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Case Studies in Nondestructive Testing and Evaluation

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Dispersion of ultrasonic surface waves in a steel-epoxy-concrete bonding layered medium based on analytical, experimental, and numerical study

Yang Shen *1, Sohichi Hirose2, Yuya Yamaguchi3

Dept. of Mechanical & Environmental Informatics, Tokyo Institute of Technology, Japan

A R T I C L E I N F 0 A B S T R A C T

Article history: The epoxy-bonded steel plate strengthening method has been widely applied in retrofitting

AraHaMe (inline 1 August 2014 reinforced concrete structures. However, using epoxy as the adhesive will bring deterio-

ration due to its aging and delamination. Ultrasonic Nondestructive Evaluation (NDE) is an advantaged approach to detect the delamination or the bonding quality of the layered medium. The elastic property of the adhesive material can also be estimated properly by obtaining the phase velocity of surface wave through NDE test. This research proposes an approach to estimate the elastic property of epoxy layer in a steel-epoxy-concrete bonding layered medium. By solving dispersion equations, the analytical dispersion curves are plotted. Then the influence factors to the modes and shapes of those dispersion curves are discussed. An ultrasonic NDE test on a specimen is conducted, by which the relation between phase velocity and frequency is obtained. Through inversion process, the elastic property of epoxy layer is estimated. Based on the estimated elastic constants, a numerical study of steel-epoxy-concrete layered medium is also conducted using Explicit Finite Element Method, from which the numerical dispersion curves are obtained. Through analytical, experimental, and numerical studies, the dispersion property of surface waves in the layered medium is well understood.

© 2014 Published by Elsevier Ltd.

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1. Introduction

As a widely used retrofitting approach applied in reinforced concrete (RC) structures, e.g. RC bridges, long span RC buildings, the epoxy-bonded steel plate strengthening method is effective and low cost [3,13]. However, using epoxy as the adhesive between concrete and steel will meet the deterioration problem due to epoxy's aging [14,19] and the delamination on the interfaces [10]. Ultrasonic Nondestructive Evaluation (NDE) is now applied frequently in multi-layered bonding materials and composite materials [4,8,11], in particular, to detect the delamination or the bonding quality of the layered medium. Furthermore, through obtaining the phase velocity of surface wave propagating in the medium using NDE, the elastic property of the adhesive material can be estimated properly [17]. For a material with aging problem like epoxy,

* Corresponding author. Tel.: +81 03 5734 2692; fax: +81 03 5734 2692. E-mail address: shen.y.aa@m.titech.ac.jp (Y. Shen).

1 Doctoral student.

2 Professor.

3 Master student.

http://dx.doi.org/10.1016/jxsndt.2014.07.002 2214-6571/© 2014 Published by Elsevier Ltd.

Fig. 1. Transducer setting and wave propagation in a steel-epoxy-concrete 3 layers model.

knowing its material property before the occurrence of severe deterioration is significant in engineering. Therefore, a feasible detection method for elastic property of epoxy layer in the steel-epoxy-concrete bonding layered medium is in demand to be developed.

In bonding layered medium, wave propagates dispersively. Wave dispersion in bonding layered medium needs to be studied well since the composite medium with different materials will have significantly different dispersion properties. [6] utilized a laser generated surface wave to obtain dispersion property of a copper-epoxy-aluminum composite specimen. Some researchers [9] conducted research and field test in multi-layer pavement structures. Through analytical calculation, [5] mfounded that even a slight alteration of layers' elastic constants can significantly affect wave's dispersion property. Therefore, before the NDE test, a comprehensive understanding of wave dispersion in steel-epoxy-concrete bonding layered medium is necessary.

Through generated surface wave test and spectral analysis, the experimental dispersion curves are able to be obtained. After an inversion process between the analytical dispersion curves and the experimental ones, layer's elastic property can be estimated. The Spectral Analysis of Surface Wave (SASW) method was originally applied in geotechnical research [15] in which the acoustic wave frequency is under 100 Hz. Also, pavement structures were usually tested through this method [9], with the frequency normally under 500 Hz. In high frequency range (500 kHz~4000 kHz), several attempts have also been made for layered metal specimens [16]. In both low and high frequency range, this method can be applied successfully, however, in the frequency range about 10 kHz~300 kHz, which is an approximately range for steel-concrete composite material, rare work has been done. The ultrasonic transducer with its working frequency around 200 kHz will be suitable for the surface wave test on the steel-epoxy-concrete bonding layered medium, but the material concrete's inhomogeneity will challenge the effect of ultrasonic NDE test. For material like concrete, which is significantly important in infrastructure construction, nevertheless, the NDE attempts are few and always not easy [12]. The ultrasonic NDE study on layered medium which consists of concrete has almost not yet been concerned, although, in many fields, it is urgently to be conducted.

