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Study on compression forming of a rotating disk considering hybrid friction
Journal of Materials Processing Technology 125–126 (2002) 421–426
Study on compression forming of a rotating disk considering hybrid friction M.N. Huanga, G.Y. Tzoub,* a
b
Department of Physics, Yung-Ta Institute of Technology, Ping-Tung 909, Taiwan, ROC Department of Mechanical Engineering, Yung-Ta Institute of Technology, Ping-Tung 909, Taiwan, ROC Received 2 December 2001; accepted 21 January 2002
Abstract Hybrid friction at the interface between die and circular disk is assumed, which indicates the Coulomb friction and the sticking friction are generated simultaneously at the contact zone. An analytical model of compression forming with consideration of the rotating effect is derived by the slab method. In this study, the effects of rotating angular speed of disk, ratio of diameter to thickness, coefficients of friction on compression characteristics are explored effectively. The limiting coefficient of friction is formulated so as to avoid the generation of hybrid friction. The analytical model can correctly calculate the compression pressure, the radial stress, the compression force and the position of radius at which the sticking friction occurs. This study utilises the analytical approach to establish a model of plastic mechanism. The characteristics of the rotating compression forming can be derived rapidly and effectively in this study and it is suitable for industry of compression forming. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Coulomb friction; Hybrid friction; Limiting frictional coefficient
1. Introduction For the compression forming of a cylinder, the compression force and the power needed increase with increasing frictional effect at the interface between the die and the material. There are also some problems that may occur, such as non-uniform deformation and cracking of the interior and exterior materials, and bulging deformation. Some researchers have begun to study these problems. Steck and Schmid [1] established a 2D admissible velocity field by means of the upper-bound method to investigate the bulging deformation of cylinder in the compression process. Kulkarni and Kalpakjian [2] examined the effect of free deformation of the outer surface in the gun forming process. Lee and Altan [3] used the upper-bound method to establish an admissible velocity field to explore the flow stress and frictional factor in the ring and cylinder compression process. The ring compression test can predict the flow stress and frictional factor. Douglas and Altan [4] used the ring compression test * Corresponding author. Present address: Department of Mechanical Engineering, University of Wollongong, 2522 NSW, Australia. Tel.: þ61-2-4221-4145; fax: þ61-2-4221-3101. E-mail addresses: [email protected] (M.N. Huang), [email protected] (G.Y. Tzou). URL: http://www.ytit.edu.tw
to predict the flow stress under consideration of various forging rates and temperatures. Sofuoglu and Rasty [5] used the ring compression test to measure friction coefficient and obtained calibration curves of frictional coefficient, and utilised the ABAQUS software to simulate it. Chen [6] adopted a numerical simulation (using Deform software) and experiment to model the plastic behaviour of the cylinder under compression. However, these papers did not consider the rotating effect of deformed material. Considering the rotating effect of deformed material. Xue et al. [7] used the finite element method to investigate the plastic deformation of cylinder in the process of twist-compression forming. The compression force and bulging deformation reduced evidently in such a deformation process. Lucchesi and Sassu [8] utilised the energy approach to discuss the deformation of elastic–plastic tube under torsion and compression. Kim et al. [9] established 3D admissible velocity field via the upper-bound method to explore the compression deformation of cylinder considering a constant shear friction and rotating cylinder. It indicated that the compression force and bulging deformation reduced significantly in such a compression process. Kim et al. [10] used the dual stream function to establish a 3D admissible velocity field to investigate the rotating compression deformation of cylinder. It was shown that this analytical model was more
0924-0136/02/$ – see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 2 ) 0 0 2 9 3 - 5
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precise. Kim et al. [11] also used the dual stream function to explore plastic behaviour when the die is rotating but the cylinder is not. The compression force, the stress distribution and the power needed must be predicted in the design of the processes of metal compression forming. In order to understand these characteristics, the flow stress and frictional coefficient (or friction factor) have to be explored. Once the flow stress and frictional coefficient (or friction factor) are given, the compression force, the stress distribution and the power needed can be calculated by this study. 2. Modelling 2.1. Coulomb friction model The authors have established an analytical model for the compression forming of a rotating circular disk considering Coulomb friction. The formulae of the compression pressure and the compression force on the circular disk are derived by the slab method. The schematic diagram of the rotating circular disk is shown in Fig. 1. Fig. 2 shows schematic diagram of a small element of the circular disk. For deriving the analytical model of the rotating circular disk, the following assumptions are employed: (1) The circular disk compressed is a rigid-plastic material. (2) Axis-symmetrical compression is assumed, so the radial stress (sr) equals the circumferential stress (sy).
