Tensional wrinkling of thin elastic sheets with two circular holes

A paradigm for the study of wrinkling in elastic sheet is the Lam\'{e} configuration, in which azimuthal wrinkles form in an annular sheet subjected to tensile loads at both edges. Since wrinkles are spatially extended, this instability provides a mechanism for stress transmission over long distances. A natural extension of this problem is wrinkling in sheets with multiple holes or broken symmetry. Here, we investigate tension-induced wrinkling in thin elastic sheets containing two circular holes by combining analytical modeling and experiments. The pre-buckled state is solved analytically using bipolar coordinates, enabling identification of the wrinkling threshold as a function of the distance between the two holes. Near-threshold wrinkling and interactions between wrinkles are analyzed, and we validate our theoretical predictions against experimental observations obtained through video imaging of spin-coated polystyrene sheets floating on liquid surfaces with controlled surface tension. Our results demonstrate that geometric symmetry breaking, such as the presence of a second hole, strongly influences wrinkle nucleation, orientation, and spatial extent. Beyond mechanics, these findings might provide a simple mechanism for cellular mechanosensing, where force transmission is amplified by mechanical instabilities.