Analysis of rubber-like materials using meshless local Petrov-Galerkin (MLPG) method

Large deformations of rubber-like materials are analyzed by the meshless local Petrov-Galerkin (MLPG) method. The method does not require shadow elements or a background mesh and therefore avoids mesh distortion difficulties in large deformation problems. Basis functions for approximating the trial solution and test functions are generated by the moving least-squares (MLS) method. A local mixed total Lagrangian weak formulation of non-linear elastic problems is presented. The deformation gradient is split into deviatoric and dilatational parts. The strain energy density is expressed as the sum of two functions: one is a function of deviatoric strains and the other is a function of dilatational strains. The incompressibility or near incompressibility constraint is accounted for by introducing the pressure field and penalizing the part of the strain energy density depending upon the dilatational strains. Unlike in the mixed finite element formulation, in the MLPG method there is no need for different sets of basis functions for displacement and pressure fields. Results computed with the MLPG method for a few sample problems are found to compare very well with the corresponding analytical solutions.