International Journal of Mechanical Sciences 155 (2019) 381–391

Contents lists available at ScienceDirect

International Journal of Mechanical Sciences

journal homepage: www.elsevier.com/locate/ijmecsci

On the determination of forming limits in thin-walled tubes

J.P. Magrinho

a , M.B. Silva

a , ∗ , G. Centeno

b , F. Moedas a , C. Vallellano

b , P.A.F. Martins a

a IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal b Department of Mechanical and Manufacturing Engineering, University of Seville, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain

a r t i c l e i n f o

Keywords:

Tube forming

Thin-walled tubes

Formability limits

Necking

Fracture

a b s t r a c t

This paper proposes a methodology to determine the formability limits of thin-walled tubes and to plot them

in principal strain space and in the space of effective strain vs. stress-triaxiality. Digital image correlation (DIC),

combined either with time-dependent methodologies or strain-force approaches, is utilized to identify the onset of

failure by necking and obtain the corresponding limit strains. Thickness measurements and determination of the

gauge length strains across the cracked regions are utilized to characterize the onset of fracture and to evaluate

the fracture limit strains.

Results show that the utilization of tube expansion with rigid punches and elastomers allow obtaining strain

loading paths and fracture loci by necking and fracture across a wide range of tube forming conditions ranging

from biaxial stretching in the first quadrant to pure tension in the second quadrant of principal strain space. The

fracture forming line (FFL) is the first time ever determined for thin-walled tubes. The forming limit curve (FLC)

and the FFL resemble those of sheet and strip materials and their use is of paramount importance in the design

and optimization of tube forming processes.

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. Introduction

In the past years, the growing interest in high-strength, lightweight

tructures for transportation vehicles, aerospace and civil engineering

as been promoting the use of tubes, namely thin-walled tubes [1] . Tube

ending and end forming operations are the most widely used to fabri-

ate thin-walled structural parts but other processes, such as tube hydro-

orming [2] , have also gained a role in mass production of lightweight

ubular parts.

Novel processes such as incremental tube forming [3] and friction-

pinning [4] also show potential to become alternative manufacturing

olutions capable of reducing energy and raw material consumption. In

his regard, there is a need to characterize the formability limits and

redict failure in tube forming.

There are two main approaches to achieve this goal: (i) establish-

ng process workability windows for a range of operation parameters

5] and (ii) setting up failure limits in the Forming Limit Diagram (FLD),

n a similar way to what is done in sheet forming.

As in metal sheets, both approaches are dependent on the material

nd the initial fabrication process, but the first is specific to each tube

orming process [6] whereas the second is independent of the tube form-

ng process by which the formability limits were determined [7] .

∗ Corresponding author.

E-mail addresses: joao.magrinho@tecnico.ulisboa.pt (J.P. Magrinho), bea

lipe.moedas@tecnico.ulisboa.pt (F. Moedas), carpofor@us.es (C. Vallellano), pmart

p

ttps://doi.org/10.1016/j.ijmecsci.2019.03.020

eceived 4 January 2019; Received in revised form 8 March 2019; Accepted 13 Marc

vailable online 14 March 2019

020-7403/© 2019 Published by Elsevier Ltd.

In sheet forming, the formability limit by necking (FLC), was orig-

nally developed by Keeler [8] and Goodwin [9] , as a general tool to

valuate the formability of a metal sheet. In tube forming, several ex-

erimental studies were performed to determine FLC’s by means of

ube bulging [10,11] and tube hydroforming with varying strain load-

ng paths [12,13] . These investigations made no distinction between

ecking and fracture and considered that the limiting strains were the

aximum strains measured on the tube surface.

The introduction of the onset of fracture in the FLD was proposed

y Embury and Duncan [14] and has been gaining importance due to

ecent developments in incremental sheet forming, which can ensure

table deformation until fracture without previous necking [15] .

However, the possibility of attaining strain values beyond the forma-

ility limit by necking is not exclusive of sheet forming. Davis et al.

16] observed that tube hydroforming of seamless AA-6260-T4 tubes

ere capable of withstanding strains significantly larger than those cor-

esponding to the onset of necking of the tubular material. More re-

ently, Song et al. [17] concluded that theoretically determined FLC’s

sing three different necking criteria also do not fit well against the ex-

erimental strains attained in tube bulging.

All these results and observations point towards the fact that tube

orming, similarly to sheet forming [18] , can also fail by fracture with

ostponing or suppression of necking. Because of this, authors have

triz.silva@tecnico.ulisboa.pt (M.B. Silva), gaceba@us.es (G. Centeno),

ins@tecnico.ulisboa.pt (P.A.F. Martins).

h 2019

J.P. Magrinho, M.B. Silva and G. Centeno et al. International Journal of Mechanical Sciences 155 (2019) 381–391

Table 1

The AA6063-T6 aluminium tube wall thickness taken every 45° of the circular perimeter.

r

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Fig. 1. Tensile testing of the commercial AA6063-T6 aluminium tubes. (a) Lon-

gitudinal tensile test apparatus with a detail of the fixing clamps. (b) Transver-

sal tensile test apparatus with identification of the (1) clamping fixture, (2) D-

shaped blocks, (3) pins, (4) polytetrafluoroethylene sheets and (5) specimen.

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ecently proposed an experimental procedure for the determination of

he critical strains and ductile damage at the onset of failure by fracture

or the special case of tube expansion undergoing pure tension [19] . The

rocedure was later extended to tube reduction and inversion [20] .

Under these circumstances, the main objective of this paper is to

resent a methodology to determine the formability limits of thin-walled

ubes by necking and fracture associated to mode I of fracture mechanics

in-plane tension). This is the first attempt, as far as the authors are

ware, to perform a simultaneous determination of the formability limits

y necking and fracture in tubular materials.

The presentation draws from tensile tests aimed at characterizing

he mechanical behaviour along the longitudinal and transversal tube

irections to a series of experiments comprising tube expansion with

igid punches and elastomers. The non-quadratic, anisotropic, Hosford

lasticity criterion is utilized to convert the strain loading paths from

rincipal strain space to the space of effective strain vs. stress-triaxiality

nd to plot the corresponding formability limits by necking and fracture.

