Surface Rust Detection Using Ultrasonic Waves in a Cylindrical Geometry by Finite Element Simulation

Surface Rust Detection Using Ultrasonic Waves in a Cylindrical Geometry by Finite Element Simulation Qixiang Tang 1, Cong Du 2, Jie Hu1, Xingwei Wang2 and Tzuyang Yu 1,* 1 Department of Civil and Environmental Engineering, One University Avenue, Lowell, MA 01854, U.S.A. 2 Department of Electric and Computer Engineering, One University Avenue, Lowell, MA 01854, U.S.A. * Correspondence: tzuyang_yu@uml.edu; Tel.: +1-978-934-2288 Version July 10, 2018 submitted to Preprints


Introduction
Slender steel members such as steel rods and bars are widely used structural components in civil infrastructure (e.g., steel truss bridges, temporary support structures, traffic signs).Unlike other construction materials such as Portland cement concrete, steel is vulnerable to corrosion.Steel corrosion can take place when suitable environmental conditions (e.g., temperature, pH, oxygen, moisture, chloride ions) are met.As a result, premature failures of steel structures can occur if one or many critical members is corroded.Corrosion of steel members will reduce the effective cross sectional area of the member by replacing steel (ferrite) with rust (ferrite oxides).Consequently, structural stiffness and bearing capacity of corroded steel members will be reduced.Furthermore, corrosion of steel members can also increase the likelihood of instability (buckling) for slender steel members, due to the change of boundary condition at the support or within each member.
Detection of early stage corrosion on slender steel members is crucial for preventing their premature failures.Various nondestructive evaluation/testing (NDE/T) and structural health monitoring (SHM) techniques have been applied to steel structures [1].Example techniques include visual testing [2], modal analysis [3], Eddy current testing [4], thermal infrared testing [5], and ultrasonic testing [6,7].Among these techniques, fiber optic sensors (FOS) represent a popular approach for long-term monitoring of steel structures [8,9].While FOS have been applied to many steel structures in the past, most of the damage detection algorithms are based on passive response of FOS.In other words, either corrosion-induced cracking or loading-induced dynamic response must be generated from monitored steel members such that FOS can passively detect the presence of corrosion.Recently, an active photoacoustic FOS system has been reported for early stage steel corrosion detection of steel plates and rebars [10].Different from traditional passive FOS techniques, active FOS can generate acoustic/ultrasonic waves to probe monitored steel members for early-stage corrosion detection.
Meanwhile, installed FOS allow engineers to assess the conditions (e.g., temperature) of structures without the use of couplant and "adapters" [11][12][13].With a minimized size, active FOS can be installed onto irregular/curved surface of structures.
In this paper, our objective is to investigate the surface rust detection problem in a cylindrical geometry (slender steel rod) using ultrasonic waves in transmission mode and to develop a surface rust detection algorithm., as a basis for the practical application of an active photoacoustic FOS system.
Steel rods are chosen as an example of slender steel members.The finite element method (FEM) is applied in simulating the propagation of ultrasonic waves at 1MHz on steel rod models.Surface rust is simulated by a rectangular prism which is characterized by its location (s 3 ), length (d), width (w), and depth (h).Transmission mode of damage detection is adopted, in which one source (transmitter) and one sensor (receiver) are considered.In this research, radial displacements (u(t)) at the receiver were simulated and analyzed by short-time Fourier transform (STFT) for detecting, locating, and quantifying a surface rust located between the transmitter and the receiver.Time domain and frequency domain analyses are conducted for developing a damage detection algorithm.In what follows, the detail of finite element (FE) simulation is first provided.

Finite element simulation
In the past, FEM had been applied for simulating ultrasonic wave propagation for damage detection [14][15][16].Among various signal processing techniques, short-time Fourier transform (STFT) has been demonstrated as an applicable approach for analyzing the transient response of ultrasonic wave propagation in the time-frequency domain [17].In this research, cylindrical geometry was numerically modeled by six steel rod models (one intact and five corroded) in a commercially available finite element package (ABAQUS 2016) [18].705, 600 linear hexahedral elements (C3D8) were used in all six models.Five corroded steel rod models were created by introducing a rectangular prism/anomaly to the surface of intact steel rod model.Transmission mode of damage detection was applied for data collection by using one transmitter (source or T) and one receiver (R) in each model, as shown in Fig. 1.Time domain radial displacement (u(t)) at the receiver was collected for all six models.Design of intact and corroded FE models are described in the following sections.

