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 δ, 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–Müller 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 types of 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 scalings on the applied forces we obtain the Γ-limit of the rescaled elastic energy. We first analyze the case of bending forces of order δ 2 which leads to a nonlinear extensible model. Smaller pure bending forces give the classical linearized model. A coupled extentional-bending model is obtained for a class of forces of order δ 2 in traction and of order δ 3 in bending.