IDDRG 2013 Conference June 2 – 5, 2013, Zurich, Switzerland

* Maysam Gorji: ETH Zurich, Institute of Virtual Manufacturing (IVP), PFA L59, Technoparkstrasse 1, 8005 Zurich,

Switzerland, Tel: +41 44 633 78 09, Fax: +41 44 633 15 96, gorji@ivp.mavt.ethz.ch

PREDICTION OF RUPTURE PHENOMENA IN SHEET METAL

FORMING

Maysam Gorji1*

, Bekim Berisha1, Pavel Hora

1

1 ETH Zurich, Institute of Virtual Manufacturing, Zurich, Switzerland

ABSTRACT: Recently, the classical FLC based prediction methods in the sheet metal forming were extended by different authors to the aspects of the fracture prediction. Prediction of fracture is based on

triaxiality diagrams, which maps the critical equivalent strain in dependency on the triaxiality and the Lode

parameter. From the experimental point of view, there is a wide range of proposals how to measure the criti-

cal strain under the different load conditions. The goal of this paper is to adapt the fracture prediction to the

needs of the sheet metal forming. For this purpose, fracture strains are measured based on microscopic pic-

tures of Nakajima specimens, which also serves to determine the parameters of a phenomenological damage

model. Finally, the damage model is coupled with the material model to predict the force-displacement be-

havior of a biaxial- stretching test.

KEYWORDS: Forming Limit Diagram, Lode-Triaxiality Diagram, damage, finite element method

1 INTRODUCTION

Beside of the typical necking instability [1] there

are additional types of fracture limits, which define

process limits. Those are specially fractures occur-

ring by extensive bending and hemming as well as

fractures occurring at sheet edges Fig. 1.

Fig. 1 Typical failure types in sheet metal forming processes [4].

Many authors like Wierzbicki [2] and Stoughton

[3] pointed out that even shear fracture can occur in

the Tension-Compression zone of the FLC with:

21

11

22

(1)

The extended diagrams thus feature an additional

shear fracture limit as it can be seen in Fig. 2.

Fig. 2 Fracture strains including shear [9].

Ductile fracture like the one visible in Fig. 1 cannot

be modeled with the localized necking criteria

[4,10,11,12]. An appropriate prediction of this type

of failure requires the definition of fracture criteria.

An excellent introduction to ductile fracture criteria

is given by Wierzbicki [2]. Hora et al. [4] proposed

a new sheet specific approach for the prediction of

necking as well as fracture failures, Fig. 3 and 4.

IDDRG 2013 Conference June 2 – 5, 2013, Zurich, Switzerland

Fig. 3 Illustration of the plastic deformation zone during necking and measurements of the strain distribution using the optical system ARAMIS, see [4].

Fig. 4 Localisation Level Forming Limit Diagram LL-FLD. B-isolines corresponds to equiva-lent localisation levels of necking zone B, see [4].

In contrast to the classical and generalized FLC

diagrams, Fig. 4, given in function of maj and min,

the so called triaxiality diagrams show that the

plastic strain at fracture depends on the Lode pa-

rameter or equivalently on the normalized Lode

angle ̅ and the triaxiality parameter

m (2)

1

2cos

2

273

3J (3)

where is the third invariant of the deviatoric stress. This corresponds to the transformation of

fracture locus to the stress space. The locus

reaches the maximum value under axial sym-

metric loading and the minimum under plane

strain conditions, see Fig. 5.

Fig. 5 Representation of the fracture locus in the space of stress triaxiality and the deviatoric state variable [2].

By Bai, Wierzbicki and other authors, about 15

different experiments were proposed to determine

the fracture locus, see [2]. Typically, the plane

stress line (Nakajima tests) in the above diagram

will be related to the sheet specific ruptures. Later

on, it will be shown that only a single point occurs. This is the reason why thin sheet specific

rupture prediction, beside the triaxiality diagrams,

should be established.

2 SHEET SPECIFIC EVALUATION

OF RUPTURE STRAINS

For the specific determination of sheet metal two

alternative methods will be discussed.

Firstly, an optical system (ARAMIS-system) has

been used to measure the strains during the defor-

mation. By reading the peak value in Fig. 3, frac-

ture strain can be captured. The method allows

detecting the strain path history from the beginning

of the test until rupture appears. Alternatively, an

experimental method for the measurement of frac-

ture strains has been applied. Fig. 6 shows micro-

scopic pictures of some Nakajima specimens after

the rupture occurs.

