EUCLID (Efficient Unsupervised Constitutive Law Identification & Discovery) utilizes local displacement and global reaction force data, but no local stress data as these are not available from experiments, to discover material models without assuming a specific functional form for the model apriori. To this end, a large catalog of candidate material models is constructed out of which dominant candidates are selected through sparse regression or the material behavior is described by a general machine learning based ansatz, e.g., a neural network. To compensate the unavailability of stress data, physics knowledge is employed by minimizing the sum of squared residuals of the weak linear momentum balance.

_{Figure 1: EUCLID workflow.}

The figure above shows the workflow of the sparse regression based EUCLID algorithm. Full-field displacement and net reaction force data that are obtained from a single experiment (a) serve as input data for EUCLID. The displacement data (b) is interpolated (c) to obtain the displacement field (d) and after differentiation the strain field (e). A general material model library is constructed, which can describe a variety of different material responses dependent on the choice of material parameters theta. Based on the model library, the stresses (f) and hence the residuals of the weak linear momentum balance (g,h) can be expressed dependent on the unknown parameters theta. The linear momentum balance serves as a physical constraint on the material parameter space. Minimizing the sum of squared residuals together with a sparsity promoting regularization term (i) yields a sparse parameter vector theta and hence an interpretable constitutive law expressed by a compact mathematical formula (j).

EUCLID was successfully demonstrated to be able to discover strain energy density functions of hyperelastic materials (see Ref. 1.). The codes and data are publically available on GitHub (see Ref. 2.) and the ETH Research Collection (see Ref. 3.), respectively. The documentation of the hyperelasticity code and an example can be found here.

In Ref. 7., the problem of material model discovery was considered from a Bayesian perspective. Through Markov Chain Monte Carlo sampling, the Bayesian-EUCLID deduces a posterior probability distribution for the unknown material parameters, which is a joined distribution of the model likelihood and a sparsity promoting spike-and-slab prior. In this way, parsimonious and interpretable material models can be discovered with quantifiable uncertainties. The codes and data are publically available on GitHub. The documentation of the Bayesian-EUCLID code and an example can be found here.

In Ref. 4., EUCLID was for the first time applied to discover path-dependent material behavior, i.e., elasto-plastic material models. In this application of EUCLID, the full-field displacement and global reaction force data are used to discover plastic yield surfaces and hardening laws as closed-form mathematical formulas (see Animation 1). The codes and data are publically available on GitHub (see Ref. 5.) and the ETH Research Collection (see Ref. 6.), respectively. The documentation of the elasto-plasticity code and an example can be found here.

_{Animation 1: Comparison between the true and discovered yield surface evolution along a given deformation path.}

1. Moritz Flaschel*, Siddhant Kumar* and Laura De Lorenzis (*contributed equally)
   Unsupervised discovery of interpretable hyperelastic constitutive laws
   Computer Methods in Applied Mechanics and Engineering, 381, p.113852 (open access)

2. Moritz Flaschel*, Siddhant Kumar* and Laura De Lorenzis (*contributed equally)
   Supplementary software for "Unsupervised discovery of interpretable hyperelastic constitutive laws"
   ETH Library
   DOI: http://doi.org/10.5905/ethz-1007-508
   GitHub: https://github.com/EUCLID-code/EUCLID-hyperelasticity

3. Moritz Flaschel*, Siddhant Kumar* and Laura De Lorenzis (*contributed equally)
   FEM Data – Unsupervised discovery of interpretable hyperelastic constitutive laws
   ETH Research Collection
   DOI: https://doi.org/10.3929/ethz-b-000505693

4. Moritz Flaschel, Siddhant Kumar and Laura De Lorenzis
   Discovering plasticity models without stress data
   npj Computational Materials, 8, 91 (2022) (open access)

5. Moritz Flaschel, Siddhant Kumar and Laura De Lorenzis
   Supplementary software for "Discovering plasticity models without stress data"
   ETH Library
   DOI: http://doi.org/10.5905/ethz-1007-509
   GitHub: https://github.com/EUCLID-code/EUCLID-plasticity

6. Moritz Flaschel, Siddhant Kumar and Laura De Lorenzis
   FEM Data - Discovering plasticity models without stress data
   ETH Research Collection
   DOI: https://doi.org/10.3929/ethz-b-000534002

7. Akshay Joshi, Prakash Thakolkaran, Yiwen Zheng, Maxime Escande, Moritz Flaschel, Laura De Lorenzis, Siddhant Kumar
   Bayesian-EUCLID: discovering hyperelastic material laws with uncertainties
   Computer Methods in Applied Mechanics and Engineering, 398, p.115225 (open access)
   GitHub: https://github.com/EUCLID-code/EUCLID-hyperelasticity-bayesian

8. Prakash Thakolkaran, Akshay Joshi, Yiwen Zheng, Moritz Flaschel, Laura De Lorenzis, Siddhant Kumar
   NN-EUCLID: deep-learning hyperelasticity without stress data
   (open access)