Thermodynamically consistent constitutive modeling of isotropic hyperelasticity based on artificial neural networks

Herein, a neural network‐based constitutive model for isotropic hyperelastic solids which makes use of a physically motivated dimensionality reduction into the invariant space is presented. In order to automatically fulfill thermodynamic consistency, gradients of the network with respect to the input quantities are considered within a customized training loop. The proposed approach is exemplarily applied to the finite element simulation of two three‐dimensional samples, while only data collected from pure two‐dimensional virtual experiments are needed for the model calibration before.


Introduction
The classical constitutive modeling and calibration in continuum solid mechanics based on experimental or synthetic data is still challenging and time consuming in many cases. Due to this, a variety of data-driven approaches which circumvent this task and can handle even noisy data are currently developed [2,5]. Likewise, artificial neural networks (ANNs) are a possibility to automate constitutive modeling by approximating stress-strain relations. Within the training of ANNs, thermodynamic consistency can be ensured by choosing a suitable network architecture and loss function [3,4,6]. In this contribution, an ANN-based model for isotropic hyperelasticity is considered. A gradient-based training process is applied for the calibration.
2 Modeling framework based on artificial neural networks Isotropic hyperelasticity In the case of hyperelasticity, the symmetric 2nd Piola-Kirchhoff stress tensor T is given by T = 2∂ C ψ. Therein, ψ denotes the Helmholtz free energy density and C := F T · F is the right Cauchy-Green deformation tensor following from the deformation gradient F. If the constitutive response is furthermore restricted to the case of isotropy, ψ can be described in terms of the three deformation type principal invariants I α with α ∈ {1, 2, 3}. By using the chain rule, the relationship T = α f α G α for the calculation of the stress tensor can be achieved, where the so called stress coefficients are given by f α := 2∂ Iα ψ and the tensor generators follow to G α := ∂ C I α .
Data processing From now on, it is assumed that a denoised data set consisting of tuples D i := ( i C, i T) which are related to an unknown hyperelastic isotropic constitutive law is given. In order to predict the stress T for a given state of deformation C with high accuracy while keeping the number of neurons as few as possible, a dimensionality reduction of the input and output variables is necessary, respectively. Thus, following the work of Shen et al. [6], where the principle invariants are chosen as input values, a lower-dimensional data tuple D red In contrast to the principal invariants i I = ( i I 1 , i I 2 , i I 3 ) T ∈ R 3×1 , an analytical expression for the free energy function ψ is generally unknown. Thus, the previously introduced stress have to be calculated from the linear least squares method given in the equation above.
Model formulation and training process In order to fulfill the 2nd law of thermodynamics a priori, the ANN is now utilized on the energy level [4], i. e. ψ ANN : R 3×1 → R, i i → i ψ ANN , where the fractal symbol i i denotes a normalized value of i I. The network architecture is restricted to only one hidden layer containing N ∈ N neurons. Furthermore, the activation function is given by a hyperbolic tangent function and the output is chosen to be affine linear. Instead of using conventional training loops to determine the neural network's weights by only inserting values of ψ in the loss function directly, a gradient-based approach is adopted. Hence, to avoid a numerical calculation of the free energy ψ for the training process, the loss function which only takes into account the normalized derivative of the ANN-based model and the corresponding true values is defined. Moreover, compared to the development of a neural network predicting the stress coefficients directly, this approach reduces the number of trainable parameters B, W , w, b while the same number of neurons are included in the hidden layer. The reduced data set is randomly divided into training and test data to allow a generalization of the neural network to unknown data ( i i, i f). An implementation of the described workflow is realized using Python and Tensorflow.

Numerical examples
Finally, to demonstrate the ability of the proposed ANN-based method, synthetic stress-strain data generated with a highly nonlinear Ogden-type constitutive model are chosen. To enable the training process of the constitutive ANN within a broad subset of the invariant space, the deformation states of the training dataset have to be as heterogeneous as possible. For this purpose, data collected from a finite element (FE) calculation in which a square specimen with several elliptical holes is loaded under uniaxial tension. To obtain perfect plane stress data which are preferable from an experimental viewpoint [1], the virtual experiment is performed in a two-dimensional setting. By using the proposed training process, the weights of a comparatively small network containing 8 neurons in the hidden layer are determined from the data set. The constitutive ANN is implemented in the open source software FEniCS and verified by using two different threedimensional validation load cases. At first, a cuboid containing several cylindrical holes is loaded up to 40% relative elongation into a uniaxial tension mode, cf. Fig. 1 (a). Secondly, a torsional sample with three cylindrical holes is loaded by specifying a distortion of 45 • , cf. Fig. 1 (b). In order to assess the approximation quality of the trained ANN, the predicted local stress fields are compared to reference solutions generated with the original Ogden-type model. Regarding the relative stress errors |P Kl − P ANN Kl |/ max |P Kl | with P := T · F T denoting the 1st Piola-Kirchhoff stress tensor, rather low deviations below 1.5% could be achieved for both examples, cf. Fig. 1 (a), (b). Thus, the proposed method enables an accurate prediction of fully three-dimensional stress fields, although only two dimensional stress states were used for the training.