In situ measurement of fatigue crack opening stresses by nonlinear vibro-acoustic modulation testing

Abstract

The difficulty of accurately measuring crack closure regions has hindered measurements of opening or closure stresses, and ultrasonic methods have shown great potentialities in evaluating closure area. Many recent researches have confirmed the advantages of nonclassical nonlinear acoustic approaches in detecting short cracks and even micro- or meso-cracks. However, there are few reports on using nonclassical nonlinear acoustic approaches to measure crack closure. The aim of this article is to verify the applicability of vibro-acoustic modulation approach in estimating opening stresses of fatigue cracks. A procedure of approximating the opening stress by the applied tensile load of the inflection point of the modulation intensity curve is proposed. To study the performance of this measuring method experimentally, in situ vibro-acoustic modulation tests were carried out on a central cracked 5052-H32 aluminum alloy fatigue sample. The experimental estimations match well with other researches. Analysis of the experimental results suggests that the vibro-acoustic modulation technology is not only capable of measuring the fatigue cracks opening stresses, but also capable of analyzing the crack closure behavior, even for physically short cracks.

Keywords

Plasticity-induced crack closure crack opening stress central fatigue crack vibro-acoustic modulation nonclassical nonlinear acoustic

Introduction

Fatigue crack closure has been recognized as one of the most influential operating mechanisms during fatigue for a long time. In the early 1970s, Elber¹ discovered that when a closed crack is loading, the crack surfaces will remain partly closed until all the compressive stresses were overcome under the corresponding crack opening stress $σ_{op}$ . During unloading, the crack will be fully open until the applied load decreases to the crack closure stress $σ_{cl}$ , which is usually assumed to be approximately equal to $σ_{op}$ . It has been testified that the effective stress range $Δ σ_{eff} = σ_{max} - σ_{op}$ is the actual driving force of fatigue crack growth rather than the applied (external) stress range $Δ σ_{eff} = σ_{\max} - σ_{\min}$ . Therefore, the crack closure parameters (e.g. $σ_{op}$ and $σ_{cl}$ ) are the key factors of crack propagation, and accurately estimating that the crack closure behavior is significant in fatigue life evaluation.

Three widely accepted crack closure mechanisms are plasticity-induced crack closure (PICC), roughness-induced crack closure (RICC), and oxide-induced crack closure (OICC).² All the mechanisms are operative and interacting. However, for Paris regime fatigue crack growth of most engineering materials, PICC is the dominant closure mechanism and PICC models successfully predict closure loads and crack growth rates.³ The well-known reason for the PICC in thin solids (the plane stress condition) is the plastic wedge in the wake of the crack front. This wedge is created by tensile necking within the crack tip plastic zone.

Extensive experiments^4,5 and numerical simulations^6,7 as well as analytical approaches⁸ have been developed to estimate PICC stresses. Among experimental approaches, compliance technique⁵ has been most widely used for crack closure measurement, due mainly to its accuracy and experimental simplicity. However, this measuring method is usually limited to measurements when strain gauges are located near the wake of the crack or crack tip. In other words, the probable location of the fatigue crack in the host structure must be known before testing. Therefore, the compliance technique is unsuitable for in situ measurement. Acoustic methods generally measure “global” effects, and therefore require less priori knowledge of defects as compared with mechanical techniques (e.g. compliance technique). This feature makes the acoustic detecting approaches suitable for in-service monitoring systems, such as prognostics and health management (PHM) and health and usage monitoring system (HUMS).

The difficulty of accurately measuring closure regions has hindered measurements of opening or closure stresses, and acoustic methods have shown great potentialities in evaluating closure area. Some linear^9–11 and classical nonlinear¹² acoustic approaches have been used to measure crack closure. Many recent researches have confirmed that the so-called nonclassical nonlinear phenomena, for example, vibro-acoustic modulation (VAM), show a higher sensitivity to a closed crack as compared to linear and classical nonlinear acoustics, especially when the crack is relatively small.^13,14 VAM technology is a convenient and effective nondestructive detecting approach.¹³ As shown in Figure 1, VAM test employs a low-frequency vibration $f_{1}$ (pumping signal) combined with a more sensitive ultrasound wave $f_{2}$ (probing signal) to stimulate the structure simultaneously, then measures the frequency domain response of the system. If the structure is damaged, one of the signals will interact with another in a way that the ultrasound is modulated by the vibration. This behavior will be presented as sum and difference frequency sidebands $f_{2} \pm f_{1}$ in the response spectrum.

