We present realistic three-dimensional numerical simulations of elastic bodies sliding on top of each other in a regime of velocities ranging from 0.1 to 10 m/s using the so-called smoothed particle hydrodynamics method. This allows us to probe in detail the response of the bodies and the nature of the friction between them. Our investigations are restricted to regimes of pressure and roughness where only elastic deformations occur between asperities at the contact surface between the slider block and the substrate. In these regimes, solid friction is due to the generation of vibrational radiations which then escape to infinity or are damped out; in which case, energy is dissipated. We study periodic commensurate and incommensurate asperities and various types of disordered surfaces. In the elastic regime studied in this paper, we report evidence of a transition from zero (or nonmeasurable μ<0.001) friction to a finite friction as the normal pressure increases above 106 Pa. For larger normal pressures (up to 109 Pa) we find a remarkably universal value for the friction coefficient of μ≈0.06, which is independent of the internal dissipation strength over 3 orders of magnitudes and independent of the detailed nature of the slider block-substrate interactions. We find that disorder may either decrease or increase μ due to the competition between two effects: (1) disorder detunes the coherent vibrations of the asperties that occur in the periodic case, leading to weaker acoustic radiation and thus weaker damping, and (2) a large disorder leads to stronger vibration amplitudes at local asperities and thus to stronger damping. Our simulations have confirmed the existence of jumps over steps or asperities of the slider blocks occurring at the largest studied velocities (10 m/s). These jumps lead to chaotic motions similar to the bouncing-ball problem. We find a velocity strengthening with a doubling of the friction coefficient as the velocity increases from 1 to 10 m/s. This reflects the increasing strength of vibrational damping. ¿ 1999 American Geophysical Union