Elastomer-assisted compression beading of tubes

Abstract

This article proposes a new tube forming process designated as ‘elastomer-assisted compression beading’ that is made from a combination of tube bulging produced by elastomer forming and conventional pressworking compression beading. The presentation includes independent determination of the mechanical properties of the tubes, analytical modelling, finite element simulation and experimentation under laboratory conditions with the aims and objectives of identifying the key operating parameters, understanding plastic flow and failure and establishing the formability limits of the elastomer-assisted compression beading process. Results and observations show that the elastomer-assisted compression beading process is capable of producing sound, large-width, compression beads in a broader range of operating parameters than those currently being achieved by means of conventional compression beading. Applications of the proposed elastomer-assisted compression beading process span a wide range of industrial uses involving attachment of tubes to sheets and damping vibrations in air-pressure lines, liquid systems or exhaust tubes.

Keywords

Tube forming elastomer forming compression beading plasticity failure

Introduction

Compression beading is a tube forming process that is accomplished by forcing one tube end towards the other (or the two tube ends towards one another) while leaving a gap opening in-between the dies that hold the tube. Figure 1(a) illustrates a tube before and after being compressed by the upper die and shows its collapse by local buckling that led to the development of a compression bead ( $l_{gap}$ refers to the initial gap opening).

Compression beads are used in a wide variety of industrial applications. For example, they are used for creating sealing beads for pressure hose applications; for attaching tubes to sheets; for damping vibrations in air-pressure lines, liquid systems or exhaust tubes and for increasing the effectiveness of a sealing by means of O-rings, among other applications.

The pioneering investigations in local buckling were performed in the late 1940s by Shanley¹ who modified the elementary theory of columns due to Euler and proposed the concept of tangent modulus for calculating the critical instability load in plastically deformed cylindrical shells under axial compression. The subject was extensively investigated by the structural mechanics community over the past decades. Hutchinson,² for example, studied the influence of geometric imperfections on the buckling behaviour and critical instability loads. Tvergaard³ analysed the influence of the ratio of the radius to the thickness of the tube $r_{0} / t_{0}$ (hereafter referred to as ‘the radius-to-thickness ratio’) on the development of axisymmetric and non-axisymmetric (e.g. diamond shaped) modes of deformation. Goto and Zhang⁴ introduced the role played by boundary conditions, and more recently, Le Grognec and Le Van⁵ performed a comprehensive discussion on the utilization of finite elements to determine the critical instability loads, the bifurcation modes and the post-critical behaviour of cylindrical shells in the elastoplastic range.

The first materials’ processing-oriented publication in local buckling of thin-walled tubes was authored by Gouveia et al.⁶ and made use of varying gap opening conditions. The work identified two main operating parameters: (1) the ratio of the initial gap opening to the reference radius of the tube $l_{gap} / r_{0}$ (hereafter referred to as ‘the tube slenderness ratio’) and (2) the aforementioned radius-to-thickness ratio $r_{0} / t_{0}$ . In concrete terms, the authors showed that by varying $r_{0} / t_{0}$ , it is possible to divide plastic flow into two different groups: for high values of $r_{0} / t_{0}$ , compression beads are steadily formed both outward and inward (refer to ‘Detail A’ in Figure 1), whereas for decreasing values of $r_{0} / t_{0}$ , compression beads gradually change to become outward dominant (refer to ‘Detail B’ in Figure 1). The work by Gouveia et al.⁶ also revealed that although the width of the compression beads can be slightly altered by setting the initial gap opening $l_{gap}$ , the reduction in the unsupported tube length due to downward movement of the upper die always results in the development of small-width (and sometimes incomplete) compression beads. Industrial application of the compression beading process for connecting tubes to sheets is comprehensively discussed by Alves et al.⁷ and Gonçalves et al.⁸

Under these circumstances, the aims and objectives of this article are to introduce a new ‘elastomer-assisted compression beading’ process (hereafter referred to as ‘the EACB process’) that makes use of the original idea by Al-Qureshi⁹ of using elastomers as pressure-transmitting medium for tube bulging in order to meet the challenge of producing outward dominant, large-width, compression beads in a broader range of operating parameters.