In this study, firstly, a model of 2 layers overlying on a solid half space will be built, based on which the dispersion equation will be deduced and analytical dispersion curves can be plotted. Then the modes and shapes of dispersion curves influenced by layers' material properties will be discussed. Meanwhile an ultrasonic NDE test on a steel-epoxy-concrete bonding specimen will be taken. The SASW method will be applied into data processing to obtain the experimental dispersion curves of the specimen. Through an inversion process based on a variance function for multi-modes between the analytical and experimental dispersion curves, the elastic property of the epoxy layer can be estimated, which will be introduced in the Explicit Finite Element Method (EFEM) model as the material property setting. Through the 3-D EFEM simulation, the surface wave propagation inside the specimen can be visualized, and the numerical dispersion curves can also be plotted through the spectrum analysis of numerical waveforms. The numerical result can be used to further verify the accuracy of the material property estimation approach.

2. Analysis of dispersion curves

Wave propagation in a semi-infinite solid covered by multi layers of uniform thickness has been studied by many researchers before. The deduction of the dispersion equation for multi-layered medium can also be found in [18]. Here, for brevity, only the final form of the dispersion equation of the model of 2 layers overlying on a half space (Fig. 1) is shown in Appendix A.

To solve the dispersion equation, an improved bisection method based algorithm is proposed for multi-roots searching. Then, the dispersion curves, which indicate the relation between wave number or frequency versus phase velocity, can be plotted.

The dispersion curves of multi-layered medium are quite sensitive to layer's material properties. The stiffening (longitude and transverse phase velocities of lower layer are larger than those of upper layer) and softening (phase velocities of lower

Table 1

Material parameters for analytical dispersion curves.

Layer's material

P-wave velocity Cl (m/s)

S-wave velocity

Ct (m/s)

Density

(kg/m3)

Thickness (mm)

M2 M3 M4

1600 2000 2500 2500

800 1000 1112 1112

1120 1120 1120 1120

Concrete

M1-M3 M4

3400 4000

2200 2450

2400 2400

Fig. 2. Analytical dispersion curves with material property assumption in Table 1. a, Model 1; b, Model 2; c, Model 3; d, Model 4. kH is a dimensionless value, where k is wavenumber, H is 1st layer's thickness.

layer are smaller than those of upper layer) media can have significantly different dispersion properties. For instance, in a copper-epoxy-aluminum composite medium [17], in which the third layer aluminum has the highest phase velocity, only one mode can be activated in the concerned frequency range, and in high frequency range, wave propagates with Rayleigh wave speed of the first layer; in a softening layered medium however, due to a low phase velocity of the half-space, when the phase velocity nearly equals the transverse velocity of the half-space, the mode will cease to propagate, which is called "cutoff effect" [2]. Here, a model of steel-epoxy-concrete layered medium is studied, which belongs to the softening medium as concrete in the bottom and softest material epoxy in the middle.

In Table 1, four models of material property are presented. In all of these models, the properties of steel layer are set to be unchangeable. To understand the effect of different material properties of epoxy layer to dispersion curves, the phase velocities of epoxy are raised in sequence from Model 1 to Model 3, where all of these settings are based on a real possibility of epoxy material. The material of concrete is approximated as homogeneous in those analytical models, and because it is difficult to be precisely qualified on its elastic constants, we propose an enhanced constants setting of concrete in Model 4 to see its influence.

Fig. 2 shows the analytical dispersion curves of the four models in Table 1. From Fig. 2, we can see that infinite modes exist in the Steel-Epoxy-Concrete layered medium. The 1st mode starts from the Rayleigh wave speed of half-space (3rd layer) CR3, and other modes start from transverse wave speed of half-space (3rd layer) CT3 with cutoff frequencies. All the modes asymptotically trend to the transverse wave speed of 2nd layer CT2 as frequency tends towards infinity. The 1st mode performs a hook-like curve in low frequency range, where the minimum point of the "hook" is determined by the 2nd layer's transverse wave speed CT2 also.

