Review Article Journal of Intelligent Material Systems and Structures 1–41 Ó The Author(s) 2020 Article reuse guidelines: sagepub.com/journals-permissions DOI: 10.1177/1045389X20916795 journals.sagepub.com/home/jim A comprehensive review on thermomechanical constitutive models for shape memory polymers Ebrahim Yarali 1 , Ali Taheri 2 and Mostafa Baghani 1 Abstract Shape memory polymers are a class of smart materials, which are capable of fixing their deformed shapes, and can return to their original shape in reaction to external stimulus such as heat. Also due to their exceptional properties, they are mostly used in four-dimensional printing applications. To model and investigate thermomechanical response of shape memory polymers mathematically, several constitutive equations have been developed over the past two decades. The purpose of this research is to provide an up-to-date review on structures, classifications, applications of shape memory polymers, and constitutive equations of thermally responsive shape memory polymers and their composites. First, a comprehensive review on the properties, structure, and classifications of shape memory polymers is conducted. Then, the proposed models in the literature are presented and discussed, which, particularly, are focused on the phase transi- tion and thermo-viscoelastic approaches for conventional, two-way as well as multi-shape memory polymers. Then, a statistical analysis on constitutive relations of thermally activated shape memory polymers is carried out. Finally, we pres- ent a summary and give some concluding remarks, which could be helpful in selection of a suitable shape memory poly- mer constitutive model under a typical application. Keywords shape memory polymer, four-dimensional printing, thermomechanical modeling, constitutive equations, thermally sensi- tive polymers 1. Introduction Smart materials are a class of materials that are pro- grammable and their properties change in reaction to external stimuli, for example, magnetic field, electrical field, moisture, pH, temperature, and light. Shape memory polymers (SMPs), shape memory alloys (SMAs), hydrogels, smart fluids, for example, electro- rheological fluid, magnetorheological fluid, ferrofluids, and self-repairing materials, are some well-known examples of smart materials (Bayat et al., 2020; Yarali et al., 2019a). Among these smart materials, during the past two decades, SMPs have drawn wide attention from mechanical point of view, and many experimental and modeling efforts have been made on these materi- als. SMPs can preserve their temporary shapes for a long time and recover their original (permanent) shape upon applying external stimuli such as heat (Zare et al., 2019), magnetic field (Testa et al., 2019), electrical field (Wan et al., 2019), light (Lendlein et al., 2005), pH (Lu et al., 2016), moisture or water (Yang et al., 2018), radiation (Wang et al., 2014a), dissolution (Xiao, 2016), focused ultrasound (Bhargava et al., 2017), external infrared (IR) light (Mailen et al., 2019), micro- waves (Chen et al., 2019), or laser light (Yenpech et al., 2019). Thanks to the shape memory effect (SME) and recovery properties of SMPs, they have wide range of potential applications in many fields, in particular, bio- medical devices. It should be noted that external stimuli can be applied to SMPs either direct (such as direct heat, magnetic field, electric fields) or indirect such as applying magnetic field to generate an induced heat in SMPs. Polymers commonly have no intrinsic considerable SMEs. In other words, a single polymer does not show significant SME by itself, but this effect is achieved by combining different polymers with different 1 School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran 2 Department of Mechanical Engineering, University of Larestan, Lar, Iran Corresponding author: Mostafa Baghani, School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran. Email: baghani@ut.ac.ir