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Friction coefficient of upsetting with a procedure of combining the inverse model and the Tikhonov method
ARTICLE IN PRESS
International Journal of Mechanical Sciences 48 (2006) 717–725 www.elsevier.com/locate/ijmecsci
Friction coefficient of upsetting with a procedure of combining the inverse model and the Tikhonov method Zone-Ching Lina,, Ven-Huei Linb a
Department of Mechanical Engineering, National Taiwan University of Science and Technology, 43, Keelung Road, Section 4, Taipei, Taiwan 106, ROC b Department of Mechanical Engineering, Hwa Hsia Institute of Technology, Taiwan, ROC Received 7 April 2005; received in revised form 25 January 2006; accepted 31 January 2006 Available online 20 March 2006
Abstract The purpose of this paper is to develop an inverse procedure, which is used to obtain the history of the Coulomb’s friction coefficient on the contact surfaces between the workpiece and the die during the AA1050 aluminum upsetting process. This procedure is based on the experimental upsetting normal loadings and combines the matrix-presentation linear least-squares errors method of inverse elastic–plastic large deformation finite element model developed in this paper with the regularization of Tikhonov method. The square of the difference of the friction coefficients between successive stages is adopted as the stabilizing function of the regularization of Tikhonov method. In this paper, the Coulomb’s friction model is adopted before the sticking effect occurs. After the sticking effect occurrence, the shear strength friction model is adopted. From the results of this paper, it can be shown and demonstrated that the inverse procedure proposed in this paper can make the history of the Coulomb’s friction coefficient more stable and reasonable. r 2006 Elsevier Ltd. All rights reserved. Keywords: Linear least-squares errors method; Tikhonov method; Upsetting; Coulomb’s friction model; Shear strength friction model
1. Introduction Upsetting is a primary forming method for metal forming. The friction coefficient on the contact surfaces between the workpiece and the die is one of the major factors affecting the upsetting workpiece quality and the die life. This factor is worthy of investigation. Most inverse problems of engineering belong to the ill-posed problems. In general, the results of the inverse method would be unstable and oscillation causing from the inevitable measurement errors. Therefore, how to overcome the unstable and the oscillation phenomena becomes one of the most concerned research topics. Although there were many researches of metal forming, they assumed that the friction coefficient on the contact surfaces between the workpiece and the die was a constant during forming process, and this assumption would induce some errors to the actual engineering application. This Corresponding author. Tel.: +886 2 2737 6455; fax: +886 2 2737 6243.
E-mail address: [email protected] (Z.-C. Lin). 0020-7403/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2006.01.015
friction coefficient relates to the forming speed, the temperature and other processing conditions. Lin and Lin [1] developed the couple thermo-elastic–plastic finite element model for upsetting. Depierre and Gurney [2] measured the changes of inner and outer diameters of ring and associate with the upper bound method to learn more about the change of the friction coefficients during the process. Lin [3] assumed the friction coefficients and the reduction increments to be in the forms of quadratic equation and logarithm function, and then measured the workpiece profile to calculate the friction coefficients during the upsetting process. Thie´baut et al. [4] used the constitution law, which determined by means of the compression test, and then the constitution law is used in a finite element code to estimate the coefficient of the Tresca or Coulomb law that fits with the experimental data. Lin [5] developed the thermo-elastic–plastic large deformation model combining with a hydrodynamic lubrication model to investigate the die-workpiece interface friction with lubrication during the upsetting process. Xincai [6] used the finite element analysis applying five
ARTICLE IN PRESS Z.-C. Lin, V.-H. Lin / International Journal of Mechanical Sciences 48 (2006) 717–725
718
Nomenclature [D]e _ fdg Fr Fz fF_ g H0 k [Kep] [KG] L LT m n
elastic stress–strain relation matrix vector of the node displacement increments friction force normal force vector of the node force increments strain hardening rate shear flow strength stiffness matrix geometric stiffness matrix experimental upsetting normal loading increment exact value of upsetting normal loading increment two times of the number of finite element meshes the number of contact nodes between workpiece and die
different friction models to experiments of upsetting of AA6082 lubricated with four lubricants. Friction parameter values are determined by fitness of data of friction area ratio from finite element analyses to experimental results. Malayappan and Narayanasamy [7] used the experimental method to analyze the upsetting forging of aluminum cylindrical billets considering the dissimilar frictional conditions at the flat die surface. Hu et al. [8] used the experimental and the numerical methods to obtain the pressure distribution on the die surface during the upsetting. In the earlier days, the inverse methods of engineering were applied to solve the unknown heat sources of heat transfer problems. By using dynamic programming and generalized cross validation (GCV), Trujillo [9] applied the L-curve to solve the inverse heat conduction problems, construction problems and vibration problems. Alifanov and Mikhailov [10] used the conjugate gradient method to solve the non-linear inverse heat conduction problems. Beck et al. [11] adopted the concept of sensitivity to solve the non-linear and multi-dimensional inverse heat conduction problems. Faurholdt [12] used the Levenberg–Marquardt method to optimize the constitutive parameters for elastoplastic problems. Conceic- a˜o Anto´nio and Magalha˜es Dourado [13] used the genetic algorithms to optimize the metal forming process. Elisabeth et al. [14] solved the optimization problems using the Gauss Newton algorithm to determine the parameters during the upsetting test. Yang and Chen [15] used the matrix rearrangement of heat transfer equation to clearly show the unknown conditions, and used the linear least-squares errors method to find out the unknown heat sources. As to the further studies and inference of the elastic–plastic large deformation finite element model of metal forming used the method of Ref. [15], not many researchers have ever investigated into these fields.
p
the number of already known node displacement increments of the fixed boundaries ¯ R regularizing operator {Xmea} vector of measurement normal loading increments of the contact nodes O(m) stabilizing function a regularizing parameter g allowance difference of friction coefficients between the successive stages of finite element analysis d difference between the L and LT {e} vector of strain m Coulomb’s friction coefficient mT friction coefficient corresponding to LT tr friction stress md friction coefficient corresponding to L {s} vector of stress s¯ equivalent stress of the element sz normal stress
The main purpose of this paper is based on the experimental upsetting loading to establish a matrixpresentation linear least-squares errors method of inverse elastic plastic large deformation finite element model. Since the experimental upsetting loading have the inevitable measurement errors, the history of Coulomb’s friction coefficient on the contact surface obtained by the inverse model proposed in this paper would be unstable and oscillation. This paper combines the inverse model with the proposed regularization of Tikhonov method to overcome the unstable and the oscillation phenomena. Furthermore, since the variation of the Coulomb’s friction coefficients on the contact surface in the successive finite element analysis stages during the upsetting process are continuous and small, the square of the difference of the friction coefficients on the contact surface in the successive stages can be used as the stabilizing function of the regularization of Tikhonov method. Finally, it will be shown and demonstrated in this paper that the inverse procedure combining the inverse model with the proposed regularization of Tikhonov method can make the history of Coulomb’s friction coefficient on the contact surface more stable and accurate. Beside, this paper uses the definition of the Coulomb’s friction law to decide whether the sticking effect occurs on the contact nodes between the workpiece and the die. If the sticking effect occurs, the shear strength friction model is adopted on these nodes.
