Evaluation of the physical characteristics of reinforced concrete subject to corrosion using a poro-elastic acoustic model inversion technique applied to ultrasonic measurements

Abstract

The use of reinforced concrete is foundational to modern infrastructure. Acknowledging this, it is imperative that health monitoring techniques be in place to study corrosion within these structures. By using a non-destructive method for detecting the early formation of cracks within reinforced concrete, the method presented in this paper seeks to improve upon traditional techniques of monitoring corrosion, within reinforced concrete structures. In this paper, the authors present a method to evaluate the physical characteristics of reinforced concrete subject to corrosion using a poro-elastic acoustic model inversion technique applied to a set of ultrasonic measurements, which constitutes a novel approach to the problem of observing the impact of corroding rebars and resulting concrete damage. A non-contact ultrasonic transducer is operated at a carrier frequency of 500 [kHz], with a layer of saltwater separating the sensor from the concrete surface. Following this non-contact measurement collection of the surface and rebar echo responses, a poro-elastic model is used to model the sound propagation, through an adapted version of the Biot-Stoll model. At first, a set of default parameters, obtained from the physical characteristics of the reinforced concrete, are used to match experimental and simulated acoustic signature of the sample. Performing statistical averaging along the corroding rebar within three samples over a period of nearly nine months, a small but monotonous increase in the distance between the concrete surface and the top of the rebar, indicating gradual corrosion of the rebar. Next, a non-linear optimization algorithm is used to optimize the match between measured and simulated echoes. Through the implementation of this model parameter optimization, the root mean square error between measured and simulated responses was reduced by 63.7% for the full signal, and 62.6% for the rebar echo.

Introduction

A key feature in modern day infrastructure is the use of bridges for transportation. In many instances, these structures are constructed using reinforced concrete, to provide support to the overlaying roadways we travel upon. With these structures being so prevalent in everyday life, it is vital that techniques are in place to monitor the health of bridges, without adding, or creating the damage, cracking, and corrosion that may be taking place.

Regarding corrosion, this process is both destructive to reinforced concrete structures, as well as costly on a federal level with each passing year. The direct costs related to bridges incurred through corrosion, were estimated to be $8.3 billion in 1998, to an estimated $13.6 billion in 2013 [1]. These values are taken from a study performed by The National Association of Corrosion Engineers (NACE) who also state that costs associated with corrosion accounted for 3.4% of the U.S. GDP in 2013 [2].

In addition, the scale at which we observe corrosion within bridges is widespread. When inspecting bridges in the United States, the U.S. Federal Highway Administration (FHWA) stated that 1/3 of bridges were either deficient, or non-functional when analyzing the structural level [1]. Observing the data presented by both NACE [2] and the FHWA [1], it becomes clear that non-invasive techniques are a more suitable option for studying these corroded structures, as the potential impacts observed through sample collection may be mitigated.

Methods for measuring reinforced concrete that eliminate the requirement for sample collection, are often referred to as non-destructive techniques (NDT). In the presented research, a non-contact ultrasonic transducer is operated at a carrier frequency of 500 [kHz], with a layer of saltwater separating the sensor from the concrete surface. The presented research aims to observe and predict pre-cracking signatures of reinforced concrete, without adding contact of a foreign material to the structure.

To analyze the acoustic signatures presented in the reinforced concrete samples studied in the laboratory, an adaptation of the Biot-Stoll model has been introduced. This adapted model uses poro-elastic modeling with three layers: saltwater, concrete, and rebar. Through comparing the trends presented in both the physical parameters of the Biot-Stoll model, in tandem with trends from electrochemical data gathered through galvanostatic pulse (GP) measurements, the presented approach to crack detection seeks to validate the resulting conclusions through correlation analysis.

The primary objectives of the research presented here are threefold. The first objective was to perform periodic acoustic data collection on the original 3 concrete samples, that were provided by the Corrosion Laboratory of Florida Atlantic University. Through cyclical testing of these samples, it allows for the acoustic response, along with Biot-Stoll parameters, to be analyzed such that changes over time can be considered. Testing for these 3 original samples was conducted from October 2021 through June 2022. Following the testing of these samples, 4 new samples of varying properties were introduced, and are currently being acoustically tested periodically.

