Geometric approach to Mechanical design principles in continuous elastic sheet

We present a systematic theoretical method for controlling and manipulating mechanical properties of slender solids. Using a geometric approach to non-Euclidean elasticity we show that the energy land-scape in configuration space can be sculpted and manipulated by controlling material geometric properties such as preferred rest lengths and curvatures. We mathematically formalize an inverse design problem, and show that by implanting geometric frustration, extreme mechanical properties can be encoded into a materials using accessible experimental techniques. To test the methodology we focus on a family of structures that present anomalous mechanical behavior such as tunable and even vanishing rigidity. The presented formalism can be discretized and thus opens a new pathway for the design of both continuum and discrete solids and structures.