Simulating physical problems with multi-time scale coupling presents a considerable challenge due to the concurrent solution of processes with different time scales. This complexity arises from the necessity to evolve large time scale processes over long physical time, while simultaneously small time step sizes are required to unveil the underlying physics in shorter time scale processes. To address this inherent conflict in the multi-time scale coupling problems, we propose an explicit multi-time step algorithm within the framework of smoothed particle hydrodynamics (SPH), coupled with a solid dynamic relaxation scheme, to quickly achieve equilibrium state in the comparatively fast solid response process. To assess the accuracy and efficiency of the proposed algorithm, a manuscript torsional example, two distinct scenarios, i.e., a nonlinear hardening bar stretching and a fluid diffusion coupled with Nafion membrane flexure, are simulated. The obtained results exhibit good agreement with analytical solution, outcomes from other numerical methods and experimental data. With this explicitly multi-time step algorithm, the simulation time is reduced firstly by independently addressing different processes being solved under distinct time step sizes, which stands in contrast to the implicit counterpart, and secondly decreasing the simulation time required to achieve a steady state for the solid by incorporating the dynamic relaxation scheme.