Intact steel rod model
An intact steel rod model (denoted by IM) was created by using a cylinder with 12.7-mm diameter (D) and 50-mm length, as shown in Fig. 1. Materials properties of steel used in the intact steel rod model was provided in Table 1.A transmitter (T) was located at mid-span and a receiver (R) was located 10 mm away from T along the longitudinal axis (z-axis) of the model.The distance between T and R was denoted as s 1 .The intact steel rod model was fixed at both ends.To suppress unnecessary reflections from both ends, ten absorbing layers [19] were used at each end of the model such that ultrasonic waves propagating into the absorbing layers can be damped out.As shown in Fig. 2), a sinusoidal pulse was introduced at T, and time domain radial displacement (u(t)) was collected at R.

Hypotheses of ultrasonic waves propagation in intact and corroded rod models
Five hypotheses on ultrasonic waves propagation in intact and corroded steel rod models were made for the damage detection problem in this paper.A Mercator projection of cylindrical geometry for steel rod models is provided in Fig. 3 to better illustrate these hypotheses.
1.In model IM, time domain radial displacement u(t) is collected at R. The first ultrasonic wave packet is the one propagating along the s 1 path at a velocity of c 1 and arriving at time t 1 .The second ultrasonic wave packet propagates along the s 2 path and arriving at time t 2 with a velocity of c 2 .
2. In corroded steel rod models (CM1 CM5), the ultrasonic waves propagating along the s waves propagates through the surface rust and arrives at time t 1 (i.e., t 1 >t 1 since the ultrasonic wave velocity in rust is slower than the one in steel).
3. Part of the ultrasonic waves is scattered from the surface rust and propagates along the s 4 path.
Time t 2 is the total time of flight (TOF) of scattered ultrasonic wave propagating along path . Propagation velocities of ultrasonic waves on path s 1 and path ( s 3 , s 4 ) are respectively c 1 and c 2 .
4. In Fig. 3 (top view), path s 8 is the path of ultrasonic waves diffracted by the surface rust ( s 8 = . TOF of these ultrasonic waves is t 8 (i.e., t 8 = t 6 + t d + t 7 ).
5. Higher frequencies are affected more than lower frequencies by the presence of surface rust.This is because that the effective depth of each frequency is approximately its wavelength [20].
With 'shallow' effective depth, higher frequencies interact with the surface rust more than lower frequencies.

Damage detection algorithm
Based on the aforementioned five hypotheses, surface rust detection, localization and quantification are carried by the following approach.

Damage detection
In this paper, detection of surface rust can be accomplished by determining the reduction of centroid frequency (∆ f c ) in the spectrogram of u(t) by using STFT.The steps of obtaining f c are reported in the following.
1. Generate/introduce ultrasonic waves at transmitter T of model IM.  5. Determine the centroid of the half-power contour for the first wave packet by finding its coordinates ( f c , t c ) in the spectrogram U(t, f ).
6.The centroid frequency f c of this FE simulation is found.For intact model (IM), f c = f c,i .
7. Repeat the steps for an artificially corroded model.For corroded models, f c = f c,c .Fig. 4 illustrates the parameters defined in the steps for damage detection, using model CM1 as an example.Eq. ( 1) shows the damage detection criterion for detecting the presence of surface rust.
where ∆ f c = difference in the centroid frequency between intact and corroded steel rod models (in MHz), f c,i = centroid frequency of model IM (in MHz), and f c,c = centroid frequency of corroded steel rod models (in MHz).In this research, a steel rod model is considered intact (no damage) if there is no reduction of centroid frequency or ∆ f c = 0 and vice versa.

