Learning solutions of thermodynamics-based nonlinear constitutive material models using physics-informed neural networks

We applied physics-informed neural networks to solve the constitutive relations for nonlinear, path-dependent material behavior. As a result, the trained network not only satisfies all thermodynamic constraints but also instantly provides information about the current material state (i.e., free energy, stress, and the evolution of internal variables) under any given loading scenario without requiring initial data. One advantage of this work is that it bypasses the repetitive Newton iterations needed to solve nonlinear equations in complex material models. Furthermore, after training, the proposed approach requires significantly less effort in terms of implementation and computing time compared to the traditional methods. The trained model can be directly used in any finite element package (or other numerical methods) as a user-defined material model. We tested this methodology on rate-independent processes such as the classical von Mises plasticity model with a nonlinear hardening law, as well as local damage models for interface cracking behavior with a nonlinear softening law. In order to demonstrate the applicability of the methodology in handling complex path dependency in a three-dimensional (3D) scenario, we tested the approach using the equations governing a damage model for a three-dimensional interface model. Such models are frequently employed for intergranular fracture at grain boundaries. However, challenges remain in the proper definition of collocation points and in integrating several non-equality constraints that become active or non-active simultaneously. As long as we are in the training regime, we have observed a perfect agreement between the results obtained through the proposed methodology and those obtained using the classical approach. Finally, we compare this new approach against available standard methods and discuss the potential and remaining challenges for future developments.