Determination of acoustic nonlinearity parameters using thermal modulation of ultrasonic waves

Hongbin Sun; Jinying Zhu

REVIEW 3 major objections 4 minor 33 references

Determination of acoustic nonlinearity parameters using thermal modulation of ultrasonic waves

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper establishes that a slow heating–cooling cycle, with a quadratic fit to relative velocity change versus temperature, yields absolute values of the acoustic nonlinearity parameters α, β, and δ for metals and concrete.

desk verdict Clever idea—thermal strain as the driver for nonlinear acoustic parameters—but the analysis ignores intrinsic temperature dependence of modulus, which likely inflates β, and the δ anomaly shows the model is incomplete; worth a serious referee, not publication as-is. read the letter →

arxiv 2608.07685 v1 pith:HWUUBQR2 submitted 2026-08-07 physics.app-ph cond-mat.mtrl-sciphysics.ins-det

Hongbin Sun , Jinying Zhu This is my paper · ORCID

classification physics.app-phcond-mat.mtrl-sciphysics.ins-det

keywords acousticnonlinearitythermalmodulationultrasonicvelocitycodawaveinterferometryhysteresisconcretedamagenonlinearparametersstrain

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes that a slow heating–cooling cycle is enough to measure the absolute values of the three acoustic nonlinearity parameters α, β, δ that describe a material's strain-dependent and hysteretic modulus. Temperature change generates a thermal strain ε = α_T ΔT that drives the nonlinear response, and the relative ultrasonic velocity change dv/v varies quadratically with ΔT. Fitting that quadratic separately for heating and cooling gives coefficients from which β and α are obtained from the average and difference of the linear slopes, and δ from the curvature. Tests on aluminum, steel, intact concrete, and concrete damaged by alkali-silica reaction produce parameter values consistent with literature, with damage increasing all three parameters. The method turns temperature from an unwanted noise source into a uniform, slow, large-strain excitation that needs only a simple ultrasonic setup.

What carries the argument

The load-bearing mechanism is a quadratic polynomial model for the dv/v–temperature curve combined with the hysteresis-capable constitutive law. The paper posits dv/v(±)=k0±+k1± ΔT+k2± ΔT² and links the coefficients to the nonlinear parameters by substituting the thermal strain ε=α_T ΔT into the strain- and strain-rate-dependent modulus expression. Coda wave interferometry supplies the precise relative velocity change (precision ~10⁻⁶) that makes the coefficient extraction meaningful.

What would settle it

Measure dv/v versus T on a sample whose modulus has a known intrinsic temperature dependence at fixed strain (for example, a single-crystal whose elastic constants are measured by Brillouin scattering), and compare the fitted β, α, δ with values from an independent mechanical-excitation method on the same sample; if they differ beyond uncertainty, the coefficients mix thermal-strain nonlinearity with a temperature-driven modulus effect.

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Extended reading notes

Core claim

The central claim is that the acoustic nonlinearity parameters α, β, and δ can be recovered directly from the correlation between relative ultrasonic velocity change and temperature in a thermal modulation test. Using a strain- and strain-rate-dependent modulus model E(ε,ε̇)=E0{1 − βε − δε² − α[Δε + ε sign(ε̇)]}, the paper derives that on heating and cooling the relative velocity change follows dv/v(±) = −(1/2){β α_T ΔT + δ α_T² ΔT² + α(α_T ΔT01 ± α_T ΔT)}, where α_T is the thermal expansion coefficient. Comparing this with the fitted quadratic dv/v = k0 + k1± ΔT + k2± ΔT² gives β = −(k1− + k1+)/α_T, α = −(k1+ − k1−)/α_T, and δ± = −2k2±/α_T². The paper validates these relations experimentally: metals yield k1+ = k1− and hence α = 0, while concrete shows hysteresis and damage-dependent increases in α, β, and δ.

Load-bearing premise

The derivation assumes that temperature changes the ultrasonic velocity only by producing thermal strain that acts through the material's strain-dependent modulus, and that the elastic modulus itself does not change with temperature at fixed strain.

