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Surface roughening analysis of cold drawn tube based on macro–micro coupling finite element method
Journal of Materials Processing Technology 224 (2015) 189–199
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Journal of Materials Processing Technology journal homepage: www.elsevier.com/locate/jmatprotec
Surface roughening analysis of cold drawn tube based on macro–micro coupling finite element method Lei Zhang a , Wujiao Xu a,∗ , Jiang Long a , Zhenzhen Lei b a b
College of Material Science and Engineering, Chongqing University, Chongqing, 400044, China College of Mechanical and Power Engineering, Chongqing University of Science & Technology, Chongqing, 401331, China
a r t i c l e
i n f o
Article history: Received 5 December 2014 Received in revised form 4 May 2015 Accepted 9 May 2015 Available online 16 May 2015 Keywords: Surface roughness Micro-scale modeling Macro-micro coupling finite element method Micro-scale material property Model size determination
a b s t r a c t The aim of this paper is to research the inner surface roughness evolution of AA6061 rectangular tube in cold drawing process. The micro-scale finite element model with initial surface roughness and reasonable model size was created firstly. Then the micro-scale finite element analysis (FEA) of surface roughness evolution was conducted based on the macro-scale analysis of tube drawing process. The comparisons between simulation and experiment results indicate the validity of the macro–micro coupling FEA method. To improve the surface quality further, the influence of micro-scale material properties (e.g., grain size) on the surface roughness evolution in the plastic deformation was investigated. For simplification, the uniaxial tension model was adopted. The research results illustrate that small initial surface roughness, fine grains and lower Taylor factor may be beneficial to improve the surface quality after plastic deformation. © 2015 Elsevier B.V. All rights reserved.
1. Introduction The high quality AA6061 rectangular tube is widely used in the aviation, radar and aircraft for microwave transmission. Due to its specialty usage conditions, high dimensional accuracy and low inner surface roughness are required (shown in Fig. 1). Cold drawing process is a feasible technology that is used for manufacturing such high quality aluminum alloy rectangular tube (Lee et al., 2007). Currently, the requirement of high dimensional accuracy can be fulfilled, while the inner surface roughness is disqualified in production. It is hard to fulfill the inner surface requirement only by optimizing the drawing process and drawing dies. Therefore, it is imperative to investigate the surface roughness evolution in plastic deformation, and then it can be targeted for surface roughness optimization. The analyses of tube drawing process in the macro-scale have been studied experimentally and theoretically for many years. Many researchers found that the dimensional accuracy and drawing force were greatly affected by die structure. In tube drawing process for automotive steering input shaft, Kim et al. (2007) researched the process parameters related with tool configuration. An advanced mandrel shape was designed to avoid the fracture and the successfully formed parts were obtained. Yoshida and Furuya
∗ Corresponding author. Tel.: +86 13368186252; fax: +86 23 65111493. E-mail address: xuwujiao [email protected] (W. Xu). http://dx.doi.org/10.1016/j.jmatprotec.2015.05.009 0924-0136/© 2015 Elsevier B.V. All rights reserved.
(2004) investigated several types of tube drawing and proposed a ball-type plug. Sheu et al. (2014) used arc and Bezier curves to optimize the drawing die profile. Kuboki et al. (2008) investigated the effect of plug on leveling of residual stress in tube drawing, and the results showed that a certain reduction in thickness by a plug was effective for leveling of the residual stress. On the other hand, the influence of process parameters on tube drawing was also discussed by a lot of scholars. In the case of the irregularshaped drawing process, Kim and Kim, 2000 found that the main process variable on the corner filling was the semi-die angle but for the regular-shaped drawing process, the reduction in area had a significant effect. To enhance the dimensional accuracy of dieless tube-drawing process, Supriadi and Manabe (2013) used visionbased fuzzy controller to control the deformation according to the desired geometries by adjusting the drawing speed. Except for dimensional accuracy, the surface quality is becoming more and more important for some special used products now, and it is closely linked with the micro-scale material properties. Because of the complexity of material deformation in the tube drawing process, most researches were conducted under simple plastic deformation conditions, e.g., uniaxial tension. Zhao et al. (2004) studied the surface roughening behavior under uniaxial and biaxial loading conditions with the aid of a 3D finite-element crystal-plasticity model. The results showed that the ensuing surface profiles were controlled by several factors: applied boundary conditions, Taylor factor and shear tendency of the individual grains and the spatial distribution of grain neighborhood orientations.
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2. Crystal plasticity model 2.1. Kinematics The rate-dependent constitutive relies on the multiplicative decomposition of the total deformation gradient (F), which is given by (Asaro and Rice, 1977): F = F∗ × Fp
Fig. 1. Accuracy requirements on rectangular tube.
Groche et al. (2010) found that the influence of single grain on surface roughness evolution could be ignored when the number of grains inside the plastic zone was higher than 20. Stoudt et al. (2011a,b) proposed that surface roughness kept a linear relationship with uniaxial plastic strain for the finest grain size, and a quadratic model became more suitable for large grain size. In the work of Lee et al. (1998), some maps and correlation function were used to show the correlation of surface texture on orange peel in AA6022, and no clear relationship was identified between the two factors. Stoudt et al. (2011a,b) analyzed the relationships between grain orientation, deformation-induced surface roughness and strain localization. The largest surface displacements were observed to be concentrated at triple junctions where a large difference between the Taylor factors of the individual grains occurred. Multi-scale model is the link between macro-scale forming process and micro-scale material properties. Madej et al. (2013) used the multi-scale model taking into account microstructure morphology to investigate inhomogeneous strain distribution during cold rolling. Obtained results were then incorporated into the discrete cellular automata model of static recrystallization. Keshavarz and Ghosh (2013) built a multi-scale crystal plasticity finite element model to represent nickel-based superalloys. The multi-scale model contains two sub-models, one is a homogenized, activation energy-based crystal plasticity (AE-CP) model for single-crystal Ni-based superalloys that can be implemented in simulations of polycrystalline aggregates. The other is a sizedependent, dislocation-density-based finite element model of the subgrain scale representative volume element with explicit depiction of the –’ morphology, which is developed as a building block for homogenization. To investigate the growth of grains and subgrain dendrites during the molten pool solidification of AISI304 in laser keyhole welding processes, Tan and Shin (2015) used a macroscale dynamic model to predict the fluid flow and heat transfer in the keyhole as well as the molten pool. Then the generated thermal history was fed to 3D and 2D Cellular Automata model to predict the meso-scale grain growth and dendrite morphology. Takuya (2001) undertook a phase field model to clarify the multi-scale deformation mechanism of polycrystalline metals. And a tensile test was conducted to verify the relationship between macroscopic loading and microstructure. To investigate the inner surface roughness evolution of the high quality AA6061 rectangular tube during cold drawing process, the micro-scale analyses with 3D crystal-plasticity model were conducted based on the macro-scale FEA results. The macro-micro coupling analysis provided a new way to analysis the inner surface roughness evolution in drawing process and could also be applied to improve the surface quality of the formed part in other plastic deformations. After that, the effects of some micro-scale material properties on surface roughness evolution were discussed under uniaxial tension condition.
