Applying Nondestructive Ultrasonic Technique in the Metrological Control of Heat Treatment of AISI 1045 Steels

Abstract

The characterization of mechanical properties in heat-treated carbon steels, which is crucial for quality control, traditionally relies on destructive testing. This study evaluated the reliability of the non-destructive ultrasonic technique as a metrological alternative for AISI 1045 steel. Samples subjected to six heat treatment conditions (Annealing, Normalizing, Quenching, and Tempering) were characterized by hardness, metallography, and ultrasound. Through linear regression analyses, the multiparametric model combining sound velocity, attenuation, and FWHM demonstrated exceptional metrological precision, resulting in a coefficient of determination of (R² = 96.687%). The metrological robustness of the model was validated by quantifying the Expanded Uncertainty (U), following the GUM (Guide to the Expression of Uncertainty in Measurement). It is concluded that the multiparametric ultrasonic methodology is an accurate, robust, and non-destructive alternative for the quantitative determination of Vickers Hardness in AISI 1045 steels, contributing to the optimization of industrial processes and metrological rigor.

1. Introduction

The quality and reliability of engineering materials are critical factors in a wide range of industrial applications, especially for the automotive, civil construction, energy, and aerospace sectors [1]. Medium carbon steels, such as AISI 1045, are widely used in industry due to their balance between mechanical strength and cost-effectiveness. To ensure their safety and structural performance, quality control following heat treatments—which critically modify the microstructure and hardness—is indispensable. Traditionally, this metrological control requires the use of destructive testing (such as large-scale Vickers hardness testing and metallography), which is costly and unsuitable for 100% inspection of production or in-service components [2,3,4,5].

In this scenario, Non-Destructive Testing (NDT) methods, particularly Ultrasonic Testing (UT), have emerged as the preferred solution due to their high sensitivity and in situ inspection capability [1]. The relevance of this field is growing, with the recent literature validating the use of advanced ultrasonic techniques in industrial safety and quality control inspections [6,7]. Current research shows a trend towards more rigorous and comprehensive methods, such as the use of ultrasonic scattering techniques for microstructural anisotropy characterization and the integration of UT with image processing to optimize alloy characterization [7,8,9,10]. The critical relevance of this work lies in the need to establish quantitative metrological traceability between mechanical properties (hardness) and ultrasonic parameters in AISI 1045 steels, as the existing literature often focuses on qualitative correlations or less common alloys. Our goal is to transform ultrasonic testing into a measurement tool with known uncertainty, rather than just a flaw detection tool.

The physical principle underlying this application is that linear ultrasonic parameters (velocity and attenuation) are measurably influenced by the material’s microstructural changes and elastic constants [9,10]. Heat treatment modifies the steel’s microstructure (formation of pearlite, ferrite, martensite), which generates variations in density and crystalline defects that, in turn, measurably alter the velocity and attenuation of the US wave [11].

Several studies point to the cross-use of non-destructive parameters as a performance criterion in the characterization and metrological control of mechanical components [12,13,14,15,16,17,18,19,20].

Carvajal et al. [13] demonstrated the capacity of the ultrasonic technique to evaluate different heat treatments performed on carbon and low-alloy steel samples, monitoring the microstructural modifications associated with these processes based on the correlation between ultrasonic parameters and the material’s elastic constants.

Freitas et al. [14] evidenced the potential of ultrasonic testing to characterize different types of microstructures in heat-treated carbon steels, based on indirect measurements of ultrasonic velocities and attenuations using the pulse-echo technique.

Lin et al. [15] evaluated the effect of microstructural variations on ultrasonic parameters. For this, they used thermally aged stainless-steel samples, evaluating the influence of the microstructure on sonic velocity and attenuation.

Neves et al. [18] correlated the variations in ultrasonic parameters with the evolution of the corrosive process in heat-treated aluminium samples. The results showed the importance of multiparametric analysis for material characterization and selection in inspection and integrity analysis activities of engineering components.

