Modelling non-local elasticity in 1D vibrating rods using Corrected Smoothed Particle Hydrodynamics method

Abstract The present study aims to assess the use of the Corrected Smoothed Particle Hydrodynamics (CSPH) method predicting non-local elastic effects in finite deformations . For this purpose, we first recall the discrete and continuum analytical solutions of a 1D bar under longitudinal harmonic vibrations with small scale effects. SPH is an integral-based non-local approach where the non-locality is introduced through the kernel convolution. Contrary to the classical use of the SPH method, our simulations are conducted with values of the smoothing length parameter that may be not infinitesimal and can be related to the characteristic size of the microstructure. The numerical results obtained from the 1D bar simulations under different boundary conditions demonstrate that the CSPH method can capture the non-local effects in dynamic conditions. In particular, it is shown that the numerical results have a good agreement with available discrete and continuum analytical solutions. Moreover, it is observed that CSPH strain-based and stress-based formulations lead to similar responses. We have also simulated hardening effects due to the increase of the vibration amplitude. Finally, we provide a discussion about the influence of finite and infinite support kernel functions.