The contribution to knowledge within materials’ processing is threefold. First, the development of a new processing technique that extends the formability limits of conventional compression beading. Second, the combination of experimental, analytical and finite element methodologies to obtain a comprehensive understanding of the behaviour of the tubular specimens across the useful range of operating conditions. Third, the analysis of potential failure modes and associated causes.

The organization of this article is as follows. Section ‘Experimental background’ summarizes the methods and procedures that were utilized in the mechanical characterization of the material, presents the fundamentals of the EACB process and describes the experimental work plan. Section ‘Finite element modelling conditions’ provides a brief overview of the finite element modelling conditions. Section ‘Results and discussion’ presents and discusses the results, namely, plastic flow and failure, and compares the formability limits of the EACB process with those of conventional pressworking compression beading. Section ‘Conclusion’ presents the conclusions and Appendix 1 includes details of the analytical derivation of the internal pressure at the onset of instability.

Experimental background

Mechanical characterization

The tubular specimens utilized in the investigation were cut from commercial S460MC (carbon steel) welded tubes with an outside radius $r_{0} = 16 mm$ and a wall thickness $t_{0} = 1.5 mm$ . The stress–strain curve of the material was obtained by means of tensile and stack compression tests performed at room temperature on a hydraulic testing machine (Instron SATEC 1200 kN) with a cross-head speed equal to 100 mm/min (1.7mm/s). The relationship between true stress and true strain was approximated by means of the Ludwik–Hollomon’s strain hardening model (Figure 2(a))

σ = 616.4 ε^{0.06} MPa

(1)

The critical instability load for the occurrence of axisymmetric local buckling was determined by compressing tubular specimens with 100 mm initial length between flat parallel platens. Two types of tube constraints with extreme conditions denoted as ‘free-ended’ and ‘fixed-ended’ were considered (Figure 2(b)). In case of free-ended conditions, both end surfaces of the tubes were neither laterally supported nor otherwise constrained, whereas in case of fixed-ended conditions, both tube ends were fixed, so that there could be neither lateral displacement nor change in slope at either end.

The experimental load–displacement curves that were obtained for each constraint condition are similar. Thus, assuming constraint conditions during both stages of the EACB process to be neither free-ended nor fixed-ended, the average critical instability load $P_{cr}$ was estimated to be approximately equal to 100 kN (refer to the dashed grey line in Figure 2(b)).

Methods and procedures

Figure 3 presents a schematic representation of the active tool components that were employed by the new proposed EACB process at the open and closed positions. As seen, the overall process is accomplished by a sequence of two different forming stages. In the first stage, a cylindrical rubber plug with 14.5 mm radius and 50 mm height, made from polyurethane with a Shore hardness of 90A, is placed inside the tube and the tube is bulged (expanded) by compressing the rubber plug with two punches moving in opposite direction with the same velocity $v$ (Figure 3(a)). The punches are then retracted, and the rubber plug is removed after returning to its original shape. In the second stage, the local bulged tube is subjected to axial compression load by means of the upper die in order to form the desired compression bead (Figure 3(b)).

The active tool components are dedicated to a specific outside radius $r_{0}$ of the tube, and the internal pressure $p$ together with the initial gap opening $l_{gap}$ between the upper and lower dies determines the final bulge width $w_{b}$ and length. The inner corner radius of the upper and lower dies $r_{c}$ is equal to 1 mm to allow material flow without damage.

During the first stage of the EACB process, the internal pressure $p$ transmitted by the rubber was experimentally determined by on-line measurement of the force applied by the upper punch using a load cell with 1200 kN capacity. The bulge width $w_{b}$ of the tube at the equatorial plane was continuously measured by means of a digital indicator with a resolution of 0.01 mm.

The second stage of the EACB process involved on-line measurement of the applied force and the vertical displacement of the upper die. The aforementioned load cell and a displacement transducer with a resolution of 0.001 mm were employed.

The active tool components were installed in a die set (not shown in Figure 3) that was mounted on the same hydraulic testing machine where the stress–strain curve of the material and the critical instability load of the tubes were determined. The characteristics of slow pressure rise of the hydraulic testing machine match the somewhat delayed deformation behaviour of the rubber plug and ensure adequate forming conditions.

