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8/13/2019 An Experimental and Numerical Studyof Necking Initiation In
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International Journal of Mechanical Sciences 46 (2004) 17271746
An experimental and numerical study of necking initiation in
aluminium alloy tubes during hydroforming
A. Kulkarnia, P. Biswasa,1, R. Narasimhana,, Alan A. Luob, Raj K. Mishrab,Thomas B. Stoughtonb, Anil K. Sachdevb
aDepartment of Mechanical Engineering, Indian Institute of Science, Bangalore, IndiabGeneral Motors Research and Development Centre, Warren, MI, USA
Received 13 January 2004; received in revised form 5 November 2004; accepted 10 November 2004
Available online 5 January 2005
Abstract
In this paper, a combined experimental and numerical investigation of free hydroforming of aluminium
alloy tubes is conducted. The tubes are subjected to different loading histories involving axial compression
and internal pressure. The circumferential and axial strains experienced by the tubes are continuously
recorded along with the pressure and axial load. The numerical simulations are carried out using both 2D
axisymmetric and 3D finite-element formulations by applying the experimentally recorded axial load and
internal pressure. In the latter, a geometric imperfection is introduced in the form of wall thickness
reduction at the tube mid-length in order to trigger necking which happens after significant bulging and
beyond the stage of peak pressure. The strain histories and peak pressures obtained from the simulations
agree well with those determined from the experiments. Further, the forming limit curve predicted by the
simulations as well as from a MK analysis incorporating the computed strain paths corroborate well with
the experimental data. The role of nonproportional straining on the mechanics of failure of the tubes due to
bulging and necking is studied in detail.
r 2004 Elsevier Ltd. All rights reserved.
Keywords: Tube hydroforming; Localized necking; Forming limits; Experiments; Finite-elements
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www.elsevier.com/locate/ijmecsci
0020-7403/$ - see front matter r 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijmecsci.2004.11.007
Corresponding author. Tel.: +91 80 2932959; fax: +91 80 3600648.
E-mail address: narasi@mecheng.iisc.ernet.in (R. Narasimhan).1Currently at General Motors India Science Laboratory, Bangalore, India.
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1. Introduction
Metal forming process is an important technological operation in manufacturing, particularly
in the automotive industry. Several components are made to shape by sheet metal forming, tube
hydroforming and tube bending processes. With the advancements in computer controls and high
pressure hydraulic systems, tube hydroforming is being increasingly used in automotiveapplications in order to achieve weight reduction, reduced part count and cost [1]. A detailed
review of tube hydroforming technology can be found in Ref. [2]. An in-depth understanding of
the formability of materials is the key to developing successful forming operations. In this context,
the forming limit diagram which defines the onset of localized necking by relating the critical
values of major to minor principal strains in 2D strain space can be used as a measure of the
maximum formability of the material. Several experimental as well as analytical methods have
been developed to generate forming limit diagrams particularly for sheet metals.
Forming limit diagrams for sheet metals are generally obtained experimentally using an in-
plane stretching test[3]or hemispherical punch stretching test[4]in which sheets are subjected to a
biaxial state of stress. In these tests, various strain states are achieved by adjusting differentparameters like the lubrication conditions between the sheet metal and the punch and the sheet
width. Graf and Hosford [5,6] applied the hemispherical punch test to investigate the effect of
strain path changes on the forming limit diagram. Several methods have been developed to predict
the forming limits theoretically. Bifurcation analysis corresponding to localized necking using
classical plasticity theory was carried out by Hill[7]but it was restricted to negative minor strains.
