When the elastic properties of structured materials become direction-dependent, the number of their descriptors increases. For example, in two-dimensions, the anisotropic behavior of materials is described by up to 6 independent elastic stiffness parameters, as opposed to only 2 needed for isotropic materials. Such high number of parameters expands the design space of structured materials and leads to unusual phenomena, such as materials that can shear under uniaxial compression. However, an increased number of properties descriptors and the coupling between shear and normal deformations render the experimental evaluation of material properties more challenging. In this paper, we propose a methodology based on the virtual fields method to identify six separate stiffness tensor parameters of two-dimensional anisotropic structured materials using just one tension test, thus eliminating the need for multiple experiments, as it is typical in traditional methods. The approach requires no stress data and uses full-field displacement data and global force data. We show the accuracy of our method using synthetic data generated from finite element simulations as well as experimental data from additively manufactured specimens.

We propose an automated computational algorithm for simultaneous model selection and parameter identification for the hyperelastic mechanical characterization of biological tissue and validate it on experimental data stemming from human brain tissue specimens. Following the motive of the recently proposed computational framework EUCLID (Efficient Unsupervised Constitutive Law Identification and Discovery) and in contrast to conventional parameter calibration methods, we construct an extensive set of candidate hyperelastic models, i.e., a model library including popular models known from the literature, and develop a computational strategy for automatically selecting a model from the library that conforms to the available experimental data while being represented as an interpretable symbolic mathematical expression. This computational strategy comprises sparse regression, i.e., a regression problem that is regularized by a sparsity promoting penalty term that filters out irrelevant models from the model library, and a clustering method for grouping together highly correlated and thus redundant features in the model library. The model selection procedure is driven by data stemming from mechanical tests under different deformation modes, i.e., uniaxial compression/tension and simple torsion. The data is acquired through conventional mechanical tests that deliver labeled one-dimensional data pairs, and thus the method can be interpreted as a supervised counterpart to the originally proposed EUCLID that is informed by full-field displacement data and global reaction forces. The proposed method is verified on synthetical data with artificial noise. In addition, we present for the first time an experimental investigation of the EUCLID framework by validating the proposed method on experimental data acquired through mechanical tests of human brain specimens, proving that the method is capable of discovering hyperelastic models that exhibit both high fitting accuracy to the data as well as concise and thus interpretable mathematical representations.

We extend the scope of our recently developed approach for unsupervised automated discovery of material laws (denoted as EUCLID) to the general case of a material belonging to an unknown class of constitutive behavior. To this end, we leverage the theory of generalized standard materials, which encompasses a plethora of important constitutive classes including elasticity, viscosity, plasticity and arbitrary combinations thereof. We show that, based only on full-field kinematic measurements and net reaction forces, EUCLID is able to automatically discover the two scalar thermodynamic potentials, namely, the Helmholtz free energy and the dissipation potential, which completely define the behavior of generalized standard materials. The a priori enforced constraint of convexity on these potentials guarantees by construction stability and thermodynamic consistency of the discovered model; balance of linear momentum acts as a fundamental constraint to replace the availability of stress–strain labeled pairs; sparsity promoting regularization enables the automatic selection of a small subset from a possibly large number of candidate model features and thus leads to a parsimonious, i.e., simple and interpretable, model. Importantly, since model features go hand in hand with the correspondingly active internal variables, sparse regression automatically induces a parsimonious selection of the few internal variables needed for an accurate but simple description of the material behavior. A fully automatic procedure leads to the selection of the hyperparameter controlling the weight of the sparsity promoting regularization term, in order to strike a user-defined balance between model accuracy and simplicity. By testing the method on synthetic data including artificial noise, we demonstrate that EUCLID is able to automatically discover the true hidden material model from a large catalogue of constitutive classes, including elasticity, viscoelasticity, elastoplasticity, viscoplasticity, isotropic and kinematic hardening.

We extend EUCLID, a computational strategy for automated material model discovery and identification, to linear viscoelasticity. For this case, we perform a priori model selection by adopting a generalized Maxwell model expressed by a Prony series, and deploy EUCLID for identification. The methodology is based on four ingredients: i. full-field displacement and net force data; ii. a very wide material model library — in our case, a very large number of terms in the Prony series; iii. the linear momentum balance constraint; iv. the sparsity constraint. The devised strategy comprises two stages. Stage 1 relies on sparse regression; it enforces momentum balance on the data and exploits sparsity-promoting regularization to drastically reduce the number of terms in the Prony series and identify the material parameters. Stage 2 relies on k-means clustering; starting from the reduced set of terms from stage 1, it further reduces their number by grouping together Maxwell elements with very close relaxation times and summing the corresponding moduli. Automated procedures are proposed for the choice of the regularization parameter in stage 1 and of the number of clusters in stage 2. The overall strategy is demonstrated on artificial numerical data, both without and with the addition of noise, and shown to efficiently and accurately identify a linear viscoelastic model with five relaxation times across four orders of magnitude, out of a library with several hundreds of terms spanning relaxation times across seven orders of magnitude.