The tubes were first bulged by compressing the rubber plug with the upper and lower punches until failure in order to determine the pressure at the onset of instability $p_{cr}$ and the maximum displacement $d_{max}$ at the onset of bursting for different values of the initial gap opening $l_{gap}$ . Then, different levels of displacement $d < d_{max}$ were applied to new tubular specimens in order to obtain the maximum allowable expansion before bursting for each value of $l_{gap}$ .

At the end of tube expansion (first stage of the EACB process), the rubber plug was removed and the upper and lower punches were retracted. An internal mandrel was then placed inside the bulged tube, which was subjected to axial compression by means of the upper die in order to produce the desired compression bead. After finishing the second stage of the EACB process and removing the internal mandrel from the tubular parts, selected specimens were halved lengthwise in order to measure the wall thickness variation along the meridional direction of the cross section.

The plan of experiments was designed in order to consider only the influence of the tube slenderness ratio $l_{gap} / r_{0}$ on the performance of the overall EACB process. The radius-to-thickness ratio $r_{0} / t_{0}$ was left out of the experimental work plan because the bulged tubes that are obtained at the end of the first stage determine that compression beads are exclusively formed outward during the second stage of the EACB process. The influence of anisotropy due to the weld seam of the tubes was not taken into consideration because experimental observations showed no influence in material flow behaviour. Table 1 summarizes the geometry of the tubular specimens that were utilized in the experiments.

Table 1.

Geometry of the tubular specimens that were utilized in the experiments with the EACB process (nomenclature according to Figure 3).

Case	$r_{0}$ (mm)	$t_{0}$ (mm)	$l_{0}$ (mm)	$l_{gap}$ (mm)
1	16	1.5	105	5
2	16	1.5	110	10
3	16	1.5	115	15
4	16	1.5	120	20
5	16	1.5	125	25
6	16	1.5	130	30
7	16	1.5	135	35

The experiments were done in a random order, and at least two replicates were produced for each test case in order to provide statistical meaning.

Finite element modelling conditions

The finite element modelling conditions that were utilized throughout the investigation made use of the in-house finite element computer program I-form. I-form was developed by the authors and is built upon the irreducible finite element flow formulation, which is based on the following extended variational principle to account frictional effects

Π = \int_{V} \bar{σ} \overset{\cdot}{\bar{ε}} dV + \frac{1}{2} K \int_{V} {\overset{\cdot}{ε}}_{v}^{2} dV - \int_{S_{T}} T_{i} u_{i} dS + \int_{S_{f}} (\int_{0}^{u_{r}} τ_{f} d u_{r}) dS

(2)

where $\bar{σ}$ is the effective stress, $\overset{\cdot}{\bar{ε}}$ is the effective strain rate, $K$ is a large positive constant imposing the incompressibility constraint, $V$ is the control volume limited by the surfaces $S_{U}$ and $S_{T}$ , $T_{i}$ are the surface tractions on $S_{T}$ and $τ_{f}$ are the friction shear stresses on the contact interface $S_{f}$ between material and tooling.

Friction is modelled as traction boundary conditions, and the additional power consumption resulting from the rightmost term in equation (2) is determined through the utilization of the law of constant friction $τ = mk$ . The friction factor $m$ was set to 0.1 after checking the finite element–predicted forming pressures and loads that best matched the experimental results. This value of $m$ is also close to that specified in the technical data on polyurethane forming elastomers supplied by Fibro GmbH.

The computer implementation of the extended variational principle $Π$ is built upon rigid-plastic material behaviour and employs a mixed formulation based on linear quadrilateral elements (in two-dimensional (2D) applications) or hexahedral elements (in three-dimensional (3D) applications) and selective reduced integration schemes. Further details on the finite element flow formulation and I-form can be found elsewhere.¹⁰

The discretization of the tubular preforms that were utilized in the investigation was performed by means of four-node axisymmetric quadrilateral elements due to rotational symmetry of the EACB process and because no anisotropy effects were taken into consideration. Convergence studies with varying arrangements of quadrilateral elements across the thickness showed that the utilization of four elements was adequate for modelling the distribution of the major field variables and for getting a proper evolution of the load–displacement curve. The tools were considered rigid and were discretized by means of contact-friction linear elements (Figure 4).