Editorial extensions

If this is right

A single thermal cycle, with temperature held at about 1 °C/h to keep strain uniform, yields absolute values of α, β, and δ without the calibration needed in harmonic-generation methods.
For non-hysteretic metals, equal heating and cooling slopes directly imply α = 0, providing an internal consistency check of the method.
Concrete damaged by alkali-silica reaction shows larger |k1±| and |k2±| than intact concrete, so the fitted coefficients themselves could serve as damage indicators.
Because the curvature terms δ+ and δ− differ in sign and magnitude for concrete, the model points to a higher-order (∼ε²) hysteretic contribution as predicted by the general nonlinear constitutive theory.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the thermal-only assumption holds, the method could be applied to any material with a known thermal expansion coefficient, turning standard temperature-cycle ultrasonic monitoring into a quantitative nonlinearity measurement without adding actuators.
The sign and magnitude difference between δ+ and δ− might be used to isolate the higher-order hysteresis term of the general model; fitting a higher-order polynomial could quantify that term directly.
The strong damage sensitivity on concrete suggests a field application: ambient daily temperature cycles, rather than lab-controlled chambers, might suffice for estimating microcrack density in structures.
The formulas assume the same thermal strain in the wave path and in the constitutive model; tests at different heating rates could reveal whether the extracted parameters are rate-independent, which the current quasi-static assumption implies but does not prove.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper proposes something genuinely new: use a slow thermal cycle to strain the material and extract the nonlinear acoustic parameters α, β, δ from the quadratic fit of dv/v versus T. The setup is simple, the derivation from the McCall–Guyer model is clean, and the concrete results are actually encouraging—the control sample gives β=100 within the literature range, and the α calculated from the gap between heating and cooling curves (40 and 73) matches well with the α from slope differences (36 and 57). That internal consistency is real evidence the method has some sensitivity to hysteresis.

The soft spots are not minor, though. The central substitution ε=α_TΔT into Eq. (2) assumes temperature affects velocity only through thermal strain. But elastic moduli of metals and concrete also depend on temperature at fixed strain, and the observed k1 values are the same order as that intrinsic temperature coefficient. If there is an additive term cΔT in dv/v, then the extracted β from −(k1⁺+k1⁻)/α_T picks up a spurious contribution of −2c/α_T. That plausibly explains why steel β=10.7 exceeds the cited 2–4.5 range. The paper mentions the discrepancy but attributes it to strain range, not to a missing physical term; that is a gap in the argument. The δ results are also a real problem: the model predicts equal δ⁺ and δ⁻, but steel and concrete show opposite signs. The paper acknowledges this and defers to unmodeled higher-order hysteresis, but without a derivation, Eq. (8) doesn't actually determine a unique δ. There are also no error bars or repeatability measurements, and the arXiv note attributing steel hysteresis to “minor temperature measurement error” is hand-wavy.

What holds up? The concept is novel and the experiments demonstrate strong, repeatable-looking correlations. For concrete, the damage sensitivity (ASR > control for α, β, δ) is consistent with known nonlinear behavior. The paper would be a useful contribution if the authors either separated the intrinsic modulus-temperature effect experimentally or reframed the method as a relative damage indicator rather than absolute parameter measurement.

For peer review: send it out. The idea deserves referee time and the flaws are identifiable and fixable, but the current absolute-accuracy claim is not supported. A serious referee should push for a control measurement of modulus versus temperature, error bars, and a proper treatment of the δ asymmetry.

Referee Report

3 major / 4 minor

Summary. The manuscript proposes a thermal modulation method for determining the acoustic nonlinearity parameters α, β, and δ. A sample is slowly heated and then cooled while ultrasonic wave velocity is monitored; the relative velocity change dv/v is fitted as a quadratic function of temperature on each branch. Using a quasistatic hysteretic constitutive model, the authors identify β and α with combinations of the linear fit coefficients k1± and δ with the quadratic coefficients k2±. Experiments on aluminum, steel, intact concrete, and ASR-damaged concrete are reported, and the extracted parameters are compared with literature values. The central claim is that this simple, uniform thermal strain field yields absolute values of α, β, and δ.