2. Elastic–plastic large deformation–large strain stiffness governing equation On the assumption that the material is isotropic, as the stress reaches the yielding condition, plastic deformation will happen to the workpiece. The elastic–plastic constitu-
ARTICLE IN PRESS Z.-C. Lin, V.-H. Lin / International Journal of Mechanical Sciences 48 (2006) 717–725
tional equation [1] is " # T ½De ðqs=qfsgÞðq s=qfsgÞ ½De ¯ ¯ dfeg, dfsg ¼ ½De 0 T H þ ðqs=qfsgÞ ½De ðqs=qfsgÞ ¯ ¯
(1)
where H0 is the strain hardening rate of material, [D]e is the elastic stress–strain relation matrix, {s} is a vector of stress, {e} is a vector of strain and s¯ is the von Mises equivalent stress of the element. Using the principle of virtual work associating with the Updated Lagrangian Formulation (ULF) and the elastic– plastic constitutional equation (1), the elastic–plastic large deformation governing equation of the finite element model can be derived [1], _ ¼ fF_ g, ð½K ep þ ½K G Þfdg
(2)
where [Kep] is the stiffness matrix, [KG] is the geometric stiffness matrix, fF_ g is a vector of the node force increments which contains the friction force increments and the _ normal force increments of the contact boundary and fdg is a vector of the node displacement increments. 3. Inverse model In this paper, the governing equation (2) of the elastic–plastic large deformation finite element model will be rearranged to clearly show the unknown parameters. The linear least-squares errors method is used to minimize the differences between the simulation results of the upsetting loading increments and the experimental upsetting loading increments. Then, a matrix presentation linear least-squares errors method of inverse elastic–plastic finite element model for upsetting can be derived. Using this inverse model, the friction coefficient on the contact surfaces between the workpiece and the die at every finite element analysis stage can be obtained. The elastic–plastic large deformation governing equation (2) can be written in a compact form as _ ¼ fF_ g, ð½KÞfdg
(3)
where ½K ¼ ð½K ep þ ½K G Þ. By the matrix partition and rearrangement, the already known node force increments and the node displacement increments can be shown clearly in a matrix form as 8 9 8 9 2 3 _ _ K 11 K 12 K 13 > > = = < d1 > < F1 > 6K 7 _ d2 ¼ F_ 2 , (4) 4 21 K 22 K 23 5 > > > ; : _ > K 31 K 32 K 33 mm : d_ 3 ; F 3 m1 m1
in which, ‘m’ is two times the number of the finite element meshes, F_ 1 ¼ f0 0 0gTðm2nÞ1 is a vector of the resultants of the node force increments which are at the internal nodes and the free surface nodes, ‘n’ is the number of the contact nodes between the workpiece and the die, F_ 2 ¼ fF_ 2 gn1 is a vector of the normal force increments of the contact nodes, F_ 3 ¼ fF_ 3 gn1 is a vector of the friction force increments of the contact nodes, d_ 2 ¼ f0 0 0gTp1 is a vector of the already known node
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displacement increments of the fixed boundaries, ‘p’ is the number of the already known node displacement increments of the fixed boundaries, d_ 3 ¼ fd_ 3 gn1 is a vector of the axial displacement increments of the contact nodes between the workpiece and the die, the axial displacement increments of the contact nodes are equal to the reduction increment and d_ 1 ¼ fd_ 1 gðmpnÞ1 is a vector of the unknown displacement increments of the nodes. By rearranging the already known terms and the unknown terms of Eq. (4) to the different sides of this equation, then, 8 9 2 3 K 11 0 0 d_ 1 > > > = < > 6 7 _2 6 K 21 1 0 7 F 4 5 > > > > K 31 0 1 mðmþnpÞ : F_ 3 ; ðmþnpÞ1 9 8 _ 1 K 12 d_ 2 K 13 d_ 3 > F > > > = < _ _ K 22 d 2 K 23 d 3 ¼ , ð5Þ > > > > ; : K 32 d_2 K 33 d_3 m1 in order to simplify Eq. (5), this paper assumes, 2 3 ½K 11 ðm2nÞðmpnÞ ½0ðm2nÞn 6 ½K 21 ½1nn 7 ¼ ½A1 , nðmpnÞ 4 5 ½K 31 nðmpnÞ ½0nn mðmpÞ
2
½0ðm2nÞn
6 ½0 4 nn ½1nn (
d_ 1 F_ 2
3 7 5
¼ ½A2 ,
mn
) ¼ fX 1 g, ðmpÞ1
fF_ 3 gn1 ¼ fX 2 g, 9 8 _ _ _ > = < F 1 K 12 d 2 K 13 d 3 > K 22 d_2 K 23 d_3 ¼ fbg. > > ; : _ _ K 32 d 2 K 33 d 3 m1 Then, Eq. (5) can be rewritten as ( ) X1 ½A1 A2 ¼ fbg. X2
(6)
{X2} is a vector of the friction force increments of the contact nodes. Eq. (6) clearly indicates the increments of friction force and normal force of the contact nodes between the workpiece and die. Eq. (6) can be rearranged as A1 X1 ¼ b A2 X2 . In this paper, both sides of this equation can be multiplied by AT1 to get a square matrix ðAT1 A1 Þ, thus this equation becomes as AT1 A1 X1 ¼ AT1 b AT1 A2 X2 .
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Since the dimension of A1 is m ðm pÞ, the inverse matrix of AT1 A1 exists generally. Therefore, the following equation can be obtained: X1 ¼
ðAT1 A1 Þ1 AT1 b
ðAT1 A1 Þ1 AT1 A2 X2 .
fX¯ 1 gn1 ¼ fðAT1 A1 Þ1 AT1 bgn1 fðAT1 A1 Þ1 AT1 A2 X2 gn1 . (7) Let {Xmea} is a vector of the measurement normal force increments of the contact nodes, and the sum of the square of the differences between the measurements and the calculations is minimum by using the linear least-squares errors method, the following relationship can be defined: ¯ 1 ÞT ðXmea X ¯ 1Þ E r ¼ ðXmea X ¼ ðXmea ðAT1 A1 Þ1 AT1 b þ ðAT1 A1 Þ1 AT1 A2 X2 ÞT ð8Þ
in which, Er is a function of X2. Let the first partial derivative of Er with respect to X2 equals to zero. That is ðqE r =qX2 Þ ¼ 0, the following equation can be obtained: X2 Xmea þ ðAT1 A1 Þ1 AT1 b ¼ 0, ððAT1 A1 Þ1 A2 ÞT ððAT1 A1 Þ1 A2 Þ X2 ¼ ððAT1 A1 Þ1 A2 ÞT ðAT1 A1 Þ1 b ð9Þ
According to the Coulomb’s friction law, X2 ¼ mXmea ,
(10)
in which, m is the friction coefficient on the contact surfaces between the workpiece and the die. Substituting the relationship of Eq. (10) into Eq. (9) yields ððAT1 A1 Þ1 AT1 A2 ÞT ððAT1 A1 Þ1 AT1 A2 ÞmXmea ¼ ððAT1 A1 Þ1 AT1 A2 ÞT ðAT1 A1 Þ1 AT1 b ððAT1 A1 Þ1 AT1 A2 ÞT Xmea . By rearranging above equation, following equation can be obtained: Xmea ¼ ½ððAT1 A1 Þ1 AT1 A2 ÞT ððAT1 A1 Þ1 AT1 A2 Þm þ ððAT1 A1 Þ1 AT1 A2 ÞT 1 ððAT1 A1 Þ1 AT1 A2 ÞT ðAT1 A1 Þ1 AT1 b.
ð11Þ
The sum of the normal force increments of the contact nodes equals to the upsetting normal loading increment. Therefore, BXmea ¼ L,
ðAT1 A1 Þ1 AT1 b ¼ L.
ð13Þ
In order to simplify Eq. (13), this paper assumes ðAT1 A1 Þ1 AT1 A2 ¼ C, ðAT1 A1 Þ1 AT1 b ¼ D, then, Eq. (13) can be rewritten as following: B½CT Cm þ CT 1 CT D ¼ L. In the compact form, then, BðmC þ IÞ1 D ¼ L.
(14)
Since B, C, D and L are the already known terms, Eq. (14) is a non-linear equation of the friction coefficient m. Eq. (14) is the matrix presentation linear least-squares errors method of inverse elastic–plastic large deformation finite element model for upsetting, which can be used to solve the friction coefficient on the contact surfaces between the workpiece and the die at every finite element analysis stages.
4. Proposed regularization of Tikhonov method for upsetting process
2ððAT1 A1 Þ1 AT1 A2 ÞT ½ððAT1 A1 Þ1 AT1 A2 Þ
ððAT1 A1 Þ1 A2 ÞT Xmea .
BXmea ¼ B½ððAT1 A1 Þ1 AT1 A2 ÞT ððAT1 A1 Þ1 AT1 A2 Þm þ ððAT1 A1 Þ1 AT1 A2 ÞT 1 ððAT1 A1 Þ1 AT1 A2 ÞT
Since X1 contains the vector of normal force increments fX¯ 1 gn1 of the contact nodes, this paper takes out the vector fX¯ 1 gn1 from X1, Therefore, fX¯ 1 gn1 can be obtained as
ðXmea ðAT1 A1 Þ1 AT1 b þ ðAT1 A1 Þ1 AT1 A2 X2 Þ,
is defined as B ¼ f1 1 1 1 1g1n . Both sides of Eq. (11) multiply by B, then,
(12)
in which, L is the upsetting normal loading increment obtained by upsetting experiment. B is a row vector, which
The right-hand side term of Eq. (14) is the experimental upsetting normal loading which has the inevitable errors, therefore, the history of the friction coefficient obtained by the inverse model of Eq. (14) will be unstable and oscillation. In order to make the history of the Coulomb’s friction coefficient on the contact surfaces between the workpiece and the die more stable and accurate, the inverse procedure that combining the inverse model with the regularization of Tikhonov method is proposed in this paper. Eq. (14) is a nth degree polynomial and it can be solved with n roots. It is also known that the values of the Coulomb’s friction coefficients are between zero and one. As the friction coefficient on the contact surfaces increases, the requirement of the upsetting normal loading increases too. The left-hand side of Eq. (14) is the function of the friction coefficient m and it is a monotone increasing function. Therefore, by using the inverse model of Eq. (14) directly, the only one value of the friction coefficient on the contact surfaces between zero and one can be obtained. The regularization of Tikhonov method [16] is used to construct the regularizing operators, and the key to the regularizing solution is to find the stabilizing function and to set the regularization parameter. Assuming LT is the exact value of the normal loading of the upsetting in Eq. (14), there is a corresponding exact solution mT existing in the interval 0pmT p1, that is BðmT C þ IÞ1 D ¼ LT ;
0pmp1.
