelength, where λf is the wavelength in the bulk aluminum at excitation frequency, f (i.e., λf = cs/f, where cs is shear wave speed). In this work, we focus on the long-wavelength regime, i.e., λfl. This regime facilitates the illustration of programmable functionality from an array of contacts as this condition reduces the lag between the wavefront and trailing reflections.

3 Eigenstrain Generation From Wave-Induced Friction

In this section, we study shear wave propagation through a single rough contact (Fig. 2(b)) and present the emergence of eigenstrains within the system.

The shear wave interaction with the contact fundamentally affects wave transmission and causes distortion of waveform (Figs. 3(a) and 3(b)). This is because when the excited wave interacts with a contact, it partially reflects and transmits due to the acoustic impedance. If the wave amplitude exceeds the friction limit, then the contact surfaces slide. This limits the maximum tangential force that can be transmitted and leads to the stress amplitude being clipped at τc (Fig. 3(a)). Such clipping effectively generates odd-order harmonics of the input frequency, as previously reported in the nondestructive evaluation of microcracks [1719].

Fig. 3
Wave evolution through a rough contact. Wave profile for ξ = 2.76, as an example, in terms of normalized shear (a) stress, τ, and (b) displacement, u, for the incident (gray) and transmitted (black) waves. Time on the x-axis is adjusted to overlap both waves for comparison. (c) Schematics with amplified deformation, illustrating the emergence of shear-polarized eigenstrains, Δ, after wave–contact interaction. Dashed rectangles indicate the envelope of reflected and transmitted waves in the waveguides carrying the static deformations with them. (d)–(f) Hysteresis loops during wave–contact interaction. Top row—contact deformation (black) and stresses (gray), and bottom row—accumulative hysteresis loops for the (d) first, (e) subsequent, and (f) last cycle of contact deformation. Markers indicate transition points of the hysteresis loops, and their instances are depicted as dashed lines with labels in the time domain. Red color indicates the start and end state of each hysteresis loop. (Color version online.)
Fig. 3
Wave evolution through a rough contact. Wave profile for ξ = 2.76, as an example, in terms of normalized shear (a) stress, τ, and (b) displacement, u, for the incident (gray) and transmitted (black) waves. Time on the x-axis is adjusted to overlap both waves for comparison. (c) Schematics with amplified deformation, illustrating the emergence of shear-polarized eigenstrains, Δ, after wave–contact interaction. Dashed rectangles indicate the envelope of reflected and transmitted waves in the waveguides carrying the static deformations with them. (d)–(f) Hysteresis loops during wave–contact interaction. Top row—contact deformation (black) and stresses (gray), and bottom row—accumulative hysteresis loops for the (d) first, (e) subsequent, and (f) last cycle of contact deformation. Markers indicate transition points of the hysteresis loops, and their instances are depicted as dashed lines with labels in the time domain. Red color indicates the start and end state of each hysteresis loop. (Color version online.)
Close modal

Interestingly, the wave interaction with the contact also causes a total tangential static deformation, Δ, at the end of the wave propagation. Note the transmitted wave measured in the ΓR waveguide (black in Fig. 3(b)) corresponds to u/U = 0 before wave–contact interaction and now at an offset u/U > 0 (i.e., ΔT ≠ 0). The same offset is created in the reflected wave (ΔR) in ΓL waveguide but with the opposite sign (results not shown). This offset physically corresponds to the change in the relative position of the contact interface, and therefore of the elastic media that the wave occupies (Fig. 3(c)). The system where both ΓL and ΓR were aligned before wave–contact interaction are now offset by Δ = ‖ΔR‖ + ‖ΔT‖. This effect is analogous to the permanent static deformation (or residual strain) of an elastic-perfectly plastic body under cyclic loading. In fact, the physical mechanisms of these two phenomena are indeed the same. They both are due to the hysteretic response and cause a change in the equilibrium state of the system at the end of loading. Further, the system attains a stress-free state at the end of wave propagation despite the presence of the static deformation (correlate the stresses in Fig. 3(a) with displacements in Fig. 3(b) for t > 35 μs). Previously, Qu et al. [41] have discussed that acoustic radiation-induced strains in nonlinear material produce no stresses and can be viewed as eigenstrains. Similarly, our study also observed shear deformations generated by wave–contact interaction, but without shear stresses. Therefore, we refer to these deformations as “shear-polarized eigenstrains”. While eigenstrains can also be generated in nonhysteretic systems with even-order nonlinearities, for example, materials with quadratic nonlinearity [4144], they are only enabled by longitudinal waves. This is because shear wave self-interaction is not supported in the isotropic nonlinear material [45,46]. Thus, the interaction of shear waves with a frictional contact causes the unique response of a shear-polarized eigenstrain.

