Neural networks meet hyperelasticity: A guide to enforcing physics

Abstract

In the present work, a hyperelastic constitutive model based on neural networks is proposed which fulfills all common constitutive conditions by construction, and in particular, is applicable to compressible material behavior. Using different sets of invariants as inputs, a hyperelastic potential is formulated as a convex neural network, thus fulfilling symmetry of the stress tensor, objectivity, material symmetry, polyconvexity, and thermodynamic consistency. In addition, a physically sensible stress behavior of the model is ensured by using analytical growth terms, as well as normalization terms which ensure the undeformed state to be stress free and with zero energy. The normalization terms are formulated for both isotropic and transversely isotropic material behavior and do not violate polyconvexity. By fulfilling all of these conditions in an exact way, the proposed physics-augmented model combines a sound mechanical basis with the extraordinary flexibility that neural networks offer. Thus, it harmonizes the theory of hyperelasticity developed in the last decades with the up-to-date techniques of machine learning. Furthermore, the non-negativity of the hyperelastic potential is numerically verified by sampling the space of admissible deformations states, which, to the best of the authors' knowledge, is the only possibility for the considered nonlinear compressible models. The applicability of the model is demonstrated by calibrating it on data generated with analytical potentials, which is followed by an application of the model to finite element simulations. In addition, an adaption of the model to noisy data is shown and its extrapolation capability is compared to models with reduced physical background. Within all numerical examples, excellent and physically meaningful predictions have been achieved with the proposed physicsaugmented neural network.

A comparative study on different neural network architectures to model inelasticity

“…Common activation functions are the hyperbolic tangent tanh(x), the rectifier ReLU(x) ∶= max(0, x) or the softplus activation SP(x) ∶= ln(1 + exp(x)). 11,60,61 Particularly, the outputs  L n of the output layer L can also be expressed with (9). Recursively, the outputs of the neurons in the previous layers  l n can be substituted with Equation ( 9) until layer 1 with its inputs  1 n = i n is reached.…”

“…To address this issue, a relatively new approach in NN-based constitutive modeling, and scientific machine learning (ML) in general, is to integrate crucial underlying physics in either a strong or weak form. These methods, known as physics-informed, 4,5 mechanics-informed, 6,7 physics-augmented, 8,9 physics-constrained, 10 or thermodynamics-based, 11 improve extrapolation capability and allow for the use of sparse training data. Thereby, various conditions such as thermodynamic consistency, material symmetry, objectivity, or also material stability can be considered.…”

“…Meanwhile, FNNs using invariants as input and the hyperelastic potential as output are a very well established approach. [8][9][10][14][15][16][17][18] Thereby, an improved training is applied that allows calibration of the network directly by tuples of stress and strain, that is, the derivative of energy with respect to the deformation is included into the loss, which is also called Sobolev training. 19,20 Alternatively, a network previously trained to predict stress coefficients can be used to construct a pseudopotential, thus ensuring thermodynamic consistency of NN-based elastic models a posteriori.…”

“…This is especially relevant for the more general 3D finite strain case. 9 In this article, conventional linear and nonlinear viscoelastic as well as elastoplastic models are used for training data generation and, later on, as reference. After training with random walk time sequences containing information on stress, strain, and-for some models-internal variables, the NN-based models are compared to the reference solution.…”

Interpretable data‐driven modeling of hyperelastic membranes

“…14,15 By interpretability we mean transparency of the mechanism by which the model works. 16 There are works that introduce physics-based constraints [17][18][19] to obtain neural networks based constitutive equations, nevertheless black-box nature of the approach raises the question of their interpretability. To overcome this problem, we propose an interpretable data-driven approach for simulation of hyperelastic materials.…”