Modelling the Inflation and Elastic Instabilities of Rubber-Like Spherical and Cylindrical Shells Using a New Generalised Neo-Hookean Strain Energy Function

Abstract

The application of a newly proposed generalised neo-Hookean strain energy function to the inflation of incompressible rubber-like spherical and cylindrical shells is demonstrated in this paper. The pressure ($P$ P ) – inflation ($\lambda $ λ or $v$ v ) relationships are derived and presented for four shells: thin- and thick-walled spherical balloons, and thin- and thick-walled cylindrical tubes. Characteristics of the inflation curves predicted by the model for the four considered shells are analysed and the critical values of the model parameters for exhibiting the limit-point instability are established. The application of the model to extant experimental datasets procured from studies across 19th to 21st century will be demonstrated, showing favourable agreement between the model and the experimental data. The capability of the model to capture the two characteristic instability phenomena in the inflation of rubber-like materials, namely the limit-point and inflation-jump instabilities, will be made evident from both the theoretical analysis and curve-fitting approaches presented in this study. A comparison with the predictions of the Gent model for the considered data is also demonstrated and is shown that our presented model provides improved fits. Given the simplicity of the model, its ability to fit a wide range of experimental data and capture both limit-point and inflation-jump instabilities, we propose the application of our model to the inflation of rubber-like materials.

On Modelling Simple Shear for Isotropic Incompressible Rubber-Like Materials

“…Indeed, the higher accuracy of the model predictions of Eq. ( 9) versus the Gent model has been previously demonstrated [26,31,37]. Building on these results, here we wish to specialise the application of this model to the problem of simple shear and demonstrate the various modes of simple shear response that the model is capable of capturing and predicting.…”

“…A detailed account of the structural derivation of the limiting chain extensibility model ( 9) based on the kinetic molecular theory of rubber elasticity has been presented in [26] and [28]. The successful application of this model to various finite deformation modes of rubberlike materials including in-plane deformations, inflation, torsion and the ensuing elastic instabilities has been presented in our previous studies [26,[30][31][32][33]. Horgan [34] demonstrated that this model is directly related to the celebrated two parameter Gent model [35], in that the first derivatives of W with respect to I 1 in these two models differ by just a constant.…”

On a new class of non-Gaussian molecular-based constitutive models with limiting chain extensibility for incompressible rubber-like materials

“…In the application of these two models to experimental data pertaining to the deformation of elastomers, the more accurate approximation of the inverse Langevin function ℓ − 1 by the Anssari-Benam and Bucchi model leads to various favourable improvements in the modelling outcomes. A detailed account of the achieved improvements using various extant deformation datasets has been presented in [15] and [35], and we hence refrain from replicating those results here.…”

“…Useful and versatile as generalised neo-Hookean functions are, datasets may be encountered at times that are not captured accurately by such models, as for example noted by Pucci and Saccomandi [37] in relation to the Gent model applied to the Treloar dataset within the small range of deformation. Another example was given in [35] in relation to the model by Anssari-Benam and Bucchi applied to the inflation data due to Rivlin and Saunders [38]. In such circumstances, the addition of an I 2 term to the generalised neo-Hookean function has proven to be advantageous in providing more accurate fits.…”

“…However, we note that not any I 2 term will always prove to be useful. As a case in point, the inflation data of Rivlin and Saunders [38] was captured accurately only when the Pucci and Saccomandi I 2 term was added to the generalised neo-Hookean model in [15]; the commonly used C 2 ( I 2 − 3 ) did not improve the fit [35].…”