The internal pressure $p$ transmitted by the rubber was assumed as a normal distributed load per unit of area acting along the edges of the quadrilateral elements that are located at the inside radius of the tubes where the rubber plugs are placed. The numerical simulation of the EACB process was accomplished through a succession of pressure (first stage) and displacement (second stage) increments each of one modelling approximately 1% of the maximum applied pressure and 0.1% of the initial tube length, respectively. The convergence of the computational procedure was stable, and the overall central processing unit (CPU) time for a typical analysis containing a total number of 600 elements was below 3 min on a standard laptop computer equipped with an Intel i7 CPU (2.7 GHz) processor and making use of a single core.

Results and discussion

Plastic flow and failure

First stage

The observation of crack morphology at bursting for the entire set of tubular specimens that are listed in Table 1 revealed the existence of two different plastic flow mechanisms during the first stage of the EACB process (Figure 5). For large values of the initial gap opening $l_{gap}$ , bursting occurs by failure along a vertical crack in close agreement with the well-known behaviour of thin-walled cylinders under internal pressure (refer to the rightmost specimens in Figure 5). On the contrary, the bursting behaviour of tubes with small values of the initial gap opening $l_{gap}$ exhibits failure along a horizontal crack (refer to the leftmost specimens in Figure 5). This last mode of failure has never been addressed before by researchers working on hydroforming or elastomer-assisted plastic deformation of tubes, as far as the authors are aware.

The experimental values of the internal pressure $p_{cr}$ at the onset of instability plotted as a function of the initial gap opening $l_{gap}$ between the upper and lower dies also revealed the existence of the two aforementioned plastic flow mechanisms (Figure 6). In fact, $p_{cr}$ was found to decrease sharply with the increase in the initial gap opening up to $l_{gap} = 20 mm$ (corresponding to a tube slenderness ratio $l_{gap} / r_{0} = 1.25$ ) and to remain approximately constant ( $p_{cr} ≅ 65 MPa$ ) throughout the largest values of $l_{gap}$ .

The constant values of the internal pressure $p_{cr}$ that were registered for the largest values of $l_{gap}$ are associated with the aforementioned tube bursting behaviour along vertical cracks, and the plastic flow mechanism may, therefore, be understood in the light of the classical solution of a thin-walled cylinder shell with open ends under internal pressure. The internal pressure $p_{cr}$ is calculated from the analytical expression included in Appendix 1 (equation (6)), and the result is plotted as a horizontal grey line in Figure 6.

Conversely, the sharp variation of the internal pressure $p_{cr}$ that was experimentally observed for the small values of the initial gap opening $l_{gap}$ is associated with subsequent tube bursting along horizontal cracks. The analytical model that is proposed by the authors for understanding plastic flow in such operating conditions is built upon an approximate solution for a thin-walled toroidal shell under internal pressure that is also included in Appendix 1. The corresponding expression for calculating $p_{cr}$ as a function of $l_{gap}$ is given by equation (10), and the result is plotted as a black curve in Figure 6.

The agreement between the experimental values and the analytical predictions of $p_{cr}$ is good and confirms the validity of the proposed plastic flow mechanisms controlling the first stage of the EACB process. Major differences are due to the simplifying assumptions that were necessary to perform in order to derive the analytical equations, to account for the influence of boundary constraints and to neglect the geometric imperfections of the tubular specimens.

In addition, the two different types of experimental results that were observed for case 3 of Table 1 (corresponding to an initial gap opening of $l_{gap} = 15 mm$ , $l_{gap} / r_{0} = 0.94$ ) allowed authors to identify the transition value of $l_{gap}$ where plastic flow and failure mechanisms associated with thin-walled cylinder and toroidal shells are likely to coexist and compete each other (Figure 7(a)). The details of the finite element–predicted distribution of ductile damage according to the normalized Cockcroft–Latham criterion $D = \int_{0}^{ε_{f}} σ_{1} / \bar{σ} d \bar{ε}$ for selected test cases with initial gap openings below and above $l_{gap} = 15 mm$ ( $l_{gap} / r_{0} = 0.94$ ) corroborate the changes in plastic flow and the corresponding location and direction of failure by cracking (Figure 7(b)). In fact, the region where the accumulated damage $D_{max}$ is the greatest and failure by cracking is likely to occur, changes from the circumferential direction along the tensile region of the inner tube surface subjected to bending and pressure (refer to ‘A’ in Figure 7(b)), to the equatorial region of the outer tube bulged surface (refer to ‘B’ in Figure 7(b)).

