Determination of acoustic nonlinearity parameters using thermal modulation of ultrasonic waves

This study presents a test method and its theoretical framework to determine the acoustic nonlinearity parameters (α, β, δ) of material using thermal modulation of ultrasonic waves. Temperature change induced thermal strain excites the nonlinear response of the material and modulates the ultrasonic wave propagating in it. Experimental results showed a strong correlation between the relative wave velocity change and the temperature change. With a quadratic polynomial model, the acoustic nonlinearity parameters were obtained from the polynomial coefficients by curve fitting the experimental curves. Their effects on thermal-induced velocity change were discussed. The parameters α, β, δ govern the hysteretic gap, average slope, and curvature of the correlation curve, respectively. The proposed theory was validated on aluminum, steel, intact and damaged concrete samples. The obtained nonlinear parameters show reasonable agreements with values reported in the literature. Compared to other nonlinear acoustic methods using vibration or acoustic excitation, the thermal modulation method generates more uniform, slow changing, and larger strain field in the test sample. Employing thermal effect as the driving force for nonlinearity instead of an undesired influencing factor, this method can measure the absolute values of α, β, δ with good accuracy using a simple ultrasonic test setup.

The paper notes that nonlinear acoustic/ultrasonic techniques show high sensitivity to microcracking damage, especially in complex materials. Following the phenomenological description of stress-strain hysteresis of rock by McCall and Guyer, the elastic modulus of a complex material can be modeled as a strain and strain rate dependent variable and described by three nonlinear parameters β, δ and α. The first two parameters represent the classical nonlinear perturbation parameters and α is the non-classical nonlinear parameter, which is introduced as a measure of material hysteresis. Meurer et al. proposed a general form of a nonlinear constitutive model, and they showed that the McCall and Guyer’s model is a special case of this general model when only the leading order effect of α is considered.

Acoustic nonlinearity of material may manifest as softening of modulus with increasing strain or higher harmonic generation. Modulus softening can be measured as the change of resonance frequency or ultrasonic wave velocity at different strain levels. The nonlinear resonance acoustic spectroscopy (NRAS) method measures the resonance frequency shift Δf when the resonance mode in a sample is excited at different strain levels. The linear slope between Δf and the strain change ε is defined as the relative nonlinear parameter α. Dynamic acousto-elastic testing (DAET) method is a recently developed nonlinear ultrasonic method that measures the relative ultrasonic wave velocity change due to a low-frequency strain modulation. Nonlinear parameters β, δ, and αDAET were derived from the correlation curves between the relative velocity change and strain. Higher harmonic generation is a phenomenon wherein the ultrasonic waveform is distorted by the nonlinear response of the material, then higher harmonic waves are generated. Second harmonic generation (SHG) measures the amplitudes of the fundamental and the second harmonics to calculate the nonlinear parameter β. Because the absolute displacement amplitudes are challenging to measure, a relative value of β is typically used for characterizing fatigue of metal materials and concrete damage.

Although these nonlinear ultrasonic testing methods show different degrees of success for extracting nonlinear parameters and characterizing material damage, some limitations hinder the application of these methods to practice. First, the absolute nonlinear parameters are not easy to measure. The NRAS method uses the acceleration instead of the strain to calculate an alternative nonlinear parameter αf. In addition, NRAS is only applicable to small samples since it needs to excite the global vibration of the test sample. Second, many nonlinear ultrasonic methods use an impact or an actuator to excite nonlinear responses of the test sample. The excitation applied at a single point, or a small area creates a non-uniform strain field in a local area of the sample. The measurement results are affected by the strain distribution in the sample.

In a recent work by the authors, they investigated ambient temperature induced ultrasonic wave velocity change in concrete samples with microcracking damage, and defined a thermal modulation coefficient to represent the sensitivity of wave velocity to the temperature effect. It is found that the thermal modulation coefficient increases with the damage level in concrete samples, which shows its potential for nondestructive evaluation of material. In this letter, they propose using the thermal modulation of nonlinear ultrasonic waves to determine the nonlinear parameters α, β, δ. Slow temperature change creates a relative uniform thermal strain field in the test sample. The thermal strain can be used as the driving force to excite nonlinear behaviors of a material and modulate ultrasonic waves propagating in the material. Based on the experimental results from the thermal modulation test, they built a theoretical model to determine the absolute values of nonlinear parameters. The proposed theory was validated on classical nonlinear materials (metals) and non-classical nonlinear materials with hysteresis (concrete).