The MK approach [8] was a major development in predicting the forming limits for positive
minor strains. This approach is based on the formation of a localized neck from an initial
geometric imperfection (which is assumed to be a linear thinned band) present in a homogeneous
material. It has been employed in many studies [912]to predict forming limit diagrams. In the
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Nomenclature
s1; s2 major and minor principal stress components, respectivelyp1;
p2 major and minor principal plastic strain components, respectively
sy; sz; sr circumferential, axial and radial stress components, respectivelypy;
pz;
pr circumferential, axial and radial plastic strain components, respectively
s equivalent stress
p equivalent plastic straina stress ratio (a s2=s1)r strain rate ratio (r dp2=dp1)R normal anisotropy parameter
p; Dp internal pressure and its increment, respectivelyU; DU actuator displacement and its increment, respectivelyFa axial load on the tube ends
L, r, t half-length, mean radius and thickness of the tube, respectively
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are recorded continuously using post-yield strain gages along with the internal pressure and axial
force data. The experimental results are analysed by conducting 2D axisymmetric and 3D finite-
element simulations. In the 3D simulations, a geometric imperfection in the form of small
reduction in the wall thickness at the tube mid-length is introduced to trigger localized necking. Inaddition, MK analyses are carried out by employing the strain paths computed in the
simulations. Further, the effect of nonproportional strain histories on the peak pressure is
investigated using a semi-analytical method and compared with simulation results. It is found that
the FLC predicted by the 3D simulations as well as the MK analyses agrees well with the
experimental data and shows little sensitivity to the axial strain (at least up to a value of0:15).
2. Experimental work
2.1. Mechanical characterization
The tubes and sheet used in this study were made from an essentially binary Al3%Mg alloy.
First, uniaxial tension tests are conducted in L;TandLTdirections with sheet specimens havinga thickness of 2.5 mm. The engineering stress versus strain curve in the L-direction is shown in
Fig. 1. The parameters representative of the tensile response along with the Lankford ratio R
(defined as the ratio of the true width to thickness strain) are summarized in Table 1. It can be
seen from this table that the behaviour of the alloy is reasonably isotropic in the plane of the sheet
(with an average R value of 0.67).
2.2. Tube hydroforming experiments
A schematic of the test setup is shown inFig. 2. The tube specimen having a length of 225 mm,
outer diameter of 70.2 mm and thickness 3.8 mm is inserted into specially designed end grips. The
assembly is placed in between two compression platens which are fixed to the actuator and load
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0 0.05 0.1 0.15 0.2 0.25 0.30
50
100
150
200
250
Engg. Strain
Engg.
Stres
s(MPa)
Fig. 1. Engineering stress versus strain curve for the aluminium alloy tested in the L direction.
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Table 1
Average values of various parameters pertaining to the aluminium alloy obtained from the tensile tests
Orientation s0:2%
0
(MPa) sU (MPa) U LR
L 80 204 0.24 R0 0:56T 75 208 0.24 R90 0:60LT 78 203 0.21 R45 0:86
Fig. 2. Schematic of the experimental setup used in the tube free hydroforming experiments (without any external tool
contact).
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cell of an INSTRON 8502 testing machine. Axial compression is applied by prescribing the
actuator displacement of the INSTRON. In addition, the tube is subjected to high internal
pressure by supplying oil pressurized using an external hydraulic power pack via an inlet in one of
the end grips (see Fig. 2). Two uni-directional strain gages (with range of 0.2), one along the
circumferential and the other along the axial direction, are installed at the tube mid-length.
Further, an array of strain grid circles (having an initial diameter of 2.5 mm) is also etched on the
tube specimen for determining the limit strains after completion of the test.The tests are carried out by subjecting the tubes to different combinations of internal pressure
and axial compression which are applied in a staggered (or stepwise) fashion. The experimental
data pertaining to axial load, actuator displacement, pressure, and strain gage signals are recorded
continuously using a multi-channel data acquisition system. In addition to the case of free
bulging, the different axial feed to internal pressure ratios, DU=Dp;employed in this work are 0.4,0.75, and 1 mm/MPa. It is observed that in all the tests, the failure mode is due to bulging,
followed by necking of the tube wall and finally by bursting. In most cases, the neck formed near
the mid-length of the tube and is aligned along its axis (see Section 4.4). In Fig. 3, the axial load
experienced by the tube is plotted against the internal pressure. It can be seen from this figure that
the magnitude of the peak axial load, as well as the pressure at which it is attained, increase asDU=Dpincreases. Also, it can be noted that the axial load drops to zero (signifying loss of contactbetween the tube and end grips) at higher pressures as DU=Dp increases.