. Experimentation

.1. Tensile tests

The investigation was carried out on commercial AA6063-T6 alu-

inium tubes with 20 mm outer radius. The average wall thickness

0 calculated from measurements taken every 45° was 1.9 mm, in or-

er to span the entire circular cross section ( Table 1 ). The 0° refer-

nce of the graphical representation of tube wall thickness variation

hown in Table 1 coincides with the most visible weld resulting from

he extrusion with porthole die that was utilized to produce the supplied

ubes.

The mechanical properties and stress–strain curve of the tubular ma-

erial were determined by means of longitudinal ( Fig. 1 a) and transver-

al (also known as ‘ring hoop’) ( Fig. 1 b) tensile tests. The specimens

ere machined out of the supplied tubes and their surfaces were elec-

rochemically etched along their active loading directions.

The longitudinal tensile test specimens with 50 mm gauge length

ere prepared in accordance to the ASTM E8/E8M standard [21] . Spe-

382

ial fixing clamps were employed not to plastically deform the tube ends

uring fixing in the universal testing machine (refer to the detail in

ig. 1 a).

The tube wall thickness variation along the circular cross section was

aken into consideration by using longitudinal tensile test specimens

ith their axis taken at every 45° along the perimeter.

The transversal tensile test specimens were prepared in accordance

o a sub-size geometry of the ASTM E8/E8M standard [21] in order

o accommodate their reduced area in one of the D-shaped blocks (or,

emi-cylinders) that hold the specimen and move freely when loading is

pplied (refer to ‘2’ in Fig. 1 b). This is necessary to prevent undesirable

ending during the tensile test [22] .

The transversal tensile test specimens were lubricated with polyte-

rafluoroethylene (teflon ®) sheets to reduce the friction between the

pecimen and the D-shaped blocks.

The tensile tests were performed at room temperature in a universal

esting machine Instron 5900R.

J.P. Magrinho, M.B. Silva and G. Centeno et al. International Journal of Mechanical Sciences 155 (2019) 381–391

Table 2

Summary of the experimental work plan for the expansion of AA6063-T6 aluminium tubes with rigid punches (notation in accordance with Fig. 2 ).

Test geometry Tube material Operating conditions

Outer radius r 0 (mm) Thickness t 0 (mm) Length l 0 (mm) Semi-angle of inclination 𝛼 (degrees) Lubricant Velocity (mm/min)

ERP 15 20 1.9 70 15 Disulphide 5

ERP 30 30 Molybdenum

ERP 45 45 MoS2 grease

Fig. 2. Tube expansion with a rigid punch. (a) Schematic representation and

notation. (b) Photograph of a test performed with a 15° semi-angle punch.

2

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2

t

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t

w

d

o

2

a

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p

383

.2. Tube formability tests

The formability limits by necking and fracture were determined by

eans of tube expansion with rigid punches and elastomers. The tests

ere performed at room temperature on a hydraulic testing machine

nstron SATEC 1200 kN in quasi-static operating conditions.

.2.1. Tube expansion with rigid punches

In the expansion tests with rigid punches, the leading edge of the

ubes undergoes pure tension up to failure due to progressive circum-

erential stretching as the conical punch surfaces are forced into the

ubes [19] .

The experimental setup and notation are shown in Fig. 2 and the

ork plan is summarized in Table 2 . The tests were performed in a ran-

om order and at least three repetitions were made for each punch ge-

metry in order to ensure reproducibility of the results.

.2.2. Tube expansion with elastomers

In the expansion tests with elastomers, the tube ends were clamped,

nd a cylindrical rubber plug was placed inside the tube and compressed

ith a punch to expand (bulge) the tubes up to fracture [23] . The ex-

erimental setup and notation are shown in Fig. 3 a and b, and the main

Fig. 3. Tube expansion with elastomers.

Schematic representation and notation in the

(a) initial and (b) final positions of the tool. A

detail of the clamping system is given in (c).

J.P. Magrinho, M.B. Silva and G. Centeno et al. International Journal of Mechanical Sciences 155 (2019) 381–391

Table 3

Summary of the experimental work plan for the expansion of AA6063-T6 aluminium tubes with elastomers (notation in accordance with

Fig. 3 ).

Test geometry Tube material Operating conditions

Outer radius r 0 (mm) Thickness t 0 (mm) Length l 0 (mm) Free length l free (mm) Fixing condition Velocity (mm/min)

EEL 10 10

EEL 20 20 1.9 150 20 Blocked 5

EEL 30 30

Fig. 4. Schematic representation of the setup utilized by the DIC system for

measuring the strains in a tube expansion with a rigid punch.

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ctive tool components are identified as: (i) the punch, (ii) the dies, (iii)

he clamping system and (iv) the cylindrical rubber plug.

The clamping system is made of three blocks ( Fig. 3 c) with knurled

urfaces to provide enough grip to hold the tubes in position and prevent

ongitudinal displacements during testing. The blocks are connected to

ach other by springs that are regulated by three fixing screws (refer to

ig. 3 c).

The cylindrical rubber plug acts as a pressure transmitting medium

or tube expansion and by changing the free length of the tubes l free (i.e.,

he gap distance between the two dies along which the tube is free to

eform) it is possible to ensure loading conditions from pure tension

eformation to equi-biaxial stretching.

The work plan is summarized in Table 3 and the overall experimental

ethods and procedures were similar to those utilized in the expansion

f tubes with rigid punches.

.3. Strain measurements

Strain measurements on the vicinity of the zones where cracks are

pened in both tensile and tube formability tests were performed with

digital image correlation (DIC) system from Dantec Dynamics –model

-400 3D ( Fig. 4 ).

The surfaces of the specimens were sprayed with a stochastic black

peckle pattern on a uniform background previously painted in white.

he tensile test specimens were illuminated with one spotlight, whereas

he tube forming specimens were illuminated with two spotlights to al-

ow a wider measurement region.