Significance. If valid, the method would be practically attractive: thermal excitation produces a large, uniform, slowly varying strain field, and CWI gives high-precision velocity measurements. The algebraic path from Eq. (2) to Eqs. (7)–(8) is transparent, and the inclusion of both classical metals and mesoscopic concrete, with an ASR damage contrast, gives the paper a plausible NDE application. However, the load-bearing identification is currently compromised by two unresolved issues: the unseparated intrinsic temperature dependence of elastic moduli, and an unexplained sign mismatch in δ+ versus δ−. The validation is also weaker than claimed because the extracted quantities are algebraic functions of the very coefficients obtained from the same fits. The paper does not provide code, machine-checked proofs, or falsifiable predictions beyond the fits themselves.

major comments (3)

[Theoretical model, Eqs. (2)–(7); CWI paragraph] The derivation substitutes ε = α_T ΔT into Eq. (2) and reads β and α from k1± via Eq. (7). This presumes that temperature affects ultrasonic velocity only through thermal strain, i.e., that the intrinsic temperature dependence of elastic modulus at fixed strain is negligible. No such term appears in Eq. (2), and the CWI correction dv/v = −(δt − α_T ΔT) removes only the thermal-expansion path contribution, not modulus softening with temperature. The reported k1 values, −1.77×10⁻⁴ /°C for aluminum and −0.93×10⁻⁴ /°C for steel, are of the same order as ordinary thermoelastic velocity-temperature coefficients, so k1± may be dominated by thermal softening rather than by strain-driven nonlinearity. Because an additive intrinsic term enters k1+ and k1− with the same sign, the β value from −(k1+ + k1−)/α_T would be systematically inflated; this is consistent with β = 10.7 for steel lying well above the cited range of 2–4.5. A control experiment separating thermal-strain nonlinearity from (∂V/∂T) at fixed strain, or an independent measurement of the intrinsic modulus-temperature term, is required before absolute β can be claimed.
[Eq. (8) and Table I] The model predicts δ+ = δ−, because the ε² contribution to Eq. (4) is identical in heating and cooling. Table I reports δ+ = 3875 and δ− = −4009 for steel, i.e., opposite signs, and Table II shows the same pattern for concrete. This sign reversal is not explained by the equations presented. The text appeals to a 'higher order hysteretic response (~ε²)' attributed to Meurer et al., but no modified constitutive equation or derivation is given. Consequently, Eq. (8) does not determine a unique δ for the materials tested, and the reported δ± values are not predictions of the model. The paper should either extend Eq. (4) to include the higher-order term explicitly and re-derive Eq. (8), or present δ as an unresolved model limitation.
[Experimental validation, Tables I–II] The validation is largely circular: β, α, and δ are defined as algebraic functions of the fitted polynomial coefficients, so a good quadratic fit does not independently confirm the model. The claimed 'reasonable agreements' with literature are also weakly supported: for aluminum β = 15.4 is above the cited upper value of 12, and for steel β = 10.7 is more than twice the cited upper value of 4.5, with no uncertainty bars or repeated-sample statistics. A stronger validation would predict at least one quantity not used in the fits, for example the closure gap Δ(dv/v) = α α_T ΔT01 or the cooling branch from the heating-branch coefficients, and then compare that prediction with the measured curve.

minor comments (4)

