- Email: [email protected]
Study on surface asperity flattening during uniaxial planar compression
Wear 271 (2011) 1778–1784
Contents lists available at ScienceDirect
Wear journal homepage: www.elsevier.com/locate/wear
Study on surface asperity flattening during uniaxial planar compression H.J. Li a,b , Z.Y. Jiang a,∗ , D.B. Wei a , J.T. Han b , A.K. Tieu a a b
School of Mechanical, Materials and Mechatronic Engineering, University of Wollongong, Wollongong NSW 2522, Australia School of Materials Sciences and Engineering, University of Sciences and Technology Beijing, Beijing 100083, PR China
a r t i c l e
i n f o
Article history: Received 1 September 2010 Received in revised form 25 November 2010 Accepted 25 November 2010
Keywords: Surface asperity flattening Channel-die Uniaxial planar compression Time-integration procedure
a b s t r a c t In order to develop the relationship between surface roughness, friction and texture in the uniaxial planar compression, Al 6061 samples are compressed in a channel-die to conduct plane-strain compression experiments. Molly bond is selected as a lubricant in a group of samples. The other group of samples is compressed without lubrication. The tool is polished smoothly; all the samples are prepared with the same method to retain the same surface roughness. Finite element modelling of the surface asperity flattening is carried out with ABAQUS under the same experimental conditions, original random textures are employed in the two dimensional model. The same surface roughness (Ra ) obtained from the experiment is also employed on the top surface of the sample. The constitutive model and time-integration procedure have been implemented into the implicit finite element code ABAQUS using user material subroutine UMAT. Both the calculated results and the experimental results show a same tendency: the lubrication can constrain the process of surface asperity flattening. When the reduction is less than 10%, the effect of lubrication on surface asperity flattening is not significant. If the reduction is 10–40%, the lubrication plays a satisfactory role during the compression. With an increase of reduction, the lubrication will be damaged, and then the sample compressed with and without lubrication keeps a similar tendency of change. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Surface asperity is an important parameter of surface quality. It is also an interesting topic in cold metal manufacturing. In general, there are many factors which result in surface roughness variation, such as the original surface roughness of the product, grain size, crystal structure, crystal orientation, texture distribution, loading path, stress–strain state (deformation mode) and tool surface [1]. There are some reports in terms of the surface asperity flattening. Wilson et al. [2,3] have investigated the effect of bulk plasticity on asperity flattening when the lay of the roughness is parallel to the bulk straining direction (longitudinal roughness). They found that the rate of asperity flattening with bulk straining was related to the spacing and pressure of asperities. Makinouchi et al. [4] have presented some elastic–plastic finite element solutions for the case of transverse roughness. Sheu and Wilson [2] also found that a large increase of contact area with bulk strain and a reduction in the load needed for bulk yielding. Sutcliffe [5] tested and developed Wilson and Sheu’s theories. He discovered that the high pressure between contacting asperities and deformation of bulk material will affect the asperity deformation. However, there are
∗ Corresponding author. Tel.: +61 2 4221 4545; fax: +61 2 4221 5474. E-mail address: [email protected] (Z.Y. Jiang). 0043-1648/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.wear.2010.11.051
few reports on the interaction between surface asperity flattening, reduction and friction by the crystal plasticity theory. In order to figure out the relationship between sample surface asperity flattening, reduction and friction by the crystal plasticity theory, a finite element model is employed in the commercial finite element software ABAQUS to simulate the surface asperity flattening of Al plate along the rolling direction during uniaxial planar compression. A finite element polycrystal model subgrain scale is used in this study.
2. Crystal plasticity model The relationship between the single crystal and polycrystal is too complicated to be expressed directly. There are some assumptions given to simplify the relationship between the single crystal and polycrystal. The most widely used approach to obtain the response of a polycrystal from the response of individual crystals is to use the extended Taylor’s assumption of iso-deformation gradient in all of the crystals composing the polycrystal. Furthermore, if all grains are assumed to be of the same size, the Cauchy stress in the polycrystal can be taken as a simple number average of the Cauchy stresses in the various crystals [6]. On the basis of finite element theory, the finite element averaging procedure used the micro–macro link of the Taylor model to acquire the polycrystal plasticity response [7]. In these calculations of finite element averaging procedure, each grain is modelled by
H.J. Li et al. / Wear 271 (2011) 1778–1784
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is the time of deformation. U˙ is the displacement rate of sample in the compression.
3.3. Development of sample surface roughness during uniaxial planar compression
Fig. 1. Schematic of sample’s surface roughness.
one or more finite elements to allow for no-uniform deformations between the grains and within the grains, and both equilibrium and compatibility are satisfied in the weak finite element sense. Following the previous research, Kalidindi developed an implicit time-integration procedure [8,9]. In this paper, Kalidindi’s implicit time-integration procedure [9] is employed in finite element calculation to make the transition from the response of a single crystal (or a region within a grain) to response of a polycrystalline aggregate. Simultaneously, a group of relevant experiments is carried out to test the simulation results. 3. Experimental 3.1. Sample preparation The sample is polished by the auto-polishing machine along the transverse direction. All the samples are ground along transverse direction with same grinding time, pressure and sand papers (grade P220). A 2D profilemeter is used to measure the sample surface roughness. Before the compression, a central line is drawn on the top surface of the sample, and surface roughness is measured along this line. After the compression, the surface roughness is measured along the same line. This line can be drawn on any position along the transverse direction. Therefore, along the transverse direction, surface roughness at any position (or line) can be determined. The roughness errors between different samples are under a certain range (less than 0.05 m). After grinding, all the roughness of samples has a value of about 0.72 m (Ra ). When the compression takes place, the sample is constrained in the transverse direction. The sample’s roughness develops along the rolling direction. The compression test has been carried out by the Instron MTS. The sample size is 10 mm × 10 mm × 6 mm. The surface roughness of samples is shown in Fig. 1. 3.2. Sample parameters After mechanical polishing, the heights of samples are measured by a micrometer. All the parameters of samples for the compression are shown in Tables 1 (with lubrication) and 2 (without lubrication). The strain rate for the compression is 0.001 s−1 . However, for the Instron MTS, it is difficult to control the strain rate. The displacement rate is thus used to replace the strain rate. The relationship between the strain rate and displacement rate is as ε = ln
H , H0
t =
ε , ε˙
U˙ =
H , t
H0 = H + H
(1)
where H0 , H, H are the original height, the height after compression and the height increment, respectively. ε is the true strain. t
It is obvious from experimental results (Table 3) that, with an increase of reduction, the surface roughness of both samples (compression with and without lubrication) decreases significantly. When the reduction is less than 5%, the two samples retain the same surface roughness. When the reduction exceeds 10%, there is an obvious difference of the surface roughness. Samples without lubrication during the compression have a larger decreasing rate of surface roughness. Finally, samples with lubrication during the compression have a comparatively rougher surface than that of samples without lubrication. From the experimental results, it is shown that the lubrication can constrain the surface asperity flattening process. In other words, friction can increase the process of surface asperity flattening.
