Int J Mater Form (2009) Vol. 2 Suppl 1:825–828 DOI 10.1007/s12289-009-0514-9 © Springer/ESAFORM 2009

MODELLING OF SPRINGBACK IN SHEET METAL FORMING Marko Vrh1, Miroslav Halilovič2, Bojan Starman2, Boris Štok2* 1

2

Kovinoplastika Lož, Stari trg pri Ložu - Slovenia Laboratory for Numerical Modelling & Simulation; Faculty of Mechanical Engineering, University of Ljubljana; Ljubljana – Slovenia

ABSTRACT: Springback, the phenomenon that is governed by strain recovery after removal of forming loads, is of great concern in sheet metal forming. There is no doubt that physically reliable numerical modelling of the forming process and predictions of springback obtained by respective computer simulations are crucial to control this problem. Unfortunately, by currently available approaches springback still cannot be adequately predicted in general. This paper is an attempt of building a corresponding constitutive model, which will simultaneously consider sheet anisotropy, damage evolution and strain path dependent stiffness degradation during metal forming. First, for the identification of the parameters in the built constitutive model a particular experimental procedure is deliberately developed. To solve the arisen inverse problem an optimization procedure is employed. The proposed approach to constitutive modelling is validated in the end by a simulation of the springback in the formed HSS steel. The simulation results prove to be in good agreement with the experimental ones. From the performed comparisons it is clearly indicated, that only simultaneous modelling of material properties can be the true key to obtain accurate prediction of springback in sheet metal forming. KEYWORDS: anisotropy, stiffness degradation, springback, damage

1 INTRODUCTION In production processes, where sheet metal forming is frequently used (vehicle production, domestic appliances production), the so-called springback effect associated with the reversible elastic strain recovery after unloading remains a major trouble for the companies’ development departments. There is no doubt that a reliable prediction of the springback based on the corresponding numerical simulations is the key for resolving this problem. In fact, a lot of work related to proper numerical modelling of the springback has been already done. Above all it turns out, that precise modelling of the material response is the most promising way. So far presented models mostly deal with the precise modelling of anisotropy [1], Bauschinger effect [2] and elastic modulus degradation [3, 4] separately. Recent theoretical results indicate however, that springback depends on a simultaneous effect of those material properties. This paper is an attempt of building a corresponding constitutive model, which will simultaneously consider sheet anisotropy, strain path dependent stiffness degradation and damage evolution in material during loading. The related work is based on a combined experimental-analytical-numerical approach in which the proven experimental evidences are tried to be captured by analytical modelling of the respective constitutive equations, and adequacy of the latter is obtained by means of numerical simulation of given experiments.

The constitutive model under development is namely implemented into a FEM based program, in our case ABAQUS/Explicit [5]. The FEM simulations are used in the development stage of the constitutive modelling purely for the purpose of constructing a constitutive model that is consistent, as much as possible, with given experimental evidence. When physical objectivity of such a model is attained it may be applied, assuming the corresponding material parameters are identified, for a wide range of similar metals. In the end the developed constitutive model is numerically validated by a simulation of the springback test.

2 EXPERIMENTAL OBSERVATIONS In this chapter some measurements, performed on the 0.66mm thick stainless steel EN 1.4301 will be elaborated. 2.1 HARDENING AND ANISOTROPY The hardening and the plastic anisotropy of the observed steel were determined by standard tensile test performed on the TIRA 2300 tensile test machine. Standard tensile specimens (width of the central part was 20mm) were cut out from the sheet in three different directions, namely in directions 0°, 90° and 45° with regard to the rolling direction. The loading force F, elongation ∆L, reduction of width ∆s and thickness reduction ∆t of the central 80mm long part of the specimen were measured during loading. From the measurements of ∆s and ∆t the Lankford values,

____________________

* Corresponding author: Prof. Boris Štok, Faculty of Mechanical Engineering, University of Ljubljana, Aškerčeva 6, SI-1000 Ljubljana e-mail: [email protected], tel.: +386-41-694-534, fax.: +386-1-2518-567

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associated with the material anisotropy, were determined to be r0 = 1.03 , r90 = 0.67 and r45 = 1.23. The F-∆L curves and the Lankford values give evidence of a significant degree of plastic anisotropy. 2.2 STIFNESS DEGRADATION At this point the reader is invited to read our work [6] presented at the ESAFORM 2008 conference, which deals with the measurements of the stiffness degradation. Here, only some results will be summarized. As the springback phenomenon is governed by the elastic strain recovery, the actual elastic properties of the plastically strained material are of greatest importance. It is widely accepted, that elastic properties drop, when sheet is being subjected to loading into plastic region. In [6] we further investigate influence of the loading direction on the degradation of Young's moduli in different directions. First a special purpose tool was constructed to prestrain rectangular sheet plates of dimensions 500mmx500mm to a specific degree of the plastic strain. From the plastically prestrained sheet plates small rectangular specimens were cut out and clamped into the tensile test machine. While being loaded uniaxially in the elastic region, the resulting force and elongation of the small specimens were precisely measured. Upon them Young's modulus of each specimen in its axial direction was calculated using Hooke's law. The basic idea of the experiment is schematically presented in Figure 1. With regard to the direction of the prestrain stretching the following four relationships of Young's modulus degradation, denoted by E11(0), E22(0) , E11(90) and E22(90) , as shown in Figure 1, can be extracted from those measurements. The subscripts denote the respective direction of the modulus measurement, while superscripts denote the direction of the plastic prestrain stretching. The results of the measurements are displayed in Figure 2.

