Residually stressed elastic ribbons are thin and narrow sheets, that cannot satisfy both the internal bending and stretching constraints at the same time, thus have no stress free configuration. Such systems exist on macro-scales (natural and man made) and on micro and nano- scales: from the growing of biological tissues, to the self assemblies of peptides and lipids. On such scales thermal fluctuations are important and one must study the statistics of ribbons configurations.

In this presentation we introduce an analytical framework to treat residually stressed ribbons of di different types. Unlike compatible ribbons, the ground state of frustrated ribbons is non trivial, exhibiting bifurcations in shape, and abnormal mechanical properties. These unique properties are reflected in the statistics of frustrated ribbons configurations, which is studied analytically.