One-dimensional equations of deformation of thin slightly curved rods. Asymptotical analysis and justification

We obtain asymptotics for the solution of the spatial problem of elasticity theory in a thin body (a rod) with a smoothly varying cross-section. Any anisotropy and any non-homogeneity of material is admitted. The ends of the a rod, which is under the action of volume forces, are rigidly fixed (clamped), and the lateral surface is under the action of forces. The small parameter $h$ is the ratio of the maximal diameter of the rod to its length. We suggest conditions on the differential properties and the structure of external load under which the solution of the one-dimensional equations yielded by asymptotical analysis provides an acceptable approximation to the three-dimensional displacement and stress fields. The error estimate is based on a special version of Korn's inequality, which is asymptotically sharp if suitable weight factors and powers of $h$ are introduced into the $L_2$-norms of displacements and their derivatives.