The second primary objective of the presented research was to refine the parameter bounds of the initial rendition of this model, and to predict the current performance of the updated version. This objective is significant as it seeks to optimize the curve fitting of our measured response with respect to a simulated response, which results in improved accuracies to the Biot-Stoll parameters outputted from the model. To satisfy this objective, several characteristics of the simulated acoustic signal are adapted to improve the curve fit between simulated and measured responses. In addition, the search interval of a set of Biot-Stoll parameters is evaluated to reduce the root mean square error between measured and artificial signals.

Beyond the work presented in this paper, the authors intend to combine the corrosion potential and galvanostatic pulse data with the acoustic results collected through poro-elastic modeling, to identify the early presence of cracks in the reinforced concrete.

Reinforced concrete corrosion

In modern infrastructure, the primary purpose of implementing bridges, is for the connection of transportation pathways over a body of water. When such structures are exposed to saltwater in particular, the corrosive process is initiated through chloride ions (contained within saltwater) that penetrate the concrete through the pore structure. Over time, the chloride ions will exceed a certain concentration and then breach the passive layer of the rebar, initiating the corrosion process on the rebar surface.

As the chloride ions make contact with the rebar surface, an oxidation-reduction (redox) reaction takes place. During this redox reaction, iron hydroxide, Fe(OH)₂, forms along the rebar surface, exerting an upward pressure on the internal concrete [3]. Over time, this pressure leads to internal cracking, which eventually reaches the surface of the reinforced concrete.

Figure 1 shows the stages of reinforced concrete corrosion, as provided in [4]. In addition, the described redox reaction may be represented mathematically through a series of half-cell reactions. First, we define the corroding segment of the rebar to be the anode, with the half-cell oxidation reaction being given in (1). Next, we state that a separate location on the same rebar with a different energy level does not corrode, acting as the cathode. For this section of rebar, we represent the cathode through a half-cell reduction reaction as provided in (2), known as the oxygen reduction reaction. Lastly, the formation of iron hydroxides is then represented by (3).

$$2Fe\to 2{Fe}^{2+}+4{e}^{-}$$

(1)

$$2{H}_{2}O+ {O}_{2}+4{e}^{-} \to 4O{H}^{-}$$

(2)

$$2F{e}^{2+}+4O{H}^{-}\to 2Fe\left(OH\right)$$

(3)

Fig. 1

Stages of Reinforced Concrete Corrosion and Iron hydroxide Formation Along Rebar Surface [3]

Non-destructive testing

Currently, Non-Destructive Testing (NDT) for monitoring corrosion in reinforced concrete structures can be divided into two primary categories. The first category of testing is the electrochemical approach which often includes open circuit potential [5,6,7] and galvanostatic pulse [8, 9] measurements.

The second category relies on ultrasonics. The detection and measurement of rebar corrosion in reinforced concrete using ultrasonic NDT has been of great interest for several decades. The benefit is obvious: using ultrasonic NDT, the operator can monitor corrosion in reinforced concrete and ideally detect early signs of corrosion, without any need to drill through or cut the concrete. The limitation of this approach resides in the ability to successfully detect early forms of corrosion, before cracks reach the surface of the reinforced concrete.

Recent publications indicate that the scientific community is testing a variety of signal processing approaches to accurately estimate the propagation of cracks and changes in the rebar diameter, using narrow-beam, broadband acoustic transducers. Such transducers are operated at hundreds of [kHz], with [mm] scale resolution. These approaches include higher-order statistics, Short-Time Fourier Transform analysis (STFT), Artificial Neural-Networks (ANN) [10,11,12], Convolutional Neural Network (CNN) [13,14,15], Wavelet Decomposition (WD) [16], Empirical Mode Decomposition (EMD) [17] and Laplace Transforms [18] among others.

Such state-of-the-art techniques have great merit but are limited in three specific ways: (i) early crack detection capability depends on spatial resolution; (ii) early crack detection capability depends on a meaningful training of the crack detection algorithm; (iii) early crack detection capability depends on the complexity of the sound propagation model.

The first limitation is that early crack detection capability depends on spatial resolution. The cited techniques rely one way or another on distance measurement, obtained from the time-of-travel of sound and an estimate of the sound speed in the concrete and rebar. Propagation of ultrasounds through concrete is subject to great attenuation, mostly due to friction and scattering [19].

Such attenuation increases significantly with frequency, while spatial resolution improves as the frequency of operation (thus the effective frequency bandwidth of the transducer) increases [20].