Figure 1.

Schematic layout demonstrating the principle of VAM used for damage detection.

VAM technology has been widely studied for nondestructive evaluation applications. Despite many efforts have been exerted, there is still very little understanding of the physical mechanisms associated with this nonlinear phenomenon. Various theories have been proposed to explain the nonlinearity resulting from crack-wave interactions. These include the nonlinear stiffness spring (NSS) mechanism,^15,16 the dissipative mechanism of elastic hysteresis,^17,18 and the contact acoustic nonlinearity (CAN) mechanism.^19,20 The damage severity characterization methods in VAM technology are varied and depend on the kind of explanation mechanism used. In 2001, Donskoy et al.¹⁶ mathematically proved the proportional relation between the crack size and the amplitude ratio of sideband components to fundamental frequency ones, based on the NSS model. This amplitude ratio was called modulation index in their paper, and the effectiveness of which was also confirmed by impact-modulation (a type of VAM test methods) experiments. Since then, the effectiveness and practicability of the modulation index have been further confirmed by many investigations extending from crack sizing^21,22 to impact damage detection²³ as well as fatigue life estimation,²⁴ and from metallic materials^22,25 to composite materials.^23,26 In this article, the NSS analysis model and modulation index mentioned above would be employed to evaluate the contact area of cracks.

Compared with linear and classical nonlinear acoustic approaches, the potential advantages of VAM technology are as follows. First, as nonlinear products, the VAM sidebands $f_{2} \pm f_{1}$ generally show a large increase in the magnitude with increasing contact area of a crack. Therefore, the sensitivity of VAM approach, in distinguishing a fully open crack from a partly open crack, is larger than that of linear acoustic approaches. Second, the pumping excitation employed by VAM technology forces the crack faces to kiss, slid, and/or tear (depending on the mode shape of structure). Such motions excavate the nonlinear features of the crack interface as much as possible, and this may be the primary reason why that VAM approach (dual-frequency excitation) shows a higher sensitivity than single frequency tests (e.g. most classical nonlinear acoustic approaches). In addition, along with the modulated response $f_{2} \pm f_{1}$ in a dual-frequency excitation, higher-order harmonics of the two individual frequencies exist (e.g. $2 f_{1}, 2 f_{2} \dots$ ), and additional sidebands such as $f_{2} \pm 2 f_{1}$ exist, which result in a large array of further components which do not exist in a single frequency system.

It should be pointed out that there are few reports on evaluating crack closure using VAM technology. This article aims to study the applicability of VAM approach in estimating opening stresses of fatigue cracks. A procedure of approximating the opening stress by the applied tensile load of the inflection point of the modulation intensity (MI) curve is proposed. This procedure is based on the fact that once the crack is fully open under a specifically applied force which is equal to or exceeds the crack opening stress, the nonlinear modulation phenomenon will be very weak. Then, the crack opening stresses can be approximated by the current applying load when the nonlinear MI decreases to a value close to zero. To study the performance of this measuring method experimentally, in situ VAM tests were carried out on a central cracked 5052-H32 aluminum alloy fatigue sample. The validity of the estimating results and analysis of the closure behavior of a physically short crack were also discussed.

Methodology

NSS model and MI of VAM

The analysis model of VAM phenomenon, which was developed from the rough-surfaces contact theory,²⁷ was first proposed in Rudenko and Vu.¹⁵ According to this model, the applied stress changes the effective contact area between the two surfaces due to deformation of asperities in contact, resulting in a nonlinear elasticity of the interface. For longitudinal stress applied normally to the interface, the wave-induced stress $Δ σ$ can be written as follows: $Δ σ = K Δ d$ , where $Δ d$ is the relative displacement of the opposite sides of the crack. The nonlinear spring stiffness $K$ is relative to effective contact area, and naturally depends on $Δ d$ : $K = K (Δ d)$ . Using Taylor’s expansion and leaving only the first-order term, and in view of the ultrasonic-wave-induced stresses are generally rather small, the nonlinear spring stiffness can be described by $K = K_{0} + α Δ d$ . Then, the internal stress $Δ σ$ can be expressed as $Δ σ = K_{0} Δ d + α (Δ d)^{2}$ , where $K_{0}$ stands for equivalent linear stiffness, and $α$ for a nonlinear coefficient.

