Fractal analysis of nonlinear ultrasonic waves in phase-space domain as a quantitative method for damage assessment of concrete structures

Journal Pre-proof Fractal analysis of nonlinear ultrasonic waves in phase-space domain as a quantitative method for damage assessment of concrete structures Sina Zamen, Ehsan Dehghan Niri PII:

S0963-8695(19)30633-4

DOI:

https://doi.org/10.1016/j.ndteint.2020.102235

Reference:

JNDT 102235

To appear in:

NDT and E International

Received Date: 24 October 2019 Revised Date:

31 January 2020

Accepted Date: 13 February 2020

Please cite this article as: Zamen S, Niri ED, Fractal analysis of nonlinear ultrasonic waves in phasespace domain as a quantitative method for damage assessment of concrete structures, NDT and E International (2020), doi: https://doi.org/10.1016/j.ndteint.2020.102235. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd.

Fractal Analysis of Nonlinear Ultrasonic Waves in Phase-Space Domain as a Quantitative Method for Damage Assessment of Concrete Structures Sina Zamen1 and Ehsan Dehghan Niri1,* 1

Department of Civil Engineering, New Mexico State University, Las Cruces 88003, USA

Highlights •

Investigating the nonlinear behavior of ultrasonic waves in phase-space domain to complement or replace traditional approaches in the time and frequency domains.

•

Leveraging fractal dimension as a mathematical tool to extract a quantitative geometric damage-sensitive feature from nonlinear ultrasonic waves in phase-space domain.

•

Comparing the reliability of this feature with the most commonly used feature extracted from the frequency domain to evaluate load-induced cracks in concrete materials while under service loading.

Abstract The present paper applies fractal dimension as a mathematical tool to extract a quantitative geometric feature from nonlinear ultrasonic waves in phase-space domain. The feature is then used for damage assessment of concrete material under different service loads after experiencing extreme compressive loads. For this purpose, concrete samples are loaded in two different scenarios: the first loading part is set to initiate micro cracks and generate macro cracks, while the second loading procedure is set to simulate service loads and change the cracks’ boundary conditions. Due to nonlinear ultrasonic waves’ sensitivity to cracks in early stages, nonlinear *

Corresponding author. E-mail address: [email protected] Postal Address: Hernandez Hall, Room 210, 3035 South Espina Street, Las Cruces, NM 88003, USA

1

ultrasound test is performed. In contrast to traditional approaches which analyze nonlinear ultrasound waves in frequency domain, in this paper, nonlinear ultrasonic waves in phase-space domain are studied. Phase-space domain is a powerful mathematical tool that allows researchers to analyze data quantitatively and qualitatively using different signal processing techniques like fractal dimension. For the first time, fractal analysis of nonlinear ultrasound waves in phasespace domain is performed to measure the nonlinearity of the waves due to interactions with loaded-induced cracks in concrete materials. In general, fractal analysis makes it possible to assign dimension for sets that do not have integer dimension. To calculate fractal dimension, the box-counting method, as a pragmatic method, is used. A two-dimensional (2D) and threedimensional (3D) box-counting method for calculating the fractal dimension of nonlinear ultrasonic waves in phase-space domain are utilized. It is shown that fractal dimension is a powerful signal processing tool for analyzing nonlinear ultrasonic signals in phase-space domain and extracting quantitative damage-sensitive features in presence of service load. In contrast, results of frequency domain analysis show a lack of noticeable trend, especially for large service loads. This observation indicates that features extracted in frequency domain may not be reliably used as damage-sensitive features for evaluating the level of cracking in concrete material under service load. Keywords: 3D box-counting, fractal dimension, ultrasonic testing, nonlinear ultrasound, phasespace, damage assessment 1. Introduction Concrete is the most widely used material in construction. Based on the importance of concrete structures, routine inspection/monitoring is required to guarantee the reliable performance of the structures. Cracking, which initiates at the micro-scale level and eventually 2