3.4. Relationship between the surface roughness and friction Firstly, during the compression, when the reduction is less than 5%, the surface roughness of sample decreases greatly from 0.7 m to 0.4 m (Fig. 2). In this stage, the rough asperity does not contact with the compressing tool completely. This means the lubrication has no obvious influence on the surface asperity flattening in this stage. Secondly, if the reduction is between 5% and 30%, the sample surface roughness will continue to decrease, but the decreasing rate of the surface roughness is much lower than that in the first stage. In this stage, a stable lubrication layer will be formed. It will limit the surface asperity flattening process. Therefore, sample surface roughness will decrease slowly. During this stage, the lubrication layer plays an important role on the surface asperity flattening. Thirdly, if the reduction exceeds 30%, sample surface roughness will decrease quickly (from about 0.3 to 0.15 m). In this stage, with an increase of reduction, the lubrication layer is destroyed by the compression. It will not affect the compression of rough surface. The surface asperity flattening will thus proceed quickly again. Finally, if the reduction exceeds 40%, the two samples will keep a certain surface, and the sample with lubrication has a much rougher surface than that of the sample without lubrication. The reason for this is that under the lubrication condition, a lubricant layer will be formed between the tool and the sample. With the lubrication of this layer, the sample will receive less constraint from the tool. Therefore, the sample can be deformed easily. Furthermore, the slip system in the sample can also be activated easily. When the reduction exceeds a certain value (about 40%), the lubrication layer will be destroyed, and the compressing tool will contact with the sample directly. However, owing to the influence of the lubrication layer, the sample with lubrication has a rougher surface than that without lubrication. On the other hand, an abnormal phenomenon is observed from the experimental results. When the reduction exceeds 60% in the sample without lubrication, there is a slight increase of surface roughness (about 0.03 m). There are three factors which may cause this phenomenon: firstly, the measuring error can cause this phenomenon; secondly, the friction and the tool surface influence can also lead to this phenomenon; thirdly, when the reduction reaches 60%, the aluminium sample cannot be further deformed easily.
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Table 1 Parameters for compression (with lubrication). Number
Cross section (mm × mm)
Height (mm)
Displacement rate (mm/min)
Strain rate (s−1 )
Reduction (%)
Reduction (mm)
Height (mm)
1 2 3 4 5
10.042 × 10.178 10.042 × 10.041 9.985 × 10.031 10.109 × 10.029 10.479 × 10.035
6.102 6.129 6.139 6.109 6.131
0.357 0.349 0.330 0.308 0.288
0.001 0.001 0.001 0.001 0.001
5 10 20 30 40
0.305 0.613 1.228 1.833 2.452
5.797 5.516 4.911 4.276 3.679
Table 2 Parameters for compression (without lubrication). Number
Cross section (mm × mm)
Height (mm)
Displacement rate (mm/min)
Strain rate (s−1 )
Reduction (%)
Reduction (mm)
Height (mm)
1 2 3 4 5
9.534 × 9.599 9.577 × 9.401 9.595 × 9.542 9.666 × 9.408 9.578 × 9.497
6.123 6.108 6.095 6.127 6.092
0.358 0.348 0.33 0.2407 0.287
0.001 0.001 0.001 0.001 0.001
5 10 20 30 40
0.306 0.6108 1.219 1.838 2.4368
5.817 5.497 4.876 4.289 3.655
Average surface roughness Ra, μm
0.8 0.7 No lubricant With lubricant
0.6 0.5 0.4 0.3 0.2 0.1 0.0 0
10
20
30
40
50
60
Reduction, %
(a) Ra
Average surface roughness Rq, μm
1.1 1.0 No lubricant With lubricant
0.9 0.8 0.7 0.6
pression. The model geometry is two dimensional. Although a 2D finite element analysis has the advantage of less computational time, the main reason for not using a 3D finite element analysis in this study is actually, that during the compression test, the transverse direction is constrained by the mould (Fig. 1), therefore there is almost no deformation of the workpiece in this direction. On the other hand, the available generalized plane strain elements in ABAQUS make it possible to impose a uniform strain on the body in the out-of-plane direction. Furthermore, as mentioned by Becker [10] and Wu and Lloyd [11], the 2D FE still allows us to simulate both the uniaxial and the biaxial deformation. For all the simulations, the 1-direction (Fig. 3) corresponds to the prior rolling direction, the 2-direction to the sample normal and the 3-direction to the transverse direction of the samples. The sample material is 6061T5 aluminium alloy. Size of the two-dimensional model is 500 m × 500 m. The initial length is denoted by L0 and the initial height by H0 . The original roughness parameters of the samples are Ra 0.72 m and Rq 0.98 m. Longer wavelength and shorter wavelength are employed to form the surface roughness. Longer wavelength (Rq 1.06 m, Ra 0.79 m) contributes about 60% to the initial surface roughness and shorter wavelength (Rq 0.82 m, Ra 0.61 m) only accounts for about 40% in the initial surface roughness. Becker’s method [10] is used to calculate the average roughness obtained from the finite element results and the experimental ones. The surface roughness of the model is determined by the experimental results. A line is drawn through the surface nodal coordinate data, therefore, the distance from this line to the surface nodes is ı [10] and
0.5
L
ı dx = 0
0.4
(2)
0
0.3
Though there are a few differences between the 2D model surface profile and sample profile, they all have the same initial roughness Ra . The model surface roughness is determined by the sample surface roughness. The average roughness, Ra , is determined from
0.2 0.1 0.0 0
10
20
30
40
50
60
Reduction, %
(b) Rq Fig. 2. Relationship between the surface roughness and friction.