The literature shows that ultrasonic parameters such as propagation velocity and attenuation are sensitive to microstructural variations such as grain size, phase presence (e.g., ferrite, pearlite, martensite), and dislocation density induced by heating and cooling. This sensitivity positions ultrasound as a promising resource for the quantitative characterization of the material’s properties [12,13,14,15,16,17,18,19,20]. Despite advances in NDT, the literature demonstrates the effectiveness of approaches that combine multiple ultrasonic parameters to overcome the limitations of each individual parameter. Recent studies have validated the application of multiparametric methodologies for structural characterization and hardness evaluation in heat-treated alloys [21,22,23,24,25,26,27,28,29,30].

The present work addresses this metrological gap by proposing a rigorous method for the metrological control of hardness in AISI 1045 steels subjected to different heat treatments. The article demonstrates the development of a multiple linear regression model that combines the ultrasonic velocity, attenuation, and full width at half maximum (FWHM), achieving a coefficient of determination (R²) of 96.89%. This multiparametric methodology proves to be highly accurate and effective, offering a non-destructive and quantitative metrological alternative for industrial quality control [31,32,33].

It is important to contextualize these findings within the broader field of ultrasonic testing. While conventional linear ultrasonic parameters (velocity and attenuation) have shown sensitivity to large microstructural changes, nonlinear ultrasonic techniques (NLT), such as second harmonic generation, have been demonstrated to offer significantly higher sensitivity to subtle microstructural features like dislocation density, precipitates, and early-stage damage in metallic alloys. Research confirms that NLT methods are crucial for characterizing critical phenomena such as thermal embrittlement and sensitization (carbide precipitation)—microstructural changes that linear methods struggle to detect [34,35,36,37,38,39]. The NLT approach is a powerful tool for academic research due to its capacity to decouple microstructural effects from elastic effects. However, this work focuses specifically on linear ultrasonic techniques which, despite their inherent sensitivity limitations compared to NLT, remain the standard for high-throughput, low-cost industrial quality control. Therefore, the methodological challenge here is to enhance the performance of the conventional linear method by combining multiple parameters and applying rigorous metrological processing.

The article is organized as follows: Section 2 details the materials and the different heat treatments used on the AISI 1045 samples and the ultrasonic testing methodology and the metrological procedure for data acquisition and processing. Section 3 provides the results and correlation analysis. Section 4 discusses the implications of the metrological model and Section 5 presents the conclusions of the work.

2. Materials and Methods

2.1. Sample Preparation and Chemical Characterization

This research investigated the applicability of the non-destructive ultrasonic technique for the metrological quality control of heat-treated steels. For this purpose, cylindrical samples of AISI 1045 steel (from Gerdau, Aracaju, Brazil) were used, with a diameter of 40 ± 1 mm and a height of 20 ± 0.5 mm (Figure 1).

First, the material was identified through chemical analysis by optical emission spectroscopy using a Foundry Master Xpert equipment from Oxford Instruments (Oxford, UK). The following composition was identified: AISI 1045 = 0.46 C, 98.4 Fe, 0.60 Mn, P and S < 0.02 [% weight].

Based on this composition, the Ac₃ temperature was calculated using Andrews’ equation (Equation (1)) [2,3,5,11].

$$\begin{gathered} \mathbf{A}{\mathbf{c}}_{\mathbf{3}}\left(°\mathbf{C}\right)=\mathbf{910}-\mathbf{203}\sqrt{\mathbf{\%}\mathbf{C}}-\mathbf{15.2}\left(\mathbf{\%}\mathbf{N}\mathbf{i}\right)+\mathbf{44.7}\left(\mathbf{\%}\mathbf{S}\mathbf{i}\right)+\mathbf{104}\left(\mathbf{\%}\mathbf{V}\right)+\mathbf{31.5}\left(\mathbf{\%}\mathbf{M}\mathbf{o}\right)+\mathbf{13.1}\left(\mathbf{\%}\mathbf{W}\right) \\ {\mathbf{A}\mathbf{c}}_{\mathbf{3}}\left(°\mathbf{C}\right)=\mathbf{910}-\mathbf{203}\sqrt{\mathbf{0.46}}=\mathbf{910}-\mathbf{137.68}=\mathbf{772.32} \end{gathered}$$

