Time reversal damage localization in concrete based on two-dimensional meso-scale modeling

Abstract

In this article, a damage localization method in concrete materials based on time reversal theory and meso-scale finite element simulation considering random heterogeneous properties is developed. In this article, concrete is regarded as a multiphase composite material consisting of cement mortar matrix, coarse aggregates, and interface transition zones. Compared to other methods, which assume that concrete is homogeneous, the meso-scale model considers the intricacies of concrete inhomogeneity and can therefore better characterize the interaction between stress waves and internal structures of concrete material. Through the meso-scale method, acoustic phenomena including reflection, transmission, and diffraction among internal structures of concrete can be modeled. Furthermore, a novel time reversal based, damage imaging method is developed using the envelope of the refocused damage scattering signal to monitor the health condition of concrete. The scattered signal received by each sensor is time reversed and reemitted via numerical computation. To decrease the dispersion effect, the autocorrelation function of the refocused signals is computed to generate an image of the estimated damage. A time correction factor is introduced to decrease the influence of the elongated wave packet. Numerical and experimental results indicate that the proposed damage imaging method can locate damage with high spatial resolution in heterogeneous concrete material. Moreover, owing to the meso-scale modeling, the propagation of high-frequency stress waves in concrete can be analyzed more accurately.

Keywords

Damage imaging method meso-scale concrete numerical simulation piezoceramic transducers time reversal method structural health monitoring

Introduction

Concrete materials are widely used in civil engineering structures due to their excellent mechanical properties and low costs.¹ However, defects may exist in a concrete structure and may compromise its integrity. Therefore, in recent decades, tremendous efforts have been made in damage detection for existing concrete structures and equipping real-time structural health monitoring systems for future constructions.^2–5

Due to its wide spectral bandwidth, piezoceramic transducers have enabled the development of various stress wave–based active or passive structural health monitoring methods for concrete structures.^6–9 Liu et al.¹⁰ proposed the use of piezoceramic smart aggregates to gauge the degree of water infiltration in concrete structures. Their experimental results demonstrated the possibility of establishing an index-based indicator that raises or declines proportionally to the depth of water seepage. Chai et al.¹¹ reported the use of surface waves for non-destructive evaluation and tomography of concrete structures. Through the sole use of surface waves, an image of the monitored area was reconstructed and embedded defects were detected. Xu et al.¹² utilized four synchronized piezoceramic patches to monitor the compactness of a concrete-filled, fiber-reinforced polymer tube structure. The half compactness indicator based on the time difference of arrival algorithm was employed to quantify the concrete infill compactness. However, concrete is a multiphase and heterogeneous material, which means that the behavior of the stress wave in concrete is quite different from those in homogeneous materials. The mere use of the experimental results may be insufficient to accurately analyze and characterize the interaction between stress waves and the complicated inner structures inside the concrete. Meanwhile, fundamental understanding of stress wave behavior in heterogeneous media is still not completely understood and is an ongoing research area.

In recent decades, researchers have explored meso-scaled modeling to obtain an accurate understanding of the interaction between stress waves and the complex and multiphase structure of concrete.^13–15 Various modeling methods, such as the lattice model, the random particle model, and the random aggregate model, have been extensively studied.^16–21 Chen et al.²² studied factors that influence the nominal tensile strength of concrete materials via a meso-scale Monte Carlosimulation of a four-phase random aggregate model. The fitting formulas used in their simulation enabled the high-precision modeling of multiple factors. Nguyen et al.²³ investigated foamed concrete materials in meso-scaleusing X-ray computed tomography. The meso-scale concrete modeling method described in their investigation helped to construct a comprehensive framework that allows further study of mechanical behavior of foamed concrete. Zhang et al.²⁴ studied the effects of reinforcement on the anti-penetration capabilities of concrete slabs using meso-scale numerical model that included random distributed aggregates. To validate the effectiveness of the created model, a numerical simulation of a ballistics-induced perforation was carried out and the results were compared with conventional empirical formulae. The main driving intention of most research about concrete meso-scale modeling is to better understand the linear elastic response and the complex evolution of concrete mechanical behavior until the structural failure. While studies on concrete material characteristics have received relatively large amounts of attention, studies on the damage detection and structural health monitoring through meso-scale modeling have comparatively lagged behind. A limited number of high spatial resolution damage localization and characterization methods, such as damage imaging methods, can be found. Most numerical studies about the health conditions of concrete are based on the assumption that concrete material is homogeneous.