[Fig. 3 caption and Table I] The preprint note states that the slight steel hysteresis is due to a temperature measurement error rather than material response, yet Table I reports α = 0 from the equality of k1+ and k1−. Please clarify how the measurement error was identified and why it does not affect the reported k1 values.
[Tables I and II] The table headers do not state units for k1 and k2, and no confidence intervals are given for the fit coefficients or for the derived β, α, and δ. Please add units and uncertainties.
[Discussion after Table I] The phrase 'slightly larger' for the β values understates the discrepancies in Table I; steel β = 10.7 is ~2.4 times the upper literature bound of 4.5. The wording should be revised to match the numerical comparison.
[Data availability] For a measurement-method paper, a data availability statement of 'available from the corresponding author upon reasonable request' is weak; please include at least the processed dv/v–T curves and fit coefficients as supplementary material.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the nonlinear parameters are obtained by explicit algebraic inversion of fitted polynomial coefficients, and the validation relies on external literature values.

full rationale

The paper's derivation chain is a straightforward model inversion rather than a circular prediction. Equations (4) and (5) are two expressions for the same measured quantity (relative velocity change versus temperature); equating coefficients of ΔT and ΔT² gives Eqs. (6)–(8). Equation (7) is literally a solved form of Eq. (6): β and α are linear combinations of the fitted slopes k1⁺ and k1⁻ divided by α_T, and δ± are scaled curvatures k2±. This is parameter estimation from a fitted curve, not a prediction of an independent observable, but it is also not circular: the parameters pre-exist in the constitutive model Eq. (2) and are inferred from data. The only 'validation' is comparison with literature ranges, which is an external check. No load-bearing step is justified solely by a self-citation; Refs. 12 and 21 are supporting background and sample-description citations. The main scientific weakness—that the model assumes temperature enters only through thermal strain, with no intrinsic modulus-temperature term—is a correctness/validity threat, not a circularity, because the derivation does not define the target parameters in terms of the fitted coefficients by construction. The paper is self-contained relative to external benchmarks, so the circularity score is 0.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central derivation assumes the McCall-Guyer strain- and strain-rate-dependent modulus model is valid under slow thermal strain, that the only relevant strain is uniform thermal strain, and that the measured velocity changes contain no intrinsic temperature dependence of modulus. The hysteresis term Delta_epsilon is assigned as the maximum cycle strain for both heating and cooling, an ad hoc choice. The quadratic polynomial fit is then inverted to define the parameters, so validation against literature is the only independent check.

free parameters (5)

k1+ (heating linear coefficient) = -1.77e-4 (Al), -0.93e-4 (steel), -6.8e-4 (control), -13.9e-4 (ASR)
Fitted slope of the heating branch, used with k1- to solve beta and alpha through Eq. (7). Its value is a direct output of the quadratic least-squares fit.
k1- (cooling linear coefficient) = -1.77e-4 (Al), -0.93e-4 (steel), -3.2e-4 (control), -8.2e-4 (ASR)
Fitted slope of the cooling branch; its difference from k1+ determines alpha, and its sum with k1+ determines beta.
k2+ (heating quadratic coefficient) = -9.3e-8 (Al), -5.8e-7 (steel), -9.5e-6 (control), -2.8e-5 (ASR)
Fitted quadratic coefficient of the heating branch; maps to delta+ through Eq. (8).
k2- (cooling quadratic coefficient) = -1.1e-7 (Al), +6.0e-7 (steel), +8.9e-6 (control), +2.9e-5 (ASR)
Fitted quadratic coefficient of the cooling branch; maps to delta- through Eq. (8).
Thermal expansion coefficient alpha_T = Al 23.0e-6, steel 17.3e-6, concrete 10.0e-6 /C
Converts temperature change to thermal strain in Eqs. (7)-(8); values are taken as known material inputs, measured for concrete and from literature for metals, with no reported uncertainty. Errors in this value propagate directly into all extracted parameters.

