Buckling of rods with spontaneous twist and curvature

We analyze stability of a thin inextensible elastic rod which has non-vanishing spontaneous generalized torsions in its stress-free state. Two classical problems are studied, both involving spontaneously twisted rods: a rectilinear rod compressed by axial forces, and a planar circular ring subjected to uniform radial pressure on its outer perimeter. It is demonstrated that while spontaneous twist stabilizes a rectilinear rod against buckling, its presence has an opposite effect on a closed ring. The history of the buckling instability in a thin elastic rod under compression goes back to the works of Euler and his contemporaries [1], who laid down the foundations of the general theory of elastic stability [2]. Although this field has long been the domain of engineers and applied mathematicians, there has recently been a renaissance of interest in problems related to the stability of thin filaments in the theoretical physics community [3]–[16] prompted by experimental advances in the art of mechanical manipulation of single DNA molecules and rodlike protein assemblies [17]–[19]. Because double stranded DNA molecules are helices that contain strongly curved regions [20], their modeling as elastic rods has to take into account their spontaneous curvature and twist. While general considerations concerning such " naturally " curved rods can already be found in the work of Kirchhoff [1], little is known about the effect of spontaneous curvature and twist on the stability with respect to buckling. The present study deals with the effect of spontaneous twist on the static stability of a thin inextensible elastic rod. Nonlinear equilibrium equations are derived for a rod under the action of external forces and moments. Two classical problems are analyzed which have well-known solutions in the absence of spontaneous twist [1]: a rectilinear rod compressed by forces applied to its ends and a ring compressed by radially inwards directed pressure uniformly distributed along its perimeter. It is demonstrated that while spontaneous twist stabilizes the straight rod against buckling, it has a destabilizing effect on the ring. In both cases buckling takes place through a three-dimensional instability. Consider an elastic rod of length L and cross-section S which has two axes of symmetry. We assume that L ≫ a where a is the largest dimension associated with the cross-section. Denote 1