4. Modelling 4.1. Model This study aims to analyse the relationship between reduction, friction and surface roughness during uniaxial plane strain com-
Ra =
1 L
l
ı dx
(3)
0
The reduction of sample ranges from 5% to 40%. Contact friction coefficient between the sample and rigid compressing tool and mould ranges from 0.001 to 0.35. The reduction is applied on the top of the sample by the rigid compressing tool. Due to symmetry, all the nodes on edge ab (Fig. 3) have no displacements in direction 1. In order to understand the relationship between surface roughness, friction and reduction by the crystal plasticity theory, a finite element polycrystal model is employed in this study. The two dimen-
H.J. Li et al. / Wear 271 (2011) 1778–1784 Table 3 Development of surface roughness under uniaxial planar compression. Reduction (%) (a) With lubrication
0
5
10
20
30
40 (b) Without lubrication
0
5
10
20
Roughness (with lubrication)
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Table 3 (Continued) Reduction (%)
Roughness (with lubrication)
30
40
sional model has 902 CPE4R reduced integration elements, and one grain set with one element. The rigid tool and mould both have 20 discrete rigid elements. Kalidindi’s method [12] is used to incorporate crystal plasticity into FEM. The constitutive model and timeintegration procedures are implemented into the implicit finite element code ABAQUS by employing the user material subroutine UMAT. The combinations of slip systems are taken into account during modelling, including 12 {1 1 0} 1 1 1. It is assumed that the shearing rate is equal on each slip system. 902 random Euler angle triplets are input into ABAQUS as the initial crystallographic condition of the model. The components of the elasticity tensor are taken as C11 = 106,750 MPa, C12 = 60,410 MPa, C44 = 28,340 MPa [13]. As hardening matrix parameters we use q˛ˇ = 1.0 for coplanar systems and q˛ˇ = 1.4 for non-coplanar slip systems [14]. The other material parameters are according to Ref. [15]. 4.2. Relationship between surface roughness, friction and reduction From the modelling results, it is obvious that with an increase of reduction, the sample surface roughness decreases quickly. In other words, the surface asperity flattening takes place quickly, and the compression with lubrication has an obvious difference from that without lubrication. With an increase of friction coefficient, the surface asperity flattening rate of sample will increase significantly. Under the condition of larger friction, the sample will obtain a lower surface roughness due to compression. The reason for this phenomenon is that the friction between the sample and the compressing tool is expressed by the penalty function
Fig. 3. Two dimensional model and mesh.
as a constraint to the sample. Therefore, during the compression the friction will block the sample deformation. It will restrain the slip system activation and action. On the other hand, the practical lubricant layer lies between the sample and tool which acts as a transfer layer to block the surface asperity flattening process. It can obviously reduce the surface asperity flattening process. Here, in this simulation, friction is only set as a coefficient. The sample will thus come into contact with the tool directly; therefore, the sample will be affected by the tool directly. Because the compression tool is smooth in this study, the sample surface will be flattened quickly. With an increase of friction coefficient, the influence of the tool on the sample will result in a speeding-up of the surface asperity flattening process. After the compression, the strain localization area will appear in the samples’ surface. This will result in a transfer layer in the sample surface. This transfer layer will play a direct role in changing the surface roughness. Other neighbouring layers will affect the surface roughness by this transfer layer. From the simulation, we observed the obvious change of crystals’ orientations in this layer. This will be further focused in the study (see Fig. 4).
4.3. Comparison of the experimental and modelling results It is shown from Fig. 5 that the experimental results are close to the simulated results. Both results show that with an increase of reduction, the surface roughness of sample decreases quickly, and lubrication can delay the process of surface asperity flattening. On the other hand, there are some differences between the experimental and simulated results. The developing tendency of surface roughness in the two results is different: the simulated result has a higher decreasing rate of surface roughness than that of the experimental results. In Fig. 5(a), when the reduction is less than 10%, the two results keep the same tendency of surface asperity flattening. On the other hand, when the reduction exceeds 10%, the decreasing rate of surface roughness from simulation is a bit larger than that from the experiment. In the experiment, when the reduction exceeds 10%, the lubrication layer can play a remarkable role in delaying surface asperity flattening. However, in the simulation, the penalty friction coefficient is used to replace the lubricant layer. There is an obvious difference between the simulation results and experimental ones. The error may result from the computational error and the frictional model error. From the calculation, the maximum error between the experimental results and the simulation results is about 0.08 m. When the maximum error appears, the sample roughness is more than 0.25 m (Fig. 5(a) and (b)). This error is about 25% of sample roughness. In the finite element simulation, the error may be acceptable. When the reduction is between 10% and 40%, the lubrication layer can perform properly. The difference between the simulated and experimental results is most obvious. When the reduction exceeds
H.J. Li et al. / Wear 271 (2011) 1778–1784
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0.8 0.7
0.6
Surface roughness, μm
Surface roughness Ra, μm
0.7
With lubricant Without lubricant
0.5 0.4 0.3 0.2
Experiment Simulation
0.6 0.5 0.4 0.3 0.2
0.1
0.1
0.0 0
10
20
30
40
50
60
0
Reduction, %
10
30
40
50
60
50
60
Reduction, %
(a) Ra
(a)
With lubrication
0.8
0.9
0.7
0.8 With lubricant Without lubricant
0.7
Surface roughness, μm
Surface roughness, Rq
20
0.6 0.5 0.4 0.3 0.2
Experiment Simulation
0.6 0.5 0.4 0.3 0.2 0.1
0.1 0.0 0
10
20
30
40
50
0
60
10
(b) Rq
It is shown in Fig. 6 that with an increase of reduction, the influence of the shorter wavelength on the sample roughness (Ra , Rq ) will increase during the compression, and the influence of the longer wavelength on the sample roughness (Ra , Rq ) decreases. In this model, longer wavelength has a larger initial roughness (Ra , Rq ), and shorter wavelength has a smaller initial roughness (Ra , Rq ). During the compression, the compressing tool will first contact the longer wavelength, and the roughness of longer wavelength will decrease quickly. Furthermore, at the beginning, under very small reduction (less than 5%), the longer wavelength will affect the sample roughness mainly, while shorter wavelength has no obvious influence on the sample roughness. With an increase of reduction, the compressing tool will contact with the shorter wavelength. The shorter wavelength will thus play a main role on the sample rough-
40
Fig. 5. Comparison of experimental and modelling results.
