Numerical study of the compression of tightly constrained slender rods

Compression of a slender rod, tightly confined in a bore, is an archetypal problem that has relevance in a wide variety of situations ranging from worm like macromolecular chains and microtubules to drill strings in wellbores. If persistence length quantifies the stiffness of the rod, these problems span over cases where persistence length is of the order of the contour length to where it is many orders larger. In spite of this wide spread, all of these problems adopt the Kirchhoff rod theory, whereby, with appropriate normalisations, the force displacement response becomes independent of stiffness. This is true for unconstrained rods but not for rods constrained in bores. In fact, when the bore dimensions are much smaller compared to the length of the rod, i.e. the rod is 'tightly' constrained, post Euler buckling responses are characterised by abrupt large configurational adjustments or so-called 'jump-instabilities'. The large and repeated configurational adjustments render the numerical simulation of the post-buckling force displacement responses difficult. We have used a geometrically exact 3-dimensional beam theory with rough constraining walls in order to successfully simulate the force displacement response of constrained rods up to large strains. In particular, we have simulated rods with stiffness values ranging over 6 orders of magnitude - starting from values comparable to microtubules in the cytoskeleton to those applicable to large scale engineering structures. Our results show that, unlike what is expected from the Kirchhoff rod theory, the geometry of the bore, and its size, play a significant role in rendering the compression response stiffness dependent. Moreover, nature and degree of confinement also dictate the stable 3-dimensional configuration to which the rod eventually evolves. In circular bores, the rod evolves to an almost perfect helical shape while in square ones, it does not.