1 Introduction

The relative motion of two surfaces with adhesion is the process where a lot of nonlinear mechanism occurs. A lot of studies on different friction laws, different contact stiffness and different contact model have been reported, but there are still a lot of questions and hypotheses related to nonlinear mechanisms of contact. Although there are many unexplained contact mechanics phenomena, many applications based on nonlinear contact effects are used for diagnostic purposes. One of the method utilizes contact related nonlinear phenomena is nonlinear acoustics. Over the last 20 years, these methods are intensively developed for damage detection and localization. There are a lot of applications where nonlinear effects are used to identify and localize faults and damages [1,2,3,4,5]. This technique based on identification of nonlinear effects generated due to contact damage related nonlinearities. The effects are produced when the two types of excitation—low and high frequency—are introduced to the structure simultaneously. Then, the interaction of contact-type damage (e.g., fatigue crack or delamination) excited by low-frequency monoharmonic excitation with high-frequency acoustic wave results in different nonlinear product, observed in the response spectra. This can be signal modulation [6,7,8], higher and sub-harmonics [9,10,11], frequency shift when excitation amplitude increase [11, 12], frequency mixing [13], modulation transfer (Luxembourg-Gorky effect) [14, 15], memory effects [9] and many others [16,17,18,19]. The above-mentioned phenomena can be related to material nonlinearities or non-symmetric thermoelastic behavior of interfaces (e.g., contact-type damage). There are a lot of works where above-mentioned effects were used for structural damage identification in metallic structures [4, 7, 15, 19], composites [20,21,22,23,24,25], rocks [11] and a lot of different materials. The main conclusion of most of the works on this method is that nonlinear acoustic techniques are very sensitive and can detect damage in very early stage. This is because damage related nonlinearities produce significant effects in signal response. According to the literature, the damage-inducted nonlinear mechanisms can be divided into three groups. The first is atomic scale where intrinsic elastic nonlinearity due to anharmonicity of interatomic potential [3], non-friction and non-hysteretic dissipation locally enhanced by thermoelastic coupling [25], hysteresis in stress–strain [26], stick-slip friction between crack surfaces or adhesion hysteresis [27] can be observe. The second is mesoscopic scale where crack induced nonlinearity (variation in elastic moduli) [27], Hertzian type nonlinearity (contact between crack surfaces) [28], local stiffness reduction leading to natural frequency shift [5] or bi-linear stiffness (closing–opening crack) [29] can occur. The last one is macroscopic scale and includes clapping mechanism [30] and wave modulation due to impedance mismatch (discontinuity due to closing–opening crack) [31]. Beyond the problems associated with the determination of nonlinearity sources, additional difficulties arise in this kind of research. Structures tested are usually elements with a complex shape. This generates problems because in many cases these elements themselves mask contact related (e.g., resulting from fatigue damage or delamination) nonlinear effects. One of reason of that are linear and nonlinear structural components resulting from the construction of the structure, specific boundary conditions and impact inducted structural components. This type of self-generating impacts causes wide-band excitation of the structure, whereby the signal response also includes high-frequency components related to structural dynamics [32,33,34]. Despite the multitude of phenomena and the possibility of applying nonlinear methods to many types of structures with contact-type damage, there are many unresolved problems. The first is related to ambiguity because the different nonlinear mechanism can be manifested by the same effects. Additionally, these mechanisms can occur simultaneously and modify structural response. This results in the problems of modeling contact-type damage. The second problem is related to experimental validation of proposed model. It is very often hard to measure the physical quantities generated as effects of nonlinear mechanisms. For these reasons, many theories about dynamic phenomena in contact-type damage remain only hypotheses. In spite of these problems, nonlinear acoustics still remain attractive in the case of detection and localization of contact damage. Understanding of nonlinear mechanisms associated with the contact phenomenon may contribute to the further development of nonlinear techniques and can help to implement it in more engineering application.

Fig. 1
figure 1

Three basic contact modes: a closing–opening mode, b sliding mode, c tearing mode

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The inspiration for elaborating the model with hysteresis stiffness and impact were phenomenon occurring in vibro-acoustic modulation tests described widely in many works [3, 16]. It should be noted that works in this field are mainly experimental. There is no model that would allow a clear explanation of nonlinear effects resulting from the interaction low-frequency excitation, ultrasonic wave and contact interface. The paper focus is on the elaborating model with hysteresis stiffness, vibro-impacts and experimental validation of the proposed model. In this paper only low-frequency excitation is investigated.