Damage localization
To locate surface rust, TOF (time-of-flight) of scattered ultrasonic waves is used.In this research, the location of surface rust is defined by the length of path s 3 or  Equivalently, where s 3 = length of path s 3 (mm), s 4 (s 3 ) = length of path s 4 (mm) = (s 1 − s 3 ) 2 + p 2 , p = perimeter of the rod model (mm), c 1 = propagation velocity on path s 1 (mm/µs), and c 4 = propagation velocity on path s 4 (mm/µs).From our previous study, a propagation velocity model (Eq.( 4)) based on the length of path on a cylindrical geometry was reportedc 4 [17].
where a and b = model parameters.By substituting Eq. ( 4) and re-arranging terms, we have where s 1 = length of path s 1 (mm).In Eq. ( 5), s 1 , p, and c 1 must be provided.Parameters a and b are from reported literature [17].Time t 2 is measured from a corroded model.Once time t 2 is measured, Eq. ( 5) can be solved by the graphic method.Eq. ( 5) also represents the damage localization criterion in our algorithm.Finding the value of s 3 locates the surface rust in this research.Peer-reviewed version available at Infrastructures 2018, 3, 29; doi:10.3390/infrastructures3030029

Damage quantification
For damage quantification, dimensions of surface rust (length d, width w, and thickness h) are to be found.In finding the length d of surface rust, TOF (t 1 ) of ultrasonic waves propagating through surface rust and arriving at receiver R is used.Time t 1 denotes the total propagation time along path s 1 which consists of path s 3 , surface rust (length d), and path s 5 .In other words, where t 1 = total TOF of ultrasonic wave propagating on path s 1 (µs), t 3 = TOF of ultrasonic wave traveling on path s 3 (µs), t 5 = TOF of ultrasonic wave traveling on path s 5 (µs)), and t r = TOF of ultrasonic wave traveling within surface rust (µs).Since s 1 = s 3 + d + s 5 , we have where c r = propagation velocity on z-axis in rust (mm/µs).By re-arranging Eq. ( 7), surface rust length d can be directly determined by Eq. ( 7) represents the length estimation criterion in our algorithm.
Once s 3 (from damage localization, Eq. ( 5)) and d (from Eq. ( 8)) are determined, the width of surface rust (w) can be obtained by using the delayed arrival time of first wave packet (t 8 ), as shown in Fig. 3 (top view).Eq. ( 9) describes the relationship between t 8 and w.
where s 5 = s 1 − d − s 3 , t 8 = TOF of ultrasonic wave propagating on path s 8 and in a corroded steel rod model (µs), s 6 = length of path s 6 (mm) and s 7 = length of path s 7 (mm).Eq. ( 9) represents the width estimation criterion in our algorithm.
From our fifth hypothesis (Section 3.1), lower frequency ultrasonic waves have 'deeper' effective depths.It suggests that more frequencies in the STFT spectrogram will be affected when increasing the thickness h of surface rust.This phenomenon is illustrated by the reduction of spectrograms' curvature or ∂ 2 U 1 ∂ f 2 and modeled by an empirical equation as shown in Eq. (11).
where h = thickness of surface rust (mm), Eqs. ( 8), ( 9) and ( 11) represent our damage quantification approach in this paper.Surface rust length d, width w, and thickness h can be estimated from the STFT spectrogram of radial displacement u(t) measured at receiver R. In the following section, FE simulation results are reported.

Time domain response
In each model, radial displacement u(t) are receiver R was collected as shown in Fig. 5.As predicted by the first hypothesis, two wave packets were observed.The first wave packet was the ultrasonic wave propagating along the longitudinal direction (z-axis).The second wave packet was the ultrasonic wave propagating along the helical direction (i.e., s 2 in Fig. 3).The waveform of the second wave packet is more complicated than the one of the first wave packet in the spectrogram, owing to the geometric dispersion (in the second wave packet) caused by the cylindrical geometry of FE models.In corroded steel rod models (CM1 -CM5), the first peak amplitude (u 1 ) was reduced after interacting with surface rust and propagating on path u 1 .While the presence of surface rust can be detected by the reduction of u 1 , quantification of surface rust using u 1 can be very difficult due to the geometric dispersion effect on u(t).In reality, peak amplitudes can also be contaminated by background noise (e.g., ambient vibration).Therefore, frequency domain analysis of u(t) is applied and described in the next section.