3. Finite-element analyses
Axisymmetric and 3D finite-element simulations of the tube hydroforming experiments are
carried out using the ABAQUS/Standard finite-element code and are based on a finite-
deformation formulation.Fig. 4(a) shows the mesh employed in the axisymmetric analyses. This
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0 5 10 15 20 250
30
60
90
120
U/p (mm/MPa)
Pressure (MPa)
Load(kN)
0.400.751.00
Fig. 3. Experimentally recorded variations of axial load experienced by the tube versus applied internal pressure for
different combination cases.
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mesh comprises of 744 four-noded quadrilateral elements and 885 nodes. Due to symmetry, only
half of the tube length (i.e., L 112:5 mm) is modeled. The loads and boundary conditionsapplied on the finite-element model are also indicated in Fig. 4(a). On the portion of the tube (of
length Lg from the end) which is encapsulated inside the grip, radial displacement is restrained.
Since rate independent material behaviour is assumed here, no physical time scale enters into the
computations. Thus the analyses are carried out by gradually incrementing the axial load Fa andinternal pressure p (as per the experimental data shown inFig. 3). In the simulation of the free
bulge test, Fa is taken to be identically zero. Some trial computations were also performed by
directly prescribing the axial end displacement of the tube in conjunction with the internal
pressure while maintaining a desired ratio ofD
U=D
p:The results obtained were almost similar tothose presented in Section 4. A consistent tangent stiffness matrix is employed and the satisfactionof equilibrium is checked at the end of each load step by examining the magnitude of the residual
force.
The mesh used in the 3D finite-element analyses is shown in Fig. 4(b). This mesh has 14,400
eight-noded hexahedral elements and 17,493 nodes. Due to symmetry, only a quarter of the tube is
modeled. The boundary conditions and loads are similar to those applied in the axisymmetric
analyses and are not shown in Fig. 4(b). The objectives of the 3D analyses are to study the
development of necking of the tube wall and predict the limit strains. In order to trigger necking
within the framework of a standard deformation analysis, a small geometric imperfection is
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Fig. 4. (a) The finite-element mesh used in the 2D axisymmetric analyses along with boundary conditions. (b) The
finite-element mesh used in the 3D simulations. Only a quarter of the tube is modelled due to symmetry. The region
where the geometric imperfection is introduced is marked as A.
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introduced over a region near the mid-length of the tube which is marked as A in Fig. 4(b). This
is accomplished by reducing the thickness gradually to a limiting value of 0:95 t0 (which isdetermined based on some trial analyses), wheret0 is the initial thickness of the remaining portion
of the tube. Since in the experiments, the neck is observed to form at the tube mid-length and to bealigned along the tube axis (see Section 4.4), the geometric imperfection is also introduced in a
similar manner. The simulation of localized necking is carried out using 3D finite-element analysis
since it is not possible to model it using a 2D axisymmetric formulation. It can be seen fromFig.
4(b) that the mesh near region A is well refined to capture development of necking accurately.
The coupling between the deformation of the fluid-filled tube and the pressure exerted by the
contained fluid on the tube inner surface is modeled using the (incompressible) hydraulic fluid
elements available in ABAQUS. The hydraulic fluid element employs a mixed formulation in
which the fluid pressure and structural displacements are treated as primary variables. Its stiffness
matrix and force vector are derived by augmenting the virtual work so as to enforce compatibility
between the volume of the fluid and that of the cavity enclosed by the tube. This approach has theimportant advantage that either the pressure inside the cavity can be specified (and the volume of
the cavity can be computed) or the fluid flux into the cavity can be prescribed (and the pressure
can be determined). In the present context, the pressure p is prescribed along with the axial end
load Fa on the tube (as perFig. 3) till the latter reduces to zero. This happens close to the stage
where the pressure attains its peak value. The subsequent part of the analysis is conducted by
specifying the fluid flux into the tube (based on the previously computed value). This enables
carrying out the analysis beyond the peak pressure until the stage when necking of the tube wall
(in the 3D simulation) occurs.