(1)

The Ac₃ temperature is the point where the transformation of ferrite into austenite is complete upon heating. This value is crucial for determining the correct austenitizing temperature for heat treatment processes like hardening and normalizing [3,5,10].

For proper hardening, the literature indicates to use temperature typically 20–40 °C above its Ac₃ (to ensure the fully austenitic transformation). For AISI 1045, the recommended hardening temperature is typically in the range of 820–850 °C [40,41,42].

2.2. Heat Treatment Cycles

The samples were subjected to six heat treatments conditions, divided into two sets:

(1)

Set 1: Annealing; Normalizing; Quenching.

(2)

Set 2: Quenching + Tempering at three distinct temperatures (300, 450 and 550 °C).

All samples were initially heated to a temperature of 850 °C for 45 min. Cooling was performed as follows: Quenched samples were cooled in an ice bath; normalized samples were cooled in air; and annealed samples were cooled inside the furnace. For the second set, quenched samples were reheated for 30 min at their respective tempering temperatures (300, 450 and 550 °C), followed by air cooling.

Figure 2 presents a graphical representation of the thermal cycles used in this research.

The heat treatments were applied with the intention of producing microstructures with distinct characteristics (according to Appendix A).

2.3. Characterization Methodology

The heat-treated samples were characterized through tests of Ultrasonic NDT, Vickers Hardness, and Microstructural Analysis. All tests were performed in triplicates (3 samples per heat treatment) and 10 points per analysis (180 hardness and ultrasound measurements).

The ultrasonic analyses were performed on a portable equipment from SIUI model CTS 59 (from Shantou, China), with a dual crystal transducer with a diameter of 10 mm and a nominal frequency of 5 MHz. The analysis was conducted in Pulse-Echo mode, determining the Velocity (SV), Attenuation (ATN) and Full Width at Half Maximum for the signal in the frequency domain (FWHM) based on the principles outlined in the Ultrasonic nondestructive testing of materials handbook [10]. More details in Appendix B.

2.3.1. Data Acquisition and Metrological Processing

The measurement procedure was designed to ensure high metrological rigor:

• Surface Preparation and Couplant Mitigation: To minimize the influence of the coupling agent (couplant) and ensure maximum acoustic energy transmission, the surface of each sample was prepared to a low roughness through sanding and final polishing.
• Measurement Points: The analysis consisted of 10 distinct measurement points distributed across the cross-section of each sample.
• Time of Flight (ToF) and Velocity Determination: The Time of Flight (t) was determined by the time interval registered between the first and second back wall echoes (interval between A₁ and A₂ in the A-Scan graph). The Ultrasonic Velocity (SV) was calculated using the conventional formula for the Pulse-Echo method (SV = 2 h/t, where h is the sample height).
• Attenuation and FWHM: The sound attenuation (ATN) was calculated based on the decrease in amplitude between consecutive backwall echoes. Signals were analyzed in the frequency domain (RF mode) to obtain the Full Width at Half Maximum (FWHM) of the pulse spectrum, a parameter particularly sensitive to microstructural changes and signal scattering.
• Signal Processing and ToF Determination: The ToF determination utilized the First Zero Crossing (FZC) method of the radio frequency (RF) pulse to ensure accuracy in calculating the time of arrival (ToA) of the A₁ and A₂ echoes. This method is robust against amplitude and noise variations. The 5 MHz radio frequency signal was digitized with a sampling rate of 100 MHz. The ultrasonic pulse employed had a nominal duration of 1.5 cycles, ensuring good temporal resolution for the accurate ToF calculation, which is crucial for velocity determination. The choice of short pulses (1.5 cycles) was a metrological compromise to optimize axial resolution at the expense of penetration, ensuring clear separation of echoes in the 20 mm sample [10].
• Frequency Domain Analysis (FWHM): The temporal signal of the second backwall echo (A₂) was isolated and subjected to Fast Fourier Transform (FFT) to obtain the frequency spectrum. The FWHM was calculated as the width of the spectrum corresponding to half the maximum peak amplitude, serving as an indicator of microstructural signal dispersion [10].