Fig. 2. The stress state of an element of the rotating circular disk.
(3) The stresses distributed within the elements are uniform. The radial stress (sr), the circumferential stress (sy), and the vertical stress (sz) are regarded as principal stresses. (4) The friction between the tool and the disk is assumed to be Coulomb friction ðt ¼ mpÞ. 2.1.1. Compression pressure Force equilibrium equations, yield criteria, geometrical conditions, boundary conditions, and friction conditions assuming Coulomb friction (t ¼ mp), are used to derive the compression characteristics such as the compression pressure distribution, the radial stress distribution and the compression force. The specific compression pressure distribution can be expressed as p hrob 2 ðR ðh=2mÞÞ ð2m=hÞðRrÞ ¼ eð2m=hÞðRrÞ e Y 2mY hrob 2 h r þ 2m 2mY
(1)
According to the yielding criteria, the radial stress distribution (q/Y) can be also expressed as q p ¼1 (2) Y Y
Fig. 1. The schematic diagram of the rotating disk.
2.1.2. Compression force When the frictional coefficient (m) and average shear yield strength (k) are known, the compression force can be calculated by integrating the compression pressure distribution over the whole surface of the circular disk. The compression
M.N. Huang, G.Y. Tzou / Journal of Materials Processing Technology 125–126 (2002) 421–426
force can be expressed as 2 Z R h Rh eð2m=hÞR h2 eð2m=hÞR P¼ p dA ¼ 2pc 4m2 2m 4m2 0 phrob 2 R3 ph2 rob 2 R2 þ (3) 3m 4m2 where c¼Ye
ð2m=hÞR
hrob 2 eð2m=hÞR h R ; 2m 2m
dA ¼ 2pr dr
The average compression pressure can be obtained via the compression force divided by the area of circular disk and it can be expressed as below: pa ¼
P pR2
(4)
2.2. Hybrid friction model Under high coefficient of friction or some geometrical condition, the interface friction must be considered as hybrid friction shown in Fig. 3. The friction near the free surface (Ic) is considered as the Coulomb friction and the friction near the centre of the circular disk (IIs) as sticking friction (sticking friction zone). It is called the hybrid friction in these conditions. In Fig. 3, rst denotes the start position generating sticking friction. Because the interface frictional shear stress cannot be larger than the shear yielding strength of the material (i.e. t k), the rst can be derived. 2.2.1. Determinations of rst and mlim By using this condition of t ¼ mp k, the start position generating sticking friction can be derived as follows: c eð2m=hÞr þ a1 r þ b1 0
(5)
When c eð2m=hÞr þ a1 r þ b1 ¼ 0, the sticking friction starts to occur. From this equation, it is easy to calculate the value of the start position (r ¼ rst ) by the bisection method. Moreover, with a view to avoid the generation of hybrid friction, it is very important to investigate some forming conditions, e.g. D/h or mlim. It is a fact that the largest compression pressure always occurs in the centre of the disk, therefore, the sticking friction firstly occurs in the centre. Thus, let r ¼ 0 in Eq. (5) and derive the following equation: Y eðD=hÞm
4km þ h2 rob 2 b1 ¼ 4m2
h2 rob 2 eðD=hÞm D 1 þ b1 0 h m 2m
(6)
where D ¼ 2R. When Y eðD=hÞm ðh2 rob 2 eðD=hÞm =2mÞððD=hÞ ð1=mÞÞþ b1 ¼ 0, indicating the sticking friction starts to occur in the centre of the disk. In such conditions, it can be firstly to determine whether the hybrid friction occurs or not. Besides the limiting frictional coefficient (mlim) under generating the hybrid friction can be derived from the following equation: ðD=hÞm
Ye
h2 rob 2 eðD=hÞm D 1 þ b1 ¼ 0 h m 2m
(7)
the limiting frictional coefficient (mlim) under geometric condition of D/h. If the limiting frictional coefficient is predicted, the hybrid friction can be controlled by lubrication. 2.2.2. Sticking friction model When the sticking friction occurs, the governing equation needs to derive again. The interface friction assumes to be t ¼ k, and the compression pressure distribution (p) can be obtained.