(1)
where, FP denotes plastic shear of the material to an intermediate reference configuration in which lattice orientation and spacing are the same as in the original reference configuration, and F* denotes stretching and rotation of the lattice. Elastic properties are assumed to be unaffected by slip, in the sense that stress is determined solely by F* . The rate of change of FP is related to the slipping rate ˙ (˛) of the ˛ slip system by: −1 F˙ p × F p =
˙ (˛) s(˛) m(˛)
(2)
˛
Where the sum ranges over all activated slip systems, unit vectors s(˛) and m(˛) are the slip direction and normal to slip plane in the reference configuration, respectively. The slip direction vector s*(˛) and the normal vector to the slip plane m*(˛) in the deformed lattice are given by:
s∗(˛) = F ∗ × s(˛)
(3)
−1
m∗(˛) = m(˛) × F ∗
The velocity gradient in the current state is L = F˙ × F −1 = D + ˝
(4)
Where the symmetric rate of stretching D and the antisymmetric spin tensor may be decomposed into lattice parts and plastic parts as follows: D = D∗ + Dp ,
˝ = ˝∗ + ˝p
(5)
2.2. Constitutive laws and hardening of rate-dependent crystalline materials The relation between the symmetric rate of stretching of the lattice, D* , and the Jaumann rate of Cauchy stress , ∇ * , is given by (Hill and Rice, 1972): ∇ ∗ + (I : D∗ ) = L : D∗
(6)
where, I is the second order identical tensor, and L is the 4 order tensor of elastic moduli. Based on the Schmid law, the slipping rate ˙ (˛) of the ˛ slip system in a rate-dependent crystalline solid is determined by the corresponding resolved shear stress (˛) as
⎧
(˛) n ⎪ ⎨ ˙ (˛) = ˙ (˛) sgn (˛) , for (˛) ≥ c(˛) 0
(˛)
c ⎪ ⎩ (˛) (˛) (˛) ˙
= 0,
for
(7)
< c
(˛)
where, sgn(x) is the signum function, ˙ 0 is the reference value of the shear strain rate and n is the rate sensitive exponent. Both (˛) (˛) ˙ 0 and n are the material parameters. c is the critical resolved shear stress of the slip system ˛. The strain hardening is characterized by the evolution of the (˛) strengths c through the incremental relation: (˛)
˙ c
=
ˇ
h˛ˇ ˙ (ˇ)
(8)
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Table 1 Material parameters for micro-scale model. Material parameter
Value C11 C12 C44
Elastic constants
108200 MPa 61300 MPa 28500 MPa 20 0.001 18.5 MPa 0.028 MPa 29.5 MPa 35.8 MPa
n 0 h0 hs 0 s Fig. 2. Stress–strain curves of macro and micro scale model.
hˇ˛ = qh˛˛ ,
/ ˛ ˇ=
(10)
where, q is a latent hardening parameter. h0 and hs are the initial hardening modulus and the hardening modulus during easy glide, respectively. 0 is the yield stress, and s is the breakthrough stress where large plastic flow initiates. is the shear strain on all slip systems. 0 is the amount of slip after which the interaction between slip systems reaches the peak strength, and each component f˛ˇ represents the magnitude of the strength of a particular slip interaction. For example, coplanar interactions tend to be weaker than non-coplanar ones, and the parameter values are according to the work of Kengo et al. (2014). The flow stress curves of the macro and micro scale model are shown in Fig. 2. The constitutive relationship for macro-scale model was obtained from the tensile test. Material parameters used for micro-scale model was determined by comparing the flow stress curves between the tensile test and the numerical data, and q = 0 (no latent hardening) was taken for simplicity (Bassani and Wu, 1991). Material parameters determined by trial and error are summarized in Table 1. Fig. 3. A 50-grain Voronoi tessellation (model size is 0.1 mm × 0.1 mm × 0.1 mm).
3. Micro-scale finite element modeling 3.1. Micro-scale geometric modeling
where, h˛ˇ are the slip hardening moduli, the sum ranges over all activated slip systems. Here h˛˛ (no sum on ˛) and h˛ˇ (˛ = / ˇ) are called self and latent hardening moduli, separately. The self and latent hardening moduli are given by (Bassani and Wu, 1991):
h˛˛ =
⎡ ⎣1 +
(h0 − hs ) sec
ˇ= / ˛
h2
(h0 − hs ) (˛) s − 0
f˛ˇ tan h (ˇ) /0
+ hs
·
⎤
⎦ , (no sum on ˛)
(9)
Geometric modeling of polycrystalline material is the precondition for micro-scale FEA. In this paper, polycrystalline morphology is described by Voronoi polyhedron. Being given a 3D spatial domain D ∈ R3 , a set of points E = {Gi (xi )} within D, and a norm d(•,•), every point Gi is associated a Voronoi polyhedron Ci as follows (Quey et al., 2011): Ci =
P(X) ∈ D|d(P, Gi ) < d(P, )Gj
∀j = / i
(11)
The norm d is taken as the Euclidean distance, and the set of points E is considered to be randomly distributed. In such a case,
Fig. 4. Surface topography and micro-scale finite element model. (a) Randomly generated surface topography with Sa 1.6 m; (b) Micro-scale finite element model with initial surface roughness (model size is 0.1 mm × 0.1 mm × 0.1 mm).
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Fig. 5. Metallographs of ND and TD surfaces.
Fig. 6. Schematic diagram of uniaxial tension model.
the tessellations are sometimes referred to as “Poisson–Voronoi tessellations”. From a physical point of view, the generation of Voronoi tessellations corresponds to a result of solidification or recrystallization where all grains nucleate at the same time and grow isotropically at the same rate. An example of 50-grain Voronoi tessellation is given in Fig. 3, and different grains are represented by different colors. The Voronoi tessellation in Fig. 3 can be seen as a micro-scale geometric model without initial surface roughness. 3.2. Surface roughness modeling The feasibility and significance of parameters in characterizing 3D surface topography are addressed by many researchers, and proposed four groups of parameters for 3D surface topography characterization based on statistics, the so-called Birmingham set of 3D parameters: amplitude, spatial, hybrid and functional (Manesh et al., 2010). The arithmetical mean height (Sa ) in amplitude parameter is chosen to evaluate the 3D surface topography in this work, which is expressed as (Stout et al., 1993):
1
E(xi , yj )
MN N
Sa =
M
j=1 i=1
(12)
Fig. 7. Statistics for the simulation results under different model size: (a) the curves of average surface roughness and standard deviation; (b) the curves of average mises stress and standard deviation; (c) the curves of average logarithmic strain and standard deviation.
where, M and N are sampling numbers in different directions, and E(x,y) is the residual surface, which is obtained by subtracting the least squares datum plane from the original surface. The surface roughness topography is assumed as a linear superposition of a series of normal distribution functions (Jiang et al., 2010). Then the surface topography can be described as follows:
Z=
n
2 2 Ai exp −[(x − i )2 /ix ] + (y − vi )2 /iy /2
(13)
i=1
where, A is the amplitude of normal distribution function; and are the summit coordinates; x and y are standard deviations that reflect the wave length or sharpness of the normal distribution function. When generating surface roughness topography, the following simplifications or assumptions are needed:
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Fig. 8. Sections change in the tube formation.
(1) The values of Ai , ix and iy in all the normal distribution functions are the same; (2) The summit coordinates (i , i ) of a normal distribution function are randomly generated within boundary limits. Fig. 4a is a randomly generated surface topography with Sa 1.6 m in 0.1 mm × 0.1 mm area. Combining the Voronoi tessellation model (e.g., Fig. 3) and surface topography (e.g., Fig. 4a), a micro-scale geometric model with initial surface roughness is obtained. Discretizing this combination model, a micro-scale FEA model with initial surface roughness is prepared (shown in Fig. 4b). 3.3. Model size determination A suitable sampling area is needed for surface roughness measurement. A small area cannot contain enough information of the surface topography; meanwhile, it will be inefficient to measure the surface roughness when the sampling area is too large. Therefore, a suitable sampling area, i.e., the size of micro-scale model,
should be determined firstly. In this section, a uniaxial tension of AA6061 sheet is taken as an example to show how to determine the suitable model size. The metallographs of ND surface (i.e., normal direction) and TD surface (i.e., transverse direction) are shown in Fig. 5. The grain sizes in RD (i.e., rolling direction), TD and ND are about 45 m, 45 m and 22 m, respectively. Fig. 6 is the schematic diagram of uniaxial tension model (without initial surface roughness). Axis X, Y and Z represent RD, TD and ND of the aluminum sheet, separately. The model stretches along X-axis and the displacement is 20% of the model size in X direction. To reduce the effects of grain numbers on analysis results, the model size ratio in three directions is approximately 1:1:0.5, which is in agreement with the grain size ratio. Surface Z1 (pointed out in Fig. 6) is used for roughness measurement. The numbers of grains under different model sizes are shown in Table 2. A theoretical experiment is developed in this paper. Firstly, we start with Model No. 1 (listed in Table 2). The orientations of grains are chosen stochastically, and the surface roughness value, Mises stress and logarithmic strain of the model are calculated
Fig. 9. Drawing dies and inner surface of the product.