As a result, ultrasonic transducers used to monitor reinforced concrete are often operated around 500 [kHz], so that sound can travel through several [cm] of concrete, reach the rebar and produce a sufficiently strong echo, all of this with a resolution of the order of 5-to-15 [mm] (depending on the transducer bandwidth and actual speed of sound in the concrete) [21, 22]. One should note that many techniques exist (shear probes, contact transducers, focused and unfocused immersion transducers) and that the angle of observation can be adjusted [23]. But it remains that the resolution is not sufficient to observe cracks a few [mm] long or less, which in turn limits the ability for an early detection.

The second limitation is that early crack detection capability depends on meaningful training of the crack detection algorithm the second limitation. The performance of the approach depends heavily on the number of training samples available and is limited to the statistical representativity of such samples. Considering the [mm] scale resolution of the transducers, the highly anisotropic nature of concrete and the strong variation in shape and composition of the corroded areas, such statistical representativity can only be achieved through very large number of concrete samples.

The third limitation is that early crack detection capability depends on the complexity of the sound propagation model, the published approaches rely on simple acoustic models to translate the acoustic signal into actual estimates of the actual damage occurring in the rebar and the concrete. Elastodynamic Finite Integration Technique (EFIT) [24], volume scattering models and similar Finite Element Analysis tools (COMSOL, ANSYS) are commonly used [25, 26]: while such techniques have great merit and produce results of great quality, they rely on a purely elastic model.

Sample observation clearly indicates that, in the vicinity of corrosion and crack, there is a complex structure combining fluid cavities and loose concrete material surrounding the corroded rebar [27, 28]. This means that a more appropriate acoustic propagation model could be a poro-elastic model, such as the well-established Biot [29, 30] and Biot-Stoll [31, 32] models developed for the propagation of sound in the seafloor. In both cases, fluid-filled cavities and a skeletal frame produce coupled vibrations induced by the incoming sound, resulting in a dispersive medium where two types of compressional waves (the fast wave and the slow wave) and shear waves propagate. Additional information regarding these various types of waves is provided in the next section. As a result, the frequency response of a poro-elastic medium is different from that of an elastic or even visco-elastic medium.

While the Biot-Stoll model has been extensively used for foundation engineering and Navy purposes over the past fifty years, it has not been applied to the problem studied here. In addition, the lead author has studied and published a specific technique, derived from the initial work by Stern [33], to handle gradual changes of the physical characteristics of the medium and produce a synthetic response of such a porous medium to a broadband acoustic impulse [34, 35]. Such a model can handle the propagation of sound through a pure fluid, a porous medium and a (corroding) solid. Therefore, this model is very relevant at the acoustic frequencies used and in material observed around the corroding bar.

The Biot-Stoll model (derived from the original Biot model), although complex, has the great benefit of tying the physical properties of the porous medium (described in the next section of this paper) to the sound propagation through the porous medium. As such, it is possible to fit a set of physical characteristics of the medium to a specific acoustic signature measured off a transducer.

In this paper, the authors present the method to evaluate the physical characteristics of reinforced concrete subject to corrosion using a poro-elastic acoustic model inversion technique applied to a set of ultrasonic measurements, which constitutes a novel approach to the problem of observing the impact of corroding rebars and resulting concrete damage.

Biot-stoll model of sound propagation in porous media

Sound propagation within porous media was introduced initially by M.A. Biot in the 1950s. In following years, this theory was expanded upon by R.D. Stoll, yielding the Biot-Stoll Model. The improved model utilizes a set of 14 physical parameters which are used in the analysis of acoustic wave propagation in porous sediments [36]. These physical parameters are tabulated in Table 1, and the adapted set of free parameters will be discussed within the Methods section. The primary changes in the model used in the presented research, is a redefining of the fluid properties to be consistent with salt water, the grain properties to be that of the concrete mix grains and dry concrete frame, and an added set of parameters to model the embedded rebar layer. In addition, the model consists of an initial acoustic pulse through the water column, which penetrates a medium to produce three separate waves.

Table 1 Biot-stoll parameter definition as provided in initial form

The waves produced are categorized as two compressional waves (one fast, one slow) and a third shear wave [37]. To further define these waves, the fast compressional wave is observed to be a dilatational wave, predicted by both visco-elastic and elastic models. The slow compressional wave results from out-of-phase vibrations of the solid portion of the medium (the skeletal frame) relative to the fluid that circulates through the pores. The fast compressional wave travels faster through the medium than its slow compressional wave counterpart. In addition, the slow compressional wave is highly attenuated. Lastly, the shear wave is defined to be a rotational wave featuring high attenuation on the response [30, 38].

