paper114nlpds

8/8/2019 paper114nlpds

http://slidepdf.com/reader/full/paper114nlpds 1/4

____________________

* Corresponding author: Div. of Material and Computational Mechanics, Dept. of Applied Mechanics, Chalmers University of

Technology, SE-412 96 Göteborg, Sweden, e-mail: [email protected], phone:+46-31-7728569

MATERIAL MODELLING FOR ACCURATE SPRINGBACK PREDICTION

Per-Anders Eggertsen1*

, Kjell Mattiasson1,2

1 Dept. of Applied Mechanics, Chalmers University of Technology, Göteborg, SWEDEN

2 Volvo Cars Safety Centre, Göteborg, SWEDEN

ABSTRACT: The magnitude of the springback depends mainly on the residual stresses in the work piece after theforming stage. An accurate prediction of residual stresses puts, in turn, high demands on the material modelling duringthe forming simulation. Among the various ingredients that make up the material model, the hardening law is one of the

most important ones for an accurate stress distribution prediction. The hardening law should be able to consider some,or all, of the phenomena that occurs during bending and unbending of metal sheets, such as the Bauschinger effect, thetransient behaviour, permanent softening and work-hardening stagnation. Five different hardening models and four

different steel grades have been evaluated in the present investigation. The unknown material parameters wereidentified by inverse modelling of a three point bending test. The model's ability the reproduce experimental force-displacement relationships were evaluated. A simple springback experiment was performed for confirmation.

KEYWORDS: Springback, Hardening law, Bauschinger effect, Inverse modelling, Three point bending

1 INTRODUCTION

The magnitude of springback is highly dependent on the prediction of residual stresses in the work piece after theforming stage. An accurate prediction of these residualstresses puts, in turn, high demands on the material

modelling during the forming simulation. Among the

various ingredients that make up the material model, thehardening law is one of the most important for anaccurate stress distribution prediction. The hardening

law should be able to consider some, or all, phenomenathat occur during bending and unbending of metalsheets, such as the Bauschinger effect, the transient behaviour, permanent softening and work-hardeningstagnation [1][6], further described in Figure 1.

Figure 1: Schematic unloading curve to illustrate the Bauschinger effect, the transient behavior, the permanent softening behaviour and the workhardening stagnation.

The main purpose of the present study has been to try toidentify a model of reasonable complexity, which can

fulfil all requirements concerning accuracy of springback predictions. Five different existing hardening laws were

implemented in LS-Dyna. The material parametersinvolved in the various hardening models were identified by inverse modelling of a three point bending test. The

differences between the predicted and the experimentalforce-displacements curves were used as evaluationcriteria for the different hardening laws.

2 MATERIALS

Four different steel grades were considered in this study:Two DP600-steels from two different suppliers, a mildDX56-steel and a 220IF-steel. The uniaxial yield stresses

and thicknesses for the considered materials are listed inTable 1.

Table 1: Material data for the materials included in the current study

Material σ0 [MPa] Thickness [mm]TKS-DP600HF 363.70 1.46SSAB-DP600 442.20 1.00TKS-220IF 226.70 0.96Voest-DX56D 173.5 0.70

Bauschinger effect

σ

pε

B

B ′

B′′ Permanent

softening

Transient behaviour Workhardening

stagnation



3 CONSTITUTIVE EQUATIONS

Five different hardening laws are considered in the present study. An isotropic law (I-H), a mixed isotropic-

kinematic hardening law attributed to Hodge [1] andfurther developed by Crisfield (M-H) [3], the

Armstrong-Frederick hardening law (A-F) [4], the Geng-

Wagoner hardening law (G-W) [5], and, finally, theYoshida-Uemori hardening law (Y-U) [6]. All thesecriteria are implemented in the finite element code LS-Dyna together with the eight parameter yield criterion byBanabic and Aretz [7][8]:

(1

12

2

M M M

σ ⎡ ⎤ ⎞= Γ + Ψ + Γ − Ψ + Λ ⎟⎢ ⎥ ⎠⎣ ⎦

(1)

where the functions Γ , Ψ and Λ are defined as

( )

( )

2

2

2

2

2

4

4

xx yy

xx yy

xy xy

xx yy

xy xy

L K

N P Q

R S T

σ σ

σ σ σ σ

σ σ σ σ

+Γ =

−Ψ = +

−Λ = +

(2)

where the parameters L,K,N,P,Q,R,S and T are identifiedfrom uniaxial- and bulge-test data.

3.1 MIXED HARDENING

The hardening law that is called mixed hardening (M-H)

in this study is a combination of isotropic and kinematichardening, where the proportion of isotropic and

kinematic hardening is weighted with a scalar m. Thescalar m represents the ratio of plastic strain associatedto isotropic hardening, whereas the ratio (1-m) is left for the kinematic hardening response. From this it followsthat:

(1 )

0 1

p p p p p

iso kin kinm m

m

= + = + −

≤ ≤

ε ε ε ε ε (3)

The evolution of the back-stress α can then be expressedas:

( ) ( ) p

ε

H' - H' m -σ

=α σ α (4)

where now H ′ is the current plastic slope (at pε ), and

H ′ is the plastic slope related to the isotropic hardening.The mixed hardening law is able to consider theBauschinger effect and the permanent softening behaviour. It can be noted that if the scalar m is set to 1, pure isotropic hardening (I-H) is recovered.

