Multiscales modeling of microstructure evolution during asymmetric cold rolling process M. Ould Ouali 1 and M. Aberkane 2 1
Université Mouloud MAMMERI de Tizi-Ouzou. Faculté du Génie de la Construction. Département de Génie Mécanique. 15000 TIZI-OUZOU. ALGERIA. 2 Laboratoire d’Etude Mécanique et Matériaux. Université Mouloud MAMMERI de Tizi-Ouzou. BP 17 RP. 15000 TIZI-OUZOU. ALGERIA.
e-mail:
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ABSTRACT: A micromechanical ductile fracture model extended to take into account thermal heating due to mechanical dissipation within the metal is used to study a 3D-asymmetric cold rolling operation. The void coalescence mechanism by internal necking is also considered by using a modified Thomason’s criteria. The described formulation is implemented into ABAQUS/Explicit finite element package using the Aravas’s schemes. The presented model is applied to simulate the microstructure evolution of a A1050P aluminum with the variation of roll radius and roll speed ratios. The confrontation with available experimental results shows the capability of the constitutive law to predict the rolling force variation for different reductions. The results show also that optimums parameters could be found in order to reduce the rolling force and that rolling parameters exert an influence on the final microstructure of the rolled metal.
KEYWORDS: Cavity shape. Cold rolling. Micromechanical modeling. Numerical implementation. Porosity.
1
INTRODUCTION
z
B b
z
The rolling operation studied in this paper consists in actuating a plate between two rollers turning in contrary directions. Two rolling types can arise in this situation : symmetric and asymmetric rolling. The symmetric rolling is obtained by using equal conditions at the roll/workpiece interfaces and by positioning the sheet at the beginning of the operation symmetrically with respect to the rollers axis. The asymmetric rolling is the result of a no-satisfaction of one or many of these conditions: lubrication mismatch, different roll speeds... Several work was devoted to analyze this process [5, 4]. Generally they have as an aim the optimization of the rolling process (force rolling, deflection of the piece rolled...) without being interested on the microstructure evolution. The influence of the rolling parameters on the aluminum microstructure is the purpose of our study. In this paper the role of roll radius and roll speed ratios on the porosity and the cavities shape change is presented. With this intention, we make use of a micromechanical model taking account for the cavities shape change extended to include the metal heating effects due the mechanical dissipation.
2
DUCTILE FRACTURE MODELING
2.1
Void growth
The proposed void growth model, namely the GLD1 , is an extension of the G URSON criterion to the case of a prolate or oblate ellipsoidal RVE containing a cavity of the same form (Fig. 1). 1 M.
Gologanu, J.-B. Leblond and J. Devaux [3]
B
c
b A .
A . a
..
c
y
y
ω a
. ω
Ω x x
a) Oblate
Ω
b) Prolate
Figure 1: Geometry of the RVE considered in the GLD model. This geometry is completely defined by two dimensionless parameters: the porosity f = ab2 / AB 2 with its classical evolution law deduced from the matrix incompressibility, and the shape parameter S = ln W = ln (a/b) which evolves with respecting the following law [3]: 0 3 p p S˙ = h1 E˙ zz + 3h2 E˙ m (1) 2 with h1 and h2 are two parameters depending on the material microstructure (f, S) and the triaxiality T . The material is assumed to harden isotropically with dependence of the flow stress on the temperature: p σ ¯˙ = ~ε ε¯˙ + ~T T˙ (2)
where ~T = ∂ σ ¯ /∂ ε¯p is the plastic module at constant temperature and ~ε = ∂ σ ¯ /∂T , the thermal module with constant deformation. This thermomechanical coupling induce the introduction of two new internal variables in the expression of plasticity criterion witch reads [7]: p
Φ = Φ (Σ, f, S, ε¯ , T )
(3)
The accumulated plastic strain evolution is obtained by writing the equivalence between the microscopic and macroscopic plastic dissipation: Σ : E˙ p p (4) ε¯˙ = (1 − f )¯ σ The material plastic dissipation is assumed transformed to temperature increase following the relation: p ρCv T˙ = ξ¯ σ ε¯˙ (5) where ρ is the metal density, Cv the specific heat capacity and ξ the coefficient of Taylor-Quinney. Using equations (4) and (5) one writes: ξ Σ : E˙ p (6) T˙ = ρcv (1 − f )
2.2
Void coalescence
The cavities coalescence is assumed initiates when the intervoid ligament reaches its plastic limit-load, as suggested by Thomason. Benzerga [2] established a full elasto-plastic coalescence model based on this plastic limit-load analysis. To be in coherence with the void growth model presented in the last section, we extend this model to take into account the thermo-hardenable behavior of the material. Consequently, the coalescence criterion is rewrited as follow: Σeq 3 | Σm | 3 + − C (1 − χ2 ) (7) Φc (Σ, χ, W, ε¯p , T ) = σ ¯ 2 σ ¯ 2 1 3 −S 3 where χ = b/B = and λ = A/B = 2 f λe λ0 exp 32 Eeq are the relative ligament size and the relative void spacing variables respectively. The limit-load factor C = C (χ, W ) was phenomenologically modified and gauged on cells calculations predictions to avoid the divergence of the denominator term in the case of a very flattened cavities. The expressions of the coefficients appearing in the various equations described before are given in [3, 8, 7].