In this model, acoustic scattering is considered as a secondary cause of signal distortion. This approximation is acceptable given the physical properties of the materials used, and because of the transducer characteristics. An in-depth justification is provided in the Experimental Results section of this paper.

Non-linear least square curve fitting

To optimize the presented model, non-linear least square curve fitting is used to match the measured acoustic response, to a simulated response. Through optimization of the curve fitting between these responses, the objective is an improvement to the accuracy of the Biot-Stoll output parameters. To implement this method of non-linear optimization, the presented research utilizes two algorithms to solve the optimization problem. The first of which is the trust region reflective (TRR) algorithm, the second being the Levenberg-Marquardt (LM) algorithm. Using these methods, a set of constraints in tandem with an objective function, is matched to a raw set of data y to fit the measured and simulated curves. Mathematically, (4) is used to solve the optimization problem as given in [39].

$$\eqalign{& {\min _x}\,{\left\| {F\left( {x,\,{x_{data}}} \right)\, - \,{y_{data}}} \right\|^2} \cr & = \,\mathop {\min }\limits_x \,\left( {\sum\nolimits_{i = 1}^n {{{\left( {F\left( {x,\,{x_{dat{a_i}}}} \right)\, - \,{y_{data}}_{_i}} \right)}^2}} } \right) \cr}$$

(4)

For the presented model, upper and lower bounds are given to each of the Biot-Stoll parameters to define a trust-region. With this, the TRR algorithm is utilized as the primary optimization algorithm for the presented model. Using a trust-region algorithm, (4) then takes the form of (5) [40].

$$X=\text{m}\text{i}\text{n}(\sum _{i=1}^{n}\parallel F\left({x,{x}_{data}}_{i}\right)-{{y}_{data}}_{i}{\parallel }^{2})$$

(5)

In (5), the term x represents one of the Biot-Stoll parameters, with a respective value assigned to both an upper and lower bound. In addition, n represents the number of Biot parameters utilized by the model. Next, we define the variable x_data to represent the given input data. Lastly, we define y_data to be the output data. The two algorithms are implemented and tested in MATLAB.

Trust region reflective algorithm

To expand upon the TRR algorithm, we first define the objective function as (6) and (7) [41]. Following a Taylor approximation of the initial objective function, (6) and (7) represent the first two terms of this computation [41]. In Eq. (6), θ represents the approximation of the objective function, using the quadrative objective function q is estimated at the location of x [42]. The term Δ defines a positive scalar, with D being the diagonal scaling matrix.

$${\theta = \text{m}\text{i}\text{n}}_{\text{s}}\{q\left(s\right), s \in {\rm N}\}$$

(6)

$$q = {\rm{min}}\left\{ {{1 \over 2}{s^T}Hs + {s^T}g{\rm{}},{\rm{}}\parallel Ds\parallel \le \Delta {\rm{}}} \right\}$$

(7)

Using the upper and lower bounds defined for each Biot-Stoll parameter, a bounded area (trust-region) is formed. Within the defined trust-region, a reflective line search is performed to locate the most optimal value for each of the Biot-Stoll parameters.

To further define the trust-region in which this line search is performed, we define the region to be a two-dimensional subspace of directionalities s₁ and s₂ [42, 43]. To define the first directionality, s₁, it is stated s₁ is used to represent q, an approximated objective functions direction [41]. Next [41], provides the second directionality, s₂, to take two forms. The first, defined in (8), shows the approximate Newton direction. In contrast, the second form is defined in (9), and represents the direction of negative curvature. It is important to observe the second directionality may only take one form, (8) or (9), but not both [41].

$$H \cdot {s_2} = - g$$

(8)

$$s_2^T \cdot H \cdot {s_2} < 0$$

(9)

In (8) and (9), H is defined to be the Hessian matrix of the objective function, while g represents the gradient of the original (non-approximated) objective function.

Levenberg-Marquardt algorithm

In the case where a conditional failure occurs while executing the TRR algorithm, the LM algorithm is used to optimize the solution. Rather than using a trust-region to optimize the defined set of Biot-Stoll parameters, the LM algorithm combines the gradient-descent method, in tandem with the Gauss-Newton method, to minimize the objective function [44].