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Fig. 10. Comparison of dimensional accuracy between simulation and experiment for (a) measuring points locations; (b) drawn tube height; (c) drawn tube width; (d) tube wall thickness.
after stretching. This process is repeated 50 times. Then the average values and the standard deviations are obtained. Afterwards this procedure will be continued until Model No. 15. Finally, the statistic results of each model are printed in Fig. 7. As seen in Fig. 7a, the standard deviation and the average value of surface roughness increase with the increasing of model size and finally become stabilized. When the model size is small, there are few grains in the surface, thus the average roughness and its distribution standard deviation are small. When the model size increases to a critical value, the model will contain enough grains. Therefore, the standard deviation and the average value of surface roughness come to convergence. According to Fig. 7, if the model size in RD is large than 0.45 mm, the standard deviation and the average surface roughness become stabilized approximately; meanwhile, the standard deviations and the average values of Mises stress and logarithmic stain come to convergence. So the suitable model size of this AA6061 sheet is decided as 0.45 × 0.45 × 0.225 mm (i.e., 1056 grains is included in the model). 4. Macro-micro coupling FEA in drawing process 4.1. Macro-scale FEA in drawing process The drawing process of the AA6061 rectangular tube consists of three passes, and the tube section change in each pass is illustrated in Fig. 8. Fig. 9 shows the drawing dies and inner surface of the product. In the first pass, drawing process is carried out without plug.
Fig. 11. Schematic diagram of multi-scale model.
For the second pass, linear type plug is adopted to transform the tube section to modified rectangular shape. To improve the contact condition between the tools and the drawn tube, the plug with lug bosses in the long side and the short side is used in the last pass to get the final product, which is indicated in Fig. 9f. The comparisons of tube dimensional accuracy between macroscale FEA and experiment are shown in Fig. 10. The measurement locations on the tube sides are illustrated in Fig. 10a. Comparison in tube height, tube width and tube thickness are shown in Fig. 10b–d. It can be seen that the drawn tube obtains eligible dimensional accuracy. The inner surface of the product is displayed in Fig. 9d, and Table 3 is the inner surface roughness at each point on side LT and SL shown in Fig. 10a (surface roughness is measured along the drawing direction). 4.2. Micro-scale FEA based on macro-scale analysis results Because of the large physical size discrepancy, surface roughness information and micro-scale material properties could not be integrated in the macro-scale model directly. Therefore micro-scale FEA model is used here to analyze the inner surface roughness evolution during drawing process, and the schematic diagram of multi-scale model is shown in Fig. 11. The macro-scale analysis was
Table 2 Numbers of grains under different model sizes (model size ratio in RD, TD and ND is 1:1:0.5). Model no.
Model size (RD)/mm
Grain numbers
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.70 0.80 0.90 1.00
12 39 93 181 313 497 741 1056 1448 1928 2502 3974 5932 8446 11,585
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Fig. 12. Comparisons of displacement, strain and stress distribution between micro-scale results and the corresponding macro-scale results in Location D. (a) and (b) are displacement along x-axis; (c) and (d) are the displacement along y-axis; (e) and (f) are the distribution of logarithmic strain; (g) and (h) are the distribution of Mises stress.
conducted firstly. Then the displacements of the concern locations on the macro-scale model were extracted. At last, the displacement boundary conditions were transferred to the micro-scale model. Micro-scale FEA are conducted at four typical locations named A, B, C and D, shown in Fig. 10a. Location A and D are the midpoint of short and long edges with lug boss; Location B and C are the rear
ends of short and long edges without boss. The results at location D are taken as example. The micro-scale results and the corresponding macro-scale results are illustrated in Fig. 12. As shown in Fig. 12a–d, the displacement distributions of micro-scale and macro-scale results are almost the same, which demonstrates that micro-scale analysis successfully inherits the
Table 3 Surface roughness at each point on side LT and SL (surface roughness is measured along the drawing direction). Location
1
C
2
3
4
5
D
6
7
8
9
Surface roughness (Ra /m)
1.423
1.393
1.340
1.401
0.912
0.723
0.705
0.754
0.431
0.841
1.326
Location Surface roughness (Ra /m)
I 0.401
II 0.597
III 0.263
A 0.125
IV 0.251
V 0.336
B 0.429
VI 0.430
10 1.262
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Fig. 13. Logarithmic strain nephograms under different initial surface roughness (initial surface roughness Sa0 is 0.0 m, 0.8 m, 1.6 m, and 3.2 m, respectively).
boundary conditions from macro-scale results. Since the microscale material properties (grain size, material orientations, etc.) are considered in micro-scale model, the stress and strain distribution will be balanced among the grains. It can be seen that the stress and strain distribution in micro-scale FEA is more uniform, in which no “layering” phenomenon exists like that in macro-scale model (shown in Fig. 12e–h). The inner surface roughness of the micro-scale FEA and experiment are shown in Table 4. The surface roughness in simulation is measured on Surface Y0 by 3D roughness parameter Sa . In experiment, the surface roughness is measured along drawing direction by 2D roughness parameter Ra . The surface roughness parameter Sa is to expand Ra from 2D to 3D, and it keeps an approximate linear relationship with Ra (Deleanu et al., 2012). As shown in Table 4, the inner surface roughness values at one specific location are different between experiment and simulation, while the tendency for both are the same. The reasons for that may be as follows:
(1) The difference between 2D parameter Ra for experiment and 3D parameter Sa for simulation; (2) Measurement error in the experiment; (3) Some simplifications in FEA simulation.
5. Influence of micro-scale material properties on surface roughness evolution The influence of micro-scale material properties, such as grain size and texture, on the surface roughness evolution is studied in this section. Due to the complexity of material deformation in the tube drawing process, it is difficult to distinguish the effect of one
Fig. 15. Relationship between grain size and surface roughness evolution under different uniaxial plastic strain.
Fig. 14. Surface roughness (Sa –Sa0 ) evolution under different initial surface roughness values (Sa is the surface roughness of the deformed part, Sa0 is the initial surface roughness).
Table 4 Comparison of inner surface roughness between the micro-scale FEA and experiment.
Location A Location B Location C Location D
Experiment (Ra /m)
Micro-scale FEA (Sa/ m)
0.125 0.429 1.393 0.705
0.314 0.441 1.603 0.994
Fig. 16. Relationship between surface roughness and uniaxial plastic strain with different textures (tension direction is RD).
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Fig. 17. The influence of Taylor factor on surface roughness evolution: (a) texture arrangement; (b) and (c) the surface topography after stretching along TD and RD; (d) and (e) the stress nephograms after stretching along TD and RD.
specific parameter on surface roughness evolution from the others. Thus the analysis is conducted under uniaxial tension condition. 5.1. Initial surface roughness and grain size Models with different initial surface roughness (the initial surface roughness Sa0 are 0.0 m, 0.8 m, 1.6 m and 3.2 m) and the same grain size and orientation (grain size in RD is 45 m; grain orientation is randomly assigned) are established to investigate the influence of initial surface roughness on roughness evolution. The logarithmic strain distribution under different initial surface roughness after uniaxial tension is shown in Fig. 13. All the strain nephograms shown in Fig. 13 are almost the same, which indicates that the initial surface roughness has little effect on the strain localization. Surface roughness topography of the deformed part is determined by the initial roughness and strain localization.