Although the above discussion has been stimulated by crack morphology at bursting, the essence of the first stage of the EACB process relies on the possibility of controlling the amount of compression of the rubber plug by means of the upper and lower punches so that sound bulged tubes can be produced without failure (Figure 8). These tubular specimens are the preforms that are used during the second stage of the EACB process.

Second stage

The second stage of the EACB process starts after extracting the upper and lower compression punches and removing the rubber plug. In general, the second stage involves axial compression of the bulged tubes by means of the upper die (refer to Figure 3(b)), and the obtained results are summarized in Figure 9.

As seen in Figure 9, the formability limits of the EACB process successfully extend those of the conventional process by allowing the formation of compression beads under initial gap openings below and above the former minimum and maximum thresholds. On the other hand, results also show that the final width of the compression beads $w_{cb}$ produced by the EACB process is 10%−20% larger than that produced by the conventional process. This is because the EACB process was successful in producing compression beads in tubes with initial gap openings of $l_{gap} = 5 mm$ ( $l_{gap} / r_{0} = 0.3$ ) and initial gap openings up to $l_{gap} = 25 mm$ ( $l_{gap} / r_{0} = 1.6$ ).

The saturation of the maximum bulge width to a value of approximately $w_{b} ≅ 4.7 mm$ for initial gap openings $l_{gap} \geq 20 mm$ prevents the formation of large bulge widths during the first stage of the EACB process and may be considered the main reason responsible for tubular specimens to fail by cracking at increased gap openings (say, $l_{gap} > 25 mm$ in Figure 9). Two types of failure are observed. One type is caused by cracking along the meridional direction (Figure 10(a)) and is attributed to high values of ductile damage at the equatorial plane of the specimens (Figure 10(b)). The other type of failure is caused by cracking along the circumferential direction (Figure 10(c)) and is attributed to meridional stresses emerging from the asymmetric propagation of compression beads whenever the initial gap opening is very large. This plastic flow mechanism is not able to be replicated by means of 2D axisymmetric finite element modelling, but it is easily understood by observing the sequence of photographs that were taken during the experiments (Figure 10(d)). In fact, when the unsupported free length of the tubes is very large, there is a chance that local buckling starts developing naturally into an asymmetric instability wave (due to the influence of geometric imperfections of the tubes, among other reasons) until it comes into contact with the upper and lower dies (refer to ‘A’ and ‘B’ in Figure 10(d)). From this moment on, the compression beads are forced to bend towards the horizontal, axisymmetric, direction, and the risk of failure along horizontal cracks increases due to exposure to high meridional tensile stresses.

The influence of plastic flow on the variation of thickness along the cross section of a tubular specimen at the end of the first and second stages of the EACB process is illustrated in Figure 11. The meridional distance is measured from the upper end to the equatorial region of the tubular specimens. The initial flat region of the graphic corresponds to nearly unstrained material placed in the cylindrical region of the specimens, and the final thickness in this region remains practically identical to the initial thickness of the tubes. The subsequent sharp decrease in thickness that is observed at the end of the first stage of the EACB process is due to tube bending around the corner radius $r_{c}$ (refer to Figure 3(a)) in order to match the contour of the upper die. Further decrease in thickness towards the equatorial region that is observed at the end of the first stage is caused by tube expansion with the rubber plug. A maximum experimental thickness reduction of 17% (corresponding to a finite element estimate of 23%) is found at the equatorial plane (refer to ‘A’ in Figure 11).

The most interesting result associated with the second stage of the EACB process is the replacement of the equatorial plane by the adjacent zone placed immediately before (refer to ‘B’ in Figure 11), as the region of the tubular specimens where thickness variation is larger. This is attributed to a combination of thinning due to bending in ‘B’ and thickening (around 9% increase) at the equatorial plane corresponding to former point ‘A’.

The result of overlapping finite element computed meshes with the cross section of the experimental tubular parts corresponding to the end of the first and second stages of the EACB process (refer to Figure 11) corroborates the overall good agreement that was found between experimentation and numerical simulation (Figure 11).