90
Percentage in the total roughness, %
4.4. Influence of longer wavelength and shorter wavelength on the surface roughness
30
(b) Without lubrication
Fig. 4. Relationship between surface roughness, friction and reduction.
40%, the lubrication layer will be damaged and cannot continue in constraining surface asperity flattening. In Fig. 5(b), if the reduction exceeds 60%, the roughness of experimental results will increase by about 0.03 m. The reasons for this phenomenon are the same as those in Fig. 2.
20
Reduction, %
Reduction, %
80 70 60
Longer wavelength Shorter wavelength
50 40 30 20 10
0
5
10
15
20
25
Reduction, % Fig. 6. Influence of longer wavelength and shorter wavelength on the sample roughness.
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H.J. Li et al. / Wear 271 (2011) 1778–1784
ness while the influence of longer wavelength on the sample can be neglected. 5. Conclusions (1) Within a certain reduction range (in this case, it is between 10% and 40%), the lubrication can limit the surface asperity flattening significantly. If the reduction is less than 10%, the lubrication layer has no obvious influence on sample surface roughness. When the reduction exceeds 40%, the lubrication layer will be damaged during the uniaxial planar compression. (2) Both the experimental and the simulation results show the same tendency: with an increase of reduction, the surface asperity flattening proceeds, and sample surface roughness will decrease greatly, and the lubrication can constrain the surface asperity flattening process. (3) In the simulation, with an increase of reduction, the influence of longer wavelength on the sample roughness will decrease quickly, while the effect of the shorter wavelength of the surface asperity on sample roughness will increase significantly. Acknowledgements The first author would like to thank UOW for Ph.D. scholarship support for current research. We also appreciate Ms. Alisa Percy for her proof reading of the paper. References [1] D. Raabe, M. Sachtleber, H. Weiland, G. Scheele, Z.S. Zhao, Grain-scale micromechanics of polycrystal surfaces during plastic straining, Acta Materialia 51 (2003) 1539–1560.
[2] S. Sheu, W.R.D. Wilson, Flattening of workpiece surface asperities in metal forming, in: Proceedings of the 11th NAMRC, 1983, pp. 172–178. [3] W.R.D. Wilson, W.M. Lee, Mechanics of surface roughening in metal forming process, Journal of Manufacturing Science and Engineering 123 (2001) 279–283. [4] A. Makinouchi, H. Ike, M. Murakawa, N. Koga, A finite element analysis of flattening of surface asperities by perfectly lubricated rigid dies in metal working processes, Wear 128 (1988) 109–122. [5] M.P.F. Sutcliffe, Surface asperity deformation in metal forming processes, International Journal of Mechanical Sciences 11 (1988) 847–868. [6] S.R. Kalidindi, S.E. Schoenfeld, On the prediction of yield surfaces by the crystal plasticity models for fcc polycrystals, Materials Science and Engineering A 293 (2000) 120–129. [7] A.J. Beaudoin, P.R. Dawson, K.K. Mathur, U.F. Kocks, A hybrid finite element formulation for polycrystal plasticity with consideration of macrostructural and microstructural linking, International Journal of Plasticity 11 (1995) 501–521. [8] S.R. Kalidindi, L. Anand, Macroscopic shape change and evolution of crystallographic texture in pre-textured fcc metals, Journal of the Mechanics and Physics of Solids 42 (1994) 459–490. [9] S.R. Kalidindi, L. Anand, An approximate procedure for predicting the evolution of crystallographic texture in bulk deformation processing of FCC metals, International Journal of Mechanical Sciences 34 (1992) 310–312. [10] R. Becker, Effect of strain localization on surface roughening during sheet forming, Acta Metallurgica 46 (1998) 1385–1401. [11] P.D. Wu, D.J. Lloyd, Modelling surface roughening with crystal plasticity, in: Proceedings of the 1st International Symposium on Metallurgical Modeling for Aluminum alloys, 2003, pp. 15–20. [12] S.R. Kalidindi, C.A. Bronkhorst, L. Anand, Crystallographic texture evolution in bulk deformation processing of FCC metals, Journal of the Mechanics and Physics of Solids 40 (1992) 552–555. [13] E.A. Brandes, Smithells Metals Reference Book, The Bath Press, Bath, 1999, p. 368. [14] R.J. Asaro, A. Needleman, Texture development and strain hardening in rate dependent polycrystals, Acta Metallurgica 33 (1985) 923–953. [15] D. Raabe, Y. Wang, F. Roters, Crystal plasticity simulation study on the influence of texture on earring in steel, Computational Materials Science 34 (2005) 221–234.