Temperature effects are often regarded as an undesired noise in most experimental studies. The dependence of elastic wave velocity on temperature has been studied by many researchers. Previous researchers have noticed a linear relationship between the relative velocity change dv/v and the temperature change ΔT on metals. For materials with hysteresis, a diagram of dv/v ∼ T correlation is shown, where a sample experienced a heating/cooling thermal cycle (T0 → T1 → T0). The relative velocity change dv/v decreases from point A to B during the heating and increases from B to C in the cooling.

A quadratic polynomial is used to model the dv/v vs. temperature relationship as: dv/v = k0 + k1 ΔT + k2 ΔT2, where k0, k1 and k2 are the coefficients of the polynomial and ΔT is the temperature change relative to a reference temperature. These coefficients are obtained from curve fitting of the correlation curves and are denoted as the thermal modulation coefficients.

For nonlinear materials with hysteresis, the modulus can be described with a strain and strain rate related model as: E(ε, ε̇) = E0 1 − β ε − δ ε2 − α [Δε + ε sign(ε̇)], where E0 is the linear modulus and β and δ are the quadratic and cubic nonlinear parameters (classical nonlinear parameters). α describes the hysteretic behavior of non-classical nonlinear materials. ε and ε̇ are the strain and strain rate, sign(ε̇) = 1 if ε̇ > 0 and sign(ε̇) = −1 if ε̇ < 0. Δε is the maximum strain experienced in the previous loading cycles.

Since the velocity is related to E, the relative velocity change is expressed as: dv/v = (1/2)(ΔE/E) = −(1/2) β ε + δ ε2 + α [Δε + ε sign(ε̇)]. Temperature change will generate thermal strain ε in a material, which can be expressed as ε = αT ΔT, where αT is the thermal expansion coefficient. If the sample experienced the temperature cycle, then Δε = αT ΔT01 represents the maximum strain the sample experienced for both the heating and the cooling processes.

For the heating process (A→B), sign(ε̇) = 1 and T0 is used as the reference temperature. For the cooling process (B→C), sign(ε̇) = −1 and T1 is the reference temperature. Then equation (3) can be written as below for the heating (+) and the cooling (-) processes: dv/v(±) = −(1/2) β (αT ΔT) + δ (αT ΔT)2 + α (αT ΔT01 ± αT ΔT). Equation (1) for heating and cooling cycle can be rewritten as: dv/v(±) = k0± + k1± ΔT + k2± ΔT2, where k± represents the coefficients for heating and cooling processes.

By comparing the first degree term ΔT in Eqs. (4-5), the following solution for k1± is obtained: k1+ = −αT (β + α)/2, k1− = −αT (β − α)/2. The nonlinear parameters β and α are solved as: β = −(k1− + k1+)/αT, α = −(k1+ − k1−)/αT. Equation (7) indicates that for classical nonlinear materials without hysteresis (α = 0), the coefficients k1+ and k1− should be equal. This statement is validated in the experiments presented in this letter. By comparing the second degree term ΔT2 in Eqs. (4-5), two solutions to the cubic nonlinear parameter δ from the heating and cooling processes are obtained, which is related to the curvatures of the correlation curves: δ± = −2k2±/αT2.

To perform the thermal modulation test, the samples were placed in an environmental chamber with a controllable temperature changing rate. The sample experienced a slow temperature changing rate (1°C/hr) to minimize the temperature gradient. Two 2.25 MHz ultrasonic transducers (Olympus A106S) were installed on the two opposite surfaces of the metal samples as the transmitter and receiver. For concrete samples, two 15 mm diameter PZT discs (STEMINC SMD15T21R111WL) were used to excite and receive ultrasonic waves around the frequency of 150 kHz. Ultrasonic signals were acquired with a digital oscilloscope (PICO 4224), and the temperature was monitored using a Type T thermocouple and a data logger.

For common structural materials (aluminum, steel, and concrete), the relative velocity change with temperature are in the order of 10−4/°C. In order to accurately measure such small velocity changes, the coda wave interferometry (CWI) method was used. If two signals differ only by a dilation, which is typically true for temperature-induced signal distortion, the dilation can be determined by stretching the disturbed signal and obtaining the stretching factor when the cross-correlation between the signals reaches maximal. In the experiments, the relative velocity change was measured with a precision of 10−6.