3.2 ARMSTRONG-FREDERICK HARDENING

The third hardening law is the law by Armstrong andFrederick (A-F) [4], which stated that the back-stressevolution is given by:

p

sat xC ε σ

α ⎥⎦

⎤⎢⎣

⎡−

−= α

ασα (5)

where α sat and C x are material parameters. The A-Fhardening law is able to consider the Bauschinger effect

and the transient behavior (see Figure 1).

3.3 GENG-WAGONER HARDENING

The Geng-Wagoner (G-W) hardening law is an

extension of the A-F law, and involves two surfaces: Ayield surface and a bounding surface. The G-W lawincludes translation of the bounding surface, in order tocapture the permanent softening effect.

p sat xC ε

σ

α ⎥

⎦

⎤⎢⎣

⎡−−−= )()( βαασα (6)

( ) p H H mε

σ

′ ′−= −β σ α (7)

where α is the center of the yield surface and β is thecenter of the bounding surface. As can be seen fromEquation 7, the hardening of the bounding surface is

governed by the M-H law (Equation 4).

3.4 YOSHIDA-UEMORI HARDENING

The final hardening law is the one by Yoshida andUemori (Y-U). The Y-U law includes both translation

and expansion of the bounding surface, while the active

yield surface only evolves kinematically. The evolutionof the back-stress is expressed as:

*= +α α β (8)

with

( )

( )

* *

x p

p

B R Y C

Y

bk

B R

ε

ε

+ −⎛ ⎞= ⋅ ⋅ − ⋅⎜ ⎟

⎝ ⎠

⎛ ⎞= ⋅ ⋅ − ⋅⎜ ⎟+⎝ ⎠

α σ - α α

β σ - β β

(9)

where *α is the relative kinematic motion of the yield

surface with respect to the bounding surface, β is thecentre of the bounding surface, B is the initial size of the bounding surface, Y is the size of the yield surface, C x and k are material parameters, and R is the isotropichardening of the bounding surface:

( ) p

sat R k R R ε = − (10)

In Eq. (10) R sat is a material parameter describing theupper limit of isotropic hardening. The Y-U law is ableto consider all the effects described in Figure 1.However, in order to consider permanent softening and

work-hardening stagnation the Y-U model includes a

third surface, called non-isotropic hardening (non-IH)surface. The meaning of the non-IH surface is that theisotropic hardening of the bounding surface takes place



only when the centre point of the bounding surface staysin the non-IH surface. This leads to, that at a certainrange of reverse deformation, no isotropic hardening of the bounding surface takes place.

4 EXPERIMENTAL PROCEDURE

The unknown material parameters in the differentkinematic hardening laws described above, wereidentified by an inverse approach from a three-point bending test. The experimental set-up of the bending testis shown in Figure 2. The bearings are free to rotate

around the bearing centre. Furthermore, the sheet is freeto slip between two rollers at the bearings. The punch in

the middle is moved with a prescribed displacement andthe applied force is measured. The resulting force-displacement curve is then used as a target curve in anoptimization procedure, where the three-point bendingtest is simulated by the finite element code LS-Dyna andthe optimization procedure is preformed by theoptimization tool LS-Opt.

Figure 2: Experimental setup used in the three-point

cyclic bending tests.

LS-Opt uses a response surface methodology to solve theoptimization problem. The objective function in the

optimization is a scalar function of the differences between the predicted and the experimental force-displacement relationships. The error is calculated in amean squared manner such that the mean squared error is defined as:

( )( )2

1

1 P

p p

p

MSE f x G P

ε

=

= = −∑ (11)

where ( ) p x , p=1, …, P are the values on the computedcurve, pG , p=1, …, P are the values on the target curve

and P is the number of measured points.

5 PERFORMANCE OF HARDENING LAWS

The performance of the different hardening laws isevaluated by comparing the MSE-value after the

optimization procedure. The lower MSE-value, the better fit to experimental data and thereby a better prediction of the material behaviour. The obtained material parametersand MSE-values for the different hardening laws and thedifferent materials are presented in Table 2 to Table 5

below.

