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Solution of rectangular bar forging with bulging of sides by strain rate vector inner-product
Solution of rectangular bar forging with bulging of sides by strain rate vector inner-product WANG Lei1, JIN Wen-zhong2, ZHAO De-wen3, LIU Xiang-hua3 1. School of Materials Science and Engineering, Shanghai University, Shanghai 200072, China; 2. Shanghai Shenhua Acoustics Equipment Co. Ltd., Shanghai 201709, China; 3. State Key Laboratory of Rolling and Automation, Northeastern University, Shenyang 110004, China Received 19 July 2010; accepted 17 November 2010 Abstract: A new linear integration solution was constructed for determining the total pressure developed during forging of a rectangular bar with flat tools. The effective strain rate for rectangular bar forging with bulging was expressed in terms of four-dimensional strain rate vector. The inner-product of the vector was termwise integrated and summed. The integral mean value theorem was applied to determining the ratio of the strain rate components and the values of direction cosine of the vector and then an analytical solution of stress effective factor was obtained. The compression experiments of pure lead bar were performed to test the accuracy of the solution. The optimized results of total pressure by golden section search were compared with those of the indicator readings of the testing machine. It indicates that the optimized total pressures are 2.60%í10.14% higher than those measured. The solution is available and still an upper-bound solution. Key words: rectangular bar forging; bulging; strain rate vector; inner-product; golden section search
effective factor is obtained.
1 Introduction 2 Velocity field Forging a bar with flat tools is one of the simplest and most widely used processes in industry. However, its analysis is far from being simple. Hence a few approximate solutions have come forth, such as the upper bound [1í2], the lower bound [3í4] and slip line [5]. With the development of computers, numerical methods have been used to simulate the forging process [6í7]. ZHANG and XIE [8] and KANG et al [9] simulated forging process by FEM. GRASS et al [10] and GAO et al [11] used 3D-FEM simulation. And UBET has been successfully applied to solving the forging process by the authors [12í13]. These methods can all be applied to solving the forging, but just the numeric results are obtained. In this work, the method so called inner-product of strain rate vector [14í15] is constructed and first applied to solving the bar forging and the analytic result of stress
The origin of Cartesian coordinates is taken as the geometric centre of the body. The axes X, Y and Z divide the deforming zone into eight identical regions. So only the eighth is considered (Fig. 1). For any point with coordinates x, y and z, the velocity field is given by [1] v = vx i v y j v z k
vx
vz
v0 x v l ʌx 2 A 0 sin , vy 2l h h S v 0z h
(1) A
v0 ʌx y cos , 2l h
(2)
where vx, vy, vz are components of particle velocity in Cartesian coordinate system; v0 is velocity of the top die; A is a constant to be determined by optimization on energy dissipation.
