Guided elastic waves informed material modelling of soft incompressible media

Abstract: Identifying a universal material constitutive law, that describes the mechanical response of rubber-like solids for all deformation fields and achievable extensions, is an outstanding challenge. Here, we propose to exploit the propagation of elastic waves and demonstrate that monitoring incremental guided wave propagation in an elastomer plate undergoing uniaxial extension reveals model sensitivities that are inaccessible in the corresponding static test. We measure the dispersion relations of the three zero-order guided modes, propagating parallel and perpendicular to the direction of imposed elongation. We compare them with predictions from the acoustoelastic theory, that also take into account material rheology, using parameters extracted from fitting the uniaxial stress-strain curve across three successive elongation regimes, following the methodical procedure of Destrade $\textit{et al.}$ (Proc. R. Soc. A 2017). We evidence that our approach lifts the degeneracy between hyperelastic models with different functional forms of the so-called $C_2$ term, which remain undistinguishable from static uniaxial tension stress-strain measurements alone. However, like their static counterpart, our dynamics measurements cannot distinguish between different generalized neo-Hookean models.

• The paper demonstrates that guided SH₀, S₀, and A₀ wave dispersion probes different derivatives of strain energy, exposing constitutive information that uniaxial stress–strain data cannot reveal.
• Wave measurements distinguish Mooney–Rivlin, Gent–Thomas, and Carroll models at small-to-moderate strains, while augmented Carroll and Gent–Thomas models generally outperform alternatives in the strain-hardening regime.
• The method cannot distinguish Gent- and Dobrynin–Carrillo-based limiting-chain models, and its conclusions are partly affected by clamp-induced biaxiality and the fitted viscoelastic coupling parameter β′.

Motivation and scope

Identifying a constitutive law for rubber-like solids that remains valid across all deformation classes and the full range of achievable extensions is a long-standing problem in nonlinear elasticity. The difficulty is well documented: Ogden and collaborators showed that multiple parameter sets can fit uniaxial tension data equally well while yielding markedly different predictions for other deformations, and Destrade et al. later formalized a fitting procedure that yields a unique optimum for simple extension data while emphasizing that static uniaxial tests alone cannot discriminate among constitutive models sharing a common structure. The paper under review proposes an alternative source of information: incremental guided elastic waves propagating in an elastomer plate subjected to large static uniaxial extension (2603.18839). The central claim is that dispersion measurements of the three zero-order guided modes reveal sensitivities of the strain-energy density that are invisible in the corresponding stress–strain curve, thereby lifting some—but not all—of the degeneracy between hyperelastic models.

Experimental characterization of guided modes

The experiments extend prior work on Ecoflex OO-30 plates (60 cm × 3 mm) held vertically between clamps, where in-plane $SH_0$ and $S_0$ mode dispersions were measured via stroboscopic imaging and digital image correlation for stretches $\lambda \in [1.03, 2.27]$. The new contribution adds the flexural $A_0$ mode, measured on a 2.5 mm-thick plate using a shaker-driven line source emitting a quadratic chirp (200 Hz down to 1 Hz) and a laser-sheet deflection technique recorded at 1 kHz. Dispersion relations are extracted from two-dimensional Fourier transforms of the displacement fields.

Two experimental caveats matter for the interpretation. First, the imposed deformation is not strictly uniaxial: marker tracking gives $\lambda_3 = \lambda^{-0.41}$ (in-plane sample) and $\lambda^{-0.39}$ (out-of-plane sample), because the clamps constrain lateral contraction; gravity also produces a small pre-stretch ($\lambda = 1.03$ at rest). Second, the $A_0$ mode proves highly sensitive to stretch: it rapidly departs from its parabolic long-wavelength shape and becomes nearly non-dispersive parallel to the elongation at large $\lambda$, whereas the in-plane mode velocities increase along the stretch direction (string-like tension behavior) and are barely affected perpendicular to it.

Modelling framework

The theoretical pipeline combines three ingredients: (i) nearly incompressible hyperelastic strain-energy densities written as deviatoric plus volumetric parts, compatible with fourth-order weakly nonlinear elasticity; (ii) acoustoelastic theory for incremental motions superposed on the finite deformation, with viscoelasticity incorporated through a fractional Kelvin–Voigt model ($\mu = \mu_0(1 + (\mathrm{i}\omega\tau)^n)$, with $S_0$0s and $S_0$1 from rheometry), introducing a coupling parameter $S_0$2 between viscosity and pre-stress; and (iii) long-wavelength approximations for the phase velocities of the $S_0$3, $S_0$4, and $S_0$5 modes, supplemented by a semi-analytical spectral collocation scheme solving for complex wavenumbers—necessary to capture the full $S_0$6 dispersion, which the long-wavelength formulas fail to reproduce beyond a limited wavenumber range.

The key structural insight is that different modes probe different derivatives of $S_0$7. In the long-wavelength limit, $S_0$8 velocities relate to first derivatives of $S_0$9 (equivalently, the stress difference $\lambda \in [1.03, 2.27]$0, independent of rheology), the $\lambda \in [1.03, 2.27]$1 velocities equal $\lambda \in [1.03, 2.27]$2 and $\lambda \in [1.03, 2.27]$3 directly, while the $\lambda \in [1.03, 2.27]$4 velocity depends on second-order derivatives of $\lambda \in [1.03, 2.27]$5 with respect to the principal stretches—the derivative of stress with stretch. This hierarchy explains why the pseudo-longitudinal mode amplifies minute differences between constitutive laws that fit the static curve identically. The price paid is the extra parameter $\lambda \in [1.03, 2.27]$6, which must be determined from dynamic data itself.

