Data-driven elastoplastic constitutive modelling with physics-informed RNNs using the Virtual Fields Method for indirect training

The increasing demand for accurate material behaviour data in engineering simulations has exposed the limitations of traditional constitutive models. Although recent advances in full-field measurement techniques provide more detailed material characterization, conventional approaches still heavily rely on explicit assumptions and labour-intensive experimentation. This paper revisits the indirect training methodology introduced by the authors [1], which integrates Recurrent Neural Networks (RNNs) with Gated-Recurrent Units (GRUs) and the Virtual Fields Method (VFM) to model material behaviour without labelled data. The earlier study demonstrated the feasibility of training a GRU-based RNN using only global force and strain data through the VFM. Building on those findings, this work presents a more robust approach featuring an improved network architecture, with residual connections to enhance gradient flow and training stability, while also incorporating physics-based constraints. Extensive hyperparameter tuning was conducted to optimize the model and a sensitivity analysis was performed to assess the impact of the virtual fields on the accuracy and training dynamics. The models were validated using additional heterogeneous mechanical tests and the Reconstructed Axial Force Ratio (RAFR), as a key performance indicator, to further assess the physical correctness. The results show enhanced predictive accuracy and improved force reconstruction when larger sets of virtual fields are employed. Additionally, normalizing the VFM loss contributed to more consistent predictions and force reconstruction across all time stages. Stress contour plots further confirm the model’s ability to accurately predict stress distributions, which are in good agreement with the reference, with low median and average absolute errors.