Foundation item: Project (51074052) supported by the National Natural Science Foundation of China; Project (20100470676) supported by the China Postdoctoral Science Foundation Corresponding author: ZHAO De-wen; Tel: +86-24-83686423; E-mail: [email protected] DOI: 10.1016/S1003-6326(11)60867-4
WANG Lei, et al/Trans. Nonferrous Met. Soc. China 21(2011) 1367í1372
1368
3 Total power 3.1 Internal deformation power The integrand of plastic power is expressed by
³ V VH dV
Nd
b0 l 2 2 V s ³ ³ Hx2 H y2 Hz2 2Hxy hdxdy 0 0 3
2 2 2 2 b0 l Hx H y Hz 2Hxy 2 dxdy V s h³ ³ 0 0 2 3 Hx2 H y2 Hz2 2Hxy
Fig. 1 Rectangular bar subjected to compression
The velocity field satisfies the boundary conditions: v at x=0, v y A 0 y ; at x=l, vy=0. and the h incompressibility condition, i.e. wvx wv y wvz wx wy wz
divv
Hx H y Hz
0
(3)
where
Hx
v0 Av0 ʌx , H y cos h h 2l
Hz
Hxz
Hzx
v0 , Hxy h
H yz
Av0 ʌx , cos h 2l
ʌAv0 ʌx y sin , 4lh 2l
Hzy
2 Vsh 3
Hx dxdy b0 l °³0 ³0 2 2 2 H H ° § y · § H · § xy · ® 1 ¨ ¸ ¨ z ¸ 2¨ ¸ H H ° © x¹ © x¹ © Hx ¹ °¯ H y dxdy b0 l ³0 ³0 2 2 2 § Hx · § Hz · § Hxy · 1 ¨ ¸ ¨ ¸ 2¨ ¨ H y ¸ ¨ H y ¸ ¨ H y ¸¸ © ¹ © ¹ © ¹ b0
where Hij is the component of strain rate tensor. From the mean value theorem of integral and Eq.(4), mean values of the strain rate can be obtained as follows:
b0
³0 ³0
1 b0 l § v0 Av0 ʌx · cos ¸dxdy b0 l ³ 0 ³ 0 ¨© h h 2l ¹ v0 2 Av0 , h ʌh 2 Av0 H y , ʌh Hz Hz ,
Hxy
1 l b0 Hxy dxdy b0 l ³ 0 ³ 0
Ab0 v0 4lh
Hii
vx vy
2ʌAb
vx
2
ʌ l 8 Al
§ H 2¨ x ¨ Hxy ©
2
· § H y ¸¸ ¨¨ ¹ © Hxy
2
· § Hz ¸¸ ¨¨ ¹ © Hxy
· ¸¸ ¹
2
½ ° ° ¾ ° °¿
b0 l 2 V s h ³ ³ H H 0 dxdy 0 0 3 b0 l b0 l 2 V s h ª ³ ³ Hx l1dxdy ³ ³ H y l2 dxdy «¬ 0 0 0 0 3
(5)
b0
l
b0
l
º
³ 0 ³ 0 Hz l3dxdy 2³ 0 ³ 0 Hxy l4dxdy »¼ 2 V s h I1 I 2 I 3 I 4 3
With the help of Eq. (2), the ratio of mean velocity can also be calculated as lv0 4 Alv0 2 , vy 2h ʌ h
2
2Hxy dxdy
0
1 l vx dx l ³0
§ Hxy · § H · § H y · 1 ¨ x ¸ ¨ ¸ 2¨ ¸ © Hz ¹ © Hz ¹ © Hz ¹
b0 l 2 V s h ³ ³ Hx l1 H y l2 Hz l3 2Hxy l4 dxdy 0 0 3
It can be seen that Eq. (5) still satisfies
Hii
2
2
l
1 b0 l Hx dxdy b0 l ³ 0 ³ 0
Hx
Hz dxdy
l
³0 ³0
(4)
0
Ab0 v0 , ʌh
(6)
where vector;
H Hx e1 H y e2 Hz e3 2Hxy e4 H 0
l1e1 l2 e2 l3e3 l4 e4 is
(7) is strain rate unit
vector;
l澠 i i 1, 2,3, 4) is direction cosines of unit vector; and I i (i 1, 2,3, 4) is termwise integration of the strain rate vector. From Eq. (7), the direction cosines are
WANG Lei, et al/Trans. Nonferrous Met. Soc. China 21(2011) 1367í1372
°l °1 ° ° ° °l2 ° ° ® ° °l3 ° ° ° ° °l4 ¯°
2 ª § H · 2 § H · 2 § H · º «1 ¨ y ¸ ¨ z ¸ 2 ¨ xy ¸ » H H « © Hx ¹ »¼ ¬ © x¹ © x¹
ª § «1 ¨ H x « ¨© H y ¬
2
· § Hz ¸ ¨ ¸ ¨ H y ¹ ©
2
· § Hxy ¸ 2¨ ¸ ¨ H y ¹ ©
1/ 2
l 2 ʌ 2 A 4 A2 l 2 ʌ 2 l 2 2
2 º 1/ 2
· ¸ » ¸ » ¹ ¼
(8)
ª § H ·2 § H ·2 § H · «1 ¨ x ¸ ¨ y ¸ 2 ¨ xy ¸ » « © Hz ¹ © Hz ¹ © Hz ¹ »¼ ¬ 2
· § H y ¸ ¨ ¸ ¨ Hxy ¹ ©
2
· § Hz ¸ ¨ ¸ ¨ ¹ © Hxy
· ¸ ¸ ¹
b0
Let W f
³0 |³
b0
2 º 1/ 2
Nf
» » ¼
l
0
2 ª § H · 2 § H · 2 § Hxy · º y z « ³ 0 H x «1 ¨© Hx ¸¹ ¨© Hx ¸¹ 2 ¨© Hx ¸¹ »» ¬ ¼
1/ 2
ª § H y H ³ 0 x ««1 ¨¨ Hx ¬ ©
1/ 2
l
l
2 2 2 · § Hz · § Hxy · º ¸ ¨ ¸ 2¨ ¸ » ¸ © Hx ¹ ¨ Hx ¸ » ¹ © ¹ ¼