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It is known that there are inevitable errors because the data of the upsetting normal loadings are obtained from the measurement. Assuming L is the measurement upsetting normal loading with errors in Eq. (14), although the difference between L and LT is less than d, it does not guarantee that the friction coefficient resulting from Eq. (14) approaches the value of mT . It means that any of md within the set of U d ¼ fmd jjBðmd C þ IÞ1 D Ljjpdg cannot guarantee to approach the value of mT. Thus, an approximate solution will be found and it continuously depends on d within Ud in this paper. A non-negative continuous function is defined as
3. Calculating Oðmdi ðaj ÞÞ, and check whether Oðmdi ðaj ÞÞpg. If Oðmdi ðaj ÞÞ4g, let ajþ1 ¼ aj þ Da and repeat step 1–2. ‘g’ is the allowance difference of the friction coefficients between the successive stages of the finite element analysis. 4. If Oðmdi ðaj ÞÞpg, then aj is the regularizing parameter. 5. Substituting this aj into Eq. (15) to obtain the friction coefficient mdi ðaj Þ, it can minimize Eq. (15) using numerical method. This mdi ðaj Þ is the friction coefficient of this finite element stage.
Oðmdi Þ ¼ jjmdi mdi1 jj2 ,
5. Friction model
where mdi is the friction coefficient resulting from the inverse procedure in the ith stage of the finite element analysis. mdi1 is the friction coefficient resulting from the inverse procedure in the (i1)th stage of the finite element analysis. The function Oðmdi Þ is defined in the compact subset U1 within the set U of the solutions of Eq. (14). The changes of the friction coefficients are continuous and small during the finite element analysis of the upsetting process. Then mT 2 U 1 and U 1;g ¼ fmdi jmdi 2 U 1 ; Oðmdi Þpgg is a compact sub-set within U1 for any g40. So that Oðmdi Þ is a stabilizing function. The solution mdi of Eq. (14) can be found by minimization of Oðmdi Þ in the set of ¯ that depends U 1;d ¼ U 1 \ U d . The regularizing operator R on the parameter d, can be constructed using the relationship of mdi and L. It can be defined as ¯ mdi ¼ RðL; dÞ. mdi is the regularizing solution that could approximate the exact solution mT. To find the solution of Eq. (14) with the experimental errors could be changed to solve the extreme value problem of the function in the set of U 1;d . Here we seek to minimize the function Oðmdi Þ under the condition of jjBðmdi C þ IÞ1 D Ljjpd, 0pmp1. This is an optimization problem with constrained condition. By using the method of ~ d ; L; bÞ ¼ Oðmd Þ þ Lagrange multiplier, the equation of Mðm i i bðjjBðmdi C þ IÞ1 D Ljj2 d2 Þ can be obtained, in which b is a multiplier to be specified. Since d is a constant, it can be neglected in finding the extreme value of the preceding equation. After d is neglected, we can multiply both sides of the equation by a constant of a ¼ ð1=bÞ, that is Mðmdi ; L; bÞ ¼ ajjmdi mdi1 jj2 þ ðjjBðmdi C þ IÞ1 D Ljj2 Þ, (15) in which, a is the regularizing parameter. The procedure of determining the regularizing parameter for the upsetting process is proposed as following: 1. The friction coefficient mdi1 of the first finite element stage can be obtained by using the inverse model of the Eq. (14) directly. 2. Substituting any aj into Eq. (15) to obtain the friction coefficient mdi ðaj Þ, it can minimize Eq. (15) by using numerical method.
In this paper, the Coulomb’s friction model and the shear strength friction model are adopted on the contact surfaces between the workpiece and the die. The interface between the workpiece and the die is divided into two zones [17]: (I) the shear strength friction of sticking occurs at the central zone; (II) the Coulomb’s friction of dry slipping occurs at the edge zone when the frictional stress is less than the shear strength. It is known that the Coulomb’s friction coefficient becomes meaningless when the sticking occurs at the contact nodes. Therefore, the shear strength friction model is adopted when the sticking effect occurs at the contact nodes in this paper. The Coulomb’s friction coefficient is defined as the ratio of the friction force to the normal force, or of the friction stress to the normal stress. For the upsetting, the direction of the friction force is in the r-direction and the direction of the normal force is in the z-direction in this paper. Therefore, the Coulomb’s friction coefficient can be shown as follows: m¼
F r tr ¼ , F z sz
(16)
in which, Fr is the friction force in the r-direction, Fz is the normal force in the z-direction, tr is the friction stress in the r-direction and sz is the normal stress in the z-direction. The Coulomb’s friction coefficient m may change at the different reduction rates during the upsetting process. However, m cannot rise indefinitely [18] and it can be used at the condition of msz pk, in which, k is the shear flow strength of the contact nodes of the workpiece material. It is much more accurate to say that the Coulomb’s friction coefficient becomes meaningless when msz 4k. After sticking effect occurrence, the shear strength friction model is adopted at the sticking nodes in this paper and the following equation is used: tr ¼ k.
(17)
This paper proposes m4ðk=sz Þ to decide the friction state of the contact nodes. If m4ðk=sz Þ, the sticking effect occurs on the contact nodes between the workpiece and the die, and the shear strength friction model of tr ¼ k is adopted on these contact nodes.
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6. Experiments and data input The material of the upsetting used in this paper is the AA1050 aluminum. After the extrusion process, it is heated under 260 1C and then annealed. The material of the die is SKD-61. In order to obtain the relation curve between the true stress and true strain and the history of the upsetting normal loading, the tensile test and the experiment of upsetting are done in a 100-tons universal testing machine respectively. Both the tensile velocity of tensile test and the reduction velocity of upsetting processing are proceeding under slow velocity for the experiment condition of this article. Therefore, the obtained flow stress equation could be considered only function of equivalent strain under the slow upsetting velocity. The contact surface between the workpiece and the die is without any lubrication during the upsetting experiment. Fig. 1 shows the equipment of the upsetting experiment of AA1050. Fig. 2 shows the scheme of the experiment of the upsetting. Fig. 3 indicates
true stress (Mpa)
100 80 60 40 20 0 0
0.05
0.1
0.15
0.2
0.25
true strain Fig. 3. The experimental result of the true stress–true strain relation curve of AA1050.
the experimental result of relation curve of AA1050. equation to fit this curve, the true stress–true strain xpressed as s ¼ 160:18e0:412 .
the true stress–true strain Using the strain-hardening the following equation of relation curve can be e (18)
The history of the experimental upsetting loading is showed in Fig. 10. The profile of the workpiece under the 28% reduction rate is showed in Fig. 11. 7. Results and discussion
Fig. 1. The equipments of the upsetting experiment of AA1050.
Z Upper die
60 mm
9 8
5
3 1
R 99 Work-piece
91
60 mm
Lower die
Fig. 2. The scheme of the upsetting experiment.
The main purpose of this paper is to develop an inverse procedure to obtain the history of the Coulomb’s friction coefficient on the contact surfaces between the workpiece and the die during the upsetting process. This inverse procedure is based on the experimental upsetting normal loading and combines the matrix-presentation linear leastsquares errors method of inverse elastic–plastic large deformation finite element model developed in this paper with the proposed regularization of Tikhonov method. In this paper, the Coulomb’s friction model is adopted before the sticking effect occurrence. After the sticking effect occurrence, the shear strength friction model will be adopted. Fig. 4 indicates the history of the Coulomb’s friction coefficient on the contact surfaces between the workpiece and the die during the upsetting process obtained by the inverse model before regularization. Since there are inevitable errors in the measurement upsetting loading, the result of Fig. 4 exists serious oscillation especial in the early stages. Fig. 5 shows the history of the friction coefficient on the contact surfaces between the workpiece and the die obtained by the inverse model combining with the regularization of Tikhonov method. It is indicated from Fig. 5, the oscillation of the history of the friction coefficient decreases clearly. By using third degree polynomial to fit, the following
ARTICLE IN PRESS Z.-C. Lin, V.-H. Lin / International Journal of Mechanical Sciences 48 (2006) 717–725 History of friction coefficient History of k σ z at node 1
1
friction coefficient
723
0.8
1
0.6
0.8
History of History of History of History of History of
k σz at node 3 k σ z at node 5 k σ z at node 6 k σ at node 7 z k σ at node 8 z
0.6
0.4
0.4
0.2
0.2 0 0
0.1 0.2 reduction rate
0.3
0 0
Fig. 4. The history of the Coulomb’s friction coefficient obtained by the inverse model of Eq. (14) before regularization.
