Analysis of the compressible, isotropic, neo-Hookean hyperelastic model

Abstract

The most widely-used representation of the compressible, isotropic, neo-Hookean hyperelastic model is considered in this paper. The version under investigation is that which is implemented in the commercial finite element software ABAQUS, ANSYS and COMSOL. Transverse stretch solutions are obtained for the following homogeneous deformations: uniaxial loading, equibiaxial loading in plane stress, and uniaxial loading in plane strain. The ground-state Poisson’s ratio is used to parameterize the constitutive model, and stress solutions are computed numerically for the physically permitted range of its values. Despite its broad application to a number of engineering problems, the physical limitations of the model, particularly in the small to moderate stretch regimes, are not explored. In this work, we describe and analyze results and make some critical observations, underlining the model’s advantages and limitations. For example, a snap-back feature of the transverse stretch is identified in uniaxial compression, a physically undesirable behavior unless validated by experimental data. The domain of this non-unique solution is determined in terms of the ground-state Poisson’s ratio and the state of stretch and stress. The analyses we perform are essential to enable the understanding of the characteristics of the standard, compressible, isotropic, neo-Hookean model used in ABAQUS, ANSYS and COMSOL. In addition, our results provide a framework for the parameter-fitting procedure needed to characterize this standard, compressible, isotropic neo-Hookean model in terms of experimental data.

Stability study of the compressible Mooney-Rivlin hyperelastic model

“…One is where C 01 << C 10 , in the limit case β = 0 , and the model reduces to the neo-Hookean model for which the stability phenomenon is discussed in. 17 The second case is C 01 >> C 10 , where in limit case β = ∞ . These two extreme scenarios are examined in the context of uniform loadings, shown in Figure 1.…”

“…In our study the conditions ( C 10 + C 01 ) > 0 and C 01 > 0 are applied to enforce stable behavior for the incompressible case in the entire domain. 17 The definition of the ground-state Poisson’s ratio ν 0 can be obtained by linearizing the Cauchy stress solution in the uniaxial loading case. Inserting Eq.…”

“…A recent study of the compressible neo-Hookean model (cNH) revealed some unstable behavior of this simple hyperelastic model. 17 Although the strain energy density function of the cNH model is relatively simple, no closed-form stress solutions can be obtained even for the simplest homogeneous deformations. Calculating stretch-stress solutions is an inevitable requirement for material parameter identification, including defining the permittable range of the model.…”

“…The methodology of this article is based on a recent study in which the authors provide stability limits for the compressible neo-Hookean model. 17…”

“…The methodology of this article is based on a recent study in which the authors provide stability limits for the compressible neo-Hookean model. 17 Three commonly used uniform deformations are examined: uniaxial loading, plane stress equibiaxial loading and plane strain uniaxial loading. It is demonstrated that transverse stretches cannot be expressed analytically and must be solved by numerical methods.…”