Learning Physically-Informed Differential Viscoelastic Constitutive Equations From Data

Kyle Lennon, Massachusetts Institute of Technology

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Although a substantial body of work has been dedicated to deriving viscoelastic constitutive equations for particular classes of materials directly from physical considerations, these models often cannot sufficiently describe the diverse response space of viscoelastic materials in common experimentally and industrially relevant conditions. Recently, the advent of widely available machine learning (ML) tools has given rise to a new approach: learning constitutive models directly from data. While current ML approaches have shown some success in very particular circumstances, they are not easily portable to different flow conditions, tend to accommodate training data taken only by specific experimental protocols, and do not enforce key physical constraints such as invariance to rotating frames of reference. Here, we present a framework for learning physically-informed differential viscoelastic constitutive equations that combines the salient features of ML and physically-informed approaches. These models, which we call “rheological universal differential equations” (RUDEs), are composed of the upper-convected Maxwell (UCM) model with an added tensor-valued neural network, which takes the rate-of-deformation and stress tensors as inputs. The UCM model provides an underlying viscoelastic skeleton for the model, and the neural network provides a tool for learning material-specific features while preserving physical constraints such as frame invariance. Moreover, because RUDEs are differential and tensorial in form, they may be trained on — and ultimately used to predict — any observable related to the stress or strain obtained in an arbitrary flow protocol. We demonstrate these predictive capabilities using RUDEs trained on synthetic data from well-known constitutive models. With increased availability of a wide breadth of experimental data for viscoelastic materials, RUDEs open new avenues for efficient and accurate data-driven rheological modeling.

Abstract Author(s): Kyle R. Lennon, Gareth H. McKinley, James W. Swan