Contents lists available at ScienceDirect
Wear journal homepage: www.elsevier.com/locate/wear
Study on surface asperity flattening during uniaxial planar compression H.J. Li a,b , Z.Y. Jiang a,∗ , D.B. Wei a , J.T. Han b , A.K. Tieu a a b
School of Mechanical, Materials and Mechatronic Engineering, University of Wollongong, Wollongong NSW 2522, Australia School of Materials Sciences and Engineering, University of Sciences and Technology Beijing, Beijing 100083, PR China
a r t i c l e
i n f o
Article history: Received 1 September 2010 Received in revised form 25 November 2010 Accepted 25 November 2010
Keywords: Surface asperity flattening Channel-die Uniaxial planar compression Time-integration procedure
a b s t r a c t In order to develop the relationship between surface roughness, friction and texture in the uniaxial planar compression, Al 6061 samples are compressed in a channel-die to conduct plane-strain compression experiments. Molly bond is selected as a lubricant in a group of samples. The other group of samples is compressed without lubrication. The tool is polished smoothly; all the samples are prepared with the same method to retain the same surface roughness. Finite element modelling of the surface asperity flattening is carried out with ABAQUS under the same experimental conditions, original random textures are employed in the two dimensional model. The same surface roughness (Ra ) obtained from the experiment is also employed on the top surface of the sample. The constitutive model and time-integration procedure have been implemented into the implicit finite element code ABAQUS using user material subroutine UMAT. Both the calculated results and the experimental results show a same tendency: the lubrication can constrain the process of surface asperity flattening. When the reduction is less than 10%, the effect of lubrication on surface asperity flattening is not significant. If the reduction is 10–40%, the lubrication plays a satisfactory role during the compression. With an increase of reduction, the lubrication will be damaged, and then the sample compressed with and without lubrication keeps a similar tendency of change. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Surface asperity is an important parameter of surface quality. It is also an interesting topic in cold metal manufacturing. In general, there are many factors which result in surface roughness variation, such as the original surface roughness of the product, grain size, crystal structure, crystal orientation, texture distribution, loading path, stress–strain state (deformation mode) and tool surface [1]. There are some reports in terms of the surface asperity flattening. Wilson et al. [2,3] have investigated the effect of bulk plasticity on asperity flattening when the lay of the roughness is parallel to the bulk straining direction (longitudinal roughness). They found that the rate of asperity flattening with bulk straining was related to the spacing and pressure of asperities. Makinouchi et al. [4] have presented some elastic–plastic finite element solutions for the case of transverse roughness. Sheu and Wilson [2] also found that a large increase of contact area with bulk strain and a reduction in the load needed for bulk yielding. Sutcliffe [5] tested and developed Wilson and Sheu’s theories. He discovered that the high pressure between contacting asperities and deformation of bulk material will affect the asperity deformation. However, there are
∗ Corresponding author. Tel.: +61 2 4221 4545; fax: +61 2 4221 5474. E-mail address: [email protected] (Z.Y. Jiang). 0043-1648/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.wear.2010.11.051
few reports on the interaction between surface asperity flattening, reduction and friction by the crystal plasticity theory. In order to figure out the relationship between sample surface asperity flattening, reduction and friction by the crystal plasticity theory, a finite element model is employed in the commercial finite element software ABAQUS to simulate the surface asperity flattening of Al plate along the rolling direction during uniaxial planar compression. A finite element polycrystal model subgrain scale is used in this study.
2. Crystal plasticity model The relationship between the single crystal and polycrystal is too complicated to be expressed directly. There are some assumptions given to simplify the relationship between the single crystal and polycrystal. The most widely used approach to obtain the response of a polycrystal from the response of individual crystals is to use the extended Taylor’s assumption of iso-deformation gradient in all of the crystals composing the polycrystal. Furthermore, if all grains are assumed to be of the same size, the Cauchy stress in the polycrystal can be taken as a simple number average of the Cauchy stresses in the various crystals [6]. On the basis of finite element theory, the finite element averaging procedure used the micro–macro link of the Taylor model to acquire the polycrystal plasticity response [7]. In these calculations of finite element averaging procedure, each grain is modelled by
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is the time of deformation. U˙ is the displacement rate of sample in the compression.
3.3. Development of sample surface roughness during uniaxial planar compression
Fig. 1. Schematic of sample’s surface roughness.
one or more finite elements to allow for no-uniform deformations between the grains and within the grains, and both equilibrium and compatibility are satisfied in the weak finite element sense. Following the previous research, Kalidindi developed an implicit time-integration procedure [8,9]. In this paper, Kalidindi’s implicit time-integration procedure [9] is employed in finite element calculation to make the transition from the response of a single crystal (or a region within a grain) to response of a polycrystalline aggregate. Simultaneously, a group of relevant experiments is carried out to test the simulation results. 3. Experimental 3.1. Sample preparation The sample is polished by the auto-polishing machine along the transverse direction. All the samples are ground along transverse direction with same grinding time, pressure and sand papers (grade P220). A 2D profilemeter is used to measure the sample surface roughness. Before the compression, a central line is drawn on the top surface of the sample, and surface roughness is measured along this line. After the compression, the surface roughness is measured along the same line. This line can be drawn on any position along the transverse direction. Therefore, along the transverse direction, surface roughness at any position (or line) can be determined. The roughness errors between different samples are under a certain range (less than 0.05 m). After grinding, all the roughness of samples has a value of about 0.72 m (Ra ). When the compression takes place, the sample is constrained in the transverse direction. The sample’s roughness develops along the rolling direction. The compression test has been carried out by the Instron MTS. The sample size is 10 mm × 10 mm × 6 mm. The surface roughness of samples is shown in Fig. 1. 3.2. Sample parameters After mechanical polishing, the heights of samples are measured by a micrometer. All the parameters of samples for the compression are shown in Tables 1 (with lubrication) and 2 (without lubrication). The strain rate for the compression is 0.001 s−1 . However, for the Instron MTS, it is difficult to control the strain rate. The displacement rate is thus used to replace the strain rate. The relationship between the strain rate and displacement rate is as ε = ln
H , H0
t =
ε , ε˙
U˙ =
H , t
H0 = H + H
(1)
where H0 , H, H are the original height, the height after compression and the height increment, respectively. ε is the true strain. t
It is obvious from experimental results (Table 3) that, with an increase of reduction, the surface roughness of both samples (compression with and without lubrication) decreases significantly. When the reduction is less than 5%, the two samples retain the same surface roughness. When the reduction exceeds 10%, there is an obvious difference of the surface roughness. Samples without lubrication during the compression have a larger decreasing rate of surface roughness. Finally, samples with lubrication during the compression have a comparatively rougher surface than that of samples without lubrication. From the experimental results, it is shown that the lubrication can constrain the surface asperity flattening process. In other words, friction can increase the process of surface asperity flattening.