where hrob 2 a1 ¼ ; 2m
423
p¼
rob 2 r 2 2kr þ c h 2
(8)
This equation is suitable for the condition of sticking friction, namely as 0 r rst , where c can be determined by the boundary condition. According to the continuity of compression pressure, i.e. at r ¼ rst , pcI ¼ psII , it can be derived: c eð2m=hÞrst þ
hrob 2 h rob 2 rst 2 2k rst rst þ c ¼ 2m h 2m 2 (9)
then c can be expressed by hrob 2 h 2k rob 2 rst 2 ð2m=hÞrst rst þ c ¼ ce þ rst 2m h 2m 2 (10) Fig. 3. Schematic representation of the compression pressure distribution in hybrid friction model.
Substituting Eq. (10) in Eq. (8), and the compression pressure distribution under the sticking friction can be
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expressed as rob 2 r 2 2kr hrob 2 h þ c eð2m=hÞrst þ rst h 2m 2 2m 2 2 2k rob rst þ rst (11) h 2 The radial stress (q/Y) can be obtained by the yield criteria.
p¼
2.2.3. Compression pressure Therefore, the compression pressure distribution in the hybrid friction model can be expressed as Zone Ic (rst r R): Coulomb friction zone hrob 2 h r pcI ¼ c eð2m=hÞr þ 2m 2m s
Zone II (0 r rst ): sticking friction zone rob 2 r 2 2kr hrob 2 h þ c eð2m=hÞrst þ psII ¼ rst h 2m 2 2m 2 2 2k rob rst þ rst h 2 2.2.4. Compression force When the hybrid friction occurs, the compression forces in two zones can be calculated using Z R Z rst P¼ pcI dA þ psII dA ¼ Ic þ IIs (12) rst
0
where
hrst eð2m=hÞrst Rh eð2m=hÞR 2m 2m h2 ð2m=hÞR 2 ðe eð2m=hÞrst Þ 4m phrob 2 R3 rst 3 hðR2 rst 2 Þ þ ; 4m m 3 rob 2 rst 4 2krst 3 c rst 2 þ IIs ¼ 2p 8 3h 2 c
I ¼ 2pc
Fig. 4. The comparisons of the compression pressure between Coulomb friction and hybrid friction model.
that the compression pressure with no rotating effect (ob ¼ 0 rad=s) is larger than that with a rotating effect (ob ¼ 5 rad=s). And considering the rotating effect the hybrid friction is generated slowly. Fig. 6 shows the compression pressure distribution with various diameter thickness ratios (D/hi). The thickness of disk is fixed at h ¼ 50 mm. The compression pressure and the sticking friction zone increase with an increase in D/hi. Itindicates that the surface is rough as the sticking friction zone is larger and the disk with a larger radius is compressed under the fixed thickness. Fig. 7 shows the compression pressure considering hybrid friction for various frictional coefficients (m). It is noted that the compression pressure and the sticking friction zone friction increase as the frictional coefficient increases. So when m is smaller, e.g. m ¼ 0:1, no hybrid friction is generated. Fig. 8 shows the comparison of compression force with different rotating angular speed between the Coulomb friction model and the hybrid friction model, where (~) denotes that there is no rotating effect ðob ¼ 0 rad=sÞ, (*) denotes
The average compression pressure is derived from the compression force divided by the circular area.