The time reversal technique, with attractive features such as its self-adapting spatial and temporal focusing characteristics, has routinely been used in the area of acoustics^25,26 and electromagnetics²⁷ and recently has also been used to locate and characterize damage in civil structures.^28–30 The probing stress wave signal used in the time reversal technique will automatically refocus at its excitation location after it is recorded, time reversed, and reemitted by an array of transducers even in highly heterogeneous materials.³¹ Thus, due to its suitability for heterogeneous materials, damage detection methods based on time reversal technique have been developed and widely researched for monitoring of concrete structures.^32,33 Zhao et al.³⁴ used the time reversal method to locate a defect along the interface between steel reinforcement and concrete with high accuracy. Tian et al.³⁵ proposed an attenuation coefficient model of stress waves in concrete based on time reversal method and the Rayleigh damping model. Experimental results agreed well with theoretical results that used the proposed attenuation coefficient in a model of concrete. Sohn et al.³⁶ developed a damage detection method for inspecting the bonding condition of the CFRP (carbon fiber–reinforced polymer) concrete structure. Leveraging the advantages of time reversal acoustics and cluster analysis, the damage detection method did not require a baseline reference to detect damage. However, despite the above research efforts, the complex behavior, such as reflection, transmission, and scattering, of stress wave in concrete is still a limiting factor in progression toward more advanced damage detection methods. For instance, the refocusing step of the time reversal method may be interfered by dispersion and attenuation. Meanwhile, the dimension of the transducers embedded in the concrete dictate the size and type of defects that can be defected. Distorted refocused time reversal signals compromise the damage detection results and worsen the resolution of the detected damage location.

To solve the above problems and overcome the limitation of recent research, a novel damage localization method is developed in this study. In this method, the material heterogeneity of concrete is simulated by the inclusion of three phases, namely, three-graded aggregates, cement paste, and interface transition zones (ITZs). The interaction between the high-frequency probing stress waves and the complicated inner structures inside concrete is analyzed in detail, including the interaction between the stress wave and the defect. In order to enhance the resolution and reduce artifacts, a new envelope-based, time reversal self-correlation algorithm is used to develop a damage imaging method. To decrease the influence from signal dispersion and distortion, the autocorrelation function of the refocused signals is computed while generating the damage image. Furthermore, a time correction factor is introduced to decrease the influence of the elongated wave packet.

The remainder of the article is organized as follows. The numerical method modeling meso-scale random aggregate with and without damage is detailed in section “Two-dimensional meso-scale modeling of concrete.” In section “Envelope-based time reversal damage imaging method,” the proposed envelope-based, time reversal self-correlation method for damage imaging is described. To validate the effectiveness of the proposed damage imaging method, numerical simulations on two-dimensional meso-scale concrete models with three-graded aggregates, cement mortar matrix, and ITZs are conducted in section “Two-dimensional meso-scale FEM models.” Damage imaging results are presented and discussed in section “Results and discussion.” Experimental verifications are shown in section “Experimental verification,” and conclusions are drawn in section “Conclusion.”

