In this investigation, the use of three different nonlinear constitutive models based on the hyper-elasticity theory with the absolute nodal coordinate formulation is considered. These three nonlinear constitutive models are based on the Neo-Hookean constitutive law for compressible materials, the Neo-Hookean constitutive law for incompressible materials, and the Mooney-Rivlin constitutive law in which the material is assumed to be incompressible. These models, which allow capturing Poisson modes, are suitable for many materials and applications, including rubber-like materials and biological tissues which are governed by nonlinear elastic behavior and are considered incompressible or nearly incompressible. Numerical examples that demonstrate the implementation of these nonlinear constitutive models in the absolute nodal coordinate formulation are presented. The results obtained using the nonlinear and linear constitutive models are compared in this study. The results show that when linear constitutive models are used in the large deformation analysis, singular configurations are encountered and basic formulas such as Nanson’s formula are no longer valid. These singular deformation configurations are not encountered when the nonlinear constitutive models are used.

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