Non-material Poincaré-Cosserat equations: application to slender sliding structures and fluid-structure interactions

The counterpart of Lagrange's equations on a noncommutative Lie group of congurations is called Poincaré's equations. Applied to the Lie group SE(3), these equations give the well-known Newton-Euler dynamic model of a rigid body. Beyond rigid bodies, Poincaré's approach can be extended to a Cosserat medium, i.e., a continuous set of rigid microstructures glued together along material dimensions. The resulting set of equations is called the Poincaré-Cosserat material equations. Applied to a Cosserat rod, it provides the Simo-Reissner model of geometrically exact rods. Recently, these equations have been extended to the case of elongated structures sliding uniformly along a non-material control domain. In this article, we extend this initial result to a new set of equations that we call non-material Poincaré-Cosserat equations. Derived from a non-material version of the principle of least action, these equations can be applied to complex situations involving several media sliding along each other with a non uniform velocity. To illustrate their use in practical situations, we will apply them to three emblematic examples taken from solid and fluid mechanics. The first is a sliding rod extending and retracting in three dimensions from a rigid sleeve. The second is the garden hose, modeled as an ideal fluid flowing inside an elastic pipe clamped in a wall. The third is a model of fish swimming developed by J. Lighthill, called LAEBT (Large Amplitude Elongated Body Theory), which has been used in recent years by fluid mechanics, biomechanics, and robotics to explain the performance of swimming sh and control bio-inspired robots resembling fish.