An asymptotic solution is presented for the singular stress and strain fields near the tip of a steadily growing crack in an elastic-viscoplastic material under Mode III loading. By taking into account the experimental study made by Clark-Duwez in which no further increase of dynamic yield stress was observed when the strain rate exceeded the critical value, an intense strain region which behaves as an elastic-perfectly plastic material is introduced in the vicinity of the crack tip where this region is surrounded by the elastic-viscoplastic material. It is shown that the size of the intense strain region measured along the crack line is proportional to the velocity of the crack growth, and the singularity of the strain distribution near the crack tip is weakened by the intense strain region.