The dilation δt = dt/t represents the relative time delay dt at the time window t of a signal. In the thermal modulation test, two effects contribute to the relative time delay δt: thermal expansion of the sample, and temperature dependence of the wave velocity. The relative velocity change is calculated as dv/v = −(δt − αT ΔT). If the thermal strain is negligible (ε = αT ΔT ≪ dv/v), the relative velocity change can be calculated as dv/v = −dt/t = δt.

The authors first tested metal materials (aluminum and steel) using the thermal modulation method. Figure 2 shows the thermal modulation results of an aluminum 6061 block (5 cm×5 cm×18 cm) and a stainless steel 304 block (3 cm×6 cm×30 cm) with the temperature histories and relative velocity change histories. The two samples were first heated to a specific temperature and then cooled back to room temperature. The temperature changing rate was about 0.96°C/hour, which was close to the designed temperature rate. The relative velocity change shows an opposite trend to the temperature change history.

Figure 3 presents the dv/v ∼ T curves for the aluminum and the steel samples. The extracted thermal modulation coefficients are summarized in Table I. The steel sample has a smaller absolute value of k1± (slope) than the aluminum sample. Both correlation curves show very good linearity, which indicates small values of k2± for aluminum and steel.

The nonlinear parameters α, β, δ can be calculated from Eqs. (7,8) by using the thermal expansion coefficients αT = 23 µε/°C for aluminum and αT = 17.3 µε/°C for steel. From the thermal modulation tests, β = 15.4 for aluminum and β = 10.7 for steel were obtained. The absolute value of β reported in literature varies from 4 to 12 for aluminum alloys and 2 to 4.5 for stainless steel. The β values obtained from the thermal modulation test are slightly larger than the reported values in literature for both aluminum and steel. A possible explanation is the differences in strain measurement and strain range. In the thermal modulation test, the samples had a very slow strain changing rate, relatively large strain level, and uniform strain distribution, while most other nonlinear acoustic methods (NRAS, DAET, SHG) generate dynamic strain at very low strain levels.

For both the aluminum and the steel curves, the heating and cooling slopes are equal k1+ = k1−, which gives the non-classical parameters α = 0. Although both curves show good linearity, the δ± values for the steel are almost 10 times higher than that of the aluminum sample. The steel sample also shows a narrow hysteresis area. Just like the difference between k1+ and k1− gives the hysteresis parameter α, the authors believe that the difference between δ+ and δ− may represent a higher order hysteretic response (∼ ε2). This higher order effect of hysteresis has been predicted by Meurer et al. in their general nonlinear model, while Eq. (3) only includes the lower order term (∼ ε). Since samples will experience large thermal strains (10−4 ∼ 10−3) in the thermal modulation test, the higher order effects of α may need to be considered in future studies.

Concrete is a typical non-classical nonlinear material. The authors tested two concrete cylinders (10 cm diameter×20 cm height), with one control sample cast with normal concrete mix, and another sample with microcracking damage induced by alkali-silica reaction (ASR). ASR is a chemical reaction in concrete that occurs between alkali hydroxides in hydrated cement and reactive silica in certain types of aggregates. ASR produces an expansive gel, which will eventually cause microcracks and degradation of concrete. The mix design and curing condition of the ASR sample were described in a previous work by the authors. At the test time, the ASR sample had moderate damage (0.1% expansion) with minor surface cracks while the control sample was in intact condition. For the thermal modulation test, the two concrete samples were first heated from 23.2°C to 44.1°C then cooled down to 23.2°C. The correlation curves between the relative velocity change and temperature are shown in Fig.4 for the two concrete samples.

The thermal modulation coefficients k1+, k2+ and k1−, k2− were obtained from curve fitting and are summarized in Table II. All fittings have the goodness-of-fit R2 larger than 0.99. For both the control and the ASR samples, there are k1+ > k1−, which means that the relative velocity change is more sensitive to temperature change in the heating process than in the cooling process. The ASR sample has larger coefficients k1± and k2± than the control sample, which indicates that microcracking damage would increase the sensitivity of velocity change to temperature. In the previous work by the authors, they also found the k1− value increased with the damage level in concrete samples.

The thermal expansion coefficients were measured for both the control and ASR samples. The results were very close, so that αT = 10 µε/°C was used for both samples. The nonlinear parameters α, β, and δ were calculated using Eqs. (7) and (8). The absolute value of β is 100 for the control sample and 221 for the ASR sample. For the control sample, the result agrees with the literature reported values in the range of 40 to 157 for concrete. The ASR sample shows a higher β value than the control, which indicates that microcracking damage increases the quadratic nonlinearity. For both samples, δ+ and δ− in the heating and the cooling processes are very close, and they are much larger than δ± of the aluminum and the steel samples. The ASR sample has larger δ± than the control sample, but the values are still on the same order of magnitude 105, and are close to the reported value of 106 for rock.