The material is assumed to obey Hills normally anisotropic plasticity theory with R 0:67 inaccordance with the experimental data. The yield function and (associated) flow rule pertaining to
this model are presented in Eqs. (A.1)(A.5) of Appendix A. It was found from trial computationsthat plastic anisotropy is an important feature which influences the processes of tube bulging and
localized necking. The true stress versus plastic strain variation, which is deduced from the
experimentally determined engineering stress versus strain curve shown inFig. 1, is employed. In
order to simulate development of large plastic strains in the tube, the above data is extrapolated
using a power-law fit. The Youngs modulus and Poissons ratio for the material are taken as
70 GPa and 0.3, respectively.
4. Results and discussion
In this section, some of the key results from the simulations are presented along with those
obtained from the experiments.
4.1. Deformed tube shapes from axisymmetric analyses
The deformed tube shapes along with fringe contours of equivalent plastic strain (p)corresponding to peak pressure, obtained from the axisymmetric analyses, are shown in
Figs. 5(a)(d). The values of the peak pressure are summarized in Table 2 for the four loading
cases pertaining to the above figures. All these figures are plotted to the same scale. It should first
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be observed that both bulging as well as contraction (of the overall tube length) has occurred. In
Figs. 5(a)(c), p increases gradually with distance from the tube end and reaches a peak level ofaround 0.230.25 at its mid-length where maximum bulging occurs. By contrast, mild wrinkling of
the tube just outside the end grip can be noticed in Fig. 5(d) which pertains to the highest
compression ratio DU=Dp 1 mm=MPa: The plastic strain level at the location of this wrinkleadjoining the tube inner wall is comparable to that prevailing at the mid-length. However, in theearly stages of deformation, the wrinkle is much more pronounced with plastic strain levels which
are almost twice that at the tube mid-length. Subsequently, the tube starts to bulge at the mid-
section leading to accumulation of large plastic strain in this region before the stage of peak
pressure is attained.
4.2. Strain and stress histories
The plastic strain histories experienced by the tube at its mid-length are presented in
Figs. 6(a)(d) for the four loading cases. Here, the circumferential and axial plastic strains are
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(a)
0.240.210.180.150.120.06
p
(b) (c) (d)
Fig. 5. The deformed tube shapes at peak pressure obtained from axisymmetric analyses along with fringe contours of
equivalent plastic strain corresponding to (a) free bulging and DU=Dp of (b) 0.4, (c) 0.75 and (d) 1 mm/MPa.
Table 2
Comparison between peak pressures as obtained from experiments, finite-element simulations and the analytical
approach given in Appendix A
DU=Dp(mm/MPa)
Experimental
(MPa)
2D Axisymmetric
(MPa)
3D FEM
(MPa)
Approximate
analytical (MPa)
Free bulge 21:8
0:2 21.99 21.96 21.96
0.4 22:8 0:3 22.79 22.77 22.820.75 24:1 0:1 24.07 24.05 24.151.0 25:1 0:2 24.95 24.99 25.08
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Appendix A) and an MK analysis. In the last two analyses mentioned above, the strain histories
computed from the 3D finite-element simulations (Fig. 6) are employed. It should first be noted
that these two analyses predict the stress histories reasonably well. Further, the stress ratio,
denoted by a
s2=s1; changes during the loading of the tube. For the case of free bulge
(Fig. 7(a)), a remains close to zero for a significant portion of the loading and then becomespositive approaching a value of around 0.3 at s1=s0 4:InFig. 7(b),a 1 initially, decreasesin magnitude with loading and becomes positive close to the stage of peak pressure. Thus, a large
rotation of the normal to the yield surface corresponding to the stress state experienced by the
tube occurs, particularly for loading cases with high axial compression ratios. This has an
important bearing on the accumulation ofp1 which leads to bulging of the tube and attainment ofa peak value for the pressure (see discussion below), since the material would offer more resistance
to nonproportional loading paths.