The remaining part of the paper is organized as follows. Section 2 contains theoretical background on stiffness characteristics related to contact. Section 3 the assumed numerical model is presented. In Sect. 4 numerical model is validated with experimental data. The discussion of results is also provided in that section. The paper is concluded in the last section.

2 Theoretical background

The most of delamination’s or fatigue cracks in engineering structures have contact nature. When the structure vibrates with given frequency, the different types of contact can occur. In general, three different crack modes are expected:

  • Closing–opening action, where the crack faces move apart from each other.

  • Sliding action, where the crack faces slide relative to each other (the movement is perpendicular to the crack leading edge).

  • Tearing action, where crack faces move relative to each other in direction parallel to crack leading edge.

Schematically, these contact modes are presented in Fig. 1.

For the first mode (closing–opening action), the bi-linear stress–strain relation is expected. This is due to fact that the structure stiffness is higher for the compression, than for tensile phase. The bi-modular contact caused by structure vibration results in pulse type modulation of material stiffness. Then, the harmonics with amplitude modulated by sinc function of fundamental frequency are expected in the response spectra. Such behavior is often called “clapping” mechanisms and can be described using piecewise stress–strain relation [35]

$$\begin{aligned} \sigma ={C}^{\mathrm{II}}\left[ {1-{H}\left( \varepsilon \right) \left( {\frac{\Delta {C}}{{C}^{\mathrm{II}}}} \right) } \right] \varepsilon \end{aligned}$$
(1)

where \(\sigma \) is stress, \(\varepsilon \) is strain, \(H\left( \varepsilon \right) \) is unit step Heaviside function, \(C^\mathrm{II}\) is linear stiffness and \(\Delta C\) is described by formula [35]

$$\begin{aligned} \Delta {C}=\left[ {{C}^{\mathrm{II}}-\left( {\frac{\hbox {d}\sigma }{\hbox {d}\varepsilon }} \right) } \right] \end{aligned}$$
(2)

for \(\varepsilon >0\). Figure 2 presents bi-linear stiffness characteristic and example of response spectra for “clapping” contact.

Fig. 2
figure 2

a Stress–strain characteristic for bi-linear stiffness function; b example of corresponding response spectra

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For the other two modes of contact, the different stress–strain characteristics are expected. There are mechanical diode effect and hysteretic stress–strain relation. If the amplitude of relative displacement between contact surfaces is low, the motion is limited to micro-slip mode due to surface roughness. The interactions of following asperities produce the friction forces and prevent surfaces form sliding. In this case, the symmetrical stress–strain relation occurs and only odd harmonics are generated. Due to pulse type stiffness modulation, the amplitude of the following harmonics is also modulated by sinc function (Fig. 3).

Fig. 3
figure 3

Stress–strain characteristic for micro-slip mode (a) and expected spectrum of the response signal (b)

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Fig. 4
figure 4

a Stress–strain characteristic for stick-slip mode; b example of corresponding response spectra

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The last contact type is stick and slip mode. When the harmonic excitation is applied to two non-bonded surfaces and resulting strain is strong enough to change friction from static to kinematic then the hysteretic stress–strain relation is expected. In this case, the contact stiffness value changes twice per cycle from \(C_\mathrm{s}\) (for stick phase) to zero (for slide phase) according to formula [36]

$$\begin{aligned} {C}\left( {t} \right) =\frac{{C}_\mathrm{s} }{2}\left\{ 1-\hbox {sign}\left( {\dot{\varepsilon }} \right) \hbox {sign}\left[ \varepsilon +\hbox {sign}\left( {\dot{\varepsilon }} \right) \varepsilon _1 \right] \right\} \end{aligned}$$
(3)

By integration Eq. (3) with respect to time, the \(\sigma \) can be determined as a function of \(\varepsilon \) [36]

$$\begin{aligned} \sigma \left( {t} \right)= & {} {C}\left( {t} \right) \varepsilon \left( {t} \right) +{C}\left( {t} \right) \varepsilon _1 \hbox {sign}\left( {\dot{\varepsilon }} \right) \nonumber \\&+\frac{{C}_\mathrm{s} }{2}\left( \varepsilon _0 -\varepsilon _1 \right) \hbox {sign}\left( {\dot{\varepsilon }} \right) \end{aligned}$$
(4)

Since the stress–strain characteristic remains symmetrical, only odd harmonic are generated in the response spectra. The amplitudes of the following harmonics are also modulated by sinc function, but amplitude ratios of the following harmonics are different than for micro-slip mode. Figure 4 presents an example of stress–strain characteristic for hysteresis nonlinearity.