The role of hysteretic nonlinearity in generating eigenstrains can be clearly understood through hysteresis loops between contact forces and deformations. These loops occur when wave cycles transition contact between the stick and slip state, and evolve as subsequent wave cycles propagate through the contact (Figs. 3(d)3(f)). The loop starts at the stress-free (τ/τc = 0) and undeformed (b/U = 0) condition (i.e., state A) during the first cycle of contact deformation (Fig. 3(d)). Initially, elastic deformations of rough asperities transmit forces as dictated by the contact stiffness, κt, until the friction limit is reached (i.e., state B). Further deformation from the wave causes contact sliding, while the force remains constant (path BC). Note, we refer to the sliding of the contact in the positive y-direction as “up” sliding (i.e., τ˙=0 and b˙>0). Once the maximum contact displacement is reached (state C), the contact starts deforming in the reverse direction owing to the oscillatory nature of the wave. Since the contact forces at each instant depend on their previous values, the contact response now takes a different path (path CDE). During this reverse motion, the contact first recovers the elastic deformation that occurred during the path AB and then starts deforming asperities in the negative direction (path DE). Eventually, when the friction limit is reached, the contact starts sliding in the other direction (path EF), which is “down” sliding (i.e., τ˙=0 and b˙<0). Due to oscillatory excitation, the contact returns to its initial configuration of b = 0 (i.e., point I) at the end of the one full cycle but with stress, τc. This is because the second cycle of the wave packet now initiates its interaction with the contact even before the system attains its original stress-free state. Effectively, the first cycle contributes to changing the state of the system by undergoing different sliding in the up (path BC) and down (path EF) directions. Note that during the path EF, the contact first has to work against the up-sliding (BC) to reach the initial configuration and then create additional sliding in the opposite direction. However, the last path HI does not fully recover the sliding that occurred in the down direction, ultimately changing the equilibrium state of the system.

Unlike the first cycle, the subsequent cycles (Fig. 3(e)) use the final state of the first cycle (state I) as the initial state and follow a hysteresis loop IC′ − D′ − E′ − F′ − G′ − H′ − I′, where the prime denotes transition points in the subsequent cycles. Therefore, the sliding in both directions (up and down) is the same for the later cycles (HC′ = EF′). Finally, the instant beyond which the contact does not transition between stick–slip regimes governs the final state of the system. For example, state Bf is the instant of the final cycle of the contact deformation beyond which contact deformation oscillates with small amplitudes only, i.e., within the stick regime (Fig. 3(f)). This state returns to its stress-free state (state Df) by recovering the elastic deformation at the contact (path BfCfDf), resulting in a shear-polarized eigenstrain, Δ.

The generation of eigenstrain is also amplitude dependent (Fig. 4(a)). The eigenstrain is zero when ξ < 1 (i.e., contact stays in the stick regime) and nonzero for ξ > 1 (i.e., when contacts undergo stick–slip transition) with a maximum eigenstrain for a specific excitation amplitude of ξ = 1.58. To get insights into how excitation amplitude governs eigenstrain generation, we present the two representative cases of hysteresis loops (Fig. 4(b)). These loops highlight that despite the same initial condition (green marker), the last instant of contact sliding (blue marker) is much closer to the first instant of contact sliding (black marker) for ξ = 4 than ξ = 1.95. These differences are due to the different amount of hysteresis that occurs at the contact for a given excitation amplitude. For ξ = 4, the amount of sliding in both the “up” and “down” directions during the first cycle are relatively closer compared to that of ξ = 1.95 for the same friction limit. Consequently, the static offset at the end of the last sliding is smaller for ξ = 4 and therefore corresponding eigenstrain.

Fig. 4
Amplitude-dependent eigenstrain generation. (a) Normalized eigenstrains for different excitation amplitudes when μ = 0.77 (circle)—as obtained through experiments and μ = 0.5 (cross)—as an alternate example. (b) Hysteresis loops for two different representative cases of ξ as shown in (a) with star markers. Contact deformations are normalized by the maximum deformation before the first sliding (b*) such that b/b* = 1 on the onset of the first contact sliding. Green, black, blue, and red dots are the instances corresponding to the start, first sliding, last sliding, and end states of the system, respectively. (Color version online.)
Fig. 4
Amplitude-dependent eigenstrain generation. (a) Normalized eigenstrains for different excitation amplitudes when μ = 0.77 (circle)—as obtained through experiments and μ = 0.5 (cross)—as an alternate example. (b) Hysteresis loops for two different representative cases of ξ as shown in (a) with star markers. Contact deformations are normalized by the maximum deformation before the first sliding (b*) such that b/b* = 1 on the onset of the first contact sliding. Green, black, blue, and red dots are the instances corresponding to the start, first sliding, last sliding, and end states of the system, respectively. (Color version online.)
Close modal

4 Leveraging Eigenstrains for Acoustically Programmable Responses

Next, we extend our analysis to two and four contacts, and those with gradients in their coefficients of friction. We exploit the generation of eigenstrains in these systems to enable advanced mechanical functions that are controlled by acoustic pulses through the system. Particularly, we demonstrate acoustically programmable responses of a mechanical switch, precision position control, and surface reconfigurability.

We first focus on the generation of eigenstrains in a configuration with two rough contacts separated by a distance, l, referred to as a “layer” hereafter. To illustrate the system behavior, a single-cycle incident shear pulse with an amplitude ξ = 2.76 at f = 0.5 MHz is simulated. To reduce the number of trailing reflections between the contacts, l is defined as approximately six times smaller than the incident wavelength (i.e., Λ = 6.2). This enables a smaller delay in the programmed output relative to the wave input. While the magnitude of eigenstrain depends on the input wave frequency and amplitude, and the relation between λ and l, the qualitative nonlinear response of the system remains the same.

