Plasmonic nanostructures integrated with soft, elastomeric substrates provide an unusual platform with capabilities in mechanical tuning of key optical properties, where the surface configurations can undergo large, nonlinear transformations. Arrays of planar plasmonic nanodiscs in this context can, for example, transform into three-dimensional (3D) layouts upon application of large levels of stretching to the substrate, thereby creating unique opportunities in wide-band tunable optics and photonic sensors. In this paper, a theoretical model is developed for a plasmonic system that consists of discrete nanodiscs on an elastomeric substrate, establishing the relation between the postbuckling configurations and the applied strain. Analytic solutions of the amplitude and wavelength during postbuckling are obtained for different buckling modes, which agree well with the results of finite-element analyses and experiment measurements. Further analyses show that increasing the nanodisc distribution yields increased 3D configurations with larger amplitudes and smaller wavelengths, given the same level of stretching. This study could serve as a design reference for future optimization of mechanically tunable plasmonic systems in similar layouts.