It is noteworthy that this type of nonlinearity should be clearly distinguished from the CAN that is related to the “opening–closing” motion of the crack faces. In the simplest form of CAN, such a “bi-modular” can be simulated by a piece-wise stress–strain relation.²⁸ The most important difference between the two explanation mechanism may be that the CAN model assumes the incident wave is powerful enough to open or close the crack faces (crack mode I). However, since the nonlinearity explained by the NSS mechanism is caused by the elastoplastic deformation of the asperities on the crack faces, NSS model has a much lower requirement on the wave energy. This exciting feature has been experimentally verified by Sohn et al.²⁹ In addition, the potential application range of NSS model may be expanded to crack mode II and/or mode III, if a tangential stiffness is introduced in the modeling processes.

Based on NSS model, Donskoy et al.¹⁶ obtained the amplitudes of the modulation sidebands $f_{2} \pm f_{1}$ induced by a contact crack in a beam of length L as

\begin{array}{l} U_{\pm} = z_{0} α k_{1}^{2} k_{2} s_{c} D^{2} \frac{A B}{S k_{\pm} \sin (k_{\pm} L)} \\ \cos [k_{\pm} (L - x)] \cos (k_{1} x_{0}) \sin (k_{2} x_{0}) \end{array}

(1)

where $α$ is the nonlinear coefficient of the crack interface; D is the effective crack thickness; $s_{c}$ is the effective contact area of the crack; $S$ is the cross-section area of the structure; $x_{0}$ and $z_{0}$ are crack coordinates; A and B are the wave amplitudes of the two fundamental signals; $k_{1}$ , $k_{2}$ , and $k_{\pm}$ are the wavenumbers of the two fundamental waves and the modulation sidebands, respectively. Equation (1) indicates the proportional dependence between sideband amplitudes $U_{\pm}$ and the fundamental wave amplitudes A and B

U_{\pm} \propto z_{0} s_{c} α D^{2} AB

(2)

Assuming the item $α z_{0} D^{2}$ is constant in a given VAM testing, the MI, which is proportional to the contact area $s_{c}$ , can naturally be defined as follows²¹

MI \overset{Δ}{=} \frac{A_{\pm}}{A_{1} A_{2}} \propto s_{c}

(3)

where $A_{\pm}$ is the average of spectrum amplitude of sidebands; $A_{1}$ and $A_{2}$ are the spectrum amplitude of two fundamental signals, respectively. It is worth noting that $s_{c}$ stands for the effective micro-contact area of crack faces rather than the macroscopic crack size $s_{m}$ . According to rough-surfaces contact theory, $s_{c}$ is always smaller than $s_{m}$ .

Procedure of approximating opening stress considering PICC

Consider a linear central through fatigue crack with length $2 a$ in an infinite plate subjected to remote tensile load $σ_{\infty}$ , as shown in Figure 2. When the PICC distributed stress $σ_{cl} (x)$ is considered, the crack surfaces are partly open instead of fully open. In addition, the contact area $s_{c}$ decreases gradually when the crack is loading. According to the discussion in subsection “NSS model and MI of VAM,” when the VAM testing is applied to a cracked structure, the separated parts of a crack have no contribution to the MI. Therefore, before the crack mouth is fully open, the MI is a decreasing function of $σ_{\infty}$ . Once the applied load reaches the crack opening stress, the surfaces are fully separated and MI decreases to a value close to zero accordingly, as illustrated in Figure 2. In addition, the MI level will not increase again during the rest of loading range. It is worth noting that the nonzero MI after the crack is fully open is attributed to the nonlinearities induced by material lattice and contact boundary.³⁰

Figure 2.

Relationship between modulation intensity and crack closing area.

From the discussion above, the general varying trend of MI versus applied load can be forecasted qualitatively as shown in Figure 3. The opening stress can be approximated by the applied load of the inflection point of the varying curve. Therefore, as an example, the opening stress estimation ${\hat{σ}}_{op}$ in Figure 3 is $σ_{\infty}^{op}$ .