dxdy
l § v0
b0
I1
dxdy
Av0 ʌx · cos ¸dxdy h 2l ¹
2 2 § ʌAb0 · § 2 A · § ʌ · 1 ¨ ¸¸ ¸ ¨ ¸ 2 ¨¨ A A ʌ 2 ʌ 2 © ¹ © ¹ © 4l ʌ 2 A ¹
2
2
1 ʌAb0 2 8
With the same procedure, the other termwise integration results are
I2
4 A2b0l 2 v0 ʌh 4 A2 l 2 l 2 ʌ 2 A ʌ 2 l 2 2
I3
2
I4
, 'v f
3
vf
vx2 v 2y , the
3
'v f ds b0
mV s 3
b0
l
³0 ³0
vx 2 v y 2 dxdy
2
l
³ 0 ³ 0 vx
§ vy · 1 ¨ ¸ dxdy © vx ¹
where IJf is the shear along surface of discontinuity; k is von Mises’ constant of yield ıs is effective flow stress; 'v f is the discontinuity. Substituting v y vx of Eq. (6) for v y vx
velocity criterion; velocity in above
Nf
mV s b0lv0 2 3ʌ
2
ʌ
2
8A
2
l2 h
2
4ʌ 2 A2
b02 h2
(10)
3.3 Shear power From Eq. (2), the velocity discontinuity at interface between deformation and external zones can be written as 'vz
vz
1 h b0 vz dzdy b0 h ³ 0 ³ 0
v0 2
So shear power becomes Ns
³ s k 'vt
kb0 hv0 2
dzdy
(11)
3.4 Stress effective factor and minimization Let applied power pb0lv0 N d N f N s , and
substitute Eqs. (9), (10) and (11) into it, then rearranging leads to 1 ʌAb0 2 8
v0 ʌb0 l 2 h
ʌ 2 l 2 l 2 ʌ 2 A 4 A2 l 2
³s W f mV s
· b0l 2 v0 § 4 A2 4 A ʌ ¸¸ ¨ ¨ h © ʌ ¹ l 2 ʌ 2 A 4 A2 l 2 ʌ 2 l 2
mV s
mk
integrand, and integrating leads to
Substituting Eqs. (4) and (5) into above integral, it follows that
³ 0 ³ 0 ¨© h
(9)
friction power is
³ 0 ³ 0 Hx l1dxdy b0
ʌ 2 A2b0 2 8
3.2 Friction power
The first termwise integration of strain rate vector inner product I1 is I1
2 V s b0 v0 3 ʌ
Nd
2 º 1/ 2
ª § «2 ¨ H x « ¨ H ¬ © xy
1369
1 ʌAb0 2 8
ʌA2 b03v0 8h
1 ʌAb0 2 l 2 ʌ 2 A 2 4 A2l 2 ʌ 2l 2 8
Substituting the results of I1, I2, I3 and I4 into Eq. (7) and rearranging yields
nV
p
Vs
2 4 A 8 A2 A2 b0 2 2 2 3 ʌ 8 l2 ʌ 2
2 h 8A · l2 § 2 b0 A ʌ 4 ¨ ¸ ʌ ¹ h2 h 2 2 3l 2 3ʌ ©
m
(12)
where nı is stress effective factor. It is obviously that the stress effective factor nı is a function of A, m, b0/h and l/h, also is a analytical solution. The optimum values of A and nı by the golden section search are shown in Fig. 2.
WANG Lei, et al/Trans. Nonferrous Met. Soc. China 21(2011) 1367í1372
1370
nV V s
F0
2l 2b1 2b0
15.48 kN
2 u 1 000
Comparing it with the measured value Fm in Table 1, the relative error can (E) be expressed as 15.48 14.10 u 100% 14.10
E
9.79%
With the same procedure, the optimized results of No. 2 and 3 specimens can also be obtained and listed in Table 2. Table 2 Optimized results for three groups of specimens
4 Experiment
b0/h
A
nı
F0/kN
E/%
1
1.62
1.10
0.78
1.22
15.48
9.79
2
1.72
2.28
0.61
1.25
30.78
2.60
3
1.65
4.40
0.34
1.32
15.97
10.14
5 Discussion
In order to validate the solution in this work, the compression test was performed on 200 kN universal material testing machine. The dimensions of the three groups of pure lead specimens and the indicator readings of the machine during compressing are listed in Table 1. The ram speed was 30 mm/min. Table 1 Specimen dimension and measured force 2h0/mm
l/h
From the results we can know that the relative errors between the optimized results and the measured ones are from 2.60% to 10.14%, all admitted in engineering.
Fig. 2 Flow chart of golden section search
No. 2b0/mm
No.
