Computational Mechanics of Rubber and Tires

Some other complications in the course of numerical simulation of rubber and tires are rubber near-incompressibUity and reinforcement in tires. The rubber nearincompressibility makes the system of finite element equations ill conditioned since in this case the volumetric stiffness greatly exceeds the shear stiffness. The nearincompressibility and incompressibility conditions are essentially constraints imposed on the solution, and depending on the ratio of the number of discrete equations and discrete number of constraints solution may or may not exist. Therefore, the design of specific finite elements to satisfy these conditions becomes very important. 

Contact conditions add even more difficulty and complexity to an already very complex and difficult analysis of rubber products and tires. Contact conditions are unilateral and need to be constantly checked during the incremental nonlinear analysis. In addition, they are not smooth, thus degrading the performance of nonlinear solvers. A number of numerical regularization parameters need to be introduced to prevent chattering and ensure robustness of a finite element analysis (FEA) with frictional contact. 

In this chapter, we will both discuss the above issues and deal with special topics of the finite element analysis of tires. 

As we have mentioned in the introduction, rubber parts typically experience large displacements and strains during their deformation history and, therefore, linearization based on the theory infinitesimal strains and small displacements that is traditionally employed for steel, reinforced concrete, and so on will produce inaccurate results. In order to retain the accuracy and realistic description of the deformation process in mbber, a fully nonlinear description of the deformation process should be considered. In the following discussion, we will obtain discretized finite element equations and outline their solution methods. 

To outline a finite element analysis approach, we will formulate a boundary value problem, transform it into a weak or variational form, and obtain discretized finite element equations. We begin with the equations of equihhrium that are written in the deformed configuration [1]  

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The chapter by PoldnefF and Heinstein ( Computational Mechanics of Rubber and Tires ) reviews the latest achievements in using finite element analysis for solving highly nonlinear problems of rubber and tire mechanics, with several illustrative examples of industrial importance. 

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