Decomposition of deformations of thin rods. Application to nonlinear elasticity

Abstract: This paper deals with the introduction of a decomposition of the deformations of curved thin beams, with section of order $\delta$, which takes into account the specific geometry of such beams. A deformation $v$ is split into an elementary deformation and a warping. The elementary deformation is the analog of a Bernoulli-Navier's displacement for linearized deformations replacing the infinitesimal 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 of the distance from gradient $v$ to SO(3). This result relies on revisiting the rigidity theorem of Friesecke-James-M\"uller in which we estimate the constant for a bounded open set star-shaped with respect to a ball. Then we use the decomposition of the deformations to derive a few asymptotic geometrical behavior: large deformations of extensional type, inextensional deformations and linearized deformations. To illustrate the use of our decomposition in nonlinear elasticity, we consider a St Venant-Kirchhoff material and upon various scaling on the applied forces we obtain the $\Gamma$-limit of the rescaled elastic energy. We first analyze the case of bending forces of order $\delta^2$ which leads to a nonlinear inextensional model. Smaller pure bending forces give the classical linearized model. A coupled extensional-bending model is obtained for a class of forces of order $\delta^2$ in traction and of order $\delta^3$ in bending.