Two-dimensional meso-scale modeling of concrete

The mechanical properties of concrete material are highly influenced by the grading of aggregates. However, most previous numerical studies considered concrete material as homogeneous for the sake of simplicity. In contrast to previous studies, this work treats concrete as a heterogeneous material consisting of three phases, namely, three-graded aggregates, cement mortar matrix, and ITZs. The Fuller curve is commonly used to describe the dimension and quantity of aggregates and can be expressed as³⁷

p (D) = 100 % \times {(\frac{D}{D_{Max}})}^{n}

(1)

where p(D) is the accumulated percentage of aggregates which can pass a sieve with the diameter of D, D_Max is the maximum diameter of aggregates, and $n$ is an exponent which will be taken as 0.5 in this research.^20,37,38

Although three-dimensional numerical simulation of concrete material would be more accurate and realistic, the computation of a three-dimensional meso-scale numerical model is highly complex and impractical. Therefore, in this study, a two-dimensional meso-scale numerical model is used to investigate the characteristics of concrete material. For this purpose, the Walraven method³⁹ that can describe the distribution of aggregates in two-dimensional space is employed, and its expression is as follows

\begin{matrix} p (D < D_{0}) = p_{k} [1.065 {(\frac{D_{0}}{D_{Max}})}^{0.5} - 0.053 {(\frac{D_{0}}{D_{Max}})}^{4} - 0.012 {(\frac{D_{0}}{D_{Max}})}^{6} - 0.0045 {(\frac{D_{0}}{D_{Max}})}^{8} + 0.0025 {(\frac{D_{0}}{D_{Max}})}^{10}] \end{matrix}

((2)

where p(D < D₀) is the volume percentage of aggregates with the diameter of D less than D₀, p_k is the ratio value between the volume of aggregates and the whole concrete, and D₀ and D_Max indicate the diameter of aggregates which can pass the objective sieve and the maximum size of aggregates, respectively.

Since the geometry of natural aggregates, such as gravels and pebbles, is nearly spherical in shape, the aggregates are assumed to be circular in this study. The diameters of the random aggregates used in this simulation are set 40–80 mm to represent large stones, 20–40 mm to represent middle stones, and 5–20 mm to represent small stones. For the sake of universality, IZTs around the random generated aggregates all have a thickness of 0.5 mm.^13,20

In this research, a rectangular two-dimensional concrete with a cross-section of 1000 mm × 1000 mm is set as an example. First, four annular lead zirconate titanate (PZT) transducers are instantiated at selected areas within the concrete block. Then, the volume percentage of aggregates in each grade is calculated based on equation (2). Aggregates in each grade are generated randomly and placed into the concrete area. Each newly generated aggregate must be completely within the concrete area and shall not overlap with the transducers or with preciously placed aggregates.

Figure 1(a)–(c) illustrates three cases of randomly generated aggregates placed within a concrete block. In Figure 1, four black spots indicate the locations of four annular PZT transducers. In each result, the aggregate ratio surpasses 52%. For clarity, the thickness of each IZT around the aggregates is not displayed in Figure 1.

Envelope-based time reversal damage imaging method

Since concrete materials are multiphase and possess heterogeneous internal structures, stress wave propagation inside the concrete is complicated. Dispersion and attenuation will reduce and limit the spatial resolution of conventional imaging-based damage detection methods. To solve this problem, an envelope-based time reversal damage imaging method augmented with a time compensation factor is developed in this section. Note that for the practical large-scale concrete, lots of transducers will be installed to form a monitoring network based on the subrange encircled by multiple transducers.⁴⁰ Thus, for simplicity, the monitoring area is limited in the region encircled by the transducers.

The scattering signals

To properly identify damaged regions, a comparison of signals before (i.e. healthy baseline) and after damage needs to be conducted.⁴¹ To obtain the response signals, spatially distributed piezoelectric transducers are arranged in the monitoring area (section “Two-dimensional meso-scale modeling of concrete”). The response signal between the transmitter-sensor pair ij in frequency domain can be expressed as

Y_{i j} = X (ω) R (ω, r_{i}, r_{j})

(3)

where $X (ω)$ is the Fourier transform version of the actuating signal $x (t)$ and $R (ω, r_{i}, r_{j})$ is the structure transfer function from location r_i to r_j.