The wave-induced contact sliding generates eigenstrains in the semi-infinite waveguides, ΓL and ΓR, as well as in the layer (Fig. 5(a)). These eigenstrains have different relative values and directions. The sign of the eigenstrains in the transmitted wave depends on the direction of the first contact sliding, which, in this simulated case, occurs in the positive (up) direction. Consequently, the transmitted wave across the first (left) contact contains a positive eigenstrain (red in Fig. 5(a)), which moves the layer up as the wave propagates through it. The reflected wave carries a negative eigenstrain (black in Fig. 5(a)) to balance out the effects in the transmitted wave, causing ΓL to deform downward. As the transmitted wave propagates through the second (right) contact, the eigenstrain carried with it deforms the waveguide ΓR upward as well. Since the amplitude of the wave transmitted across the first contact is capped to the friction limit, it prevents sliding at the second contact (Fig. 5(b)). Accordingly, the relative eigenstrain between the layer and ΓR is zero, while the relative eigenstrain between the layer and ΓL is Δ.

Fig. 5
Generation of eigenstrains in a dual-contact system for uni-directional (left) and bi-directional (right) pulse excitation. (a) and (c) show wave displacements recorded in reflected (black), transmitted (blue) waveguides, and inside the layer (red), whereas (b) and (d) show corresponding contact deformations. Schematics with amplified deformations show system reconfiguration from its initial state. (Color version online.)
Fig. 5
Generation of eigenstrains in a dual-contact system for uni-directional (left) and bi-directional (right) pulse excitation. (a) and (c) show wave displacements recorded in reflected (black), transmitted (blue) waveguides, and inside the layer (red), whereas (b) and (d) show corresponding contact deformations. Schematics with amplified deformations show system reconfiguration from its initial state. (Color version online.)
Close modal

To enable programmable functionalities, we isolate the generation of eigenstrains to the layer and cancel out the eigenstrains in the semi-infinite waveguides by exciting identical wave pulses from both ends of the system (Fig. 5(c)). The pulse propagating left to right generates a positive deformation in the transmitted wave, causing ΓR to move up, and the wave propagating right to left generates a negative deformation in the reflected wave, causing ΓR to move down. These deformations cancel each other, maintaining the initial state of ΓR. The same dynamics occur in ΓL waveguide (Fig. 5(c)).

However, the eigenstrain remains in the layer when waves are excited from both ends, resulting in an amplified effect (compare red curves in Figs. 5(a) and 5(c)). This is due to three simultaneous effects: (1) sliding of both contacts (Fig. 5(d)), (2) a change in the boundary conditions of the contacts, and (3) subsequent interactions of the reflected and transmitted waves. When waves propagate from each direction, they cause the sliding of the first contact in their path once they exceed the friction limit, and the wave transmitted through the first contact interferes with the wave propagating backward as it approaches the second contact. This wave mixing inside the layer changes the overall dynamics at the contact, leading to a nonzero eigenstrain in the layer. This demonstrates that acoustic pulses can control the position of the layer without affecting the state of the end waveguides.

4.1 Mechanical Switch.

We leverage the idea of controlling the offset of the layer from two-sided wave excitation to enable a mechanical switch function by programming the input pulses and their phases. Since the phase of the pulses dictates the direction of contact sliding and therefore the eigenstrain, we can control whether the layer shifts up or down using the input pulse phase. We further highlight the versatility of the system by controlling the instances of pulse generation, which allow us to vary the duration of the specific state of the mechanical switch.

This functionality is numerically demonstrated by exciting a sequence of four single-cycle pulses with programmed phases (Fig. 6(a)). Overall, the approach is to excite a pulse to change the switch’s state and use subsequent pulses with the same amplitude but opposite phases to reverse the change. The first pulses with phase 0 deg move the layer to the up (ON) position as contacts slides in the positive direction generating a positive eigenstrain. The second and third pulses with a phase of 180 deg return the layer to the initial (neutral) position and move it further down in the opposite direction, respectively. The switch is now in its OFF state. The final pulses with a phase 0 deg bring the layer back to its original state. This is an instantaneous activation as eigenstrain is generated in the layer as soon as the wave interacts with the contacts, and is potentially advantageous in reducing the time lag between actuation and response. Additionally, the switch is remotely tailored using shear waves, which makes it useful in extreme environments where electronic switches often malfunction. By controlling the time gap between pulses, the duration of a specific state of the switch can also be adjusted, providing more versatile control (note tOFF > tON in Fig. 6(a)).

Fig. 6
Leveraging friction-induced eigenstrains for programmable (a) mechanical switch and (b) precision position control functionalities. Each subplot contains programmed temporal patterns of input and output. Inputs are acoustic pulses with programmed amplitudes, where positive and negative values correspond to 0 deg and 180 deg phases, respectively, while outputs are (a) switch states and (b) physical positions of the layer. System schematics with amplified deformations are also shown for representative time instants. Numerically obtained spatiotemporal plots are shown for the region in the vicinity of two contacts, denoted by C1 and C2. The sets of parallel arrows show input pulses propagating toward the contacts. (Color version online.)
Fig. 6
Leveraging friction-induced eigenstrains for programmable (a) mechanical switch and (b) precision position control functionalities. Each subplot contains programmed temporal patterns of input and output. Inputs are acoustic pulses with programmed amplitudes, where positive and negative values correspond to 0 deg and 180 deg phases, respectively, while outputs are (a) switch states and (b) physical positions of the layer. System schematics with amplified deformations are also shown for representative time instants. Numerically obtained spatiotemporal plots are shown for the region in the vicinity of two contacts, denoted by C1 and C2. The sets of parallel arrows show input pulses propagating toward the contacts. (Color version online.)
Close modal