In Model 1 (Fig. 2a), phase velocities of all the modes decrease faster than they do in other models as the lowest phase velocity of the 2nd layer is assumed. An interesting phenomenon is that the dispersion curve of the 4th mode seems to be

cut when its phase velocity reaches CT3, and be excited again when frequency increases. This phenomenon again confirms that the phase velocity of guided waves must be less than the shear wave velocity of the last layer in a stratified half-space. Otherwise, the energy of this guided wave will be infinite when the depth z tends towards infinity [5]. In Model 2 (Fig. 2b), due to the increment of elastic constants of epoxy layer, the "hook" of the 1st mode curve is pulled up to meet the 2nd mode closely. Then in Model 3 (Fig. 2c), with a relative high elastic constants of epoxy, the 1st mode has crossed over the 2nd mode and reaches closely to CT3, which is the limitation of the wave speed in this medium. In Model 4 (Fig. 2d), the high elastic constants of concrete has clearly lifted the curves of all the modes up as the maximum values of those curves are determined by the material property of half-space, however, the minimum values, determined by the second layer, are almost same as the ones in Model 3.

Different frequency ranges of dispersion curves are sensitive or insensitive to different parameters. In relative high frequency range (above 230 kHz), theoretically, the dispersion curves are sensitive to the change of elastic properties: CL and CT of epoxy layer. However, due to the decrease of 1st mode's velocity, and more energy contribution of other higher modes, those multi-modes are hard to be detected and distinguished. When the frequency goes even higher, for a wave with very short wavelength, the layered half-space is equivalent to a homogeneous half-space composed of the first layer's material only. Hence, although theoretically, there is no mode can be excited whose phase velocity is greater than CT3, it still should be considered that the limit of the phase velocity in high frequency range is CR1, which is the Rayleigh wave speed of the first layer. The practical measurement also shows that, in high frequency range, the detectable wave's speed is quite close to the Rayleigh wave speed of steel. Therefore, for the parameter of epoxy's elastic constants, we found that the range of 20 kHz~160 kHz, which is around the "hook" shape part of the 1st mode, is the most distinguishable and sensitive part of dispersion curves with even small change of elastic constants values. After knowing each parameter's effect on dispersion curves and its sensitivity to those parameters, we decide to use ultrasonic transducer of 200 kHz central frequency as transmitter and receiver, and we can roughly predict the shape of the analytical dispersion curves and the range of phase velocities before conducting calculation, which can efficiently improve the inversion process in the later work.

3. Ultrasonic nondestructive test

A steel-epoxy-concrete 3 layer bonding specimen has been manufactured approximating to the composite structure on RC bridge strengthened by steel plates. Firstly a concrete block (400 mm x 400 mm x 150 mm) was casted, then it was supported above a steel plate (500 mm x 500 mm x 4.5 mm) with a gap of 5 mm, after that the epoxy was injected into the gap evenly, as shown in Fig. 3a. When the epoxy finished hardening, the ultrasonic transducers can be set on the steel plate (Fig. 3b).

Through compression test conducted on sample concrete cylinders, with diameter of 100 mm, length of 200 mm, and density of 2400 kg/m3, the compressive elastic modulus (Ec) of the concrete is obtained as Ec = 28.92Gpa. However, in this study, the dynamic modulus (Ed) of concrete rather than the static compressive modulus (Ec) should be used. Several attempts have been made to correlate static compressive (Ec) and dynamic (E¿) moduli for concrete. The simplest of these empirical relations is proposed by Lydon and Balendran [1]:

Ec = 0.83Ed (1)

According to Eq. (1), the dynamic modulus of the concrete used in the specimen is E¿ = 34.84Gpa. With a given Poisson's ratio of 0.2, the longitudinal and transverse phase velocity of the concrete used in the specimen is about: CL = 4000 m/s; CT = 2450 m/s. These values will be used in the following inversion process, as the known parameters of the analytical dispersion curves.