When the dimension of the defect is less than or is comparable with the wavelength of the actuating signal, stress waves will be scattered at the location of the defect. Therefore, the response signal between transmitter-sensor pair $ij$ in the presence of a defect can be expressed as

Y_{ijd}^{d} = X (ω) (R (ω, r_{i}, r_{j}) + D (ω, r_{r}, r_{j}))

(4)

where $D (ω, r_{i}, r_{j})$ denotes the transfer function for the stress wave propagation path from location r_i to r_j through the defect.

Note that most environmental disturbances and noise can be minimized by the data acquisition equipment.⁴² Meanwhile, the focus in this research is not noise suppression. Therefore, the only changes between the current condition and the baseline are assumed to be caused by defects. Therefore, the baseline subtraction method can be used to obtain the scattering signal, and its expression is as follows

\begin{matrix} Y_{ijd} = Y_{ijd}^{d} - Y_{ij} \\ = X (ω) (R (ω, r_{i}, r_{j}) + D (ω, r_{i}, r_{j}) - R (ω, r_{i}, r_{j})) \\ = X (ω) D (ω, r_{i}, r_{j}) \end{matrix}

(5)

Equation (5) provides a convenient method to obtain the scattering signal and can be used to identify the health condition of the tested structure. If there is no damage in the concrete, the value of equation (5) will be zero.

The envelope-based damage imaging method

To explain the proposed damage imaging method, $N$ annular PZT transducers which can generate and detect radially uniform stress waves are assumed to be embedded in a two-dimensional concrete. To obtain the health condition of the tested concrete, one transducer works as an actuator to emit the probing signal, and other transducers in turn work as sensors to receive the transmitted probing signals. All the transducers in the monitoring area need to work as actuators in turn to reveal the characteristic of damage from different directions. Therefore, when the $i th$ transducer at the location of r_i works as actuator, the received scattering signal by the $j th$ transducer at the location of r_j can be expressed as

y_{ijd} (t, r_{i}, r_{j}) = x (t) \otimes g_{e} (t, r_{i}, r_{j})

(6)

where $g_{e} (t, r_{i}, r_{j})$ is the time-domain channel response between the actuator at r_i and the sensor at r_j and $\otimes$ represents the convolution operation. The subscript “e” indicates that this time-domain channel response corresponds to the forward scattered fields, which can be obtained by simulation results or experimental data.

Then, the received scattered signal between transducers pair $ij$ is time reversed, and it can be represented as

y_{ijd} (- t, r_{i}, r_{j}) = x (- t) \otimes g_{e} (- t, r_{i}, r_{j})

(7)

To achieve self-adapting spatial and temporal focusing, the time reversed version of the received signal is retransmitted from the transducer at r_j. Meanwhile, the refocused signal received by the transducer at r_i can be expressed as

\begin{matrix} y_{ijd}^{TR} (t) = y (- t, r_{i}, r_{j}) \otimes g_{c} (t, r_{i}, r_{j}) \\ = x (- t) \otimes g_{e} (- t, r_{i}, r_{j}) \otimes g_{c} (t, r_{i}, r_{j}) \end{matrix}

(8)

where $g_{c} (t, r_{i}, r_{j})$ is the computational time-domain channel response, and the subscript “c” indicates that the channel response is obtained by numerical computation. In this study, the computed time-domain channel response employs the expression described in research.⁴³

To decrease dispersion and waveform distortion, the refocused signals between each transmitter-sensor pair should be modified by a time compensation factor. Then, the envelope of the time reversal refocused signal can be computed by

y_{ijd}^{TRER} (t) = {〈 y_{ijd}^{TR} (t - t_{cor}) 〉}_{env}

(9)

where t_cor is the time compensation factor which will be introduced in the following sections, and ${〈〉}_{env}$ represents the envelope-detection operation.⁴⁴

Then, the image of the monitoring area will be constructed as

I (r_{k}) = \sum_{i = 1}^{N} \sum_{j = 1}^{N, j \neq i} {(y_{ijd}^{TRER} (t), y_{ijd}^{TRER} (- t))}_{corr} | t = 0

(10)

where $()_{corr}$ represents the cross-correlation operation.