a)

b)

Figure 2: Degradation of Young's moduli after plastic prestraining (a) in the rolling direction (b) perpendicular to it

2.3 DAMAGE Voids in the material of the plastically prestrained specimens can be noticed by observation of the microstructure. For that purpose the small specimens of dimensions 5mmx5mm were cut out from the stretched steel with plate shears. The two photos in Figure 3 show the edge of the small specimens, while being inspected with the JEOL KSM-5610 electronic microscope at two different degrees of the plastic prestrain ε p . The growth of number and size of the voids in the material during the evolution of plastic strains is out of doubt.

a)

Figure 1: Measurement of Young's moduli after plastic prestraining (a) in the rolling direction (b) perpendicular to it

Figure 3_a: Observations of microstructure of plastically prestrained specimens at ε p = 0.2

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3.2 ELASTICITY For the calculation of the elastic strains the orthotropic Hooke's law is used

b)

σ ij = Cijklε kle

(4)

The orthotropic elastic tensor Cijkl is defined with nine constants {E1, E2, E3, G12, G23, G31, ν12, ν23, ν31} and their variation is modelled in accordance with the damage and microstructure evolutions in the following way

dEi = Ei0 ( −α df − g (ε mp ) dε mm ) i = 1,3 dGij = Gij0 ( −β df − g (ε mp ) dε mm ) i = 1,3 ; j = 1,3 (5) dν ij = ν ij0 ( −γ df − g (ε mp )dε mm ) i = 1,3 ; j = 1,3 Figure 3_b: Observations of microstructure of plastically prestrained specimens at ε p = 0.386

3 CONSTITUTIVE MODELLING 3.1 PLASTICITY To consider both anisotropy and damage simultaneously we propose a yield criterion based on an upgrade of the Gurson type (GTN) potential in the following form

Φ=

 3q σ + 2 f q1 Cosh  2 H  2σ y (σ y ) 

(σ eq ) 2 2

 2  − (1 + q3 f ) 

(1)

where the anisotropy is modelled by a corresponding definition of the equivalent stress σ eq

(

2

2

σ eq = a F1 (σ 22 − σ 33 ) + G1 (σ 33 − σ 11 ) + 2

H1 (σ 11 − σ 22 ) + 2 ( L1 σ 122 + M 1 σ 132 + N1 σ 232 )

(

4

2 2

)

1 2

+

(2)

4

2 2

2 (1 − a ) F (σ 22 − σ 33 ) + G (σ 33 − σ 11 ) + 4

4 H 22 (σ 11 − σ 22 ) + 2 ( L22 σ 124 + M 22 σ 134 + N 22 σ 23 )

)

1 4

The parameters of the introduced model can be grouped with regard to their action in two subsets. Constants F1 , , 1 , H 1 , L1 , M1 , N1 , a , F2 , G2 , H 2 , L2 , M 2 , N 2, G describe the anisotropy, while constants q1 , q2 , q3 describe damage. In this work the yield stress σ y is defined as a function of the equivalent strain of matrix material ε eqM , constructed from 10 cubic splines up to ε eqM = 0.47 . f is further the void volume fraction and σ H = σ kk / 3 is the hydrostatic stress. The constitutive model is considered completed, when also evolution equations for the state variables are defined

df = (1 − f ) dε kkp + An dε mp ; dε ijp =

∂Φ dλ ∂σ ij

σ dε   ∂Φ (3) dσ ij = Cijkl  dε kl − dλ  ; dε mp = ij ∂σ kl (1 − f ) σ y    1  An = f n /( sn 2π ) exp  − (ε mp − ε n ) / sn  2   p ij

For small values of the porosity f the damage influence constants are α = 2.00 , β = 0.414 , γ = 1.91 [3], whereas function g (ε mm ) is approximated in our case with three third order splines as a function of the substitutive plastic strain ε mm . The variation of the latter is governed by .dε mm = σ ij Pijkl dε klp / (σ y (1 − f )) where Pijkl denotes the strain path influence tensor. Tensor Pijkl must be identified according to the case, in general. Here, it is assumed that P1111 = P3333 = P1212 = P1313 = P2323 = 1 and P2222 is defined as P2222 = Pa + Pbε mm + Pc (ε mm )2 , whereas the remaining components of Pijkl are zero.