The DIC system is equipped with 2 cameras with 6 megapixels of

esolution and 50.2 mm of focal lenses with an aperture of f/11. The

requency of image acquirement was set to 20 frames per second and

he correlation algorithm was performed with the INSTRA 4D software.

facet size of 13 px with a spacing grid of 7 px was considered.

. Methodology

The proposed methodology is aimed at providing the formability lim-

ts by necking and fracture in the principal strain space and in the space

f effective strain vs. stress-triaxiality.

384

.1. Formability limits

The formability limits by necking (FLC) are determined by means

f DIC. The instant of time corresponding to the onset of necking is

dentified by means of a time-dependent approach or, if not possible,

y means of an alternative approach that combines the evolution of the

trains and forces with punch displacement ( Fig. 5 ).

The time-dependent approach utilized by the authors ( Fig. 5 a) makes

se of the experimental strains measured by DIC and of the maximum

train rate value to identify the instant of time corresponding to the

nset of failure necking. The approach is based on the original work of

artínez-Donaire et al. [24] in sheet forming.

The alternative approach combining the experimental strains mea-

ured by DIC and the force-displacement evolutions ( Fig. 5 b) is based

n the work on tube expansion with rigid punches by Cristino et al.

25] and had been originally developed by Magrinho et al. [26] for bulk

orming tests. The approach (that will be hereafter denoted as ‘combined

train-force’) is based on the identification of the amount of displace-

ent after which the force drops and the evolution of major and minor

trains remain stationary.

Utilization of DIC to obtain strains in the necking region after it forms

nd, therefore, close to the fracture, provides strain values that cannot

e considered the fracture strains. Moreover, such measurements suffer

rom sensitivity to the location of the selected points where measure-

ents are to be performed owing to the inhomogeneous deformation in

he neighbourhood of the crack. As a result of this, the experimental pro-

edure for constructing the formability limits by fracture (FFL) requires

easuring of thickness at cracking in order to obtain the ‘gauge length’

trains. These strains were obtained from individual measurements of

hickness along the cracks with a stereomicroscope Nikon SMZ800 with

magnification of 20x using KAPPA Image Base Metreo 2.7.2.

The procedure is schematically shown in Fig. 6 a and the thickness

trains at fracture 𝜀 tF are calculated by,

𝑡𝐹 = 𝑙𝑛

(

𝑡 𝐹

𝑡 0

)

(1)

here t O is the initial thickness and 𝑡 𝐹 =

1 𝑛

∑𝑛

𝑖 =0 𝑡 𝑖 𝐹

the average fracture

hickness of the specimens.

In case fracture is preceded by necking, the minor strain 𝜀 2 (merid-

onal strain), is assumed to remain constant after the last measurement

y DIC and the major strain 𝜀 1 (circumferential strain), is obtained by

ncompressibility under plane strain deformation conditions ( Fig. 6 b),

1 𝐹 + 𝜀 2 𝐹 + 𝜀 𝑡𝐹 = 0 ⇔ 𝜀 1 𝐹 = −

(𝜀 2 𝐹 + 𝜀 𝑡𝐹

)(2)

.2. Graphical representation of the formability limits

Graphical representation of the strain loading paths in principal

train space is obtained by merging the time (or displacement) evo-

utions of the major and minor strains determined by DIC. If test ge-

metries are properly selected, it is possible to plot strain loading paths

anging from the first to the second quadrant and to represent the on-

et of failure by necking as a V-shaped FLC, similar to that commonly

epicted in sheet forming.

The formability limits by fracture are determined from the gauge

ength strains along the crack regions of the specimens after testing by

J.P. Magrinho, M.B. Silva and G. Centeno et al. International Journal of Mechanical Sciences 155 (2019) 381–391

Fig. 5. Schematic representation of the two

approaches to identify the onset of failure by

necking and determine the corresponding neck-

ing strains. (a) Time-dependent approach [24] .

(b) Combined strain and force displacement ap-

proach [25] .

Fig. 6. Schematic representation of the approach

to determine the formability limits by fracture.

(a) Measuring the specimen’s thickness along the

crack. (b) Fracture strain pair may result from lo-

calization after necking or from progressive re-

duction of thickness.

m

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𝑟

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a

9

t

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a

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𝑑

w

t

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𝑑

∶

d

𝑟

l

𝛽

i

𝜎

d

𝑑

a

c

𝑑

(

eans of the experimental technique that was previously described in

ection 3.1 .

The plot of both formability limits (necking and fracture) in principal

train space for the AA6063-T6 aluminium tubes under investigation is

isclosed in Section 4 .

The transformation of the experimental strain loading paths and of

he corresponding formability limits from the principal strain space to

he space of effective strain vs. stress-triaxiality requires determining

he effective strain �� , the effective stress �� and the average stress 𝜎m

for

ll the points belonging to the experimental strain loading paths. The

ransformation to be presented in what follows, is built upon the non-

uadratic plasticity criterion proposed by Hosford [27] and assumes pro-

ortional loading and tube deformation under plane stress deformation

onditions ( 𝜎t = 𝜎3 = 0) along the thickness direction.

The non-quadratic plasticity criterion proposed by Hosford [27] was

riginally derived for anisotropic sheet metals and considers planar

nisotropy and the principal axes of the stress and strain tensors to co-

ncide at each point,

0 (𝜎2 − 𝜎3

)𝑎 + 𝑟 90 (𝜎3 − 𝜎1

)𝑎 + 𝑟 0 𝑟 90 (𝜎1 − 𝜎2

)𝑎 = 𝑟 90 (1 + 𝑟 0

)��𝑎 (3)

here, the subscripts 1, 2, and 3 denote the principal axis of anisotropy,

nd r 0 and r 90 are the anisotropy coefficients at 0° (longitudinal) and

0° (transversal) with the rolling direction. The exponent a is related to

he crystal structures of materials and experimental results by Hosford

28] suggest a = 6 and a = 8 for metals with body centred cubic (BCC)

nd face centred cubic (FCC) crystal structures, respectively.