As shown in Fig. 14, if the initial surface is very flat (e.g., Sa0 is 0.0 m), the strain localization resulted from plastic strain plays a dominant role on the final surface roughness. Whereas if the initial surface is rough (e.g., Sa0 is 3.2 m), the initial surface roughness plays the leading role in roughness evolution. It can be concluded that in the uniform plastic deformation stage, with the increasing of initial surface roughness (Sa0 ), the effect of strain localization resulted from plastic strain on the final surface roughness (Sa ) is receding. The model with a little larger initial surface roughness may have a smaller surface roughness after plastic deformation than that with too small initial surface roughness. The uniformity of the grain deformation has a close relationship with the grain size. With the decreasing of the grain size, the strain difference between grain interior and grain boundary is reduced. Models with different grain size (grain size is 45 m, 100 m, 150 m and 200 m in RD, separately), the same grain
Fig. 18. {1 1 1}, {2 0 0}, {2 2 0} pole figures: (a) sample No. 1, (b) sample no.2 and no. 3.
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Table 5 Taylor factors calculated from full constraints Taylor theory (stretching along RD). Texture
Copper
S
Brass
Random
Goss
Cube
Taylor factor
3.7
3.33
3.17
3.07
2.45
2.45
Table 6 Material parameters in uniaxial tension experiment. Sample no.
Initial surface roughness values (Ra /m)
Grain size (TD × RD × ND/m)
Orientation
1 2 3
0.11 0.68 1.42
45 × 45 × 22 100 × 300 × 50 100 × 300 × 50
Shown in Fig. 18a Shown in Fig. 18b Shown in Fig. 18b
orientation and surface roughness (random orientation is assigned to models and without the initial surface roughness) are used to investigate the relationship between surface roughness evolution and grain size. A linear relationship between grain size and surface roughness is shown in Fig. 15. The larger grain size will result in a rougher surface roughness at the same deformation condition. To minimize the surface roughness after deformation, the initial roughness and grain size should be appropriately reduced. 5.2. Crystal orientation and texture Grains with different orientations under the same deformation conditions have distinct deformation abilities, which result in the strain localization and different surface topographies (Liao and Chen, 2010). Several common textures, such as Cube texture, Brass texture, Copper texture and Gross texture, are chosen in this analysis, the result of which is shown in Fig. 16. To some extent, Taylor factor can represent the deformation coordination of the material (Zhao et al., 2004). The Taylor factors of some textures under RD stretching are shown in Table 5. According to Fig. 16 and Table 5, it can be found that with the decreasing of Taylor factor, the surface roughness reduces as well. The model with Cube texture has the smallest surface roughness. A model with the mixture of Cube texture and Goss texture (seen Fig. 17a) is used to further analyze the influence of Taylor factor on surface roughness and surface topography. The red lines along grain boundaries in Fig. 17a are the boundaries between Cube texture and Goss texture. The ridging profile shown in Fig. 17b is obtained after tensioning along TD. However, the surface will remain flat when stretching along RD (shown in Fig. 17c). The reason is that Goss texture presents a large plastic anisotropy, i.e., its Taylor factor changes from 2.45 under RD stretching to 4.9 under TD stretching, whereas the Cube orientation preserves the same value of 2.45 under either loading direction (Zhao et al., 2004). The texture with smaller Taylor factor may have better deformation coordination, so plastic deformation mainly occurs at lower Taylor factor zone and stress localization mainly occurs around the texture with larger Taylor factor. 5.3. Uniaxial tension experiment Uniaxial tension experiment was conducted on CMT5000 series floor-standing electromechanical universal testing machine and the samples were stretched along RD. Some material parameters of the AA6061 sheet were shown in Table 6 and Fig. 18. During the tension test, the surface roughness Ra was measured along RD and TD by TR200 portable surface roughness tester at five randomly chosen points every 2% of the fiducial length increase in tension displacement. Then the average of the 10 data was taken as the final surface roughness Ra at that displacement. After measurement, the sample
Fig. 19. Surface roughness comparison of uniaxial tension experiment and simulation.
was reloaded to the next tension step. The results of simulation and experiment are illustrated in Fig. 19. As shown in Fig. 19, the decreasing of grain size can slow down the increasing of surface roughness, and the influence of uniaxial plastic strain is more significant in the model with a fine surface finish. The laws of surface roughness evolution during the uniaxial tension experiment are similar with that obtained by simulations in the previous sections. The surface roughness during plastic deformation is codetermined by grain size, initial surface roughness, textures and other factors. 6. Conclusion In this paper, the macro–micro coupling method is used to analyze the evolution of surface roughness in the drawing process. To improve the surface quality further, the impacts of some factors, such as initial surface roughness, grain size, on the surface roughness are subsequently researched under uniaxial tension. The main conclusions are summarized as follows: (1) The macro–micro coupling method can be used to predict the surface roughness evolution in the tube drawing process; (2) With the increasing of initial surface roughness, the effect of strain localization resulted from plastic strain on the surface roughness is receding; (3) A linear dependence of surface roughness on grain size is found, and the larger grain size may result in a rougher final surface under the same deformation condition; (4) Plastic deformation mainly occurs at lower Taylor factor zone, while stress localization mainly occurs around the texture with larger Taylor factor. Lower Taylor factor may be beneficial to improve the surface quality after plastic deformation. Acknowledgments The authors gratefully acknowledge the funding for this work from National Natural Science Foundation of China under (grant no. 51205427), Scientific and Technological Research Program of Chongqing Municipal Education Commission (grant no. KJ1401320) and Fundamental Research Funds for the Central Universities (grant no. CDJZR14130005). References Asaro, R.J., Rice, J.R., 1977. Strain localization in ductile single crystals. J. Mech. Phys. Solids 25, 309–338. Bassani, J.L., Wu, T.Y., 1991. Latent hardening in single crystals II. Analytical characterization and predictions. Proc.: Math. Phys. Sci. 435, 21–41. Deleanu, L., Georgescu, C., Suciu, C., 2012. A comparison between 2D and 3D surface parameters for evaluating the quality of surfaces. Ann. “Dunarea de Jos” Univ. Galati- Fascicle V 1, 5–12.
L. Zhang et al. / Journal of Materials Processing Technology 224 (2015) 189–199 Groche, P., Schafer, R., Justinger, H., Ludwig, M., 2010. On the correlation between crystallographic grain size and surface evolution in metal forming processes. Int. J. Mech. Sci. 52, 523–530. Hill, R., Rice, J.R., 1972. Constitutive analysis of elastic–plastic crystals at arbitrary strain. J. Mech. Phys. Solids 20, 401–413. Jiang, Z.Y., Tang, J., Sun, W., Tieu, A.K., Wei, D., 2010. Analysis of tribological feature of the oxide scale in hot strip rolling. Tribol. Int. 43, 1339–1345. Kengo, Y., Asato, I., Yuichi, T., 2014. Work-hardening behavior of polycrystalline aluminum alloy under multiaxial stress paths. Int. J. Plast. 53, 17–39. Keshavarz, S., Ghosh, S., 2013. Multi-scale crystal plasticity finite element model approach to modeling nickel-based superalloys. Acta Mater. 61, 6549–6561. Kim, S.W., Kwon, Y.N., Lee, Y.S., Lee, J.H., 2007. Design of mandrel in tube drawing process for automotive steering input shaft. J. Mater. Process. Technol. 187–188, 182–186. Kim, Y.C., Kim, B.M., 2000. A study on the corner filling in the drawing of a rectangular rod from a round bar. Int. J. Mach. Tool Manuf. 40, 2099–2117. Kuboki, T., Nishida, K., Sakaki, T., Murata, M., 2008. Effect of plug on levelling of residual stress in tube drawing. J. Mater. Process. Technol. 204, 162–168. Lee, P.S., Piehler, H.R., Adams, B.L., Jarvis, G., Hampel, H., Rollett, A.D., 1998. Influence of surface texture on orange peel in aluminum. J. Mater. Process. Technol. 80–81, 315–319. Lee, S.K., Ko, D.C., Kim, B.M., Lee, J.H., Kim, S.W., Lee, Y.S., 2007. A study on monobloc tube drawing for steering input shaft. J. Mater. Process. Technol. 191, 55–58. Liao, K.C., Chen, C.L., 2010. Investigation of surface roughness of aluminum alloy sheet based on crystalline plasticity model. Comput. Mater. Sci. 49, S47–S53. Madej, L., Sieradzki, L., Sitko, M., Perzynski, K., Radwanski, K., Kuziak, R., 2013. Multi scale cellular automata and finite element based model for cold deformation and annealing of a ferritic–pearlitic microstructure. Comput. Mater. Sci. 77, 172–181.
