New Compressible Hyper-elastic Models

Soft materials, such as rubberlike materials, are highly elastic and give rise to large recoverable deformations under relatively low stress levels. Toward reasonable design in engineering and efficient applications of highly elastic soft materials, we need to establish hyper-elastic constitutive models for general multi-axial compressible deformation.This contribution proposes a new explicit approach toward obtaining a hyper-elastic multi-axial potential for general compressible multi-axial deformation directly from test data for uniaxial stress-strain relationship. This new approach is composed of several procedures. Firstly, the elastic potential based on logarithmic strain is formulated. Secondly, the Poisson ratio at infinitesimal strain is extended to large deformation case for the purpose of characterizing the compressibility behavior at finite strain. Thirdly, a uniaxial stress-strain relation is obtained, which can excellently match uniaxial data by using a simple form of rational functions with the two poles, and then a single-variable potential will be derived from the uniaxial stress-strain relation. Fourthly, the single-variable potential obtained is extend to two multi-axial potentials by simply replacing the uni-axial Hencky strain with the two bridging invariants for general multi-axial deformations. Finally, a unified potential is obtained by combining the two multi-axial potentials through Hermite interpolation.Novelties in three respects are incorporated in the new constitutive model:(i)The new constitutive model for general compressible deformation is free of the commonly assumed constraint of incompressibility and thus bypasses possible issues resulting from this constraint. Meanwhile, the incompressibility can be derived as a natural limit.(ii)Constitutive parameters of direct physical meanings may be introduced to represent features of hyper-elastic materials.(iii) The new constitutive model can achieve excellently agreement with benchmark data for several deformation modes, including uniaxial extension, equi-biaxial extension as well as plane-strain extension and others.