Studying the Gauss-Newton method, we first assume the initial values defining the Biot-Stoll parameters, to be near the optimized solution. In addition, we state the Biot-Stoll parameters to be quadratic, with the update rate at each iteration being defined in (10). To declare the variables given in (10), W represents the weighting matrix, J represents the Jacobian matrix of the function $\widehat{y}$ with respect to the model parameters, y is the initial data, $\widehat{y}$ is the function used for curve fitting, and the denotation of gn on the term h indicates the Gauss-Newton method.

$$\left[{J}^{T}WJ\right]{h}_{gn}={J}^{T}W(y-\widehat{y})$$

(10)

While the Gauss-Newton method is given in (10), the update rate governing the gradient-descent method is defined in (11). Observing (11), the directionality of the update rate is opposite to objective function [44] with α representing a positive scalar giving the steepest direction step length, and h_gd representing the gradient-descent method for the update rate.

$${h}_{gd}=\alpha {J}^{T}W(y-\widehat{y})$$

(11)

Through the combination of the Gauss-Newton (10) and gradient-descent (11) methods, the general update equation for the LM algorithm takes the form provided in (12). In (12) the term I represents the identity matrix, while the damping parameter is given by λ. When observing the damping parameter, the value for λ fluctuates based on which method (Gauss-Newton or gradient-descent) is applied [44].

$$\left[{J}^{T}WJ+\lambda I\right]{h}_{lm}={J}^{T}W(y-\widehat{y})$$

(12)

Overview of non-destructive method

Figure 2 provides a visual overview of the method utilized in the presented research [45]. The presented method begins with the acquisition of a raw acoustic response. The response collected presents two main echoes, the first being the concrete surface, with the second being the rebar. Following this, the two echoes of the raw response are isolated using time gating. With the time gated signal, the raw data is passed into a MATLAB simulation which runs an adapted version of the Biot-Stoll model. In this adapted model, each of the Biot-Stoll parameters is given an initial value, along with an upper and lower bound. Using these inputs, the non-linear least square solver utilizes a TRR algorithm to optimize the parameter values, producing a new set of Biot-Stoll parameters.

Fig. 2

Overview of Acoustic Data Acquisition and Processing [30]

Next the model produces a simulated response, which is matched to the raw response both in time alignment and echo amplitude. Following the generation of the simulated response, we identify two echoes in the simulated curve, the first being the concrete echo, with the second representing the rebar echo. After verifying a match between raw and measured signals, the new set of Biot-Stoll parameter values output from the model, are studied to identify trends. Ultimately, the combination of trends identified in parameter values along with trends produced during the electrochemical measurements to evaluate will be used to evaluate the presence of cracks in the concrete due to corrosion. The focus of this paper solely focuses on the acoustics modeling and measurements.

Acoustic testing overview

Figure 3 shows a depiction of the testing setup used for the presented acoustic data acquisition. To begin, a 7 [µs] pulse centered at 500 [kHz], with a peak voltage of 6 [V], transmitted every 100 [ms], is passed through a signal generated. In the context of the presented research, an HP 8112 A pulse generator was used to generate the desired signal. Following this, an Olympus Panametrics transducer oriented above the embedded rebar transmits the signal, receiving the echoes. The Panametrics V389 unfocused immersion transducer has a beamwidth of 14 [deg] at -6 [dB] when operated at 500 [kHz]. The returned echoes are passed through a transmit/receive switch connected to a digital oscilloscope. For the presented research, a Tektronix MDO3024 Mixed Domain Oscilloscope was used to display and save the raw acoustic data at a sampling frequency of 1.25 [GHz]. Next, the saved data was passed to a user laptop to run the adapted Biot-Stoll model through a MATLAB simulation.

Fig. 3

Graphic Depiction of Acoustic Testing Setup [31]

Reinforced concrete samples

Figure 4 shows the initial 3 samples used in the presented research. Each of the initial samples had the same composition. The mix design consisted of 213 [kg/m³] of cement, 106.5 [kg/m³] of water (i.e., 0.5 w/c ratio), 652 [kg/m³] of fine aggregate (silica), 753 [kg/m³] of coarse aggregate (limestone).

Fig. 4

Initial Three Reinforced Concrete Samples with Two Embedded Rebars Each

In addition, 2 rods of rebar 15.875 [mm] in diameter, 254 [mm] in length, were inserted underneath a concrete cover thickness of approximately 3.56 [cm]. The degree of corrosion varied across samples, with sample 1 showing the least amount of corrosion, sample 2 showing moderate cracking (observed at the surface of the concrete block), and sample 3 showing severe cracking (extending across the entire length of the concrete block), as shown on the picture in Fig. 4. Samples 1 through 3 were tested from October 2021 through June of 2022.

