A mechanical model of the pressure-sensitive dilatant material is established in order to investigate the viscous effect in mode I quasi-static growing crack-tip field. The constitutive equations on the pressure-sensitive dilatant material are deducted. Through asymptotic analysis, it is shown that in the stable creep growing stage, the elastic-deformation and the visco-deformation are equally dominant in the near-tip field, as . The asymptotic solutions of separative variable in the crack-tip field of plane stress mode I quasi-static are aslo obtained. According to numerical calculation, the curves of stress, strain and displacement in terms of various parameters are given. The asymptotic solutions of quasi-static growing crack-tip field gained here can conveniently degenerate the incompressible case, when the Poisson ratio , named as HR field. The conclusions can provide the references for further studying the dynamic growing crack-tip field in the pressure-sensitive dilatant material.