A review on multiscale constitutive modeling of hard magnetic soft materials

Hard-magnetic soft materials (HMSMs) are smart-responsive composites consisting of a hard magnetic particle-reinforced phase embedded in a soft polymer matrix. Due to their unique magneto-mechanical coupling behavior, HMSMs can undergo precisely controllable and programmable deformations under external magnetic fields. These materials have broad application prospects in soft robotics, biomedicine, and microfluidics. To better understand the mechanical behavior of HMSMs and support their design and application, researchers have extensively investigated their constitutive theories and deformation characteristics. Several comprehensive review articles on these topics are available in the literature. It is important to note that HMSMs possess inherently multiscale structures. Their macroscopic mechanical properties depend on factors such as the spatial distribution of magnetic particles, the properties of the soft matrix, and other microstructural mechanical parameters. This paper provides a comprehensive review of hard magnetic soft materials (HMSMs) from a multiscale perspective. It first introduces the fundamental characteristics of HMSM components and subsequently categorizes existing constitutive models into three main types: microscale and nanoscale models, phenomenological macroscopic continuum models, and multiscale models based on homogenization theory. By comparing their theoretical foundations, applicable domains and limitations, this review highlights the critical role of multiscale correlation mechanisms in accurately predicting material behavior. Finally, the paper summarizes current theoretical developments and discusses potential directions for future research in this field.