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Guided elastic waves informed material modelling of soft incompressible media

Pierre Chantelot, Samuel Croquette, Fabrice Lemoult

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.

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 (Proc. R. Soc. A 2017). We evidence that our approach lifts the degeneracy between hyperelastic models with different functional forms of the so-called 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.
Paper Structure (13 sections, 21 equations, 6 figures, 3 tables)

This paper contains 13 sections, 21 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Measuring guided elastic waves in a plate
  3. In-plane modes
  4. Out-of-plane mode
  5. Material modelling
  6. Non-linear elasticity for hyperelastic materials
  7. Fitting the strain energy density to uniaxial data
  8. Guided waves dispersion relations
  9. Results
  10. Small to moderate strain regime
  11. Strain-hardening regime
  12. Large deformation regime
  13. Conclusion

Figures (6)

  • Figure 1: (a) Sketch of the experimental setup used to measure the dispersion curves of the in-plane modes in a soft elastomer plate. (b-c) Dispersion relations of the in-plane $S\!H_0$ (green circles) and $S_0$ (pink triangles) modes propagating parallel and perpendicular to the direction of elongation for $\lambda \in [1.03,2.27]$. (d) Schematic of experimental setup used to obtain the dispersion relation of the out-of-plane $A_0$ mode. (e-f) Dispersion curves of the $A_0$ mode propagating parallel and perpendicular to the imposed stretch for $\lambda \in [1.11,2.22]$.
  • Figure 2: Uniaxial extension test of an Ecoflex OO-30 sample. (a) Engineering stress $\sigma^e$ as a function of the stretch ratio in the direction of elongation, $\lambda$. (b) Same data as in (a) represented in the Mooney-space (equation \ref{['eq:mooneyspace']}).
  • Figure 3: (a) Rheological measurement of the complex shear modulus of Ecoflex OO-30. We plot the real and imaginary parts of the shear modulus $\mu'$ and $\mu"$, respectively. The dashed lines show the fit of the fractional Kelvin-Voigt model to the data. (b-c) Dispersion relations of the in-plane $S\!H_0$ and $S_0$ modes propagating parallel or perpendicular to the direction of elongation for $\lambda = 1.03$. The solid lines represent the long-wavelength approximations defined in equations \ref{['eq:SH0']} and \ref{['eq:S0']}. (d-e) Dispersion relations of the out-of-plane $A_0$ mode propagating parallel or perpendicular to the direction of elongation for $\lambda = 1.11$. The blue lines stand for the long-wavelength predictions of equation \ref{['eq:A0']} and the black lines show the full semi-analytical predictions.
  • Figure 4: (a) Mooney plot for Ecoflex OO-30 showing the fits of the Mooney-Rivlin, Gent-Thomas and Carroll models in the small to moderate stretch regime (solid lines). The greyed-out region evidences the points for which $\lambda > 1.5$ that are excluded from the fitting procedure. (b) Maximum relative error $\mathcal{E}$ as a function of the fit endpoint for the three models considered in (a). (c-d) Normalized wave velocities for the $S\!H_0$ (dots) and $S_0$ (triangles) modes propagating parallel and perpendicular to the direction of elongation as a function of $\lambda$ for $f = \qty{170}{\hertz}$. The solid lines are long-wavelength predictions computed from equations \ref{['eq:SH0']} and \ref{['eq:S0']}. (e-f) Normalized wave velocities in the directions $\boldsymbol{e}_1$ and $\boldsymbol{e}_3$ for the $A_0$ mode for $f = \qty{50}{\hertz}$. Parallel to the direction of propagation, the predictions come from the approximation \ref{['eq:A011']} while perpendicular to the direction of propagation we represent the results of the semi-analytical theory.
  • Figure 5: (a) Uniaxial elongation test for Ecoflex OO-30 represented in the Mooney space. The solid lines are the fits of equation \ref{['eq:SHI1']} for $\lambda < 2.5$ when considering functional forms of the $I_2$ term corresponding to the Mooney-Rivlin, Gent-Thomas, and Carroll models. (b-c) Normalized wave velocities for the $S\!H_0$ (dots) and $S_0$ (triangles) modes propagating parallel and perpendicular to the direction of elongation as a function of $\lambda$ for $f = \qty{170}{\hertz}$. The solid lines are long-wavelength predictions computed from equations \ref{['eq:SH0']} and \ref{['eq:S0']}. (d-e) Normalized wave velocities in the directions $\boldsymbol{e}_1$ and $\boldsymbol{e}_3$ for the $A_0$ mode. Parallel to the direction of propagation, the predictions come from the approximation \ref{['eq:A011']}, while perpendicular to the direction of propagation we represent the results of the semi-analytical theory.
  • ...and 1 more figures