4.2 Precision Position Control.

By programming a sequence of pulses, we can generate eigenstrains in series and precisely control the position of the layer by incrementally increasing or decreasing the eigenstrains. We numerically demonstrate this by exciting two groups of pulses of the same amplitude, where pulses within the same group have the same phase but are opposite to that of the other group (Fig. 6(b)). The first group of pulses with 0 deg phases generates eigenstrain in the positive direction, causing the layer to increase its position incrementally (Δ), where Δ depends on the pulse amplitude. When the second group of pulses of 180 deg phases is excited, these pulses counteract the effects of the previous group of pulses and return the layer to its initial state. We note that the generated static deformation is of the order of input wave amplitude, which for the simulated case is of the order of micrometers. Therefore, despite the permanent reconfiguration, the contact area does not change considerably. Thus, we assume that the acoustic impedance at the contacts does not change for the subsequent pulses.

4.3 Surface Reconfigurability.

In this section, we demonstrate how shear-polarized eigenstrains can serve as a platform for programmable surface reconfigurability. Surface reconfigurability, or morphing, has applications in soft robotics and fluid–structure interactions. Examples include designing soft robots capable of gripping arbitrary-shaped objects and tuning the aerodynamic response by modulating turbulent drag. Here, we explore a more complex system consisting of an array of four contacts with a gradient in their coefficient of friction. We demonstrate how this gradient in μ allows control over the position of an arbitrary layer between successive contacts, highlighting the potential for precise surface manipulation.

We consider a system with four contacts (or three layers) (schematic of Fig. 7(a)). This configuration is chosen to show how the position of the inner layer (I), which is not directly accessible from either side, can be controlled by programming the input pulses. To do so, we take advantage of the gradient of frictional properties by setting the coefficient of friction at the outermost contacts (μO) smaller than the one at the innermost contacts (μI), i.e., μO < μI. More specifically, we consider μO = 2/3μI as a test case. The gradient is expected to cause different responses at the inner and outer contacts, and therefore different eigenstrains at the inner (I) and outer (O) layers. Further, note that the wave interaction with an array of contacts results in multiple reflections and transmissions at each contact. All these waves interfere with each other, and depending on their phase difference, the effective contact deformation is either amplified or truncated. Therefore, we first study the eigenstrains generated in these layers for different excitation amplitudes (Fig. 7(a)).

Fig. 7
Leveraging friction-induced eigenstrains for programmable surface reconfigurability in a four-contact system with a gradient of coefficients of friction. (a) Generated eigenstrains in outer (ΔO) and inner (ΔI) layers, and their relative difference (ΔO − ΔI) for different excitation amplitudes. The plot is normalized by the maximum amplitude (Umax) of the simulated range (star marker). (b) Numerical validation of programmed input to shift the inner layer only. Inputs are acoustic pulses with programmed amplitudes (marked in (a)), where positive and negative values correspond to 0 deg and 180 deg phases, respectively, while outputs are the physical positions of the layers. System schematics with amplified deformations are also shown for representative time instants. The spatiotemporal plot is shown for the region adjacent to the contacts, denoted by CO and CI as outer and inner contacts, respectively. The sets of parallel arrows show input pulses propagating toward the contacts. (c) Inner contact deformation (bI) for two different programmed inputs, where U2 corresponds to ΔO − ΔI = 0 (black) and ΔO − ΔI ≠ 0 (blue). In both cases, U1 is the same, and U3 is different but corresponds to ΔO − ΔI = 0. (d) Low-pass filtered response of (c) illustrating eigenstrains. Dashed lines indicate approximate time instances when input pulses interact with the inner contacts. (Color version online.)
Fig. 7
Leveraging friction-induced eigenstrains for programmable surface reconfigurability in a four-contact system with a gradient of coefficients of friction. (a) Generated eigenstrains in outer (ΔO) and inner (ΔI) layers, and their relative difference (ΔO − ΔI) for different excitation amplitudes. The plot is normalized by the maximum amplitude (Umax) of the simulated range (star marker). (b) Numerical validation of programmed input to shift the inner layer only. Inputs are acoustic pulses with programmed amplitudes (marked in (a)), where positive and negative values correspond to 0 deg and 180 deg phases, respectively, while outputs are the physical positions of the layers. System schematics with amplified deformations are also shown for representative time instants. The spatiotemporal plot is shown for the region adjacent to the contacts, denoted by CO and CI as outer and inner contacts, respectively. The sets of parallel arrows show input pulses propagating toward the contacts. (c) Inner contact deformation (bI) for two different programmed inputs, where U2 corresponds to ΔO − ΔI = 0 (black) and ΔO − ΔI ≠ 0 (blue). In both cases, U1 is the same, and U3 is different but corresponds to ΔO − ΔI = 0. (d) Low-pass filtered response of (c) illustrating eigenstrains. Dashed lines indicate approximate time instances when input pulses interact with the inner contacts. (Color version online.)
Close modal