4.3. Variation of tube mean radius with pressure
The variations of applied internal pressure with tube mean radius at its mid-length, as
computed from simulations along with the experimental data for the four loading cases, are
shown inFigs. 8(a)(d). Due to the peeling of the strain gages from the tube as mentioned above,
the experimental curves are terminated at a certain stage which is earlier than that pertaining to
bursting of the tube. However, the pressure is recorded continuously in the experiments till it
attains a peak value. The corresponding level is indicated on the simulation curves shown in
Figs. 8(a)(d) by point A. These peak pressures are summarized in Table 2 along with those
predicted by the approximate analytical approach outlined in Appendix A. It should be noted
fromFigs. 8(a)(d), as well asTable 2, that the simulation results are in good agreement with the
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2 1 0 1 2 30
1
2
3
4
Free bulge
Simulation 3DApprox. analyticalMK analysis
(a) (b)
Simulation 3DApprox. analyticalMK analysis
1/0
0
1
2
3
4
1/0
2/0
2 1 0 1 2 3
2/0
U/p = 0.4 mm/MPa
Fig. 7. Comparison of stress history obtained from 3D simulation with that predicted by the approximate analytical
approach (given in Appendix A) as well as by MK analysis corresponding to: (a) free bulging; and (b) DU=Dp 0:4 mm=MPa:
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experimental data. Further, the approximate analytical estimates of the peak pressure are close to
the computed values.
On examining Table 2, it can be observed that the peak pressure increases with axial
compression ratio DU=Dp: This is caused by the greater resistance to plastic deformation offeredby the material owing to strong nonproportional loading as explained earlier. By contrast,
previous studies (see, for example, Ref. [17]) which assume proportional strain histories (i.e., a
constant value ofr), predict that the peak pressure decreases with increase in r in the negative
direction. In order to illustrate this trend, the analytical method detailed in Appendix A isemployed to determine the peak pressure corresponding to various r values and the result is
plotted inFig. 9. It can be seen from this figure that the peak pressure decreases from about 22 to
15 MPa as the value ofr (assumed as constant during the loading history) changes from 0 to 1:Thus, it is important to emphasize that the assumption of proportional strain histories during
tube hydroforming can lead to grossly erroneous estimates of plastic instability.
The finite-element simulations are carried out beyond the stage of peak pressure using the
procedure mentioned in Section 3. The point of incipient necking of the tube wall in the 3D
simulations is marked as point B on the curves presented inFigs. 8(a)(d). The criterion used to
identify this phenomenon will be discussed in Section 4.4. It can be seen from these figures that for
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30 40 50 60 700
10
20
30
Mean radius (mm)
p(MPa)
A B
ExperimentSimulation2DSimulation3D
30 35 40 45 500
10
20
30
Mean radius (mm)
p(MPa)
A B
Experiment
Simulation2DSimulation3D
30 35 40 45 500
10
20
30
Mean radius (mm)
p(MPa)
Free bulge
A B
ExperimentSimulation2DSimulation3D
30 40 50 600
10
20
30
Mean radius (mm)
p(MPa)
U/p = 0.75 mm/MPa
A B
Experiment
Simulation2DSimulation3D
(a) (b)
(c) (d)
U/p = 0.4 mm/MPa
U/p = 1.0 mm/MPa
Fig. 8. Variation of pressure with tube mean radius at mid-length corresponding to (a) free bulging and DU=Dpof (b)0.4, (c) 0.75 and (d) 1 mm/MPa. The computed variations are shown along with experimental recordings.
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all the loading cases, necking of the tube wall occurs only beyond the stage of peak pressure. This
corroborates with the earlier work of Larsson et al. [13]. InFigs. 8(b) and (c), the pressure drops
dramatically once the neck forms with the plastic deformation localizing in the neighborhood of
the neck.
