Large deformation and stability of an extensible elastica with an unknown length

Abstract

a b s t r a c tA beam resting on spatially fixed supports may slide relatively to these as soon as external forces are applied. Consequently, the length of the portion of the reference configuration, which is currently located between the supports, depends on the loading and therefore is not known in advance. In the present paper, the problem of a slender beam under a uniformly distributed force is investigated, which is clamped at one side but may slide through another clamping device in axial direction at the opposite side. In combination with a suitable coordinate transformation by which the numerical treatment is simplified, a finite element approach is utilized to determine the equilibrium shape for the maximum critical load that can be imposed on the beam. In the course of this, the influence of the extensibility of the beam axis is studied. A theory based on Reissner's geometrically exact relations for the plane deformation of beams is adapted such that it allows constitutive relations on stress-strain level to be integrated consistently. In addition to the classical equations of the extensible elastica, a constitutive model derived from the St. Venant-Kirchhoff material of non-linear continuum mechanics is studied. The results obtained in this survey are finally compared to those from the linear beam theory, which turns out to be incapable of describing the problem under consideration in a satisfactory manner.

Geometrically exact Cosserat rods with Kelvin–Voigt type viscous damping

“…In general the tensorÛ has to be constructed via the spectral decomposition ofĈ, which in 3-D cannot be expressed easily 17 in closed form. For special simplified problems, like the plane deformation of an extensible Kirchhoff rod as discussed by Irschik and Gerstmayr (2009) and Humer and Irschik (2011), it is possible to derive simple, kinematically exact closed form expressions 18 forÛ andR pd by inspection of the deformation gradient. Also in the more general case ofĈ given by Eq.…”

Effects of deformation in the dynamics of belt drive

“…The difference from the previously considered case of a timing belt is that only one condition needs to be fulfilled: the velocities of the material points at the end of a straight path must coincide with the velocity of the pulley surface, but there is no condition on the strain of the belt. Consider the equation of balance in the tangential direction (16) with the kinematic relations (12) and (15) for the acceleration w τ . Substituting now the force Q from the constitutive relation (4) and the expression for the strain (14), for a straight path (q τ = 0), we arrive at a homogeneous partial differential equation for S(σ, t):S…”

“…This step, which was originally suggested by one of the authors [14], brings the dynamical equations of the belt to a domain with fixed boundaries, and preserves the level of generality sufficient for a consistent analysis of unsteady operation of the drive; for another example of coordinate transformation in a static problem of deformation of a rod with an unknown length see Humer and Irschik [15]. In contrast to the approaches mentioned above, in which the solutions in different regions are combined using some conditions of continuity and compatibility, we are considering belt dynamics both on the pulleys and on the straight paths simultaneously with the same system of equations.…”

Variable-Length Link-Spring Model of Wire-Bonding Looping Process

“…Some researchers have studied the arc length of a beam, either with one end subjected to a moment at a hinge and able to slide freely over a support [22]; with one support elevated above the other [23]; or as a strip that has a specified length and deformation due to its own weight [24]. A more closely related problem is the slender beam that is clamped at one side but can slide along the axial direction at the opposite side [25], [26]. However, compared with the variable-length gold wire problem, these variablearc-length beam problems are relatively simple because the support is static and there is no complex opening end force, moment loading or plastic deformation.…”