assumptions (5)

domain assumption McCall-Guyer constitutive model E(epsilon, epsilon_dot) = E0{1 - beta*epsilon - delta*epsilon^2 - alpha[Delta_epsilon + epsilon sign(epsilon_dot)]} describes metals and concrete under slow thermal strain.
The entire derivation starts from this model; no evidence is given that its hysteresis term, developed for dynamic loading of rock, applies to quasistatic thermal strain.
standard math Velocity change is dv/v = (1/2) Delta E / E.
First-order relation from v proportional to sqrt(E), used in Eq. (3); ignores density change and assumes the correction for thermal expansion is complete.
domain assumption Thermal strain is uniform, equals alpha_T Delta T everywhere, and no intrinsic temperature dependence of modulus exists.
The paper states slow temperature change at 1 C/hr minimizes gradients but does not quantify them, and it does not separate intrinsic modulus-temperature dependence from strain-induced nonlinearity.
ad hoc to paper Maximum strain in the cycle Delta_epsilon equals alpha_T Delta_T01 for both heating and cooling branches.
During heating the maximum strain has not yet been experienced, so the 'previous loading cycle' interpretation fails; this choice is required to obtain the closed-form Eq. (4).
domain assumption The dv/v versus Delta T curve is exactly quadratic.
Eq. (1) is assumed; the paper reports R^2 > 0.99 for concrete but provides no residual analysis or justification for excluding higher-order terms.

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Pith. "Pith review of Determination of acoustic nonlinearity parameters using thermal modulation of ultrasonic waves." pith.science (2026). https://pith.science/paper/HWUUBQR2

@misc{pith2026260807685,
  author       = {Pith},
  title        = {Pith review of: Determination of acoustic nonlinearity parameters using thermal modulation of ultrasonic waves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HWUUBQR2}},
  note         = {Machine review of arXiv:2608.07685}
}

abstract

This study presents a test method and its theoretical framework to determine the acoustic nonlinearity parameters ($\alpha,\beta,\delta$) 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 $\alpha,\beta,\delta$ 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 $\alpha,\beta,\delta$ with good accuracy using a simple ultrasonic test setup.

Figures

Figures reproduced from arXiv: 2608.07685 by the authors.

**Figure 2.** FIG. 2. Thermal modulation test results of aluminum and stee [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3. Thermal modulation test results on aluminum 6061 and [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. Thermal modulation test results of concrete [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Journal of Geophysical Research: Solid Earth , volume=

Equation of state and wave propagation in hysteretic nonlinear elastic materials , author=. Journal of Geophysical Research: Solid Earth , volume=. 1994 , publisher=

work page 1994

[2] [2]

Physical review letters , volume=

Hysteresis, discrete memory, and nonlinear wave propagation in rock: A new paradigm , author=. Physical review letters , volume=. 1995 , publisher=

work page 1995

[3] [3]

and Guyer, Robert , year =

Mccall, K. and Guyer, Robert , year =. Hysteresis, discrete Memory and nonlinear elastic wave propagation in rock: a new theoretical paradigm , volume =

work page

[4] [4]

The Journal of the Acoustical Society of America , volume=

On the quasi-analytic treatment of hysteretic nonlinear response in elastic wave propagation , author=. The Journal of the Acoustical Society of America , volume=. 1997 , publisher=

work page 1997

[5] [5]

and Carmeliet, Jan and Ten Cate, James A and Johnson, Paul A , journal=

Van Den Abeele, K.E.-A. and Carmeliet, Jan and Ten Cate, James A and Johnson, Paul A , journal=. Nonlinear elastic wave spectroscopy (. 2000 , publisher=

work page 2000

[6] [6]

Journal of applied physics , volume=

Pump and probe waves in dynamic acousto-elasticity: Comprehensive description and comparison with nonlinear elastic theories , author=. Journal of applied physics , volume=. 2013 , publisher=

work page 2013

[7] [7]

Ultrasonics , volume=

Dynamic acousto-elastic testing of concrete with a coda-wave probe: comparison with standard linear and nonlinear ultrasonic techniques , author=. Ultrasonics , volume=. 2017 , publisher=

work page 2017

[8] [8]

Journal of Applied Physics , volume=

Ultrasonic studies of the nonlinear behavior of solids , author=. Journal of Applied Physics , volume=. 1965 , publisher=

work page 1965