4.4. Necking of the tube wall
Figs. 10(a) and (b) show the deformed tube and fringe contours of equivalent plastic strain
corresponding to peak pressure, obtained from the 3D simulations, for the loading case with
DU=Dp 1 mm=MPa: Similar plots after necking of the tube wall occurs are presented inFigs. 11(a) and (b). In all these figures, the deformed tube shape is reflected twice (about the
diameter and mid-section) in order to obtain the full view of the tube. The region where the
geometric imperfection has been introduced is marked in Figs. 10(a) and 11(a) by A. In
Fig. 11(c), a photograph of the failed tube specimen in the experiment is presented.
At the stage of peak pressure, bulging of the tube at its mid-length, as well as occurrence ofwrinkles just outside the portion encased in the end grips, can be observed (see Fig. 10(a)).
Fig. 10(b) depicts gradual increase in level of plastic strain from the tube end to its mid-section.
Also, a small patch of marginally higher plastic strain at the location of the geometric
imperfection can be perceived. These features are similar to those noted from the axisymmetric
simulations (compareFigs. 10(b) and 5(d)).
On examiningFig. 11(a), it can be seen that further bulging of the tube has taken place and the
wrinkles noted inFig. 10(a) are much less pronounced. Also, necking of the tube wall has clearly
occurred near the region A and is aligned along the axis of the tube. Similar features are
observed from the deformed tube specimens in the corresponding experiments (see Fig. 11(c)).
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1 0.8 0.6 0.4 0.2 00
5
10
15
20
25
Peak
Pressure(MPa)
Fig. 9. Variation of peak pressure withr assuming proportional strain paths obtained from the analytical approach
given in Appendix A.
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occurred (in comparison to the uniform values measured elsewhere on the tube surface) are taken
to be in the necked zone. The data determined from the remaining circles examined as mentioned
above are classified as safe strain combinations.
The strain histories obtained from the 3D simulations corresponding to the four loading casesare superposed on the above data as dotted lines in Fig. 12. Also shown by the thick solid line is
the FLC predicted by the 3D finite-element simulations. In addition to these simulations, MK
analyses are conducted using the computed strain histories shown in Fig. 12. The MK
formulation adopted here is based on the work of Hutchinson and Neale [9] and Graf and
Hosford [12]. In particular, the initial orientation of the imperfection band is varied and the
correct one is determined as that which yields the minimum value for p1: The MK analyses arecarried out for two values of the imperfection parameter x (defined as the ratio of the initial
thickness reduction inside the band to the uniform thickness outside it) of 0.01 and 0.04. The
FLCs predicted by the MK analyses are shown by dashed and dash-dot lines for x 0:01 and0.04, respectively, inFig. 12. It can be seen from this figure that these two curves provide upperand lower bounds to the transition region between the necked and safe strain data
extracted from the tube specimens. The FLC obtained by the 3D simulation falls in between
these two curves (i.e., it passes through the middle of the transition zone). Thus, both the
simulations and MK analyses predict the failure behaviour of the tubes due to localized necking
quite accurately.
All the FLCs shown in Fig. 12 are flat in the sense that the major (or circumferential) limit
strain component appears to be fairly constant over a wide range of axial limit strain values. This
behaviour is traced to the particular (strongly nonproportional) strain histories experienced
by the tube specimens tested in the present work. In particular, it must be recalled that the
strain rate ratio r has a large negative value initially and decreases in magnitude (i.e., becomes
close to 0) at later stages (see dotted curves inFig. 12). By contrast, MK calculations carried outusing proportional strain paths predict a FLC which increases strongly with negative
minor principal strain (as in the work of Hutchinson and Neale [9] and Chan et al. [10]).
Another interesting observation made from the present MK analyses was that the critical
band orientation is close to the tube axis (with the limiting band angle c with respect to the
axis lying between 101 and 141). This also corroborates with the experimental observations.
By contrast, the band would be inclined significantly with respect to the tube axis if pro-
portional strain path had been assumed in the MK calculations. For example, corresponding to
a fixed strain ratio ofr 0:5; which would represent the extreme limit of the data presented inFig. 12, it is found that c 341: The above discussion clearly shows that the strain pathexperienced by the tube influences both the orientation of the neck as well as the nature ofthe FLC.