3. Results and discussions Fig. 4 shows comparisons of the compression pressure between Coulomb friction and hybrid friction model. It is indicated that the forming zone range (116 mm r 116 mm) is the sticking friction zone and the rest is the Coulomb friction under the angular speed (ob ¼ 5 rad=s). If we only consider the Coulomb friction, the compression pressure distribution is larger than the hybrid friction, and which is not true for the real forming situation. Fig. 5 shows the compression pressure distributions under the hybrid friction for various rotating speeds. It indicates
Fig. 5. The compression pressure distributions under the hybrid friction for various rotating speeds.
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425
Fig. 9. The relationship mlim and D/h with various rotating angular speeds. Fig. 6. The compression pressure distribution with various diameter thickness ratios.
Fig. 7. The compression pressure considering hybrid friction for various frictional coefficients.
that there is a rotating effect ðob ¼ 5 rad=sÞ. The dashed line represents the Coulomb friction model and the solid line represents the hybrid friction model. The compression force always increases with increasing D/h. The compression force with the hybrid friction is smaller than that with
Fig. 10. The comparison of the compression force between the two models with various densities.
Coulomb friction, and the difference becomes larger as the value of D/h increases. Fig. 9 shows the relationship mlim and D/h with various rotating angular speeds, where mlim denotes the frictional coefficient firstly generating the sticking friction. The hybrid friction is more difficult to occur as mlim is larger. The value of mlim with the rotating effect is larger than that without the rotating effect. Then the value of mlim becomes smaller as D/ h increases, however, the curve with the rotating effect decreases slowly. Fig. 10 shows the comparison of the compression force between the two models with various densities. It indicates that the compression force in the hybrid friction model is much smaller than that in the Coulomb friction model under the fixed density. When the density of the material increases, the compression forces both decrease for the two models.
4. Conclusions Fig. 8. The comparison of compression force with various rotating angular speed between the Coulomb friction and the hybrid friction models.
Because the sticking friction makes the surface of the material rough, and influences the quality of the product.
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Table 1 Summary of compression characteristics for the hybrid friction modela Compression condition
p/Y
q/Y
P
m ob D/h r
% & % &
% & % &
% & % &
The advice and financial support of the NSC are gratefully acknowledged.
mlim
References % & %
a The symbol ‘‘%’’ indicates that the compression characteristics increase with increasing the compression conditions; symbol ‘‘&’’ indicates that it is decreasing.
It should be avoided. The effects of D/h, m and ob on the compression pressure, radial stress and compression force are investigated effectively as shown in Table 1. The main conclusions are as follows: (1) The rotating of the circular disk can reduce the compression pressure, radial stress and compression force. (2) The hybrid friction can easily occur under high frictional coefficient and high diameter thickness ratio. (3) The limiting friction coefficient is larger with the rotating effect and higher density; it indicates and the hybrid friction is harder to occur.
Acknowledgements This work was sponsored bytheNationalScience Council of Republic of China under grant No. NSC-90-2212-E-132-005.