Figure 3.

An example of opening stress estimation.

Experiments

Specimen and fatigue test

The fatigue sample was a $400 \times 100 \times 2 m m^{3}$ 5052-H32 aluminum alloy sheet with 193 MPa the yield stress, and 12% elongation. A 10 mm long initial notch was machined by a wire electrical discharge machine (WEMD) at the center of the plate to initiate the crack. As illustrated in Figure 5, four spacers which were made of the same material of the sample were bonded at two ends of the plate, in order to reinforce the specimen and reduce the unexpected nonlinearities induced by clamping force.³⁰

A constant-amplitude uniform-tension cyclic fatigue load was applied to the specimen by a MTS 810 material testing system. The frequency of the fatigue load was 10 Hz; maximum stress $σ_{\infty}^{\max}$ was 60 MPa (12 kN for the corresponding force), and stress ratio R was 0.1. Load profile versus cycles with indicated cycles of VAM tests are schematically shown in Figure 4. The fatigue experiment procedure was divided into a preparatory stage and five segments. Each of them consisted of two parts: Part I was the cyclic fatigue load, during which no ultrasonic signals were taken; Part II was the quasi-static testing process where the tensile load varied from 0 to 60 MPa with steps 15 MPa. The excitation and data acquisition of ultrasonic wave were performed in each quasi-static step. It is worth noting that, to avoid possible compressive buckling, the thin plate was always kept in tensile state. In addition, the maximum quasi-static stress was held at the maximum fatigue stress $σ_{\infty}^{\max}$ to avoid retardation effect caused by single overload.

Figure 4.

Fatigue loading and quasi-static testing sequence for opening stress estimation.

The 6000 cycles (10 min for the corresponding time duration) vibratory stress relief (VSR)³¹ fatigue load in the preparatory stage was employed to relief the residual stress induced by the machining process and to initialize the entire testing system. Vibration response tests (VRT), which will be discussed in subsection “Pumping frequency selection,” were carried out in Part II of the preparatory stage, aiming to select out the reasonable pumping frequency for the following VAM tests. At the end of all the five fatigue processes, the produced fatigue cracks were sized by a digital camera through a 15× magnifying lens. VAM tests were then performed in each quasi-static step to estimate the opening stresses of the five sized cracks. The results of crack propagation are shown in Table 1.

Table 1.

Fatigue crack propagation results.

Experimental process	Number of fatigue cycles	Half-crack length c (mm)
Preparatory stage	6 k (VSR)	not sized
Segment 1	80 k	1.20
Segment 2	97 k	2.55
Segment 3	102 k	3.22
Segment 4	107 k	3.78
Segment 5	112 k	4.33

VSR: vibratory stress relief.

Experimental set-up

Figure 5 illustrates the schematic representation of the ultrasonic testing setup. Testing equipment consisted of four low-profile piezoelectric transducers (PZTs), a low frequency power amplifier (PI E-505 LVPZT piezo-amplifier), a band-pass (500 Hz–1 MHz) signal filter amplifier, an oscilloscope (Tektronix MSO 2014B), a signal generator (Tektronix AFG3022B), and a computer controller. All the PZTs were cemented on the right part of the plate using the X60 fast-curing two-component adhesive glue. PZT actuator VA was employed to vibrate the plate at different resonance frequency selected according to the VRT processes; sensor S2, which was located closely to the expected crack, was used to measure the vibration response; UE was used as an ultrasonic probing signal exciter; and ultrasonic sensor S1 was used to gather the VAM response signal. All the PZTs except VA (PI PL055.31, 5 mm × 5 mm × 2 mm) were PIC155 transducers (diameter 10 mm, thickness 0.5 mm). Figure 6 shows the pictorial view of the experimental setup.

Figure 5.

A schematic representation of the experimental setup.

Figure 6.

Pictorial view of the experimental setup.

