Dozens of hyperelastic models have been formulated and have been extremely handy in understanding the complex mechanical behavior of materials that exhibit hyperelastic behavior (characterized by large nonlinear elastic deformations that are completely recoverable) such as elastomers, polymers, and even biological tissues. These models are indispensable in the design of complex engineering components such as engine mounts and structural bearings in the automotive and aerospace industries and vibration isolators and shock absorbers in mechanical systems. Particularly, the problem of vibration control in mechanical system dynamics is extremely important and, therefore, knowledge of accurate hyperelastic models facilitates optimum designs and the development of three‐dimensional finite element system dynamics for studying the large and nonlinear deformation behavior. This review work intends to enhance the knowledge of 15 of the most commonly used hyperelastic models and consequently help design engineers and scientists make informed decisions on the right ones to use. For each of the models, expressions for the strain‐energy function and the Cauchy stress for both arbitrary loading assuming compressibility and each of the three loading modes (uniaxial tension, equibiaxial tension, and pure shear) assuming incompressibility are provided. Furthermore, the stress–strain or stress–stretch plots of the model's predictions in each of the loading modes are compared with that of the classical experimental data of Treloar and the coefficient of determination is utilized as a measure of the model's predictive ability. Lastly, a ranking scheme is proposed based on the model's ability to predict each of the loading modes with minimum deviations and the overall coefficient of determination.

Shape-memory polymers (SMPs) and their composite materials are stimuli-responsive materials that have the unique characteristics of lightweight, large deformation, variable stiffness, and biocompatibility. This paper reviews the research status of the mechanical models for SMPs, shape-memory nanocomposites and shape-memory polymer composites (SMPCs); it also introduces some spatially deployable structures, such as hinges, beams, and antennae based on SMPCs. In addition, the deformation types of 4D printing structures and the potential applications of this technology in robots and medical devices are also summarized.

Electromechanical instability may occur in dielectric elastomer films due to the coupling between mechanical forces and electric fields. According to Zhao and Suo [Appl. Phys. Lett. 91, 061921 (2007)], free-energy in any form, which consists of elastic strain energy and electric energy, can be used to analyze the electromechanical stability of dielectric elastomer. By taking the permittivity as a variable depending on the deformation in a free energy function, a relationship is established among critical nominal electric field, critical real electric field, nominal stress, and principal stretch ratios. The accurate expressions of these parameters are presented for a special equal biaxial stretch case. All the results obtained by utilizing the single material constant neo-Hookean elastic strain energy model coincide with the conclusions by Zhao and Suo.

Shape memory polymers (SMPs) and SMP composites (SMPCs) have been widely employed in several fields and exhibit excellent self-actuation, deformation, and self-adaption. Establishing reasonable constitutive models is vital for understanding the shape memory mechanism and expanding its applications. Moreover, the mechanical response of SMPs under different conditions can be predicted, facilitating their precise control. The internal mechanism for the shape memory behavior in most SMPs is thermal actuation. This study reviews the theories of thermally actuated SMPs, rheological and phase transition concept models, and models combining the rheology and phase transition concepts. Furthermore, the constitutive models of particulate-reinforced SMPCs, carbon-fiber-reinforced SMPCs, and the buckling behavior of SMPCs are summarized. This study is expected to help solve the remaining issues rapidly and contribute to the establishment of rational constitutive models for SMPs and SMPCs.