Experimental and numerical study on microcrack detection using contact nonlinear acoustics

Abstract

This paper introduces a non-classical nonlinear acoustic theory for microcrack detection in materials, comparing contact nonlinearity with material nonlinearity. The paper’s main work concentrates on the experimental and numerical verification of the effectivity of contact nonlinear acoustic detection by using the contact nonlinear parameter {\beta }^{\prime }, which can be represented by the ratio of the second-harmonic amplitude to the square of the first-harmonic amplitude. Both experiments and numerical tests are performed. The results show that {\beta }^{\prime } is sensitive to the initiation of microcracks and varies with the development of the microcracks. The numerical test illustrates the decline of {\beta }^{\prime } when microcracks penetrate each other.

Keywords

microcrack detection / contact nonlinearity / numerical analysis

Introduction

Structural materials will be damaged to some extent during the process of application, which will contribute to the loss of structural service performance and safe reliability. How to evaluate structural damage level at some time, quantificationally and nondestructively, has great social and economic significance.

The damage process of structural materials is an evolutive process, reflected as the formation, development, and penetration of microcracks. The ultrasonic wave detection method is a relatively mature method in project construction. However, the ultrasonic wave method used presently is based on linear acoustic theory, which can identify a large defect but is not sensitive to microcracks. Actually, the formation of microcracks may occupy the longest time in the structural service life. The detection of microcracks is more important. It has been proven that the nonlinear acoustic theory developed in recent years, based on the microcracks’ contact action, has a good identification capability for microcracks [1-6].

Basic nonlinear acoustic detection theory

Linear and nonlinear acoustic detection

For linear acoustic theory, it is assumed that the material stress-strain relationship is linear elastic. A single longitudinal wave propagates in the materials because of particle vibration, with the same velocity in different particles. When the particle vibration reaches a large defect interface, the vibration cannot continue but reflects and diffracts, which alters the acoustical wave propagation and leads to a change in the acoustic parameters, such as longer wave propagation time, more energy dissipation, etc. All the features are considered as linear acoustical characters. The NDE methods based on the linear acoustical characters are called linear acoustic detection methods.

When the material stress-strain relationship is not linear elastic, an inputting longitudinal wave will have a different velocity in different particles, which may cause acoustical wave interference and wave shape deformation. Higher harmonic components may appear, which result in energy redistribution in the frequency domain. The nonlinear acoustic detection method is based on nonlinear acoustic characters, typically as the appearance of higher harmonic components.

It is proven by experiments that for materials with microcracks, even if the material stress-strain relationship is linear elastic, the output wave signal may have higher harmonic components, as shown in atomic nonlinear elastic materials. It is because of this that the contact action in the microcrack is highly nonlinear; that is, border stage nonlinearity [7,8].

The acoustic theory based on atomic material nonlinearity is called classical nonlinear acoustic theory. The acoustic theory based on contact nonlinearity, or border stage nonlinearity, is named as contact nonlinear acoustic theory or nonclassical nonlinear acoustic theory, which is certificated as an effective way of microcrack detection.

Classical nonlinear acoustic theory and nonlinear parameter

Classical nonlinear acoustic theory can be explained by deducing a one-dimensional nonlinear elastic longitudinal wave equation. We can define the stress-strain relationship of nonlinear elastic material as \sigma =Eϵ+F{ϵ}^{\mathrm{2}}; here, \sigma is stress, ϵ is strain, E is the first-order elastic constant, and F is the second-order elastic constant. \beta =\mathrm{2}F/E is defined as a nonlinear parameter that expresses the level of nonlinearity. Then, a one-dimensional nonlinear elastic longitudinal wave equation can be described as follows:

(1){\displaystyle \frac{{\partial }^{\mathrm{2}}u}{\partial t^{\mathrm{2}}}}-c^{\mathrm{2}}{\displaystyle \frac{{\partial }^{\mathrm{2}}u}{\partial x^{\mathrm{2}}}}=\beta c^{\mathrm{2}}{\displaystyle \frac{\partial u}{\partial x}}{\displaystyle \frac{{\partial }^{\mathrm{2}}u}{\partial x^{\mathrm{2}}}}.

In the above equation, u is the particle displacement, t is time, x is the coordinate axis, and c is the one-dimensional linear elastic longitudinal wave velocity, which relates with the first-order elastic constant and density of the material as c=\sqrt{{\displaystyle \frac{E}{\rho }}}; here, \rho is the material density.

If the initial condition of the wave equation is

(2)u(x,t)=A_{\mathrm{0}}\cos (\omega \tau ),

in which A_{\mathrm{0}} is the initial signal amplitude of vibration, \omega is the wave circular frequency, and \tau =t-x/c.

By using the perturbation method, we can obtain the approximate solution of the wave equation:

(3)u(x,t)=A_{\mathrm{0}}\cos (\omega \tau )-\beta x{\displaystyle \frac{A_{\mathrm{0}}^{\mathrm{2}}{k}^2}{\mathrm{8}}}\cos (\mathrm{2}\omega \tau ).