Fig. 6 Evaluation of rupture strains on Nakajima tests for AC170 aluminium alloy.

IDDRG 2013 Conference June 2 – 5, 2013, Zurich, Switzerland

By measuring the current thickness t,

can be computed. The fracture occurs under plane

strain conditions with

. The addition-

al increase of can then be evaluated under the assumption of the plane strain condition

with . The final strain is iden-tical to the last point evaluated with the optical

measurement method.

FLC, calculated by time dependent method

fracture strain based on optical measurement system

fracture strain based on thinning

Fig. 7 FLD and fracture strains for the material AC170. Comparison of the optical and microscopical evaluation method

Differences between these two methods are

shown in Fig. 7. The fracture strains measured

with the presented method are much higher than

those measured with the optical system. Differ-

ences come mainly from the following source:

The measured strains by the optical system are an

average over a certain region, whereas the strains

measured based on the microscopic pictures are

local values (pointwise). Therefore, it is recom-

mended to use the data measured based on micro-

scopic pictures to determine critical stresses

(stress limits) and to use the data measured by the

optical system to validate the computed strains of

FE-Simulations, where an element size specific

averaging occurs.

3 CORRELATION BETWEEN THE

“FLD” AND “LODE-

TRIAXIALITY FRACTURE

CURVE”

The transformation from the strain space to the

triaxiality-Lode diagram can be done as follows.

By calculating ( ) ( ) (see Fig. 8) [13]

( ) ̅ (4)

( ( )) ( ) ( ) (5)

dependent on stress ratio

, and ̅ can be

rewriten to:

( )

(6)

̅

[ ( )] ( ) ( )

( )

(7)

Fig. 8 Relation between stress ratio and ( ) [up] ( ) [down].

The algorithm for the evaluation of the critical

(failure) equivalent plastic strain ,̅ based on the equations (4-7) is given in Table 1.

Evaluation of the fucntions ( ) ( ) ( ( ))

( ) ( )

( )

( ( )); ( )

Evaluation of and ̅ for the different strain paths: and of strain path captured from optical measure-ment system:

̅ ( ( )) ̅ ̅

( )

̅

[ ( )] ( ) ( )

( )

Table 1: Algorithm for the computation of the triaxi-ality parameter

IDDRG 2013 Conference June 2 – 5, 2013, Zurich, Switzerland

For the later graphical representation of the critical

states, and ̅ als well as average values will be used:

∫ ( ̅) ̅

(8)

̅

∫ ̅( )̅ ̅

(9)

3.1 EVALUATION FOR THE MATERIAL

AC170

These realtions are demonstated for the necking

and fracture strains as depicted in Fig. 9, 10 and 11

for the AC170 material. The hardening data are

approximated using a combination of the Hockett-

Sherby and Gosh equations as given in equation

(10), where ̅ and ̅ are the equivalent stress and equivalent strain, respectively, see Table 2.

̅ { ( )̅ }

( ){ ( )

̅ }

(10)

0. 812 0. 973 233.2952 0.1597 0.3783

122.0863 802.8718 11.7175 0.9291

Table 2: Hardening curve parameters

The yield locus description Yld2000-2d as pro-

posed by Barlat [5] has been used. It reads

(

)

̅ (11)

where

| |

| | | |

(12)

S1 and S2 are the principal deviatoric stresses and

“a” is an exponent determined based on the crystal-

lographic structure of the material, which for FCC

materials is considered as 8 [5]. Yld2000-2d pa-

rameters for the investigated material are presented

in Table 3.

0. 9783 0.9511 0.9159 0.9969

0.9974 0.9136 0.9351 1.1882

Table 3: Yld2000-2d parameters

Instability and fracture values measured by

conventional Nakajima tests and thinning method

explained in previous section (see Fig. 7) are

converted to ( ) values by the algorithm given in Table 1, see Fig. 9.

Fig. 9 Converted experimental principal strains to ( ) space.

The triaxiality parameter and ̅ are

computed as an integral value with respect to the

deformation path (see equations 8 and 9) [2]. As

expected, the fracture line lies much higher than

the necking instability line of the forming limit

curve.

Due to the fact that under plane strain conditions

the ̅ and values are directly related, Fig. 9 can be alternatively expressed in function of ̅ . Numerical values of the fracture line ( ̅ ) are given in Table 4.

̅ -0.92 -0.39 -0.38 -0.25 0.05 0.21

0.55 0.71 0.65 0.59 0.69 0.81

Table 4: Converted experimental principal strains

to ̅ space.