2l/mm 2b1/mm
2h/mm Fm/kN
1
20.33
20.33
30.0
21.70
18.50
14.10
2
39.86
19.70
30.0
42.16
17.49
30.00
3
39.97
10.12
15.0
40.75
9.08
14.50
In order to examine the characteristics of the present solution, the optimum values of A and nı were optimized for the values of l/h varying from 0.5 to 5.5 and b0/h equal to 1.10, 2.28 and 4.40 and m from 0.2 to 1.0. The value of parameter A signifies the extent of bulging. From Fig. 3 it can be seen that bulging parameter A increases with increasing l/h or decreasing b0/h for a given m. Figure 4 shows that nı increases as m increases for given values of b0/h and l/h, but always exists a minimum value of nı in each curve. Fig. 5 shows that the value of nı increases with increasing b0/h and m for a given l/h.
Taking No.1 specimen as an example, the calculating procedure is given in detail as follows. From Table 1, l/h=1.62, b0/h=1.10, and m=0.2 (for quenched steel and lapped face). Then inputting the values of l/h, b0/h, and m into Eq. (12) and optimizing by flow chart in Fig. 2, the optimum result becomes: A=0.78, nı=1.22. From Table 1, the strain and the strain rate for 20.33 H 0.094 ; H No.1 specimen are H ln 18.50 t 30 60 20.33 ln 0.026 sí1. 20.33 18.50 18.50 According to the values of İ and H , the yield stress for pure lead is ıs=20.05 MPa [16]. Then the total compression force for No.1 sample is
Fig. 3 Influence of l/h and b0/h on optimum values of A
WANG Lei, et al/Trans. Nonferrous Met. Soc. China 21(2011) 1367í1372
1371
6 Conclusions 1) With the help of strain rate vector inner-product, the analytical solution of stress effective factor nı for rectangular bar forging is obtained. And nı is a function of A, m, b0/h and l/h. 2) Through pure lead compression test, the optimized results by above solution are 2.60%í10.14% higher than those measured. 3) The relationship among A, l/h, b0/h and m can be described. Bulging parameter A increases with increasing l/h or decreasing b0/h for a given m.
References [1]
SAGAR R, JUNEJA B L. An upper bound solution for flat tool forging taking into account the bulging of sides [J]. International Journal of Machine Tool Design and Research, 1979, 19(4): 253í258.
[2]
WANG Zhen-fan. Upper bound solution and its application in metal forming [M]. Shenyang: Northeast University of Technology Press, 1991: 200í203. (in Chinese)
[3]
HASHMI M S J. A lower bound solution for the plane strain extrusion forging process [J]. Mathematical and Computer Modelling, 1988, 11: 1183í1188.
[4]
SINGH D N, BASUDHAR P K. Optimal lower-bound solutions of plane-strain metal indentation and extrusion problems [J]. Journal of Materials Processing Technology, 1995, 49(1í2): 75í84.
[5]
SAMOLYK G, PATER Z. Application of the slip-line field method to the analysis of die cavity filling [J]. Journal of Materials Processing Technology, 2004, 153í154: 729í735.
[6]
KOPP R. Some current development trends in metal-forming technology [J]. Journal of Materials Processing Technology, 1996, 60(1í4): 1í9.
[7]
PITTMAN J F T. Numerical analysis of forming processes [M]. New
[8]
ZHANG Zhi-hao, XIE Jian-xin. A numerical simulation of
York: John Wiley and Sons, 1984: 32í33. super-plastic die forging process for Zr-based bulk metallic glass spur
Fig. 4 Influence of l/h, b0/h and m on nı for optimum values of A
gear [J]. Material Science and Engineering A, 2006, 433(1í2): 323í328. [9]
KANG G J, SONG W J, KIM J, KANG B S, PARK H J. Numerical approach to forging process of a gear with inner cam profile using FEM [J]. Journal of Material Processing Technology, 2005, 164í165: 1212í1217.
[10]
GRASS H, KREMPASZKY C, WERNER E. 3-D FEM-simulation of hot forming processes for the production of a connecting rod [J]. Computational Materials Science, 2006, 36(4): 480í489.
[11]
GAO Tao, YANG He, LIU Yu-li. Backward tracing simulation of precision forging process for blade based on 3D FEM [J]. Transactions of Nonferrous Metals Society of China, 2006, 36(S2): 639í644.
[12]
HWANG B C, HONG S J, BAE W B. An UBET analysis of the non-axisymmetric extrusion/forging process [J]. Journal of Material
Fig. 5 Influence of b0/h on nı for given l/h
Processing Technology, 2001, 111(1í3): 135í141.
WANG Lei, et al/Trans. Nonferrous Met. Soc. China 21(2011) 1367í1372
1372 [13]
MOLLER J, TRAN L, SHRADER D, DOUGLAS J R, KUHLMAN
[15]
G W. Augmented upper bound element technique for prediction of
Inner-product of strain rate vector through direction cosine in
temperature and strain in forgings [J]. Journal of Material Processing
coordinates for disk forging [J]. Transactions of Nonferrous Metals
Technology, 2004, 152(2): 162í175. [14]
ZHAO De-wen, JIN Wen-zhong, WANG Lei, LIU Xiang-hua.
