  We develop a geometric approach to understand the mechanics of perforated
thin elastic sheets, using the method of strain-dependent image elastic
charges. This technique recognizes the buckling response of a hole under
external load as a geometrically tuned mechanism of stress relief. We use a
diagonally pulled square paper frame as a model system to quantitatively test
and validate our approach. Specifically, we compare nonlinear force-extension
curves and global displacement fields in theory and experiment. We find a
strong softening of the force response accompanied by curvature localization at
the inner corners of the buckled frame. Counterintuitively, though in complete
agreement with our theory, for a range of intermediate hole sizes, wider frames
are found to buckle more easily than narrower ones. Upon extending these ideas
to many holes, we demonstrate that interacting elastic image charges can
provide a useful kirigami design principle to selectively relax stresses in
elastic materials.