transforms to macro cracks, is the main reason for performance degradation in concrete structures. Micro crack generation in concrete, which is a quasi-brittle material, is inevitable [1]. These cracks range from 10 to 100 µm and are hard to detect using typical visual inspection methods. Ultrasonic testing, however, has been shown to be a powerful tool to detect macro and micro cracks due to its ability to achieve higher resolutions at higher frequencies in contrast to vibration based methods [2]. Because of the unique sensitivity of ultrasound to small defects such as cracks, it has been used not only for routine inspection, but also for real-time monitoring of concrete structures [3–8]. Traditional ultrasound testing, based on linear theory that relies on time domain analysis of ultrasound waveforms, has acceptable sensitivity when interacting with macro cracks. However, for detecting micro cracks or closed macro cracks, the linear ultrasound method is not suitable since ultrasonic waveforms do not noticeably change while interacting with these cracks [9,10]. Hence, since around 1960s, researchers have shifted toward implementation of the nonlinear ultrasound technique, which relies on nonlinear material behavior that does not have the limitations of linear ultrasonic testing [9–12]. Most researchers study nonlinear ultrasound waves’ behavior in frequency domain. The main principle of using nonlinear ultrasound method is to use frequency change between injected signal and received signal to extract quantitative damage-sensitive features. In general, when transforming data to frequency domain, valuable information can be lost. Since the Fourier transform of a nonlinear system turns complex differential equations into integral equations. These integral equations consist of convolutions between Fourier transforms of dependent variables [13] which may result in information loss. Recently it was shown that one of the main problems associated with studying nonlinear

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ultrasonic waves in frequency domain is that different defects may cause the same spectral change in the frequency domain [14]. Phase-space is a strong mathematical tool that can replace or complement frequency domain analysis. Generating phase-space representation is possible both from available experimental data (e.g. response of a system) and having states of a system simulated from a mathematical model that represents the system [15–17]. By constructing a phase-space representation of a system’s response, valuable information can be extracted using different signal processing techniques [15,16] that cannot be leveraged in frequency and time domains. Recently, researchers have started to analyze nonlinear ultrasonic waves’ behavior in the phase-space domain and results are promising [14,18–20]. It has been shown that phase-space is a powerful domain to analyze complex nonlinear ultrasonic waves’ behavior, such as chaotic motion [20]. Furthermore, having phase-space representation will provide researchers and practitioners with a domain that makes it possible to analyze data qualitatively as well as quantitatively. The main contribution of this article is to apply fractal dimension as a mathematical tool to extract a quantitative geometric feature from nonlinear ultrasonic waves in phase-space domain. In other words, for the first time, fractal dimension of nonlinear ultrasonic waves in phase-space domain is leveraged to measure the nonlinearity of ultrasonic waves while interacting with cracks in different loading conditions. This feature is then used for damage assessment of concrete material under different service loads while it has experienced extreme compressive loads. External compressive loads initiate cracking due to external loading, while the service load simulates a real case scenario for testing concrete structures. To achieve this goal, first, phase-space representation of nonlinear ultrasonic waves will be constructed numerically from

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ultrasound waveforms. In the second step, as a damage-sensitive feature, fractal dimension of received ultrasound waves in phase-space domain will be calculated. Fractal dimension provides a simple yet effective way of assigning dimension to a set which does not have an integer dimension [16]. Mandelbrot named these sets Fractals [21] but in dynamics, these sets are referred to as attractors. Fractal analysis has the power of describing irregular shapes or complex object geometries, while other methods like Euclidean geometry fail to do so [22]. Fractal dimension of one dimensional objects falls between 0 and 1 while for a two dimensional objects takes value between 1 and 2. Finally, for three dimensional objects, fractal dimension would fall between 2 and 3. Furthermore, it has been reported that noise barely affect the fractal structure of a strange attractor when the behavior of a dynamical system is deterministic [23]. Fractal analysis has been successfully implemented in different research fields like medical science, computer science, image processing, geodesy, etc. [22,24,25]. Also, in the structural health monitoring field, researchers have used fractal analysis to characterize surface cracks’ behavior and their evolution pattern in concrete structures [26–30]. With the rise of powerful computational systems and growing interest in analyzing real time three-dimensional objects, three-dimensional fractal analysis is also utilized in different research fields like geodesy, medicine, image processing etc. when dealing with three-dimensional objects [31–34]. Typically, researchers use 2D fractal dimension to analyze 3D objects, where they simply add 1 to 2D fractal values [31]. This simplified method lacks strong mathematical basis and is not a pragmatic approach. As a result, optimized algorithms for calculating 3D fractal dimension have been proposed and results are promising [35,36]. In this research, both 2D and 3D fractal dimension as a quantitative tool for analyzing nonlinear ultrasonic wave behavior in