Society of China, 2006, 16(6): 1320í1324.
ZHAO De-wen. Mathematical solutions of continuum forming force
[16]
LOIZOU N, SIMS R B. The yield stress of pure lead in compression
[M]. Shenyang: Northeastern University Press, 2003: 421í425. (in
[J]. Journal of the Mechanics and Physics of Solids, 1953: 1(4):
Chinese)
234í243.
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effective factor is obtained.
1 Introduction 2 Velocity field Forging a bar with flat tools is one of the simplest and most widely used processes in industry. However, its analysis is far from being simple. Hence a few approximate solutions have come forth, such as the upper bound [1í2], the lower bound [3í4] and slip line [5]. With the development of computers, numerical methods have been used to simulate the forging process [6í7]. ZHANG and XIE [8] and KANG et al [9] simulated forging process by FEM. GRASS et al [10] and GAO et al [11] used 3D-FEM simulation. And UBET has been successfully applied to solving the forging process by the authors [12í13]. These methods can all be applied to solving the forging, but just the numeric results are obtained. In this work, the method so called inner-product of strain rate vector [14í15] is constructed and first applied to solving the bar forging and the analytic result of stress
The origin of Cartesian coordinates is taken as the geometric centre of the body. The axes X, Y and Z divide the deforming zone into eight identical regions. So only the eighth is considered (Fig. 1). For any point with coordinates x, y and z, the velocity field is given by [1] v = vx i v y j v z k
vx
vz
v0 x v l ʌx 2 A 0 sin , vy 2l h h S v 0z h
(1) A
v0 ʌx y cos , 2l h
(2)
where vx, vy, vz are components of particle velocity in Cartesian coordinate system; v0 is velocity of the top die; A is a constant to be determined by optimization on energy dissipation.
Foundation item: Project (51074052) supported by the National Natural Science Foundation of China; Project (20100470676) supported by the China Postdoctoral Science Foundation Corresponding author: ZHAO De-wen; Tel: +86-24-83686423; E-mail: [email protected] DOI: 10.1016/S1003-6326(11)60867-4
WANG Lei, et al/Trans. Nonferrous Met. Soc. China 21(2011) 1367í1372
1368
3 Total power 3.1 Internal deformation power The integrand of plastic power is expressed by
³ V VH dV
Nd
b0 l 2 2 V s ³ ³ Hx2 H y2 Hz2 2Hxy hdxdy 0 0 3
2 2 2 2 b0 l Hx H y Hz 2Hxy 2 dxdy V s h³ ³ 0 0 2 3 Hx2 H y2 Hz2 2Hxy
Fig. 1 Rectangular bar subjected to compression
The velocity field satisfies the boundary conditions: v at x=0, v y A 0 y ; at x=l, vy=0. and the h incompressibility condition, i.e. wvx wv y wvz wx wy wz
divv
Hx H y Hz
0
(3)
where
Hx
v0 Av0 ʌx , H y cos h h 2l
Hz
Hxz
Hzx
v0 , Hxy h
H yz
Av0 ʌx , cos h 2l
ʌAv0 ʌx y sin , 4lh 2l
Hzy
2 Vsh 3
Hx dxdy b0 l °³0 ³0 2 2 2 H H ° § y · § H · § xy · ® 1 ¨ ¸ ¨ z ¸ 2¨ ¸ H H ° © x¹ © x¹ © Hx ¹ °¯ H y dxdy b0 l ³0 ³0 2 2 2 § Hx · § Hz · § Hxy · 1 ¨ ¸ ¨ ¸ 2¨ ¨ H y ¸ ¨ H y ¸ ¨ H y ¸¸ © ¹ © ¹ © ¹ b0
