Disclosed herein are articles and processes for making elastic composite structures which can be used as or converted into a component of disposable hygiene products or articles of apparel. At least one relatively inelastic substrate, for example a nonwoven substrate, is adhesively bonded with a hot melt adhesive to a certain selected type of elongated polyurethane material in the form of a film or one or more fibers or filaments. The elongated polyurethane material is then allowed to relax which provides an elastic composite structure which is gathered or puckered.