Within the scope of our recent approach for Efficient Unsupervised Constitutive Law Identification and Discovery (EUCLID), we propose an unsupervised Bayesian learning framework for discovery of parsimonious and interpretable constitutive laws with quantifiable uncertainties. As in deterministic EUCLID, we do not resort to stress data, but only to realistically measurable full-field displacement and global reaction force data; as opposed to calibration of an a priori assumed model, we start with a constitutive model ansatz based on a large catalog of candidate functional features; we leverage domain knowledge by including features based on existing, both physics-based and phenomenological, constitutive models. In the new Bayesian-EUCLID approach, we use a hierarchical Bayesian model with sparsity-promoting priors and Monte Carlo sampling to efficiently solve the parsimonious model selection task and discover physically consistent constitutive equations in the form of multivariate multi-modal probabilistic distributions. We demonstrate and validate the ability to accurately and efficiently recover isotropic and anisotropic hyperelastic models like the Neo-Hookean, Isihara, Gent–Thomas, Arruda–Boyce, Ogden, and Holzapfel models in both elastostatics and elastodynamics. The discovered constitutive models are reliable under both epistemic uncertainties — i.e. uncertainties on the true features of the constitutive catalog – and aleatoric uncertainties – which arise from the noise in the displacement field data, and are automatically estimated by the hierarchical Bayesian model.

We propose a new approach for unsupervised learning of hyperelastic constitutive laws with physics-consistent deep neural networks. In contrast to supervised learning, which assumes the availability of stress–strain pairs, the approach only uses realistically measurable full-field displacement and global reaction force data, thus it lies within the scope of our recent framework for Efficient Unsupervised Constitutive Law Identification and Discovery (EUCLID) and we denote it as NN-EUCLID. The absence of stress labels is compensated for by leveraging a physics-motivated loss function based on the conservation of linear momentum to guide the learning process. The constitutive model is based on input-convex neural networks, which are capable of learning a function that is convex with respect to its inputs. By employing a specially designed neural network architecture, multiple physical and thermodynamic constraints for hyperelastic constitutive laws, such as material frame indifference, material stability, and stress-free reference configuration are automatically satisfied. We demonstrate the ability of the approach to accurately learn several hidden isotropic and anisotropic hyperelastic constitutive laws – including e.g., Mooney–Rivlin, Arruda–Boyce, Ogden, and Holzapfel models – without using stress data. For anisotropic hyperelasticity, the unknown anisotropic fiber directions are automatically discovered jointly with the constitutive model. The neural network-based constitutive models show good generalization capability beyond the strain states observed during training and are readily deployable in a general finite element framework for simulating complex mechanical boundary value problems with good accuracy.

We propose an approach for data-driven automated discovery of material laws, which we call EUCLID (Efficient Unsupervised Constitutive Law Identification and Discovery), and we apply it here to the discovery of plasticity models, including arbitrarily shaped yield surfaces and isotropic and/or kinematic hardening laws. The approach is unsupervised, i.e., it requires no stress data but only full-field displacement and global force data; it delivers interpretable models, i.e., models that are embodied by parsimonious mathematical expressions discovered through sparse regression of a potentially large catalog of candidate functions; it is one-shot, i.e., discovery only needs one experiment. The material model library is constructed by expanding the yield function with a Fourier series, whereas isotropic and kinematic hardening is introduced by assuming a yield function dependency on internal history variables that evolve with the plastic deformation. For selecting the most relevant Fourier modes and identifying the hardening behavior, EUCLID employs physics knowledge, i.e., the optimization problem that governs the discovery enforces the equilibrium constraints in the bulk and at the loaded boundary of the domain. Sparsity promoting regularization is deployed to generate a set of solutions out of which a solution with low cost and high parsimony is automatically selected. Through virtual experiments, we demonstrate the ability of EUCLID to accurately discover several plastic yield surfaces and hardening mechanisms of different complexity.

We propose a new approach for data-driven automated discovery of isotropic hyperelastic constitutive laws. The approach is unsupervised, i.e., it requires no stress data but only displacement and global force data, which are realistically available through mechanical testing and digital image correlation techniques; it delivers interpretable models, i.e., models that are embodied by parsimonious mathematical expressions discovered through sparse regression of a large catalogue of candidate functions; it is one-shot, i.e., discovery only needs one experiment — but can use more if available. The problem of unsupervised discovery is solved by enforcing equilibrium constraints in the bulk and at the loaded boundary of the domain. Sparsity of the solution is achieved by ℓ_p regularization combined with thresholding, which calls for a non-linear optimization scheme. The ensuing fully automated algorithm leverages physics-based constraints for the automatic determination of the penalty parameter in the regularization term. Using numerically generated data including artificial noise, we demonstrate the ability of the approach to accurately discover five hyperelastic models of different complexity. We also show that, if a “true” feature is missing in the function library, the proposed approach is able to surrogate it in such a way that the actual response is still accurately predicted.