Mixed finite element methods for elastic rods of arbitrary geometry

Abstract

The boundary-value problem for rods having arbitrary geometry, and subjected to arbitrary loading, is studied within the context of the small-strain theory. The basic assumptions underlying the rod kinematics are those corresponding to the Timoshenko hypotheses in the plane rectilinear case: that is, plane sections normal to the line of centroids in the undeformed state remain plane, but not necessarily normal. The problem is formulated in both the standard and mixed variational forms, and after establishing the existence and uniqueness of solutions to these equivalent problems, the corresponding discrete problems are studied. Finite element approximations of the mixed problem are shown to be stable and convergent. It is shown that the equivalence between the mixed problem and the standard problem with selective reduced integration holds only for the case of rods having constant curvature and torsion, though. The results of numerical experiments are presented; these confirm the convergent behaviour of the mixed problem.