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Manesh, K.K., Ramamoorthy, B., Singaperumal, M., 2010. Numerical generation of anisotropic 3D non-Gaussian engineering surfaces with specified 3D surface roughness parameters. Wear 268, 1371–1379. Quey, R., Dawson, P.R., Barbe, F., 2011. Large-scale 3D random polycrystals for the finite element method: generation, meshing and remeshing. Comput. Methods Appl. Mech. Eng. 200, 1729–1745. Sheu, J.-J., Lin, S.-Y., Yu, C.-H., 2014. Optimum die design for single pass steel tube drawing with large strain deformation. Procedia Eng. 81, 688–693. Stout, K.J., Sullivan, P.J., Dong, W.P., Mainsah, E., Luo, N., Mathia, T., Zahouani, H., 1993. The Development of Methods for the Characterization of Roughness in Three Dimensions. Commission of the European Communities, Luxembourg. Stoudt, M.R., Hubbard, J.B., Leigh, S.D., 2011a. On the relationship between deformation-induced surface roughness and plastic strain in AA5052— is it really linear. Metall. Mater. Trans. A 42, 2668–2679. Stoudt, M.R., Levine, L.E., Creuziger, A., Hubbarda, J.B., 2011b. The fundamental relationships between grain orientation, deformation-induced surface roughness and strain localization in an aluminum alloy. Mater. Sci. Eng. A 530, 107–116. Supriadi, S., Manabe, K., 2013. Enhancement of dimensional accuracy of dieless tubedrawing process with vision-based fuzzy control. J. Mater. Process. Technol. 213, 905–912. Takuya, U., 2011. A phase field modelling for multi-scale deformation mechanics of polycrystalline metals. Procedia Eng. 10, 1779–1784. Tan, W., Shin, Y.C., 2015. Multi-scale modeling of solidification and microstructure development in laser keyhole welding process for austenitic stainless steel. Comput. Mater. Sci. 98, 446–458. Yoshida, K., Furuya, H., 2004. Mandrel drawing and plug drawing of shape memoryalloy fine tubes used in catheters and stents. J. Mater. Process. Technol. 153–154, 145–150. Zhao, Z., Radovitzky, R., Cuitino, A., 2004. A study of surface roughening in fcc metals using direct numerical simulation. Acta Mater. 52, 5791–5804.
Contents lists available at ScienceDirect
Journal of Materials Processing Technology journal homepage: www.elsevier.com/locate/jmatprotec
Surface roughening analysis of cold drawn tube based on macro–micro coupling finite element method Lei Zhang a , Wujiao Xu a,∗ , Jiang Long a , Zhenzhen Lei b a b
College of Material Science and Engineering, Chongqing University, Chongqing, 400044, China College of Mechanical and Power Engineering, Chongqing University of Science & Technology, Chongqing, 401331, China
a r t i c l e
i n f o
Article history: Received 5 December 2014 Received in revised form 4 May 2015 Accepted 9 May 2015 Available online 16 May 2015 Keywords: Surface roughness Micro-scale modeling Macro-micro coupling finite element method Micro-scale material property Model size determination
a b s t r a c t The aim of this paper is to research the inner surface roughness evolution of AA6061 rectangular tube in cold drawing process. The micro-scale finite element model with initial surface roughness and reasonable model size was created firstly. Then the micro-scale finite element analysis (FEA) of surface roughness evolution was conducted based on the macro-scale analysis of tube drawing process. The comparisons between simulation and experiment results indicate the validity of the macro–micro coupling FEA method. To improve the surface quality further, the influence of micro-scale material properties (e.g., grain size) on the surface roughness evolution in the plastic deformation was investigated. For simplification, the uniaxial tension model was adopted. The research results illustrate that small initial surface roughness, fine grains and lower Taylor factor may be beneficial to improve the surface quality after plastic deformation. © 2015 Elsevier B.V. All rights reserved.
1. Introduction The high quality AA6061 rectangular tube is widely used in the aviation, radar and aircraft for microwave transmission. Due to its specialty usage conditions, high dimensional accuracy and low inner surface roughness are required (shown in Fig. 1). Cold drawing process is a feasible technology that is used for manufacturing such high quality aluminum alloy rectangular tube (Lee et al., 2007). Currently, the requirement of high dimensional accuracy can be fulfilled, while the inner surface roughness is disqualified in production. It is hard to fulfill the inner surface requirement only by optimizing the drawing process and drawing dies. Therefore, it is imperative to investigate the surface roughness evolution in plastic deformation, and then it can be targeted for surface roughness optimization. The analyses of tube drawing process in the macro-scale have been studied experimentally and theoretically for many years. Many researchers found that the dimensional accuracy and drawing force were greatly affected by die structure. In tube drawing process for automotive steering input shaft, Kim et al. (2007) researched the process parameters related with tool configuration. An advanced mandrel shape was designed to avoid the fracture and the successfully formed parts were obtained. Yoshida and Furuya
∗ Corresponding author. Tel.: +86 13368186252; fax: +86 23 65111493. E-mail address: xuwujiao [email protected] (W. Xu). http://dx.doi.org/10.1016/j.jmatprotec.2015.05.009 0924-0136/© 2015 Elsevier B.V. All rights reserved.
(2004) investigated several types of tube drawing and proposed a ball-type plug. Sheu et al. (2014) used arc and Bezier curves to optimize the drawing die profile. Kuboki et al. (2008) investigated the effect of plug on leveling of residual stress in tube drawing, and the results showed that a certain reduction in thickness by a plug was effective for leveling of the residual stress. On the other hand, the influence of process parameters on tube drawing was also discussed by a lot of scholars. In the case of the irregularshaped drawing process, Kim and Kim, 2000 found that the main process variable on the corner filling was the semi-die angle but for the regular-shaped drawing process, the reduction in area had a significant effect. To enhance the dimensional accuracy of dieless tube-drawing process, Supriadi and Manabe (2013) used visionbased fuzzy controller to control the deformation according to the desired geometries by adjusting the drawing speed. Except for dimensional accuracy, the surface quality is becoming more and more important for some special used products now, and it is closely linked with the micro-scale material properties. Because of the complexity of material deformation in the tube drawing process, most researches were conducted under simple plastic deformation conditions, e.g., uniaxial tension. Zhao et al. (2004) studied the surface roughening behavior under uniaxial and biaxial loading conditions with the aid of a 3D finite-element crystal-plasticity model. The results showed that the ensuing surface profiles were controlled by several factors: applied boundary conditions, Taylor factor and shear tendency of the individual grains and the spatial distribution of grain neighborhood orientations.
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2. Crystal plasticity model 2.1. Kinematics The rate-dependent constitutive relies on the multiplicative decomposition of the total deformation gradient (F), which is given by (Asaro and Rice, 1977): F = F∗ × Fp
Fig. 1. Accuracy requirements on rectangular tube.