The application of the Hosford’s plasticity criterion to tube forming

as been successfully applied by other researchers [29, 30] , this plastic-

ty criterion is appropriated to anisotropic tubes because only considers

he anisotropy coefficients at 0° and 90° ( r 0 and r 90 ), and these can be

asily determined in thin-wall tubes in the longitudinal and transversal

irection with relation to the extrusion.

Considering the general flow rule,

𝜀 𝑖𝑗 =

𝜕𝑓 (𝜎𝑖𝑗

)𝜕 𝜎𝑖𝑗

𝑑𝜆 (4)

385

here the plastic potential f ( 𝜎ij ) is taken as the Hosford’s plasticity cri-

erion and d 𝜆 is an instantaneous, nonnegative, proportionality factor

elated to the material stress–strain curve, the following ratios of strain

ncrement are obtained,

𝜀 1 ∶ 𝑑 𝜀 2 ∶ 𝑑 𝜀 3 = − 𝑟 90 (𝜎3 − 𝜎1

)𝑎 −1 + 𝑟 0 𝑟 90 (𝜎1 − 𝜎2

)𝑎 −1 ∶ 𝑟 0

(𝜎2 − 𝜎3

)𝑎 −1 − 𝑟 0 𝑟 90 (𝜎1 − 𝜎2

)𝑎 −1 ∶ − 𝑟 0 (𝜎2 − 𝜎3

)𝑎 −1 + 𝑟 90 (𝜎3 − 𝜎1

)𝑎 −1 (5)

Considering thin-walled tubes to be loaded under plane stress con-

itions ( 𝜎3 = 0), the plasticity criterion (3) can be rewritten as follows,

0 𝜎𝑎 2 + 𝑟 90 𝜎

𝑎 1 + 𝑟 0 𝑟 90

(𝜎1 − 𝜎2

)𝑎 = 𝑟 90 (1 + 𝑟 0

)��𝑎 (6)

If the strain loading path 𝛽 and the stress ratio 𝛼 are defined as fol-

ows,

=

𝑑 𝜀 2 𝑑 𝜀 1

=

𝜀 2 𝜀 1

𝛼 =

𝜎2 𝜎1

(7)

t follows from Eqs. (6) and (7) that the effective stress, �� is given by,

=

[

𝑟 90 + 𝑟 0 𝛼𝑎 + 𝑟 0 𝑟 90 ( 1 − 𝛼) 𝑎

𝑟 90 (1 + 𝑟 0

)] 1∕ 𝑎

𝜎1 (8)

Now, considering the incremental plastic work per unit of volume

w

p to be expressed as,

𝑤

𝑝 = 𝜎1 𝑑 𝜀 1 + 𝜎2 𝑑 𝜀 2 = ��𝑑 𝜀 (9)

nd substituting Eqs. (9) and (10) , the effective strain increment 𝑑 𝜀 be-

omes,

𝜀 =

𝜎1 𝑑 𝜀 1 + 𝜎2 𝑑 𝜀 2 ��

=

1 + 𝛼𝛽[𝑟 90 + 𝑟 0 𝛼𝑎 + 𝑟 0 𝑟 90 ( 1− 𝛼) 𝑎

𝑟 90 ( 1+ 𝑟 0 ) ]1∕ 𝑎 𝑑 𝜀 1 (10)

Then, considering the flow rules with plane stress loading conditions

𝜎3 = 0) and taking the strain loading path 𝛽 and the stress ratio 𝛼 into

J.P. Magrinho, M.B. Silva and G. Centeno et al. International Journal of Mechanical Sciences 155 (2019) 381–391

Table 4

Summary of the average mechanical properties and stress–strain curves parameters derived from tensile tests

of the AA6063-T6 aluminium tubes performed along the longitudinal and transversal directions.

Test geometry Thickness (mm) 𝜎Y (MPa) 𝜎UTS (MPa) A (%) K (MPa) 𝜀 0 n r avg

TL – Longitudinal 1.90 164.4 204.1 44.10 300.0 0.002 0.125 0.58

TT – Transversal 1.90 163.5 238.2 58.60 408.5 0.008 0.216 1.19

c

t

𝑑

w

𝛼

𝛽

i

(

𝜎

𝑑

𝛽

(

𝜎

i

r

𝜂

p

o

H

4

4

s

s

t

e

s

s

d

s

o

𝜎

Fig. 7. Average true stress - true strain curves obtained from longitudinal and

transversal tensile tests.

T

i

t

f

(

4

a

d

w

w

t

p

i

i

m

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i

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p

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b

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e

b

r

onsideration [31] , the ratios of strain increment ( Eq. (5) ) may be writ-

en as,

𝜀 1 ∶ 𝑑 𝜀 2 ∶ 𝑑 𝜀 3 = 1 ∶ 𝛽 ∶ − ( 1 + 𝛽) = 1 + 𝑟 0 ( 1 − 𝛼) 𝑎 −1 ∶ 𝑟 0

𝑟 90 𝛼𝑎 −1 − 𝑟 0 ( 1 − 𝛼) 𝑎 −1 ∶ −

𝑟 0 𝑟 90

𝛼𝑎 −1 − 1 (11)

ith the relation between the strain loading path 𝛽 and the stress ratio

given by,

=

𝑑 𝜀 2 𝑑 𝜀 1

=

𝑟 0 𝛼𝑎 −1 − 𝑟 0 𝑟 90 ( 1 − 𝛼) 𝑎 −1

𝑟 90 + 𝑟 0 𝑟 90 ( 1 − 𝛼) 𝑎 −1 (12)

Considering that the AA6063-T6 aluminium tubes utilized in the

nvestigation have a FCC crystal structure ( a = 8), Eqs. (8) , (10) and

12) become,

=

[

𝑟 90 + 𝑟 0 𝛼8 + 𝑟 0 𝑟 90 ( 1 − 𝛼) 8

𝑟 90 (1 + 𝑟 0

)] 1∕8

𝜎1 (13)