As shown in Fig. 3b, two normal-type ultrasonic transducers of frequency of 200 kHz are used in this NDE test. A pitch-catch method is applied, with one transducer as transmitter and another one as receiver. To keep good signal coherence, the transmitter and receiver are the same type. The diameter of the transducer's contact surface is 34.2 mm. They are vertically set on the plane of the specimen, to produce normal-type ultrasonic wave. As conducting medium, Glycerine is pasted between transducers and specimen. The contact pressure of transmitting and receiving transducer is produced by their self-weight of 325 g. An integrated high power ultrasonic pulser&receiver is utilized in this test. The generated frequency range of pulser is 30 kHz~10 MHz, and the detectable frequency range of receiver is 300 Hz~30 MHz. In this test, one cycle of pulse wave (rectangular wave) with frequency of 200 kHz is employed as incident wave and will be generated by the pulser. The pulser&receiver is controlled by a portable computer with parameter setting and pulser's trigger releasing. The received signal is also recorded in the PC with real-time wave form presenting and FFT spectrum analyzing. This ultrasonic testing system is light-weight, portable for field testing; meanwhile, it has a wide working frequency range for different objective to be detected, as long as corresponding working frequencies' transducers are utilized.

To know the transducers' characteristic more specifically, a pulse-echo test has been conducted on a homogeneous Poly-methylmethacrylate (PMMA) block, with the dimension of 400 mm x 152 mm x 152 mm. Identical with the following tests, one cycle of 200 kHz pulse wave is generated by the pulser, which is transmitted by the transducer and reflected by the opposite face (152 mm) of the block, then received by the same transducer. Fig. 4a shows the waveform of the 1st reflected wave; Fig. 4b shows its Fourier spectrum, from which we can see that the effective frequency range of this type of transducer is from 50 kHz to 350 kHz, as the incident wave is 200 kHz.

As shown in Fig. 5, the receiver is positioned on two spots symmetrically off the center line with a distance D apart. These two spots are marked as R1 and R2, and the position of transmitter is marked as T. During the test, the receiver is firstly positioned on R1, detecting the surface wave signal from R1, then it is shifted to R2, keeping the same pulse wave from the transmitter on T. Although the signals from R1 and R2 are not recorded simultaneously, by keeping the pulser's parameters and distance between transmitter and receiver identical, this method of single receiver shifting can be equivalent to the detection using double receivers simultaneously.

Fig. 5 shows the plan of transducer setting in the experiment. Because our interested frequency range is about 20 kHz~160 kHz, and especially 20 kHz~60 kHz, in which the most distinguishable (sensitive) part: the bottom of the 1st mode's "hook" exists, so a relatively large spacing between receivers is adopted. In low frequency range (below 50 kHz), the corresponding long wavelength requires a relative large receiver spacing, that can probe the wave signal distinctly. For example, at f = 20 kHz, the wavelength is about 75 mm, to which a comparative receiver spacing distance is needed. However, based on practical experience from repeatedly testing, a too large spacing may cause significant signal attenuation from R1 to R2, which is a negative effect in spectral analysis afterwards. In the experiment, the spacing D between R1 and R2 is shifted from 15 mm, with every 15 mm's interval, until 75 mm to find a most proper distance from multiple concerns. Meanwhile, to compare the data collected from two opposite directions, the transmitter is also positioned on two symmetrical spots. Theoretically, if the material is isotropic and homogeneous, and the epoxy layer has uniform thickness, the results obtained from the opposite direction should be the same.

4. Spectral analysis of surface waves

From the NDE test, the waveform data in time domain can be obtained. However, it is difficult to extract information regarding to the dispersion properties directly from the time domain signals. The Fast Fourier Transform (FFT) and Spectral Analysis of Surface Waves (SASW) are needed in the signal processing. Then, in frequency domain, the phase difference of

Fig. 4. Pulse-echo test on a homogeneous PMMA block: a, received waveform; b, Fourier spectrum.

Fig. 5. Transducers' position setting on the specimen. (TL: Transmitter on LHS, TR: Transmitter on RHS Ri: Receiver 1, R2: Receiver 2.)

the two receivers can be obtained, from which, the relation between phase velocity and frequency or wavenumber, namely, dispersion curves can be plotted experimentally.

The transducers' coordinates are already shown in Fig. 1. If we transform the time domain wave form into frequency domain using FFT, we can have the cross-power spectrum, Sx1x2 (f), between the two signals from R1 and R2 with distance D, which is defined as

Sx1,x2 (f) = SE{[R1( f)] i ■ [R2( f)] J (2)

where, R1(f) and R2(f) correspond to the Fourier transforms of time records from two receivers located a distance D apart. The bar above Sx1x2(f) corresponds to the frequency-domain average of several records. Parameter n is the number of records averaged, in this study, n = 3 is adopted. The asterisk above R2 (f) corresponds to the complex conjugate operator.