[1] E. Steck, K. Schmid, The application of the upper bound principle upon upsetting and forging operation, Industrie-Anzeiger 87 (1965) 1751. [2] K.M. Kulkarni, S. Kalpakjian, A study of barrelling as an example of free deformation in plastic working, J. Eng. Ind. 91 (1969) 743. [3] C.H. Lee, T. Altan, Influence of flow stress and friction upon metal flow in upset forging of ring and cylinders, J. Eng. Ind. 94 (1972) 775. [4] J.R. Douglas, T. Altan, Flow stress determination for metals at forging rates and temperatures, J. Eng. Ind. 97 (1975) 66. [5] H. Sofuoglu, J. Rasty, On the measurement of friction coefficient utilizing the ring compression test, Tribol. Int. 32 (1999) 327. [6] Z.J. Chen, Stress analysis of non-uniform cylindrical compression test, Master Thesis, Department of Mechanical Engineering at National Taiwan University, 1996. [7] K.M. Xue, Y. Lu, H.H. Liu, X.M. Zhao, Numerical simulation and experimental research into the process of twist-compression forming, Adv. Technol. Plasticity II (1993) 1065. [8] M. Lucchesi, M. Sassu, Energy dissipation of a thin elastoplastic tube under torsion and compression, Int. J. Solids Struct. 32 (19) (1995) 2891. [9] Y.H. Kim, J.H. Park, Y.E. Jin, An analysis of plastic deformation process for twist-assisted upset forging of cylindrical billets, in: Proceedings of the International Conference on Advances in Materials and Processing Technologies (AMPT’99), Vol. I, 1999, p. 79. [10] Y.H. Kim, J.H. Park, Y.E. Jin, An upper bound analysis for torsional upset forging of cylindrical billets, Adv. Technol. Plasticity I (1999) 859. [11] Y.H. Kim, Y.E. Jin, J.H. Park, Y. Lee, An analysis of the torsional forming process using the dual stream function, in: Proceedings of the Eighth International Conference on Metal Forming, 2000, p. 741.
Study on compression forming of a rotating disk considering hybrid friction M.N. Huanga, G.Y. Tzoub,* a
b
Department of Physics, Yung-Ta Institute of Technology, Ping-Tung 909, Taiwan, ROC Department of Mechanical Engineering, Yung-Ta Institute of Technology, Ping-Tung 909, Taiwan, ROC Received 2 December 2001; accepted 21 January 2002
Abstract Hybrid friction at the interface between die and circular disk is assumed, which indicates the Coulomb friction and the sticking friction are generated simultaneously at the contact zone. An analytical model of compression forming with consideration of the rotating effect is derived by the slab method. In this study, the effects of rotating angular speed of disk, ratio of diameter to thickness, coefficients of friction on compression characteristics are explored effectively. The limiting coefficient of friction is formulated so as to avoid the generation of hybrid friction. The analytical model can correctly calculate the compression pressure, the radial stress, the compression force and the position of radius at which the sticking friction occurs. This study utilises the analytical approach to establish a model of plastic mechanism. The characteristics of the rotating compression forming can be derived rapidly and effectively in this study and it is suitable for industry of compression forming. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Coulomb friction; Hybrid friction; Limiting frictional coefficient
1. Introduction For the compression forming of a cylinder, the compression force and the power needed increase with increasing frictional effect at the interface between the die and the material. There are also some problems that may occur, such as non-uniform deformation and cracking of the interior and exterior materials, and bulging deformation. Some researchers have begun to study these problems. Steck and Schmid [1] established a 2D admissible velocity field by means of the upper-bound method to investigate the bulging deformation of cylinder in the compression process. Kulkarni and Kalpakjian [2] examined the effect of free deformation of the outer surface in the gun forming process. Lee and Altan [3] used the upper-bound method to establish an admissible velocity field to explore the flow stress and frictional factor in the ring and cylinder compression process. The ring compression test can predict the flow stress and frictional factor. Douglas and Altan [4] used the ring compression test * Corresponding author. Present address: Department of Mechanical Engineering, University of Wollongong, 2522 NSW, Australia. Tel.: þ61-2-4221-4145; fax: þ61-2-4221-3101. E-mail addresses: [email protected] (M.N. Huang), [email protected] (G.Y. Tzou). URL: http://www.ytit.edu.tw