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phase-space domain is implemented. The outcome of this research can provide a valuable tool to assess and quantify structural conditions under different loads. This paper is organized as follow: in section 2, mathematical formulation of constructing phase-space from recorded data is described briefly. Section 3 gives a brief overview about fractal dimension fundamentals and introduces different methods and algorithms for calculating fractal dimension. Experimental setup is described thoroughly in section 4. Results and discussions are provided in section 5. Finally, conclusions are made in section 6. 2. Phase-space Phase-space is a domain that represents all the possible states of a dynamic system while time is implicit. It provides a geometric representation of a systems’ states. Considering a case that only one state of a system is measurable, it is possible to numerically construct a phase-space representation of the system that can represent all the states. When dealing with ultrasonic testing, the received signal s(t) is one state of a system S (e.g. wave displacement) which has a scalar form with respect to time: (1) s(t ) = S[ x(t )], where x(t) is the input. By defining the delay coordinate vector one can derive a phase-space representation of the signal [16,37] as

where

= [ −

(2) W = (w1 , w2 ,..., wE ), − 1 ], E is the embedding dimension, and τ is the discrete time delay

or time lag. Choosing proper time lag and dimension is crucial in generating phase-space domain representation [15,16,37]. For numerically constructing the phase-space, an optimum time lag should be determined. Choosing a small time delay value would result in having highly correlated states, which is unfavorable. On the other hand, choosing a large time lag can result is 6

generating almost independent states that may not have a meaningful mathematical or physical basis [37]. Recently the authors successfully have used the mutual information theory proposed by Fraser and Swinney [38] to calculate optimal time lag [14,20]. In this regard, mutual information theory for a signal and its delayed form can be expressed by following equation:

P( s(t ), s(t + τ )) (3 ), ) P1 ( s(t )) P2 ( s(t + τ )) where P1 and P2 are marginal probability distributions and P represents two-dimensional I (t , t + τ ) = log 2 (

probability density function. The first minimum value from the average mutual information equation can be selected as an optimum time delay. The following equation expresses average mutual information: I av (τ ) = ∫

t

∫

t +τ

P ( s (t ), s (t + τ )) I (t , t +τ ) dtd (t + τ ).

(4)

The embedding dimension (E) has to be numerically determined from a single response that is an ultrasound waveform. To determine minimum embedding dimension (E), false nearest neighbor algorithm proposed by Kennel et. al. has been used [39]. This algorithm tracks the change in the number of neighbors of a point along a phase-space trajectory by increasing the embedding dimension from E to E+1. False or true neighbors of a point can be identified using an error function established based on Euclidian distance between point s(ti) and its rth nearest neighbor s(ti)(r) as [39]: ,

for

=

+

−

+

,

(5)

= 2,3, … , # . By adding a dimension, a new coordinate will be added to the phase-space +

vector W that is %

. In this case the new Euclidian distance can be determined as: ,

=

,

+

+ 7

−

+

.

(6)

An increase in a distance between W(i) and W(r)(i) is large if the initial assumption for the nearest neighbor r is “false”. This error or change in distance can be calculated as: − , (7) > '. , is a threshold that usually set between 0.10-0.15. In this research to identify the false %

The

'

,

neighbors, this threshold is set to 0.15. This criterion as it was discussed by Kennel [39] is not enough because in some cases it can result in inaccurate estimation of the embedding dimension E (e.g. this criterion can estimate a low value for high dimensional case such as random noise). To fix this problem, an additional criterion is proposed as: % (

> )'

where )' is the second threshold that is 0.02 in this study and (

where

=

1 *

+

[

− ] ,

(8) (

is:

(9)

= + ∑+ . (10) The false neighbor can be identified if both criteria are met. The goal is to find the minimum

E (embedding dimension) that gives the minimum false neighbors percentage (e.g. close to zero) among all the initial points that are selected to check these criteria. The algorithm starts with randomly selecting *- points in the phase-space domain. For these points using the criteria that are given in Eq. (5) and Eq. (6) one can identify all of the false neighbors. For each dimension E, starting from 1, the percentage of false neighbors with respect to the total points *- is calculated. The smallest dimension that provides false neighbors of very low percentage (close to zero) is the optimum embedding dimension (E). Although the selection of higher order dimension may result in less error, it can cause further computations. In this study the acceptable level of error to select dimension E is 0.5%. Although the embedding dimension is determined numerically in 8

