Guided elastic waves informed material modelling of soft incompressible media

Technical Deep Dive: Guided Elastic Waves Inform Material Modeling of Soft Incompressible Media

Overview

This research aims to demonstrate how measuring the propagation of guided elastic waves in a soft, incompressible elastomer plate undergoing uniaxial extension can reveal insights about the material's constitutive behavior that are inaccessible from static stress-strain tests alone. The authors show that the dispersion relations of the three fundamental guided modes (shear horizontal, pseudo-longitudinal, and flexural) provide additional constraints on the form of the strain energy density function compared to a traditional uniaxial tension experiment.

Problem & Context

• Modeling the mechanical response of rubber-like solids and soft tissues is challenging, as finding a universal constitutive law that accurately describes behavior across all deformation fields and extensions remains an outstanding problem.
• Existing approaches have followed two paths: (1) refining statistical chain models to capture strain-stiffening, and (2) introducing phenomenological continuum models with additional terms beyond the neo-Hookean law.
• However, fitting these models to uniaxial tension data alone can lead to non-unique parameter sets that make different predictions for other deformation modes.

Methodology

• The authors measured the dispersion relations of the three fundamental guided elastic wave modes (shear horizontal, pseudo-longitudinal, and flexural) propagating in an Ecoflex OO-30 elastomer plate under uniaxial extension.
• They compared the experimental dispersion curves to predictions from the acoustoelastic theory, using strain energy density parameters extracted by fitting the uniaxial stress-strain data.
• Three hyperelastic models were considered: Mooney-Rivlin, Gent-Thomas, and Carroll, each with different functional forms for the I₂ (second strain invariant) contribution.

Data & Experimental Setup

• The plate was held vertically and subjected to a nearly uniaxial deformation by adjusting the distance between clamps.
• In-plane modes (shear horizontal, pseudo-longitudinal) were measured using stroboscopic imaging and digital image correlation.
• The out-of-plane flexural mode was measured using a laser sheet and camera.
• Experiments were performed for stretch ratios λ between 1.03 and 2.27.

Results

• In the small to moderate strain regime (λ < 1.5):
  • All three hyperelastic models fit the uniaxial stress-strain data equally well.
  • However, the models exhibited small but significant differences in their predictions for the pseudo-longitudinal mode velocity perpendicular to the stretch direction.
• In the strain-hardening regime (1 < λ < 2.5):
  • Adding a power-law term to the strain energy density significantly improved the ability of all models to capture the experimental data at large stretches.
  • The Gent-Thomas and Carroll forms of the I₂ term performed better than Mooney-Rivlin, especially for the flexural mode perpendicular to the stretch.
• In the large deformation regime (full range):
  • Generalized neo-Hookean models like Gent-Gent and Dobrynin-Carrillo-Gent, with only 3 parameters, were able to accurately describe both the uniaxial stress-strain and guided wave dispersion data across all stretches.
  • The dynamic measurements did not distinguish between these limiting-chain models, which capture the singular strain energy behavior at large extensions.

Interpretation

• Guided wave measurements provide additional constraints on the constitutive model beyond what can be obtained from static uniaxial tests alone.
  • The pseudo-longitudinal mode is sensitive to second derivatives of the strain energy with respect to stretch, while the flexural mode probes the stress perpendicular to the stretch direction.
• This extra information allows the authors to discriminate between hyperelastic models that fit the uniaxial stress-strain data equally well.
• The results echo previous findings that the Mooney-Rivlin model performs well for uniaxial loading but has limitations for other deformation modes.
• A generalized neo-Hookean model with a Gent-Thomas or Carroll I₂ term provides a consistent description of both the static and dynamic measurements.

Limitations & Uncertainties

• The experiments did not impose a strictly uniaxial deformation, leading to biaxial stress states that favored certain constitutive models over others.
• Only first-order viscoelastic effects were considered, which may become insufficient at large deformations.
• Higher-order guided modes or other static deformation modes (e.g. biaxial) could provide further insight, but were not explored in this work.

What Comes Next

• Exploiting higher-order guided modes, for example by using a strip geometry, could extract richer information from dispersion measurements.
• Investigating incremental wave propagation under other static deformation modes, such as biaxial loading, could provide additional constraints on the constitutive model.
• Developing wave-based inversion procedures to directly extract the constitutive parameters from the dynamic data is a promising direction.
• Overall, the authors conclude that incremental wave propagation is a powerful tool for the mechanical characterization of soft materials.