𝜀 =

1 + 𝛼𝛽[𝑟 90 + 𝑟 0 𝛼8 + 𝑟 0 𝑟 90 ( 1− 𝛼) 8

𝑟 90 ( 1+ 𝑟 0 ) ]1∕8 𝑑 𝜀 1 (14)

=

𝑟 0 𝛼7 − 𝑟 0 𝑟 90 ( 1 − 𝛼) 7

𝑟 90 + 𝑟 0 𝑟 90 ( 1 − 𝛼) 7 (15)

Finally, writing the mean stress 𝜎m

for plane stress loading conditions

𝜎3 = 0) as,

𝑚 =

𝜎1 + 𝜎2 + 𝜎3 3

=

𝜎1 + 𝜎2 3

=

( 1 + 𝛼) 3

𝜎1 (16)

t is possible to use Eqs. (13) and (16) to express the stress triaxiality

atio,

𝜂 = 𝜎𝑚 ∕ 𝜎 as follows,

=

𝜎𝑚

��=

( 1 + 𝛼) [𝑟 90

(1 + 𝑟 0

)]1∕8 3 [𝑟 90 + 𝑟 0 𝛼

8 + 𝑟 0 𝑟 90 ( 1 − 𝛼) 8 ]1∕8 (17)

Eqs. (14) and (17) will be later used to transform the strain loading

aths and the formability limits from principal strain space to the space

f effective strain vs. stress-triaxiality using, for the first time ever, the

osford’s plasticity criterion [27] .

. Results and discussion

.1. Material characterization

The average mechanical properties obtained for the entire set of ten-

ile tests performed along the longitudinal and transverse directions are

ummarized in Table 4 .

In Table 4 , the symbol 𝜎Y denotes the yield stress, 𝜎UTS the ultimate

ensile strength and A the area reduction at fracture. The anisotropy co-

fficients ( r = 𝜀 width / 𝜀 thickness ) were directly obtained from the principal

trains provided by DIC, for a point located at the center of the tensile

pecimens. The r 0 and r 90 coefficients correspond to r avg in the longitu-

inal (TL) and transversal (TT) directions of Table 4 , respectively.

The stress–strain curves resulting from the longitudinal and transver-

al tensile tests are shown in Fig. 7 and can be approximated by means

f the Swift material strain hardening model with pre-strain, 𝜀 0 ,

= 𝐾

(𝜀 + 𝜀 0

)𝑛 (18)

386

The parameters, K , 𝜀 0 and n of the above equation are listed in

able 4 .

The differences between the average stress–strain curves shown

n Fig. 7 are attributed to frictional effects in the transversal tensile

est [22] and to differences in the anisotropy coefficient r avg obtained

rom the tensile tests along the longitudinal and transversal directions

Table 4 ).

.2. Formability limits by necking

The principal strain pairs at the onset of necking for longitudinal

nd transversal tensile testing were determined by means of the time-

ependent methodology proposed by Martínez-Donaire et al. [24] . This

as the first time ever that necking strains defining the FLC of tubes

ere determined from transversal tensile tests.

Fig. 8 a presents the experimental evolutions of the major strain at

he onset of necking (points A and B) and of the major strain rate for

oint B. The evolutions were obtained from DIC and Fig. 8 b shows an

mage of the distribution of the major strains at the onset of necking,

ndicating the location of points A and B.

The main conclusion arising from Fig. 8 is that the time-dependent

ethodology can be successfully applied to transversal tensile tests. Sim-

lar conclusion is obtained for the longitudinal tensile tests.

However, the attempts made by the authors to extend the application

f the time-dependent methodology to tube expansion were not success-

ul due to oscillations of the major strain rate that make impossible the

dentification of the maximum strain rate value ( Fig. 9 ).

Alternative time-dependent methodologies based on other deriva-

ive procedures of the main variables would also suffer from the same

roblem and, therefore, their straight application would also reveal in-

ppropriate for tube expansion.

In addition to this, it is worth mentioning that alternative approaches

ased on the observation and analysis of the displacements on the tube

urface during expansion would also experience difficulties due to the

xistence of non-negligible strain gradients along the tube end caused

y non-uniform frictional conditions.

All this prevented authors from identifying the instant of time cor-

esponding to the onset of necking by means of the above mentioned

J.P. Magrinho, M.B. Silva and G. Centeno et al. International Journal of Mechanical Sciences 155 (2019) 381–391

Fig. 8. Time-dependent methodology applied to the transversal tensile tests. (a) Experimental evolution with time of the major strain in points A and B, and of the

major strain rate in point B. (b) Distribution of the major strain at the onset of necking obtained from DIC.

Fig. 9. Unsuccessful application of the time-dependent methodology to tube expansion (Case ERP30 of Table 2 ). (a) Experimental evolution with time of the major

strain in points A, B and C, and of the major strain rate in point B. (b) Distribution of the major strain at the onset of necking obtained from DIC.

w

f

b

t

p

p

h

d

l

a

b

t

d

s

p

a

t

p

u

c

s

t

s

m

M

m

t

t

c

ell-established procedures and required the use of the combined strain-

orce approach proposed by Cristino et al. [25] . This approach was

riefly introduced in Section 3.1 and requires the experimental evolu-

ions of major and minor strains with displacement to be measured in

oints located at the vicinity of the necked regions.

Fig. 10 discloses the application of the combined strain-force ap-

roach to case ‘ERP30’ of Table 2 (tube expansion with a rigid punch

aving a 30° semi-angle of inclination). As seen, the force increases with

isplacement up to a peak after which drops as a result of a sudden re-

ief of stresses at necking and subsequent cracking. The peak in force is

ligned with the maximum values of major or minor strains measured

y the DIC system. After the peak both strains remain stationary.

As a result of this, the amount of displacement (and the instant of

ime) corresponding to the onset of necking is clearly identified by the

ashed vertical line included in Fig. 10 a.