2.3.2. Physical and Metrological Justification

• Transducer Justification (Far-Field) and Frequency: The choice of 5 MHz frequency was a metrological compromise to achieve a balance between high sensitivity (to detect microstructural variations) and sufficient penetration depth. The decision to use the 10 mm transducer was based on its focal relationship. The Far-Field zone (N) for a 10 mm transducer at 5 MHz in steel (with SV ≅ 5900 m/s) is calculated as N = D²f/(4SV) ≅ 2.1 mm. Given that the sample thickness is 20 mm, all measurements were carried out well within the Far-Field zone, as required by the conventional Pulse-Echo method. This geometry ensures that the wave propagates as a plane wave, metrologically validating the velocity measurement (SV = 2 h/t) and attenuation. While measurements with smaller diameters or different pulse durations are of scientific interest, they were not performed as the current configuration meets the fundamental metrological requirements for characterizing the sample volume. The theoretical basis for the correlation is that ultrasonic velocity (SV) is a function of the material’s elastic constants and density. The microstructural changes induced by heat treatment modify these constants, establishing the physical correlation between the US parameters and Vickers Hardness (HV) [1,10,30,43].
• Velocity Uncertainty: The apparent discrepancy between the dimensional tolerance of the sample height (20 ± 0.5 mm, suggesting an uncertainty of 2.5% in the dimension h) and the dispersion reported in

  Table 1. Experimental data presentation.

  Properties/ Heat Treatment		Hardness [HV]	Sound Velocity [m/s]	Sound Atennuation [dB]	Widht at Half [MHz]
  		H	SV	ATN	FWHM
  Annealing	A	170 ± 10	5920.65 ± 4.777	0.148 ± 0.005	0.91 ± 0.073
  Normalizing	N	210 ± 10	5911.96 ± 3.706	0.155 ± 0.004	1.13 ± 0.090
  Quenching	Q	660 ± 30	5879.93 ± 3.933	0.175 ± 0.005	0.79 ± 0.063
  Tempering 300 °C	T300	375 ± 20	589476 ± 2.768	0.168 ± 0.004	1.07 ± 0.086
  Tempering 450 °C	T450	355 ± 15	5905.42 ± 1.215	0.165 ± 0.005	1.08 ± 0.086
  Tempering 550 °C	T550	340 ± 15	5913.323 ± 5.145	0.160 ± 0.006	1.10 ± 0.088

  (approximately 0.1% Type A uncertainty) is explained by the principle of uncertainty propagation (GUM). The velocity calculation is given by SV = 2 h/t. The ultrasonic equipment has a high time-of-flight resolution (Δt) of 0.01 ≅ µs. The combined uncertainty (uSV) is dominated by the high precision of the ToF measurement, which proved to be statistically much more accurate than the Type B uncertainty of the dimensional measurement h (uh) [4,10,43,44]. The equipment’s high temporal precision compensates for the sample’s dimensional tolerance, justifying the low dispersion (Type A uncertainty) of the results in Table 1.

2.3.3. Vickers Hardness

Vickers hardness measurements (HV) were performed on a digital micro-hardness tester from TIME model HVS-1000 (from Jinan, Shandong, China). A diamond pyramidal indenter was used, with a penetration time of 10 s and an applied force of 9.8 N.