Discriminating the $\lambda \in [1.03, 2.27]$7 term: small to moderate strains

Following Destrade et al.'s procedure, the authors identify the extent of the small-to-moderate regime via the maximum relative residual in Mooney space, finding a sharp error increase beyond $\lambda \in [1.03, 2.27]$8. Three two-parameter models differing only in the functional form of the $\lambda \in [1.03, 2.27]$9-dependent term—Mooney–Rivlin, Gent–Thomas, and Carroll—are fitted to the uniaxial data. All three fit the Mooney plot excellently and yield essentially identical shear moduli ($A_0$0–23.4 kPa), confirming that the static test cannot discriminate them.

The wave data can. At fixed frequency (170 Hz for in-plane modes, 50 Hz for $A_0$1), the predicted phase velocity of the $A_0$2 mode propagating perpendicular to the stretch differs measurably between the three models even within the fitted range, with Mooney–Rivlin performing best. Notably, $A_0$3 is recovered for all three models in this regime, indicating that viscous effects do not couple to pre-stress at moderate elongations—in contrast with the non-zero value reported in earlier work covering larger stretches.

Strain-hardening regime

To capture the upturn in the Mooney plot ($A_0$4 up to 2.5), a power-law term proportional to $A_0$5 with $A_0$6 is appended to each model. All augmented models superimpose almost perfectly on the static data, again leaving the Mooney plot non-discriminating. The dynamic comparison is more informative: the maximum relative velocity error across all modes and directions is 12.7% for Carroll+$A_0$7, 16.9% for Gent–Thomas+$A_0$8, but 23.7% for Mooney–Rivlin+$A_0$9, with the largest Mooney–Rivlin error occurring for the $\lambda_3 = \lambda^{-0.41}$0 mode perpendicular to the stretch. This failure is traced to the fact that this mode's velocity equals $\lambda_3 = \lambda^{-0.41}$1: the clamped geometry imposes a slightly biaxial stress state ($\lambda_3 = \lambda^{-0.41}$2), a deformation class where Mooney–Rivlin is known to perform poorly. The authors state plainly that had the deformation been strictly uniaxial, the $\lambda_3 = \lambda^{-0.41}$3 mode would not be discriminating—and excluding it drops the Mooney–Rivlin error to 7.0%, the best of the three. This is a genuine caveat on the claimed discrimination: part of it stems from the imperfect boundary conditions rather than from intrinsic model ranking. The coupling parameter becomes non-zero in this regime ($\lambda_3 = \lambda^{-0.41}$4–0.32) but is only weakly dependent on the choice of constitutive law, so conclusions are robust to its determination.

Large deformation regime: limits of the method

For the limiting-chain regime, the authors fit the Gent model (freely jointed chain) and the Dobrynin–Carrillo model (worm-like chain), each augmented with a Gent–Thomas $\lambda_3 = \lambda^{-0.41}$5 term, using only three parameters. Both describe the full static curve and all measured phase velocities accurately, and—critically—their wave predictions are indistinguishable from each other, even extrapolated beyond the experimentally accessed range. Like their static counterparts, the dynamic measurements therefore cannot distinguish generalized neo-Hookean models representing different microscopic chain statistics. The practical recommendation is that these three-parameter models advantageously replace the four-parameter power-law constructions.

Limitations and open questions

Several limitations are acknowledged or implicit. The discrimination of $\lambda_3 = \lambda^{-0.41}$6 forms relies partly on the unintended biaxiality of the loading; a truly uniaxial setup would weaken the $\lambda_3 = \lambda^{-0.41}$7-based conclusion. Only first-order viscous terms in the pre-stress coupling are retained, an approximation the authors note may fail at large elongation where the response is strongly nonlinear. The value of $\lambda_3 = \lambda^{-0.41}$8 must be extracted from the same dynamic data used for validation, so it is not an independent prediction. The choice of comparison frequencies (170 Hz, 50 Hz) is admittedly arbitrary, though an interactive tool allows exploration over the full spectrum. Open questions left by the paper include whether higher-order guided modes (accessible, e.g., in strip geometries) provide richer discriminating information, whether attenuation measurements can constrain the hyperelastic–viscoelastic coupling beyond the single $\lambda_3 = \lambda^{-0.41}$9 parameter, and whether a full wave-based inversion of constitutive parameters—rather than the static-fit-plus-validation scheme used here—is feasible.

Conclusion

This work demonstrates that incremental guided-wave dispersion in a pre-stretched soft plate carries constitutive information absent from the uniaxial stress–strain curve: sensitivity to second derivatives of the strain energy (via the $\lambda^{-0.39}$0 mode) and to transverse stress (via the $\lambda^{-0.39}$1 mode). Guided-wave measurements lift the degeneracy among models with different $\lambda^{-0.39}$2 functional forms—favoring Gent–Thomas and Carroll forms overall—while remaining blind to differences among generalized neo-Hookean limiting-chain models. The results echo the classical lesson from multi-deformation-class static testing, obtained here from a single specimen geometry, and suggest that wave-based mechanical characterization could complement standard traction protocols for soft incompressible media (2603.18839).