Application Of The Meshless Local Petrov-Galerkin Method To The Problems Of Hyperelastic Materials

Meshless methods are developed based on the finite element method, which establish trial function independent of elements, thus they have some advantages to solve nonlinear mechanical and crack propagation problems, because of effectively abstaining from complex meshing and disadvantageous influence of elements distorting. The meshless local Petrov-Galerkin(MLPG) method is a new numerical technique presented in recent years, for it doesn't need any element or mesh for the energy integral or the purpose of interpolation, and it's termed as a truly meshless method. In recent years, Atluri and Long SY et al have made a lot of investigations on the MLPG approach and its applications. On the basis of their work, applications of MLPG method to large deformation analysis of rubber hyperelasticity are presented in this dissertation, due to the rubber and rubber-like materials are frequently used in the automotive industry, mechanical, civil, electrical and electronic engineering.At the beginning of the dissertation, recent developments of meshless methods are briefly summarized. Several kinds of meshless methods are reviewed and commented. And the advantages and disadvantages for a variety of meshless methods are outlined. Especially, the research and development on the numerical method of the hyperelastic materials are introduced in detail. Then, 4 types of trial functions and 6 types of test functions are explored based on the research of Atluri et al. A radial basis function with polynomials interpolation method and a Heaviside step function are chosen for establishing local Petrov-Galerkin formulations of hyperelasticity. Further, strain energy function is separated into volumetric and deviatoric parts, and the relationship between the Green-Lagrange strain tensor and second Piola-Kirchhoff stress tensor is given.There exist a lot of papers about the MLPG method, which are based on the hypothesis of linear-elasticity and small strain, but the papers for solving the nonlinear and large deformation problems are rarely presented. In this dissertation, the MLPG method is developed to hyperelastic material problems, and the local Petrov-Galerkin formulations are employed for static, dynamic problems, steady contact and transient impact problems, respectively. In the process of establishing equations, a simple Heaviside test function is chosen for eliminating or simplifying domain integrals. The condition of plane stress and a pressure projection method are used to overcome the incompressibility or nearly incompressibility in the plane stress...