ABSTRACT
The general problem of a slender elastic body constrained to deform inside a straight or sinuous conduit is widely encountered in medical and engineering applications. Examples include the insertion of miniature instruments into blood vessels for the treatment of vascular and cerebrovascular pathologies (e.g. abdominal and thoracic aortic aneurism, stroke), or applications concerning the oil and petroleum industries which rely on several kilometers long drillstrings to transmit the drilling efforts necessary to crush the rock formations and reach deep hydrocarbon reservoirs. The identification of the number of contacts between the rod and the conduit (e.g. blood vessel or borehole) as well as their position and extent constitutes the central concern of this relatively broad class of problems. The nonlinearities associated with the large deflection of the rod and the unilateral contact condition as well as the a priori unknown number of contacts, however, make the use of conventional numerical tools rather inefficient. Additionally, the commonly adopted division of the problem in rod segments bounded by two contacts (Chen and Li 2007, Denoël 2008) requires to solve the governing equations on domains which are initially unknown and, therefore, leads to the establishment of integral constraints on the unknown length of a rod forced to go through fixed points in space. Finally, the assessment of the unilateral contact condition, which requires in principle the comparison of two curves parametrized by distinct curvilinear coordinates (e.g. the rod centroid and the conduit axis), prove to be a rather intensive computational task.