Application of the combined strain-force approach to tube expan-

ion with elastomer is shown in Fig. 11 . As seen, not only the approach

387

roved successful in the determination of the onset of failure by necking

s it shows a strong dependency of strains from the free length of the

ubes l free . For example, a tube with l free = 30 mm is deformed under near

lane strain deformation whereas a tube with l free = 10 mm is expanded

nder biaxial strain loading conditions.

This last result is very important because it opens the possibility of

ombining tube expansion with rigid punches and elastomers to obtain

train loading paths and strains at the onset of necking across the en-

ire principal strain space. In this sense, the utilization of tube expan-

ion with rigid punches and elastomers to determine the FLC of tubular

aterials bears a resemblance to the well-known use of Nakajima or

arciniak stretch forming tests to determine the FLC of sheet and strip

aterials [32] .

The major and minor strain-displacement evolutions for expansion

ests with elastomers showed a non-uniform evolution ( Fig. 11 b) due

o non-uniform descendent movement of the cylindrical rubber plug

aused by high friction.

J.P. Magrinho, M.B. Silva and G. Centeno et al. International Journal of Mechanical Sciences 155 (2019) 381–391

Fig. 10. Combined strain-force methodology applied to tube expansion with a rigid punch (Case ERP30 of Table 2 ). (a) Experimental force-displacement and major

and minor strain-displacement evolutions at point A located at the vicinity of the necked region (b) Distribution of the major strain at the onset of necking obtained

from DIC.

Fig. 11. Combined strain-force methodology applied to tube expansion with elastomers. Experimental force-displacement and major and minor strain-displacement

evolutions at point A for (a) Case EEL10 and EEL30 of Table 3 . (b) Distribution of the major strain at the onset of necking obtained from DIC for test cases corresponding

to the evolutions shown in (a).

4

p

e

r

o

d

m

t

S

p

g

o

b

.3. Formability limits by fracture

Fig. 12 presents the results obtained from the application of the pro-

osed methodology to the determination of the fracture strains in tube

xpansion with a rigid punch. As seen, a point located in the cracked

egion (point B of Fig. 12 b) undergoes pure tension up to the onset

f necking, after which experiences localization and rapid plane strain

eformation until failure by fracture. The strains at fracture are deter-

388

ined by measuring the gauge length strains in the cracked region after

esting and employing the procedure that was previously described in

ection 3.1 .

The strain loading path for point B undergoes pure tension for a long

eriod after the onset of diffuse necking ( Fig. 12 a) due to the rigid punch

eometry. The conical geometry of the punch causes a stabilizing effect

n higher strains at the tube end due to the lower strains experienced

y the nearest regions above it, as claimed by Daxner et al. [33] .

J.P. Magrinho, M.B. Silva and G. Centeno et al. International Journal of Mechanical Sciences 155 (2019) 381–391

Fig. 12. Tube expansion with a rigid punch (Case ERP30 of Table 2 ). (a) Strain loading path for points A and B located at the vicinity of necking and in the cracked

region, respectively. (b) Distribution of the major strain at the onset of necking obtained from DIC.

Fig. 13. Strain loading paths and formability limits of the AA6063-T6 alu-

minium tubes by necking (FLC) and fracture (FFL) in principal strain space.

Tensile and formability tests according to Tables 2–4 .

4

S

b

l

t

p

c

s

g

s

v

p

s

o

p

d

fl

e

t

m

l

d

d

s

w

−

T

c

t

[

4

m

r

t

a

s

f

A

s

s

.4. Formability limits in principal strain space

The methodology described in Section 3.1 and applied in previous

ections 4.2 and 4.3 allowed plotting the strain loading paths obtained

y DIC for all the test cases listed in Tables 2–4 , as well as the formability

imits by necking (FLC) and fracture (FFL) of the AA6063-T6 aluminium

ube material ( Fig. 13 ).

The FLC was built from the open markers corresponding to the strain

airs at the onset of necking. The FFL was built from the solid markers

orresponding to the gauge length strains at fracture and required mea-

uring the thickness of the specimens along the cracked regions.

389

As seen in Fig. 13 , the slope of the strain loading paths of the lon-

itudinal and transversal tensile tests are close to pure tension and the

mall differences found between the two paths are attributed to different

alues of the anisotropy coefficient in both directions ( Table 4 ).

The experimental strain loading paths of tube expansion with rigid

unches are also close to pure tension and the main reason why there are

lightly deviated to the left side of the principal strain space is because

f friction. In fact, friction at the contact interface between the tube and

unch leads to slightly more compressive strains along the meridional

irection.

The differences in the punch semi-angles of inclination reveal no in-

uence in the strain loading paths, as previously reported by Centeno

t al. [19] . In contrast, tube expansion with elastomers is very sensitive

o the free length l free of the tubes. Larger values of l free originate defor-

ation under plane strain loading conditions whereas small values of

free originate deformation under biaxial strain. This means that by re-

ucing the free length of the tubes, the meridional strain increases, and

eformation is able to span across the entire first quadrant of principal

train space.

The FLC depicted in Fig. 13 is approximated by V-shaped curve

hereas the FFL is approximated by a straight line with a negative slope

0.9. Both results are similar to those observed in sheet metal forming.

his is in good agreement with critical ductile damage at fracture being

ontrolled by stress-triaxiality and the FFL being a line falling from left

o right with a negative − 1 theoretical slope in principal strain space

18] .

.5. Formability limits in the space of effective strain vs. stress-triaxiality

The space of effective strain vs. stress-triaxiality relates ductility of a

aterial, which is commonly defined by the strain at fracture, with the

atio of the first invariant of the stress tensor (hydrostatic stress) and

he second invariant of the deviatoric stress tensor (effective stress). It

llows characterizing the formability limits as a function of the stress

tate [34] and provides a direct link between critical ductile damage,

racture toughness and formability limits at fracture [35] .

