Moritz Flaschel

Prakash Thakolkaran, Akshay Joshi, Yiwen Zheng et al.

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, (poly-)convexity, 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.

Moritz Flaschel, Hagen Holthusen, Denisa Martonová et al.

We recently proposed a method called Material Fingerprinting for the rapid discovery of mechanical material models that avoids solving continuous optimization problems. Material Fingerprinting assumes that each material exhibits a unique response when subjected to a standardized experimental setup, which is interpreted as the material's mechanical fingerprint. If a database of fingerprints is generated in an offline phase, a model for an unseen experimental measurement can be discovered in real time by comparing the experimentally measured fingerprint to the fingerprints in the database. In our original contributions, the database comprised a fixed number of material models, each with a fixed number of parameters. To increase the fitting flexibility of Material Fingerprinting, we propose an adaptive model database coupled with an iterative pattern recognition algorithm that refines the material model in each step. This strategy enables Material Fingerprinting to discover arbitrary linear combinations of material models from the database, rather than being restricted to selecting a single model from a predefined set. In comparison to previous works on Material Fingerprinting, this enables the discovery of more complex models, such as multi-term Ogden models or the anisotropic Holzapfel-Gasser-Ogden model. To design the adaptive database, we leverage sums of strain energy density feature functions that depend on isotropic and anisotropic invariants. All modeling features satisfy fundamental physical constraints, and polyconvexity can be optionally enforced via a simple user-controlled switch. We test the method on experimental data stemming from mechanical tests of isotropic rubber materials and anisotropic animal skin tissue.

Jan Niklas Fuhg, Govinda Anantha Padmanabha, Nikolaos Bouklas et al.

This review article highlights state-of-the-art data-driven techniques to discover, encode, surrogate, or emulate constitutive laws that describe the path-independent and path-dependent response of solids. Our objective is to provide an organized taxonomy to a large spectrum of methodologies developed in the past decades and to discuss the benefits and drawbacks of the various techniques for interpreting and forecasting mechanics behavior across different scales. Distinguishing between machine-learning-based and model-free methods, we further categorize approaches based on their interpretability and on their learning process/type of required data, while discussing the key problems of generalization and trustworthiness. We attempt to provide a road map of how these can be reconciled in a data-availability-aware context. We also touch upon relevant aspects such as data sampling techniques, design of experiments, verification, and validation.