The ASR sample also shows a larger value of α = 57 than the control sample α = 36, which suggests that the damage in concrete increases the hysteresis. In Eqs. (4-5), α can also be solved from the constant term k0±. In a closed thermal cycle with the same starting and ending temperature, α is related to the gap in the dv/v cur…

Improvements to AI Systems Based on This Paper:

1. Physics-informed curve fitting for nonlinear parameter extraction: An AI system can be trained to automatically fit the quadratic polynomial model (dv/v = k0 + k1ΔT + k2ΔT2) to experimental thermal modulation data, then extract the acoustic nonlinearity parameters (α, β, δ) using the derived equations (β = −(k1− + k1+)/αT, α = −(k1+ − k1−)/αT, δ± = −2k2±/αT2). This eliminates manual curve fitting errors and enables real-time material characterization.

2. Automatic hysteresis detection and classification: The AI can analyze the difference between heating and cooling slopes (k1+ vs k1−) and the gap in dv/v curves to automatically classify materials as classical (α ≈ 0, e.g., metals) or non-classical/hysteretic (α > 0, e.g., damaged concrete). This provides a quantitative, automated damage assessment tool.

3. Damage quantification in concrete via nonlinear parameter trends: The AI can correlate increases in β, δ, and α with microcracking damage levels (e.g., ASR-induced expansion). For instance, the paper shows β increases from 100 (control) to 221 (ASR-damaged), and α from 36 to 57. An AI system can build predictive models linking these parameters to damage severity, enabling non-destructive structural health monitoring.

4. Temperature-compensated ultrasonic velocity measurements: Since temperature changes cause velocity shifts (order 10−4/°C), an AI can use the extracted nonlinear parameters to correct ultrasonic measurements for thermal effects in real-world applications, improving accuracy of other NDE techniques that are currently confounded by temperature drift.

5. Automated coda wave interferometry (CWI) analysis: The AI can automate the stretching factor calculation for CWI, determining relative velocity changes with 10−6 precision. It can also distinguish between thermal expansion effects (αT ΔT) and true velocity changes, improving signal processing robustness.

6. Optimal experimental design: The AI can recommend optimal temperature ranges, heating/cooling rates (e.g., 1°C/hr to minimize gradients), and transducer frequencies (2.25 MHz for metals, 150 kHz for concrete) based on material type and expected nonlinearity, reducing trial-and-error in experimental setup.

7. Higher-order hysteresis modeling: The paper notes discrepancies between δ+ and δ−, suggesting higher-order hysteretic effects (∼ε2). An AI can be trained to extend the current model (Eq. 3) by incorporating higher-order α terms, improving accuracy for large thermal strains (10−4–10−3) and providing more precise parameter extraction.

8. Cross-method validation and uncertainty quantification: The AI can compare thermal modulation results with literature values from NRAS, DAET, and SHG methods, and quantify uncertainties due to strain range differences, providing confidence intervals for β, α, and δ. This enables more reliable material characterization across different testing conditions.

9. Automated damage prognosis: By tracking nonlinear parameter evolution over multiple thermal cycles, the AI can predict the progression of microcracking damage (e.g., ASR expansion from 0% to 0.1% and beyond), enabling early warning systems for infrastructure degradation.

10. Transfer learning for new materials: The AI can be pre-trained on the aluminum, steel, and concrete data from this paper, then fine-tuned for new materials (e.g., composites, rocks, ceramics) using minimal experimental data, accelerating nonlinear parameter extraction for diverse applications.

What the improved AI system can do:

• Provide real-time, automated extraction of absolute acoustic nonlinearity parameters (α, β, δ) from simple thermal modulation experiments.

• Non-destructively assess and quantify microcracking damage in concrete, metals, and other materials without complex calibration.

• Distinguish between classical and non-classical nonlinear behavior, enabling material classification.

• Correct ultrasonic measurements for thermal artifacts, improving accuracy of other NDE methods.

• Predict damage progression and remaining service life of infrastructure.

• Optimize experimental protocols for new materials and conditions.

• Extend the theoretical model to capture higher-order hysteretic effects, improving physical understanding.