3
NUMERICAL ASPECTS
The model discussed above was implemented into ABAQUS Explicit solver through a VUMAT user material subroutine. This displacement-based finite element code allows the resolution of the mechanical equilibrium equation by both spatial and time discretization. The weak form of the principal of virtual work obtained by using the classical nodal approximation can be written: ¨ + Fint − Fext = 0 MU (8) where M is the consistent mass matrix. Fint and Fext are the internal and external forces vectors respectively. These quantities are the assemblies of the elementary ones calculated for each element: 2 The
Me
Z = Ve
Feint Feext
Z
T
= Ve
Z = V
ρ NT n Nn dV
e
(Ben ) Σe dV Z e NT F dV + n v
Γes
(9) e NT n Fs dΓ +
Z Γec
e NT n Fc dΓ
where Nn is the nodal interpolation functions of the element (e) and (Ben ) the elementary strain-displacement matrix defined in the current configuration. Fs et Fc are respectively the surface and contact forces vectors applied on the boundaries parts Γes and Γec of Γ. The resolution of equation (8) accordingly to the Dynamic Explicit scheme available in ABAQUS/Explicit is performed by resolving the motion equations witch begin by computing the ac¨ n = M−1 celeration vector : U n (Fint − Fext ). The computation of this quantity at any time tn needs the estimation of the stress tensor appearing in the expression of the internal force vector (9-b). This is made by solving the constitutive equations presented in section (2). The implementation of this model is realized by following the ”elastic predictor-plastic correction” algorithm proposed by Aravas [1]. However, the application of this algorithm in our case needs a modification of the elastic predictor expression Σpred to take for account the thermohypoelasticity form of the law describing the elastic domain: e Σpred n+1 = Σn + Λ : (∆E − ∆Eth ) − Σn .∆Ω − Σn .∆Ω (10) ˇ is the Jaumann co-rotational rate of the cauchy stress where Σ tensor, Λe is the fourth order tensor of the elastic moduli and α is the thermal dilation coefficient. Details on the implementation and the validation of the model with unit cell model predictions using an explicit calculation technique can be found in [7, 9].
4 4.1
APPLICATION TO A ROLLING OPERATION Description of the rolling operation
The sheet with initial dimensions : thickness e0 = 6 mm, width l0 = 80 mm and length L0 = 300 mm; is positioned between two rigid2 rollers with radius Ru = Rl = 105 mm and rotating with a constant rotational speed ωu = ωl = 3 rad/s. In this paper, the indexes u and l are used for upper and lower rolls respectively. The initial velocity of the sheet was assumed to be equal to the x-component of the roll surface velocity. The sheet material is a A1050P aluminum (E = 69 GP a and σ0 = 69 GP a) with isotropic hardening described by the law σ ¯ (¯ ε) = 130, 1 ε¯0,0372 . The temperature dependence of the flow stress is assumed linear according to the relation: σ ¯ (¯ ε, T ) = σ ¯ (¯ ε) [1 − β (T − T0 )] (11) where β is a material parameter and T0 = 25◦ C the initial sheet temperature. The thermals characteristics of the metal are Cv = 486 J/kgK and α = 1, 2 10−5 K −1 . In the absence of experimental data on the microstructure of the material, we have analyzed a backscattered electron image of a same material taken by Marui and al. [6]. We obtained the values f0 = 0, 862% and
rigidity assumption is reasonable if the stiffness and the yield strength of the rolls are significantly higher than those of the sheet.
S0 = 0, 853. The friction between the sheet and the two rollers is modeled by a combined Coulomb and sticking friction law with a friction coefficient µl = µu = 0, 14. 12 500 C3D8R elements with reduced integration are chosen to mesh the sheet.
4.2
on the reduction of the rolling force and the deflection of the piece at the exit. But less importance was given to the material microstructure evolution. In what follows we study the influence of the rolls radii and rolls speeds ratios on the final microstructure of four elements A, B, C and D (Fig. 4) which are supposed to represent their neighboring areas.
Results and discussion 240
200
D
A
B
C
160 Numerical Experiment
Figure 4: Position of the fourth elements within the sheet section.
120
80 8
12
16
20
24
28
Thickness Reductions
32
36
40
∆e (%) e0
Figure 2: Variation of the rolling force for different thickness reductions. 240
Fig. 5-8 represent the final normalized porosity f /f0 and shape parameter S/S0 for a thickness reduction ∆e/e0 = 17, 53%. As expected in the case of a symmetric rolling (Ru /Rl = 1 and ωl /ωu = 1), the areas (A and C) in contact with the two cylinders have the same final microstructure (f, S) because of the same deformation conditions.