The presented above stress–strain characteristic are of course ideal. It means, for example, that after every loading and unloading cycle, structure returns to the same state at the same point. For real structures, the characteristics can be modified due to additional phenomena like cyclic hardening and creeping. The real hysteretic characteristics can also deviate from the theoretical and can change for example depending on the strain rate sign.

The complexity of phenomena occurring in presented types of contact can result from many factors. It can be:

  • the different dimensions of the contact (small for thin fatigue cracked plate and large for delaminated composite materials)

  • the impulse excitation of the structure by roughness inducted vibration

  • the time-varying size of the contact area due to low and high frequency excitation

  • the friction and adhesion forces acting at the contact interface and provide micro-slip and stick and slip processes

  • the strain and temperature dependence of material properties.

All of the above-mentioned phenomena can cause different distortion is stiffness characteristics, including non-symmetricity. Knowledge on the nonlinear phenomena occurring in the case of contact is important not only from the scientific point of view. Many damage detection methods are based on the effects produced by nonlinear mechanisms related to contact. Process of extraction of nonlinear components from signal response is complicated because very often is impossible to define where the effect should appear at the spectra. Additionally, the nonlinear damage detection methods are very sensitive for nonlinearities coming from other than damage related sources like nonlinearities of measurement chain, boundary conditions or material nonlinearities. One of the main problem in such cases is to separate damage and non-damage related nonlinear sources.

3 Numerical model

Numerical model description

This section provides simulation of numerical model with amplitude dependent stiffness model. This model is implemented as 1DOF lumped parameter system presented in Fig. 5.

Fig. 5
figure 5

Scheme of the investigated model

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Due to the hysteretic stiffness, expected behavior is such that for the low excitation level, when the displacement will be in range of stick motion, the overall behavior will be different than in case of high excitation level, when the displacement will be in micro-slip and stick and slip motion. Additionally, when the motion will change from stick to slip phase, the assumed roughness inducted impact is introduced to generate additional components related to the structure transfer function. The time-dependent system equations in tangential direction are

$$\begin{aligned} m{\ddot{x}} + c{\dot{x}} + {k_\mathrm{L}}x + {k_\mathrm{P}}x + {k_\mathrm{H}}x + T = {F_\mathrm{exc}} \end{aligned}$$
(5)

where \(x,{\dot{x}}\) and \({\ddot{x}}\) are displacement, velocity and acceleration in tangential direction respectively, \(k_\mathrm{L}\) is linear stiffness of the system and \(F_\mathrm{exc}\) is excitation force

$$\begin{aligned} {F}_{\mathrm{exc}} ={f}_\mathrm{o} \hbox {sin}\left( \upomega {t} \right) \end{aligned}$$
(6)

The \(k_\mathrm{P} \) represents piecewise linear stiffness described by formula (according to Fig. 5.)

$$\begin{aligned} {k}_\mathrm{P} = \left\{ {{\begin{array}{lll} {{k}_1 \qquad \quad \hbox {for}\qquad \quad {x}\le {x}_{\mathrm{k}1} } \\ {{k}_1 +{k}_2 \quad \quad \hbox {for}\quad \quad x>\left| {{x}_{\mathrm{k}2} } \right| } \\ \end{array} }} \right. \end{aligned}$$
(7)

When \(x\ge \left| {x_\mathrm{H} } \right| \) then the hysteresis stiffness is introduced. The friction force T related to the contact force (vertical) is assumed as

$$\begin{aligned} {T}=\mu \left( \Delta {\dot{{x}}} \right) {F} \end{aligned}$$
(8)

The normal contact force F is given by

$$\begin{aligned} {F}={k}_\mathrm{C} \Delta {yH}\left( {\Delta {y}} \right) \end{aligned}$$
(9)

where y is displacement in vertical direction and \(\Delta y\) is displacement difference in vertical direction between upper and lower steel blocks

$$\begin{aligned} \Delta {y}={y}_{\mathrm{upblock}} -{y}_{\mathrm{loblock}} \end{aligned}$$
(10)

and \(k_\mathrm{C} \) is contact stiffness parameters. \(H\left( y \right) \) is Heaviside function and prevents the upper block to be lifted when \(\Delta y\) is negative. The displacement in vertical direction for upper block is calculated using Green’s function (convolution of impulse response and contact force [37])

$$\begin{aligned} {y}_{\mathrm{upblock}} =-{F}*{g}_{\mathrm{upblock}} \end{aligned}$$
(11)

For stiffness characteristics, the following scenarios are assumed. For low excitation level (\(x>\left| {x_{k2} } \right| )\), the pricewise stiffness characteristics is used (micro-slip, without vertical displacement). When amplitude of excitation reach assumed level (\(x\ge \left| {x_\mathrm{H} } \right| )\), then the stiffness characteristic become hysteresis (Fig. 6) and additionally the impulse response of the structure is also included [according to Eq. (9)]. The displacements \(x_{k2} \) and \(x_\mathrm{H} \) have been identified based on experimental data. Figure 6 presents stiffness characteristics of the investigated model.