2.3.4. Microstructural Analysis

Microstructural analyses were performed in two stages. First, metallographic preparation involved sanding and polishing, followed by chemical etching in 2% Nital solution for 10 s. Second, microscopic analysis was conducted using an optical microscope from LEICA model DM2500 (from São Paulo city, São Paulo, Brazil) and a Scanning Electron Microscope from JEOL model JSM-5700 (from Peabody, MA, USA) (10 kV voltage, up to 3000× magnification).

Statistical regression analyses were performed, seeking to verify the existence of a correlation between the ultrasonic parameters and the variations in material hardness [25,41].

3. Results

3.1. Microstructural Analysis

The heat treatments (Annealing, Normalizing, Quenching, and Tempering) were successful in producing a spectrum of distinct microstructures in the AISI 1045 steel samples. The as-quenched condition (Q) resulted in a predominantly martensitic microstructure. The annealed condition (A) resulted in a ferrite/pearlite microstructure [2,3,11].

Figure 3 and Figure 4 present the microstructures obtained for the samples subjected to the first and second heat treatment cycles. The variation in tempering temperatures (T300, T450, T550) also induced the expected microstructural evolutions, with the progressive reduction in martensite brittleness and the precipitation of carbides [11]. The correlation between heat treatment and microstructure is fundamental for the performance of AISI 1045 steel in various applications, such as the determination of mechanical properties and the investigation of thermomechanical phenomena in machining [3,11,40,41,42].

3.2. Vickers Hardness

The Vickers Hardness results are presented in Table 1, serving as the destructive metrological reference for the present study. The maximum hardness was observed in the Quenched condition (Q) (660 ± 30 HV), and the minimum in the Annealed condition (A) (170 ± 10 HV). The test successfully recorded the variation between the microstructures (Annealing, Normalizing, and Quenching) and the gradual reduction in the values obtained for the tempered samples (T300, T450 and T550).

3.3. Ultrasound and Metrological Model

The ultrasonic parameters Sound Velocity (SV), Sound Attenuation (ATN), and Full Width at Half Maximum (FWHM) were measured for all heat treatment conditions, as detailed in Table 1.

Metrological Clarification: The dispersion values presented in Table 1 (e.g., 170 ± 10 HV; 5920.65 ± 4.777 m/s) correspond to the Standard Deviation (SD) of the measurements (n = 10 for US and n = 30 for Hardness) taken across the three samples of each condition. These values represent the Type A Standard Uncertainty (statistical) of the collected data [25]. The final Expanded Uncertainty (U) of the model is addressed in Section 4.3.

Addressing Velocity Uncertainty: The high precision in the Sound Velocity (SV) measurement, with a dispersion of approximate 0.1% (Type A uncertainty), is justified by the rigor of the metrological procedure:

(1)

The uncertainty of the sample height (20 ± 0.5 mm) is combined with the high precision of the Time of Flight (ToF) measurement [43].

(2)

The final uncertainty is calculated using the Propagation of Uncertainty method (GUM), which combines the uncertainties of all variables (Δh and Δt), demonstrating that the precision of the US equipment’s time resolution outweighs the dimensional tolerance [43].

Model Acquisition: Based on the collected ultrasonic parameters and the Vickers hardness values, Multiple Linear Regression analyses were performed to evaluate the correlation between the properties, considering different combinations of ultrasonic parameters. The statistical analysis of the data’s normality was confirmed by the Shapiro–Wilk test, validating the use of linear regression [25,43]. The analysis utilized a total dataset derived from 180 Vickers Hardness measurements (30 points × 6 conditions) and 60 Ultrasonic measurements (10 points × 6 conditions). The results are presented in

Table 2. Ultrasonic data linear regression evaluation.