For the studied range of amplitudes, an increasing excitation amplitude increases the eigenstrains of both inner and outer layers (square and circle markers in Fig. 7(a), respectively). As expected, larger eigenstrains are generated in the outer layers (ΔO) compared to the inner layer (ΔI) under the same excitation. This is because of the lower stick limit of the outer contacts, which results in a stronger hysteresis response. More importantly, the difference between their eigenstrains (ΔO − ΔI) also grows with excitation amplitude (cross markers in Fig. 7(a)). At low amplitude, sliding occurs only at the outer contacts, resulting in zero relative eigenstrain between the inner and outer layers. However, as wave amplitude is increased, the friction limit of inner contact is also exceeded, causing sliding at both inner and outer contacts but of different magnitudes. This results in nonzero relative eigenstrains between the inner and outer layers. We take advantage of this difference in their static deformation to shift only the inner layer while keeping the outer layers aligned with the semi-infinite waveguides. The principle here is to (1) first excite a pulse with an amplitude high enough to cause sliding at all contacts, and (2) then excite relatively low-amplitude pulses with opposite phases in at least two steps to bring back the outer layers to their initial state. Due to varying differences between the relative eigenstrains for different excitation amplitudes, the inner layer is expected to be at an offset with the entire system despite outer layers returning to their original position.

To demonstrate this approach, we conducted numerical simulations on an array of four contacts with programmed input pulses of varying amplitudes (U1, U2, and U3) and phases (180 deg, 0 deg, and 0 deg) (Fig. 7(b)). The amplitudes were selected based on two conditions: (1) the eigenstrains generated in the outer layers by the first pulses (U1) should be equal to the total strains generated by the second (U2) and third (U3) pulses, i.e., ΔO(1)=ΔO(2)+ΔO(3), where the superscripts denote the pulse indices, and (2) the relative eigenstrain between the inner and outer layers for U1 should be greater than the total relative eigenstrain for U2 and U3, i.e., |ΔO − ΔI|(1) > |ΔO − ΔI|(2) + |ΔO − ΔI|(3). The required pulse amplitudes for numerical simulations are obtained by best fitting the data points of Fig. 7(a). The outcome of the simulation confirms that only the inner layer is shifted with respect to the rest of the system after wave propagation, as seen in the system configuration at t > 140 μs in Fig. 7(b).

Additionally, the magnitude of the offset of the inner layer can also be controlled by programming the amplitudes of the subsequent pulses only (Figs. 7(c) and 7(d)). Two cases are presented, where U1 is the same, but U2 is selected such that it causes sliding either at the outer contacts only (black) or at both inner and outer contacts (blue). The temporal deformation patterns of inner contacts reveal that the eigenstrain generated from U1 is reduced only when U2 can cause sliding at inner contacts as well (blue in Figs. 7(c) and 7(d)). While U3 is different for these two cases, it generates no strain in the inner layer as it corresponds to sliding of outer contacts only. Hence, the final offset of the inner layer differs for both cases. This demonstration illustrates the potential of contact-based materials and friction-induced eigenstrains to achieve a variety of remotely actuated surface reconfigurations by selecting appropriate combinations of input pulse amplitudes and phases. For example, the fluid flow along the surface of an object can be obstructed or altered by protruding a part of the surface in its flow path or reconfiguring the surface to an optimized profile. Further, this could be used to deploy or modify the spacing of riblets [47,48], which have been shown to reduce turbulent drag by resisting cross-flow and displacing vortices away from the wall of wall-bounded aerodynamic flows, or deploy surface textures that have been shown to delay separation [49].

5 Conclusion

In this article, we investigated strongly nonlinear shear wave propagation through single and arrays of rough contacts with friction. By using the experimentally derived Jenkin friction model, we incorporated frictional properties into a numerical wave propagation model to elucidate how shear waves drive frictional instability at contacts. We discussed resulting memory-dependent (hysteretic) nonlinear responses, particularly, shear-polarized eigenstrain. We also showed how eigenstrain generation in multiple and periodic contacts combined with programmed acoustic pulses enables smart mechanical functionalities, including switching, precision position control, and surface reconfigurability.

Our study contributes fundamentally by providing insights into wave–contact interaction in the presence of friction. This includes the understanding of how dynamic excitation generates static deformations within the system due to local hysteretic nonlinearity. The study also provides a unique approach of combining hysteretic nonlinearity with periodicity and using loading history to develop novel wave responses. These findings have implications for various engineering applications as well, including nondestructive evaluation, signal processing, and fluid–structure interaction. Eigenstrains may be a better tool for detecting cracks, and the contact-based programmable switch can be used to design acoustic sensors and actuators. Controlling the position of periodic layers may be used for remote tuning of the surface topology of aerodynamic objects such as turbine blades, riblets, and airfoils in extreme and inaccessible environments. Additionally, increasing the input wave frequency can trigger the modes of the finite system and may enable frequency-dependent surface reconfigurability.

Our study has some limitations due to the complexity of the problem, but it also suggests new research directions. For example, partial slip at rough contacts, which we ignored in our model, may lead to stronger memory-dependent responses and reveal new phenomena. Additionally, a steady-state contact response and eigenstrains of at least three orders smaller than the contact area are assumed. This means that the contact response should be approximately identical under subsequent wave actuation. However, investigating the wave evolution and hysteresis effects during regimes where the frictional properties evolve with wave actuation may lead to additional forms of wave–contact interactions. Considering wavelengths near asperity size may also result in other shear wave responses from gap resonances and frequency-dependent contact properties. Finally, exploring contact orientations where both normal and shear displacements are coupled may reveal new insights into the mechanics of their interaction, for example, between friction and stegotons [33,34], beyond the pure shear wave interactions examined in this study.