Fig. 6
figure 6

Stiffness characteristic of the investigated model

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The simulation was prepared in Matlab/Simulink environment, sampling frequency was 50 kHz, and total simulation time was 10 s. Schematic diagram of the model used is presented in Fig. 7.

Fig. 7
figure 7

Schematic diagram of nonlinear model with hysteresis stiffness

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4 Experimental setup

Description of the test samples

Two test samples made from C45 steel were prepared for testing. The overall dimensions of the test sample are \(40\times 40\times 45\) mm with the contact surface of the area \(10\times 10\) mm. The contact surfaces are grounded with a sand paper with the grit size P40 to obtain rough surface. Figure 8 presents sample used in test.

Analysis of transfer function of the system

To pre-identify the parameters of the analyzed system, the modal analysis test was carried out. The investigated object has been freely suspended and piezo-electric actuator NOLIAC NAC 2013 is attached to object to excite the structure in high-frequency regions. Excitation signal was a chirp signal with frequency from 5 to 20 kHz. Fourier transform of the signal is shown in Fig. 9 with blue solid line. The three modes have been identified based on Frequency Response Function (FRF). For this modes damping ratio (using half-power method) and natural frequency has been estimated and the results are summarized in Table 1. These modes have been introduced to the model. The comparison of FRF for both numerical model and tested sample for the same excitation parameters is presented in Fig. 9.

It is worth noting that the estimated parameters were used in the simulation as initial parameters. Due to boundary conditions during the test, the values of frequency and damping can significantly change. In subsequent stages, they were tuned on the basis of experimental data.

Test rig description

In order to investigate phenomena related to the nonlinear stiffness contact, an experimental setup is created. It consists of modal shaker K2007E01 as an excitation source. One sample was mounted at the end of threaded rod connected to the shaker and the other was firmly attached to the steel base plate. In this configuration, the relative motion in a single axis could occur between the samples. The top sample is loaded by an extender mass 200 g. The experimental setup is shown in Fig. 10a. The data measurement was conducted using impedance head PCB 288D01 to obtain acceleration and force data and the Doppler laser vibrometer to record displacement and velocity data. The excitation signal is supplied by the means of Agilent 33500B Series generator controlled by Matlab script and the acquisition was performed using Polytec PSV software. Sampling frequency is 51.2 kHz with total acquisition time 9 s. A series of measurements are conducted with sine excitation with frequency ranging from 10 to 500 Hz, with step of 10 Hz, so in total 50 measurements are obtained. Amplitude is linearly increasing during each experiment, to observe the nonlinear behavior.

Results and discussion

At first, time domain signals are analyzed. Detailed analysis is limited to signal that was excited with frequency 10 Hz. Displacement and velocity signals are analyzed as shown in Figs. 11 and 12, respectively. It is observed that the entire motion could be divided into two parts, i.e., up to 2.6 s when the displacement amplitude increase very slowly, and above 2.6 s when displacement linearly increases. Figure 13 presents comparison of displacement signals zoomed at this time interval.

Fig. 8
figure 8

Investigated sample

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Fig. 9
figure 9

Frequency response function for numerical model (red dashed line) and experimental data (blue solid line). (Color figure online)

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To analyze the particular stick and slip phase, the time domain signal was zoomed to one period of motion and is presented in Fig. 14. Then, the one period window of the signal was investigated.

The four different motion phases are visible when Fig. 14 is analyzed—two for stick and two for phase motion. Additionally, the high-frequency structural components can be observed (Fig. 15a). The nature of this component is different than in the numerical model. The reason for this may be that the measurement data is collected at a certain distance from the contact interface. This means that there is a transfer function between the contact area and the measuring point that can modify the response signal. The model does not take into account the geometry of the structure.

In order to analyze the high-frequency components (related to the vibro-inducted impacts), spectrograms for the velocity data have been calculated as shown in Fig. 15. It is clearly shown that at time 2.4 s, there are additional components visible in high-frequency range for both numerical example and experimental data. Additionally, for experimental data, the harmonics of the structural modes are also visible.

