The Cosserat director theory is used to formulate the problem of a long thin weightless rod constrained, by suitable distributed forces, to lie on a cylinder while being held by end tension and twisting moment. Applications of this problem are found, for instance, in the buckling of drill strings inside a cylindrical hole. In the case of a rod of isotropic cross‐section the equilibrium equations can be reduced to those of a one‐degree‐of‐freedom oscillator in terms of the angle that the local tangent to the rod makes with the axis of the cylinder. Depending on the radius of the cylinder and the applied load, the oscillator has several fixed points, each of which corresponds to a different helical solution of the rod. More complicated shapes are also possible, and special attention is given to localized configurations described by homoclinic orbits of the oscillator. Heteroclinic saddle connections are found to play an important role in the post‐buckling behaviour by defining critical loads at which a straight rod may coil up into a helix.

The static deformation of a twisted elastic rod constrained to lie on a cylinder

The transformation of the straight embryonic heart tube into a helically wound loop is named cardiac looping. Such looping is regarded as an essential process in cardiac morphogenesis since it brings the building blocks of the developing heart into an approximation of their definitive topographical relationships. During the past two decades, a large number of genes have been identified which play important roles in cardiac looping. However, how genetic information is physically translated into the dynamic form changes of the looping heart is still poorly understood. The oldest hypothesis of cardiac looping mechanics attributes the form changes of the heart loop (ventral bending → simple helical coiling → complex helical coiling) to compressive loads resulting from growth differences between the heart and the pericardial cavity. In the present study, we have tested the physical plausibility of this hypothesis, which we call the growth-induced buckling hypothesis, for the first time. Using a physical simulation model, we show that growth-induced buckling of a straight elastic rod within the confined space of a hemispherical cavity can generate the same sequence of form changes as observed in the looping embryonic heart. Our simulation experiments have furthermore shown that, under bilaterally symmetric conditions, growth-induced buckling generates left- and right-handed helices (D-/L-loops) in a 1:1 ratio, while even subtle left- or rightward displacements of the caudal end of the elastic rod at the pre-buckling state are sufficient to direct the buckling process towards the generation of only D-loops or L-loops, respectively. Our data are discussed with respect to observations made in biological ‘models’. We conclude that compressive loads resulting from unequal growth of the heart and pericardial cavity play important roles in cardiac looping. Asymmetric positioning of the venous heart pole may direct these forces towards a biased generation of D- or L-loops.

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Cardiac looping may be driven by compressive loads resulting from unequal growth of the heart and pericardial cavity. Observations on a physical simulation model.

The post-buckling behavior of a beam subjected to lateral constraints is of practical importance in a variety of applications, such as stent procedures, filopodia growth in living cells, endoscopic examination of internal organs, and deep drilling. Even though in reality the constraining surfaces are often deformable, the literature has focused mainly on rigid and fixed constraints. In this paper, we make a first step to bridge this gap through a theoretical and experimental examination of the post-buckling behavior of a beam constrained by a fixed wall and a springy wall, i.e. one that moves laterally against a spring. The response exhibited by the proposed system is much richer compared to that of the fixed-wall system, and can be tuned by choosing the spring stiffness. Based on small-deformation analysis, we obtained closed-form analytical solutions and quantitative insights. The accuracy of these results was examined by comparison to large-deformation analysis. We concluded that the closed-form solution of the small-deformation analysis provides an excellent approximation, except in the highest attainable mode. There, the system exhibits non-intuitive behavior and non-monotonous force–displacement relations that can only be captured by large-deformation theories. Although closed-form solutions cannot be derived for the large-deformation analysis, we were able to reveal general properties of the solution. In the last part of the paper, we present experimental results that demonstrate various features obtained from the theoretical analysis.

The post-buckling behavior of a beam constrained by springy walls

In this paper, we study the deformation of a thin elastic rod constrained inside a cylindrical tube and under the action of an end twisting moment. The ends of the rod are clamped in the lateral direction. Unlike the previous works of others, in which only the fully developed line-contact spiral was considered, we present a complete analysis on the deformation when the dimensionless twisting moment M x is increased from zero. It is found that the straight rod buckles into a spiral shape and touches the inner wall of the tube at the midpoint when M x reaches 8.987. As M x increases to 11.472, the contact point in the middle splits into two, leaving the midpoint floating in the air. As M x increases to 13.022, the midpoint returns to touch the tube wall and the two-point-contact deformation evolves to a three-point-contact deformation. Starting from M x = 13.098, the point contact in the middle evolves to a line contact, and the deformation becomes a point-line-point contact configuration and remains so thereafter. In the case when the line-contact pattern is fully developed, it is possible to predict the spiral shape analytically. The numerical results are found to agree very well with those predicted analytically. Finally, an experimental setup is constructed to observe the deformation evolution of the constrained rod under end twist.

On an Elastic Rod Inside a Slender Tube Under End Twisting Moment

In this paper we use elastica model to calculate the deformation of a clamped–clamped rod under end twist and constrained inside a straight tube. Unlike most of the previous works, in which only the fully-developed line-contact spiral from end to end was considered, we study the case when both ends of the rod are at the center of the tube cross section. As a consequence, free of contact and point contact may occur in the deformation. The results are compared with those predicted from a previous work using small-deformation theory. Ten deformation patterns from deformation 1–10 are calculated by shooting method, with a possibility of finding more. The deformation sequence forms a smooth load-deflection locus. It is found that the small-deformation theory is capable of finding only the early stage of the deformation sequence from deformation 1 to 5. The elastica model, on the other hand, predicts that the constrained elastica may undergo snapping jump and self-contact when it is under load or displacement control. These deformations cannot be found from a small-deformation theory.

A twisted elastica constrained inside a tube

Buckling of drill and tubing strings in drill or casing holes will affect the life of the string and cause difficulties in drilling and in oil production. A number of theoretical models have been developed to study the buckling load and the post-buckling configuration of the strings under some idealized conditions. However, verification of these theories in practice or in laboratory has not been extensively reported.

X.C. Tan

Papers

Buckling of slender string in cylindrical tube under axial load: experiments and theoretical analysis