A variational upper-bound method for analysis of upset forging of rings

Journal of Materials Processing Technology 170 (2005) 392–402

A variational upper-bound method for analysis of upset forging of rings Wei-Ching Yeh ∗ , Ming-Chang Wu Department of Mechanical Engineering, National Central University, Chung-Li 32054, Taiwan, ROC Received 17 October 2003; received in revised form 17 October 2003; accepted 3 June 2005

Abstract In this investigation, a variational upper-bound (VUB) method is employed. It is the method that determines an upper-bound solution using variational calculus. By using the method, the natural boundary condition, which was ignored in the traditional upper-bound (UB) method, can be theoretically derived for the stream function assumed to be arbitrary in prior. The VUB method has shown to be applicable to the problem of upset forgings of ring, and it presents an improvement on the UB method in general. From the result, we can clearly indicate that the natural boundary condition, which constrains plastic flow of the upsetting ring on the contact interface, significantly affects the upper-bound solution not only in predicting the bulge profile of the upset ring, but also in calculating the total forming energy rate. Theoretical predictions obtained using the VUB, the UB and the FEM methods were compared and discussed. Some experimental results were employed for verification of the theories as well. © 2005 Elsevier B.V. All rights reserved. Keywords: Variational upper-bound method; Natural boundary condition; Stream function; Upset forging

1. Introduction For problems of metal forming, there are no exact solutions that can be used for practical purposes. Therefore, methods of analysis, which gives results with various degrees of approximations, must be used. Among various theoretical methods available for metal forming problems, the upperbound method is known to be a limiting approach to predict the maximum energy and assures a material to plastically deform into a desired shape. When this method is applied, an admissible velocity field that satisfies the incompressibility, continuity, and the velocity boundary conditions is usually an import element to the upper-bound solution. Based on this velocity field and limit theorems [1], the total forming energy rate and also the forming load can be computed to represent an upper-bound to the actual forming energy rate or actual forming load. Thus, the lower this upper-bound load, the better the prediction is. Often the velocity field considered includes some parameters that are determined by minimizing the total forming energy with respect to those parameters, and then the ∗

Corresponding author. Tel.: +886 3 426 3904; fax: +886 3 425 4501. E-mail address: [email protected] (W.-C. Yeh).

0924-0136/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2005.06.007

determined values of the parameters give a somewhat better upper-bound velocity field. In general, with increasing number of parameters in the velocity field, the solution improves while the computation becomes more complicated. As a consequence, a practical compromise is made in selecting an admissible velocity field for practical use of the upper-bound method. In an effort to obtain a better upper-bound solution of metal forming problems, a variety of mathematical models, which define admissible velocity fields as well as rigid/plastic interfaces, have been widely investigated. Among these models include, for instances, the single triangle velocity field proposed by Green [2], the spherical velocity field proposed by Avitzur [3], the multi-triangle velocity field extended from the Green’s model by Kobayashi [4] and the trapezoidal velocity field proposed by Feng and Pan [5]. However, the approach for selecting general kinematically admissible velocity fields was investigated by Stepanskii [6] and Nagpal [7], respectively. When any of these aforementioned models is applied in the upper-bound analysis [8–14], solution is often determined by minimizing the total forming energy rate with respect to one or more parameters of an admissible function that defines the velocity field. Specifically, since the admissible function

W.-C. Yeh, M.-C. Wu / Journal of Materials Processing Technology 170 (2005) 392–402

