A Local Frame Approach for Line-to-Line Contact between thin Beams with circular Cross-Sections

Flexible slender structures like cables have a variety of applications in high-performance engineering
systems. The complex mechanical behaviour of such components, resulting from interactions between
individual fibers, cannot be captured by means of global constitutive laws [1]. In mesoscopic models a
cable is seen as a collection of fibers, modeled as 1D Cosserat beams, experiencing frictional contact.
In the mathematical framework of Lie groups, beams are described through a space curve augmented
by a frame attached to each cross section. The mechanical behaviour of a beam is the result of frame
transformations belonging to the Lie group of rigid body motions SE(3). Discretization methods con-
sistent with the framework implicitly conserve the intrinsic geometric characteristics of Cosserat curves
and hence lead to interesting numerical properties [2].
In this contribution the modeling of cables made of elastic fibers with circular cross sections undergo-
ing frictionless contact interactions is addressed. A continuous formulation as Cosserat beams with line
to line contact conditions [3] modeled as unilateral constraints is established in SE(3). A local frame
formulation of contact kinematics emerges from the formalism. The spatial discretization is performed
using helicoidal shape functions resulting from the geodesic interpolation of elements in SE(3). At the
minisymposium an outline of the problem with its scientific challenges, first numerical results, as well
as an outlook on future work will be presented.