Following this, two additional samples were studied, as depicted in Fig. 5. In the newer samples, each had a singular embedded rod of rebar spanning the entirety of the sample length, 304.8 [mm]. In addition, sample 16X has 2 cm concrete cover, while sample MD1 was mortar (no coarse aggregate) and thinner concrete cover of 1 cm.

Fig. 5

Current Reinforced Concrete Samples 16X and MD1

A gradation of #89 was selected for the coarse aggregate in samples 16X and samples 1, 2 and 3. The gradation #89 was done as per FDOT Handbook Specifications (Sect. 901 – coarse aggregates, Table 1 [47]). The maximum aggregate size was 0.95 [mm].

Acoustic data acquisition

To acquire the raw acoustic response, the setup shown in Fig. 6 was used. Prior to the transmission of the acoustic pulse, a reinforced concrete sample was first submerged into a NaCl solution of 10%. Although seawater is known to be 3.5% NaCl solution, our mix utilized 10% to stay consistent with other laboratory measurements. In addition, the increase in salinity has no meaningful impact on our acoustic data given the proximity of the sensor with respect to the concrete surface.

Fig. 6

Transducer Orientation over Reinforced Concrete Sample

Note that the samples were only exposed to 10% NaCl while the ultrasonic test took place, while the samples were under lab humidity environment the rest of the time. A 3.5% or 5% NaCl concentration could be used for the ultrasonic testing. Samples 1, 2 and 3 (shown in Fig. 4) were exposed outdoors prior to the ultrasonic testing and the samples remained in the laboratory since 2021. The 10% NaCl concentration was chosen due to the initial scope to initiate corrosion via electromigration on sample 16X. Concentrations as high as 16.5% are used to measure chloride transport properties (e.g. NT492 [48]). The 10% NaCl was chosen to have a high concentration gradient. The chloride concentration was later changed to 5% during the monitoring/propagation stage.

Next, the transducer head was oriented above the concrete surface at a distance of 12 [cm], centered above the embedded rebar. Once the placement of the transducer was correct, a 7 [µs] pulse centered at 500 [kHz] was transmitted. The returned echoes of the concrete and embedded rebar were then passed to the digital oscilloscope. At each measurement location, a minimum of 10 records were collected to reduce noise both in and out of band. Next, the transducer was moved along the track system 5 [mm] for the initial samples, or 10 [mm] for the current samples, to the next position. At the new location along the rebar, the process of collecting 10 records of the raw response is repeated. This process continues for the entire span of the embedded rebar within each sample.

Table 2 Initial values for biot-stoll physical parameters

Defining biot-stoll parameter inputs

The tabulated initial values for the Biot-Stoll model can be seen in Table 2. As previously stated, the adapted version of this model includes three categories of physical properties: saltwater, concrete, and rebar. The saltwater layer replaces the fluid properties, to better represent our environment. In addition, the frame and grain properties are combined to model the concrete properties. Lastly, an additional set of physical parameters is defined for the embedded rebar layer. The values were taken from literature and represent common values for each of the given physical properties. The exception to this is the depth below concrete surface, which was measured in lab to be the concrete cover thickness.

Results

To verify proper time alignment was met, along with the matching of the concrete and rebar echoes in both the measured and simulated curve, the presented model was studied to determine if the model produced accurate results (as discussed in subsequent sections). Early evaluation of the presented model focused on the bounds of the Biot-Stoll physical parameters, and how the optimized solution value fluctuated across measurement location, as well as bound ranges.

Following this, the model was reviewed to improve the overall curve fitting between the measured and simulated response. In doing this, aspects of the signal characteristics such as bandwidth and oversampling rate were optimized to improve the shaping of the simulated curve. In addition, the parameter bounds were adjusted to minimize the error between measured and simulated echoes.

Following these changes to the model characteristics, the root mean square error (RMSE) was computed comparing the original and updated model conditions, to determine if the curve fitting of the model was improved. Completing this evaluation, it was shown that the RMSE for the full signal was reduced up to 63.7%, with the largest improvement of the rebar echo being a 62.6% reduction in the RMSE.

Echo location

Using the 16x sample shown in Fig. 5, the measured and simulated echoes were studied to verify if the time alignments matched. In addition, this time alignment was compared to a calculated theoretical location. To obtain the theoretical location, the thickness of the concrete cover, 0.0375 [m], was multiplied by 2 to account for the 2-way travel of the signal. Next, this result was divided by the speed of sound in undamaged concrete, 4500 [m/s]. This computation yielded 1.66e-5 [s] which was then added to the theoretical location of the concrete echo, 16e-5 [s]. Figure 7 provides an overlay of the measured and simulated echoes for the 16x sample at the 240 [mm] location.