Table 2: Parameters and MSE for the TKS-DP600HF

m Cx α sat /B b R sat k MSE

I-H 0.3456

M-H 0.26 0.0911

A-F 49 248 0.0046G-W 0.93 128 211 0.0021

Y-U 56 628 181 558 1.97 0.0017

Table 3: Parameters and MSE for the SSAB-DP600


I-H 0.4370

M-H 0.22 0.0981

A-F 12 194 0.0864

G-W 0.87 41 67 0.0773

Y-U 38 229 31 0 263 0.0723

Table 4: Parameters and MSE for the TKS-220IF


I-H 0.5389

M-H 0.06 0.0632

A-F 151 250 0.0086

G-W 0.95 146 279 0.0085

Y-U 30 539 332 426 13 0.0084

Table 5: Parameters and MSE for the Voest-DX56D


I-H 0.2796M-H 0.09 0.0293

A-F 2.39 548 0.0258

G-W 0.36 6.51 215 0.0210

Y-U 21 173 8.54 314 19 0.0180

The results from the three point bending fit for all thedifferent kinematic hardening laws and materials showthat the Y-U model results in the best fit for all materialsThe G-W model gives in general almost as good fit. TheI-H model gives, as can be expected, the largest MSE-value in all cases. For the M-H and the A-F models nodirect conclusions can be established. For some of the

materials these laws show good fit, but for other materials the deviation compared to experimental data is

more pronounced. An interesting thing is that the A-Fmodel works very well for the SSAB-DP600, but for thesame material from another supplier (TKSDP600) andwith another thickness the fit is worse compared to theother kinematic hardening laws.



6 SPRINGBACK

To confirm the influence of hardening law upon thespringback magnitude, a simple springback experiment

was simulated. The NUMISHEET’93 benchmark problem has been used. The experimental set-up is

described in Figure 3 below.

Punch

50Blank holder force Blank holder force

Die

Blank lenght: 300 mm

R5

R5

Stroke

depht:66 mm

Figure 3: Experimental set-up for the NUMISHEET’93 benchmark problem.

For the springback predictions the FE-code LS-Dynawas used. The results from the predictions are presentedin Figure 4. The predictions were performed both withelastic unloading behaviour and with an unloadingmodulus according to the work by Yoshida [6].

0

5

10

15

20

2530

35

I-H M-H A-F G-W Y-U

T i p d e f l e c t i o n [ m m

]

Elastic modulus Unloading modulus

Figure 4: Results from the springback predictions for the

TKSDP600 steel and various hardening law.

It can be seen from Figure 4 that there is a deviation inthe springback magnitude for the different hardening

laws. Furthermore, the result shows the importance of the elastic degradation with plastic strain [9], here

accounted for with the unloading modulus.The springback predictions show also a clear relationship between the predicted springback magnitudeand the MSE-value from the material parameter identification (Table 2). The lower MSE-value, the better the springback prediction is.

7 CONCLUDING REMARKS

For accurate prediction of springback it is necessary touse a hardening model that can properly describe the

Bauschinger effect, the transient behaviour and the permanent softening effect, such as the Geng-Wagoner

hardening law or the Yoshida-Uemori hardening law.

Furthermore, it is of great importance to include theeffect of elastic degradation with plastic strain.

ACKNOWLEDGEMENT

The characterization of the materials used in this studywere conducted by Per Thilderkvist and JörgenHertzman at the Industrial Development Center in

Olofström, Sweden. The three-point bending tests were performed by Bertil Enquist at Växjö University. Their contribution to this work is gratefully acknowledged.The work has been performed within the Swedishnational research program MERA (ManufacturingEngineering Research Area). Financial support has been provided by Vinnova.

REFERENCES

[1] Boger R. K. Non-Monotonic strain hardening and itsconstitutive representation. Dissertation, The Ohiostate University, 2006.

[2] Hodge P. G. A new method of analyzing stresses

and strains in work hardening solids. Journal of Applied Mechanics, 25:482-483, 1957.

[3] Crisfield M. A. More plasticity and other materialnon-linearity-II. In M.A. Crisfield, editor, Non- Linear finite element analysis of solids and

structures, Vol.2, pages 158-187, Wiley, Chichester,England, 1997.

[4] Armstrong P. J., Frederick C. O. A mathematicalrepresentation of the multiaxial Bauschinger effect.G.E.G.B. report RD/B/N 731, 1966.

[5] Geng L., Wagoner R. H. SRole of plastic anisotropyand its evolution on springback. InternationalJournal of Mechanical Sciences, 44:123-148, 2002.

[6] Yoshida F., Uemori T. A model of large-straincyclic plasticity and its application to springback simulation. International Journal of MechanicalSciences, 45:1687-1702, 2003.

[7] Banabic D., Aretz H., Comsa D. S., Paraianu L. Animproved analytical description of orthotropy inmetallic sheets. International Journal of Plasticity,21:493-512, 2005.

[8] Aretz H. A non-quadratic plane stress yield functionfor orthotropic sheet metals. Journal of MaterialProcessing Technology, 168:1-9, 2005.

[9] Morestin F., Boivin M. On the necessity of taking

into account the variation in the Young moduluswith plastic strain in elastic plastic software. Nuclear

Engineering Design 162:107-116, 1996.

Experimental Springback

paper114nlpds

Documents