Another Special Case of Vibrations of a Rolling Tire

We investigate a special case of vibrations of a loaded tire rolling at constant speed. A previously proposed analytical model of a radial tire is considered. The surface of the tire is a flexible tread combined with elastic sidewalls. In the undeformed state, the tread is a circular cylinder. The tread is reinforced with inextensible cords. The tread is the part of the tire that makes actual contact with the ground plane. In the undeformed state, the sidewalls are represented by parts of two tori and consist of incompressible rubber described by the Mooney –Rivlin model. The previously obtained partial differential equation which describes the tire radial in-plane vibrations about steady-state regime of rolling is investigated. Analyzing the discriminant of the quartic polynomial, which is the function of the frequency of the tenth degree and the function of the angular velocity of the sixth degree, the rare case of a root of multiplicity three is discovered. The angular velocity of rotation, the tire speed and the natural frequency, corresponding to this case, are determined analytically. The mode shape of vibration in the neighborhood of the singular point is determined analytically.

CLASSIC STRAIN ENERGY FUNCTIONS AND CONSTITUTIVE TESTS OF RUBBER-LIKE MATERIALS

ABSTRACT Four classic strain energy density (SED) functions for incompressible rubber-like materials, neo-Hookean, Mooney-Rivlin, Yeoh, and Ogden forms, are briefly reviewed. The strain–stress relations of the above-mentioned SED functions for uniaxial, planar (pure shear), and equibiaxial deformation modes and formulas for transforming tensile data to compressive data of these three special deformation modes are given. Three approximate criteria are proposed for judging the validity of test data obtained from various methods. It is found that Treloar's data are the most reasonable among six groups of published data judged using the proposed criteria. Different combinations of Treloar's data were used to determine the parameters of these SED functions. The effects of three special tests on determining the parameters of various SED functions using the curve-fitting tools provided by ABAQUS were analyzed. Uniaxial tension data were found to be sufficient for determining the neo-Hookean, Yeoh, and first-order Ogden models, whereas both uniaxial and equibiaxial data were necessary for determining the Mooney-Rivlin and Ogden (N = 2, 3) models, and planar data were unnecessary to determine the parameters for all reviewed SED functions.

A Finite Element Procedure with Poisson Iteration Method Adopting Pattern Approach Technique for Near-Incompressible Rubber Problems

A finite element procedure is presented for the analysis of rubber-like hyperelastic materials. The volumetric incompressibility condition of rubber deformation is included in the formulation using the penalty method, while the principle of virtual work is used to derive a nonlinear finite element equation for the large displacement problem that is presented in a total-Lagrangian description. The behavior of rubber deformation is represented by hyperelastic constitutive relations based on a generalized Mooney-Rivlin model. The proposed finite element procedure using analytic differentiation exhibited results that matched very well with those from the well-known commercial packages NISA II and ABAQUS. Furthermore, the convergence of equilibrium iteration is quite slow or frequently fails in the case of near-incompressible rubber. To prevent such phenomenon even for the case that Poisson's ratio is very close to 0.5, Poisson's ratio of 0.49000 is used, first, to get an approximate solution without any difficulty; then the applied load is maintained and Poisson's ratio is increased to 0.49999 following a proposed pattern and adopting a technique of relaxation by monitoring the convergence rate. For a given Poisson ratio near 0.5, with this approach, we could reduce the number of substeps considerably.

The Study on the Mechanical Properties of Closed-Cell Foam Rubber Based on Microstructure Model

This paper analyses the mechanical behavior of matrix material and cell structure of closed-cell foam rubber by Gibson-Ashby model and the strain energy function of incompressible rubber-like material. The constitutive relation of closed-cell foam rubber is established by the overlay analysis of the two impact factors. The constitutive data for the new model is fitted and the uniaxial compression experiment is utilized to prove the feasibility of the constitutive equation produced in this paper. The model is in good agreement with the experimental results.

The stress field in a pulled cork and some subtle points in the semi-inverse method of nonlinear elasticity

In an attempt to describe cork-pulling, we model a cork as an incompressible rubber-like material and consider that it is subject to a helical shear deformation superimposed onto a shrink fit and a simple torsion. It turns out that this deformation field provides an insight into the possible appearance of secondary deformation fields for special classes of materials. We also find that these latent deformation fields are woken up by normal stress differences. We present some explicit examples based on the neo-Hookean, the generalized neo-Hookean and the Mooney–Rivlin forms of the strain-energy density. Using the simple exact solution found in the neo-Hookean case, we conjecture that it is advantageous to accompany the usual vertical axial force by a twisting moment, in order to extrude a cork from the neck of a bottle efficiently. Then we analyse departures from the neo-Hookean behaviour by exact and asymptotic analyses. In that process, we are able to give an elegant and analytic example of secondary (or latent) deformations in the framework of nonlinear elasticity.