  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. Additionally, strategies are provided to
reduce the required order of derivative for obtaining the tangent operator. The
trained model can be directly used in any finite element package (or other
numerical methods) as a user-defined material model. However, challenges remain
in the proper definition of collocation points and in integrating several
non-equality constraints that become active or non-active simultaneously. 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. We have observed a perfect agreement between the results obtained
through the proposed methodology and those obtained using the classical
approach. Furthermore, the proposed approach requires significantly less effort
in terms of implementation and computing time compared to the traditional
methods.
