Insight into constitutive theories of 4D printed polymer materials: a review

Abstract Four-dimensional (4D) printing has emerged as a branch of additive manufacturing that utilizes stimuli-responsive materials to generate three-dimensional structures with functional features. In this context, constitutive models play a paramount role in designing engineering structures and devices using 4D printing, as they help understand mechanical behavior and material responses to external stimuli, providing a theoretical framework for predicting and analyzing their deformation and shape-shifting capabilities. This article thoroughly discusses available constitutive models for single-printed and multi-printed materials. Later, we explore the role of machine learning (ML) algorithms in inferring constitutive relations, particularly in viscoelastic problems and, more recently, in shape memory polymers. Moreover, challenges and opportunities presented by both approaches for predicting the mechanical behavior of 4D printed polymer materials are examined. Finally, we concluded our discussion with a summary and some future perspectives expected in this field. This review aims to open a dialogue among the mechanics community to assess the limitations of analytical models and encourage the responsible use of emerging techniques, such as ML. By clarifying these aspects, we intend to advance the understanding and application of constitutive models in the rapidly growing field of 4D printing.