A Modified Meshless Local Petrov-Galerkin Method for Nearly Incompressible Rubber Materials

The analysis of rubber materials is a challenging task in computational mechanics due to extremely large deformation and the nearly incompressible property of rubber. Despite the finite element methods (FEM) dealing with nonlinear structures have been well developed and a significant amount of work has been accomplished. But the widely used finite element methods are still ineffective in handling extreme material distortions due to the regularity requirement of interpolation functions and meshes. In recent years, a number of meshless methods have been developed based on the finite element methods, which effectively overcoming the mesh entanglement in solving the excessive large deformation problems. In meshless methods, the domains of interest are discretized by a scattered set of points. The successes of meshless methods are due to the development of new shape functions that allow the interpolation of field variables to be accomplished at a global level or a local level and therefore avoid the use of a mesh. These methods are ideal for modeling refinement, adaptivity, fracture problems, and large deformation problems, etc. Nonlinear formulations of the meshless local Petrov-Galerkin method (MLPG) are presented for a large deformation analysis of rubber materials which are considered to be hyperelastic and nearly incompressible. The method requires no explicit mesh in computation and therefore avoids mesh distortion difficulties in the large deformation analysis. In this paper, a simple Heaviside test function is chosen for reducing the computational effort by simplifying domain integrals for hyperelasticity problems. Trial functions are constructed using the radial basis function (RBF) coupled with a polynomial basis function. A pressure projection method is employed by locally projecting the pressure onto a lower-order space to overcome the nearly incompressibility in the plane strain problems. Several examples are presented to demonstrate the effectiveness of this method in the large deformation rubber materials analysis.