Using neural networks to represent von Mises plasticity with isotropic hardening

Neural networks are universal function approximators that form the backbone of most modern machine learning based models. Starting from a conventional return-mapping scheme, the algorithmic description of von Mises plasticity with isotropic hardening is mathematically reformulated such that the relationship between the strain and stress histories may be modeled through a neural network function. In essence, the neural network provides an estimate of the instantaneous elasto-plastic tangent matrix as a function of the current stress and plastic work density. For plane stress conditions, it thus describes a non-linear mapping from R-4 to R-6. The neural network function is first developed for uniaxial stress conditions including loading histories with tension-compression reversal. Special attention is paid to the identification of the network architecture and artifacts related to overfitting. Furthermore, the performance of networks featuring the same number of total parameters, but different levels of non-linearity, is compared. It is found that a fully-connected feedforward network with five hidden layers and 15 neurons per layer can describe the plane stress plasticity problem with good accuracy. The results also show that a high density of training data (of the order of ten to hundred thousand points) is needed to obtain reasonable estimates for arbitrary loading paths with strains of up to 0.2. The final neural network model is implemented into finite element software through the user material subroutine interface. A simulation of a notched tension experiment is performed to demonstrate that the neural network model yields the same (heterogeneous) mechanical fields as a conventional J2 plasticity model. The present work demonstrates that it is feasible to describe the stress-strain response of a von Mises material through a neural network model without any explicit representation of the yield function, flow rule, hardening law or evolution constraints. It is emphasized that the demonstration of feasibility is the focus of the present work, while the assessment of potential computational advantages is deferred to future research.