Fig. 7

Sample 16X at 240 [mm] Simulated and Raw Response with Labeled Rebar Echo Starting Location

In Fig. 7, the dashed vertical line corresponds to the theoretical start of the rebar echo, while the circled region defines the region of the rebar echo. This result signifies that not only do the echoes align properly in the measured signal, but also in what the Biot-Stoll model produces. This result verifies the approximate solution, showing that the Biot model produces echoes in the correct location.

At this stage, a discussion regarding acoustic scattering is needed. While scattering is indeed present between 200 and 700 [kHz] (the operating band centered at 500 [kHz] of the transducer) as sound travels through concrete, the elastic response of the material (truly the poro-elastic response of the material in this case) and the resulting specular reflection at the concrete-steel interface remains very dominant.

The characteristic impedance of a steel rebar is approximately 40e6 [Rayleigh or kg/m²s] [49]. In comparison, the characteristic impedance of concrete is of the order of 8e6 [Rayleigh], that of quartz (which would constitute an extreme case) is 10e6 [Rayleigh], and that of cement is about 9e6 [Rayleigh] [49]. Therefore, the difference in characteristic impedance between concrete and steel is around 32e6 [Rayleigh], vs. 1e6 [Rayleigh] of difference in characteristic impedance between cement and quartz. This is a factor of 32, which will translate into far stronger echoes at the concrete-rebar interface.

In addition, The Panametrics V389 unfocused immersion transducer has a beamwidth of 14 [deg] at -6 [dB] when operated at 500 [kHz], so that the scattering cross-section of the transducer is very small at the short range of operation (approximately 12 [cm]) [23]. As a result, most of the scattered sound will not be measured as backscattering but will be dissipated in the material. The acoustic signatures shown in Fig. 7 and in other published presentations [37, 50, 51] clearly indicate that scattering is very small indeed in comparison to the specular response at the concrete-steel interface. As a result, acoustic scattering is considered as a secondary cause of signal distortion in this paper.

Comparative analysis of initial acoustic measurements

Using the original samples shown in Fig. 4, the optimized parameter values were studied across a parameter bound variation from 3.1 to 15.1% using 0.1% increments. At this time, 5 Biot-Stoll parameters were being used in the TRR algorithm. The parameters used were as follows: porosity, concrete bulk modulus, compressional log decrement, shear log decrement, and water concrete interface depth (concrete cover thickness).

Based on the data shown in Table 3, it is observed that no significant changes to the parameter values occurred during the expansion of the parameter bounds from 3.1 to 15.1%. This result is inclusive of the concrete bulk modulus, which held a high magnitude for standard deviation. Despite this large magnitude, this value only differed from the initial guess by 2.8%, meaning that changes to this parameter were still relatively low.

Following this, the water concrete interface depth, depth of rebar below the water, and concrete rebar interface depth were studied across each measurement location and measurement date to identify trends in the data. Figure 8 shows the result of this analysis for sample 1, rebar 1, as in Fig. 4. Following the review of these listed parameters across each measurement date showed that on average, the concrete cover thickness increased as more corrosion took place on the sample.

Fig. 8

Signed Transducer to Rebar Interface Depth, Signed Transducer to Concrete Interface Depth and Signed Concrete Cover for Sample 1 Rebar 1 from October 2021 to June 2022 Measurement

Table 3 Tabulated Parameter Values for a Bound Variation of 3.1 to 15.1%

This may be the result of the concrete being expanded away from the rebar during corrosion or rebar cross-section loss, which would suggest an identifiable corrosive trend in the model data. The tabulated results corresponding to Fig. 8 are provided in Table 4.

An initial observation of Fig. 8 shows that the depth of rebar undergoing corrosion seems to only exhibit non-monotonic variation, making it difficult to assess the corrosion condition. However, through statistical averaging along the corroding rebar within three samples over a period of nearly nine months (from October 2021 to June 2022), a small but monotonous increase in the distance between the concrete surface and the top of the rebar is observed (column 3, rows 1 to 3 in Table 4), which indicates a gradual corrosion of the rebar. Clearly, additional experimentation is underway to further study such increase over a longer period of time and will lead to a following publication.