Groche et al. (2010) found that the influence of single grain on surface roughness evolution could be ignored when the number of grains inside the plastic zone was higher than 20. Stoudt et al. (2011a,b) proposed that surface roughness kept a linear relationship with uniaxial plastic strain for the finest grain size, and a quadratic model became more suitable for large grain size. In the work of Lee et al. (1998), some maps and correlation function were used to show the correlation of surface texture on orange peel in AA6022, and no clear relationship was identified between the two factors. Stoudt et al. (2011a,b) analyzed the relationships between grain orientation, deformation-induced surface roughness and strain localization. The largest surface displacements were observed to be concentrated at triple junctions where a large difference between the Taylor factors of the individual grains occurred. Multi-scale model is the link between macro-scale forming process and micro-scale material properties. Madej et al. (2013) used the multi-scale model taking into account microstructure morphology to investigate inhomogeneous strain distribution during cold rolling. Obtained results were then incorporated into the discrete cellular automata model of static recrystallization. Keshavarz and Ghosh (2013) built a multi-scale crystal plasticity finite element model to represent nickel-based superalloys. The multi-scale model contains two sub-models, one is a homogenized, activation energy-based crystal plasticity (AE-CP) model for single-crystal Ni-based superalloys that can be implemented in simulations of polycrystalline aggregates. The other is a sizedependent, dislocation-density-based finite element model of the subgrain scale representative volume element with explicit depiction of the –’ morphology, which is developed as a building block for homogenization. To investigate the growth of grains and subgrain dendrites during the molten pool solidification of AISI304 in laser keyhole welding processes, Tan and Shin (2015) used a macroscale dynamic model to predict the fluid flow and heat transfer in the keyhole as well as the molten pool. Then the generated thermal history was fed to 3D and 2D Cellular Automata model to predict the meso-scale grain growth and dendrite morphology. Takuya (2001) undertook a phase field model to clarify the multi-scale deformation mechanism of polycrystalline metals. And a tensile test was conducted to verify the relationship between macroscopic loading and microstructure. To investigate the inner surface roughness evolution of the high quality AA6061 rectangular tube during cold drawing process, the micro-scale analyses with 3D crystal-plasticity model were conducted based on the macro-scale FEA results. The macro-micro coupling analysis provided a new way to analysis the inner surface roughness evolution in drawing process and could also be applied to improve the surface quality of the formed part in other plastic deformations. After that, the effects of some micro-scale material properties on surface roughness evolution were discussed under uniaxial tension condition.
(1)
where, FP denotes plastic shear of the material to an intermediate reference configuration in which lattice orientation and spacing are the same as in the original reference configuration, and F* denotes stretching and rotation of the lattice. Elastic properties are assumed to be unaffected by slip, in the sense that stress is determined solely by F* . The rate of change of FP is related to the slipping rate ˙ (˛) of the ˛ slip system by: −1 F˙ p × F p =
˙ (˛) s(˛) m(˛)
(2)
˛
Where the sum ranges over all activated slip systems, unit vectors s(˛) and m(˛) are the slip direction and normal to slip plane in the reference configuration, respectively. The slip direction vector s*(˛) and the normal vector to the slip plane m*(˛) in the deformed lattice are given by:
s∗(˛) = F ∗ × s(˛)
(3)
−1
m∗(˛) = m(˛) × F ∗
The velocity gradient in the current state is L = F˙ × F −1 = D + ˝
(4)
Where the symmetric rate of stretching D and the antisymmetric spin tensor may be decomposed into lattice parts and plastic parts as follows: D = D∗ + Dp ,
˝ = ˝∗ + ˝p
(5)
2.2. Constitutive laws and hardening of rate-dependent crystalline materials The relation between the symmetric rate of stretching of the lattice, D* , and the Jaumann rate of Cauchy stress , ∇ * , is given by (Hill and Rice, 1972): ∇ ∗ + (I : D∗ ) = L : D∗
(6)
where, I is the second order identical tensor, and L is the 4 order tensor of elastic moduli. Based on the Schmid law, the slipping rate ˙ (˛) of the ˛ slip system in a rate-dependent crystalline solid is determined by the corresponding resolved shear stress (˛) as
⎧
(˛) n ⎪ ⎨ ˙ (˛) = ˙ (˛) sgn (˛) , for (˛) ≥ c(˛) 0
(˛)
c ⎪ ⎩ (˛) (˛) (˛) ˙
= 0,
for
(7)
< c
(˛)
where, sgn(x) is the signum function, ˙ 0 is the reference value of the shear strain rate and n is the rate sensitive exponent. Both (˛) (˛) ˙ 0 and n are the material parameters. c is the critical resolved shear stress of the slip system ˛. The strain hardening is characterized by the evolution of the (˛) strengths c through the incremental relation: (˛)
˙ c
=
ˇ
h˛ˇ ˙ (ˇ)
(8)
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Table 1 Material parameters for micro-scale model. Material parameter
Value C11 C12 C44
Elastic constants
108200 MPa 61300 MPa 28500 MPa 20 0.001 18.5 MPa 0.028 MPa 29.5 MPa 35.8 MPa
n 0 h0 hs 0 s Fig. 2. Stress–strain curves of macro and micro scale model.
hˇ˛ = qh˛˛ ,
/ ˛ ˇ=
(10)
where, q is a latent hardening parameter. h0 and hs are the initial hardening modulus and the hardening modulus during easy glide, respectively. 0 is the yield stress, and s is the breakthrough stress where large plastic flow initiates. is the shear strain on all slip systems. 0 is the amount of slip after which the interaction between slip systems reaches the peak strength, and each component f˛ˇ represents the magnitude of the strength of a particular slip interaction. For example, coplanar interactions tend to be weaker than non-coplanar ones, and the parameter values are according to the work of Kengo et al. (2014). The flow stress curves of the macro and micro scale model are shown in Fig. 2. The constitutive relationship for macro-scale model was obtained from the tensile test. Material parameters used for micro-scale model was determined by comparing the flow stress curves between the tensile test and the numerical data, and q = 0 (no latent hardening) was taken for simplicity (Bassani and Wu, 1991). Material parameters determined by trial and error are summarized in Table 1. Fig. 3. A 50-grain Voronoi tessellation (model size is 0.1 mm × 0.1 mm × 0.1 mm).
3. Micro-scale finite element modeling 3.1. Micro-scale geometric modeling
where, h˛ˇ are the slip hardening moduli, the sum ranges over all activated slip systems. Here h˛˛ (no sum on ˛) and h˛ˇ (˛ = / ˇ) are called self and latent hardening moduli, separately. The self and latent hardening moduli are given by (Bassani and Wu, 1991):
h˛˛ =
⎡ ⎣1 +
(h0 − hs ) sec
ˇ= / ˛
h2
(h0 − hs ) (˛) s − 0
f˛ˇ tan h (ˇ) /0
+ hs
·
⎤
⎦ , (no sum on ˛)
(9)
Geometric modeling of polycrystalline material is the precondition for micro-scale FEA. In this paper, polycrystalline morphology is described by Voronoi polyhedron. Being given a 3D spatial domain D ∈ R3 , a set of points E = {Gi (xi )} within D, and a norm d(•,•), every point Gi is associated a Voronoi polyhedron Ci as follows (Quey et al., 2011): Ci =
P(X) ∈ D|d(P, Gi ) < d(P, )Gj
∀j = / i
(11)
The norm d is taken as the Euclidean distance, and the set of points E is considered to be randomly distributed. In such a case,
Fig. 4. Surface topography and micro-scale finite element model. (a) Randomly generated surface topography with Sa 1.6 m; (b) Micro-scale finite element model with initial surface roughness (model size is 0.1 mm × 0.1 mm × 0.1 mm).
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Fig. 5. Metallographs of ND and TD surfaces.
Fig. 6. Schematic diagram of uniaxial tension model.
the tessellations are sometimes referred to as “Poisson–Voronoi tessellations”. From a physical point of view, the generation of Voronoi tessellations corresponds to a result of solidification or recrystallization where all grains nucleate at the same time and grow isotropically at the same rate. An example of 50-grain Voronoi tessellation is given in Fig. 3, and different grains are represented by different colors. The Voronoi tessellation in Fig. 3 can be seen as a micro-scale geometric model without initial surface roughness. 3.2. Surface roughness modeling The feasibility and significance of parameters in characterizing 3D surface topography are addressed by many researchers, and proposed four groups of parameters for 3D surface topography characterization based on statistics, the so-called Birmingham set of 3D parameters: amplitude, spatial, hybrid and functional (Manesh et al., 2010). The arithmetical mean height (Sa ) in amplitude parameter is chosen to evaluate the 3D surface topography in this work, which is expressed as (Stout et al., 1993):
1
E(xi , yj )
MN N
Sa =
M
j=1 i=1
(12)
Fig. 7. Statistics for the simulation results under different model size: (a) the curves of average surface roughness and standard deviation; (b) the curves of average mises stress and standard deviation; (c) the curves of average logarithmic strain and standard deviation.
where, M and N are sampling numbers in different directions, and E(x,y) is the residual surface, which is obtained by subtracting the least squares datum plane from the original surface. The surface roughness topography is assumed as a linear superposition of a series of normal distribution functions (Jiang et al., 2010). Then the surface topography can be described as follows:
Z=
n
2 2 Ai exp −[(x − i )2 /ix ] + (y − vi )2 /iy /2
(13)
i=1
where, A is the amplitude of normal distribution function; and are the summit coordinates; x and y are standard deviations that reflect the wave length or sharpness of the normal distribution function. When generating surface roughness topography, the following simplifications or assumptions are needed:
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Fig. 8. Sections change in the tube formation.