0.1 0.2 reduction rate
0.3
Fig. 6. The comparison between the history of the Coulomb’s friction coefficient and the history of the value of k=sz .
1 History of coefficient by inverse procedure
8
History of friction coefficient by inverse procedure after curve fitting
No. of sticking node
friction coefficient
0.8 0.6 0.4 0.2 0 0
0.1 0.2 reduction rate
6 4 2 0 0.268
0.3
0.272
0.276
0.28
0.284
reduction rate
Fig. 5. The history of the Coulomb’s friction coefficient obtained by the inverse procedure proposed in this paper.
Fig. 7. The number of the sticking nodes of the contact surface between the workpiece and the die during upsetting process.
equation can be derived:
process. As the curve of the history of the friction coefficient intersects the curve of the history of k=sz , that is m4ðk=sz Þ. It is known from the Coulomb’s friction law, as the friction stress is large than the shear strength of material, i.e. tr ¼ msz 4k, it means the contact surface of the workpiece has been yielding and the sticking effect will occur. At that time, the Coulomb’s friction model cannot be used and the friction model changes to the shear strength friction model on the sticking nodes. It can be found from Fig. 6 that the sticking effect begins from the nearly center node (node 8) as the reduction rate reaches 27%. The latest sticking effect occurs on the edge node as the reduction rate reaches 28.1%. Fig. 7 indicates the variation of the number of the sticking nodes of the contact surface as the reduction rate increases. It is known from Fig. 7, the sticking effect begins from the nearly center of the contact surface as the reduction rate reaches 27%, the sticking effect occurs on the whole contact surface as the reduction rate reaches 28.1%. Fig. 8 shows the normal stress distributions of the contact surface under the various reduction rates. Since
2 DZ DZ m ¼ 0:134055 0:18828 1:9557 Z Z 3 DZ þ 44:7616 , Z
ð19Þ
in which, jDZ=Zj is the reduction rate of the upsetting process. From the curve of the third degree polynomial in Fig. 5, as the reduction rate increases, some elements of the workpiece reach plastic state and the plastic deformation increases, the contact condition between the workpiece and the die will change from point contact to surface contact, therefore, the friction coefficients will increase steeply. According to qualitative analysis, the trend of the history of the friction coefficient on the contact surfaces during the upsetting process obtained by the inverse procedure proposed in this paper is very similar to the steel obtained by Lin [3]. Fig. 6 indicates the comparison between the history of the Coulomb’s friction coefficient m obtained by the inverse procedure and the history of k=sz during the upsetting
ARTICLE IN PRESS Z.-C. Lin, V.-H. Lin / International Journal of Mechanical Sciences 48 (2006) 717–725
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500 reduction rate 40% reduction rate 35% reduction rate 30% reduction rate 25% reduction rate 20%
40
simulation result experiment result
400 loading (kN)
normal stress (Mpa)
60
20
300 200 100
0
0 0
10
20
30
40
0.1
0
0.2
R (mm) Fig. 8. The distribution of the normal stress of the contact surface between the workpiece and the die.
0.5
Simulation result (µ=0.1) Simulation result by the inverse procedure
20
Z (mm)
shear stress (Mpa)
20
0.4
Fig. 10. The comparison of the upsetting normal loadings between the experimental results and the simulation results.
reduction rate 40% reduction rate 35% reduction rate 30% reduction rate 25% reduction rate 20%
25
0.3
reduction rate
15
15 10
10 5 5 0 0
0 0
10
20
30
40
R (mm) Fig. 9. The distribution of the shear stress of the contact surface between the workpiece and the die.
5
10
15 20 R (mm)
25
30
Fig. 11. The comparison of the workpiece profiles under the 28% reduction rate.
10.00 5.00 0.00 0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 R (mm)
Fig. 12. The distribution of the effective strain under the 40% reduction rate.
15.00 Z (mm)
there is no node of the contact surface occurred sticking effect as the reduction rate is less than 25%, the normal stress of the contact surface is almost homogenous. As the reduction rate increases, the normal stress increases steeply and forms a stress peak nearly the center of the contact surface since the nearly center of the contact surface occurs sticking effect. Fig. 9 shows the distributions of the shear stress of the contact surface at the various reduction rates. The shear stress vanishes at the center of the contact surface since the axial-symmetric workpiece is analyzed. As the reduction is less than 25%, the same holds true for the Coulomb’s friction coefficient m ¼ ðtr =sz Þ. As the reduction rate increases, the sticking effect occurs near the center of the contact surface and the shear stress increases. When the sticking effect occurs on the whole contact surface (40%), the shear stress distribution is almost homogeneous of the contact surface between the workpiece and the die. Fig. 10 shows a comparison of the upsetting normal loadings between the experimental results and the simulation results. It is clearly showed that the results obtained by this paper coincide with the experimental results. Fig. 11 shows the comparison of the workpiece profiles under the
Z (mm)
15.00
10.00 5.00 0.00 0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 Unit: Mpa R (mm)
Fig. 13. The distribution of the effective stress under the 40% reduction rate.
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28% reduction rate. It is clearly known from Fig. 11, the workpiece profile obtained by the inverse procedure proposed in this paper is very coinciding with the experimental result. Figs. 12 and 13 show the distributions of the effective strain and the effective stress, respectively under the 40% reduction rate. It is known from these figures, the trend of the effective strain distribution is similar to the trend of the effective stress distribution. Both of maximum values of effective strain and the effective stress occur in the center of the workpiece.
8. Conclusion The main purpose of this paper is to develop an inverse procedure, which can obtain the history of the Coulomb’s friction coefficient during the upsetting process. This procedure is based on the experimental upsetting normal loading and combines a matrix-presentation linear leastsquares errors method of inverse elastic–plastic large deformation finite element model developed in this paper with the proposed regularization of Tikhonov method. Beside, this paper uses the definition of the Coulomb’s friction model to decide whether the sticking effect occurs on the contact surfaces between the workpiece and the die. After the sticking effect occurs, the shear strength friction model is adopted at the sticking nodes. It is also known from the results, the sticking effect will occur from the center to the edge of the contact surfaces between the workpiece and the die gradually. The inverse procedure proposed in this paper can precisely obtain the history of the Coulomb’s friction coefficient and the simulation time can be greatly reduced. Beside, this inverse procedure can make the history of the Coulomb’s friction coefficient more stable and reasonable. Therefore, the inverse procedure which combining the inverse model with the regularization of Tikhonov method is valuable regarding the academic originality of engineering inverse problems, as well as practical industrial applications.
Acknowledgments The accomplishment of the study is supported by the project no. NSC91-2122-E-011-037 of the National Science Council, Taiwan, ROC.