Pumping frequency selection

It is well known that, when applying VAM tests to a structure, the structure resonance frequency shifting caused by external load, damage propagation, and so on, may change the MI significantly. Therefore, the VRT processes were carried out in the preparatory stage mentioned in subsection “Specimen and fatigue test.” The VRT processes were performed to select out the reasonable pumping frequency to match the structure resonance frequency which is shifting during quasi-static loading. As shown in Figure 4, VR tests were not performed after the crack spotted because the results of previous preparatory experiments suggested that the resonance frequency shifting caused by crack growth is quite little. This experimental simplification is reasonable when considering that the maximum crack length (e.g. 4.33 mm in half-length) is significantly smaller than the specimen width (e.g. 50 mm in half-width).

During each VRT, a 20 $V_{p - p}$ sweep sine chirp signal starting at 500 Hz and linearly crossing 5000 Hz in 2 s was injected into VA to excite the plate. The response signal from S2, together with the input signal, was then analyzed in the frequency domain to obtain the frequency response function. Frequency response functions were estimated according to

T_{xy} (ω) = \frac{P_{xy} (ω)}{P_{xx} (ω)}

(4)

where $P_{xx} (ω)$ is the auto power spectrum of input signal; $P_{xy} (ω)$ is the cross power spectrum of the input and response signals.

As an example, response signals and frequency response functions of the sample under the applied load of 0 and 60 MPa are given in Figure 7. It is clearly shown in Figure 7 that, some resonant modes disappeared and the resonance frequency ascended 20 Hz when the structure was loaded to 60 MPa. On the other hand, the response function curve presents as a series of sharp peaks at different resonance frequency, so the vibration behavior of the structure and also the crack changes a lot, if the pumping frequency mismatch the current resonance frequency. The mode with maximum mode amplitude was selected as the expected pumping frequency. Table 2 shows the pumping frequency selecting results at each quasi-static load.

Figure 7.

Vibration response testing results: (a) response signal and (b) frequency response function at 0 MPa; (c) response signal and (d) frequency response function at 60 MPa.

Table 2.

Selected pumping signal frequency at different applied load.

Load (MPa)	Resonance frequency (Hz)
0	3099
15	3107
30	3111
45	3115
60	3119

It should be explained that the reasons why the pumping frequency was chosen by finding the maximum of frequency response function rather than by modal shape analyzing which was employed in other researches^23,32 generally aiming to detecting or sizing cracks in freely suspended structures, were as follows. First, as described in the second paragraph of subsection “NSS model and MI of VAM,” according to the RSS modeling theory, the nonlinearity of a rough interface seems relatively less sensitive to the energy of pumping excitation as well as the mode of the crack motion controlled by the modal shape of structure. Lastly and most importantly, the selected resonant mode (refer to Table 2) is relatively stable in all the quasi-static loading steps, while other modes change a lot in modal amplitude or even disappear (refer to Figure 7). This characteristic of the chosen resonant mode ensures the crack stress conditions as consistent as possible during following VAM tests.

VAM tests

VAM experiments were executed during each step of Part II in all the five fatigue segments. All the pumping and probing signals were sinusoidal and with a constant amplitude of 20 V peak-to-peak level. During each testing, the frequency of vibration (i.e. pumping signal) excited by the transducer UE was selected according to Table 2, and the probing signal frequency was set to 50 kHz. The nonlinear responses gathered by S1 was filtered and amplified and then fast Fourier transform (FFT) analyzed for crack opening stress estimation.

Results and discussion

Sensitivity of MI to crack contact condition

Figure 8 shows the FFT spectrum of received modulation response when the half-crack length was 4.33 mm. It is clear that the amplitudes of modulation sidebands caused by a closed crack are significantly larger than an open crack. From Figure 8 and equation (3), it can be figured out that when the applied load was 0 MPa, the MI is over six times higher than the intensity when the applied force was 60 MPa. The sensitivity of MI to crack closure confirmed the feasibility of estimating crack opening stress using VAM testing.

Figure 8.

The spectrum of nonlinear modulation caused by a crack with half-length 4.33 mm under a tensile load of (a) 0 MPa and (b) 60 MPa.

Crack opening stress estimation

Once the FFT analysis was conducted, MI was calculated through equation (3). Then, the crack opening stress was estimated according to the procedure mentioned in subsection “Procedure of approximating opening stress considering PICC.”Figure 9 shows the normalized MI versus applied tensile load when the half-crack length varies from 2.55 to 4.33 mm. The closure behavior of the physically short (e.g. 1.2 mm the half-length) crack produced in segment 1 is far different from the behaviors of those relatively long cracks. Therefore, in order to study the short crack closure separately, the MI calculation results of the short crack are depicted in Figure 10 in the next subsection.