to predict the flow stress under consideration of various forging rates and temperatures. Sofuoglu and Rasty [5] used the ring compression test to measure friction coefficient and obtained calibration curves of frictional coefficient, and utilised the ABAQUS software to simulate it. Chen [6] adopted a numerical simulation (using Deform software) and experiment to model the plastic behaviour of the cylinder under compression. However, these papers did not consider the rotating effect of deformed material. Considering the rotating effect of deformed material. Xue et al. [7] used the finite element method to investigate the plastic deformation of cylinder in the process of twist-compression forming. The compression force and bulging deformation reduced evidently in such a deformation process. Lucchesi and Sassu [8] utilised the energy approach to discuss the deformation of elastic–plastic tube under torsion and compression. Kim et al. [9] established 3D admissible velocity field via the upper-bound method to explore the compression deformation of cylinder considering a constant shear friction and rotating cylinder. It indicated that the compression force and bulging deformation reduced significantly in such a compression process. Kim et al. [10] used the dual stream function to establish a 3D admissible velocity field to investigate the rotating compression deformation of cylinder. It was shown that this analytical model was more
0924-0136/02/$ – see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 2 ) 0 0 2 9 3 - 5
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precise. Kim et al. [11] also used the dual stream function to explore plastic behaviour when the die is rotating but the cylinder is not. The compression force, the stress distribution and the power needed must be predicted in the design of the processes of metal compression forming. In order to understand these characteristics, the flow stress and frictional coefficient (or friction factor) have to be explored. Once the flow stress and frictional coefficient (or friction factor) are given, the compression force, the stress distribution and the power needed can be calculated by this study. 2. Modelling 2.1. Coulomb friction model The authors have established an analytical model for the compression forming of a rotating circular disk considering Coulomb friction. The formulae of the compression pressure and the compression force on the circular disk are derived by the slab method. The schematic diagram of the rotating circular disk is shown in Fig. 1. Fig. 2 shows schematic diagram of a small element of the circular disk. For deriving the analytical model of the rotating circular disk, the following assumptions are employed: (1) The circular disk compressed is a rigid-plastic material. (2) Axis-symmetrical compression is assumed, so the radial stress (sr) equals the circumferential stress (sy).
Fig. 2. The stress state of an element of the rotating circular disk.
(3) The stresses distributed within the elements are uniform. The radial stress (sr), the circumferential stress (sy), and the vertical stress (sz) are regarded as principal stresses. (4) The friction between the tool and the disk is assumed to be Coulomb friction ðt ¼ mpÞ. 2.1.1. Compression pressure Force equilibrium equations, yield criteria, geometrical conditions, boundary conditions, and friction conditions assuming Coulomb friction (t ¼ mp), are used to derive the compression characteristics such as the compression pressure distribution, the radial stress distribution and the compression force. The specific compression pressure distribution can be expressed as p hrob 2 ðR ðh=2mÞÞ ð2m=hÞðRrÞ ¼ eð2m=hÞðRrÞ e Y 2mY hrob 2 h r þ 2m 2mY
(1)
According to the yielding criteria, the radial stress distribution (q/Y) can be also expressed as q p ¼1 (2) Y Y
Fig. 1. The schematic diagram of the rotating disk.
2.1.2. Compression force When the frictional coefficient (m) and average shear yield strength (k) are known, the compression force can be calculated by integrating the compression pressure distribution over the whole surface of the circular disk. The compression
