Hierarchical Bayesian constitutive model selection for high-strain-rate soft material characterization

TL;DR

The work tackles the challenge of characterizing soft hydrogel rheology at ultra-high strain rates ($>10^3\,\mathrm{s}^{-1}$) where measurement noise and model-form uncertainty hinder conventional approaches. It introduces a scalable hierarchical Bayesian Inertial Microcavitation Rheometry (IMR) framework that couples forward simulations of bubble dynamics via the Keller–Miksis model with a stress integral $S(t)$ to perform model selection among viscoelastic constitutive forms and to estimate material parameters. Key innovations include a strain-rate–weighted, heteroscedastic likelihood, a Half-Cauchy prior for the noise scale $\beta$, a redundancy penalty to discourage overparameterization, and grid-based marginalization to yield MAP estimates with uncertainty diagnostics. Across synthetic tests and experiments on gelatin, fibrin, polyacrylamide, and agarose (including cross-institution gelatin data), the KV model is most often selected, with UT2 favoring the nonlinear qKV form; the inferred parameters and $\beta$-posteriors provide interpretable measures of material behavior and data quality, demonstrating a principled, uncertainty-aware approach to dynamic rheometry of soft matter at high strain rates.

Abstract

The high-fidelity characterization of soft, tissue-like materials under ultra-high-strain-rate conditions is critical in engineering and medicine. Still, it remains challenging due to limited optical access, sensitivity to initial conditions, and experimental variability. Microcavitation techniques (e.g., laser-induced microcavitation) have emerged as a viable method for determining the mechanical properties of soft materials in the ultra-high-strain-rate regime (higher than 10^3 s^{-1}); however, they are limited by measurement noise and uncertainty in parameter estimation. A hierarchical Bayesian model selection method is employed using the Inertial Microcavitation Rheometry (IMR) technique to address these limitations. With this method, the parameter space of different constitutive models is explored to determine the most credible constitutive model that describes laser-induced microcavitation bubble oscillations in soft, viscoelastic, transparent hydrogels. The target data/evidence is computed using a weighted Gaussian likelihood with a hierarchical noise scale, which enables the quantification of uncertainty in model plausibility. Physically informed priors, including range-invariant, stress-based parameter priors, a model-redundancy prior, and a Bayesian Information Criterion motivated model prior, penalize complex models to enforce Occam's razor. Using a precomputed grid of simulations, the probabilistic model selection process enables an initial guess for the Maximum A Posteriori (MAP) material parameter values. Synthetic tests recover the ground-truth models and expected parameters. Using experimental data for gelatin, fibrin, polyacrylamide, and agarose, MAP simulations of credible models reproduce the data. Moreover, a cross-institutional comparison of 10% gelatin indicates consistent constitutive model selection.

Figures (6)

• Figure 1: A spherical bubble is nucleated in a soft material and grows to a maximum radius before collapsing and oscillating until it reaches mechanical equilibrium. Identification of the stress integral formulation enables the characterization of the material.
• Figure 2: Experimental maximum stretch ratio v. maximum radius. UM1: blue circle, UM2: black square, UM3: red up triangle, UT1: cyan right triangle, UT2: green down triangle.
• Figure 3: Bubble radius (left), velocity (middle), and magnitude of strain rates at the bubble wall (right) histories for nine sets of gelatin LIC experimental data.
• Figure 4: Schematic diagram of the hierarchical relationship among the constitutive models. Arrows indicate limiting cases obtained by setting parameters to zero, thereby reducing the model's dimensionality. The first column of models corresponds to three parameters, the second column has two parameters, and the third column has one parameter.
• Figure 5: Normalized bubble radius experimental cloud $R^*$ versus non-dimensional time $t^*$. Cyan region: experimental cloud across trials, showing variability in collapse and rebound dynamics. Solid red curve: the simulation corresponding to the MAP parameter set $\boldsymbol{\theta}_{\text{MAP}}$ for the most plausible models(see \ref{['tab:expdata-fit']}).