5. Conclusions
In this paper, a combined experimental and computational study of free hydroforming of
aluminium alloy tubes has been carried out. In addition, an approximate analytical approach has
been adopted to obtain the peak internal pressure and MK analyses have been conducted to
predict the limit strains corresponding to onset of necking of the tube wall. In both these analyses,
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Appendix A. Approximate analysis of tube instability during free hydroforming
In this appendix, an approximate semi-analytical method for calculating the peak pressure
encountered during a free tube hydroforming operation is presented. A thin-walled cylindricaltube of mean radius r and thickness t is considered. The axial, circumferential and thickness
directions of the tube are denoted by z, y and r, respectively. Assuming that sr is negligible
compared to sy and sz; the equivalent Cauchy stress and true plastic strain increment can beexpressed as follows[17]:
s 1ffiffiffiffiffiffiffiffiffiffiffiffi1 Rp
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2z s2y Rsz sy2
q ; (A.1)
dp 1 Rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2Rp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid
pz2 dpy2
2R
1
R
dpzd
pys ; (A.2)
where R is the anisotropy parameter. On employing the associated flow rule, the true principal
plastic strain increments are given by
dpzdp
s sz
R
1 R sy
; (A.3)
dpydp
s sy
R
1 R sz
; (A.4)
dpr dpz dpy; (A.5)where the last equation follows from plastic incompressibility.
The true plastic strain increments (on neglecting the elastic counterparts) during an increment
of loading are given by,
dpydr
r ; dpz rdpy; dpr dpy dpz
dt
t ; (A.6)
where dr and dt are increments in the (mean) radius and thickness of the tube, respectively.
Further,r represents the principal plastic strain rate ratio experienced by the tube which is taken
to be prescribed as a function of the true circumferential plastic strain py: The objective of thisappendix is to analytically determine the corresponding stress history, as well as the peak pressure
to which the tube is subjected.
The circumferential stress is given in terms of the applied pressure p bysy pr=t: Now, giventhe plastic strain rate ratio, dpz =d
py; at a given stage of deformation as rpy; one can employ the
flow rule (Eqs. (A.3) and (A.4)) to show that
sz
sy apy
r1 R R1 R rR : (A.7)
The Hill equivalent stress given by Eq. (A.1) can then be expressed in terms of the circumferential
stress as
s Aasy; (A.8)
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where
Aa ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 a2
2aR
1 Rs : (A.9)Similarly the equivalent plastic strain increment can be related to dpy by
dp 1 arA
dpy: (A.10)
The increment in circumferential stress during an increment in loading is given by
dsy dpr
tp dr
tpr dt
t2 : (A.11)
On employing Eq. (A.6), the above equation can be rewritten as
dpdpy
dsydpy
sy2 r
tr
: (A.12)
The condition for peak pressure requires that the right-hand side of the above equation should be
zero.
On employing Eqs. (A.8) and (A.10) along with a given equivalent stress versus plastic strain
relationship, s hp; the above condition for tube instability can be expressed as
h0p 11 ar hp A2 r A0 da
dr
dr
dpy
: (A.13)
It must be noted that the function h
p
is deduced from the uniaxial stress versus strain curve
shown inFig. 1. The second term in the right-hand side of the above equation can be omitted forthe special case of a proportional strain path (i.e., constantr). Alternately, for a prescribed strain
path (or given rpy ), one can obtain p by (numerically) integrating Eq. (A.10) and express thepressure as
p syt
r h
pA
t0
r0e
prpy; (A.14)
wheret0 andr0 are the initial tube thickness and (mean) radius, respectively. Thus, the pressure
can be plotted as a function of py and its peak value can be determined. Further, with thecomputed pressure along with a given by Eq. (A.7), the circumferential and axial stresses can be
obtained as a function of p
y
: The stress history experienced by the tube can then be determined.
References
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