Frustrated Euclidean ribbons: a new class of geometric frustration

Geometrical frustration in thin sheets is ubiquitous across scales in biology and becomes increasingly relevant in technology. It may indeed lead to mechanical instabilities, anomalous mechanics and shape-morphing abilities that can be harnessed in engineering systems. It is widely accepted that such frustration stems from violation of Gauss's Theorema Egregium, i.e. "Gauss frustration". Here we report on a new type of geometrical frustration, one that exists in sheets that satisfy Gauss's theorem. We show that the origin of the frustration is the violation of Mainardi-Codazzi-Peterson compatibility equations. Combining experiments, simulations and theory, we study the specific case of an Euclidean ribbon with radial and geodesic curvatures. Experiments, conducted using different materials and techniques, reveal shape transitions, symmetry breaking and spontaneous stress focusing. These observations are quantitatively rationalized using analytic solutions and geometrical arguments. We argue that this type of frustration is as general as the Gauss frustration and is, thus, expected to appear in natural and engineering systems, specifically in slender 3D printed sheets.