The Virtual Fields Method to Indirectly Train Artificial Neural Networks for Implicit Constitutive Modelling

Abstract

Artificial Neural Networks (ANNs) have the potential to provide a different approach to constitutive modelling, with the main advantage that these do not require to postulate a mathematical formulation or identify empirical parameters. Currently, the training of an ANN for implicit constitutive modelling mostly relies on paired data, usually stress-strain however, stress cannot be directly measured in a real experiment. As such, the training should be carried out indirectly using measurable variables from the experimental setting, such as displacements and the applied force. In the current work, displacements and global force data are used to indirectly train an ANN to predict the stress state of a material. An experimental test is recreated numerically in order to obtain displacement and global force data for different load distributions, i.e. obtaining synthetic data using a virtual experiment. The strain from the current and previous time increments are obtained from the corresponding displacements and used as inputs for the ANN to predict the current state of stress. Training is carried out without stress labels to compute the loss. Instead, the local and global equilibrium conditions, corresponding to the application of the Virtual Fields Method (VFM) to the physical model, are employed to compute the loss and update the network parameters, until the predicted stress state is accurate.