A band made from a material normally elastically non-stretchable in its longitudinal direction to which elastic stretchability is imparted by one or more sections which include lateral non-rectilinear band portions on both sides of the central longitudinal band plane. The lateral non-rectilinear band portions are thereby made by cut-outs in the band along its longitudinal sides and in the center area. In one embodiment, the cut-outs along the sides are segmentally shaped while a window in the central area is made of a shape resembling an hourglass. Adjacent sections are connected with each other by transversely extending web portions delimited in the longitudinal direction by end surfaces of the central cut-outs. A clamp structure utilizing a band of the aforementioned type, especially a clamping band made from galvanized or stainless steel, offers numerous advantages, especially if a so-called "Oetiker" ear is used to tighten the clamp structure about the object to be fastened thereby. This is so because changes in the circumferential length of the clamp structure will be compensated thereby in the first instance by the elastically stretchable clamping band so that greater latitude exists in the design and installation of such a plastically deformable ear which can now be designed and closed to maximize its holding ability.