3.4. Relationship between the surface roughness and friction Firstly, during the compression, when the reduction is less than 5%, the surface roughness of sample decreases greatly from 0.7 m to 0.4 m (Fig. 2). In this stage, the rough asperity does not contact with the compressing tool completely. This means the lubrication has no obvious influence on the surface asperity flattening in this stage. Secondly, if the reduction is between 5% and 30%, the sample surface roughness will continue to decrease, but the decreasing rate of the surface roughness is much lower than that in the first stage. In this stage, a stable lubrication layer will be formed. It will limit the surface asperity flattening process. Therefore, sample surface roughness will decrease slowly. During this stage, the lubrication layer plays an important role on the surface asperity flattening. Thirdly, if the reduction exceeds 30%, sample surface roughness will decrease quickly (from about 0.3 to 0.15 m). In this stage, with an increase of reduction, the lubrication layer is destroyed by the compression. It will not affect the compression of rough surface. The surface asperity flattening will thus proceed quickly again. Finally, if the reduction exceeds 40%, the two samples will keep a certain surface, and the sample with lubrication has a much rougher surface than that of the sample without lubrication. The reason for this is that under the lubrication condition, a lubricant layer will be formed between the tool and the sample. With the lubrication of this layer, the sample will receive less constraint from the tool. Therefore, the sample can be deformed easily. Furthermore, the slip system in the sample can also be activated easily. When the reduction exceeds a certain value (about 40%), the lubrication layer will be destroyed, and the compressing tool will contact with the sample directly. However, owing to the influence of the lubrication layer, the sample with lubrication has a rougher surface than that without lubrication. On the other hand, an abnormal phenomenon is observed from the experimental results. When the reduction exceeds 60% in the sample without lubrication, there is a slight increase of surface roughness (about 0.03 m). There are three factors which may cause this phenomenon: firstly, the measuring error can cause this phenomenon; secondly, the friction and the tool surface influence can also lead to this phenomenon; thirdly, when the reduction reaches 60%, the aluminium sample cannot be further deformed easily.
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Table 1 Parameters for compression (with lubrication). Number
Cross section (mm × mm)
Height (mm)
Displacement rate (mm/min)
Strain rate (s−1 )
Reduction (%)
Reduction (mm)
Height (mm)
1 2 3 4 5
10.042 × 10.178 10.042 × 10.041 9.985 × 10.031 10.109 × 10.029 10.479 × 10.035
6.102 6.129 6.139 6.109 6.131
0.357 0.349 0.330 0.308 0.288
0.001 0.001 0.001 0.001 0.001
5 10 20 30 40
0.305 0.613 1.228 1.833 2.452
5.797 5.516 4.911 4.276 3.679
Table 2 Parameters for compression (without lubrication). Number
Cross section (mm × mm)
Height (mm)
Displacement rate (mm/min)
Strain rate (s−1 )
Reduction (%)
Reduction (mm)
Height (mm)
1 2 3 4 5
9.534 × 9.599 9.577 × 9.401 9.595 × 9.542 9.666 × 9.408 9.578 × 9.497
6.123 6.108 6.095 6.127 6.092
0.358 0.348 0.33 0.2407 0.287
0.001 0.001 0.001 0.001 0.001
5 10 20 30 40
0.306 0.6108 1.219 1.838 2.4368
5.817 5.497 4.876 4.289 3.655
Average surface roughness Ra, μm
0.8 0.7 No lubricant With lubricant
0.6 0.5 0.4 0.3 0.2 0.1 0.0 0
10
20
30
40
50
60
Reduction, %
(a) Ra
Average surface roughness Rq, μm
1.1 1.0 No lubricant With lubricant
0.9 0.8 0.7 0.6
pression. The model geometry is two dimensional. Although a 2D finite element analysis has the advantage of less computational time, the main reason for not using a 3D finite element analysis in this study is actually, that during the compression test, the transverse direction is constrained by the mould (Fig. 1), therefore there is almost no deformation of the workpiece in this direction. On the other hand, the available generalized plane strain elements in ABAQUS make it possible to impose a uniform strain on the body in the out-of-plane direction. Furthermore, as mentioned by Becker [10] and Wu and Lloyd [11], the 2D FE still allows us to simulate both the uniaxial and the biaxial deformation. For all the simulations, the 1-direction (Fig. 3) corresponds to the prior rolling direction, the 2-direction to the sample normal and the 3-direction to the transverse direction of the samples. The sample material is 6061T5 aluminium alloy. Size of the two-dimensional model is 500 m × 500 m. The initial length is denoted by L0 and the initial height by H0 . The original roughness parameters of the samples are Ra 0.72 m and Rq 0.98 m. Longer wavelength and shorter wavelength are employed to form the surface roughness. Longer wavelength (Rq 1.06 m, Ra 0.79 m) contributes about 60% to the initial surface roughness and shorter wavelength (Rq 0.82 m, Ra 0.61 m) only accounts for about 40% in the initial surface roughness. Becker’s method [10] is used to calculate the average roughness obtained from the finite element results and the experimental ones. The surface roughness of the model is determined by the experimental results. A line is drawn through the surface nodal coordinate data, therefore, the distance from this line to the surface nodes is ı [10] and
0.5
L
ı dx = 0
0.4
(2)
0
0.3
Though there are a few differences between the 2D model surface profile and sample profile, they all have the same initial roughness Ra . The model surface roughness is determined by the sample surface roughness. The average roughness, Ra , is determined from
0.2 0.1 0.0 0
10
20
30
40
50
60
Reduction, %
(b) Rq Fig. 2. Relationship between the surface roughness and friction.