Table 1 Frequency and damping ratio values for identified high-frequency modes
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Fig. 10
figure 10

Experimental test rig: a schematic view and b photo

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Fig. 11
figure 11

Time domain signals of displacement from numerical model (a) and experiment (b) with excitation frequency 10 Hz

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Fig. 12
figure 12

Time domain signals of velocity from numerical model (a) and experiment (b) with excitation frequency 10 Hz

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Fig. 13
figure 13

Zoomed time domain displacement signals from numerical model (red dashed line) and experiment (blue solid line) with excitation frequency 10 Hz. (Color figure online)

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Fig. 14
figure 14

One period of the time domain response signal (a) and corresponding stiffness characteristics for different motion phases (b)

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Fig. 15
figure 15

Spectrograms of the velocity signals from numerical model (a) and experimental data (b)

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In order to visualize the difference between the frequency content of both phase of motions, Fourier transform has been calculated. For the piecewise stiffness part of the signal in vicinity of time \(t=0{-}2\) s is considered. For the hysteretic stiffness part of the signal in vicinity of time \(t=3{-}9\) s is considered. Figures 16 and 17 present comparison of spectra calculated for model and experimental data.

Fig. 16
figure 16

Fourier transform of velocity signal from model (a) and experimental example (b) at time interval \(t=0{-}2\) s

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Fig. 17
figure 17

Fourier transform of velocity signal from model (a) and experimental example (b) at time interval \(t=3{-}9\) s

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It can be also noticed that there is a good agreement between model and experimental data results. For both the two phase of motion can be clearly recognized. The frequency components are very similar when Figs. 16 and 17 are compared. Moreover, the experimental data includes additional components (about 20 kHz), probably related to harmonics of natural frequency. This problem is not investigated in this paper.

Frequency dependence of motion types

It is shown in previous section that the type of nonlinearities present actually depend on the type of motion occurring, and this is related to relative displacement of two surfaces in contact. As it is shown in Fig. 14a, there are different phases of motion that depend on the value of relative displacement, i.e., additional high-frequency components appear in the signal, when certain displacement threshold value has been crossed, as seen in Fig. 15b.

In many examples of nonlinear acoustic tests, the effects associated with contact depend on excitation frequency. To verify this, the analyses have been perform to find the threshold level at which the high-frequency components appear in the response signals depending on the low-frequency excitation frequency. In the investigated model, the constant value of displacement specifying the change in motion phase has been assumed. For this purpose, the experiment with set of excitation frequency values has been performed. The values were changed from 10 to 500 Hz with step 10 Hz. The amplitude of excitation was assumed as in previous experimental works. Next, the displacement data were used to find the moment when the high-frequency structural components appear at the signal spectra. Figure 18a presents the characteristic calculated for given frequency range. This figure shows that the moment of motion change is different for the whole frequency range and it is not constant.

Fig. 18
figure 18

Frequency characteristics of motion type changes (a), comparison with FRF for low frequencies (b)

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Theoretical model assumes constant value of \(\varepsilon _{1}\) (according to Fig. 4), on the basis of which type of motion changes, while the analysis of Fig. 18a suggests frequency dependent behavior. To investigate this problem, the additional modal analysis test was performed to find low-frequency structural components. Figure 18b presents FRF (red solid line) for frequency range up to 500 Hz. When the displacement threshold and FRF are compared, then the consistency of both can be noticed. It follows that the structural resonance of the sample (about 150 Hz) affects the moment of motion changes. Generated resonant vibrations cause additional displacement and in this way changes the moment of appearance of high-frequency structural components. Hence, the conclusion that in order to properly adjust the excitation level in nonlinear acoustics tests, both high- and low-frequency structural components should be taken into account.

5 Conclusions

Experimental and numerical investigation of hysteretic stiffness for contact-type nonlinearity was presented. The numerical model with non-symmetrical, amplitude dependent hysteresis stiffness was developed. The two steel samples with contact were used for experimental work. The monoharmonic signals with a linearly increasing amplitude were used for both cases as excitation. Additionally, the structural dynamics was included in numerical model. The study involved the analysis of two motion types: micro-slip and stick and slip. The experimental results demonstrate that the displacement level caused motion phase change is closely related to the resonance vibrations of the structure. For these reasons, it is important to consider the appropriate frequency of excitation when nonlinear effects related to different types of motion/contact are analyzed. The time and frequency domain analyses confirm that inclusion of additional high-frequency structural components improves the quality of the model. In particular, this applies to the frequency domain. Further work assumes to consider high-frequency acoustic wave and observation of vibro-acoustic modulation effects.