Overall, we anticipate that our study will stimulate the exploration of elastic wave propagation through local and periodic hysteretic nonlinearities, an area that has received limited attention so far. The results reveal novel nonlinear wave properties emerging from memory-dependent responses, which suggest that metamaterials and phononic materials with hysteretic nonlinearity have the potential to manipulate high-amplitude waves beyond the capability of classical reversible nonlinearities.

Acknowledgment

This work was partially supported by the Army Research Office, USA, and was accomplished (Grant No. W911NF-20-1-0250). The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Office or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.

Conflict of Interest

There are no conflicts of interest.

Data Availability Statement

The authors attest that all data for this study are included in the paper.

References

1.
Hussein
,
M. I.
,
Leamy
,
M. J.
, and
Ruzzene
,
M.
,
2014
, “
Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook
,”
Appl. Mech. Rev.
,
66
(
4
), p.
040802
.
2.
Nesterenko
,
V. F.
,
1983
, “
Propagation of Nonlinear Compression Pulses in Granular Media
,”
J. Appl. Mech. Technical Phys.
,
24
(
5
), pp.
733
743
.
3.
Boechler
,
N.
,
Theocharis
,
G.
,
Job
,
S.
,
Kevrekidis
,
P. G.
,
Porter
,
M. A.
, and
Daraio
,
C.
,
2010
, “
Discrete Breathers in One-Dimensional Diatomic Granular Crystals
,”
Phys. Rev. Lett.
,
104
(
24
), p.
244302
.
4.
Narisetti
,
R. K.
,
Leamy
,
M. J.
, and
Ruzzene
,
M.
,
2010
, “
A Perturbation Approach for Predicting Wave Propagation in One-Dimensional Nonlinear Periodic Structures
,”
ASME J. Vib. Acoust.
,
132
(
3
), p.
031001
.
5.
Silva
,
P. B.
,
Leamy
,
M. J.
,
Geers
,
M. G.
, and
Kouznetsova
,
V. G.
,
2019
, “
Emergent Subharmonic Band Gaps in Nonlinear Locally Resonant Metamaterials Induced by Autoparametric Resonance
,”
Phys. Rev. E
,
99
(
6
), p.
063003
.
6.
Patil
,
G. U.
, and
Matlack
,
K. H.
,
2022
, “
Review of Exploiting Nonlinearity in Phononic Materials to Enable Nonlinear Wave Responses
,”
Acta Mechanica
,
233
(
1
), pp.
1
46
.
7.
Khajehtourian
,
R.
, and
Hussein
,
M. I.
,
2014
, “
Dispersion Characteristics of a Nonlinear Elastic Metamaterial
,”
AIP Adv.
,
4
(
12
), p.
124308
.
8.
Carretero-González
,
R.
,
Khatri
,
D.
,
Porter
,
M. A.
,
Kevrekidis
,
P.
, and
Daraio
,
C.
,
2009
, “
Dissipative Solitary Waves in Granular Crystals
,”
Phys. Rev. Lett.
,
102
(
2
), p.
024102
.
9.
Martakis
,
P.
,
Aguzzi
,
G.
,
Dertimanis
,
V. K.
,
Chatzi
,
E. N.
, and
Colombi
,
A.
,
2021
, “
Nonlinear Periodic Foundations for Seismic Protection: Practical Design, Realistic Evaluation and Stability Considerations
,”
Soil Dyn. Earthquake Eng.
,
150
, p.
106934
.
10.
Fortunati
,
A.
,
Bacigalupo
,
A.
,
Lepidi
,
M.
,
Arena
,
A.
, and
Lacarbonara
,
W.
,
2022
, “
Nonlinear Wave Propagation in Locally Dissipative Metamaterials via Hamiltonian Perturbation Approach
,”
Nonlinear Dyn.
,
108
(
2
), pp.
765
787
.
11.
Sepehri
,
S.
,
Mashhadi
,
M. M.
, and
Fakhrabadi
,
M. M. S.
,
2022
, “
Wave Propagation in Nonlinear Monoatomic Chains with Linear and Quadratic Damping
,”
Nonlinear Dyn.
,
108
(
1
), pp.
457
478
.
12.
Burgoyne
,
H. A.
, and
Daraio
,
C.
,
2015
, “
Elastic-Plastic Wave Propagation in Uniform and Periodic Granular Chains
,”
ASME J. Appl. Mech.
,
82
(
8
), p.
081002
.
13.
Fantetti
,
A.
,
Tamatam
,
L.
,
Volvert
,
M.
,
Lawal
,
I.
,
Liu
,
L.
,
Salles
,
L.
,
Brake
,
M.
,
Schwingshackl
,
C.
, and
Nowell
,
D.
,
2019
, “
The Impact of Fretting Wear on Structural Dynamics: Experiment and Simulation
,”
Tribol. Int.
,
138
, pp.
111
124
.
14.
Yevtushenko
,
A.
, and
Ivanyk
,
E.
