Computer Methods in Applied Mechanics and Engineering
Instability phenomena during the conical expansion of circular cylindrical shells
Introduction
This paper discusses instability phenomena that may be encountered during the expansion of the open end of a circular cylindrical shell into a cone. This process is referred to as conical tube flaring. It employs a conical punch for enlarging the end diameter of a tube by driving the punch into the open end of a tube that is supported on the opposite end. As the punch is pressed into the tube along the axis of rotational symmetry, the tube walls are forced to expand in the radial direction and to assume the contour of the punch.
Fig. 1 shows the kinematics of tube flaring in an idealized way; a cone-shaped punch, characterized by the angle γ, is driven into a tube with the initial inner radius R0 and the initial wall thickness h0. The displacement of the cone relative to the position of first contact between the cone and the inner edge of the tube is denoted as H0. Following the contour of the cone the tube wall material is radially displaced, the new radial position being R. For fulfilling the condition of plastic volume conservation the plastic straining in the circumferential direction has to be compensated for by a reduction of the wall thickness h and a retraction of the material in the meridional direction. The latter causes the expanded section of the cylinder, marked by the distance H, to be shorter than any simple bending deformation mode would suggest.
Tubes with a cone-shaped end section can be applied to design fittings, for triggering collapse in crash-elements, for use in test procedures for the determination of limits of tube expansion, and for shaping axisymmetric vessels in general. Accounting for this wide field of applications, several studies dealt with the mechanical description of this process. Most of these studies aim at providing relationships between the tool stroke depth and the force driving the tool. Furthermore, analytical solutions for the resulting tube end radius and the tube end strain were derived. In [1], [2] analytical relationships are developed and compared to both finite element simulations and experimental results. In [3], [4] results of finite element analyses are presented. Failure due to plastic buckling or necking is not considered in these papers.
In this study we want to explore instabilities which place forming limits on the tube flaring process. These instabilities manifest themselves as global, progressive buckling in the so-called ‘Concertina’ mode and bifurcation due to local necking. Both phenomena are investigated by means of the finite element method. Experimental evidence for both kinds of instabilities is presented, too.
Section snippets
Experimental Setup
The experimental setup for the presented tube flaring experiments followed the principal kinematics described in Fig. 1, as can be seen from Fig. 2. A lubricant was used for reducing the friction between the tube and the cone surface.
The cone angle γ was chosen as γ = 30°. All samples for which quantitative results are presented were cut from a long tube with an initial wall thickness of h0 = 3.2 mm and an initial inner radius of R0 = 41.3 mm. With respect to the undeformed length of the tube specimens
Experimental results
The long cylinder (A) and the two short cylinders (B) and (C) showed markedly different mechanical behavior owing to their different geometrical parameters.
For the long tube (A) the peak driving force was reached with the loss of stability due to elasto-plastic buckling. The limit load of approximately 330 kN can be identified in the force–displacement diagram presented in Fig. 4. In this figure the measured data for experiment (A) is indicated by a solid line with additional dots. After the
Simulation methods
For the numerical investigations of the tube flaring process the finite element code ABAQUS/Standard [5] was employed. The simulation setup was similar to the one in Fig. 1, because the cone, which was modeled as a rigid, analytical surface of rotation, was moving relative to the tube. The supported tube end was restrained in the axial direction, but no constraints were put on the radial displacements.
As long as the necking behavior is not of interest axisymmetric finite element models are
Simulation results
Deformation controlled finite element simulations were performed with the axisymmetric finite element model of the long tube (A). The predicted force–displacement relationship for the cone is shown in Fig. 4 alongside the experimental results. Excellent qualitative and quantitative agreement was obtained for a chosen coefficient of friction of μ = 0.1. The onset and progress of elasto-plastic buckling were readily observable from the deformation plots. Fig. 9 shows a 3D rendering of the
Discussion
The finite element method was successfully applied for simulating two different instability mechanisms occurring in the course of a tube flaring process. The agreement of the predicted force–displacement relationships with experimental measurements was excellent, especially in the sub-critical loading regime, see Fig. 4, Fig. 5.
For a cylinder with a length to diameter ratio of about 2 the failure mode was found to be elasto-plastic buckling involving the formation of axisymmetric folds that are
Conclusions
In the present study experimental evidence and numerical simulations are presented for two instability phenomena which affect the achievable degree of deformation in the tube flaring process, namely axisymmetric Concertina buckling and periodic necking at the free end. These mechanisms are related in so far as the drop of the driving force during Concertina buckling shields the necking regions from further loading, so that necking is suppressed by premature Concertina buckling.
With respect to
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The second author is most grateful to Prof. E. Werner, Head of the Chair of Materials Science and Mechanics of Materials, for valuable discussions and for the provision of personnel and equipment of the laboratory of his Institute during the stay of the author as visiting professor at TU München.