4 MATERIAL CHARACTERIZATION By material characterization we mean the evaluation of material parameters that for a particular material describe the material behaviour in accordance with the adopted constitutive model. The parameters identification is in fact an inverse problem, which is solved by means of an optimisation procedure. Accordingly, those values of the parameters that bring a corresponding cost function, defined as a sum of squares of the deviations between the measured and the respective computed quantities, to a minimum may be considered the correct choice. Table 1: Values of the parameters F1

G1

H1

L1

M1 *

0.665

0.536

0.464

1.83

1.5

1.5

L2 *

M2 *

1.5

1.5 fn *

a

F2

G2

H2

0.194 N2 *

0.512 q1 *

0.314 q2 *

0.512 q3 *

εn *

N1 *

1.5

1.5

1

2.25

0.3

0.05

sn *

σ 0y

σ 0.05 y

σ 0.1 y

σ 0.15 y

σ 0.2 y

308

500

573

682

775

σ 0.23 y

0.1

σ 0.27 y

σ 0.3 y

σ 0.32 y

σ 0.37 y

σ 0.47 y

827

892

941

973

0

0.2

0.4

1.

1049 Pa

1186 Pb

2.19

8.75

g 0.63

g 0.16

Pc

E10

E20

E30

10.2

208800

204800

208800

Gij0

g 0.46

; i = 1,3 ; j = 1,3 80670

g 0

ν ij0 ; i = 1,3 ; j = 1,3 0.3

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After removing the lever the angle α is reduced due to springback. The measured angle is then compared with the results from a numerical simulation of the presented springback test (Fig. 5c). The simulations are based on FEM code Abaqus/Explicit with an implemented VUMAT subroutine. In the simulation 300 shell elements with a reduced integration (S4R) and having 11 through thickness section points are used to model the sheet, while the tool is assumed to be rigid with the maximum unevenness of the surfaces being 0.05mm. The results of computed and measured springback are presented in Table 2. Figure 4: Identification of parameters; comparison between experimental results and optimised model

The parameters identification is performed by considering a cost function in which the F-∆L curves of the standard tensile specimens cut from the sheet plate in three directions (0°, 90°, 45°), together with the variation of the elastic moduli (as a function of ∆L) and Lankford coefficients were included. The identified parameters are given in Table 1 (parameters marked with * were adopted from literature) and prove good agreement between the experimental and computed response, which is clearly shown in Figure 4.

Table 2: Comparison of exp. vs. cal. springback (angle)

Prestrain in the rolling direction Prestrain angle α ( °) p ε Exp. Cal. 0 79.7 79.8 0.053 76.1 75.8 0.097 72.6 72.4 0.144 69.2 69.2 0.203 65.2 65.4 0.244 63.2 62.0 0.300 59.6 57.0

Prestrain in the direction perpendicular to rolling Prestrain angle α ( °) εp Exp. Cal. 0 80.1 80.0 0.056 74.7 76.1 0.090 72.2 72.4 0.147 69.2 69.9 0.200 66.0 66.4 0.234 63.4 62.8 0.300 59.1 61.3

5 VERIFICATION

6 CONCLUSIONS

The developed constitutive model is verified on a springback test, consisting of bending and subsequent release of plastically prestrained rectangular specimens. Cut from the sheet plate either in the rolling direction or perpendicular to it, the specimens of dimensions 240mmx20mm were first longitudinally stretched to a specified level of the plastic strain ε p = Ln( L / L0 ) on the tensile test machine. Then they were clamped into a special bending tool (Fig. 5a) and bent for a specified amount α = 90o (Fig. 5b).

We have presented a constitutive model, which considers simultaneously sheet anisotropy, strain path dependent stiffness degradation and damage evolution in the material during plastic deformation. Some parameters of the model, which is implemented in Abaqus/EXPLICIT code via VUMAT subroutine, were identified from the obtained experimental data, while some of them were adopted from literature. The performed simulations of springback prove good agreement with the experiments.

REFERENCES

Figure 5: Springback test;bending of prestrained specimens

[1] C. Gomes, O. Onipede, M. Lovell, Investigation of springback in high strength anisotropic steels. Journal of Material Processing Technology, 159:91-98, 2005. [2] M.C. Oliveira, J.L. Alves, B.M. Chaparro and L.F. Menezes, Study on the influence of work-hardening modelling in springback prediction. International Journal of Plasticity, 23:516-543,2006. [3] M. Halilovic, M. Vrh, B. Stok, Prediction of a elastic recovery of a formed stell sheet considering stiffness degradation, Meccanica, 2008. http://dx.doi.org/10.1007/s11012-008-9169-8 [4] M. Vrh, M. Halilovic, B. Stok, Impact of Young's modulus degradation on springback calculation in steel sheet drawing. Stroj. vestn. 54:288-296. 2008 [5] ABAQUS/Explicit V6.6 (Theory manual, user’s manual, example problems manual) (2006) Simulia, Providence [6] M. Vrh, M. Halilovic, M. Misic, B. Stok, Strain path dependent stiffness degradation of a loaded sheet. International journal of material forming, ESAFORM Conference on Material Forming, 2008. http://dx.doi.org/10.1007/s12289-008-0343-2.