Figure 9.

Normalized modulation intensity against applied load of long cracks with different length: (a) c = 2.55 mm, (b) c = 3.22 mm, (c) c = 3.78 mm, and (d) c = 4.33 mm.

Figure 10.

Normalized modulation intensity against applied stress of a physically short crack.

Three primary results suggested by Figure 9 can be summarized as follows:

It is a general trend that the MI decreases with rising applied tensile load, and then plateaus to a level close to zero.

The opening stress estimations of the relatively long cracks are in the range of 30–45 MPa, according to the estimating procedure discussed in subsection “Procedure of approximating opening stress considering PICC.”

All the opening stress estimations are in the same range. That is to say, the crack opening stress is relatively stable during the crack propagation.

The first result is in consistence with the theoretical analysis discussed in subsection “Procedure of approximating opening stress considering PICC.” The second one matches well with the studies of Tong and Wu.³³ In their study, a modified crack opening stress equation is obtained for the center-crack tension infinite plate, by taking the crack tip element into account in a numerical analysis. The equation is as follows

\begin{array}{l} {\hat{σ}}_{o p} = (A_{0} + A_{1} R + A_{2} R^{2} + A_{3} R^{3}) σ_{\infty}^{\max}, \\ \frac{σ_{\infty}^{\max}}{σ_{y}} \leq 0.4 and R \geq 0 \end{array}

(5)

where

{\begin{matrix} A_{0} = (0.9585 - 0.4169 α + 0.0618 α^{2}) {[\cos (\frac{π σ_{\infty}^{\max}}{2 σ_{y}})]}^{1 / α} \\ A_{1} = (0.5645 - 0.0566 α) \frac{σ_{\infty}^{\max}}{σ_{y}} \\ A_{2} = 1 - A_{0} - A_{1} - A_{3} \\ A_{3} = 2 A_{0} + A_{1} - 1 \end{matrix}

(6)

where $α$ is the constraint factor, $α = 1$ for plane stress, and $α = 3$ for plane strain; $σ_{y}$ is the material yield stress. The calculation result through equation (5) and equation (6) in condition of this study is about 33 MPa, which is in the range estimated by VAM testing mentioned above. It is worth noting that the estimating accuracy will be improved if a finer sampling to the applied stress is chosen. The last result shown above also agrees well with other investigations.^34,35

Behavior of the physically short crack closure

The MI calculation results of the crack with half-length 1.2 mm are shown in Figure 10. In comparison with Figure 9, Figure 10 illustrates the far difference of closure behavior between the physically short crack and the relatively long cracks.

Figure 10 shows that there is an apparent increase in MI before the applied stress reaches 30 MPa. When the applied stress is larger than 30 MPa, the MI gradually decreases with increasing applied load. However, the inflection point does not appear up till the crack is loaded to the maximum fatigue stress $σ_{\infty}^{\max}$ , that is, 60 MPa. The possible reasons of these two phenomena are as follows:

When the crack is just initiated at the root of the initial notch, the tiny crack surfaces will carry a huge closure stress, that is, the residual stress caused by stress concentration, which can represent nearly 40% of the yield stress.³⁶ This powerful compressive stress significantly increases the stiffness of the small crack and may keep it partly closing, even if the applied stress reaches $σ_{\infty}^{\max}$ .

When the applied tensile load is removed, the crack will show the maximum stiffness. As a result, the pumping and probing waves may not be able to deform the micro-contact asperities on the crack faces. Then, the crack nonlinearity and also the MI will be very weak. As the crack is loading, MI will increase until the applied load is powerful enough to overcome the huge compressive stress, then the surfaces begin to open. After that, if the applied load is continuing to rise, the MI will turn to a decreasing function of applied load. The mathematical proof of the increasing relationship between the MI and the applied tension for a contact interface under an internal compressive stress is given in Appendix 1.

From the discussion above, it can be deduced from Figure 10 that the physically short crack began to partly open at about 30 MPa. In addition, the conservative opening stress estimation of the crack is in the range of 45–60 MPa. That is to say, the opening stress estimation of the physically short crack is higher than that of those relatively long cracks.