Another important function here is the coherence function, y2 (f), which characterizes the signals' reliability. It is calculated from:

2sc\ |Sx1,x2( f )|2

Y2(f) = t T ^ (3)

Ax1( f) ■ Ax2 (f)

where, Ax1(f) and Ax2(f) correspond to the averaged auto power spectra of records from receiver 1 and receiver 2, respectively. The auto power spectrum for a record Axk(f) is defined as: - 1 n

Axk(f) = ^{[Rf • [Ri(/)LJ (k = 1 2) (4)

From RHS of Eq. (4), we can see that everything is averaged by n times, which means the value of coherence function Y2( f) is a measurement of experimental repeatability. If the recorded signals are reliable, then only tiny difference exists among n times repeat, and the value of Y2( f) will approach to the unity.

For each frequency f, the phase shift m can be picked from the cross-power spectrum. The cross-power spectrum Sx1,x2(f) is a complex-valued parameter. Therefore, the phase is calculated from:

_1 ¡mag[SX1x2( f)] .c.

m = tan 1-=--(5)

^ Real[ Sx1,x2 (f)]

Knowing the phase, the travel time t can be calculated by:

t = ni (g)

and the phase velocity c can be obtained by:

D X1 - X2 X1 - X2

c =- = —m— = 2n f--(7)

t 2Ff m

where, D = x1 — x2 is the distance between the receivers.

From Eq. (7), the relation between phase velocity c and frequency f is revealed, according to which the experimental dispersion curves can be plotted.

5. Estimation of elastic properties

Fig. 6a shows the earlier arrived waveforms observed by the receiver located on R1 and R2 when the transmitter is located on the left and right hand side respectively (see Fig. 5 as reference). Fig. 6b shows the later arrived waveforms observed in the two opposite settings. From both Figs. 6a and b, we can clearly see that the waveforms obtained from the two opposite position settings of the transducer are quite close, which reveals a good "homogeneity" from different areas of the specimen. In order to focus on the surface wave only but not the reflected wave, a waveform of a short time period (about 130 |as here) is cropped. The attenuation of the surface wave after 60 mm's propagation is clear through the amplitude comparison between waveforms in Fig. 6a and b.

Fig. 7a and b show the Fourier spectra of the waveforms with transmitter on LHS. The Fourier spectrum gives a clear sight that main energy of those waves is distributed between 0 kHz and 300 kHz, among which, around very low frequency near 0 kHz and 150 kHz, the energy distribution is low with some notable amplitude drops. Usually, for ultrasonic transducer, the performance in very low frequency part is tricky, limited by the working range of the transducer. In the spectral analysis, the waveform with a general even energy distribution on considered frequency range is preferred. The amplitude drop in frequency domain will bring phase information loss, which will be discussed later.

With the Fourier spectrum, according to Eq. (2), we can have the cross-power spectrum, Sx1x2( f), from which the phase difference of each frequency can be obtained (Fig. 9a). With Eq. (3), the coherence function, y2(f) (Fig. 8) can be calculated. For each position setting, the same test will be repeated for n times, in this study, n = 3. Since the spectrum analysis is based on the averaged result of these n times tests, the function value of Y2( f) is a judge of coherence of those tests. If the test is repeatable, namely, the measured data is stable, the coherence value will converge towards one. In Fig. 8, the coherence values of both the opposite settings are close to 1 below 300 kHz, and become unstable above 300 kHz. Thus, only frequency components lower than 300 kHz are adopted. However, even in this range, we can also see that at the

Fig. 6. Received waveforms when D = 60 mm. a, earlier arrived waveforms; b, later arrived waveforms.

frequencies near 0 kHz, and 150 kHz, the coherence value drops by 10% more or less, due to the relatively low energy distribution around there.