200
0
8.33% 17.53% 26.94% 36.44%
ff
160
D
120
80 0.95
1
0.8
1
1.05
1.1
1.15
ω Roll Speed Ratio l ωu
1.2
0.6
A
B
C
0.4
Figure 3: Variation of the rolling force for different roll speed ration and thickness reductions. To asses the capability of the model to describe correctly the response of the material, we show in Fig. 2 comparison between the rolling force calculated using the model described in section 2 and the force measured experimentally [4]. Good agreements were found between experimental and numerical results for the forth thickness reductions used: ∆e/e0 = 8, 33%, 17, 53%, 26, 94% and 36, 44%. As an illustration, Fig. 3 shows the rolling force variations with the roll speed ratio ωl /ωu for the forth reductions. Globally, the rolling force decreases with the increase of the roll speed ratio. However in the case of a ∆e/e0 = 8, 33% reduction, the rolling force presents a less sensitivity to the velocity ratio. This can be explained by the small material quantity deformed in this situation. Several studies has shown that the quality of the rolling process depends on different parameters : roll radius ratio, roll speed ration, workpiece-roll friction coefficient ratio ... This studies have generally an interest on the influence of those parameters
1
1.1
1.2
1.3
Roll Radius Ratio R u
1.4
Rl
Figure 5: Variation of normalized porosity f /f0 with the roll radius ration Ru /Rl (∆e/e0 = 17, 53%).
In all simulations, the porosities in the center of sheet are lowest in comparison to those of the areas close to the rollers. In a sheet cross-section, the free side zone (D) contains the highest porosity. That can be explained by the traction loading in this area. The porosity increases with the two ratios Ru /Rl (1; 1, 24 and 1, 4) and ωl /ωu (1; 1, 1 and 1, 176). It appears on the diagrams that increasing the speed or radius of the roll has for consequence obtaining higher porosities in the zones which are in contact to him. It also causes more elongation in the rolling direction to those cavities
ff
0
5
A ductile fracture constitutive law extended to account for thermal heating under adiabatic condition is presented from both theoretical and numerical aspects. The model was confronted successfully with experimental results. This micromechanical approach offers the possibility to follow the material microstructure evolution during the deformation process. As preliminary results we can state:
D
0.8
C 0.6
B
A
0.4
1
1.05
1.1
1.15
Roll Speed Ratio
1.2
ωl ωu
Figure 6: Variation of normalized porosity f /f0 with the roll speed ration ωl /ωu (∆e/e0 = 17, 53%).
• The porosity increases with the roll radius and roll speed ratios, at the same time as the shape parameter decreases.
0
ss
Normalized Shape Parameter
• The rolling force increases with the thickness reduction and decreases with roll speed ratio.
• The increase of the roll velocity or the roll radius increases the porosity level and the cavity elongation in the rolling direction on the close sheet areas.
1
0.8
B
References
C A
[1] N. Aravas. On the numerical integration of a class of pressure-dependent plasticity models. International Journal for Numerical Methods in Engineering, 24:1395–1416, 1987.
D 0.6
0.4 1
1.1
1.2
1.3
[2] A.A. Benzerga. Micromechanics of coalescence in ductile fracture. Journal of the mechanics and physics of Solids, 50(6):1331–1362, 2002.
1.4
Roll Radius Ratio R u
Rl
Figure 7: Variation of normalized shape parameter S/S0 with the roll radius ration Ru /Rl (∆e/e0 = 17, 53%).
[5] W. Johnson and G. Needham. Further experiments in asymmetrical rolling. International Journal of Mechanical Sciences, 8:443, 1966.
0
0.9
ss
[3] Mihai Gologanu. Etude de quelques problèmes de rupture ductile des métaux. PhD thesis, Université de Paris VI, 1997. [4] Y.M. HWANG and G.Y.P. TZOU. Analytical and experimental study on asymmetrical sheet rolling. Int. J. Mech. Sci., 39 (3-6):289–303, 1997.
1
Normalized Shape Parameter
CONCLUSIONS
1
0.8
B A
[6] Etsuo Marui, Mitsuru Handa, Hiroki Endo, and Norihiko Hasegawa. In-plane deformation observation of indentation along indenter surface. Tribology International, 32:255– 263, 1999.
C
0.7
D 0.6
0.5 1
1.05
1.1
Roll Speed Ratio
1.15
ωl ωu
1.2
[7] M. Ould Ouali. Approche micromécanique de la rupture ductile dans les procédés de mise en forme des matériaux. Prise en compte de l’effet de forme des cavités. PhD thesis, Université de Reims Champagne-Ardenne, April 20, 2007.
Figure 8: Variation of normalized shape parameter S/S0 with the roll speed ration ωl /ωu (∆e/e0 = 17, 53%).
[8] T. Pardoen and J.W. Hutchinson. An extended model for void growth and coalescence. J. Mech. Phys. Solids, 48:2467–2512, 2000.
Globally, the cavities elongation in the rolling direction is accentuated with the increase of Ru /Rl and ωl /ωu . However, the shape parameter presents less sensitivity to these two ratios in the side zone (D).
[9] Larbi Siad, Mohand Ould Ouali, and Anouar Bennabes. Comparison of explicit and implicit finite element simulations of void growth and coalescence in porous ductile materials. Materials and Design., 29(2):319–329, 2008.