Conclusion

This article presents a method to measure central through fatigue crack opening stresses using nonlinear VAM testing. This method takes the advantage of the sensitivity of the vibro-acoustic MI to the crack contact area change. The crack opening stress is approximated by the current applying load when the nonlinear MI decreases to a value close to zero.

The in situ VAM tests on a 5052-H32 aluminum alloy plate under constant-amplitude uniform-tension fatigue load are carried out to experimentally verify the capability of VAM in opening stress measurement. The FFT analysis results show that when a crack is fully close, the MI is significantly higher than that of a fully open crack. Those results also confirm the sensitivity of MI to crack contact condition. The opening stress estimating results of cracks with half-length of 2.55–4.33 mm are all in the range of 30–45 MPa, and such results agree well with other approaches. The MI calculation results of a crack with half-length 1.2 mm indicate that the physically short crack begins to partly open at 30 MPa. In addition, the associated opening stress estimation is in the range of 45–60 MPa, which is higher than that of the relatively long cracks.

All the experimental results suggest that the VAM technology is not only capable of measuring the fatigue crack opening stresses, but also capable of analyzing the crack closure behavior, even for physically short cracks.

Footnotes

Appendix 1 Acknowledgements

The authors thank Dr Zhiwen Xian and Dr Yong Xiao for proofreading the manuscript. This work has been done in collaboration with all authors. Jun Jia contributed significantly to the designing, the performing as well as the result analysis of the experiments, and contributed to the writing of the manuscript. Haifeng Hu and Limin Tao contributed significantly to the procedure and the design of the proposed method and helped revise the manuscript. Yongpan Hu helped to design the experiments and revise the manuscript.

Academic Editor: Michal Kuciej

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China No. 51305449.

References

Elber

. The significance of fatigue crack closure. Damage Toler Aircr Struct 1971; 486: 230–242.

Newman

Riddell

Piascik

. Modeling the interactions between multiple crack closure mechanisms at threshold (NASA TM 2003-212402), 2003, https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20030053073.pdf

Newman

Elber

. Mechanics of fatigue crack closure. Philadelphia, PA: ASTM, 1988.

Renshaw

Holland

Thompson

. Measurement of crack opening stresses and crack closure stress profiles from heat generation in vibrating cracks. Appl Phys Lett 2008; 93: 081914–081913.

Song

Chung

. A review of crack closure measurement by compliance technique and the normalized-extended ASTM method as a currently most refined, practical and simple one. Proced Eng 2010; 2: 777–786.

Solanki

Daniewicz

Newman

. Finite element analysis of plasticity-induced fatigue crack closure: an overview. Eng Fract Mech 2004; 71: 149–171.

Antunes

Correia

Ramalho

. A parameter for quantitative analysis of plasticity induced crack closure. Int J Fatigue 2013; 71: 87–97.

Codrington

Kotousov

. The distributed dislocation technique for calculating plasticity-induced crack closure in plates of finite thickness. Int J Fracture 2007; 144: 285–295.

Resch

Nelson

. An ultrasonic method for measurement of size and opening behavior of small fatigue cracks. In: Larsen

Allison

(eds) Small-crack test methods (STP1149). Conshohocken, PA: ASTM, 1992, pp.169–196.

10.

Rokhlin

Kim

. In situ ultrasonic measurement of crack closure. Int J Fatigue 2003; 25: 51–58.

11.

Akanda

MAS

Saka

. Relationship between closure stress of small fatigue crack and ultrasonic response. J Nondestruct Eval 2004; 23: 37–47.

12.

Akino

Mihara

Yamanaka

. Fatigue crack closure analysis using nonlinear ultrasound. Symp Eval Diagn 2004; 700: 104–106.

13.

Broda

Staszewski

Martowicz

. Modelling of nonlinear crack–wave interactions for damage detection based on ultrasound—a review. J Sound Vib 2014; 333: 1097–1118.

14.

Zagrai

Donskoy

. Micro/meso scale fatigue damage accumulation monitoring using nonlinear acoustic vibro-modulation measurements. In: Proceedings of the SPIE testing, reliability, and application of micro- and nano-material systems IV, San Diego, CA, 28 February–02 March 2006, vol. 6175, p.617506. SPIE.