If we unfold the phase difference in Fig. 9a from periodic phase to continuous one, we can redraw the continuous phase difference as shown in Fig. 9b. After unfolding, the data points on A (98 kHz) and B (195 kHz) in Fig. 9a are then located on a continuous phase difference curve as in Fig. 9b. With the continuous phase difference curve, the experimental dispersion curves then can be plotted (Fig. 10) according to Eq. (14). In Fig. 9b and Fig. 10, the results agree well when the transmitter is set oppositely, which shows a good homogeneity of the specimen. From Fig. 10, we can see that in the frequency range near 0 kHz and around 150 kHz, the results do not behave as expected, caused by the amplitude drop in frequency domain (Fig. 7) and coherence value drop in Fig. 8.

When both the analytical (Fig. 2) and experimental (Fig. 10) dispersion curves have been obtained, an inversion process based on a variance function for multi-modes can be applied to find the most appropriate analytical curve, which has minimum difference with the experimental one. The variance function can be expressed as:

Fvar —

EN=1 [Cexp (i) - cZde" (i)]2

EN— 1 [Cexp (i )]2

where, i represents each data point in the calculated frequency range and N is the total number of data points utilized in the inversion process. The cexp (i) is the experimental phase velocity obtained from spectrum analysis, and the cimotdex (i) is the analytical phase velocity of the most adjacent mode with the experimental value. When there are multi-modes existing in the frequency range, the algorithm will calculate and compare the distances of those modes' value to the experimental value, and pick up the most adjacent one into the calculation of Fvar. The value of cmf™(i) is also affected by several variables such as phase velocity of materials, and layer's depth, as we discussed in Section 2. Here, as the epoxy layer's depth is already known, we only consider two parameter variables: the epoxy layer's longitudinal velocity CL2 and transverse velocity C T2. Namely, the variance function can be described as:

Fvar — f (CL2, CT2)

Fig. 7. Fourier spectra of waveforms. a, receiver on R1, transmitter on left; b, receiver on R2, transmitter on left.

Fig. 8. Coherence function values in the cases of transmitter on left and right.

A simplex method [7] based minimization process is then applied to find the minimum value of Fvar from an initial guess of CL2 and CT2. A very small critical value will be set to cease the minimum value searching. Fig. 11 gives the inversion result of one of specimens. An initial guess has been set as CL2 = 1600 m/s and CT2 = 800 m/s. After 32 loops of iteration, the most appropriate analytical curves have been found. The variance function value of the last loop, namely, the error of the inversion process is 9.0e—4. The frequency range of the inversion process is from 15 kHz to 230 kHz, excluding the very low frequency points with low coherence value. Fig. 11 uses the same experimental data as previous figures. From this figure, we can see that the interference of the 2nd wave mode has influenced the measured data around 150 kHz, in other

50 100 150 200 250 300 350 400 450 500 Frequency (kHz)

a - ft ° a

B i •a 3 o o □ 0 □ o 0 B ° a

8 6 -

A ° ° | Q 0 ^ G § 8

s g o Transmitter on Left

o ° Transmitter on Right

9 0 1 1 1 1 b

0 50 100 150 200 250 300

Frequency (kHz)

Fig. 9. a, Periodic phase difference; b, continuous phase difference.

Fig. 10. The experimental dispersion curves.

words, in this frequency range, the 2nd mode is more easily to be excited than the 1st mode. Hence, the phase velocity measured around 150 kHz is mainly from the 2nd mode, which is faster than the 1st mode.

Before we conduct the inversion process, we need to check the experimental dispersion curve manually and get rid of those obviously deviated phase velocity points refer to the coherence value spectrum and phase difference spectrum. This procedure can efficiently increase the accuracy of the inversion and reduce the convergence time. Hence, obtaining a well-recognized experimental dispersion curve is the basic premise of the following inversion process.

Fig. 11. Comparison of the experimental and analytical dispersion curves.

Table 2

Estimated material property of two specimens.

Layer's material

Estimated CL (m/s)

Estimated CT (m/s)

Density (kg/m3)

Measured CL (m/s)

Steel Epoxy

3000 2000

1700 1000

1126 1121

2903 (+3.34%) 1977 ( + 1.16%)

Concrete

Table 2 lists the estimated material property of epoxy layer in two different specimens with different types of epoxy through ultrasonic NDE test, spectral analysis and an inversion process between analytical curves and experimental data. The measured longitudinal wave velocities are also obtained through pitch-catch ultrasonic test on cylinder samples of epoxy. A good agreement has been made between estimated and measured data, and the estimated values are slightly larger than the measured values by 3.34% and 1.16% in the two samples respectively. Here, the thickness of the epoxy layer is assumed to be known. In field test, however, the layer's thickness is generally unknown, at least partially, and so are the material properties of concrete. These factors need to be considered together in the further research.

