Large deformation analysis of rubber based on a reproducing kernel particle method

A nonlinear formulation of the Reproducing Kernel Particle Method (RKPM) is presented for the large deformation analysis of rubber materials which are considered to be hyperelastic and nearly incompressible. In this approach, the global nodal shape functions derived on the basis of RKPM are employed in the Galerkin approximation of the variational equation to formulate the discrete equations of a boundary-value hyperelasticity problem. Existence of a solution in RKPM discretized hyperelasticity problem is discussed. A Lagrange multiplier method and a direct transformation method are presented to impose essential boundary conditions. The characteristics of material and spatial kernel functions are discussed. In the present work, the use of a material kernel function assures reproducing kernel stability under large deformation. Several of numerical examples are presented to study the characteristics of RKPM shape functions and to demonstrate the effectiveness of this method in large deformation analysis. Since the current approach employs C-m global shape functions, the method demonstrates a superior performance to the conventional finite element methods in dealing with large material distortions.