Under the assumption that the artificial viscosity coefficient at the propagating crack tip is in inverse proportion to power law of the plastic strain rate, a rate-sensitive constitutive relationship is derived for perfect elastic-plastic material. With the adoption of the rate-sensitive constitutive relationship, it is asymptotically investigated the propagating tip fields of plane strain mode I. And the quasi-static equations are obtained separately governing the stress and strain fields at the crack-tip by means of Airy stress function. Numerical calculations of governing equations are carried out by double parameters shooting, with selections of appropriate values of each characteristic parameter by combinations of boundary, and the fully continuous stress-strain fields are obtained at the crack-tip. The analytical and computational results indicate that viscosity effect is an important factor in crack-tip fields.