6. Explicit FEM simulation and numerical dispersion curves

With the estimated material property, an FEM model of the steel-epoxy-concrete 3-layered specimen can be built. The wave propagation in the 3-layered medium can be simulated and compared with the experimental waveform. The numerical results can help us to verify the feasibility of the NDE test and the accuracy of the experimental data. Through the dispersion curves' comparison from the three different approaches, we can more clearly identify the most distinguishable and sensitive frequency range of the dispersion curves, which can be helpful in the choosing of the proper range of inversion process. The Explicit Finite Element Method (EFEM) is applied to reduce the computational cost per increment when the model is large on space and time approximation. In this EFEM model, we neglect the inhomogeneity of concrete from aggregates. The wavelength of the ultrasonic wave we used in the test is generally larger than the aggregates' dimension; hence the insignificant scattering between aggregates can be neglected without accuracy less.

A 3-dimensional FEM model of steel-epoxy-concrete specimen has been built (Fig. 12). The dimension of steel plate (500 mm x 500 mm x 4.5 mm) is larger than epoxy layer (400 mm x 400 mm x 5 mm) and concrete block (400 mm x 400 mm x 150 mm), which is the same as real specimen. The 8-node solid cube element is used in this simulation, with the horizontal size of 1 mm, the vertical size of 0.9 mm in steel plate, 1 mm in epoxy layer and 2 mm in concrete block. Totally 1.405e7 elements are included. For the two interfaces among steel plate, epoxy layer and concrete block, a fully tied condition is applied, on which the nodes in pair on the same interface is restricted to have the same performance. The explicit method is applied in time analysis, which requires a small time increment but a relatively small computational cost in per increment. Hence, for time discretization, time step of 1e-8 s is adopted in whole analysis time of 3.05e-4 s. To create a virtual experiment as close as the real experiment, concentrated force loads are applied on several nodes as an area force as the simulation of the transmitting transducer and its contacted area, as marked by "T" in Fig. 12. One period of sine wave (200 kHz, same as the real pulser) is adopted as the input pulse. The calculated vertical displacement of two nodes distanced the transmitter with 60 mm and 120 mm, as marked by "R1" and "R2" in Fig. 12, will be approximately used as signals received by R1 and R2. The material constants setting in this model is the same as Specimen 2 in Table 2, which is the estimated material property of the layered specimen.

Fig. 12. FEM model of steel-epoxy-concrete specimen.

Fig. 13. Simulation of wave propagation (u3) on the section along x-axis: a, at 20 |is; b, at 60 |is.

Through 3-D EFEM simulation, the wave propagation inside the 3-layer medium can be visualized. Fig. 13 shows the waveform on the center section when the 3-D model has been cut by half vertically along the x-axis. In Fig. 13a, the surface wave front has just passed through the spot of R1, but not yet reached R2. Also, the bulk wave front is clear shown at the 1/2 depth of the concrete, which has already passed through two interfaces but not yet reached the bottom or the edge of the specimen, therefore no reflection wave from there. In Fig. 13b, 40 ^s after the time in a, the surface wave has traveled through both R1 and R2, meanwhile, the bulk wave has reached the bottom and edge and been reflected by those boundaries. However, from the color map, we can see that the amplitudes of the reflected bulk waves are far smaller than the amplitudes of surface waves and guided waves propagating in steel and epoxy layer. It is revealed that the amplitude's attenuation in the concrete is much more distinct than that in steel and epoxy.

Fig. 14 shows the waveforms on R1 and R2. Compared to Fig. 6, they are quite familiar both on shape and relative amplitude. In both numerical and experimental signals, the maximum amplitude of waveform of R2 is almost half of that of R1, which means the attenuations of the surface wave in both cases are about the same.

Through the same procedures as Section 5, the numerical dispersion curves are obtained, as the points marked by circles in Fig. 15. From 20 kHz to nearly 160 kHz, they are quite agree with the experimental phase velocities and the analytical dispersion curves (the 1st Mode). In low frequency part (0 < f < 20 kHz), the matching is not so good, similar as the experimental data performs. In relative high frequency part (160 kHz < f), the difference between numerical and experimental data shows: the numerical curve is more close to the 2nd Mode of analytical curve; the experimental curve is still rising, where the Rayleigh wave of 1st layer in high frequency starts to contribute the dispersion. From the comparison of the dispersion curves of these 3 types, a clear knowledge has been achieved that the most reliable frequency range for inversion process is (20 kHz < f < 160 kHz) in this case, almost the same as the "hook" part of the 1st Mode, the most

Fig. 14. Vertical displacement of: a, node representing Ri; b, node representing R2.

