Abstract
The analysis of the mechanical behaviour of rods subject to large displacements
and rotations constitutes a research ¯eld where signi¯cant advances from the view-
point of both, formulation of physical models and ¯nding of computational solutions
have taken place over the last three decades. The realm of applications of this kind of
models di®ers from the scope of problems traditionally treated by civil engineering,
approaching other technologies as aeronautics, robotics or biomechanics.
Among the variety of one dimensional models available for non linear analysis of
rods, the one proposed by Sim¶o as a extension of the work of Reissner is capable to
treat arbitrarily large rotations of the cross{sections. Its conceptual simplicity and the
model{based powerful numerical solutions have turned it into a main reference and
a starting point for many recent research works. Sim¶o coined the label geometrically
exact rod model to refer to his model. Nevertheless the Reissner{Sim¶o model is not free
of di±culties. These are mainly caused by the exact treatment of rotations, requiring
to work in a non linear, non commutative con¯guration space.
This dissertation examines the foundations of the geometrically exact model and
its connections to the non linear theory of elasticity. The relation between material
and spatial variables through the transformation de¯ned by the rotation of the cross{
sections determines the approach method. As a ¯rst step, model kinematics and
¯eld equations in both forms {material and spatial{ have been fully developed. This
allowed to systematize the deductive process and to contribute some new theoretical
results. The problem analysis from the variational viewpoint has shown the formal
connections between the equations of rod elastostatics and the equations of rigid body
dynamics, and has led to the generalization and update of Kirchho®'s kinetic analogy
employing the modern language of classical mechanics.
Prior to the development of numerical solutions, the expression of the spatial
tangent operator has been derived from a consistent linearization of the virtual work
equation. When linearization is carried on before discretization, new additional terms
in the operator appear; their in°uence on the numerical results has been shown to
be negligible. The material form of the operator (still unpublished) has also been
obtained. The fact that the expressions of the tangent operator are independent of
the selected parameterization of rotations is a remarkable feature of the formulation.
The ¯nal part of the dissertation focuses on the model numerical solution. In a
¯rst stage a ¯nite element based on the spatial form of the operator has been develo-
ped, which is essentially the one proposed by Sim¶o and Vu{Quoc with some changes
introduced by Ibrahimbegovi¶c and Taylor. The analysis of several examples shows the
power of this model, but also poses some numerical problems. As a last contribution,
a new ¯nite element using the material tangent operator and the spherical interpola-
tion proposed by Cris¯eld and Jeleni¶c was developed. Processing of several examples
has led to the conclusion that solutions obtained using the material element reach
the same precision as with the spatial one, but the convergence rate is much slower.