(1) The values of Ai , ix and iy in all the normal distribution functions are the same; (2) The summit coordinates (i , i ) of a normal distribution function are randomly generated within boundary limits. Fig. 4a is a randomly generated surface topography with Sa 1.6 m in 0.1 mm × 0.1 mm area. Combining the Voronoi tessellation model (e.g., Fig. 3) and surface topography (e.g., Fig. 4a), a micro-scale geometric model with initial surface roughness is obtained. Discretizing this combination model, a micro-scale FEA model with initial surface roughness is prepared (shown in Fig. 4b). 3.3. Model size determination A suitable sampling area is needed for surface roughness measurement. A small area cannot contain enough information of the surface topography; meanwhile, it will be inefficient to measure the surface roughness when the sampling area is too large. Therefore, a suitable sampling area, i.e., the size of micro-scale model,
should be determined firstly. In this section, a uniaxial tension of AA6061 sheet is taken as an example to show how to determine the suitable model size. The metallographs of ND surface (i.e., normal direction) and TD surface (i.e., transverse direction) are shown in Fig. 5. The grain sizes in RD (i.e., rolling direction), TD and ND are about 45 m, 45 m and 22 m, respectively. Fig. 6 is the schematic diagram of uniaxial tension model (without initial surface roughness). Axis X, Y and Z represent RD, TD and ND of the aluminum sheet, separately. The model stretches along X-axis and the displacement is 20% of the model size in X direction. To reduce the effects of grain numbers on analysis results, the model size ratio in three directions is approximately 1:1:0.5, which is in agreement with the grain size ratio. Surface Z1 (pointed out in Fig. 6) is used for roughness measurement. The numbers of grains under different model sizes are shown in Table 2. A theoretical experiment is developed in this paper. Firstly, we start with Model No. 1 (listed in Table 2). The orientations of grains are chosen stochastically, and the surface roughness value, Mises stress and logarithmic strain of the model are calculated
Fig. 9. Drawing dies and inner surface of the product.
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Fig. 10. Comparison of dimensional accuracy between simulation and experiment for (a) measuring points locations; (b) drawn tube height; (c) drawn tube width; (d) tube wall thickness.
after stretching. This process is repeated 50 times. Then the average values and the standard deviations are obtained. Afterwards this procedure will be continued until Model No. 15. Finally, the statistic results of each model are printed in Fig. 7. As seen in Fig. 7a, the standard deviation and the average value of surface roughness increase with the increasing of model size and finally become stabilized. When the model size is small, there are few grains in the surface, thus the average roughness and its distribution standard deviation are small. When the model size increases to a critical value, the model will contain enough grains. Therefore, the standard deviation and the average value of surface roughness come to convergence. According to Fig. 7, if the model size in RD is large than 0.45 mm, the standard deviation and the average surface roughness become stabilized approximately; meanwhile, the standard deviations and the average values of Mises stress and logarithmic stain come to convergence. So the suitable model size of this AA6061 sheet is decided as 0.45 × 0.45 × 0.225 mm (i.e., 1056 grains is included in the model). 4. Macro-micro coupling FEA in drawing process 4.1. Macro-scale FEA in drawing process The drawing process of the AA6061 rectangular tube consists of three passes, and the tube section change in each pass is illustrated in Fig. 8. Fig. 9 shows the drawing dies and inner surface of the product. In the first pass, drawing process is carried out without plug.
Fig. 11. Schematic diagram of multi-scale model.
For the second pass, linear type plug is adopted to transform the tube section to modified rectangular shape. To improve the contact condition between the tools and the drawn tube, the plug with lug bosses in the long side and the short side is used in the last pass to get the final product, which is indicated in Fig. 9f. The comparisons of tube dimensional accuracy between macroscale FEA and experiment are shown in Fig. 10. The measurement locations on the tube sides are illustrated in Fig. 10a. Comparison in tube height, tube width and tube thickness are shown in Fig. 10b–d. It can be seen that the drawn tube obtains eligible dimensional accuracy. The inner surface of the product is displayed in Fig. 9d, and Table 3 is the inner surface roughness at each point on side LT and SL shown in Fig. 10a (surface roughness is measured along the drawing direction). 4.2. Micro-scale FEA based on macro-scale analysis results Because of the large physical size discrepancy, surface roughness information and micro-scale material properties could not be integrated in the macro-scale model directly. Therefore micro-scale FEA model is used here to analyze the inner surface roughness evolution during drawing process, and the schematic diagram of multi-scale model is shown in Fig. 11. The macro-scale analysis was
Table 2 Numbers of grains under different model sizes (model size ratio in RD, TD and ND is 1:1:0.5). Model no.
Model size (RD)/mm
Grain numbers
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.70 0.80 0.90 1.00
12 39 93 181 313 497 741 1056 1448 1928 2502 3974 5932 8446 11,585
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Fig. 12. Comparisons of displacement, strain and stress distribution between micro-scale results and the corresponding macro-scale results in Location D. (a) and (b) are displacement along x-axis; (c) and (d) are the displacement along y-axis; (e) and (f) are the distribution of logarithmic strain; (g) and (h) are the distribution of Mises stress.
conducted firstly. Then the displacements of the concern locations on the macro-scale model were extracted. At last, the displacement boundary conditions were transferred to the micro-scale model. Micro-scale FEA are conducted at four typical locations named A, B, C and D, shown in Fig. 10a. Location A and D are the midpoint of short and long edges with lug boss; Location B and C are the rear
ends of short and long edges without boss. The results at location D are taken as example. The micro-scale results and the corresponding macro-scale results are illustrated in Fig. 12. As shown in Fig. 12a–d, the displacement distributions of micro-scale and macro-scale results are almost the same, which demonstrates that micro-scale analysis successfully inherits the
Table 3 Surface roughness at each point on side LT and SL (surface roughness is measured along the drawing direction). Location
1
C
2
3
4
5
D
6
7
8
9
Surface roughness (Ra /m)
1.423
1.393
1.340
1.401
0.912
0.723
0.705
0.754
0.431
0.841
1.326
Location Surface roughness (Ra /m)
I 0.401
II 0.597
III 0.263
A 0.125
IV 0.251
V 0.336
B 0.429
VI 0.430
10 1.262
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Fig. 13. Logarithmic strain nephograms under different initial surface roughness (initial surface roughness Sa0 is 0.0 m, 0.8 m, 1.6 m, and 3.2 m, respectively).
boundary conditions from macro-scale results. Since the microscale material properties (grain size, material orientations, etc.) are considered in micro-scale model, the stress and strain distribution will be balanced among the grains. It can be seen that the stress and strain distribution in micro-scale FEA is more uniform, in which no “layering” phenomenon exists like that in macro-scale model (shown in Fig. 12e–h). The inner surface roughness of the micro-scale FEA and experiment are shown in Table 4. The surface roughness in simulation is measured on Surface Y0 by 3D roughness parameter Sa . In experiment, the surface roughness is measured along drawing direction by 2D roughness parameter Ra . The surface roughness parameter Sa is to expand Ra from 2D to 3D, and it keeps an approximate linear relationship with Ra (Deleanu et al., 2012). As shown in Table 4, the inner surface roughness values at one specific location are different between experiment and simulation, while the tendency for both are the same. The reasons for that may be as follows:
(1) The difference between 2D parameter Ra for experiment and 3D parameter Sa for simulation; (2) Measurement error in the experiment; (3) Some simplifications in FEA simulation.