A further relation between and ̅ is shown in Fig. 10.

Fig. 10 Converted experimental principal strains to ( ̅ ) space.

If alternatively in Fig. 10, the averaged values

and ̅ will be replaced by the effective (current) time dependent and ̅, all different strain paths comes together to identical ( , ̅) - point, corresponding to the plane strain behavior.

In this sense, the triaxiality diagram, reduces to one

point. The question, which representation Fig. 10

or Fig 11, is physically more correct, has still to be

investigated. In contrast to Fig. 10, Fig. 11 shows a

IDDRG 2013 Conference June 2 – 5, 2013, Zurich, Switzerland

strain path dependency of the failure strains.

Fig. 11 Influence of the strain path on fracture strains for different Nakajima specimens

Two characteristics of the diagram can be high-

lighted: firstly, the linear strain path ( ) is represented by the vertical lines. Secondly, at the fracture strain, is equal to zero, for all strain paths. This means that for the case of

sheet metal forming the triaxiality diagram reduces

to one point ( ̅) corresponding to .

4 NUMERICAL FRACTURE

PREDICTION BASED ON

DEPENDENT DAMAGE MODELING

In the framework of this study, the change of the

mechanical properties due to damage is represented

by a scalar variable “D” such that ( ), where D = 0 and D = 1 represent the undamaged

(initial) and fully damaged states (complete loss of

material strength), respectively. The damage D can be expressed as a function of

the equivalent plastic strain and triaxiality. For

constant values of failure strain the following rela-

tionship between damage and equivalent plastic

strain is given in the literature [6,7]

( ̅

)

(13)

By choosing an exponent n = 1, aforementioned

equation is simplified to the linear Johnson-Cook

[8] criterion.

Fig. 12 Growing of the damage for the Johnson-

Cook model (n=1) and for an exponent n=5

Differentiating equation (13) results to a rate form

of the non-linear damage evolution:

̇

̅̇ (14)

The effective stresses are then calculated in func-

tion of D:

( ) (15)

where denotes the Cauchy stress tensor and

the resulting stress due to damage.

It is immediately clear from equation (15) that no

stresses can be transferred for D = 1. In the local

approach to fracture this critical state is used to

represent a crack by a region of completely dam-

aged material. In the remaining part of the domain,

and particularly next to the crack, some (noncriti-

cal) damage may have been developed, while other

areas may still be unaffected by damage.

The damage accumulation is given by equation

(13) and Fig. 12. The exponent has been found to

be 5 in this experiment for the studied material,

where the fracture strain is taken from the biaxi-

al experiment shown in Fig. 10. Fig. 13 represents

the true stress-strain of the investigated material

with and without introducing damage.

Fig. 13 Hardening curve with and without consider-ing damage along percentage of damage in equi-biaxial loading.

Fig. 13 shows the damage state at different equiva-

lent plastic strains. The initiation of the damage

starts at diffuse necking. The example shows how

fast damage increases.

To control the behaviour, biaxial-stretching test

was applied. Fig. 14 demonstrates that for a correct

prediction of the failure state, the introduction of

damage is indispensable. Without the damage

influence the virtually predicted failure arises to

late.

IDDRG 2013 Conference June 2 – 5, 2013, Zurich, Switzerland

Fig. 14 Punch force-displacement in different FE model compared to the experiment

Fig. 15 demonstrates the corresponding accumula-

tion of the damage function during the forming

process.

Fig. 15 Punch force-displacement and percentage of damage in equi-biaxial loading.

5 CONCLUSIONS AND OUTLOOK

The presented paper deals with the prediction of

ruptures in sheet metal forming. It demonstrates

that the general application of the triaxiality dia-

gram may be a correct approach, but for typical

application covered with the plane stress Nakajima

tests, the rupture states may be described as well

with an extended forming limit diagram as demon-

strated in Fig. 7 (strain space) and Fig.11 (stress

space). The paper has additional discussed two

different methods for the determination of the

critical rupture strains. The first one based on the

well-known optical evaluation methods and the

other one by a local evaluation method based on

thinning measurements. The paper closes with a

damage based material model, using the experi-

mental specified critical strain ̅ ( ̅) or ̅ ( ) which helps to improve the mapping of the real behaviour.

ACKNOWLEDGEMENT

The authors are very grateful to the CTI (The Swiss

Innovation Promotion Agency) for the financial

support of this work. In addition to, the technical

support of Mr Samuel Staub during the execution

of experiments is appreciated.

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