Independent Variables	R2	R2 Adjusted	Standard Error	Equation
SV	86.914	83.642	69,778	H1 = 64,559.42 − 10.8747 × SV
ATN	86.444	83.055	71.021	H2 = 16,672.67 × ATN − 2346.53
FWHM	32.845	16.056	158.073	H3 = 1101.269 − 739.739 × FWHM
SV; ATN	88.206	82.010	73.177	H4 = 33,556.859 − 5.848 × SV + 8190.1985 × ATN
SV; FWHM	88.388	80.647	75.900	H5 = 59,888.089 − 10.0524 × SV − 181.2105 × FWHM
SV; ATN; FWHM	96.687	91.717	49.654	H6 = 3.6483 × SV + 19,824.7952 × ATN − 519.3172 × FWHM – 23,871.094

, and the regression plots are shown in Figure 5.

The analyses were conducted considering univariate (Models 1, 2, and 3), bivariate (Models 4 and 5) and multiparametric (Model 6) approaches. Model 6 (Equation (2)), which combined SV, ATN and FWHM (Table 2), resulted in a coefficient of determination (R²) of 96.687% [43].

The Figure 5 presents the scatter plots obtained from the linear regression analysis, comparing the destructively measured hardness values (metrological reference) with the hardness values predicted by the non-destructive ultrasonic (NDT) models.

Subfigure Figure 5a shows the dispersion of all calculated hardness data relative to the measured hardness. This graph includes the performance of all tested regression models (H1 to H6), visually demonstrating the correlation between the properties and the models’ prediction capability. The concentration of data points close to the ideal correlation line indicates the precision achieved by the multiparametric model H6.

Subfigure Figure 5b is exclusively dedicated to the metrological validation of Model 6 (H6), which demonstrated superiority with the highest Coefficient of Determination (R²) of 96.687%. The H6 plot is presented with the error bars, which represent the Expanded Uncertainty (U). This uncertainty was calculated as U = ±99.31 HV (using a coverage factor of k = 2), following the principles of the GUM (Guide to the Expression of Uncertainty in Measurement). The inclusion of error bars quantifies the metrological robustness and predictive power of Model H6, confirming its acceptability for the quantitative classification of different heat treatment conditions. This result validates the multiparametric methodology for the non-destructive determination of Vickers Hardness.

$${\mathrm{H}}_{\mathrm{N}\mathrm{D}\mathrm{T}}=3.6483 \times \mathrm{S}\mathrm{V}+19824.7952 \times \mathrm{A}\mathrm{T}\mathrm{N}-519.3172 \times \mathrm{F}\mathrm{W}\mathrm{H}\mathrm{M}-23871.094$$

(2)

4. Discussion

4.1. Validation of Destructive Reference and Microstructure

The Vickers Hardness results (Section 3.2), which served as the destructive metrological reference for this study, confirmed the expected and well-established trends in materials science [4,10]. The maximum hardness in the Quenched (Q) condition is a direct reflection of the formation of the highly stressed, metastable martensitic phase, while the minimum hardness in the Annealed (A) condition is characteristic of the equilibrium ferrite/pearlite microstructure. The gradual reduction in hardness in the tempered samples (T300, T450, T550) is a direct result of martensite decomposition and the controlled precipitation of carbides, a primary mechanism controlling the evolution of strength and hardness in steels [3,41,42]. The importance of precise measurement of these properties is crucial, as they are intrinsically linked to the material’s application and final performance [43]. It is acknowledged that, as elastic waves, wave velocity (SV) is primarily governed by elastic moduli and is generally insensitive to fine microstructural features associated with hardness or plasticity, and attenuation (ATN) is expected to exhibit greater sensitivity at frequencies above 10 MHz [36,37,38,39]. However, the measurable correlations obtained here are justified by the scale of the microstructural variations, as discussed below.

4.2. Ultrasonic Sensitivity and Microstructural Complexity

The ultrasonic parameters (Sound Velocity, Attenuation, and Full Width at Half Maximum) proved to be sensitive to the microstructural variations induced by the heat treatments, a finding consistent with other studies on heat-treated steels [9,13,15].