this research, it has a physical meaning. For example, if the minimum dimension is two in a received ultrasound wave, strain and strain rate can be considered as two states of the system. The ordinary second order partial differential equations, either nonlinear or linear, might be a good choice for modeling ultrasound wave propagation. If the dimension of the system is more than two, then the regular, both linear and nonlinear, second order partial differential equations might not be an adequate mathematical model for ultrasound propagation. Thus, the embedding dimension can help us with identifying or discovering a mathematical model that can include all of the independent state variables. It is worth mentioning that different algorithms for calculating time lag and minimum embedding dimension exist. Regardless of the implemented method, if these values are calculated properly, the method selection would barely affect the phase-space reconstruction and properties [18]. The delay construction method has been successfully implemented on ultrasound data by different researchers for constructing ultrasonic waves’ behavior in phase-space domain [14,18–20]. After determining optimum time lag and minimum embedding dimension, one can construct the phase-space representation of a signal. Fig. 1 demonstrates a schematic view of the phase-plane (2-dimensional phase-space can be called phase-plane) representation of a sinusoidal function and its delayed form. For generating phase-plane, a sinusoidal signal regarding its delayed form is plotted. While it has been shown that phase-space can provide valuable qualitative information about the behavior of dynamical systems, this qualitative information cannot be used for quantitative evaluation of damage. There are numerous methods to analyze these behaviors quantitatively in phase-space domain. In this research, fractal dimension is used to extract a quantitative feature from ultrasound waveforms in phase-space domain. 9

τ

Fig. 1. Schematic representation of phase-space construction, for sinusoidal signal (E=2, and τ =0.25T (T is the period of the sinusoidal signal). 3. Fractals, dimensions and computing method For calculating fractal dimension, numerous algorithms have been proposed. Each of these methods have their own theoretical basis. Hence, fractal dimension of a single object would depend on the implemented method. However, the foundation of these methods is similar to each other and can be summarized in the following three main steps [22]: 1) Using several step sizes, quantities of an object should be evaluated. 2) Then, data points (measured quantities versus step size) should be plotted in logarithmic scale format. A line should be fitted through the plotted points. 3) The slope of the regression line is the fractal dimension. Box-counting methods [40], fractional Brownian motion methods [41], and area measurement methods [22] are three main methods for calculating fractal dimension of an object. In this research article, the box-counting method for estimating fractal dimension of ultrasonic waves in

10

phase-space domain is used. For a comprehensive literature review of other methods, readers are encouraged to refer to a review paper by Lopes and Betrouni [22]. The box-counting method first proposed by Russel et al. [40] is the most frequently implemented method by researchers, although it has some drawbacks [42]. In this method, by covering a geometric object with squares (boxes for 3D cases) of length ε, the fractal dimension (FD) is calculated as:

log( N (ε )) , log(ε ) where N(ε) is the number of boxes required to cover the binary signal completely. FD = − lim

(11)

Limacon function is used to show the effectiveness and sensitivity of fractal dimension in quantifying small geometric changes. The polar equation for Limacon is: r = a + b cos θ , where a and b are constants. Fig. 2 represents Limacon in the Cartesian plane.

(12)

Fig. 2(a) shows large mesh sizes and Fig. 2(b) demonstrate small mesh sizes. There exists a large box which can contain the entire object. As the mesh size decreases, the number of boxes required for covering the entire object increases. The smallest mesh size depends on the computational power and the incremental size (sampling size) of the discretized data. As shown in Fig. 2(c), fractal dimension calculation based on the box-counting method is sensitive to slightest changes in object form and shape (i.e. morphology). This is important because these features can help with detecting small geometric changes in ultrasonic wave behavior in phasespace domain.

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r = cosθ ,

r = 0.5 + cos θ ,

r = 1 + cosθ ,

(a)

(b)

(c)

Fig. 2. Schematic representation of implemented box-counting method for Limacon: (a) large box size, (b) small box size, (c) fractal dimension calculation. 4. Experiment To evaluate the interaction of micro cracks and macro cracks with ultrasonic waves inside a concrete medium in the presence of service load, standard cubic samples of 101.6 mm were tested. Tested cubic samples reached a compressive strength of 20 MPa in 7 days. Table 1 summarizes the admixture proportion as well as aggregates’ sizes.