5. Influence of micro-scale material properties on surface roughness evolution The influence of micro-scale material properties, such as grain size and texture, on the surface roughness evolution is studied in this section. Due to the complexity of material deformation in the tube drawing process, it is difficult to distinguish the effect of one
Fig. 15. Relationship between grain size and surface roughness evolution under different uniaxial plastic strain.
Fig. 14. Surface roughness (Sa –Sa0 ) evolution under different initial surface roughness values (Sa is the surface roughness of the deformed part, Sa0 is the initial surface roughness).
Table 4 Comparison of inner surface roughness between the micro-scale FEA and experiment.
Location A Location B Location C Location D
Experiment (Ra /m)
Micro-scale FEA (Sa/ m)
0.125 0.429 1.393 0.705
0.314 0.441 1.603 0.994
Fig. 16. Relationship between surface roughness and uniaxial plastic strain with different textures (tension direction is RD).
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Fig. 17. The influence of Taylor factor on surface roughness evolution: (a) texture arrangement; (b) and (c) the surface topography after stretching along TD and RD; (d) and (e) the stress nephograms after stretching along TD and RD.
specific parameter on surface roughness evolution from the others. Thus the analysis is conducted under uniaxial tension condition. 5.1. Initial surface roughness and grain size Models with different initial surface roughness (the initial surface roughness Sa0 are 0.0 m, 0.8 m, 1.6 m and 3.2 m) and the same grain size and orientation (grain size in RD is 45 m; grain orientation is randomly assigned) are established to investigate the influence of initial surface roughness on roughness evolution. The logarithmic strain distribution under different initial surface roughness after uniaxial tension is shown in Fig. 13. All the strain nephograms shown in Fig. 13 are almost the same, which indicates that the initial surface roughness has little effect on the strain localization. Surface roughness topography of the deformed part is determined by the initial roughness and strain localization.
As shown in Fig. 14, if the initial surface is very flat (e.g., Sa0 is 0.0 m), the strain localization resulted from plastic strain plays a dominant role on the final surface roughness. Whereas if the initial surface is rough (e.g., Sa0 is 3.2 m), the initial surface roughness plays the leading role in roughness evolution. It can be concluded that in the uniform plastic deformation stage, with the increasing of initial surface roughness (Sa0 ), the effect of strain localization resulted from plastic strain on the final surface roughness (Sa ) is receding. The model with a little larger initial surface roughness may have a smaller surface roughness after plastic deformation than that with too small initial surface roughness. The uniformity of the grain deformation has a close relationship with the grain size. With the decreasing of the grain size, the strain difference between grain interior and grain boundary is reduced. Models with different grain size (grain size is 45 m, 100 m, 150 m and 200 m in RD, separately), the same grain
Fig. 18. {1 1 1}, {2 0 0}, {2 2 0} pole figures: (a) sample No. 1, (b) sample no.2 and no. 3.
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Table 5 Taylor factors calculated from full constraints Taylor theory (stretching along RD). Texture
Copper
S
Brass
Random
Goss
Cube
Taylor factor
3.7
3.33
3.17
3.07
2.45
2.45
Table 6 Material parameters in uniaxial tension experiment. Sample no.
Initial surface roughness values (Ra /m)
Grain size (TD × RD × ND/m)
Orientation
1 2 3
0.11 0.68 1.42
45 × 45 × 22 100 × 300 × 50 100 × 300 × 50
Shown in Fig. 18a Shown in Fig. 18b Shown in Fig. 18b
orientation and surface roughness (random orientation is assigned to models and without the initial surface roughness) are used to investigate the relationship between surface roughness evolution and grain size. A linear relationship between grain size and surface roughness is shown in Fig. 15. The larger grain size will result in a rougher surface roughness at the same deformation condition. To minimize the surface roughness after deformation, the initial roughness and grain size should be appropriately reduced. 5.2. Crystal orientation and texture Grains with different orientations under the same deformation conditions have distinct deformation abilities, which result in the strain localization and different surface topographies (Liao and Chen, 2010). Several common textures, such as Cube texture, Brass texture, Copper texture and Gross texture, are chosen in this analysis, the result of which is shown in Fig. 16. To some extent, Taylor factor can represent the deformation coordination of the material (Zhao et al., 2004). The Taylor factors of some textures under RD stretching are shown in Table 5. According to Fig. 16 and Table 5, it can be found that with the decreasing of Taylor factor, the surface roughness reduces as well. The model with Cube texture has the smallest surface roughness. A model with the mixture of Cube texture and Goss texture (seen Fig. 17a) is used to further analyze the influence of Taylor factor on surface roughness and surface topography. The red lines along grain boundaries in Fig. 17a are the boundaries between Cube texture and Goss texture. The ridging profile shown in Fig. 17b is obtained after tensioning along TD. However, the surface will remain flat when stretching along RD (shown in Fig. 17c). The reason is that Goss texture presents a large plastic anisotropy, i.e., its Taylor factor changes from 2.45 under RD stretching to 4.9 under TD stretching, whereas the Cube orientation preserves the same value of 2.45 under either loading direction (Zhao et al., 2004). The texture with smaller Taylor factor may have better deformation coordination, so plastic deformation mainly occurs at lower Taylor factor zone and stress localization mainly occurs around the texture with larger Taylor factor. 5.3. Uniaxial tension experiment Uniaxial tension experiment was conducted on CMT5000 series floor-standing electromechanical universal testing machine and the samples were stretched along RD. Some material parameters of the AA6061 sheet were shown in Table 6 and Fig. 18. During the tension test, the surface roughness Ra was measured along RD and TD by TR200 portable surface roughness tester at five randomly chosen points every 2% of the fiducial length increase in tension displacement. Then the average of the 10 data was taken as the final surface roughness Ra at that displacement. After measurement, the sample
Fig. 19. Surface roughness comparison of uniaxial tension experiment and simulation.
was reloaded to the next tension step. The results of simulation and experiment are illustrated in Fig. 19. As shown in Fig. 19, the decreasing of grain size can slow down the increasing of surface roughness, and the influence of uniaxial plastic strain is more significant in the model with a fine surface finish. The laws of surface roughness evolution during the uniaxial tension experiment are similar with that obtained by simulations in the previous sections. The surface roughness during plastic deformation is codetermined by grain size, initial surface roughness, textures and other factors. 6. Conclusion In this paper, the macro–micro coupling method is used to analyze the evolution of surface roughness in the drawing process. To improve the surface quality further, the impacts of some factors, such as initial surface roughness, grain size, on the surface roughness are subsequently researched under uniaxial tension. The main conclusions are summarized as follows: (1) The macro–micro coupling method can be used to predict the surface roughness evolution in the tube drawing process; (2) With the increasing of initial surface roughness, the effect of strain localization resulted from plastic strain on the surface roughness is receding; (3) A linear dependence of surface roughness on grain size is found, and the larger grain size may result in a rougher final surface under the same deformation condition; (4) Plastic deformation mainly occurs at lower Taylor factor zone, while stress localization mainly occurs around the texture with larger Taylor factor. Lower Taylor factor may be beneficial to improve the surface quality after plastic deformation. Acknowledgments The authors gratefully acknowledge the funding for this work from National Natural Science Foundation of China under (grant no. 51205427), Scientific and Technological Research Program of Chongqing Municipal Education Commission (grant no. KJ1401320) and Fundamental Research Funds for the Central Universities (grant no. CDJZR14130005). References Asaro, R.J., Rice, J.R., 1977. Strain localization in ductile single crystals. J. Mech. Phys. Solids 25, 309–338. Bassani, J.L., Wu, T.Y., 1991. Latent hardening in single crystals II. Analytical characterization and predictions. Proc.: Math. Phys. Sci. 435, 21–41. Deleanu, L., Georgescu, C., Suciu, C., 2012. A comparison between 2D and 3D surface parameters for evaluating the quality of surfaces. Ann. “Dunarea de Jos” Univ. Galati- Fascicle V 1, 5–12.
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