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Discovery of Hyperelastic Constitutive Laws from Experimental Data with EUCLID

Arefeh Abbasi, Maurizio Ricci, Pietro Carrara, Moritz Flaschel, Siddhant Kumar, Sonia Marfia, Laura De Lorenzis

TL;DR

The study evaluates EUCLID, a sparse-regression framework that automates model selection and parameter identification for hyperelastic constitutive laws, using experimental data from natural rubber across simple and complex geometries. By combining global (force–elongation) and local (DIC-derived full-field) measurements, the authors compare EUCLID with conventional parameter identification across UT, PS, and TT tests. EUCLID consistently achieves accuracy comparable to or better than fixed-model fits, including in unseen geometries, and demonstrates robust generalization while producing interpretable, sparse constitutive representations. The work highlights the practical value of automated constitutive-law discovery for experimental material characterization and multi-geometry generalization, with potential implications for more data-driven, interpretable material modeling.

Abstract

We assess the performance of EUCLID, Efficient Unsupervised Constitutive Law Identification and Discovery, a recently proposed framework for automated discovery of constitutive laws, on experimental data. Mechanical tests are performed on natural rubber specimens spanning simple to complex geometries, from which we collect both global, force elongation, and local, full-field displacement, measurements. Using these data, we obtain constitutive laws via two routes, the conventional identification of unknown parameters in a priori selected material models, and EUCLID, which automates model selection and parameter identification within a unified model-discovery pipeline. We compare the two methodologies using global versus local data, analyze predictive accuracy, and examine generalization to unseen geometries. Moreover, we quantify the experimental noise, investigate the coverage of the material state space achieved by each approach and discuss the relative performance of different datasets and different a priori chosen models versus EUCLID.

Discovery of Hyperelastic Constitutive Laws from Experimental Data with EUCLID

TL;DR

The study evaluates EUCLID, a sparse-regression framework that automates model selection and parameter identification for hyperelastic constitutive laws, using experimental data from natural rubber across simple and complex geometries. By combining global (force–elongation) and local (DIC-derived full-field) measurements, the authors compare EUCLID with conventional parameter identification across UT, PS, and TT tests. EUCLID consistently achieves accuracy comparable to or better than fixed-model fits, including in unseen geometries, and demonstrates robust generalization while producing interpretable, sparse constitutive representations. The work highlights the practical value of automated constitutive-law discovery for experimental material characterization and multi-geometry generalization, with potential implications for more data-driven, interpretable material modeling.

Abstract

We assess the performance of EUCLID, Efficient Unsupervised Constitutive Law Identification and Discovery, a recently proposed framework for automated discovery of constitutive laws, on experimental data. Mechanical tests are performed on natural rubber specimens spanning simple to complex geometries, from which we collect both global, force elongation, and local, full-field displacement, measurements. Using these data, we obtain constitutive laws via two routes, the conventional identification of unknown parameters in a priori selected material models, and EUCLID, which automates model selection and parameter identification within a unified model-discovery pipeline. We compare the two methodologies using global versus local data, analyze predictive accuracy, and examine generalization to unseen geometries. Moreover, we quantify the experimental noise, investigate the coverage of the material state space achieved by each approach and discuss the relative performance of different datasets and different a priori chosen models versus EUCLID.

Paper Structure

This paper contains 24 sections, 38 equations, 16 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Overview of EUCLID
  3. Problem setting and model library
  4. Equilibrium and boundary conditions
  5. Constitutive law discovery
  6. Discovery from local data
  7. Discovery from global data
  8. Experimental campaign
  9. Experimental setup
  10. Uniaxial tension test
  11. Pure shear test
  12. Tensile tests
  13. Acquisition and processing of DIC data
  14. Assessment of DIC noise
  15. Coverage of the state space
  16. ...and 9 more sections

Figures (16)