Velocity (SV): The correlation between SV and HV is physically justified because the ultrasonic velocity is a function of the material’s elastic constants and density. The microstructural changes induced by the heat treatments directly alter these constants, establishing a robust physical mechanism for the SV measurement to track the hardening process. The formation of martensite and the accompanying high dislocation density and internal residual stresses measurably modify the elastic constants, thereby justifying the SV’s observed correlation with the macroscopically determined hardness [1,10,13,18,21]. Its success in monitoring the tempering process, in particular, corroborates studies that use ultrasound to characterize the properties of heat-treated steels, such as AISI 4340 [16].

Attenuation (ATN): The highest ATN in the Quenched (Q) sample is justified by the high density of crystalline defects and internal stresses in the martensite, which act as powerful scatterers of ultrasonic wave energy. Despite the low frequency of 5 MHz, the observed attenuation is primarily attributed to scattering and absorption caused by the significant micro-heterogeneity and the high density of defects resulting from the non-equilibrium phase transformations (martensite) and is not solely dependent on grain-size effects (which dominate at higher frequencies). It is recognized that attenuation sensitivity increases with frequency (above 10 MHz), but studies show that linear attenuation, even at lower frequencies, can effectively track major microstructural changes such as static recrystallization and grain growth in superalloys [37]. This supports the validity of using ATN at 5 MHz, in conjunction with other parameters, to track the large microstructural changes present in AISI 1045. The literature confirms that the heat treatment generates defects and alters the grain structure, which increases the ultrasonic scattering and the measured attenuation [7,8,9,11,13,16,22]. This confirms the direct correlation between the hardness/defect density and the attenuation, a sensitivity that is fundamental for the multiparametric model. The amplitude of the ultrasonic signal (related to attenuation) is sensitive to changes in mechanical properties, such as hardness and internal stresses. Studies such as that by Cao et al. [7] on magnetostrictive magnetoacoustic conversion reveal that mechanical properties (including hardness) are critical parameters that directly affect the structural reliability of ferromagnetic materials.

Comparison with Previous Studies: The small variation in velocity and attenuation observed across the different heat treatment conditions is consistent with results reported in previous studies for similar carbon and low-alloy steels [7,8,9,13,14,15,16,17,18,19,20,21,22,23,24]. This consistency validates the sensitivity of the multiparametric approach to detect both macroscopic (Annealed vs. Quenched) and subtle alterations (Tempering).

4.3. Superiority of the Multiparametric Model and Advanced Metrology

The main finding of this study lies in the decisive superiority of the multiparametric approach. Although univariate and bivariate models showed significant correlations, the optimized combination of SV, ATN, and FWHM in Model 6 resulted in a Coefficient of Determination (R²) of 96.687% (Table 2).

Advanced Metrology and Validation: This superior performance is crucial and is aligned with the principles of advanced NDE metrology. The model’s robustness is ensured by rigorous metrological control:

Uncertainty Quantification: The result is presented with Expanded Uncertainty (U), calculated according to the principles of the Guide to the Expression of Uncertainty in Measurement (GUM) [25]. The final Expanded Uncertainty for the Model 6 prediction was calculated as U = ±99.31 HV (using a coverage factor of k = 2). The Error Bars presented in Figure 5b represent this value, providing a complete metrological picture of the model’s predictive power.

Acceptability of Uncertainty: Despite the absolute margin of ±99.31 HV, this value is fully acceptable and robust for the metrological objective of the study. Given the wide range of hardness (from 170 to 660 HV), this uncertainty is highly effective for distinguishing and quantitatively classifying the different heat treatment conditions (e.g., distinguishing between Annealed and Quenched samples), which is the primary requirement for industrial quality control. The high R² confirms that the model is fundamentally robust for the characterization of mechanical properties [43].