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force can be expressed as 2 Z R h Rh eð2m=hÞR h2 eð2m=hÞR P¼ p dA ¼ 2pc 4m2 2m 4m2 0 phrob 2 R3 ph2 rob 2 R2 þ (3) 3m 4m2 where c¼Ye
ð2m=hÞR
hrob 2 eð2m=hÞR h R ; 2m 2m
dA ¼ 2pr dr
The average compression pressure can be obtained via the compression force divided by the area of circular disk and it can be expressed as below: pa ¼
P pR2
(4)
2.2. Hybrid friction model Under high coefficient of friction or some geometrical condition, the interface friction must be considered as hybrid friction shown in Fig. 3. The friction near the free surface (Ic) is considered as the Coulomb friction and the friction near the centre of the circular disk (IIs) as sticking friction (sticking friction zone). It is called the hybrid friction in these conditions. In Fig. 3, rst denotes the start position generating sticking friction. Because the interface frictional shear stress cannot be larger than the shear yielding strength of the material (i.e. t k), the rst can be derived. 2.2.1. Determinations of rst and mlim By using this condition of t ¼ mp k, the start position generating sticking friction can be derived as follows: c eð2m=hÞr þ a1 r þ b1 0
(5)
When c eð2m=hÞr þ a1 r þ b1 ¼ 0, the sticking friction starts to occur. From this equation, it is easy to calculate the value of the start position (r ¼ rst ) by the bisection method. Moreover, with a view to avoid the generation of hybrid friction, it is very important to investigate some forming conditions, e.g. D/h or mlim. It is a fact that the largest compression pressure always occurs in the centre of the disk, therefore, the sticking friction firstly occurs in the centre. Thus, let r ¼ 0 in Eq. (5) and derive the following equation: Y eðD=hÞm
4km þ h2 rob 2 b1 ¼ 4m2
h2 rob 2 eðD=hÞm D 1 þ b1 0 h m 2m
(6)
where D ¼ 2R. When Y eðD=hÞm ðh2 rob 2 eðD=hÞm =2mÞððD=hÞ ð1=mÞÞþ b1 ¼ 0, indicating the sticking friction starts to occur in the centre of the disk. In such conditions, it can be firstly to determine whether the hybrid friction occurs or not. Besides the limiting frictional coefficient (mlim) under generating the hybrid friction can be derived from the following equation: ðD=hÞm
Ye
h2 rob 2 eðD=hÞm D 1 þ b1 ¼ 0 h m 2m
(7)
the limiting frictional coefficient (mlim) under geometric condition of D/h. If the limiting frictional coefficient is predicted, the hybrid friction can be controlled by lubrication. 2.2.2. Sticking friction model When the sticking friction occurs, the governing equation needs to derive again. The interface friction assumes to be t ¼ k, and the compression pressure distribution (p) can be obtained.
where hrob 2 a1 ¼ ; 2m
423
p¼
rob 2 r 2 2kr þ c h 2
(8)
This equation is suitable for the condition of sticking friction, namely as 0 r rst , where c can be determined by the boundary condition. According to the continuity of compression pressure, i.e. at r ¼ rst , pcI ¼ psII , it can be derived: c eð2m=hÞrst þ
hrob 2 h rob 2 rst 2 2k rst rst þ c ¼ 2m h 2m 2 (9)
then c can be expressed by hrob 2 h 2k rob 2 rst 2 ð2m=hÞrst rst þ c ¼ ce þ rst 2m h 2m 2 (10) Fig. 3. Schematic representation of the compression pressure distribution in hybrid friction model.
Substituting Eq. (10) in Eq. (8), and the compression pressure distribution under the sticking friction can be
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expressed as rob 2 r 2 2kr hrob 2 h þ c eð2m=hÞrst þ rst h 2m 2 2m 2 2 2k rob rst þ rst (11) h 2 The radial stress (q/Y) can be obtained by the yield criteria.
p¼
2.2.3. Compression pressure Therefore, the compression pressure distribution in the hybrid friction model can be expressed as Zone Ic (rst r R): Coulomb friction zone hrob 2 h r pcI ¼ c eð2m=hÞr þ 2m 2m s
Zone II (0 r rst ): sticking friction zone rob 2 r 2 2kr hrob 2 h þ c eð2m=hÞrst þ psII ¼ rst h 2m 2 2m 2 2 2k rob rst þ rst h 2 2.2.4. Compression force When the hybrid friction occurs, the compression forces in two zones can be calculated using Z R Z rst P¼ pcI dA þ psII dA ¼ Ic þ IIs (12) rst
0
where
hrst eð2m=hÞrst Rh eð2m=hÞR 2m 2m h2 ð2m=hÞR 2 ðe eð2m=hÞrst Þ 4m phrob 2 R3 rst 3 hðR2 rst 2 Þ þ ; 4m m 3 rob 2 rst 4 2krst 3 c rst 2 þ IIs ¼ 2p 8 3h 2 c
I ¼ 2pc
Fig. 4. The comparisons of the compression pressure between Coulomb friction and hybrid friction model.
