Plane Bending of a Curved Rod
We present a combination of the asymptotic, direct, and numerical methods on the example plane problem of finite deformations of a thin curved strip. The study includes both static and dynamic analyses; the inhomogeneity of the strip is taken into account. The method of asymptotic splitting allows for a consistent dimensional reduction of the equations of the original two-dimensional continuous problem into a one-dimensional formulation of the reduced theory and a problem in the cross section. It also provides a consistent way to recover the distributions of stresses, strains, and displacements in two dimensions. The direct approach to a material line extends the results to the geometrically nonlinear range. Several analytical solutions and the implementation of a finite element scheme are demonstrated with the Mathematica computer system. We investigate the convergence of solutions of various problems in the original (two-dimensional) and reduced (one-dimensional) models with respect to the thickness. This justifies the analytical conclusion that the classical Kirchhoff theory of rods remains asymptotically accurate irrespective from both the curvature of the strip and the variation of the material properties over the thickness. This chapter quotes extensively from Vetyukov (Acta Mech., 223(2):371–385, 2012) with kind permission from Springer Science and Business Media.
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