Here, k is the wave number and k=\omega /c. If the first-harmonic amplitude is A_{\mathrm{1}} and the second-harmonic amplitude is A_{\mathrm{2}}, then

(4)A_{\mathrm{1}}=A_{\mathrm{0}},A_{\mathrm{2}}=\beta x{\displaystyle \frac{A_{\mathrm{0}}^{\mathrm{2}}{k}^2}{\mathrm{8}}}.

The nonlinear parameter can be described as \beta =\mathrm{8}{A}_{\mathrm{2}}/(x{A}_{\mathrm{1}}^{\mathrm{2}}{k}^2). Apparently, the nonlinear parameter \beta has a linear relationship with the ratio of the second-harmonic amplitude to the square of the first-harmonic amplitude, as

(5)\beta {\displaystyle \sim \frac{A_{\mathrm{2}}}{A_{\mathrm{1}}^{\mathrm{2}}}}.

By the deduction of classical nonlinear acoustic theory, we can consider the nonlinear parameter \beta as \beta =A_{\mathrm{2}}/A_{\mathrm{1}}^{\mathrm{2}}, which can show the level of nonlinearity relatively. For the material with contact nonlinearity, a similar form also can be assumed for the contact nonlinear parameter [9], {\beta }^{\prime }=A_{\mathrm{2}}/A_{\mathrm{1}}^{\mathrm{2}}.

Experimental verification

Testing system

Figure 1 shows the nonlinear ultrasonic testing system configuration. A narrow band ultrasonic transmitting transducer (Panametrics X1021) with 50 kHz center frequency was used as transmitter, a broad band ultrasonic transmitting transducer (Ultran RD100) with 100 kHz center frequency was used as receiver. The wave form of the transmitting signal was a tone of a burst sinusodial wave with 200 cycles and a frequency of 36 kHz. In order to reduce the nonlinear influence of the ultrasonic transducer itself, the level of input signal was controlled to 15 Vpp, and the wave time domain signal equaled the average value of sampling 300 times, with the sampling frequency being 5 MHz and with a sampling length equal to 200000.

Samples preparation

To experimentally investigate damage in cement-based materials, microcracks were induced by an alkali-silica reaction (ASR). Two groups of one-dimensional mortar specimens, S0 and S1, with a water to cement to aggregate ratio of 1 to 2.16 to 4.85 were prepared. There were 6 specimens in all, and each group had 3 specimens, numbered as S0-1/2/3 and S1-1/2/3. The dimensions of the specimens are 25.4 mm × 25.4 mm × 254 mm. In order to induce microcracks in the samples, an alkali-reactive cherty sand was used as aggregate, consisting of a 50/50 blend of aggregate retained on the No.30 and No.50 sieves, according to ASTM C1260.

According to the experiment method defined in ASTM C1260, an alkali-silica reaction damage of different grades can be induced into the S1 samples as follows. After a curing period, the samples in group S1 were exposed to 1 N NaOH solution in an 80°C oven, keeping the S1-1/2/3 sample in NaOH solution, respectively, for 2 d, 3 d, and 5 d; the samples in group S0 were made for comparison, being kept in pure water in an 80°C oven for 2 d, 3 d, and 5 d, respectively. Then, the extension of every sample was measured by a micrometer gauge, and the extension value was used as a parameter for describing the material damage caused by the alkali-silica reaction. Table 1 shows the extension of the two group samples with and without ASR damage.

According to ASTM C1260, if the extension of the samples kept in the NaOH solution for 14 d reaches 0.2%, then the aggregate must be sensitive to the alkali-silica reaction. In the experiment, the max extension of the testing sample kept in the NaOH solution for 5 d reaches 0.32%. It is shown that the aggregate used is sensitive to the alkali-silica reaction, and there should be many microcracks in the samples in group S1.

Testing results and discussion

The tests were repeated on every sample with the mode of increasing signal strength in each sample, so that with the increasing amplitude of first-harmonic A_{\mathrm{1}}, the correspondent amplitude of second-harmonic A_{\mathrm{2}} can be obtained. Figure 2 shows the relationship between A_{\mathrm{1}}^{\mathrm{2}} and A_{\mathrm{2}}, in which each line is the linear fitting for all the test data of the samples.

From Fig. 2, it can be seen that the square of the amplitude of the first-harmonic A_{\mathrm{1}} and the amplitude of the second-harmonic A_{\mathrm{2}} maintain a good linear relationship. Using linear fitting, the slope k of the line, which represents the ratio A_{\mathrm{2}}/A_{\mathrm{1}}^{\mathrm{2}}, the nonlinear parameter, shows more nonlinearity in the damaged group. So the method of nonlinear acoustic detection based on contact action is proven to be sensitive to microcracks.