where Hij is the component of strain rate tensor. From the mean value theorem of integral and Eq.(4), mean values of the strain rate can be obtained as follows:
b0
³0 ³0
1 b0 l § v0 Av0 ʌx · cos ¸dxdy b0 l ³ 0 ³ 0 ¨© h h 2l ¹ v0 2 Av0 , h ʌh 2 Av0 H y , ʌh Hz Hz ,
Hxy
1 l b0 Hxy dxdy b0 l ³ 0 ³ 0
Ab0 v0 4lh
Hii
vx vy
2ʌAb
vx
2
ʌ l 8 Al
§ H 2¨ x ¨ Hxy ©
2
· § H y ¸¸ ¨¨ ¹ © Hxy
2
· § Hz ¸¸ ¨¨ ¹ © Hxy
· ¸¸ ¹
2
½ ° ° ¾ ° °¿
b0 l 2 V s h ³ ³ H H 0 dxdy 0 0 3 b0 l b0 l 2 V s h ª ³ ³ Hx l1dxdy ³ ³ H y l2 dxdy «¬ 0 0 0 0 3
(5)
b0
l
b0
l
º
³ 0 ³ 0 Hz l3dxdy 2³ 0 ³ 0 Hxy l4dxdy »¼ 2 V s h I1 I 2 I 3 I 4 3
With the help of Eq. (2), the ratio of mean velocity can also be calculated as lv0 4 Alv0 2 , vy 2h ʌ h
2
2Hxy dxdy
0
1 l vx dx l ³0
§ Hxy · § H · § H y · 1 ¨ x ¸ ¨ ¸ 2¨ ¸ © Hz ¹ © Hz ¹ © Hz ¹
b0 l 2 V s h ³ ³ Hx l1 H y l2 Hz l3 2Hxy l4 dxdy 0 0 3
It can be seen that Eq. (5) still satisfies
Hii
2
2
l
1 b0 l Hx dxdy b0 l ³ 0 ³ 0
Hx
Hz dxdy
l
³0 ³0
(4)
0
Ab0 v0 , ʌh
(6)
where vector;
H Hx e1 H y e2 Hz e3 2Hxy e4 H 0
l1e1 l2 e2 l3e3 l4 e4 is
(7) is strain rate unit
vector;
l澠 i i 1, 2,3, 4) is direction cosines of unit vector; and I i (i 1, 2,3, 4) is termwise integration of the strain rate vector. From Eq. (7), the direction cosines are
WANG Lei, et al/Trans. Nonferrous Met. Soc. China 21(2011) 1367í1372
°l °1 ° ° ° °l2 ° ° ® ° °l3 ° ° ° ° °l4 ¯°
2 ª § H · 2 § H · 2 § H · º «1 ¨ y ¸ ¨ z ¸ 2 ¨ xy ¸ » H H « © Hx ¹ »¼ ¬ © x¹ © x¹
ª § «1 ¨ H x « ¨© H y ¬
2
· § Hz ¸ ¨ ¸ ¨ H y ¹ ©
2
· § Hxy ¸ 2¨ ¸ ¨ H y ¹ ©
1/ 2
l 2 ʌ 2 A 4 A2 l 2 ʌ 2 l 2 2
2 º 1/ 2
· ¸ » ¸ » ¹ ¼
(8)
ª § H ·2 § H ·2 § H · «1 ¨ x ¸ ¨ y ¸ 2 ¨ xy ¸ » « © Hz ¹ © Hz ¹ © Hz ¹ »¼ ¬ 2
· § H y ¸ ¨ ¸ ¨ Hxy ¹ ©
2
· § Hz ¸ ¨ ¸ ¨ ¹ © Hxy
· ¸ ¸ ¹
b0
Let W f
³0 |³
b0
2 º 1/ 2
Nf
» » ¼
l
0
2 ª § H · 2 § H · 2 § Hxy · º y z « ³ 0 H x «1 ¨© Hx ¸¹ ¨© Hx ¸¹ 2 ¨© Hx ¸¹ »» ¬ ¼
1/ 2
ª § H y H ³ 0 x ««1 ¨¨ Hx ¬ ©
1/ 2
l
l
2 2 2 · § Hz · § Hxy · º ¸ ¨ ¸ 2¨ ¸ » ¸ © Hx ¹ ¨ Hx ¸ » ¹ © ¹ ¼
dxdy
l § v0
b0
I1
dxdy
Av0 ʌx · cos ¸dxdy h 2l ¹
2 2 § ʌAb0 · § 2 A · § ʌ · 1 ¨ ¸¸ ¸ ¨ ¸ 2 ¨¨ A A ʌ 2 ʌ 2 © ¹ © ¹ © 4l ʌ 2 A ¹
2
2
1 ʌAb0 2 8
With the same procedure, the other termwise integration results are
I2
4 A2b0l 2 v0 ʌh 4 A2 l 2 l 2 ʌ 2 A ʌ 2 l 2 2
I3
2
I4
, 'v f
3
vf
vx2 v 2y , the
3
'v f ds b0
mV s 3
b0
l
³0 ³0
vx 2 v y 2 dxdy
2
l
³ 0 ³ 0 vx
§ vy · 1 ¨ ¸ dxdy © vx ¹
where IJf is the shear along surface of discontinuity; k is von Mises’ constant of yield ıs is effective flow stress; 'v f is the discontinuity. Substituting v y vx of Eq. (6) for v y vx
velocity criterion; velocity in above
Nf
mV s b0lv0 2 3ʌ
2
ʌ
2
8A
2
l2 h
2
4ʌ 2 A2
b02 h2
(10)
3.3 Shear power From Eq. (2), the velocity discontinuity at interface between deformation and external zones can be written as 'vz
vz
1 h b0 vz dzdy b0 h ³ 0 ³ 0
v0 2
So shear power becomes Ns
³ s k 'vt
kb0 hv0 2
dzdy
(11)
3.4 Stress effective factor and minimization Let applied power pb0lv0 N d N f N s , and