,
1996
, “
Effect of the Rough Surface on the Transient Frictional Temperature and Thermal Stresses Near a Single Contact Area
,”
Wear
,
197
(
1–2
), pp.
160
168
.
15.
Brake
,
M. R.
,
2017
,
The Mechanics of Jointed Structures: Recent Research and Open Challenges for Developing Predictive Models for Structural Dynamics
,
Springer
,
Cham
.
16.
Argatov
,
I.
, and
Sevostianov
,
I.
,
2010
, “
Health Monitoring of Bolted Joints via Electrical Conductivity Measurements
,”
Int. J. Eng. Sci.
,
48
(
10
), pp.
874
887
.
17.
Meziane
,
A.
,
Norris
,
A. N.
, and
Shuvalov
,
A. L.
,
2011
, “
Nonlinear Shear Wave Interaction at a Frictional Interface: Energy Dissipation and Generation of Harmonics
,”
J. Acoustical Soc. America
,
130
(
4
), pp.
1820
1828
.
18.
Blanloeuil
,
P.
,
Croxford
,
A. J.
, and
Meziane
,
A.
,
2014
, “
Numerical and Experimental Study of the Nonlinear Interaction between a Shear Wave and a Frictional Interface
,”
J. Acoustical Soc. America
,
135
(
4
), pp.
1709
1716
.
19.
Li
,
X.
, and
Dwyer-Joyce
,
R. S.
,
2020
, “
Measuring Friction at an Interface using Ultrasonic Response
,”
Proc. Royal Soc. A
,
476
(
2241
), p.
20200283
.
20.
Vikas
,
V.
,
Cohen
,
E.
,
Grassi
,
R.
,
Sözer
,
C.
, and
Trimmer
,
B.
,
2016
, “
Design and Locomotion Control of a Soft Robot Using Friction Manipulation and Motor–Tendon Actuation
,”
IEEE Trans. Robotics
,
32
(
4
), pp.
949
959
.
21.
Garland
,
A. P.
,
Adstedt
,
K. M.
,
Casias
,
Z. J.
,
White
,
B. C.
,
Mook
,
W. M.
,
Kaehr
,
B.
,
Jared
,
B. H.
,
Lester
,
B. T.
,
Leathe
,
N. S.
,
Schwaller
,
E.
, and
Boyce
,
B. L.
,
2020
, “
Coulombic Friction in Metamaterials to Dissipate Mechanical Energy
,”
Extreme Mech. Lett.
,
40
, p.
100847
.
22.
Li
,
J.
,
Chen
,
Z.
,
Li
,
Q.
,
Jin
,
L.
, and
Zhao
,
Z.
,
2022
, “
Harnessing Friction in Intertwined Structures for High-Capacity Reusable Energy-Absorbing Architected Materials
,”
Adv. Sci.
,
9
, p.
2105769
.
23.
Theocharis
,
G.
,
Boechler
,
N.
, and
Daraio
,
C.
,
2013
, “Nonlinear Periodic PhononicStructures and Granular Crystals,”
Acoustic Metamaterials and Phononic Crystals
,
P.
Deymier
, ed.,
Springer
,
Berlin/Heidelberg
, pp.
217
251
.
24.
Allein
,
F.
,
Tournat
,
V.
,
Gusev
,
V.
, and
Theocharis
,
G.
,
2020
, “
Linear and Nonlinear Elastic Waves in Magnetogranular Chains
,”
Phys. Rev. Appl.
,
13
(
2
), p.
024023
.
25.
Merkel
,
A.
,
Tournat
,
V.
, and
Gusev
,
V.
,
2011
, “
Experimental Evidence of Rotational Elastic Waves in Granular Phononic Crystals
,”
Phys. Rev. Lett.
,
107
(
22
), p.
225502
.
26.
Zhang
,
Q.
,
Umnova
,
O.
, and
Venegas
,
R.
,
2019
, “
Nonlinear Dynamics of Coupled Transverse-Rotational Waves in Granular Chains
,”
Phys. Rev. E
,
100
(
6
), p.
062206
.
27.
Wang
,
C.
,
Zhang
,
Q.
, and
Vakakis
,
A. F.
,
2021
, “
Wave Transmission in 2D Nonlinear Granular-Solid Interfaces, Including Rotational and Frictional Effects
,”
Granular Matter
,
23
(
2
), p.
21
.
28.
Cabaret
,
J.
,
Béquin
,
P.
,
Theocharis
,
G.
,
Andreev
,
V.
,
Gusev
,
V. E.
, and
Tournat
,
V.
,
2015
, “
Nonlinear Hysteretic Torsional Waves
,”
Phys. Rev. Lett.
,
115
(
5
), p.
054301
.
29.
Charan
,
H.
,
Chattoraj
,
J.
,
Ciamarra
,
M. P.
, and
Procaccia
,
I.
,
2020
, “
Transition from Static to Dynamic Friction in an Array of Frictional Disks
,”
Phys. Rev. Lett.
,
124
(
3
), p.
030602
.
30.
Banerjee
,
A.
,
Sethi
,
M.
, and
Manna
,
B.
,
2022
, “
Vibration Transmission through the Frictional Mass-in-Mass Metamaterial: An Analytical Investigation
,”
Int. J. Non-Linear Mech.
,
144
, p.
104035
.
31.
Patil
,
G. U.
, and
Matlack
,
K. H.
,
2021
, “
Wave Self-Interactions in Continuum Phononic Materials with Periodic Contact Nonlinearity
,”
Wave Motion
,
105