Time-frequency domain response
By applying STFT to u(t), frequency change in u(T) over time was shown on its spectrograms.
Frequency range on the STFT spectrogram between 0.1 MHz and 2 MHz was determined, since this frequency range included most of the kinetic energy of transmitted ultrasonic waves.wave propagating along the helical direction or path s 2 .Fig. 8 shows the contours at the half-power level of the first wave packet in each spectrogram.In corroded steel rod models (CM1 CM5), higher frequencies in the first wave packet were reduced due to smaller effective depth.In addition, shape of the second wave packet changed due to the size change of surface rust, as shown in Fig. 8.

Surface rust detection
Detection of surface rust in a corroded steel rod model was accomplished by comparing its centroid frequency f c with the one of an intact model (IM).3. Based on Eq. ( 1), the presence of surface rust in these models were detected.

Surface rust localization
Eq. ( 3) was used to locate surface rust in our simulations.In view of the presence of geometric dispersion in u(t), measuring TOF in the time domain became challenging.To avoid the problem of chasing multiple frequencies at a time, the center frequency of transmitted ultrasonic wave in the STFT spectrogram (i.e., 1 MHz in this paper) was chosen.were measured from the time t 0 when the ultrasonic wave was introduced at transmitter T; in this paper, t 0 = 6.13µs.Since the peak amplitude of the first wave packet was 9.01 µs, t 1 = (9.01− 6.13)µs = 2.88µs.With traveling distance s 1 being 10 mm, the wave velocity c 1 can be calculated by ⇒ c 1 = 10 2.88 = 3.47 mm/µs, ( With E = 210,000 MPa, ρ = 7,850 kg/m 3 , and ν = 0.3, approximated theoretical c t value was found to be 3.03mm/µs.Consequently, theoretical TOF t t for the first wave packet was found to be where s 1 = distance from center of transmitter T to receiver R (mm) (= 10 mm) and z l = distance from center of transmitter T to the edge of loading area at T (mm) (= 0.5 mm).An error of 8.2% was obtained between the approximated theoretical c t value and numerical c 1 value.
A subtracted/differential 1-MHz curve (subtract model IM from model CM1) was generated and shown in Fig. 10 (b) from where differential TOF values of the first wave packet t 1 and of the second packet t 2 were determined to be 15.16 − t 0 = 9.03 µs and 22.28 − t 0 = 22.28 − 6.13 = 16.15 µs, respectively.In our algorithm, differential TOF of the second wave packet t 2 was used for surface rust localization.
From the differential 1-MHz curve in Fig. 10 (b), a propagation velocity model from literature [17] for elastic waves on a cylindrical geometry was used.In all six FE models, p = 12.7π = 39.9 mm.From the Mercator projection shown in Fig. 3, it is clear that With s 1 = 10 mm, p = 39.9 mm, c 1 = 3.47 mm/µs, a = 3.47, b = −0.8348,Eq. ( 5) could be written as Eq. ( 21) provides the condition between s 3 and t 2 .With differential TOF t 2 , surface rust location s 3 can be found from Eq. ( 21).Since Eq. ( 21) cannot be solved analytically, the graphic method was applied with its result shown in Fig. 11.Eq. ( 21) represents a model for locating the surface rust in our algorithm.Following the same procedure, 1-MHz curves of models CM2 and CM3 were generated Estimated surface rust locations (s 3 ) in corroded steel rod models was reported in Table 4.For surface rust quantification, Eq. ( 8) was used to determine surface rust length d for models CM1 and CM2 by using measured time t 1 .Eq. ( 9) was applied to determine surface rust width w for models CM1 and CM4 by using measured time t 8 .Eq. ( 11) was utilized to determine surface rust depth h for models CM1 and CM5 by using measured curvature For determining surface rust length d by using Eq. ( 8), s 1 = 10 mm and propagation velocity in steel c 1 = 3.47 mm/µs (from Eq.( 13)).Propagation velocity in rust c r was calculated by 0.08454c t = 0.257 mm/µs from [20].Therefore, Eq. ( 8) became  9) with measured t 8 (TOF of the first wave wave packet as shown in Fig. 12 ).