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References [1] Lin ZC, Lin SY. An investigation of a coupled analysis of a thermoelastic–plastic model during warm upsetting. International Journal of Machine Tools and Manufacture 1990;30(4):599–612. [2] Depiere V, Gurney F. A method for determination of constant and varying friction factors during ring compression tests. Transactions of the ASME Journal of Lubrication Technology 1974;96:482–8. [3] Lin SY. An investigation of die–workpiece interface friction during the upsetting process. Journal of Materials Processing Technology 1995;54:239–48. [4] Thie´baut C, Bonnet C, Morey JM. Evolution of the friction factor of a molybdenum workpiece during upsetting tests at different temperatures. Journal of Materials Processing Technology 1998; 77:240–5. [5] Lin SY. Investigation of die–workpiece interface friction with lubrication during the upsetting process. International Journal of Advanced Manufacturing Technology 1999;15:666–73. [6] Xincai T. Comparison of friction models in bulk metal forming. Tribology International 2002;35:385–93. [7] Malayappan S, Narayanasamy R. An experimental analysis of upsetting forging of aluminium cylindrical billets considering the dissimilar friction conditions at flat die surfaces. International Journal of Advanced Manufacturing Technology 2004;23:636–43. [8] Hu XL, Hai JT, Chen WM. Experimental study and numerical simulation of the pressure distribution on the die surface during upsetting. Journal of Materials Processing Technology 2004;151: 367–71. [9] Trujillo DM. Application of dynamic programming to the general inverse problem. International Journal for Numerical Methods in Engineering 1978;12:613–24. [10] Alifanov OM, Mikhailov VV. Solution of the nonlinear inverse thermal conductivity problem by the iteration method. High Temperature 1978;36(6):1501–6. [11] Beck JV, Blackwell B, Haji-Sheikh A. Comparison of some inverse heat conduction methods using experimental data. International Journal of Heat and Mass Transfer 1996;39(17):3649–57. [12] Faurholdt TG. Inverse modeling of constitutive parameters for elastoplastic problems. Journal of Strain Analysis 2000;35(6):471–8. [13] Conceic- a˜o Anto´nio CA, Magalha˜es Dourado N. Metal-forming process optimization by inverse evolutionary search. Journal of Materials Processing Technology 2002;121:403–13. [14] Elisabeth M, Be´atrice B, Romain F. Inverse analysis of thermomechanical upsetting tests using gradient method with semi-analytical derivatives. International Journal of Thermal Sciences 2002;41: 557–63. [15] Yang C-Y, Chen C-K. The boundary estimation in two-dimensional inverse heat conduction problems. Journal of Physics D-Applied Physics 1996;29:333–9. [16] Tikhonov AN. Solution of incorrectly posed problems and the regularization method. Soviet Math Doklady 1963;4(4):1035–8. [17] Orowan E. The calculation of roll pressure in hot and cold flat rolling. Proceedings of the Institution of Mechanical Engineers 1943; 150:140–67. [18] Shaw MC, Ber A, Mamin PA. Friction characteristics of sliding surface undergoing subsurface plastic flow. Transactions of the Journal of Basic Engineering 1960;82:342–6.
International Journal of Mechanical Sciences 48 (2006) 717–725 www.elsevier.com/locate/ijmecsci
Friction coefficient of upsetting with a procedure of combining the inverse model and the Tikhonov method Zone-Ching Lina,, Ven-Huei Linb a
Department of Mechanical Engineering, National Taiwan University of Science and Technology, 43, Keelung Road, Section 4, Taipei, Taiwan 106, ROC b Department of Mechanical Engineering, Hwa Hsia Institute of Technology, Taiwan, ROC Received 7 April 2005; received in revised form 25 January 2006; accepted 31 January 2006 Available online 20 March 2006
Abstract The purpose of this paper is to develop an inverse procedure, which is used to obtain the history of the Coulomb’s friction coefficient on the contact surfaces between the workpiece and the die during the AA1050 aluminum upsetting process. This procedure is based on the experimental upsetting normal loadings and combines the matrix-presentation linear least-squares errors method of inverse elastic–plastic large deformation finite element model developed in this paper with the regularization of Tikhonov method. The square of the difference of the friction coefficients between successive stages is adopted as the stabilizing function of the regularization of Tikhonov method. In this paper, the Coulomb’s friction model is adopted before the sticking effect occurs. After the sticking effect occurrence, the shear strength friction model is adopted. From the results of this paper, it can be shown and demonstrated that the inverse procedure proposed in this paper can make the history of the Coulomb’s friction coefficient more stable and reasonable. r 2006 Elsevier Ltd. All rights reserved. Keywords: Linear least-squares errors method; Tikhonov method; Upsetting; Coulomb’s friction model; Shear strength friction model
1. Introduction Upsetting is a primary forming method for metal forming. The friction coefficient on the contact surfaces between the workpiece and the die is one of the major factors affecting the upsetting workpiece quality and the die life. This factor is worthy of investigation. Most inverse problems of engineering belong to the ill-posed problems. In general, the results of the inverse method would be unstable and oscillation causing from the inevitable measurement errors. Therefore, how to overcome the unstable and the oscillation phenomena becomes one of the most concerned research topics. Although there were many researches of metal forming, they assumed that the friction coefficient on the contact surfaces between the workpiece and the die was a constant during forming process, and this assumption would induce some errors to the actual engineering application. This Corresponding author. Tel.: +886 2 2737 6455; fax: +886 2 2737 6243.
E-mail address: [email protected] (Z.-C. Lin). 0020-7403/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2006.01.015
friction coefficient relates to the forming speed, the temperature and other processing conditions. Lin and Lin [1] developed the couple thermo-elastic–plastic finite element model for upsetting. Depierre and Gurney [2] measured the changes of inner and outer diameters of ring and associate with the upper bound method to learn more about the change of the friction coefficients during the process. Lin [3] assumed the friction coefficients and the reduction increments to be in the forms of quadratic equation and logarithm function, and then measured the workpiece profile to calculate the friction coefficients during the upsetting process. Thie´baut et al. [4] used the constitution law, which determined by means of the compression test, and then the constitution law is used in a finite element code to estimate the coefficient of the Tresca or Coulomb law that fits with the experimental data. Lin [5] developed the thermo-elastic–plastic large deformation model combining with a hydrodynamic lubrication model to investigate the die-workpiece interface friction with lubrication during the upsetting process. Xincai [6] used the finite element analysis applying five
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718
Nomenclature [D]e _ fdg Fr Fz fF_ g H0 k [Kep] [KG] L LT m n
elastic stress–strain relation matrix vector of the node displacement increments friction force normal force vector of the node force increments strain hardening rate shear flow strength stiffness matrix geometric stiffness matrix experimental upsetting normal loading increment exact value of upsetting normal loading increment two times of the number of finite element meshes the number of contact nodes between workpiece and die
different friction models to experiments of upsetting of AA6082 lubricated with four lubricants. Friction parameter values are determined by fitness of data of friction area ratio from finite element analyses to experimental results. Malayappan and Narayanasamy [7] used the experimental method to analyze the upsetting forging of aluminum cylindrical billets considering the dissimilar frictional conditions at the flat die surface. Hu et al. [8] used the experimental and the numerical methods to obtain the pressure distribution on the die surface during the upsetting. In the earlier days, the inverse methods of engineering were applied to solve the unknown heat sources of heat transfer problems. By using dynamic programming and generalized cross validation (GCV), Trujillo [9] applied the L-curve to solve the inverse heat conduction problems, construction problems and vibration problems. Alifanov and Mikhailov [10] used the conjugate gradient method to solve the non-linear inverse heat conduction problems. Beck et al. [11] adopted the concept of sensitivity to solve the non-linear and multi-dimensional inverse heat conduction problems. Faurholdt [12] used the Levenberg–Marquardt method to optimize the constitutive parameters for elastoplastic problems. Conceic- a˜o Anto´nio and Magalha˜es Dourado [13] used the genetic algorithms to optimize the metal forming process. Elisabeth et al. [14] solved the optimization problems using the Gauss Newton algorithm to determine the parameters during the upsetting test. Yang and Chen [15] used the matrix rearrangement of heat transfer equation to clearly show the unknown conditions, and used the linear least-squares errors method to find out the unknown heat sources. As to the further studies and inference of the elastic–plastic large deformation finite element model of metal forming used the method of Ref. [15], not many researchers have ever investigated into these fields.
p
the number of already known node displacement increments of the fixed boundaries ¯ R regularizing operator {Xmea} vector of measurement normal loading increments of the contact nodes O(m) stabilizing function a regularizing parameter g allowance difference of friction coefficients between the successive stages of finite element analysis d difference between the L and LT {e} vector of strain m Coulomb’s friction coefficient mT friction coefficient corresponding to LT tr friction stress md friction coefficient corresponding to L {s} vector of stress s¯ equivalent stress of the element sz normal stress
The main purpose of this paper is based on the experimental upsetting loading to establish a matrixpresentation linear least-squares errors method of inverse elastic plastic large deformation finite element model. Since the experimental upsetting loading have the inevitable measurement errors, the history of Coulomb’s friction coefficient on the contact surface obtained by the inverse model proposed in this paper would be unstable and oscillation. This paper combines the inverse model with the proposed regularization of Tikhonov method to overcome the unstable and the oscillation phenomena. Furthermore, since the variation of the Coulomb’s friction coefficients on the contact surface in the successive finite element analysis stages during the upsetting process are continuous and small, the square of the difference of the friction coefficients on the contact surface in the successive stages can be used as the stabilizing function of the regularization of Tikhonov method. Finally, it will be shown and demonstrated in this paper that the inverse procedure combining the inverse model with the proposed regularization of Tikhonov method can make the history of Coulomb’s friction coefficient on the contact surface more stable and accurate. Beside, this paper uses the definition of the Coulomb’s friction law to decide whether the sticking effect occurs on the contact nodes between the workpiece and the die. If the sticking effect occurs, the shear strength friction model is adopted on these nodes.
