Tensile residual stress in a plate or membrane clamped at the perimeter can be measured by either applying a uniform hydrostatic pressure or a central load via a cylindrical punch (with several different loading configurations). Analytical constitutive relations are derived here based on an average membrane stress approximation and are compared to finite element analysis results. The thickness and flexural rigidity of the film are not confined to a small range but will span a wide spectrum. The elastic responses of the blistering films are shown to be linear when the film is thick, relatively rigid, or subjected to a large residual stress, and cubic when the film is thin, flexible, or under a small residual stress. The linear-to-cubic transition is formulated.