4. Modelling 4.1. Model This study aims to analyse the relationship between reduction, friction and surface roughness during uniaxial plane strain com-
Ra =
1 L
l
ı dx
(3)
0
The reduction of sample ranges from 5% to 40%. Contact friction coefficient between the sample and rigid compressing tool and mould ranges from 0.001 to 0.35. The reduction is applied on the top of the sample by the rigid compressing tool. Due to symmetry, all the nodes on edge ab (Fig. 3) have no displacements in direction 1. In order to understand the relationship between surface roughness, friction and reduction by the crystal plasticity theory, a finite element polycrystal model is employed in this study. The two dimen-
H.J. Li et al. / Wear 271 (2011) 1778–1784 Table 3 Development of surface roughness under uniaxial planar compression. Reduction (%) (a) With lubrication
0
5
10
20
30
40 (b) Without lubrication
0
5
10
20
Roughness (with lubrication)
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Table 3 (Continued) Reduction (%)
Roughness (with lubrication)
30
40
sional model has 902 CPE4R reduced integration elements, and one grain set with one element. The rigid tool and mould both have 20 discrete rigid elements. Kalidindi’s method [12] is used to incorporate crystal plasticity into FEM. The constitutive model and timeintegration procedures are implemented into the implicit finite element code ABAQUS by employing the user material subroutine UMAT. The combinations of slip systems are taken into account during modelling, including 12 {1 1 0} 1 1 1. It is assumed that the shearing rate is equal on each slip system. 902 random Euler angle triplets are input into ABAQUS as the initial crystallographic condition of the model. The components of the elasticity tensor are taken as C11 = 106,750 MPa, C12 = 60,410 MPa, C44 = 28,340 MPa [13]. As hardening matrix parameters we use q˛ˇ = 1.0 for coplanar systems and q˛ˇ = 1.4 for non-coplanar slip systems [14]. The other material parameters are according to Ref. [15]. 4.2. Relationship between surface roughness, friction and reduction From the modelling results, it is obvious that with an increase of reduction, the sample surface roughness decreases quickly. In other words, the surface asperity flattening takes place quickly, and the compression with lubrication has an obvious difference from that without lubrication. With an increase of friction coefficient, the surface asperity flattening rate of sample will increase significantly. Under the condition of larger friction, the sample will obtain a lower surface roughness due to compression. The reason for this phenomenon is that the friction between the sample and the compressing tool is expressed by the penalty function
Fig. 3. Two dimensional model and mesh.
as a constraint to the sample. Therefore, during the compression the friction will block the sample deformation. It will restrain the slip system activation and action. On the other hand, the practical lubricant layer lies between the sample and tool which acts as a transfer layer to block the surface asperity flattening process. It can obviously reduce the surface asperity flattening process. Here, in this simulation, friction is only set as a coefficient. The sample will thus come into contact with the tool directly; therefore, the sample will be affected by the tool directly. Because the compression tool is smooth in this study, the sample surface will be flattened quickly. With an increase of friction coefficient, the influence of the tool on the sample will result in a speeding-up of the surface asperity flattening process. After the compression, the strain localization area will appear in the samples’ surface. This will result in a transfer layer in the sample surface. This transfer layer will play a direct role in changing the surface roughness. Other neighbouring layers will affect the surface roughness by this transfer layer. From the simulation, we observed the obvious change of crystals’ orientations in this layer. This will be further focused in the study (see Fig. 4).
4.3. Comparison of the experimental and modelling results It is shown from Fig. 5 that the experimental results are close to the simulated results. Both results show that with an increase of reduction, the surface roughness of sample decreases quickly, and lubrication can delay the process of surface asperity flattening. On the other hand, there are some differences between the experimental and simulated results. The developing tendency of surface roughness in the two results is different: the simulated result has a higher decreasing rate of surface roughness than that of the experimental results. In Fig. 5(a), when the reduction is less than 10%, the two results keep the same tendency of surface asperity flattening. On the other hand, when the reduction exceeds 10%, the decreasing rate of surface roughness from simulation is a bit larger than that from the experiment. In the experiment, when the reduction exceeds 10%, the lubrication layer can play a remarkable role in delaying surface asperity flattening. However, in the simulation, the penalty friction coefficient is used to replace the lubricant layer. There is an obvious difference between the simulation results and experimental ones. The error may result from the computational error and the frictional model error. From the calculation, the maximum error between the experimental results and the simulation results is about 0.08 m. When the maximum error appears, the sample roughness is more than 0.25 m (Fig. 5(a) and (b)). This error is about 25% of sample roughness. In the finite element simulation, the error may be acceptable. When the reduction is between 10% and 40%, the lubrication layer can perform properly. The difference between the simulated and experimental results is most obvious. When the reduction exceeds
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0.8 0.7
0.6
Surface roughness, μm
Surface roughness Ra, μm
0.7
With lubricant Without lubricant
0.5 0.4 0.3 0.2
Experiment Simulation
0.6 0.5 0.4 0.3 0.2
0.1
0.1
0.0 0
10
20
30
40
50
60
0
Reduction, %
10
30
40
50
60
50
60
Reduction, %
(a) Ra
(a)
With lubrication
0.8
0.9
0.7
0.8 With lubricant Without lubricant
0.7
Surface roughness, μm
Surface roughness, Rq
20
0.6 0.5 0.4 0.3 0.2
Experiment Simulation
0.6 0.5 0.4 0.3 0.2 0.1
0.1 0.0 0
10
20
30
40
50
0
60
10
(b) Rq
It is shown in Fig. 6 that with an increase of reduction, the influence of the shorter wavelength on the sample roughness (Ra , Rq ) will increase during the compression, and the influence of the longer wavelength on the sample roughness (Ra , Rq ) decreases. In this model, longer wavelength has a larger initial roughness (Ra , Rq ), and shorter wavelength has a smaller initial roughness (Ra , Rq ). During the compression, the compressing tool will first contact the longer wavelength, and the roughness of longer wavelength will decrease quickly. Furthermore, at the beginning, under very small reduction (less than 5%), the longer wavelength will affect the sample roughness mainly, while shorter wavelength has no obvious influence on the sample roughness. With an increase of reduction, the compressing tool will contact with the shorter wavelength. The shorter wavelength will thus play a main role on the sample rough-
40
Fig. 5. Comparison of experimental and modelling results.