Due to the self-adapting space-time focusing characteristic of the time reversal method, if the computed time-domain channel response in equation (8) is conducted at the location of the damage, $y_{ijd}^{TRER} (t)$ is similar to $y_{ijd}^{TRER} (- t)$ near the time reference $t = 0$ . Therefore, the cross-correlation function achieves the maximum. However, if the computed time-domain channel response in equation (8) is conducted within the location of the undamaged area, $y_{ijd}^{TRER} (t)$ is dissimilar to $y_{ijd}^{TRER} (- t)$ . Thus, when the imaging map is built using equation (10), the point with the maximum image value is the location of the damage.

The time compensation factor

In this study, the back-propagation process of the time reversal method is implemented on a separate computer, instead of reemitting the time reversed signal at the location of the corresponding sensor. The refocused time is influenced by the wavelength of the actuating signal. However, the waveform of the stress wave is distorted and elongated during propagation in concrete. The distorted and elongated waveform will influence the refocusing time. Thus, a time compensation factor is proposed to decrease the dispersion effect.

The time compensation factor t_cor is calculated as

t_{cor} = \frac{L_{s} - L_{a}}{f}

(11)

where L_a is half the length of the enveloped actuating signal, L_s is the horizontal distance between the starting point and the highest point of the enveloped first arriving scattered signal, and $f$ is the sampling frequency.

Two-dimensional meso-scale FEM models

To validate the effectiveness of the proposed damage imaging method and study the characteristic of the high-frequency stress wave propagation in concrete, two-dimensional meso-scale finite element models are established in ABAQUS. Meanwhile, for the sake of improving the generality, the procedures of the random aggregate generation and placement described in section “Two-dimensional meso-scale modeling of concrete” are programmed using the open-source language Python to establish the meso-scale concrete models in ABAQUS.

After inputting the material parameters, three two-dimensional meso-scale concrete models (Model 1, Model 2, and Model 3 corresponding to the samples illustrated in Figure 1(a)–(c), respectively) are generated semi-automatically. Except for the distribution of aggregates, all simulation parameters are identical. Therefore, for simplicity, only the finite element model for Model 1 is detailed in Figure 2(a). Herein the ITZs (the purple areas) between the aggregates and the cement paste are considered, as shown in Figure 2(b). To actuate and receive radially uniform stress waves, four piezoelectric transducers with the internal and external diameter of 6 and 8 mm, respectively, are set in the concrete models. Meanwhile, to reduce the influence of the transducers from their shapes in generating and sensing stress waves, annular PZT transducers (the green areas) are used,⁴² as shown in Figure 2(c). The red ring (DP1) and the blue ring (DP0) outside and inside the PZT transducer in Figure 2(c) indicate the loading-voltage surface and the zero-voltage surface, respectively. A structured grid was used for meshing the aggregates, piezoelectric transducers, and ITZs to ensure the simulation quality of stress waves propagating in the concrete models. Meanwhile, a free grid is used for the cement mortar matrix. Quadrilateral meshes are employed instead of the typical triangular mesh²⁰ due to the superior computational efficiency, high accuracy, and high convergence rates of quadrilateral meshes. Eight-node biquadratic reduced-integration plane stress quadrilateral elements are employed for the aggregates, the cement paste, and the ITZs to improve the computational accuracy. Meanwhile, eight-node biquadratic reduced-integration plane stress piezoelectric quadrilateral elements are adopted for the annular PZT transducers. The meso-scale numerical models are analyzed based on the implicit analysis module of ABAQUS (ABAQUS/Standard). The detailed material properties of the concrete and the piezoelectric transducers are listed in Table 1.