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Table 1 Mixture proportion, aggregates’ size and compressive strength of cubic sample. Unit Weight (Kg/m3) Water

Cement

Fine Aggregate

Coarse Aggregate

180

300

900

830

Strength (MPa) 20

Coarse Aggregate Size (mm) 19.05

Fineness Modules 2.9

Two different loading scenarios are implemented. First, to generate micro cracks and later transform them into macro cracks, a compressive machine was used to compress cubic samples incrementally until the samples failed. The load increments as percentage of ultimate load is applied. Due to the brittle nature of concrete, it is safe to say that at lower loads, micro cracks start to initiate inside concrete samples, and by compressing them with large forces, micro cracks transform into macro cracks. At each increment of load history, the concrete sample was moved to another compressive tool (Fig. 3) to perform ultrasound testing while it is under service load. The compressive tool is designed in a way that makes it possible to perform ultrasonic testing while the concrete sample is steadily compressed. Using the compressive tool, the concrete sample was then compressed with loads which equaled a small portion of the ultimate loading capacity of the concrete sample. The secondary load simulates service loads that can potentially exist during the ultrasound testing on a real structure. Service load is an important factor, since it can change the boundary condition of cracks. For instance, it can close the gaps between open cracks, which makes crack detection a more challenging task while performing ultrasound testing. It is worth mentioning

13

that although a compressive strength of 20 MPa is reported, due to extensive loading and unloading cycles, the tested samples failed at lower compressive strength. The sample used in the next section failed at 14 MPa (Fu = 145 KN). The compressive tool shown in Fig. 3 can provide access to transducers at both ends of the samples during service load simulation. After loading the sample with a specific amount of compressive load, ultrasonic testing parallel to applied load direction was performed, and this process continued throughout the experiment with different compressive loads. For ultrasound testing, an arbitrary wave generator and high voltage Panametrics transducer (X1020) were used to create a continuous harmonic signal with 100 KHz center frequency. On the receiver side, a Panametrics transducer (X1019) with broader frequency range (180 KHz center frequency) is used to help with detecting higher harmonics. For inducing nonlinear ultrasonic behavior, higher amplitude is essential. Hence, by using linear amplifier, generated waves are amplified to 200 volts peak-to-peak. Fig. 4 represents a schematic sketch of the implemented experimental setup.

Fig. 3. Exploited compressive tool for compressing the sample and ultrasonic testing.

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Load Cell

Treaded Rod

Data Acquisition

Wave Generator

Oscilloscope

Linear Amplifier

Concrete Sample Receiver Transducer

Transmitter Transducer

Fig. 4. Schematic presentation of experimental setup. From an intact sample, 80 cycles from a steady part of the received signal’s tale were selected to be used in constructing the phase-space domain as shown in Fig. 5. The same number of cycles was used to construct the phase-space from all signals received from the sample with different loading conditions. To remove random noises, a low-pass filter with a cutting frequency of 1 MHz was considered while sampling frequency is set to 10 MHz.

Amplitude [Volt]

(a)

(b)

(c)

Time [msec]

15

Fig. 5. Part of transmitted signal (a) and received signal (b) and selected signal for phasespace construction (c) 5. Results and discussions After selecting the proper signal range (e.g. 80 cycles), the phase-space response can be numerically constructed from received ultrasound waves using the delay method. As previously mentioned, when generating phase-space, two parameters are important: 1. Time delay/lag, and 2. Embedding dimension. The dimension of the phase-space for ultrasonic waves would vary between 2 and 3, depending on the sample’s condition. For an intact sample, the dimension is 2, while it increases to 3 as macro cracks and service loads increase. The 2D and 3D box-counting method is used for 2D and 3D phase-space, respectively. The results of the 2D box-counting of some specific load histories and service load is tabulated in Table 2 and comprehensive results are displayed in Fig. 6, where 2 is considered as an embedding dimension for all phase-space reconstructions. From Table 2 and Fig. 6, it is discernible that fractal analysis of ultrasonic waves is extremely sensitive to the sample’s loading condition. An increase in load history as well as an increase in service load alter the fractal dimension tremendously, whether acting alone or effecting the sample’s condition simultaneously. Knowing the service load, one can use this damage-sensitive feature to estimate the load history of concrete materials locally. This is important, since it can be used to evaluate the current condition of concrete materials due to local over-loading. Fractal dimensions of the ultrasound waveforms in phase-space domain increase at load history larger than 40% of ultimate strength. This observation can be because of cracks transforming from micro cracks to macro cracks. Additionally, increasing the service load value