Multiparametric Rationale: The combination of multiple parameters allows the model to effectively decouple and quantify the complex microstructural changes (phase variations, internal stresses, grain size) more effectively than isolated parameters. This approach is validated in recent NDT literature for structural characterization of materials with complex microstructures [13,18,19,20,21,22,23,24].

Optimization and Precision: Ultrasonic precision is strongly influenced by the optimization of testing and signal processing parameters [26,27,28]. The principle that “algorithm performance is better when the number of features is higher” [26] justifies the choice of our multiparametric model, which uses FWHM (extracted from the frequency domain) to capture signal characteristics that SV and ATN in the time domain cannot isolate.

Statistical Validation: The high correlation obtained, along with the use of statistical significance tests (p-value) in regression models, fulfills the same metrological function as ANOVA in similar studies, statistically validating that the precision of our multiparametric method is robust and reliable, overcoming the uncertainties inherent in non-destructive metrology [4,10,25,29].

4.4. Implications and Scientific Contribution

Our work contributes to the field of non-destructive metrology by establishing a quantitative model that can be applied to finished components, where destructive testing is unfeasible. The success of Model 6 in AISI 1045 steels, a widely used structural alloy, reinforces that ultrasound is a reliable tool and applicable to a wide range of steel microstructures [30]. The high metrological precision achieved (R² = 96.687%) establishes a new standard for quality control and the optimization of heat treatment processes, aligning with the concepts of Nondestructive Evaluation 4.0 (NDE 4.0) [31,32,33]. While the current scope focused on metrological validation under a standard industrial configuration that demonstrated compliance with the Far-Field criteria, future investigations should focus on expanding the model’s generality by exploring the influence of parameters such as frequency dependence, sample height, and the use of smaller transducer diameters. The study of these parameters will be addressed in subsequent work.

Figure 6 presents a comparison of the regression curves obtained in this analysis.

5. Conclusions

The evaluation of the potential of ultrasonic testing for the metrological control of heat-treated steels was successfully conducted, resulting in a robust quantitative model for the non-destructive characterization of mechanical properties.

1. 

Microstructural Sensitivity: Ultrasonic parameters are sensitive to microstructural variations induced by heat treatments (annealing, normalizing, quenching, and tempering). It is recognized that the frequency of 5 MHz represents a limitation in sensitivity compared to higher frequencies (>10 MHz) or nonlinear techniques. However, the sensitivity achieved is justified by the measurable changes in elastic constants (SV) and the strong scattering mechanisms (ATN) resulting from the wide range of microstructures produced. This sensitivity confirms the validity of the linear ultrasound technique, when used multiparametrically, as an effective tool for monitoring the microstructural evolution of carbon steels. Furthermore, the sensitivity of the measured amplitude (attenuation) to properties like hardness and internal stress is consistent with advanced physical models in the literature.

2. 

Multiparametric Superiority and Metrological Precision: The correlation model that utilized the multiparametric approach, combining SV, ATN, and FWHM, demonstrated exceptional metrological precision, achieving a Coefficient of Determination (R²) of 96.89%. This result is significantly superior to the univariate and bivariate models, validating the optimization of parameter combination for property quantification. The model’s robustness was confirmed by the rigorous quantification of the Expanded Uncertainty (U), calculated according to the principles of the Guide to the Expression of Uncertainty in Measurement (GUM), providing a high level of confidence in the model’s predictive capability.

3. 

Metrological Contribution and Future Work: The NDT model establishes a robust and non-destructive alternative for the quantitative determination of Vickers Hardness in AISI 1045 steels, overcoming the limitations of destructive testing. This methodology aligns directly with the principles of Non-Destructive Evaluation 4.0 (NDE 4.0) and meets industry demands for faster, reliable, and scientifically grounded quality control solutions. Given that the metrological validation was performed under a standard industrial configuration (ensuring Far-Field compliance), future work will focus on expanding the general applicability of the model by investigating the influence of parameters such as frequency, sample height, and alternative transducer diameters.