Revisiting Preparation of Phase Space for Learning Path-Dependent Behavior via Deep Neural Networks

This technical note investigates the preparation of a phase space of two simple constitutive laws, von Mises (J2) and Drucker-Prager (DP) models, for deep neural networks with special attention to elastic and plastic data points. The phase space is referred to as a database describing the material behavior via data points instead of mathematical formulation. To examine the effect of the proportions of elastic and plastic data points on the prediction quality, two sets (for J2 and DP models) including three equal phase spaces with different proportions of elastic and plastic data points are considered. To make a fair comparison, the only difference in phase spaces is the proportion of elastic and plastic data points in each set. We further study two different data types for constitutive behaviors, the tensorial space and the invariant space (volumetric-deviatoric space). The Nash-Sutcliffe error (NSE) is calculated for an unbiased comparison of prediction with different phase spaces in both tensorial and volumetric-deviatoric spaces. The results reveal that the distribution of elastic and plastic points may affect the accuracy of prediction for simple constitutive laws (isotropic, elasto-perfect plastic) via artificial neural networks; when an equal portion of elastic and plastic data points are used, the more robust prediction is achieved in this study.