Temperature-dependent elastic viscoplastic material model was used for the numerical simulation of the friction stir welding process. The non-elastic response of the rate-dependent material in the large deformation problems was calculated by using the closest point algorithm. The numerical results show that the shape of the equivalent plastic strain looks like onion rings and the spacing of the rings is approximately equal to the forward movement of the tool in one rotation. The equivalent plastic strain is increased with the increase of viscosity coefficient due to the increase of friction stress in the pin-plate interface. The region which is influenced by the rotating tool is decreased with the decrease of viscosity coefficient. The radial and circumferential stresses in front of the pin are greater than the ones behind the pin. This difference can be reduced with the decrease of viscosity.