The Ogden model and the natural neighbour radial point interpolation method for hyperelastic analyses

Hyperelastic behaviour can be observed in rubber-like materials and biological tissues. Different hyperelastic models have been developed, the neo-Hookean, Mooney-Rivlin, and Ogden are among the most common. Furthermore, the Ogden model can represent these three material behaviours, so its numerical implementation is more versatile. Although the finite element method (FEM) is well known for numerical modelling, meshless methods like the natural neighbour radial point interpolation method (NNRPIM) have matching accuracy even with coarser discretisations, making meshless methods an alternative for modelling soft materials like adhesives. However, the Ogden model is not available in the NNRPIM or mesh less methods alike. In this work, the Ogden model was implemented into the NNRPIM. The implementation was validated with four 2D examples from the literature: two with Mooney-Rivlin material properties, one with neo-Hookean, and one with Ogden. The results were compared against available literature data and FEM solutions. For the three material models, the NNRPIM models were slightly stiffer (mean 2.5%) than their FEM counterparts; nevertheless, for strains below 10%, the difference dropped (< 1% ). The stress contours obtained with FEM and NNRPIM were equivalent in all cases, assuring the stress transformations were correct, mostly for shear., consequently validating the implementation. Also, the mean solution time for the NNRPIM models was 1.25 s/node including preprocessing. In conclusion, the implementation presented here is suitable to model three hyperelastic material models: neo-Hookean, Mooney-Rivlin, and Ogden in applications where strains are below 10% with accuracy.