Coupling the virtual fields method with ANNs for implicit constitutive modelling

Training an ANN for implicit constitutive modelling mostly relies on paired data, usually stress-strain. However, stress is not measurable in a real experiment. As such, training is carried out indirectly using only measurable variables, such as displacements and global force. In this work, displacements and force data are used to train an ANN to predict the stress state of a material. A numerical experiment was created to obtain displacement and global force data for different load distributions. The strain from two subsequent time increments are 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.