In the petroleum engineering, the nonlinear bulk strain or bulk strain rate must be considered in rock deformation, adopting the viscous-elastic theory, a constitutive equation of the pressure-sensitive dilatant materials was established. The governing eqations were deducted in crack-tip field of quasi-static growing crack in pressure-sensitive dilatant materials. And for the mechanical model of a rigid-viscoelastic material interfacial crack, through numerical calculate, the asymptotic solution of the continuous separate variable formal of stress, strain and displacement. The significance of the material parameter in the crack-tip field is discussed. The study of the paper will provide theoretical references for preventing the glide of the mud rock and reducing the arises of the failure of casings of well.