This paper presents a critical review of the nonlinear dynamics of hyperelastic structures. Hyperelastic structures often undergo large strains when subjected to external time-dependent forces. Hyperelasticity requires specific constitutive laws to describe the mechanical properties of different materials, which are characterised by a nonlinear relationship between stress and strain. Due to recent recognition of the high potential of hyperelastic structures in soft robots and other applications, and the capability of hyperelasticity to model soft biological tissues, the number of studies on hyperelastic structures and materials has grown significantly. Thus, a comprehensive explanation of hyperelastic constitutive laws is presented, and different techniques of continuum mechanics, which are suitable to model these materials, are discussed in this literature review. Furthermore, the sensitivity of each hyperelastic strain energy density function to coefficient variation is shown for some well-known hyperelastic models. Alongside this, the application of hyperelasticity to model the nonlinear dynamics of polymeric structures (e.g., beams, plates, shells, membranes and balloons) is discussed in detail with the assistance of previous studies in this field. The advantages and disadvantages of hyperelastic models are discussed in detail. This present review can stimulate the development of more accurate and reliable models.

In this article, a comprehensive study on mechanical behaviors of nonhomogenous nonuniform size-dependent beams is presented. Beam is modeled with axially functionally graded materials with different types of materials and functions for material variation. Also, the beam is assumed with nonuniform cross-section and scale effects are presented by having both strain gradient and nonlocal effects. Nonuniformity is modelled in several ways by having tapered, parabolic and higher-order cross-section variation in both thickness and width directions. By having these assumptions, a great majority of small-scale beams from uniform homogeneous beams to nonuniform axially functionally graded beams are modeled and formulated. For this general beam model, static deformation, stability, and free vibration response, which are the three most important mechanical behaviors of such structures, are investigated. Finite element method in conjunction with numerical integration, Gaussian quadrature method, and Wilson's Lagrangian multiplier are employed to solve different problems investigated in this study by combining different assumptions. In order to show the influence of different parameters on mechanical behavior of such structures, a parametric study is presented and the effects by varying material, cross-section, using different type of functions, etc. on bending, buckling, and free vibration response of size-dependent beams are illustrated. It is seen that by combining different cross-section and material variations through the length, mechanical behavior of nanobeams change significantly which could be a step-forward in optimize designing nanostructures.

In this paper, dynamic behavior of multi-layered viscoelastic nanobeams resting on a viscoelastic medium with a moving nanoparticle is studied. Eringens nonlocal theory is used to model the small scale effects. Layers are coupled by Kelvin-Voigt viscoelastic medium model. Hamilton's principle, eigen-function technique and the Laplace transform method are employed to solve the governing differential equations. Analytical solutions for transverse displacements of double-layered is presented for both viscoelastic nanobeams embedded in a viscoelastic medium and without it while numerical solution is achieved for higher layered nanobeams. The influences of the nonlocal parameter, stiffness and damping parameter of medium, internal damping parameter and number of layers are studied while the nanoparticle passes through. Presented results can be useful in analysing and designing nanocars, nanotruck moving on surfaces, racing nanocars etc.