Weighted radial basis collocation method for large deformation analysis of rubber-like materials

A nonlinear formulation, based on the total Lagrange description of the weighted radial basis collocation method (WRBCM), is proposed for the large deformation analysis of rubber-like materials where the materials are hyperelastic and nearly incompressible. The WRBCM based on the strong form collocation is a genuinely meshfree method that eliminates the need for meshing. As a result, it effectively circumvents challenges associated with mesh distortion during large deformation analysis. The proper weights that should be imposed on the boundary collocation equations are first derived to achieve the optimal convergence in hyperelasticity. In the WRBCM, the support of the shape function remains unchanged throughout material deformation, thereby guaranteeing the absence of tension instability during large deformation analysis. Additionally, by combining WRBCM with a least-squares solution, volumetric locking in nearly incompressible hyperelastic problems can be suppressed. This is due to the infinite continuity possessed by the radial basis approximation, which ensures the divergence-free condition. Several numerical examples are examined, demonstrating the high accuracy and exponential convergence of WRBCM in hyperelastic large deformation analysis. Moreover, no volumetric locking can be observed which further substantiates the effectiveness of applying nonlinear WRBCM to nearly incompressible hyperelastic problems.