Table 1.

Material parameters of concrete and PZT.

Parameters	PZT	Aggregates	Mortar	ITZ	Concrete
Young’s modulus (GPa)	53.2	55.5	26.0	25.0	30.0
Poisson’s ratio	0.261	0.16	0.22	0.16	0.17
Density (kg/m³)	7800	2700	2100	2400	2400
Piezoelectric strain coefficientsd₃₁/d₃₂/d₃₃/d₂₄/d₁₅ (×10⁻¹⁰ m/V)	−2.10/-2.10/5.00/5.80/5.80	–	–	–	–
Electric permittivity $ε_{11}^{T}$ / $ε_{22}^{T}$ / $ε_{33}^{T}$ (×10⁻⁸ F/m)	1.75/1.75/2.12	–	–	–	–

PZT: lead zirconate titanate; ITZ: interface transition zones.

To compromise between the spatial resolution and the monitoring range, a symmetrical modulated five-cycle sine burst with the central frequency of 100 kHz is used as the probing signal.⁴³ After obtaining the healthy signals from the monitoring area as a baseline, a circular hole is created to represent a defect. Without loss of generality, the defective area simultaneously crosses the aggregate, the IZT, and the cement mortar matrix, as shown in Figure 2(d). Meanwhile, three different diameters of circular hole are created to study the influence of the defects’ size on the proposed damage imaging method. The blue area in Figure 2(d) indicates the defect with a diameter of 10 mm. The yellow area and the pink area indicate the defects with the diameter of 10–20 mm and 20–30 mm, respectively. The central locations of each PZT transducer and the defect are displayed in Table 2.

Table 2.

Coordinates of PZT transducers and defect.

PZT transducers and defect	Coordinates (x, y) (mm)
Transducer 1 (T1)	(300, 300)
Transducer 2 (T2)	(700, 300)
Transducer 3 (T3)	(700, 700)
Transducer 4 (T4)	(300, 700)
Defect	(575, 450)

PZT: lead zirconate titanate.

Results and discussion

Stress wave propagation analysis at meso-level and macro-level

To demonstrate the advantage of the meso-scale concrete model in analyzing stress wave propagation, wave fields obtained from the meso-scale concrete model and the macro-scale concrete model, which assume a homogeneously structured concrete, are shown in Figure 3. The material properties of the homogeneous concrete model are listed in Table 1. In Figure 3(a), wave fields including the wave front and the reflected wave from the boundary can be observed in the homogeneous concrete model. In the figure, the third PZT transducer acts as the actuator. Furthermore, since the dimension of the defect is less than or comparable with the wavelength of the probing stress wave, the scattering stress wave induced by the defect can be observed as a secondary wave source when the wave front of the probing stress wave impinges the defect. In contrast, as shown in Figure 3(b), the wave field obtained from the meso-scale concrete model (Model 1) is highly complex. The propagation of stress waves is significantly influenced by the heterogeneous internal structures inside the meso-scale concrete model. Each of the random aggregates can be regarded as a secondary wave source. When the wave front of the probing stress wave goes through these aggregates, scattered waves induced by the aggregates add a level of complexity to the wave fields. However, these details of stress wave propagation in concrete will not be observed from the homogeneous model (Figure 3(a)).

Interaction between the stress wave and the defect at the meso-level

The advantage of using FEM to analyze the characteristics of the stress wave propagation is the ability to reveal the interaction between the stress waves and the complex internal structures, including the defects. Figure 4 shows the wave fields obtained from the meso-scale concrete model (Model 1) under the healthy condition and three defective conditions, respectively. From the authors’ previous study,⁴² the strength of the scattering stress wave induced by the defect is much lower than that of the probing stress wave. However, the wave field in the meso-scale concrete model is made complicated by the heterogeneous internal structures. The scattering stress waves are superimposed with the probing stress wave. However, a variety of the wave fields can be observed when the probing stress wave traverses the defect, as shown in Figure 4. The simulation result also indicates that the intensity of the varied wave fields at the location of the defect increases in proportion to the size of the defect (Figure 4(a)–(d)).