Fig. 14 presents the loading paths and the formability limits of the

A6063-T6 aluminium tubes by necking and fracture in this space. The

train values for each point were obtained by DIC and the effective

tress and stress-triaxiality were calculated by means of the analytical

J.P. Magrinho, M.B. Silva and G. Centeno et al. International Journal of Mechanical Sciences 155 (2019) 381–391

Fig. 14. Loading paths and formability limits

of the AA6063-T6 aluminium tubes by necking

(FLC) and by fracture (FFL) in the space of ef-

fective strain vs. stress-triaxiality. Tensile and

formability tests according to Tables 2–4 .

p

i

M

I

t

𝐷

d

a

o

p

t

n

c

c

d

5

m

i

t

o

s

t

t

f

t

q

b

m

l

a

o

t

a

f

A

F

M

U

c

R

rocedure based on the Hosford plasticity criterion that was presented

n Section 3.2 (refer to Eqs. (14) and (17) ).

Considering the void-growth damage-based criterion proposed by

cClintock [36] that is associated with crack opening by tension (mode

of fracture mechanics) [18] , it is possible to link the fracture limit with

he critical value of damage,

𝐹 𝑐𝑟𝑖𝑡

= ∫�� 𝐹

0

𝜎𝑚

��𝑑 𝜀 (19)

This is performed by determining the critical experimental value of

uctile damage at the onset of failure by fracture 𝐷

𝐹 𝑐𝑟𝑖𝑡

by calculating the

rea between the FFL and the effective strain axis for a specific value

f stress triaxiality, under the assumption of proportional loading. The

rocedure should be understood as an approximation, because in real

ests loading is not proportional and, therefore, stress triaxiality ratio is

ot constant.

Such approximation is shown in Fig. 14 for a strain loading path

orresponding to plane strain deformation (refer to the grey area asso-

iated to the longitudinal tensile tests) and the associated critical ductile

amage 𝐷

𝐹 𝑐𝑟𝑖𝑡

is equal to 0.225.

. Conclusions

A new methodology to determine the formability limits of tubular

aterials by necking and fracture was successfully applied to tube form-

ng. The onset of failure by necking was determined by application of

ime-dependent and strain-force procedures on DIC measurements. The

nset of failure by fracture involved determination of the gauge length

trains along the cracked regions and was, for the first time ever, plot in

he principal strain space and in the space of effective strain vs. stress-

riaxiality.

The transformation of the formability limits and strain loading paths

rom principal strain space to the space of effective strain vs. stress-

riaxiality by means of an analytical procedure based on the non-

uadratic plasticity criterion due to Hosford [27] revealed appropriate,

ecause the anisotropy coefficients of tubes can only be easily deter-

ined at 0° and 90° with relation to the extrusion direction.

390

The choice of tube expansion with rigid punches and elastomers al-

ows determining the strain loading paths and the fracture loci across

wide range of tube forming conditions. This result resembles the use

f Nakajima tests to determine the failure limits in sheet and strip ma-

erials across the entire range of strain loading conditions and provides

n easy and straightforward procedure to be applied in thin-walled tube

orming.

cknowledgments

The authors would like to acknowledge the support provided by

undação para a Ciência e a Tecnologia of Portugal, through ID-

EC, under LAETA project PDTC/EMS-TEC/0626/2014 and project

ID/EMS/50022/2019. The work of João Miguel Gago on the material

haracterization is greatly acknowledged.

eferences

[1] Jeswiet J, Geiger M, Engel U, Kleiner M, Schikorra M, Duflou J, Neugebauer R,

Bariani P, Bruschi S. Metal forming progress since 2000. CIRP Ann Manuf Technol

2008;1:2–17. https://doi.org/10.1016/j.cirpj.2008.06.005 .

[2] Ahmetoglu M, Altan T. Tube hydroforming: state-of-the-art and future

trends. J Mater Process Technol 2000;98(1):25–33. https://doi.org/10.1016/

S0924-0136(99)00302-7 .

[3] Becker C, Tekkaya AE, Kleiner M. Fundamentals of the incremental tube form-

ing process. CIRP Ann Manuf Technol 2014;63(1):253–6. https://doi.org/10.1016/

j.cirp.2014.03.009 .

[4] Lossen B, Andreiev A, Stolbchenko M, Homberg W, Schaper M. Friction-

spinning —grain structure modification and the impact on stress/strain be-

haviour. J Mater Process Technol 2018;261:242–50. https://doi.org/10.1016/

j.jmatprotec.2018.06.015 .

[5] Dohmann F, Hartl Ch. Tube hydroforming - research and practical ap-

plication. J Mater Process Technol 1997;71:174–86. https://doi.org/10.1016/

S0924-0136(97)00166-0 .

[6] Almeida BPP, Alves ML, Rosa PAR, Brito AG, Martins PAF. Expansion and reduction

of thin-walled tubes using a die: experimental and theoretical investigation. Int J

Mach Tool Manu 2006;46(12–13):1643–52. https://doi.org/10.1016/j.ijmachtools.

2005.08.018 .

[7] Liu S-D, Meuleman D, Thompson K. Analytical and experimental examination

of tubular hydroforming limits. SAE Tech Pap 1998:980449. https://doi.org/

10.4271/980449 .

[8] Keeler SP. Circular grid system – a valuable aid for evaluation sheet metal formabil-

ity. SAE Tech Pap 1968:680092. https://doi.org/10.4271/680092 .

J.P. Magrinho, M.B. Silva and G. Centeno et al. International Journal of Mechanical Sciences 155 (2019) 381–391

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[9] Goodwin G. Application of strain analysis to sheet metal forming problems in the

press shop. SAE Tech Pap 1968:680093. https://doi.org/10.4271/680093 .

10] Yoshida K , Kuwabara T , Narihara K , Takahashi S . Experimental verification of the

path-independence of forming limit stresses. Int J Form Process 2005;8:283–98 .

11] Hu G, Yang L, Liu J. Determination of forming limit diagram for tube hydroform-

ing based on the strain rate change criterion. AIP Conf Proc 2013;1567:390–3.

http://dx.doi.org/10.1063/1.4850455 .

