Inextensible Elastica Research Articles

Equilibria and vibration of a heavy pinched loop

A thin strip is bent such that the two ends are brought together and clamped (pinched) to form a teardrop shape. The clamped end is held at various angles with the loop either upright, horizontal, downward, or halfway between these positions. The length of the loop is increased, and the resulting equilibrium shapes, as well as small in-plane vibrations about equilibrium, are investigated analytically and experimentally. When the loop is held upright, in-plane buckling occurs at a critical length, and subsequent postbuckling deflections can be large. For the other orientations, except the hanging one, deflections also become large as the length is increased. In the analysis, the strip is assumed to be an inextensible elastica which is unstrained when straight, and its self-weight is included. A shooting method is applied to obtain numerical solutions to the nonlinear equilibrium boundary value problem and the linear vibration boundary value problem. Polycarbonate strips are used in the experiments, and data are acquired with a laser vibrometer. The experimental deflections, frequencies, and mode shapes exhibit excellent agreement with the analytical solutions.

エラスティカの変分原理

An elastica is a mathematical model of a beam undergoing the large deformations and rotations. The central line of beam is assumed to be inextensible in order to make the analysis of the beam feasible. Since the extensional deformation is not included in the strain energy of beam, the variational theory for the inextensible elastica has not been developed. In the present paper, the principle of virtual work and the principle of stationary potential energy for the elastica, exclusively expressed in terms of bending deformations, are derived from the principle of virtual work in the three dimensional elasticity. And it is shown that the developed variational principles yield the exact equilibrium equations for the beam in the large deformations and rotations.

DIVERGENCE INSTABILITY OF VARIABLE-ARC-LENGTH ELASTICA PIPES TRANSPORTING FLUID

A flexible elastic pipe transporting fluid is held by an elastic rotational spring at one end, while at the other end, a portion of the pipe may slide on a frictional support. Regardless of the gravity loads, when the internal flow velocity is higher than the critical velocity, large displacements of static equilibrium and divergence instability can be induced. This problem is highly nonlinear. Based on the inextensible elastica theory, it is solved herein via the use of elliptic integrals and the shooting method. Unlike buckling with stable branching of a simply supported elastica pipe with constant length, the variable arc-length elastica pipe buckles with unstable branching. The friction at the support has an influence in shifting the critical locus over the branching point. Alteration of the flow history causes jumping between equilibrium paths due to abrupt changes of direction of the support friction. The elastic rotational restraint brings about unsymmetrical bending configurations; consequently, snap-throughs and snap-backs can occur on odd and even buckling modes, respectively. From the theoretical point of view, the equilibrium configurations could be formed like soliton loops due to snapping instability.

Two-dimensional elastica analysis of equilibrium shapes of single-anchor inflatable dams

Several types of inflatable dams are considered. These are long, air-inflated, cylindrical structures on a rigid foundation. Sometimes one of the long edges of a sheet is folded back to the other edge, and then the two edges are clamped to the foundation along a single anchoring line. A second configuration can be modeled as two sheets attached along two long edges, with one edge anchored and the other free to lift as air pressure is applied between the sheets. Another device treated here is a hinged spillway gate lifted by an inflatable bladder. The cross section of the dam or bladder is analyzed as an inextensible elastica. The governing equations and boundary conditions are formulated for each case, and shooting methods are utilized to obtain numerical solutions for the equilibrium shapes. The effects of the internal air pressure and the external water height are investigated.

Symmetric and unsymmetric buckling of circular arches

A nonlinear geometrically exact inextensible elastica theory is used to derive a mathematical system which models a clamped circular arch of central angle 2 α 2\alpha under the action of a vertical force field of amplitude P P (e.g., gravity). The equilibria of the arch are studied for various values of α , 0 > α > π \alpha , 0 > \alpha > \pi . The existence of a solution of symmetric form for all fixed values of P P and α \alpha is proved analytically by arguments based on variational principles. Numerical solutions are calculated for a variety of choices of α \alpha , and in each case buckling (nonuniqueness) is shown to occur when P P is sufficiently large. In some cases, both symmetric and unsymmetric configurations are found, but each unsymmetric configuration obtained is found to be an unstable equilibrium, having energy greater than that of the symmetric configuration. Implications concerning the relative strengths and weaknesses of the various arches are discussed.

A bifurcation problem involving elastica

A flexible inextensible elastica has been considered, with fixed end points and fixed equal end slopes, and varying length. The existence of at least one elastic curve has been proved when the elastica length is greater than the distance between the end points. A critical value of the length has been found, as a function of the given end slope only, corresponding to some elastic curves. From each of them two new elastic curves branch when the length is greater than the critical value.