2. Elastic–plastic large deformation–large strain stiffness governing equation On the assumption that the material is isotropic, as the stress reaches the yielding condition, plastic deformation will happen to the workpiece. The elastic–plastic constitu-
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tional equation [1] is " # T ½De ðqs=qfsgÞðq s=qfsgÞ ½De ¯ ¯ dfeg, dfsg ¼ ½De 0 T H þ ðqs=qfsgÞ ½De ðqs=qfsgÞ ¯ ¯
(1)
where H0 is the strain hardening rate of material, [D]e is the elastic stress–strain relation matrix, {s} is a vector of stress, {e} is a vector of strain and s¯ is the von Mises equivalent stress of the element. Using the principle of virtual work associating with the Updated Lagrangian Formulation (ULF) and the elastic– plastic constitutional equation (1), the elastic–plastic large deformation governing equation of the finite element model can be derived [1], _ ¼ fF_ g, ð½K ep þ ½K G Þfdg
(2)
where [Kep] is the stiffness matrix, [KG] is the geometric stiffness matrix, fF_ g is a vector of the node force increments which contains the friction force increments and the _ normal force increments of the contact boundary and fdg is a vector of the node displacement increments. 3. Inverse model In this paper, the governing equation (2) of the elastic–plastic large deformation finite element model will be rearranged to clearly show the unknown parameters. The linear least-squares errors method is used to minimize the differences between the simulation results of the upsetting loading increments and the experimental upsetting loading increments. Then, a matrix presentation linear least-squares errors method of inverse elastic–plastic finite element model for upsetting can be derived. Using this inverse model, the friction coefficient on the contact surfaces between the workpiece and the die at every finite element analysis stage can be obtained. The elastic–plastic large deformation governing equation (2) can be written in a compact form as _ ¼ fF_ g, ð½KÞfdg
(3)
where ½K ¼ ð½K ep þ ½K G Þ. By the matrix partition and rearrangement, the already known node force increments and the node displacement increments can be shown clearly in a matrix form as 8 9 8 9 2 3 _ _ K 11 K 12 K 13 > > = = < d1 > < F1 > 6K 7 _ d2 ¼ F_ 2 , (4) 4 21 K 22 K 23 5 > > > ; : _ > K 31 K 32 K 33 mm : d_ 3 ; F 3 m1 m1
in which, ‘m’ is two times the number of the finite element meshes, F_ 1 ¼ f0 0 0gTðm2nÞ1 is a vector of the resultants of the node force increments which are at the internal nodes and the free surface nodes, ‘n’ is the number of the contact nodes between the workpiece and the die, F_ 2 ¼ fF_ 2 gn1 is a vector of the normal force increments of the contact nodes, F_ 3 ¼ fF_ 3 gn1 is a vector of the friction force increments of the contact nodes, d_ 2 ¼ f0 0 0gTp1 is a vector of the already known node
719
displacement increments of the fixed boundaries, ‘p’ is the number of the already known node displacement increments of the fixed boundaries, d_ 3 ¼ fd_ 3 gn1 is a vector of the axial displacement increments of the contact nodes between the workpiece and the die, the axial displacement increments of the contact nodes are equal to the reduction increment and d_ 1 ¼ fd_ 1 gðmpnÞ1 is a vector of the unknown displacement increments of the nodes. By rearranging the already known terms and the unknown terms of Eq. (4) to the different sides of this equation, then, 8 9 2 3 K 11 0 0 d_ 1 > > > = < > 6 7 _2 6 K 21 1 0 7 F 4 5 > > > > K 31 0 1 mðmþnpÞ : F_ 3 ; ðmþnpÞ1 9 8 _ 1 K 12 d_ 2 K 13 d_ 3 > F > > > = < _ _ K 22 d 2 K 23 d 3 ¼ , ð5Þ > > > > ; : K 32 d_2 K 33 d_3 m1 in order to simplify Eq. (5), this paper assumes, 2 3 ½K 11 ðm2nÞðmpnÞ ½0ðm2nÞn 6 ½K 21 ½1nn 7 ¼ ½A1 , nðmpnÞ 4 5 ½K 31 nðmpnÞ ½0nn mðmpÞ
2
½0ðm2nÞn
6 ½0 4 nn ½1nn (
d_ 1 F_ 2
3 7 5
¼ ½A2 ,
mn
) ¼ fX 1 g, ðmpÞ1
fF_ 3 gn1 ¼ fX 2 g, 9 8 _ _ _ > = < F 1 K 12 d 2 K 13 d 3 > K 22 d_2 K 23 d_3 ¼ fbg. > > ; : _ _ K 32 d 2 K 33 d 3 m1 Then, Eq. (5) can be rewritten as ( ) X1 ½A1 A2 ¼ fbg. X2
(6)
{X2} is a vector of the friction force increments of the contact nodes. Eq. (6) clearly indicates the increments of friction force and normal force of the contact nodes between the workpiece and die. Eq. (6) can be rearranged as A1 X1 ¼ b A2 X2 . In this paper, both sides of this equation can be multiplied by AT1 to get a square matrix ðAT1 A1 Þ, thus this equation becomes as AT1 A1 X1 ¼ AT1 b AT1 A2 X2 .
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Since the dimension of A1 is m ðm pÞ, the inverse matrix of AT1 A1 exists generally. Therefore, the following equation can be obtained: X1 ¼
ðAT1 A1 Þ1 AT1 b
ðAT1 A1 Þ1 AT1 A2 X2 .
fX¯ 1 gn1 ¼ fðAT1 A1 Þ1 AT1 bgn1 fðAT1 A1 Þ1 AT1 A2 X2 gn1 . (7) Let {Xmea} is a vector of the measurement normal force increments of the contact nodes, and the sum of the square of the differences between the measurements and the calculations is minimum by using the linear least-squares errors method, the following relationship can be defined: ¯ 1 ÞT ðXmea X ¯ 1Þ E r ¼ ðXmea X ¼ ðXmea ðAT1 A1 Þ1 AT1 b þ ðAT1 A1 Þ1 AT1 A2 X2 ÞT ð8Þ
in which, Er is a function of X2. Let the first partial derivative of Er with respect to X2 equals to zero. That is ðqE r =qX2 Þ ¼ 0, the following equation can be obtained: X2 Xmea þ ðAT1 A1 Þ1 AT1 b ¼ 0, ððAT1 A1 Þ1 A2 ÞT ððAT1 A1 Þ1 A2 Þ X2 ¼ ððAT1 A1 Þ1 A2 ÞT ðAT1 A1 Þ1 b ð9Þ
According to the Coulomb’s friction law, X2 ¼ mXmea ,
(10)
in which, m is the friction coefficient on the contact surfaces between the workpiece and the die. Substituting the relationship of Eq. (10) into Eq. (9) yields ððAT1 A1 Þ1 AT1 A2 ÞT ððAT1 A1 Þ1 AT1 A2 ÞmXmea ¼ ððAT1 A1 Þ1 AT1 A2 ÞT ðAT1 A1 Þ1 AT1 b ððAT1 A1 Þ1 AT1 A2 ÞT Xmea . By rearranging above equation, following equation can be obtained: Xmea ¼ ½ððAT1 A1 Þ1 AT1 A2 ÞT ððAT1 A1 Þ1 AT1 A2 Þm þ ððAT1 A1 Þ1 AT1 A2 ÞT 1 ððAT1 A1 Þ1 AT1 A2 ÞT ðAT1 A1 Þ1 AT1 b.
ð11Þ
The sum of the normal force increments of the contact nodes equals to the upsetting normal loading increment. Therefore, BXmea ¼ L,
ðAT1 A1 Þ1 AT1 b ¼ L.
ð13Þ
In order to simplify Eq. (13), this paper assumes ðAT1 A1 Þ1 AT1 A2 ¼ C, ðAT1 A1 Þ1 AT1 b ¼ D, then, Eq. (13) can be rewritten as following: B½CT Cm þ CT 1 CT D ¼ L. In the compact form, then, BðmC þ IÞ1 D ¼ L.
(14)
Since B, C, D and L are the already known terms, Eq. (14) is a non-linear equation of the friction coefficient m. Eq. (14) is the matrix presentation linear least-squares errors method of inverse elastic–plastic large deformation finite element model for upsetting, which can be used to solve the friction coefficient on the contact surfaces between the workpiece and the die at every finite element analysis stages.
4. Proposed regularization of Tikhonov method for upsetting process
2ððAT1 A1 Þ1 AT1 A2 ÞT ½ððAT1 A1 Þ1 AT1 A2 Þ
ððAT1 A1 Þ1 A2 ÞT Xmea .