Analysis of experimental data collected from new samples

To improve the matching between simulated and measured responses, signal characteristics were optimized based on sample. For the MD1 sample shown in Fig. 5, the bandwidth was reduced from the initial band of 100–900 [kHz] to 200–700 [kHz]. In addition, the oversampling rate was increased from 400 to 500. During these changes to signal characteristics, it was found that changes to bandwidth improved curve matching of the concrete echo side lobes, while changes to the oversampling rate improved the curve matching following the concrete echo. An overlay of the curve matching for simulated curves (both old and new conditions) with the measured signal can be seen in Fig. 9. Looking at Fig. 9, the phasing error of the concrete echo is reduced in the side lobes of the concrete echo. In addition, the matching of the rebar echo is significantly improved as seen in the windowed view.

Fig. 9

MD1 280 [mm] Position Overlay of Measured Response with Simulated Response under Initial and Current Signal Characteristics

Table 4 Sample 1 rebar 1 parameter value statistical results from october 2021 to june 2022 measurements

Though improvements were observed through these changes, it is important to note the MD1 sample, which is mortar, was likely not undergoing a corrosive process. With this in mind, it is possible that observed improvements may be associated with the homogeneous layer of the mortar. With a more homogeneous layer (mortar) the raw signal produces cleaner echoes than that of a non-homogeneous material such as concrete.

Analysis of parameter bounds for new samples

In addition to changes to the bandwidth and oversampling rate, both the upper and lower bounds of the Biot-Stoll parameters were optimized. To do so, the first order optimality (FOO) was selected as the metric defining model improvement. The reasoning behind selecting the FOO to monitor model improvement, is that this value is updated following each iteration of the TRR algorithm and signifies how optimal the solution is. Within the TRR algorithm, the FOO is calculated by taking the absolute value of the current point of a Lagrangian gradient, minus the bound value of a Biot parameter.

Using this difference, it is written that the closer the FOO is to zero, the more optimal the solution has become [47]. Using this method, Fig. 10 shows the FOO for the MD1 sample at each measurement location under initial and current parameter bounds. Trends within Fig. 10 signify that on average, the FOO was reduced across measurement location. The result of the changes to parameter bounds are significant, as the result can be used to infer how optimal the solution matrix of Biot-Stoll parameters is following the simulation.

A visual representation of different FOO values can be seen in Fig. 11. Looking at Fig. 11, the curve labeled as initial guess is correspondent to a simulated curve with a large FOO. In contrast, the curve labeled new fitted produces a low FOO. Comparing both simulated curves, curve matching is improved following a reduction of FOO through the implementation of the TRR algorithm.

Fig. 10

MD1 First Order Optimality at Each Measurement Location for Original and Updated Bounds

Fig. 11

MD1 Simulated and Measured Rebar Echo Before (Initial Guess) and After (New Fitted) TRR Algorithm

Root mean square error results

To quantify the overall improvements to the curve matching between simulated and measured responses under initial and current conditions, the root mean square error (RMSE) was computed. This metric was selected as it provides the error between the simulated and measured responses. When analyzing the results of the MD1 sample (Fig. 9), the RMSE showed a reduction up to 63.7% for the full signal, and 62.6% for the rebar echo. Though not every measurement location produced improvements between new and current model conditions. Most locations showed improvement to model fit. In addition, in locations from which improvements were not observed, the RMSE difference was small, being only 0.53% of a difference for the full signal. The tabulated RMSE values for the MD1 sample when comparing the full measured signal to the full simulated signal can be seen in Table 5.

Table 5 MD1 Sample RMSE Table for Full Signal

Conclusions and future work

Ultrasonic testing of provided reinforced concrete samples is being conducted periodically to establish trends present in each sample over time. In addition, through optimizing signal characteristics and parameter bounds, the current version of the presented model produces a series of Biot-Stoll parameters with increased accuracy to the output solution. For the full signal of the MD1 sample, these improvements were seen as a 63.7% reduction in RMSE, with the rebar echo having a 62.6% reduction in RMSE. Prior to correlation analysis of the model data in tandem with the electrochemical data, the simulated signal still needs further optimization. A side effect of the TRR algorithm in the current model is an emphasis of the concrete echo when optimizing the solution. The result is a very strong fit between measured and simulated curves at the main concrete echo, however this weakens the fit of these curves at the rebar echo. Current research of this model seeks to address this problem through reducing the impact of the concrete echo on the TRR algorithm. Through this, an auxiliary objective would be to evenly distribute the TRR matching across each of the presented echoes.