15.

Rudenko

. Nonlinear acoustic properties of a rough surface contact and acoustodiagnostics of a roughness height distribution. Acoust Phys+ 1994; 40: 593–596.

16.

Donskoy

Sutin

Ekimov

. Nonlinear acoustic interaction on contact interfaces and its use for nondestructive testing. Ndt&E Int 2001; 34: 231–238.

17.

Zaitsev

Sas

. Nonlinear response of a weakly damaged metal sample: a dissipative modulation mechanism of vibro-acoustic interaction. J Vib Control 2000; 6: 803–822.

18.

Klepka

Staszewski

Jenal

. Nonlinear acoustics for fatigue crack detection—experimental investigations of vibro-acoustic wave modulations. Struct Health Monit 2011; 11: 197–211.

19.

Duffour

Cawley

Morbidini

. Vibro-acoustic modulation NDE technique. Part 1: theoretical study. In: Thompson

Chimenti

(eds) Review of quantitative nondestructive evaluation. Melville, NY: American Institute of Physics, 2005, pp.608–615.

20.

Duffour

Morbidini

Cawley

. A study of the vibro-acoustic modulation technique for the detection of cracks in metals. J Acoust Soc Am 2006; 119: 1463.

21.

Lim

Song

Park

. Noncontact fatigue crack visualization using nonlinear ultrasonic modulation. Ndt&E Int 2015; 73: 8–14.

22.

Zagrai

Donskoy

Chudnovsky

. Nonlinear acoustic structural health monitoring. In: Proceedings of the 47th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference, Newport, Rhode Island, 1–4 May 2006. American Institute of Aeronautics and Astronautics.

23.

Klepka

Pieczonka

Staszewski

. Impact damage detection in laminated composites by non-linear vibro-acoustic wave modulations. Compos Part B: Eng 2014; 65: 99–108.

24.

Gian

PMF

Meo

. Residual fatigue life estimation using a nonlinear ultrasound modulation method. Smart Mater Struct 2015; 24: 1473–1484.

25.

Sun

Jiao

. Quantitative evaluation of micro-cracks using nonlinear ultrasonic modulation method. Ndt&E Int 2016; 79: 63–72.

26.

Chen

Soh

Lee

. A vibro-acoustic modulation method for the detection of delamination and kissing bond in composites. J Compos Mater 2015; 50: 3089–3104.

27.

Greenwood

Williamson

JBP

. Contact of nominally flat surfaces. P R Soc London 1966; 295: 300–319.

28.

Solodov

Krohn

Busse

. Can: an example of nonclassical acoustic nonlinearity in solids. Ultrasonics 2002; 40: 621–625.

29.

Sohn

Lim

DeSimio

. Nonlinear ultrasonic wave modulation for online fatigue crack detection. J Sound Vib 2014; 333: 1473–1484.

30.

Aymerich

Staszewski

. Impact damage detection in composite laminates using nonlinear acoustics. Compos Part A: Appl S 2010; 41: 1084–1092.

31.

Walker

Waddell

Johnston

. Vibratory stress relief—an investigation of the underlying processes. Proc IMechE, Part E: J Process Mechanical Engineering 1995; 209: 51–58.

32.

Parsons

Staszewski

. Nonlinear acoustics with low-profile piezoceramic excitation for crack detection in metallic structures. Smart Mater Struct 2006; 15: 1110–1118.

33.

Tong

. Analysis of crack opening stresses for center- and edge-crack tension specimens. Chinese J Aeronaut 2014; 27: 291–298.

34.

Newman

. A crack-closure model for predicting fatigue crack growth under aircraft spectrum loading. ASTM STP 1981; 748: 53–84.

35.

Antunes

Chegini

Correia

. Numerical study of contact forces for crack closure analysis. Int J Solids Struct 2014; 51: 1330–1339.

36.

Williams

Stouffer

. An estimate of the residual stress distribution in the vicinity of a propagating fatigue crack. Eng Fract Mech 1979; 11: 547–557.

37.

Biwa

Nakajima

Ohno

. On the acoustic nonlinearity of solid-solid contact with pressure-dependent interface stiffness. J Appl Mech 2004; 71: 508–515.