16

increases the fractal dimension of ultrasonic waves regardless of the sample’s load history status. Another advantage of using the proposed damage-sensitive feature is that it shows an almost linear trend. This is very important for practical applications where a simplified equation is needed to calibrate and evaluate the local condition of concrete materials. For illustrative purposes, phase-plane representations of the sample in different conditions are illustrated Fig. 7. Also, the squares required to contain the signal in the phase-plane have been displayed in Fig. 7. From Fig. 7, it is clear that by increasing the service load, the phase-space map tends to occupy more space; therefore, the fractal dimension of these phase-planes increases accordingly. Table 2 2D Fractal dimension of ultrasound waves in phase-space domain for a sample with different

Load history

loading condition.

Intact 0.36 Fu 0.45 Fu 0.63 Fu 0.8 Fu

0 1.09 1.14 1.20 1.28 1.23

Service load 0.072 Fu 1.09 1.16 1.31 1.30 1.28

17

0.148 Fu 1.12 1.37 1.59 1.46 1.51

0.164 Fu 1.12 1.49 1.64 1.46 1.60

Fig. 6. Phase-space domain fractal dimension (2D) evolution with loading condition of concrete sample. Another interesting point, which is clear from Table 2 and Fig. 6, is that when the load history of the sample reaches 63% of its ultimate loading capacity, the fractal dimension of nonlinear ultrasound waves decreases slightly compared to 45%. While load history generated cracks in the concrete sample, the service load implemented by the compressive tool affects the boundary condition of cracks (e.g. gap between crack interfaces). The main reason for this observation might be due to the ratio of displacement induced by ultrasound to cracking gaps. As the load history increases cracking size and gap between crack interfaces increase while the displacement field due to ultrasound amplitude stays constant (in this research constant excitation amplitude is used). This ratio as explained by Yamanaka [43] can significantly affect the dynamic response of each individual crack. This might be a physical explanation for the decreasing trend in fractal dimension’s value at force equal to 63% of the concrete’s ultimate loading capacity. As previously mentioned, by increasing load history and service load values, the embedding dimension for constructing phase-space shifts from 2 to 3. This means that when analyzing 18

systems’ behavior in phase-plane, valuable geometric information may be lost. Hence, for signals with embedding dimension 3, the 3D box-counting method for calculating fractal dimension was utilized to provide a more reliable and accurate analysis. It is worth mentioning that calculating 3D fractal dimension requires a powerful computational system and it is a timeconsuming task. In Table 3, results for the 3D box-counting method for cases with embedding dimension of 3 are tabulated. When objects have three dimensions, the fractal dimension would fall between 2 and 3. The results provided in Table 3 are more accurate in comparison to the same result in Table 2 because the 3D box-counting method considers all possible states of the data when calculating fractal dimension. Also, the same increase and decrease trend in the 3D fractal dimension are observable. For illustrative purposes, Fig. 8 represents the evolution of ultrasonic waves in phase-space domain for a concrete sample that experienced load history equal to 63 percent of ultimate load capacity. Additionally, the evolution of boxes required to contain these signals in phase-space have been displayed in Fig. 8. Table 3 3D Fractal dimension of ultrasound waves in phase-space domain for a sample with different

Load history

loading condition.

Intact 0.36 Fu 0.45 Fu 0.63 Fu 0.8 Fu

0 2.30 2.26

Service load 7.2 Fu 2.31 2.33 2.29 19

14.8 Fu 2.46 2.53 2.34

16.4 Fu 2.76 2.53 2.68

Fig. 7. Evolution of nonlinear ultrasonic waves in phase-plane domain for a concrete sample in different service loads experienced 0.63 Fu load history: (a) service load = 0, (b) service load = 14.8 Fu, and (c) service load = 16.4 Fu.

