An Experimental and Numerical Studyof Necking Initiation In

8/13/2019 An Experimental and Numerical Studyof Necking Initiation In

1/20

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

ARTICLE IN PRESS

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: [email protected] (R. Narasimhan).1Currently at General Motors India Science Laboratory, Bangalore, India.
http://www.elsevier.com/locate/ijmecscihttp://www.elsevier.com/locate/ijmecsci


2/20

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

ARTICLE IN PRESS

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

A. Kulkarni et al. / International Journal of Mechanical Sciences 46 (2004) 172717461728


3/20


4/20

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

ARTICLE IN PRESS

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.



5/20

ARTICLE IN PRESS

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).

A. Kulkarni et al. / International Journal of Mechanical Sciences 46 (2004) 17271746 1731


6/20

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

ARTICLE IN PRESS

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.



7/20

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

ARTICLE IN PRESS

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.



8/20

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

ARTICLE IN PRESS



9/20

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

ARTICLE IN PRESS

(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



10/20


11/20

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

ARTICLE IN PRESS

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:



12/20

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

ARTICLE IN PRESS

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.



13/20

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)).

ARTICLE IN PRESS

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.



14/20


15/20


16/20

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,

ARTICLE IN PRESS



17/20


18/20

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)

ARTICLE IN PRESS



19/20

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

[1] Dohmann F, Hartl Ch. Hydroforminga method to manufacture light-weight parts. Journal of Materials

Processing Technology 1996;60:66976.

[2] Koc M, Altan T. An overall review of the tube hydroforming (THF) technology. Journal of Materials Processing

Technology 2001;108:38493.

[3] Azrin M, Backofen WA. The deformation and failure of a biaxially stretched sheet. Metallurgical Transactions

1970;1:2857.

ARTICLE IN PRESS



20/20

[4] Hecker SS. Formability of aluminium alloy sheets. Journal of Engineering Materials and Technology

(Transactions of the ASME, H) 1975;97:6673.

[5] Graf A, Hosford W. Effect of changing strain paths on forming limit diagrams of Al 2008-T4. Metallurgical

Transactions A 1993;24A:250312.[6] Graf A, Hosford W. The influence of strain-path changes on forming limit diagrams of Al 6111 T4. International

Journal of Mechanical Sciences 1994;36-10:897910.

[7] Hill R. On discontinuous plastic states, with special reference to localized necking in thin sheets. Journal of the

Mechanics and Physics of Solids 1952;1:1930.

[8] Marciniack Z, Kuczynski K. Limit strains in the processes of stretch-forming sheet metal. International Journal of

Mechanical Sciences 1967;9:60920.

[9] Hutchinson JW, Neale KW. Sheet neckingII. Time-independent behavior, mechanics of sheet metal forming.

New York: Plenum Press; 1978. p. 12753.

[10] Chan KS, Koss DA, Ghosh AK. Localized necking of sheet at negative minor strains. Metallurgical Transactions

A 1984;15A:3239.

[11] Graf A, Hosford W. Calculations of forming limit diagrams. Metallurgical Transactions A 1990;21A:8793.

[12] Graf A, Hosford W. Calculations of forming limit diagrams for changing strain paths. Metallurgical Transactions

A 1993;24A:2497501.

[13] Larsson M, Needleman A, Tvergaard V, Storakers B. Instability and failure of internally pressurized ductile metal

cylinders. Journal of the Mechanics and Physics of Solids 1982;3033:12154.

[14] Xia ZC. Failure analysis of tubular hydroforming. Journal of Engineering Materials and Technology

(Transactions of the ASME) 2001;123:4239.

[15] Koc M, Altan T. Prediction of forming limits and parameters in the tube hydroforming process. International

Journal of Machine Tools & Manufacture 2002;42:12338.

[16] Chow CL, Yang XJ. Bursting for fixed tubular and restrained hydroforming. Journal of Materials Processing

Technology 2002;130131:10714.

[17] Kim S, Kim Y. Analytical study for tube hydroforming. Journal of Materials Processing Technology

2002;128:2329.

[18] Carleer B, van der Kevie G, d Winter L, van Veldhuizen B. Analysis of the effect of material properties on the

hydroforming process of tubes. Journal of Materials Processing Technology 2000;104:15866.[19] Asnafi N, Skogsgardh A. Theoretical and experimental analysis of stroke-controlled tube hydroforming. Materials

Science and Engineering A 2000;279:95110.

[20] Davies R, Grant G, Herling D, Smith M, Evert B, Nykerk S, Shoup J. Formability investigation of aluminium

extrusions under hydroforming conditions. SAE Technical Paper 2000;2000-01-2675.

[21] Varma NSP, Narasimhan R. Prediction of forming limits for tube hydroforming using an anisotropic continuum

damage model. 2004, in preparation.

ARTICLE IN PRESS


Documents

An Experimental and Numerical Studyof Necking Initiation In