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We develop a geometric approach to understand the mechanics of perforatedthin elastic sheets, using the method of strain-dependent image elasticcharges. This technique recognizes the buckling response of a hole underexternal load as a geometrically tuned mechanism of stress relief. We use adiagonally pulled square paper frame as a model system to quantitatively testand validate our approach. Specifically, we compare nonlinear force-extensioncurves and global displacement fields in theory and experiment. We find astrong softening of the force response accompanied by curvature localization atthe inner corners of the buckled frame. Counterintuitively, though in completeagreement with our theory, for a range of intermediate hole sizes, wider framesare found to buckle more easily than narrower ones. Upon extending these ideasto many holes, we demonstrate that interacting elastic image charges canprovide a useful kirigami design principle to selectively relax stresses inelastic materials.

Kirigami mechanics as stress relief by elastic charges