Fig. 15. Comparison of analytical, experimental and numerical dispersion curves.

distinguishable part of the analytical curves. The FEM simulation validates the experimental data in accuracy and also gives a support to the approach of material property estimation based on SASW method.

7. Conclusions

The dispersion property of a steel-epoxy-concrete layered medium has been studied from analytical, experimental, and numerical aspects. By solving the dispersion equation, the plotting of analytical dispersion curves can be achieved, in which the frequency range of 20 kHz~160 kHz, around the "hook" shape part of the 1st mode, is the most distinguishable and sensitive part of dispersion curves for variation of elastic constants. In the case of steel-epoxy-concrete layered medium, those curves are cropped by the transverse wave speed of the third layer CT3 and tend toward to the transverse wave speed of the second layer CT2 when frequency goes to infinity. Meanwhile, the ultrasonic NDE test and spectral analysis have been applied in constructing of experimental dispersion curves. During this test, cases with different spacings between R1 and R2 have been conducted, and a relatively large spacing (60 mm) between receivers is recommended, for a low frequency range of 20 kHz~160 kHz is the most interested. Through an inversion process based on a variance function for multi-modes, the matched analytical dispersion curves have been found for two different specimens with a good accuracy. However, the inversion process is highly related with the quality of experimental dispersion curves we obtained. A manual check is needed to exclude those obviously deviated phase velocity points with low coherence values. Based on the estimated material property, the numerical study by a 3-D Explicit FEM model is completed. From the comparison of analytical, experimental, and numerical dispersion curves, a comprehensive understanding of dispersive property of this steel-epoxy-concrete layered medium has been obtained. The results reveal that the measured experimental dispersion curves contain multi-dispersion modes, but the most distinguishable and sensitive mode in inversion process is the first mode. It should be believed that the estimation approach of the elastic property proposed in this study is feasible.

Acknowledgements

The authors sincerely acknowledge KOMAIHALTEC Inc. for the preparation of the specimens and the cooperation during the experiments.

Appendix A

The dispersion equation of the model of 2 layers overlying on a half space is: A = 0

where,

(2k2 -k2 1)eV1H 2kv1 eV1H (2k2 - kj 1)e-V1 H -2kv'1 e-V1H 0 0 0 0 0 0

2kVieVI H (2k2 - k2T1)eV1H -2kv1e-V1 H (2k2 - k2T1)e-V1H 0 0 0 0 0 0

-k -Vi1 -k V11 k V21 k -V21 0 0

-Vi -k V1 -k V2 k -V2 k 0 0

2kVi 2k2 - k2T1 -2kV1 2k2 - k2T1 -2kV2 f2 - f- (2k2 - k=2) 2kV2 f- - f2 (2k2 - k=2) 0 0

2k2 - k2T1 2kv{ 2k2 - k2T 1 -2kV11 - & (2k2 - k2 2) -2kv2 ff - f2 (2k2 - k2 2 ) 2kv2 f2 0 0

0 0 0 0 -ke-V2h -V2 e-V2h -keV2h V2 eV2h ke-V3h V3 e-V3h

0 0 0 0 -V2e-V2h -ke-V2h V2eV2h -ke'i h V3e-V3h ke-V3h

0 0 0 0 2kV2e-V2h (2k2 - k2r2)e-Vih -2kV2eV2h (2k2 - k^e'ih -2kv3 » e-V'h - f3 (2k2 - kj3)e-Vih

0 0 0 0 (2k2 - k2T2)e-V2h 2kv2 e-V2h (2k2 - k22)eV2h -2kv'2 eV2h - (2k2 - k2r3)e-V3h -2kv3 f3 e-V3h

where, H and h are the 1st and 2nd layers' thicknesses accordingly; v and V are the simplified form of the following expressions:

k2 - k2Li = vf, k2 - k2Ti = Vj2, i = 1, 2,3 where, kL and kT are the wave number of longitudinal wave and transverse wave accordingly.

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