substitute Eqs. (9), (10) and (11) into it, then rearranging leads to 1 ʌAb0 2 8
v0 ʌb0 l 2 h
ʌ 2 l 2 l 2 ʌ 2 A 4 A2 l 2
³s W f mV s
· b0l 2 v0 § 4 A2 4 A ʌ ¸¸ ¨ ¨ h © ʌ ¹ l 2 ʌ 2 A 4 A2 l 2 ʌ 2 l 2
mV s
mk
integrand, and integrating leads to
Substituting Eqs. (4) and (5) into above integral, it follows that
³ 0 ³ 0 ¨© h
(9)
friction power is
³ 0 ³ 0 Hx l1dxdy b0
ʌ 2 A2b0 2 8
3.2 Friction power
The first termwise integration of strain rate vector inner product I1 is I1
2 V s b0 v0 3 ʌ
Nd
2 º 1/ 2
ª § «2 ¨ H x « ¨ H ¬ © xy
1369
1 ʌAb0 2 8
ʌA2 b03v0 8h
1 ʌAb0 2 l 2 ʌ 2 A 2 4 A2l 2 ʌ 2l 2 8
Substituting the results of I1, I2, I3 and I4 into Eq. (7) and rearranging yields
nV
p
Vs
2 4 A 8 A2 A2 b0 2 2 2 3 ʌ 8 l2 ʌ 2
2 h 8A · l2 § 2 b0 A ʌ 4 ¨ ¸ ʌ ¹ h2 h 2 2 3l 2 3ʌ ©
m
(12)
where nı is stress effective factor. It is obviously that the stress effective factor nı is a function of A, m, b0/h and l/h, also is a analytical solution. The optimum values of A and nı by the golden section search are shown in Fig. 2.
WANG Lei, et al/Trans. Nonferrous Met. Soc. China 21(2011) 1367í1372
1370
nV V s
F0
2l 2b1 2b0
15.48 kN
2 u 1 000
Comparing it with the measured value Fm in Table 1, the relative error can (E) be expressed as 15.48 14.10 u 100% 14.10
E
9.79%
With the same procedure, the optimized results of No. 2 and 3 specimens can also be obtained and listed in Table 2. Table 2 Optimized results for three groups of specimens
4 Experiment
b0/h
A
nı
F0/kN
E/%
1
1.62
1.10
0.78
1.22
15.48
9.79
2
1.72
2.28
0.61
1.25
30.78
2.60
3
1.65
4.40
0.34
1.32
15.97
10.14
5 Discussion
In order to validate the solution in this work, the compression test was performed on 200 kN universal material testing machine. The dimensions of the three groups of pure lead specimens and the indicator readings of the machine during compressing are listed in Table 1. The ram speed was 30 mm/min. Table 1 Specimen dimension and measured force 2h0/mm
l/h
From the results we can know that the relative errors between the optimized results and the measured ones are from 2.60% to 10.14%, all admitted in engineering.
Fig. 2 Flow chart of golden section search
No. 2b0/mm
No.
2l/mm 2b1/mm
2h/mm Fm/kN
1
20.33
20.33
30.0
21.70
18.50
14.10
2
39.86
19.70
30.0
42.16
17.49
30.00
3
39.97
10.12
15.0
40.75
9.08
14.50
In order to examine the characteristics of the present solution, the optimum values of A and nı were optimized for the values of l/h varying from 0.5 to 5.5 and b0/h equal to 1.10, 2.28 and 4.40 and m from 0.2 to 1.0. The value of parameter A signifies the extent of bulging. From Fig. 3 it can be seen that bulging parameter A increases with increasing l/h or decreasing b0/h for a given m. Figure 4 shows that nı increases as m increases for given values of b0/h and l/h, but always exists a minimum value of nı in each curve. Fig. 5 shows that the value of nı increases with increasing b0/h and m for a given l/h.
