# Parameter Identification Methods for Hyperelastic and Hyper-Viscoelastic Models
## 1 Introduction  
With the rapid development of modern industrialization, rubbers are one of the most remarkable materials having a wide range of applications in civil engineering, aerospace engineering, mechanical engineering, automotive engineering, etc. In order to meet various requirements of the industry, special fillers, like carbon black or silica, with different proportions are usually added during vulcanization for improving the strength and toughness properties, which in turn makes it difficult to accurately characterize the mechanical a properties of rubbers [1]. Rubbers usually present a number of interesting features like hyperelasticity and hyper-viscoelasticity, many studies and experiments have been performed to investigate these features. For the former one, the stress-strain relationship of the hyperelasticity can be illuminated by a strain energy function W, and some mathematical models have been developed for the function, among which the strain-invariant-based models mainly include the Mooney-Rivlin model [2], the Yeoh model [3], and the Gent model [4], the principal-stretch-based models mainly include the Ogden model [5]. All the models have been verified by experiment results, for example [PERSON]'s experiment [6], and the Ogden model breaks through the limitation of the strain energy function being the even power of the stretches, which is capable of more accurately fitting the experimental data when rubbers undergo extremely large deformations. Comparisons of theperformances of 20 hyperelastic models are presented in the article from [PERSON] and [PERSON] [7]. More detailed reviews on the hyperelasticity of rubber can be accessed in several literatures [8; 9; 10]. The identification of the parameters in theses hyperelastic models has also gained much attention. For example, [PERSON] and [PERSON] [11] employed the Levenberg-Marquardt optimization algorithm to determine the material constants in Ogden models, [PERSON] et al. [12; 13] developed a professional scheme named COMPARE to get the parameters by using uniaxial tension, biaxial tension, and planar tension data. The constitutive relations for the viscoelasticity can be characterized by combining the elastic components with viscous components. In the past decades, numerous studies focused on investigating the viscoelasticity properties of rubber-like materials and many models explaining such behaviors were proposed. In the range of small strain, the basis of linear viscoelasticity theory was presented by [PERSON] et al. [14]. A relaxation-modulus constitutive function for the incompressible isotropic materials based on the results of uniaxial and equal biaxial stress relaxation tests was developed by [PERSON] [15]. The phenomenological theory of linear viscoelasticity was proved powerful for modelling the static and dynamic responses of rubber when subjected to small strain [16; 17]. For the finite deformation of viscoelastic materials, two important hypotheses were assumed. One is a split of the free energy of viscoelastic solid into an equilibrium and a non-equilibrium part, and the other one is a multiplicative decomposition of the deformation gradient tensor into elastic and inelastic parts [18]. In such a framework, Huber and Tsakmak is generalized the application range of the so-called three-parameter solids to the finite deformation [19]. [PERSON] et al. proposed a fairly simple identification method for the constants in a finite viscoelasticity model [20]. [PERSON] et al. presented a constitutive model comprised by an elastoplastic body with a strain-dependent isotropic hardening law and a hyperelastic body with a damage model [21]. [PERSON] et al. proposed an improved hyperelastic constitutive equation for rubbers in compression and shear regimes [1; 22; 23], the parameters in the equation were identified by experiment data and a nonlinear viscous coefficient was introduced to represent the rate-dependent properties.