Exploiting symmetries in the modeling and analysis of tires

Abstract

Three basic types of symmetry (and their combinations) exhibited by tire response are identified. A simple and efficient computational strategy is presented for reducing both the size of the model and the cost of the analysis of tires in the presence of symmetry-breaking conditions (e.g., unsymmetry of the tire material, geometry, and/or loading). The strategy is based on approximation of the unsymmetric response of the tire with a linear combination of symmetric and antisymmetric global approximation vectors ( or modes). The three main elements of the computational strategy are as follows: (1) use of three-field mixed finite element models having independent shape functions for stress resultants, strain components, and generalized displacements, with the stress resultants and the strain components allowed to be discontinuous at interelement boundaries; (2) use of operator splitting (additive decomposition of some of the matrices and vectors in the finite element model) to delineate the symmetric and antisymmetric contributions to the response; and (3) successive use of the finite element method and the classic Rayleigh-Ritz technique to substantially reduce the number of degrees of freedom. The finite element method is first used to generate a few global approximation vectors (or modes). Then the amplitudes of these modes are computed with the Rayleigh-Ritz technique. The proposed computational strategy is applied to three quasi-symmetric problems of tires, namely, (1) linear analysis of anisotropic tires through the use of semianalytic finite elements, (2) nonlinear analysis of anisotropic tires through the use of two-dimensional shell finite elements, and (3) nonlinear analysis of orthotropic tires subjected to unsymmetric loading. In the first two applications, the anisotropy (nonorthotropy) of the tire is the source of the symmetry breaking; in the third application, the quasi-symmetry is due to the unsymmetry of the loading. The effectiveness of the proposed computational strategy is also demonstrated with numerical examples, and its potential for handling practical tire problems is outlined.