Finite element analysis for three-dimensional hyper-elastic problems

Abstract

Hyper-elastic materials are special elastic material that the stress can be derived by the strain energy density function. The most attractive property of these rubber-like materials is their ability to undergo large strains with small loads and recover their initial shape after unloading. Hyper-elastic materials are widely used in engineering such as tire, elastomeric bearing pad, belt drives, rail pad and so on. Due to the large strain state in most of application of this type of material, the behavior of hyper-elastic is often considered in finite deformation analysis. In comparison with linear material, analysis for hyper-elastic material which have nonlinearities is more complicated because both material nonlinearity and geometric nonlinearity should be considered. Along with the development of computer and numerical methods, the finite element method (FEM) is known as a powerful numerical computation tool that can helps to solve engineering problems in many fields such as structural, thermal, electronic, and biomedical analyses. Actually, most of practical engineering problems appear in three-dimensional (3D) state, especially in nonlinear analyses. So, the development for an effective numerical method for 3D nonlinear problems is very necessary. In this study, a finite element approach with 8-node hexahedron element is presented for analyses of large deformation problems of hyper-elastic material. The compressible neo-Hookean is used as the constitutive model for 3D hyper-elastic problems and the total Lagrange formulation is applied for the discretization of large deformation problems. To obtain the nonlinear solutions, the standard Newton-Raphson algorithm with constant load step is chosen as the iteration method. The computing programs are built with Matlab language. To verify the accuracy of the present approach, the obtained results are compared with the reference solutions given by Ansys program.