Figure 3 demonstrates that the slope varies with the different degrees of ASR damage, showing that with the microcrack development, the initial single microcrack may combine into another, which results in the decrease of the ratio A_{\mathrm{2}}/A_{\mathrm{1}}^{\mathrm{2}}.

Numerical analyses

The process of acoustical wave transmission in materials is actually a process of stress wave transmission. A numerical analysis based on ABAQUS was performed to stimulate the effects of the microcracks, especially multi-microcracks penetration.

Numerical model

A 200 mm × 400 mm plain block model was established with the material Young’s module E=100 GPa and the Poisson’s ratio \upsilon=0.3. The sinusoidal load P, with amplitude equaling 500 MPa (about 72 ksi) and frequency equaling 25 kHz, was applied to one side of the model. On the other side, restraint was applied and the counterforce was recorded. Here, the large load amplitude was applied for analysis.

Two microcracks were initially defined in the model for simulating microcrack development and penetration, as shown in Fig. 4. The microcrack was diamond shaped, simulating the contact action of the crack tips. And it was assumed that when the microcrack developed, the crack tips would keep the same shape; that is to say, the angle of the crack tips would not change. The center crack width was adjusted according to the previous assumption when the microcrack changed. Table 2 shows all the models for numerical analysis.

In the numerical stimulation, the General-Static step in ABAQUS was used for analysis. It can be found out that if the gap of the sampling data is short enough, the result of the static analysis is near to that of a dynamic analysis for stimulating stress wave transmission. Here, a sampling frequency of 2.5 MHz and a sampling length of 10000 were adopted; the frequency resolution was 0.25 kHz. The mesh in each model was in the size of 1 mm×1 mm, which was small enough for analysis by a numerical check.

Numerical results and discussion

Figure 5 shows the typical frequency-domain signal of the restraint counterforce. For each model, the average value of the first-harmonic amplitude A_{\mathrm{1}} and the second-harmonic amplitude A_{\mathrm{2}} was summed up and averaged along the edge. Then, the relationship between the ratio A_{\mathrm{2}}/A_{\mathrm{1}}^{\mathrm{2}} and the total crack length is shown in Fig. 6.

In Fig. 5, it can be seen that higher harmonics appear in the restraint counterforce signal because of the microcrack contact action. The contact nonlinear acoustic detection method is proven to be sensitive to microcrack damage.

In Fig. 6, it can be seen that the numerical test has shown a similar tendency to an experimental test for the relationship between the ratio A_{\mathrm{2}}/A_{\mathrm{1}}^{\mathrm{2}} and damage development. The assumption that the microcracks’ penetration would cause a decrease in the ratio A_{\mathrm{2}}/A_{\mathrm{1}}^{\mathrm{2}} is proven by numerical analysis.

Conclusions

From the experiment and numerical test above, some qualitative conclusions for microcrack detection using contact nonlinear acoustic theory can be obtained:

The microcracks in materials may come into contact with each other, which can result in higher harmonics. The contact nonlinear parameter {\beta }^{\prime }=A_{\mathrm{2}}/A_{\mathrm{1}}^{\mathrm{2}} can be used to measure material damage level. And it has been found out that the contact nonlinear parameter {\beta }^{\prime } increases with the microcracks’ development and decreases with the microcracks’ penetration. The tendency has been proven by the experiment and numerical test above.

References

[1] Yost W T, Cantrell J H. The effects of fatigue on acoustic nonlinearity in aluminum alloys. In: Ultrasonics Symposium, 1992. Proceedings, IEEE,1992, 947–955

[2] Nagy P B. Fatigue damage assessment by nonlinear ultrasonic materials characterization. Ultrasonics, 1998, 36(1-5): 375–381

[3] Frouin J. Acoustical linear and nonlinear behavior of fatigued titanium alloys. OH: Dayton University, 2001

[4] Shah A A, Ribakov Y, Hirose S. Nondestructive evaluation of damaged concrete using nonlinear ultrasonics. Materials and Design, 2008, 30(3): 775–782

[5] Donskoy D, Sutin A, Ekimov A. Nolinear acoustic interaction on contact interfaces and its use for nondestructive testing. NDE & International, 2001, 34(4): 231–238

[6] Chen X J, Kim J Y, Qu J, Kurtis K E, Wo S C, Jacobs L J. Microcrack identification in cement-based materials using nonlinear acoustic waves. Review of Progress in Quantitative Nondestructive Evaluation, AIP Proceedings, 2007, 894: 1361–1367

[7] Solodov I Y. Ultrasonic of non-linear contacts: propagation, refection and NDE application. Ultrasonics, 1998, 36(1-5): 383–390

[8] Solodov I Y, Krohn N, Busse G. CAN: an example of nonclassical acoustic nonlinearity in solids. Ultrasonics, 2002, 40(1-8): 621–625

[9] Chen Xiajia. Study on initial damage detection and evaluation of concrete based on nonlinear acoustic features. Wuhan: Wuhan University of Technology, 2007 (in Chinese)

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