12] Chen X, Yu Z, Hou B, Li S, Lin Z. A theoretical and experimental study on

forming limit diagram for a seamed tube hydroforming. J Mater Process Technol

2011;211:2012–21. http://dx.doi.org/10.1016/j.jmatprotec.2011.06.023 .

13] Li S, Chen X, Kong Q, Yu Z, Lin Z. Study on formability of tube hydroform-

ing through elliptical die inserts. J Mater Process Technol 2012;212:1916–24.

http://dx.doi.org/10.1016/j.jmatprotec.2012.04.016 .

14] Embury JD, Duncan JL. Formability maps. Annu Rev Mater Sci 1981;11:505–21.

https://doi.org/10.1146/annurev.ms.11.080181.002445 .

15] Silva MB, Skjoedt M, Atkins AG, Bay N, Martins PAF. Single point incremen-

tal forming & formability/failure diagrams. J Strain Anal Eng 2008;43:15–36.

https://doi.org/10.1243/03093247JSA340 .

16] Davis R, Grant G, Herling D, Smith M, Evert B, Nykerk S, Shoup J. Formability

investigation of aluminum extrusions under hydroforming conditions. SAE Tech Pap

2000. 2000-01-2675 https://doi.org/10.4271/2000-01-2675 .

17] Song W-L, Heo S-C, Ku T-W, Kim J, Kang B-S. Evaluation of effect of flow stress

characteristics of tubular material on forming limit in tube hydroforming pro-

cess. Int J Mach Tool Manu 2010;50:753–64. https://doi.org/10.1016/j.ijmachtools.

2010.05.010 .

18] Martins PAF, Bay N, Tekkaya AE, Atkins AG. Characterization of fracture loci in

metal forming. Int J Mech Sci 2014;83:112–23. https://doi.org/10.1016/j.ijmecsci.

2014.04.003 .

19] Centeno G, Silva MB, Alves LM, Vallellano C, Martins PAF. Towards the charac-

terization of fracture in thin-walled tube forming. Int J Mech Sci 2016;119:12–22.

https://doi.org/10.1016/j.ijmecsci.2016.10.001 .

20] Centeno G, Silva MB, Alves LM, Vallellano C, Martins PAF. On the utilization of

circle grid analysis in thin-walled forming of tubes: experimental and numerical

evaluation. International conference on the technology of plasticity; 2017 Sep 17–

22 Cambridge, UK; 2017 . https://doi.org/10.1016/j.proeng.2017.10.937 .

21] ASTM E8/E8M. Standard test methods for tension testing of metallic materials.

ASTM international, West Conshohocken USA; 2013 .

22] Dick C, Korkolis YP. Mechanics and full-field deformation study of the Ring

Hoop Tension Test. Int J Solids Struct 2014;51:3042–57. https://doi.org/10.1016/

j.ijsolstr.2014.04.023 .

391

23] Alves LM, Leitão PMF, Martins PAF. Elastomer-assisted compression beading of

tubes. Proc Inst Mech Eng Part B 2014;228(7):744–56. https://doi.org/10.1177/

0954405413510154 .

24] Martínez-Donaire AJ, García-Lomas FJ, Vallellano C. New approaches to detect the

onset of localised necking in sheets under through-thickness strain gradients. Mater

Des 2014;57:135–45. https://doi.org/10.1016/j.matdes.2014.01.012 .

25] Cristino VAM, Magrinho JPG, Silva MB, Centeno G, Martins PAF. A digital image

correlation based methodology to characterize formability in tube forming. J Strain

Anal Eng 2019. first published https://doi.org/10.1177/0309324718823629 .

26] Magrinho JP, Silva MB, Alves LM, Atkins AG, Martins PAF. New methodology

for the characterization of failure by fracture in bulk forming. J Strain Anal Eng

2018;53:242–7. https://doi.org/10.1177/0309324718758842 .

27] Hosford WF . On yield loci of anisotropic cubic metals. In: Proceedings of the seventh

North American metalworking research conference May 13–16. Ann Arbor; 1979.

p. 191–7 .

28] Hosford WF. On the crystallographic basis of yield criteria. Texture Microstruct

1996;26-27:479–93. https://doi.org/10.1177/0309324718758842 .

29] Korkolis YP, Kyriakides S. Inflation and burst of anisotropic aluminium tubes for

hydroforming applications. Int J Plasticity 2008;24(3):509–43. https://doi.org/

10.1016/j.ijplas.2007.07.010 .

30] Cui XL, Yang ZP, Wang XS. Characterization of multiaxial stress-strain response of

tube metal from double-sided hydro-bulging test based on Hosford’s 1979 yield cri-

terion. JOM 2017;69(5):930–6. https://doi.org/10.1007/s11837-017-2260-7 .

31] Morales-Palma D, Martínez-Donaire AJ, Vallellano C. On the use of maximum force

criteria to predict localized necking in metal sheets under stretch-bending. Metals

2017;7:469–85. https://doi.org/10.3390/met7110469 .

32] ISO 12004-2. Metallic materials — sheet and strip — determination of forming-limit

curves — part 2: determination of forming-limit curves in the laboratory; 2008 .

33] Daxner T, Rammerstorfer FG, Fischer FD. Instability phenomena during the con-

ical expansion of circular cylindrical shells. Comput Methods Appl Mech Eng

2005;194:2591–603. https://doi.org/10.1016/j.cma.2004.07.047 .

34] Vujovic V, Shabaik AH. A new workability criterion for ductile metals. J Eng Mater

Technol 1986;108(3):245–9. https://doi.org/10.1115/1.3225876 .

35] Atkins AG. The importance of toughness in manufacturing. J Mater Process Technol

2018;261:280–94. https://doi.org/10.1016/j.jmatprotec.2018.04.042 .

36] McClintock FA. A criterion for ductile fracture by the growth of holes. J Appl Mech

1968;35(2):363–71. https://doi.org/10.1115/1.3601204 .