, p.
102763
.
32.
Patil
,
G. U.
,
Cui
,
S.
, and
Matlack
,
K. H.
,
2022
, “
Leveraging Nonlinear Wave Mixing in Rough Contacts-Based Phononic Diodes for Tunable Nonreciprocal Waves
,”
Extreme Mech. Lett.
,
55
, p.
101821
.
33.
Patil
,
G. U.
, and
Matlack
,
K. H.
,
2022
, “
Strongly Nonlinear Wave Dynamics of Continuum Phononic Material with Periodic Rough Contacts
,”
Phys. Rev. E
,
105
(
2
), p.
024201
.
34.
Patil
,
G. U.
, and
Matlack
,
K. H.
,
2022
, “
Nonlinear Wave Disintegration in Phononic Material With Weakly Compressed Rough Contacts
,”
Proceedings of 10th European Nonlinear Dynamics Conference (ENOC2022)
,
Lyon, France
,
July 17–22
, p. 382267. https://enoc2020.sciencesconf.org/382267
35.
Drinkwater
,
B.
,
Dwyer-Joyce
,
R.
, and
Cawley
,
P.
,
1996
, “
A Study of the Interaction Between Ultrasound and a Partially Contacting Solid–Solid Interface
,”
Proc. Royal Soc. London. A
,
452
(
1955
), pp.
2613
2628
.
36.
Bograd
,
S.
,
Reuss
,
P.
,
Schmidt
,
A.
,
Gaul
,
L.
, and
Mayer
,
M.
,
2011
, “
Modeling the Dynamics of Mechanical Joints
,”
Mech. Syst. Signal. Process.
,
25
(
8
), pp.
2801
2826
.
37.
Medina
,
S.
,
Nowell
,
D.
, and
Dini
,
D.
,
2013
, “
Analytical and Numerical Models for Tangential Stiffness of Rough Elastic Contacts
,”
Tribology Lett.
,
49
(
1
), pp.
103
115
.
38.
O’connor
,
J.
, and
Johnson
,
K.
,
1963
, “
The Role of Surface Asperities in Transmitting Tangential Forces between Metals
,”
Wear
,
6
(
2
), pp.
118
139
.
39.
Fantetti
,
A.
, and
Schwingshackl
,
C.
,
2020
, “
Effect of Friction on the Structural Dynamics of Built-up Structures: An Experimental Study
,”
Proceedings of the ASME Turbo Expo 2020: Turbomachinery Technical Conference and Exposition
,
Virtual Online
,
Sept. 21–25
, Vol. 11, American Society of Mechanical Engineers, p. V011T30A021.
40.
Delrue
,
S.
,
Aleshin
,
V.
,
Truyaert
,
K.
,
Matar
,
O. B.
, and
Abeele
,
K. V. D.
,
2018
, “
Two Dimensional Modeling of Elastic Wave Propagation in Solids Containing Cracks with Rough Surfaces and Friction—Part II: Numerical Implementation
,”
Ultrasonics
,
82
, pp.
19
30
.
41.
Qu
,
J.
,
Jacobs
,
L. J.
, and
Nagy
,
P. B.
,
2011
, “
On the Acoustic-Radiation-Induced Strain and Stress in Elastic Solids with Quadratic Nonlinearity (L)
,”
J. Acoustical Soc. America
,
129
(
6
), pp.
3449
3452
.
42.
Yost
,
W. T.
, and
Cantrell
,
J. H.
,
1984
, “
Acoustic-Radiation Stress in Solids. II. Experiment
,”
Phys. Rev. B
,
30
(
6
), pp.
3221
3227
.
43.
Jacob
,
X.
,
Takatsu
,
R.
,
Barrìre
,
C.
, and
Royer
,
D.
,
2006
, “
Experimental Study of the Acoustic Radiation Strain in Solids
,”
Appl. Phys. Lett.
,
88
(
13
), p.
134111
.
44.
Nagy
,
P. B.
,
Qu
,
J.
, and
Jacobs
,
L. J.
,
2013
, “
Finite-Size Effects on the Quasistatic Displacement Pulse in a Solid Specimen with Quadratic Nonlinearity
,”
J. Acoustical Soc. America
,
134
(
3
), pp.
1760
1774
.
45.
Norris
,
A. N.
,
1991
, “
Symmetry Conditions for Third Order Elastic Moduli and Implications in Nonlinear Wave Theory
,”
J. Elasticity
,
25
(
3
), pp.
247
257
.
46.
Jiang
,
W.
, and
Cao
,
W.
,
2004
, “
Second Harmonic Generation of Shear Waves in Crystals
,”
IEEE Trans. Ultrason. Ferroelectr. Freq. Control
,
51
(
2
), pp.
153
162
.
47.
Bushnell
,
D. M.
, and
Hefner
,
J. N.
,
1990
,
Viscous Drag Reduction in Boundary Layers
, Vol.
123
,
AIAA
,
Washington, DC
.
48.
Garcia-Mayoral
,
R.
, and
Jiménez
,
J.
,
2011
, “
Drag Reduction by Riblets
,”
Philosoph. Trans. Royal Soc. A
,
369
(
1940
), pp.
1412
1427
.
49.
Lang
,
A. W.
,
Motta
,
P.
,
Hidalgo
,
P.
, and
Westcott
,
M.
,
2008
, “
Bristled Shark Skin: A Microgeometry for Boundary Layer Control
?”
Bioinspiration Biomimetics
,
3
(
4
), p.
046005
.