Effect of the time compensation factor

Imaging results of Model 1 with and without the time compensation factor are shown in Figure 5(a) and (b), respectively. Due to the dispersion effect, the waveform of the stress wave is distorted and elongated during the propagation in concrete. Namely, the peak value of the refocused time reversed stress wave will deviate from its theoretical location. Without the time compensation factor, the image of the damage was affected by the distorted and elongated waveform. As shown in Figure 5(a), the pixel value reaches the maximum at the location (624,472). However, the actual location of the damage is (575,450). The deviation is unacceptably large (about 53.7 mm). On the other hand, with the time compensation factor, the influence of the dispersion effect in concrete materials is limited. The localization errors caused by the distorted and elongated scattered wave signals are reduced. The error between the image of the defect and its actual location is about 6.7 mm, as shown in Figure 5(b). The diameter of the defect is 10 mm, which means that the distance between the edge of defect and its image is 1.7 mm. Such a small error is acceptable.

Simulation imaging results

To construct the image of the monitoring area, the signals scattered by the defect are obtained using the baseline subtraction method, as shown in equation (5). Then, the damage is located using the proposed method. Based on the meso-scale simulation results, the average velocities of the stress waves in Model 1, Model 2, and Model 3 are 3868.8, 3897.8, and 3906.1 m/s, respectively.

The imaging results of the 10-mm defect are shown in Figure 6, where the white circular area at the location (575,450) denotes the actual dimension of defect. As shown in Figure 6(a)–(c), the defect in Model 1, Model 2, and Model 3 can be detected by the proposed method. Through the time reversal method, the image of defect is refocused on a bright red spot. The time compensation factor and the envelope-detection operation reduced the influence of the expanded and distorted stress wave on the accuracy of the location. Therefore, even in complex concrete, the damage can still be imaged with high resolution and accuracy. To display the defect more clearly, the −1.5 dB areas of the image are shown in Figure 6(d)–(f), respectively.

Figure 7 shows the imaging results of Model 1, Model 2, and Model 3, when the diameter of the defect is 20 mm. As predicted, the defect in each model can be located with high spatial resolution and satisfactory accuracy. As shown in Figure 7(a)–(c), the pixel value reaches the maximum at the location (573,447), (571,448), and (581,446), respectively. Due to the increased diameter of the defect, the intensity of the scattered signals around the defect becomes higher. Therefore, the maximal pixel value of the defect with 20 mm diameter in the image increases. Similar results are also found when the defect is 30 mm in diameter (Figure 8).

In addition, a comparison among the imaging results (Figures 6 –8) obtained from each meso-scale model suggests that the imaging method can reveal information about the dimensions of the defect. The bright red areas become larger when the defect diameter increases, as shown in Figure 6(d)–(f), Figure 7(d)–(f), and Figure 8(d)–(f). At the same time, the dimensions of defect also increase with the maximal pixel value. It means that the obtained images from the proposed method can be used to characterize the extension of the defect based on the common damage index method.^9,10 In future work, the relationship between the defect dimensions and the maximum pixel value will be further examined. The ability of the proposed damage imaging method for characterizing multiple defects and cracks in concrete will also be investigated. Furthermore, the effects of the aggregates’ shape and the steel reinforcement inside the actual concrete structure on the propagation of the stress waves will be theoretically studied using the meso-scale concrete model.