BXmea ¼ B½ððAT1 A1 Þ1 AT1 A2 ÞT ððAT1 A1 Þ1 AT1 A2 Þm þ ððAT1 A1 Þ1 AT1 A2 ÞT 1 ððAT1 A1 Þ1 AT1 A2 ÞT
Since X1 contains the vector of normal force increments fX¯ 1 gn1 of the contact nodes, this paper takes out the vector fX¯ 1 gn1 from X1, Therefore, fX¯ 1 gn1 can be obtained as
ðXmea ðAT1 A1 Þ1 AT1 b þ ðAT1 A1 Þ1 AT1 A2 X2 Þ,
is defined as B ¼ f1 1 1 1 1g1n . Both sides of Eq. (11) multiply by B, then,
(12)
in which, L is the upsetting normal loading increment obtained by upsetting experiment. B is a row vector, which
The right-hand side term of Eq. (14) is the experimental upsetting normal loading which has the inevitable errors, therefore, the history of the friction coefficient obtained by the inverse model of Eq. (14) will be unstable and oscillation. In order to make the history of the Coulomb’s friction coefficient on the contact surfaces between the workpiece and the die more stable and accurate, the inverse procedure that combining the inverse model with the regularization of Tikhonov method is proposed in this paper. Eq. (14) is a nth degree polynomial and it can be solved with n roots. It is also known that the values of the Coulomb’s friction coefficients are between zero and one. As the friction coefficient on the contact surfaces increases, the requirement of the upsetting normal loading increases too. The left-hand side of Eq. (14) is the function of the friction coefficient m and it is a monotone increasing function. Therefore, by using the inverse model of Eq. (14) directly, the only one value of the friction coefficient on the contact surfaces between zero and one can be obtained. The regularization of Tikhonov method [16] is used to construct the regularizing operators, and the key to the regularizing solution is to find the stabilizing function and to set the regularization parameter. Assuming LT is the exact value of the normal loading of the upsetting in Eq. (14), there is a corresponding exact solution mT existing in the interval 0pmT p1, that is BðmT C þ IÞ1 D ¼ LT ;
0pmp1.
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It is known that there are inevitable errors because the data of the upsetting normal loadings are obtained from the measurement. Assuming L is the measurement upsetting normal loading with errors in Eq. (14), although the difference between L and LT is less than d, it does not guarantee that the friction coefficient resulting from Eq. (14) approaches the value of mT . It means that any of md within the set of U d ¼ fmd jjBðmd C þ IÞ1 D Ljjpdg cannot guarantee to approach the value of mT. Thus, an approximate solution will be found and it continuously depends on d within Ud in this paper. A non-negative continuous function is defined as
3. Calculating Oðmdi ðaj ÞÞ, and check whether Oðmdi ðaj ÞÞpg. If Oðmdi ðaj ÞÞ4g, let ajþ1 ¼ aj þ Da and repeat step 1–2. ‘g’ is the allowance difference of the friction coefficients between the successive stages of the finite element analysis. 4. If Oðmdi ðaj ÞÞpg, then aj is the regularizing parameter. 5. Substituting this aj into Eq. (15) to obtain the friction coefficient mdi ðaj Þ, it can minimize Eq. (15) using numerical method. This mdi ðaj Þ is the friction coefficient of this finite element stage.
Oðmdi Þ ¼ jjmdi mdi1 jj2 ,
5. Friction model
where mdi is the friction coefficient resulting from the inverse procedure in the ith stage of the finite element analysis. mdi1 is the friction coefficient resulting from the inverse procedure in the (i1)th stage of the finite element analysis. The function Oðmdi Þ is defined in the compact subset U1 within the set U of the solutions of Eq. (14). The changes of the friction coefficients are continuous and small during the finite element analysis of the upsetting process. Then mT 2 U 1 and U 1;g ¼ fmdi jmdi 2 U 1 ; Oðmdi Þpgg is a compact sub-set within U1 for any g40. So that Oðmdi Þ is a stabilizing function. The solution mdi of Eq. (14) can be found by minimization of Oðmdi Þ in the set of ¯ that depends U 1;d ¼ U 1 \ U d . The regularizing operator R on the parameter d, can be constructed using the relationship of mdi and L. It can be defined as ¯ mdi ¼ RðL; dÞ. mdi is the regularizing solution that could approximate the exact solution mT. To find the solution of Eq. (14) with the experimental errors could be changed to solve the extreme value problem of the function in the set of U 1;d . Here we seek to minimize the function Oðmdi Þ under the condition of jjBðmdi C þ IÞ1 D Ljjpd, 0pmp1. This is an optimization problem with constrained condition. By using the method of ~ d ; L; bÞ ¼ Oðmd Þ þ Lagrange multiplier, the equation of Mðm i i bðjjBðmdi C þ IÞ1 D Ljj2 d2 Þ can be obtained, in which b is a multiplier to be specified. Since d is a constant, it can be neglected in finding the extreme value of the preceding equation. After d is neglected, we can multiply both sides of the equation by a constant of a ¼ ð1=bÞ, that is Mðmdi ; L; bÞ ¼ ajjmdi mdi1 jj2 þ ðjjBðmdi C þ IÞ1 D Ljj2 Þ, (15) in which, a is the regularizing parameter. The procedure of determining the regularizing parameter for the upsetting process is proposed as following: 1. The friction coefficient mdi1 of the first finite element stage can be obtained by using the inverse model of the Eq. (14) directly. 2. Substituting any aj into Eq. (15) to obtain the friction coefficient mdi ðaj Þ, it can minimize Eq. (15) by using numerical method.
In this paper, the Coulomb’s friction model and the shear strength friction model are adopted on the contact surfaces between the workpiece and the die. The interface between the workpiece and the die is divided into two zones [17]: (I) the shear strength friction of sticking occurs at the central zone; (II) the Coulomb’s friction of dry slipping occurs at the edge zone when the frictional stress is less than the shear strength. It is known that the Coulomb’s friction coefficient becomes meaningless when the sticking occurs at the contact nodes. Therefore, the shear strength friction model is adopted when the sticking effect occurs at the contact nodes in this paper. The Coulomb’s friction coefficient is defined as the ratio of the friction force to the normal force, or of the friction stress to the normal stress. For the upsetting, the direction of the friction force is in the r-direction and the direction of the normal force is in the z-direction in this paper. Therefore, the Coulomb’s friction coefficient can be shown as follows: m¼
F r tr ¼ , F z sz
(16)
in which, Fr is the friction force in the r-direction, Fz is the normal force in the z-direction, tr is the friction stress in the r-direction and sz is the normal stress in the z-direction. The Coulomb’s friction coefficient m may change at the different reduction rates during the upsetting process. However, m cannot rise indefinitely [18] and it can be used at the condition of msz pk, in which, k is the shear flow strength of the contact nodes of the workpiece material. It is much more accurate to say that the Coulomb’s friction coefficient becomes meaningless when msz 4k. After sticking effect occurrence, the shear strength friction model is adopted at the sticking nodes in this paper and the following equation is used: tr ¼ k.
(17)
This paper proposes m4ðk=sz Þ to decide the friction state of the contact nodes. If m4ðk=sz Þ, the sticking effect occurs on the contact nodes between the workpiece and the die, and the shear strength friction model of tr ¼ k is adopted on these contact nodes.
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6. Experiments and data input The material of the upsetting used in this paper is the AA1050 aluminum. After the extrusion process, it is heated under 260 1C and then annealed. The material of the die is SKD-61. In order to obtain the relation curve between the true stress and true strain and the history of the upsetting normal loading, the tensile test and the experiment of upsetting are done in a 100-tons universal testing machine respectively. Both the tensile velocity of tensile test and the reduction velocity of upsetting processing are proceeding under slow velocity for the experiment condition of this article. Therefore, the obtained flow stress equation could be considered only function of equivalent strain under the slow upsetting velocity. The contact surface between the workpiece and the die is without any lubrication during the upsetting experiment. Fig. 1 shows the equipment of the upsetting experiment of AA1050. Fig. 2 shows the scheme of the experiment of the upsetting. Fig. 3 indicates
true stress (Mpa)
100 80 60 40 20 0 0
0.05
0.1
0.15
0.2
0.25
true strain Fig. 3. The experimental result of the true stress–true strain relation curve of AA1050.
the experimental result of relation curve of AA1050. equation to fit this curve, the true stress–true strain xpressed as s ¼ 160:18e0:412 .
the true stress–true strain Using the strain-hardening the following equation of relation curve can be e (18)
The history of the experimental upsetting loading is showed in Fig. 10. The profile of the workpiece under the 28% reduction rate is showed in Fig. 11. 7. Results and discussion
Fig. 1. The equipments of the upsetting experiment of AA1050.