  • Figure 1: Experimental setup and samples: (a) UT test setup with a doubly clamped dogbone specimen under tension in a mechanical testing machine. The deformation in the region of interest (ROI, highlighted with the hatched area) is analyzed. (b) PS test setup with a long rectangular specimen subjected to tension. It produces a deformation equivalent to pure shear in the ROI (hatched region). (c) Stress-stretch response from the UT test. (d) Stress-stretch response from the PS test. (e) Rectangular specimen geometries with circular holes (TTa, TTb, TTc). (f) Wide dogbone specimen geometries with elliptical holes (TTd, TTe, TTf).
  • Figure 2: DIC workflow. (a) Sample preparation, including the application of a high-contrast speckle pattern and accurate positioning within the DIC system’s field of view. (b) Image acquisition using the DIC system, where a sequence of images is captured and subsets (facets) are defined across the region of interest. (c) Displacements are determined by tracking each facet between the reference and deformed images using cross-correlation techniques. (d) Following correlation, a discrete set of nodal points with corresponding displacement data is obtained. (e) A DIC supporting mesh is generated based on these nodal points, and (f) interpolation is used to reconstruct the continuous displacement field. (g) The strain field is then computed from the displacement data using differentiation of the FE interpolation. This strain field is subsequently used as input for EUCLID.
  • Figure 3: DIC noise assessment. (a) The TTf sample is used for noise assessment, capturing 600 still images over a 300-second duration. A confined facet of $250 \times 250$ pixels (marked by the red square) is selected for the analysis. (b) A reference average facet, reconstructed as the average of all 600 facets, is created to represent the region with the lowest noise level. (c) The temporal evolution of noise is evaluated by plotting (c1) the distribution of the standard deviation of grayscale residuals, $\sigma_{r}$, relative to the reference average facet. (c2) The temporal variation of $\sigma_{r}$ over the 600 time steps is illustrated by red markers. (c3) Distribution of the average of grayscale residuals, $\bar{r}$, over time. (c4) The temporal variation of $\bar{r}$ over 600 time steps is illustrated by blue markers. (d) The spatial distribution of noise is evaluated by plotting (d1) the distribution of the standard deviation of grayscale values for each pixel, $\sigma_{p}$. (d2) Heatmap of $\sigma_{p}$ across the selected region, with the color bar indicating the standard deviation at each pixel.
  • Figure 4: Test-induced strain heterogeneity across different geometries and loading conditions with a similar visualization as in promma2009applicationguelon2009new. (a) Strain invariants $(I_{1}-3)-(I_{2}-3)$ plane for UT and PS tests. Experimental data points are shown with markers alongside the theoretical boundaries for UT (red line), PS (green line), and ET (blue line). The red shaded region indicates the domain where $I_{1} > I_{2}$, while the blue shaded region represents $I_{2} > I_{1}$. (b) Strain invariants for TT tests on samples TTa, TTb, and TTc. (c) Strain invariants for TT tests on samples TTd, TTe, and TTf. (d1, d2, d3) Color-coded representation of the loading state at each Gauss point in the final loading step for the UT (d1), PS (d2), and TTf (d3) specimens. Colors indicate the proximity of each point to pure loading modes in the invariant plane. (d4, d5, d6) Corresponding maximum principal stretch $\lambda_{1}$ at each Gauss point vs. the $X_1$-axis for the UT (d4), PS (d5), and TTf (d6) specimens. Combined probability distribution of $\lambda_{1}$ from both (e1) UT and PS global data, (e2) UT and PS local data, and (e3) UT and TTf local data. Insets in (e1-e3) depict the corresponding sample geometries.
  • Figure 5: Comparison of the a priori chosen models upon parameter identification and the model discovered by EUCLID, whereby identification/discovery uses combined UT and PS global data. Panels (a)–(g) show the stress-stretch response in UT and PS tests for different material models, compared with experimental data. Specifically, panels (a)–(f) show results for a priori chosen models: (a) 1st-order GMR, (b) 2nd-order GMR, (c) 3rd-order GMR, (d) GT, (e) 1-term Ogden, and (f) 2-term Ogden. Panel (g) shows the response obtained through automated model discovery using EUCLID. (h1) Pareto analysis for the automated selection of the hyperparameter $\lambda$, showing the MSE and the $L_1$ norm of $\boldsymbol{\theta}$ as functions of $\lambda$. (h2) A close-up around the automatically selected hyperparameter, with the chosen threshold for MSE and the selected solution indicated by cyan and green dashed lines, respectively.
  • ...and 11 more figures