Slender rods are ubiquitous in nature, devices and engineering structures. At length scale separated by orders of magnitude, these one dimensional structures are typically constrained and mechanically supported: microtubules by the dense cellular matrix, plant roots by the surrounding soil and oilfield tubulars by the borehole walls. Despite the support, the rods can still buckle under large compressive forces, leading to catastrophic structural failure. Differently from the research projects I described above, here we are investigating instabilities with the precise goal of delaying them.
(A) Undeformed and (B) deformed configurations of an elastic rod compressed in a horizontal cylindrical channel. (C): Cross-section view of the horizontal channel. (D) Under a displacement control, the rod is compressed and the reaction force is recorded in the simulations. This figure shows the force-displacement relation for four different cases. The force peak at which the linear force-displacement relation breaks down is recorded as the critical buckling force. (D) Numerically predicted critical buckling force as a function of the initial velocity perturbation amplitude.
When a sufficiently high compressing force is applied to a constrained cylinder, there are two scenario one can think of: the cylinder cab bend back-and-forth staying in the diameter plane (sinusoidal buckling), or it can become a helix touching the outer cylinder wall (helical buckling). In the coiled tubing community it has been recognized for a long time that it is the latter case that eventually happens. Moreover, helical buckling appears to play an important role also in the investigation and manipulation of DNA, where nanoscale rods are confined within micro- and nano-channels.
Motivated by these phenomena we are investigating elastic buckling instabilities of a cylinder within a confining cylindrical channel, using an approach based on a powerful combination of numerical/analytical analyses and experiments. While the experiments are performed at MIT by Prof. Reis, in our group at Harvard we focus on the development of analytical and numerical tools to analyze the problem.
We started by developing a theory to reveal the distinct mechanisms by which a compressed rod confined in a channel buckles in the presence of dry friction. Contrary to the case of a frictionless contact, with friction the system can bear substantially enhanced compressive load without buckling after its stiffness turns negative, and the onset of instability is strongly affected by the amount of perturbation set by the environment. Our theory, confirmed by simulations, shows that friction enhances stability by opening a wide stable zone in the perturbation space. Buckling is initiated when the applied compressive force is such that the boundary of the stable zone touches a point set by the environment, at a much higher critical load. The proposed theory clearly demonstrates why frictional systems are reported to have much higher critical loads. It also shows why the onset of their instability is strongly affected by the amount of imperfections/perturbations set by the surrounding environment.
We are currently exploring the transition between sinusoidal and helical buckling and the effect of geometric non-linearity. Results thus far are preliminary but highly encouraging. One of the tools used to investigate this behavior is transient dynamic software originally developed at Schlumberger. This software has been modified jointly (with input from Harvard, MIT, and Schlumberger) with a refined friction model (including anisotropic friction) and including injection functionality. We have begun using this software to model the buckling behavior and to provide insight into the most effective strategies for suppressing the helical buckling.
- T. Su, N. Wicks, J. Pabon and K. Bertoldi. Mechanism by which a frictionally confined rod loses stability. International Journal of Solids and Structures. 2013; 50:2468-2476