Z Upper die
60 mm
9 8
5
3 1
R 99 Work-piece
91
60 mm
Lower die
Fig. 2. The scheme of the upsetting experiment.
The main purpose of this paper is to develop an inverse procedure to obtain the history of the Coulomb’s friction coefficient on the contact surfaces between the workpiece and the die during the upsetting process. This inverse procedure is based on the experimental upsetting normal loading and combines the matrix-presentation linear leastsquares errors method of inverse elastic–plastic large deformation finite element model developed in this paper with the proposed regularization of Tikhonov method. In this paper, the Coulomb’s friction model is adopted before the sticking effect occurrence. After the sticking effect occurrence, the shear strength friction model will be adopted. Fig. 4 indicates the history of the Coulomb’s friction coefficient on the contact surfaces between the workpiece and the die during the upsetting process obtained by the inverse model before regularization. Since there are inevitable errors in the measurement upsetting loading, the result of Fig. 4 exists serious oscillation especial in the early stages. Fig. 5 shows the history of the friction coefficient on the contact surfaces between the workpiece and the die obtained by the inverse model combining with the regularization of Tikhonov method. It is indicated from Fig. 5, the oscillation of the history of the friction coefficient decreases clearly. By using third degree polynomial to fit, the following
ARTICLE IN PRESS Z.-C. Lin, V.-H. Lin / International Journal of Mechanical Sciences 48 (2006) 717–725 History of friction coefficient History of k σ z at node 1
1
friction coefficient
723
0.8
1
0.6
0.8
History of History of History of History of History of
k σz at node 3 k σ z at node 5 k σ z at node 6 k σ at node 7 z k σ at node 8 z
0.6
0.4
0.4
0.2
0.2 0 0
0.1 0.2 reduction rate
0.3
0 0
Fig. 4. The history of the Coulomb’s friction coefficient obtained by the inverse model of Eq. (14) before regularization.
0.1 0.2 reduction rate
0.3
Fig. 6. The comparison between the history of the Coulomb’s friction coefficient and the history of the value of k=sz .
1 History of coefficient by inverse procedure
8
History of friction coefficient by inverse procedure after curve fitting
No. of sticking node
friction coefficient
0.8 0.6 0.4 0.2 0 0
0.1 0.2 reduction rate
6 4 2 0 0.268
0.3
0.272
0.276
0.28
0.284
reduction rate
Fig. 5. The history of the Coulomb’s friction coefficient obtained by the inverse procedure proposed in this paper.
Fig. 7. The number of the sticking nodes of the contact surface between the workpiece and the die during upsetting process.
equation can be derived:
process. As the curve of the history of the friction coefficient intersects the curve of the history of k=sz , that is m4ðk=sz Þ. It is known from the Coulomb’s friction law, as the friction stress is large than the shear strength of material, i.e. tr ¼ msz 4k, it means the contact surface of the workpiece has been yielding and the sticking effect will occur. At that time, the Coulomb’s friction model cannot be used and the friction model changes to the shear strength friction model on the sticking nodes. It can be found from Fig. 6 that the sticking effect begins from the nearly center node (node 8) as the reduction rate reaches 27%. The latest sticking effect occurs on the edge node as the reduction rate reaches 28.1%. Fig. 7 indicates the variation of the number of the sticking nodes of the contact surface as the reduction rate increases. It is known from Fig. 7, the sticking effect begins from the nearly center of the contact surface as the reduction rate reaches 27%, the sticking effect occurs on the whole contact surface as the reduction rate reaches 28.1%. Fig. 8 shows the normal stress distributions of the contact surface under the various reduction rates. Since
2 DZ DZ m ¼ 0:134055 0:18828 1:9557 Z Z 3 DZ þ 44:7616 , Z
ð19Þ
in which, jDZ=Zj is the reduction rate of the upsetting process. From the curve of the third degree polynomial in Fig. 5, as the reduction rate increases, some elements of the workpiece reach plastic state and the plastic deformation increases, the contact condition between the workpiece and the die will change from point contact to surface contact, therefore, the friction coefficients will increase steeply. According to qualitative analysis, the trend of the history of the friction coefficient on the contact surfaces during the upsetting process obtained by the inverse procedure proposed in this paper is very similar to the steel obtained by Lin [3]. Fig. 6 indicates the comparison between the history of the Coulomb’s friction coefficient m obtained by the inverse procedure and the history of k=sz during the upsetting
ARTICLE IN PRESS Z.-C. Lin, V.-H. Lin / International Journal of Mechanical Sciences 48 (2006) 717–725
724
500 reduction rate 40% reduction rate 35% reduction rate 30% reduction rate 25% reduction rate 20%
40
simulation result experiment result
400 loading (kN)
normal stress (Mpa)
60
20
300 200 100
0
0 0
10
20
30
40
0.1
0
0.2
R (mm) Fig. 8. The distribution of the normal stress of the contact surface between the workpiece and the die.
0.5
Simulation result (µ=0.1) Simulation result by the inverse procedure
20
Z (mm)
shear stress (Mpa)
20
0.4
Fig. 10. The comparison of the upsetting normal loadings between the experimental results and the simulation results.
reduction rate 40% reduction rate 35% reduction rate 30% reduction rate 25% reduction rate 20%
25
0.3
reduction rate
15
15 10
10 5 5 0 0
0 0
10
20
30
40
R (mm) Fig. 9. The distribution of the shear stress of the contact surface between the workpiece and the die.
5
10
15 20 R (mm)
25
30
Fig. 11. The comparison of the workpiece profiles under the 28% reduction rate.
10.00 5.00 0.00 0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 R (mm)
Fig. 12. The distribution of the effective strain under the 40% reduction rate.
15.00 Z (mm)
there is no node of the contact surface occurred sticking effect as the reduction rate is less than 25%, the normal stress of the contact surface is almost homogenous. As the reduction rate increases, the normal stress increases steeply and forms a stress peak nearly the center of the contact surface since the nearly center of the contact surface occurs sticking effect. Fig. 9 shows the distributions of the shear stress of the contact surface at the various reduction rates. The shear stress vanishes at the center of the contact surface since the axial-symmetric workpiece is analyzed. As the reduction is less than 25%, the same holds true for the Coulomb’s friction coefficient m ¼ ðtr =sz Þ. As the reduction rate increases, the sticking effect occurs near the center of the contact surface and the shear stress increases. When the sticking effect occurs on the whole contact surface (40%), the shear stress distribution is almost homogeneous of the contact surface between the workpiece and the die. Fig. 10 shows a comparison of the upsetting normal loadings between the experimental results and the simulation results. It is clearly showed that the results obtained by this paper coincide with the experimental results. Fig. 11 shows the comparison of the workpiece profiles under the
Z (mm)
15.00
10.00 5.00 0.00 0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 Unit: Mpa R (mm)
Fig. 13. The distribution of the effective stress under the 40% reduction rate.
ARTICLE IN PRESS Z.-C. Lin, V.-H. Lin / International Journal of Mechanical Sciences 48 (2006) 717–725
28% reduction rate. It is clearly known from Fig. 11, the workpiece profile obtained by the inverse procedure proposed in this paper is very coinciding with the experimental result. Figs. 12 and 13 show the distributions of the effective strain and the effective stress, respectively under the 40% reduction rate. It is known from these figures, the trend of the effective strain distribution is similar to the trend of the effective stress distribution. Both of maximum values of effective strain and the effective stress occur in the center of the workpiece.
8. Conclusion The main purpose of this paper is to develop an inverse procedure, which can obtain the history of the Coulomb’s friction coefficient during the upsetting process. This procedure is based on the experimental upsetting normal loading and combines a matrix-presentation linear leastsquares errors method of inverse elastic–plastic large deformation finite element model developed in this paper with the proposed regularization of Tikhonov method. Beside, this paper uses the definition of the Coulomb’s friction model to decide whether the sticking effect occurs on the contact surfaces between the workpiece and the die. After the sticking effect occurs, the shear strength friction model is adopted at the sticking nodes. It is also known from the results, the sticking effect will occur from the center to the edge of the contact surfaces between the workpiece and the die gradually. The inverse procedure proposed in this paper can precisely obtain the history of the Coulomb’s friction coefficient and the simulation time can be greatly reduced. Beside, this inverse procedure can make the history of the Coulomb’s friction coefficient more stable and reasonable. Therefore, the inverse procedure which combining the inverse model with the regularization of Tikhonov method is valuable regarding the academic originality of engineering inverse problems, as well as practical industrial applications.
Acknowledgments The accomplishment of the study is supported by the project no. NSC91-2122-E-011-037 of the National Science Council, Taiwan, ROC.
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