Statically and kinematically exact nonlinear theory of rods and its numerical verification

Abstract

The entirely general nonlinear theory of rods is developed. Two vector equilibrium equations, valid on the rod axis are derived exactly, by direct integration of the mechanical balance laws of continuum mechanics over the rod cross section. Then the energetically conjugate exact 6-field kinematics of the rod axis is implied in an integral identity, which expresses the principle of the virtual work. The kinematics consists of the displacement vector of the axis, describing the average translation, and the proper orthogonal tensor, representing the average rotation of the cross section. The integral identity also suggests the required general form of the constitutive relations. They are already approximate from the physical side, are then the best place to make geometric approximations and abbreviations to a certain constrained theory. The constitutive relations, the only nonexact segment of the theory, are treated here as an interchangeable element, supplementing the exactly formulated principle of virtual work.The solution of the nonlinear rod problem becomes nonstandard here, for the configuration space is the nonlinear manifold containing the rotation group SO(3). Applying the finite element method, the proper parameterization, linearization, interpolation, accumulation and discretization procedures in SO(3) group are discussed. A library of 2-, 3-, 4-node 1-D isoparametric elements is developed, and the computer program RAM is worked out.The presented numerical tests contain several problems of highly nonlinear plane and spatial deformations of rods with constitutive relations taken, at the moment, from the classical isotropic elastic beams. In all the tests the developed elements, in comparison with the solutions known in the literature, show a good performance.