that the compression pressure with no rotating effect (ob ¼ 0 rad=s) is larger than that with a rotating effect (ob ¼ 5 rad=s). And considering the rotating effect the hybrid friction is generated slowly. Fig. 6 shows the compression pressure distribution with various diameter thickness ratios (D/hi). The thickness of disk is fixed at h ¼ 50 mm. The compression pressure and the sticking friction zone increase with an increase in D/hi. Itindicates that the surface is rough as the sticking friction zone is larger and the disk with a larger radius is compressed under the fixed thickness. Fig. 7 shows the compression pressure considering hybrid friction for various frictional coefficients (m). It is noted that the compression pressure and the sticking friction zone friction increase as the frictional coefficient increases. So when m is smaller, e.g. m ¼ 0:1, no hybrid friction is generated. Fig. 8 shows the comparison of compression force with different rotating angular speed between the Coulomb friction model and the hybrid friction model, where (~) denotes that there is no rotating effect ðob ¼ 0 rad=sÞ, (*) denotes
The average compression pressure is derived from the compression force divided by the circular area.
3. Results and discussions Fig. 4 shows comparisons of the compression pressure between Coulomb friction and hybrid friction model. It is indicated that the forming zone range (116 mm r 116 mm) is the sticking friction zone and the rest is the Coulomb friction under the angular speed (ob ¼ 5 rad=s). If we only consider the Coulomb friction, the compression pressure distribution is larger than the hybrid friction, and which is not true for the real forming situation. Fig. 5 shows the compression pressure distributions under the hybrid friction for various rotating speeds. It indicates
Fig. 5. The compression pressure distributions under the hybrid friction for various rotating speeds.
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425
Fig. 9. The relationship mlim and D/h with various rotating angular speeds. Fig. 6. The compression pressure distribution with various diameter thickness ratios.
Fig. 7. The compression pressure considering hybrid friction for various frictional coefficients.
that there is a rotating effect ðob ¼ 5 rad=sÞ. The dashed line represents the Coulomb friction model and the solid line represents the hybrid friction model. The compression force always increases with increasing D/h. The compression force with the hybrid friction is smaller than that with
Fig. 10. The comparison of the compression force between the two models with various densities.
Coulomb friction, and the difference becomes larger as the value of D/h increases. Fig. 9 shows the relationship mlim and D/h with various rotating angular speeds, where mlim denotes the frictional coefficient firstly generating the sticking friction. The hybrid friction is more difficult to occur as mlim is larger. The value of mlim with the rotating effect is larger than that without the rotating effect. Then the value of mlim becomes smaller as D/ h increases, however, the curve with the rotating effect decreases slowly. Fig. 10 shows the comparison of the compression force between the two models with various densities. It indicates that the compression force in the hybrid friction model is much smaller than that in the Coulomb friction model under the fixed density. When the density of the material increases, the compression forces both decrease for the two models.
4. Conclusions Fig. 8. The comparison of compression force with various rotating angular speed between the Coulomb friction and the hybrid friction models.
Because the sticking friction makes the surface of the material rough, and influences the quality of the product.
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M.N. Huang, G.Y. Tzou / Journal of Materials Processing Technology 125–126 (2002) 421–426
Table 1 Summary of compression characteristics for the hybrid friction modela Compression condition
p/Y
q/Y
P
m ob D/h r
% & % &
% & % &
% & % &
The advice and financial support of the NSC are gratefully acknowledged.
mlim
References % & %
a The symbol ‘‘%’’ indicates that the compression characteristics increase with increasing the compression conditions; symbol ‘‘&’’ indicates that it is decreasing.
It should be avoided. The effects of D/h, m and ob on the compression pressure, radial stress and compression force are investigated effectively as shown in Table 1. The main conclusions are as follows: (1) The rotating of the circular disk can reduce the compression pressure, radial stress and compression force. (2) The hybrid friction can easily occur under high frictional coefficient and high diameter thickness ratio. (3) The limiting friction coefficient is larger with the rotating effect and higher density; it indicates and the hybrid friction is harder to occur.
Acknowledgements This work was sponsored bytheNationalScience Council of Republic of China under grant No. NSC-90-2212-E-132-005.
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