Abstract:Hyperelastic materials are commonly used in practical engineering with the prominent feature that a very large deformation may be produced under a force but the initial state can be completely recovered when the force is removed. Hyperelastic materials are typically nonlinear elastic ones, whose behaviors are in general characterized by their strain energy functions. For several decades, a lot of mathematical models and physical models have been proposed to study their constitutive relations through constructing the form of energy functions. However, a complete constitutive relation suitable for varied deformation modes and the entire deformation range is still the significant issue to expect in this field. This paper summarizes and analyzes the latest research status of constitutive relations of hyperelastic materials from three perspectives: (1) volume change modes including incompressible and compressible ones; (2) deformation modes such as uniaxial tension, shearing, biaxial tension and combined stretch and shear; (3) the entire range of deformation including small deformation, moderate deformation and large deformation. The latest progresses indicate that, in order to comprehensively describe experimental data of a given hyperelastic material and to apply it in practical problems, it is necessary to establish a complete constitutive relationship of compressible hyperelastic materials, which is suitable for varied deformation modes and the entire range of deformation. The authors suggest an implementation procedure for establishing the complete constitutive relationship of an actual hyperelastic material and an approach to construct the strain energy function of a compressible material.