This paper deals with the introduction of a decomposition of the deformationsof curved thin beams, with section of order $\delta$, which takes into accountthe specific geometry of such beams. A deformation $v$ is split into anelementary deformation and a warping. The elementary deformation is the analogof a Bernoulli-Navier's displacement for linearized deformations replacing theinfinitesimal rotation by a rotation in SO(3) in each cross section of the rod.Each part of the decomposition is estimated with respect to the $L^2$ norm ofthe distance from gradient $v$ to SO(3). This result relies on revisiting therigidity theorem of Friesecke-James-M\"uller in which we estimate the constantfor a bounded open set star-shaped with respect to a ball. Then we use thedecomposition of the deformations to derive a few asymptotic geometricalbehavior: large deformations of extensional type, inextensional deformationsand linearized deformations. To illustrate the use of our decomposition innonlinear elasticity, we consider a St Venant-Kirchhoff material and uponvarious scaling on the applied forces we obtain the $\Gamma$-limit of therescaled elastic energy. We first analyze the case of bending forces of order$\delta^2$ which leads to a nonlinear inextensional model. Smaller pure bendingforces give the classical linearized model. A coupled extensional-bending modelis obtained for a class of forces of order $\delta^2$ in traction and of order$\delta^3$ in bending.
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