Pressure and forming loads

Figure 12 shows a typical experimental and finite element–predicted evolution of the internal pressure with the radius of the bulged tube during the first stage of the EACB process. As seen, the internal pressure $p$ increases until it reaches a critical pressure $p_{cr}$ where instability starts and bursting follows with further deformation caused by the upper and lower punches compressing the rubber plug. The insets showing 3D details of the finite element meshes at different instants of time $t$ reveal that the increase in tube radius is negligible during most of the first stage and only grows suddenly very close to its end (say, $t > 0.95 t_{f}$ ).

Finally, Figure 13 shows the experimental and finite element–predicted load–displacement curves corresponding to entire set of test cases listed in Table 1, for the second stage of the EACB process. As seen, all test cases apart from case 1 show three different plastic deformation stages (refer to the regions labelled as ‘A’, ‘B’ and ‘C’ in case 3).

The first stage (region ‘A’) is characterized by a steep increase in the forming load up to values of 40–80 kN as the bulged tube is axially loaded and the compression bead starts to develop. In the second stage (region ‘B’), the compression bead grows under a falling load for all test cases apart from case 1. Case 1 presents a monotonic increase in the forming load due to a small value of the initial gap opening that leads to the development of small-width compression beads. Results also show that the load–displacement curve corresponding to case 6 presents the lowest drop in load and the largest overall displacement. This is attributed to the fact that case 6 is shaped under the largest initial gap opening $l_{gap}$ .

The third stage (region ‘C’) is characterized by a sharp increase in load requirements caused by the enclosure of the compression beads between the upper and lower dies. The agreement between experimental and finite element–predicted evolutions of the load–displacement curve is generally good for the entire set of operating parameters corresponding to Table 1.

Conclusion

This article is about combining tube bulging produced by elastomer forming and compression beading produced by conventional pressworking into a new materials’ processing technique. The proposed EACB process extends the formability limits of compression beads by allowing the utilization of initial gap openings $l_{gap}$ between the upper and lower dies that are beyond the minimum and maximum thresholds of conventional compression beading. Experimental results also show that compression bead widths produced by the EACB process are 10%−20% larger than those obtained by means of the conventional process.

The deformation mechanics of the first and second stages of the EACB process were investigated by means of analytical modelling, numerical simulation and experimentation. Plastic flow and failure during the first stage of the EACB process were explained by the classical solution of a thin-walled cylinder shell with open ends under internal pressure and by a new approximate solution of a thin-walled toroidal shell under internal pressure. The new solution was appropriate for understanding the critical values of the internal pressure $p_{cr}$ and for understanding the development of circumferential cracks at bursting, when using small values of the initial gap opening $l_{gap}$ . The formability limits of the overall EACB process also were dependent on failure by cracking during the second stage. In general, the cracks occurring in the second stage are due to high values of ductile damage in the equatorial region of the tubes, but large values of $l_{gap}$ may trigger asymmetric modes of deformation and failure by circumferential cracking.

Footnotes

Appendix 1 Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

This work was partially supported by the Portuguese Foundation for Science and Technology under the research contract PEst-OE/EME/LA0022/2011. This work was also supported by MCG – Mind for Metal, Carregado, Portugal.

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Case	$r_{0}$ (mm)	$t_{0}$ (mm)	$l_{0}$ (mm)	$l_{gap}$ (mm)
1	16	1.5	105	5
2	16	1.5	110	10
3	16	1.5	115	15
4	16	1.5	120	20
5	16	1.5	125	25
6	16	1.5	130	30
7	16	1.5	135	35

Case	$r_{0}$ (mm)	$t_{0}$ (mm)	$l_{0}$ (mm)	$l_{gap}$ (mm)
1	16	1.5	105	5
2	16	1.5	110	10
3	16	1.5	115	15
4	16	1.5	120	20
5	16	1.5	125	25
6	16	1.5	130	30
7	16	1.5	135	35

Case	$r_{0}$ (mm)	$t_{0}$ (mm)	$l_{0}$ (mm)	$l_{gap}$ (mm)
1	16	1.5	105	5
2	16	1.5	110	10
3	16	1.5	115	15
4	16	1.5	120	20
5	16	1.5	125	25
6	16	1.5	130	30
7	16	1.5	135	35