Buckling of a spinning elastic cylinder: linear, weakly nonlinear and post-buckling analyses

An elastic cylinder spinning about a rigid axis buckles beyond a critical angular velocity, by an instability driven by the centrifugal force. This instability and the competition between the different buckling modes are investigated using analytical calculations in the linear and weakly nonlinear regimes, complemented by numerical simulations in the fully post-buckled regime. The weakly nonlinear analysis is carried out for a generic incompressible hyperelastic material. The key role played by the quadratic term in the expansion of the strain energy density is pointed out: this term has a strong effect on both the nature of the bifurcation, which can switch from supercritical to subcritical, and the buckling amplitude. Given an arbitrary hyperelastic material, an equivalent shear modulus is proposed, allowing the main features of the instability to be captured by an equivalent neo-Hookean model.

Wrinkles and creases in the bending, unbending and eversion of soft sectors

We study what is clearly one of the most common modes of deformation found in nature, science and engineering, namely the large elastic bending of curved structures, as well as its inverse, unbending, which can be brought beyond complete straightening to turn into eversion. We find that the suggested mathematical solution to these problems always exists and is unique when the solid is modelled as a homogeneous, isotropic, incompressible hyperelastic material with a strain-energy satisfying the strong ellipticity condition. We also provide explicit asymptotic solutions for thin sectors. When the deformations are severe enough, the compressed side of the elastic material may buckle and wrinkles could then develop. We analyse, in detail, the onset of this instability for the Mooney–Rivlin strain energy, which covers the cases of the neo-Hookean model in exact nonlinear elasticity and of third-order elastic materials in weakly nonlinear elasticity. In particular, the associated theoretical and numerical treatment allows us to predict the number and wavelength of the wrinkles. Guided by experimental observations, we finally look at the development of creases, which we simulate through advanced finite-element computations. In some cases, the linearized analysis allows us to predict correctly the number and the wavelength of the creases, which turn out to occur only a few per cent of strain earlier than the wrinkles.

CAE-Based application for identification and verification of hyperelastic parameters

The main objective of this study is to develop a CAE-based application with a convenient GUI for identification and verification of material parameters for hyperelastic models available in the current release of the FE code ANSYS Mechanical APDL. This Windows application implements a two-step procedure: (1) fitting of experimental stress–strain curves provided by the user; (2) verification of the obtained material parameters by the solution of a modified benchmark problem. The application, which was developed using the Visual Basic.NET language, implements a two-way interaction with ANSYS as a single loop using the APDL script as input and text, graphical and video files as output. With this application, nine isotropic incompressible hyperelastic material models are compared by fitting them to the conventional Treloar’s experimental dataset (1944) for vulcanised rubber. A ranking of hyperelastic models is constructed according to model efficiency, which is estimated using fitting quality criteria. The model ranking is done based upon the complexity of their mathematical formulation and their ability to accurately reproduce the test data. Recent hyperelastic models (Extended Tube and Response Function) are found to be more efficient compared to conventional ones. The verification is done by the comparison of an analytical solution to an FEA result for the benchmark problem of a rubber cylinder under compression proposed by Lindley (1967). In the application, the classical formulation of the benchmark is improved mathematically to become valid for larger deformations. The wide applicability of the proposed two-step approach is confirmed using stress–strains curves for seven different formulations of natural rubber and seven different grades of synthetic rubber.

A comparison of hyperelastic constitutive models applicable to brain and fat tissues

In some soft biological structures such as brain and fat tissues, strong experimental evidence suggests that the shear modulus increases significantly under increasing compressive strain, but not under tensile strain, whereas the apparent Young's elastic modulus increases or remains almost constant when compressive strain increases. These tissues also exhibit a predominantly isotropic, incompressible behaviour. Our aim is to capture these seemingly contradictory mechanical behaviours, both qualitatively and quantitatively, within the framework of finite elasticity, by modelling a soft tissue as a homogeneous, isotropic, incompressible, hyperelastic material and comparing our results with available experimental data. Our analysis reveals that the Fung and Gent models, which are typically used to model soft tissues, are inadequate for the modelling of brain or fat under combined stretch and shear, and so are the classical neo-Hookean and Mooney–Rivlin models used for elastomers. However, a subclass of Ogden hyperelastic models are found to be in excellent agreement with the experiments. Our findings provide explicit models suitable for integration in large-scale finite-element computations.

On the global stability of two-dimensional, incompressible, elastic bars in uniaxial extension

When a rectangular bar is subjected to uniaxial tension, the bar usually deforms (approximately) homogeneously and isoaxially until a critical load is reached. A bifurcation, such as the formation of shear bands or a neck, may then be observed. One approach is to model such an experiment as the in-plane extension of a two-dimensional, homogeneous, isotropic, incompressible, hyperelastic material in which the length of the bar is prescribed, the ends of the bar are assumed to be free of shear and the sides are left completely free. It is shown that standard constitutive hypotheses on the stored-energy function imply that no such bifurcation is possible in this model due to the fact that the homogeneous isoaxial deformation is the unique absolute minimizer of the elastic energy. Thus, in order for a bifurcation to occur either the material must cease to be elastic or the stored-energy function must violate the standard hypotheses. The fact that no local bifurcations can occur under the assumptions used herein was known previously, since these assumptions prohibit the load on the bar from reaching a maximum value. However, the fact that the homogeneous deformation is the absolute minimizer of the energy appears to be a new result.