Mechanical design principles in frustrated thin elastic sheets

Using a geometric formalism of elasticity theory we develop a systematic theoretical framework for shaping and manipulating the energy landscape of slender solids, and consequently their mechanical response to external perturbations. We formally express global mechanical properties associated with non-Euclidean thin sheets in terms of their local rest lengths and rest curvatures, and we interpret the expressions as both forward and inverse problems for designing the desired mechanical properties. We show that by wisely designing geometric frustration, anomalous mechanical properties can be encoded into a material using accessible experimental techniques. To test the methodology we derive a family of ribbon-springs with extreme mechanical behavior such as tunable, anharmonic, and even vanishing rigidities. The presented formalism can be discretized, offering a new methodology for designing mechanical properties and thus opens a new pathway for the design of both continuum and discrete solids and structures.