Taking No.1 specimen as an example, the calculating procedure is given in detail as follows. From Table 1, l/h=1.62, b0/h=1.10, and m=0.2 (for quenched steel and lapped face). Then inputting the values of l/h, b0/h, and m into Eq. (12) and optimizing by flow chart in Fig. 2, the optimum result becomes: A=0.78, nı=1.22. From Table 1, the strain and the strain rate for 20.33 H 0.094 ; H No.1 specimen are H ln 18.50 t 30 60 20.33 ln 0.026 sí1. 20.33 18.50 18.50 According to the values of İ and H , the yield stress for pure lead is ıs=20.05 MPa [16]. Then the total compression force for No.1 sample is
Fig. 3 Influence of l/h and b0/h on optimum values of A
WANG Lei, et al/Trans. Nonferrous Met. Soc. China 21(2011) 1367í1372
1371
6 Conclusions 1) With the help of strain rate vector inner-product, the analytical solution of stress effective factor nı for rectangular bar forging is obtained. And nı is a function of A, m, b0/h and l/h. 2) Through pure lead compression test, the optimized results by above solution are 2.60%í10.14% higher than those measured. 3) The relationship among A, l/h, b0/h and m can be described. Bulging parameter A increases with increasing l/h or decreasing b0/h for a given m.
References [1]
SAGAR R, JUNEJA B L. An upper bound solution for flat tool forging taking into account the bulging of sides [J]. International Journal of Machine Tool Design and Research, 1979, 19(4): 253í258.
[2]
WANG Zhen-fan. Upper bound solution and its application in metal forming [M]. Shenyang: Northeast University of Technology Press, 1991: 200í203. (in Chinese)
[3]
HASHMI M S J. A lower bound solution for the plane strain extrusion forging process [J]. Mathematical and Computer Modelling, 1988, 11: 1183í1188.
[4]
SINGH D N, BASUDHAR P K. Optimal lower-bound solutions of plane-strain metal indentation and extrusion problems [J]. Journal of Materials Processing Technology, 1995, 49(1í2): 75í84.
[5]
SAMOLYK G, PATER Z. Application of the slip-line field method to the analysis of die cavity filling [J]. Journal of Materials Processing Technology, 2004, 153í154: 729í735.
[6]
KOPP R. Some current development trends in metal-forming technology [J]. Journal of Materials Processing Technology, 1996, 60(1í4): 1í9.
[7]
PITTMAN J F T. Numerical analysis of forming processes [M]. New
[8]
ZHANG Zhi-hao, XIE Jian-xin. A numerical simulation of
York: John Wiley and Sons, 1984: 32í33. super-plastic die forging process for Zr-based bulk metallic glass spur
Fig. 4 Influence of l/h, b0/h and m on nı for optimum values of A
gear [J]. Material Science and Engineering A, 2006, 433(1í2): 323í328. [9]
KANG G J, SONG W J, KIM J, KANG B S, PARK H J. Numerical approach to forging process of a gear with inner cam profile using FEM [J]. Journal of Material Processing Technology, 2005, 164í165: 1212í1217.
[10]
GRASS H, KREMPASZKY C, WERNER E. 3-D FEM-simulation of hot forming processes for the production of a connecting rod [J]. Computational Materials Science, 2006, 36(4): 480í489.
[11]
GAO Tao, YANG He, LIU Yu-li. Backward tracing simulation of precision forging process for blade based on 3D FEM [J]. Transactions of Nonferrous Metals Society of China, 2006, 36(S2): 639í644.
[12]
HWANG B C, HONG S J, BAE W B. An UBET analysis of the non-axisymmetric extrusion/forging process [J]. Journal of Material
Fig. 5 Influence of b0/h on nı for given l/h
Processing Technology, 2001, 111(1í3): 135í141.
WANG Lei, et al/Trans. Nonferrous Met. Soc. China 21(2011) 1367í1372
1372 [13]
MOLLER J, TRAN L, SHRADER D, DOUGLAS J R, KUHLMAN
[15]
G W. Augmented upper bound element technique for prediction of
Inner-product of strain rate vector through direction cosine in
temperature and strain in forgings [J]. Journal of Material Processing
coordinates for disk forging [J]. Transactions of Nonferrous Metals
Technology, 2004, 152(2): 162í175. [14]
ZHAO De-wen, JIN Wen-zhong, WANG Lei, LIU Xiang-hua.
Society of China, 2006, 16(6): 1320í1324.
ZHAO De-wen. Mathematical solutions of continuum forming force
[16]
LOIZOU N, SIMS R B. The yield stress of pure lead in compression
[M]. Shenyang: Northeastern University Press, 2003: 421í425. (in
[J]. Journal of the Mechanics and Physics of Solids, 1953: 1(4):
Chinese)
234í243.
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