Experimental verification

The practicality of the proposed damage imaging method is verified experimentally via the concrete slab as shown in Figure 9. The experimental setup is identical with the authors’ previous work,^42,45 including the dimension of the concrete slab, the data acquisition system, the concrete mix, and the location of the defect (a through-hole with a diameter of 15 mm). Therefore, for the sake of conciseness, the experimental setup will not be detailed in this article. Meso-scale and macro-scale numerical models corresponding to the experimental setup are established to demonstrate the advantages of the stress wave simulation based on the method described in section “Two-dimensional meso-scale FEM models.”^13,20

The image (Figure 10) of the artificial through-hole defect is constructed based on the scattering signals obtained from the concrete slab and its corresponding meso-scale simulation. Using the self-adapting spatial and temporal focusing characteristics of time reversal technique, the through-hole defect on the concrete slab can be isolated with relatively high clarity, and it is represented by a bright red spot as shown in Figure 10(a) and (c). Compared to the authors’ previous results,^42,45 the image constructed by the proposed method has a higher resolution. Under identical experimental conditions, the through-hole imaged by the proposed method covers a 5.9 cm × 8.0 cm area, as shown in Figure 11(b). However, the size of the defect constructed by the conventional damage imaging method⁴² is 16.0 cm × 16.7 cm. In addition, the proposed method can locate the defect with higher precision. As shown in Figure 11(b) and (d), the pixel value reaches the maximum at locations (622,509) and (626,501). The error between the image of the artificial through-hole defect and its actual location is 9.5 and 1.4 mm, respectively. Given that the diameter of the through-hole is 15 mm, the small deviation is acceptable. However, the error of the conventional damage imaging method⁴² is 15.2 mm.

Figure 11 compares direct responses between the identical transmitter and sensor pair of the homogeneous concrete model, the meso-scale model with random aggregates, and the experiment as normalized by the first arrived signals. Figure 11(b) shows that the normalized received signal at the annular PZT transducer is elongated and distorted due to the heterogeneous material properties of concrete. Due to the heterogeneous internal structures, the waveform becomes more complicated. However, these effects are not observed from the homogeneous concrete model, as shown in Figure 11(a). In contrast to the response obtained from the experiment in Figure 11(c), the stress waves obtained from the response of the meso-scale model agree more closely with the realistic conditions. Hence, the meso-scale FEM of concrete provides a more accurate way to analyze the propagation of stress waves in concrete. It should be noted that the distribution of the aggregates in the meso-scale model is random, and the distribution may differ from the actual distribution in the concrete slab. As mentioned above, the propagation of stress waves is significantly influenced by the random aggregates. Therefore, the signals obtained from the meso-scale model and the experiment will have inevitable slight differences.

Conclusion

In this research, the characteristic of the stress wave propagation in concrete is investigated using a meso-scale finite element simulation that considers the random heterogeneous properties of concrete. A novel time reversal damage imaging method is developed using the envelope of the refocused signal from the damage-induced scattered signal to locate the defect of concrete. To verify the effectiveness of the proposed method, the meso-scale two-dimensional finite element models with embedded, annular PZT transducers are simulated in ABAQUS. The numerical results indicate that compared to conventional methods, which assume that concrete is homogeneous, the meso-scale model can better characterize the interactions (e.g. reflection, transmission, and diffraction) between the stress waves and the internal structures in concrete material. Furthermore, due to the introduction of the time correction factor and the autocorrelation function of the refocused time reversed signals, defects of three different sizes can be imaged with high spatial resolution and accuracy. Meanwhile, the dimension of defect can be revealed by the maximal pixel value of its corresponding image. Finally, the practicality of the proposed method is verified experimentally on a fabricated concrete slab and its corresponding simulation. Through the comparison of the experimental and simulation results, the stress wave propagation simulated from meso-scale model of concrete agrees more closely with experimental observations.

Footnotes

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Major State Basic Research Development Program of China (973 Program, Grant Number 2015CB057704). The authors would like to thank this financial supporter.

ORCID iD

Weihang Gao

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