Abstract: A rivet member comprising a flange part (12) and a shank part (16) protruding upright from the flange part, with a circular recess (18) formed on the inner side (14) of the flange, which is used in combination with a burr member (5, 5′, or 7) having a barrel (13, 13′, or 15) of a desired shape, the circular recess (18) being adapted to receive the barrel that has pierced through a fabric or other sheet material (11) or to receive the barrel with the sheet material interposed between the bottom of barrel and the flange of the sheet. This arrangement overcomes the problem of formation of gaps between the flanges of the rivet and the barrel of the burr member.