Conclusion
This paper reports a finite element study of utilizing point-source generated ultrasonic waves for detecting surface rust in steel rod models.Methods of detecting, locating and quantifying the surface rust are achieved by using the STFT (short time Fourier transform) spectrogram of radial displacement collected on the surface of corroded steel rod models.We have concluded the following.
• Presence of surface rust can be detected by the reduction of centroid frequency of the first wave packet in the STFT spectrogram of corroded steel rod models.
• Location of surface rust is estimated by finding the difference in arrival time (TOF) between helically propagating ultrasonic waves and scattered ultrasonic waves (due to surface rust).
• Length of surface rust can be predicted by calculating the difference in TOF between longitudinally propagating ultrasonic waves of intact and corroded steel rod models.This difference in TOF is related to the longitudinal dimension (length) of surface rust.
• Width of surface rust can be determined by calculating the difference in TOF of the first wave packet between intact and corroded steel rods in the STFT spectrogram at a fixed frequency (e.g., 1 MHz in this paper).
• Thickness of surface rust can be estimated by utilizing the second-order derivative of the first wave packet of corroded steel rod models.In conclusion, this paper presents our finite element analysis of ultrasonic waves on intact and corroded steel rod models for detecting, locating, and quantifying surface rust in a systematically approach.
While research result is obtained in several empirical equations, it is believed that our proposed damage detection algorithm can be applied to other corrosion detection problems using distributed photoacoustic fiber optic sensors on steel rods or steel rebars.

PreprintsFigure 3 .
Figure 3. Mercator projection of intact and corroded surface for ultrasonic wave propagation paths

4 .
In the spectrogram U(t, f ), show the half-power contour at -3 dB from the maximum amplitude of the first wave packet.

Figure 4 .
Figure 4. Procedure of obtaining f c value

Preprints
(www.preprints.org)| NOT PEER-REVIEWED | Posted: 10 July 2018 doi:10.20944/preprints201807.0170.v1 order partial derivative of the first wave packet's frequency domain projection, and e and g = model parameters.∂ 2 U 1 ∂ f 2 approximates the curvature of the first wave packet.Eq. (11) represents the thickness estimation criterion in our algorithm.

Figure 5 .
Figure 5.Time domain radial displacement in intact and corroded steel rod models

Fig. 6
Fig.6shows the STFT spectrogram of u(t) at transmitter T of model IM.In Fig.6, the first wave packet (white-colored vertical shape) represented the transmitted ultrasonic wave traveling in the longitudinal direction or path s 1 (without geometric dispersion), whose amplitudes confirm our choice

Figure 6 .
Figure 6.Radial displacement and spectrogram of the intact steel rod model at transmitter T

Fig. 9
compares the half-power contours of intact (model IM) and corroded (models CM1 CM5) FE models in individual STFT spectrogram at receiver R. Center of the half-power contour of model IM was denoted by centroid frequency f c,i .For five other corroded models, their centroid frequency was denoted by f c,c with different values.After finding f c,i and f c,c , their difference ∆ f c was calculated and reported in Table

Fig. 10
Fig. 10 shows the STFT spectrogram (at 1MHz) of model IM at receiver R to demonstrate how to calculate the TOF of the scattered wave (t 2 ) traveling through paths s 3 and s 4 .In Fig. 10 (a), the 1-MHz curves on the STFT spectrogram of model IM and model CM1 were extracted.Time t 1 denoted the TOF of the first wave packet and time t 2 the second wave packet for model IM.In Fig. 10, t 1 and t 2

Figure 8 .
Figure 8. Half-power contours in STFT spectrogram for six models at receiver R

c 4 (Figure 9 .Figure 10 .
Figure 9. Half-power contours of the first wave packet at receiver R

Figure 12 .
Figure 12.Time delay of the first wave packet

Figure 13 .Figure 14 .
Figure 13.Ridge of the first wave packets and their second-order derivatives

Table 2 .
Four attributes were used to characterize the corroded

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 10 July 2018 doi:10.20944/preprints201807.0170.v1
Time domain radial displacement (u(t)) of each model at receiver R was collected from six FE simulation cases (one for intact model and five for corroded models).Spectrogram (U( f , t)) of each u(t) was obtained by STFT.Comparison of u(t) and U( f , t) between intact and corroded steel rod models was made to study the effects of surface rust on u(t) and U( f , t).
was applied to determine surface rust length d for models CM1 and CM2.With s 3 (from surface rust localization) and d found, surface rust width w values for models CM1 and CM4 were determined by solving Eq. (