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Fig. 8. Evolution of nonlinear ultrasonic waves in 3-dimensional phase-space domain for a concrete sample in different. service loads experienced 0.63 Fu load history: (a) service load = 0, (b) service load = 14.8 Fu, and (c) service load = 16.4 Fu. As previously mentioned, frequency analysis is the main tool when it comes to nonlinear ultrasonic testing. As a practical approach, researchers are using the plane wave theory of nonlinear elasticity to calculate the acoustic nonlinear parameter β and use it as a damagesensitive feature. Based on the nonlinear harmonic generation technique, Zarembo and Krasil’nikov proposed [44]:

β=

4 A2 2

2

k x A1

,

(13)

where k is the wave number, ) and ) are the displacement amplitude of the first and second harmonic component of received signal, and x is propagation distance. Eq. (13) is based on a plane wave assumption and cannot consider complex acoustic wave behavior. Furthermore, in literature, it is seldom mentioned that this analysis should be performed under adiabatic conditions [45]. In finite amplitude-based methods, researchers use different approximations to consider absorption phenomena, which can be considered as a drawback in these methods [45]. For the (

sake of comparison, . ∝ (00 values, as the most frequently used quantitative damage-sensitive 1

feature in frequency domain, are calculated. Table 4 asserts that an acoustic nonlinear parameter 21

calculated from Eq. (13) is not a practical feature, especially when the concrete material is under (

service load. According to Table 4 and Fig. 9, (00 values are not following any logical trend while 1

the number and size of cracks in concrete material are increasing, which results in intense nonlinear behavior. However, a steady increasing trend with SL = 0% and different LH values are observable.

Fig. 9. Relative values of acoustic nonlinearity parameter in presence of service load and load history. Table 4 (0

values for concrete sample in different loading conditions.

Load history

(01

Intact 0.36 Fu 0.45 Fu

0 0.02 3.86

Service load 0.072 Fu 2.87 10.04 22

0.148 Fu 11.62 110.42

0.164 Fu 10.14 213.46

0.63 Fu 0.8 Fu

29.143 3.66

38.83 4.10

67.24 6.03

45.39 11.51

A lack of trend, especially for larger service loads, indicates that the β is not a reliable damage-sensitive feature for evaluating the level of cracking in concrete material under service load. The proposed damage-sensitive feature from phase-space domain, however, was shown to be a reliable value to evaluate load-induced cracking in concrete materials under service load. The successful application of the proposed quantitative feature extracted from phase-space domain is a strong justification for investigating nonlinear ultrasound wave behavior in phasespace domain. This approach considers all possible states of ultrasonic waves while interacting with defects. 6. Conclusion In this paper, fractal dimension as a quantitative tool for assessing concrete materials’ condition under service loads is proposed. Fractal analysis of ultrasonic waves in phase-space domain for measuring their nonlinearity while interacting with micro cracks and macro cracks is considered. To generate micro cracks and macro cracks, a compressive force to standard concrete blocks was applied in incremental steps. Then, to change the cracks’ boundary condition and simulate service loads, a specialized compressive tool was utilized. During compression with the compressive tool, an ultrasonic test was performed. In contrast with the traditional approach to analyze ultrasound data in the frequency domain which results in valuable data loss, ultrasonic waves in phase-space domain were examined. Having a phase-space representation of ultrasonic waves makes it possible to analyze data with numerous signal processing techniques without losing any data regarding the system’s states.

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Fractal dimension as a mathematical tool to extract a quantitative geometric feature from nonlinear ultrasonic waves in phase-space domain was applied. The feature was then used for damage assessment of concrete material under different service loads after experiencing extreme compressive loads. The box-counting method, as one of the most popular methods for calculating fractal dimension was utilized. Initially 2D box-counting was applied to the signals in phase-plane. Later, since by increasing macro cracks in concrete samples, embedding dimension changed from 2 to 3, the 3D box-counting method was applied. Results verify that nonlinear ultrasonic waves start to change their behavior in phase-space domain at early stages of crack growth, and fractal analysis of these data is sensitive to these changes. As macro cracks grow inside the concrete sample by increasing the applied compressive force, the fractal dimension of nonlinear ultrasonic waves approaches 2 in 2D analyses and 3 in 3D analyses, since the phase-space representation of ultrasonic waves takes more space in 2D and 3D space. This linear increasing trend in fractal dimension while the concrete sample’s condition became severe is ubiquitous in 2D and 3D analyses. This is important because it provides researchers and practitioners with a reliable and sensitive quantitative tool to assess civil structures’ conditions under service loads. Particularly, it will be an effective tool for online monitoring of different civil structures under different conditions. This method potentially can be used for real-time monitoring of structures under service load since fractal dimension value is sensitive to system’s states trajectories in phase-space domain. The real practical application of this method for full scale structural testing and monitoring, however, may face several challenges and requires a thorough investigation. Conflict of interest 24

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Acknowledgments This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References [1]

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Conflict of interest The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.