90
Percentage in the total roughness, %
4.4. Influence of longer wavelength and shorter wavelength on the surface roughness
30
(b) Without lubrication
Fig. 4. Relationship between surface roughness, friction and reduction.
40%, the lubrication layer will be damaged and cannot continue in constraining surface asperity flattening. In Fig. 5(b), if the reduction exceeds 60%, the roughness of experimental results will increase by about 0.03 m. The reasons for this phenomenon are the same as those in Fig. 2.
20
Reduction, %
Reduction, %
80 70 60
Longer wavelength Shorter wavelength
50 40 30 20 10
0
5
10
15
20
25
Reduction, % Fig. 6. Influence of longer wavelength and shorter wavelength on the sample roughness.
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ness while the influence of longer wavelength on the sample can be neglected. 5. Conclusions (1) Within a certain reduction range (in this case, it is between 10% and 40%), the lubrication can limit the surface asperity flattening significantly. If the reduction is less than 10%, the lubrication layer has no obvious influence on sample surface roughness. When the reduction exceeds 40%, the lubrication layer will be damaged during the uniaxial planar compression. (2) Both the experimental and the simulation results show the same tendency: with an increase of reduction, the surface asperity flattening proceeds, and sample surface roughness will decrease greatly, and the lubrication can constrain the surface asperity flattening process. (3) In the simulation, with an increase of reduction, the influence of longer wavelength on the sample roughness will decrease quickly, while the effect of the shorter wavelength of the surface asperity on sample roughness will increase significantly. Acknowledgements The first author would like to thank UOW for Ph.D. scholarship support for current research. We also appreciate Ms. Alisa Percy for her proof reading of the paper. References [1] D. Raabe, M. Sachtleber, H. Weiland, G. Scheele, Z.S. Zhao, Grain-scale micromechanics of polycrystal surfaces during plastic straining, Acta Materialia 51 (2003) 1539–1560.
[2] S. Sheu, W.R.D. Wilson, Flattening of workpiece surface asperities in metal forming, in: Proceedings of the 11th NAMRC, 1983, pp. 172–178. [3] W.R.D. Wilson, W.M. Lee, Mechanics of surface roughening in metal forming process, Journal of Manufacturing Science and Engineering 123 (2001) 279–283. [4] A. Makinouchi, H. Ike, M. Murakawa, N. Koga, A finite element analysis of flattening of surface asperities by perfectly lubricated rigid dies in metal working processes, Wear 128 (1988) 109–122. [5] M.P.F. Sutcliffe, Surface asperity deformation in metal forming processes, International Journal of Mechanical Sciences 11 (1988) 847–868. [6] S.R. Kalidindi, S.E. Schoenfeld, On the prediction of yield surfaces by the crystal plasticity models for fcc polycrystals, Materials Science and Engineering A 293 (2000) 120–129. [7] A.J. Beaudoin, P.R. Dawson, K.K. Mathur, U.F. Kocks, A hybrid finite element formulation for polycrystal plasticity with consideration of macrostructural and microstructural linking, International Journal of Plasticity 11 (1995) 501–521. [8] S.R. Kalidindi, L. Anand, Macroscopic shape change and evolution of crystallographic texture in pre-textured fcc metals, Journal of the Mechanics and Physics of Solids 42 (1994) 459–490. [9] S.R. Kalidindi, L. Anand, An approximate procedure for predicting the evolution of crystallographic texture in bulk deformation processing of FCC metals, International Journal of Mechanical Sciences 34 (1992) 310–312. [10] R. Becker, Effect of strain localization on surface roughening during sheet forming, Acta Metallurgica 46 (1998) 1385–1401. [11] P.D. Wu, D.J. Lloyd, Modelling surface roughening with crystal plasticity, in: Proceedings of the 1st International Symposium on Metallurgical Modeling for Aluminum alloys, 2003, pp. 15–20. [12] S.R. Kalidindi, C.A. Bronkhorst, L. Anand, Crystallographic texture evolution in bulk deformation processing of FCC metals, Journal of the Mechanics and Physics of Solids 40 (1992) 552–555. [13] E.A. Brandes, Smithells Metals Reference Book, The Bath Press, Bath, 1999, p. 368. [14] R.J. Asaro, A. Needleman, Texture development and strain hardening in rate dependent polycrystals, Acta Metallurgica 33 (1985) 923–953. [15] D. Raabe, Y. Wang, F. Roters, Crystal plasticity simulation study on the influence of texture on earring in steel, Computational Materials Science 34 (2005) 221–234.