From F\"{o}ppl-von K\'{a}rm\'{a}n shells to enhanced one-dimensional rods: localization phenomena and multistability

Starting from the Foppl-von Karman model of thin shells, we deduce two 1D models of elastic rods enhanced by additional kinematical descriptors that keep explicit track of the compatibility condition requested in the 2D parent continua, that in the classical rods models are identically satisfied after the dimensional reduction. These enhanced models allow to describe some phenomena of preeminent importance even in 1D bodies, such as formation of singularities and localization (d-cones), otherwise inaccessible by the classical 1D models. Moreover, the effects of the compatibility translate into the possibility to obtain multiple stable equilibrium configurations.