used has been mathematically expressed as an explicit function before the total forming energy rate is minimized, the velocity field to be determined can only satisfy its kinematic boundary conditions. However, the upper-bound solution obtained in this manner, as previously described, has been improved significantly, since Yeh and Yang [15] proposed a variational upper-bound (VUB) method. In the work of Yeh and Yang [15–17], different kinds of plane–strain problems, the cup ironing and tube extrusion problems, were solved by the VUB method. Moreover, an assumed admissible function with unknown constants is replaced by an arbitrary function. Rather than minimizing the unknown constants in the assumed function, the arbitrary function is found using variational calculus. As a result, a number of boundary conditions, including both kinematic and natural boundary conditions, can be theoretically derived, and then imposed as constraints in the VUB analysis. Due to the natural boundary conditions which were ignored in the traditional UB analysis, the upper-bound solution reflects more precisely and reasonably in predicting the pattern of metal flow as well as the calculated total forming load for the plane–strain forming problems [15–17]. In this investigation, in order to extend the domain of capability and validity of the VUB method, the problem of upset forging of rings, which is no longer belong to the category of plane–strain problems, is analyzed. Although such a problem has been widely investigated on the basis of upperbound theorem [18–24], the method we employed, also based on the upper-bound theorem, is significantly distinguishable. Specifically, an arbitrary function, which governs the velocity field in deforming material at each step of the upsetting process, is associated with the upper-bound theorem to formulate the total forming energy rate equation. The method of variational calculus is then applied to extremize the equation with respect to the arbitrary function. Accordingly, both kinematic and natural boundary conditions can be mathematically derived. Also, in order to determine the upper-bound solution, a mathematical model that satisfies all of these boundary conditions is proposed for the stream function, before the total forming rate equation is numerically minimized with respect to the free parameters of the stream function. In this investigation, the result determined by the present method and by the others (the UB method and FEM) are discussed and compared with experimental data [18,20–22,25,26]. From the result, we can clearly indicate that in calculating total forming load the VUB method considerably presents an improvement on the traditional UB method. Among various models employed in the upperbound analysis, the model proposed in this investigation can be regarded as the best one for prediction of bulged profiles of the upset ring and disks, and it is in conformity with the calibration curves obtained from ring tests. In capability of predicting bulged profiles, the present model arrives at the result better than the FEM, which is according to the commercial software package MARC [27], for the upset forgings of disk, and vice versa for the upset forgings of ring.

393

2. Theoretical derivation The problem to be analyzed is schematically shown in Fig. 1, where two modes of deformation occurring in ring upsetting, along with the employed cylindrical coordinate system (r, θ, z), are illustrated. The ring is upset between two flat, parallel platens, which vertically move toward each other with the same absolute velocity, V0 . With continue motion of these two platens, the initial height (T0 ) of the ring keep decreasing while its initial external diameter (R0 ) keep moving outward non-uniformly. The mode of deformation for the initial internal diameter (Ri ) of the ring, however, is primarily dominated by the interfacial friction condition. More specifically, when a ring is upset, its internal diameter increases in the case of low friction (represented by the dashed curve in Fig. 1) and decreases in the case of high friction (represented by the solid curve in Fig. 1). This dependency of deformation behavior has been extensively used for testing lubricants, lubrication conditions and for determining the interfacial friction factor or the coefficient of friction [28,29]. In theoretical, the neutral surface at radius r = Rn , which does not move in radial direction at given time during deformation, is introduced to make a distinction between these two modes of deformation. 2.1. Basic assumptions When applying the upper-bound method, the following assumptions are usually made: (1) effect of strain hardening as well as the rate and temperature effects are neglected for simplifying analysis; (2) the yielding of materials obeys von-Mises criterion; (3) the deforming material is isotropic and plastically incompressible and (4) constant shear stress due to interface friction effect is considered. In addition, the assumption we have made is that the neutral radius (Rn ), is always considered to be independent of z (see Fig. 1), but may vary at each step of the upsetting process, considered in small steps for numerical computation. Accordingly, the neutral surface, defined by r = Rn , always beomes to be cylindrical, although bulged profiles of the lateral surfaces of the upset ring and disk are experimentally observed. 2.2. The stream function, the velocity and the strain-rate fields The theory of stream function is basically derived from the principle of conservation of mass. Under the consideration of axisymmetric characteristics, the stream function (ψ), when defined in the cylinderical coordinate system (r, θ, z), has been readily shown to be independent of the coordinate θ. Furthermore, due to symmetry, only the top right quadrant of the ring as shown in Fig. 1 is considered, and the potential of the stream function on the symmetric plane, z = 0, and at the neutral surface, r = Rn , is defined to be zero. That is, ψ(r, 0) = ψ(Rn , z) = 0

(1)

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Fig. 1. Schematic representation of the upset ring before and after deformation.

Due to the assumption of plastic incompressibility, the velocity field in the plastic deformation zone can be expressed in terms of the stream function as 1 ∂ψ(r, z) Vr = , (2a) 2πr ∂z −1 ∂ψ(r, z) Vz = , 2πr ∂r and Vθ = 0,

Vz (r, 0) = 0,

ε˙ zz (2c)

V0 4T


 r− 

+

r−

R2n r 2

Rn r



g(r)h (z) −

  g (r) h(z)

(6b) (6c)

  R2 3 + 2n g (r) r (6d)

and (3c)

πV0 2 (r − R2n )[z + h(z)g(r)] (4) T where both h(z) and g(r) can be arbitrary functions with the following constraints:

ψ(r, z) =

(5a)

and h(T ) = h(−T ) = 0.

    R2n   −2 − r − g (r) + 2g(r) h (z) r   R2 V0 1 − 2n {1 + g(r)h (z)} = 2Ti r

(6a)




V0 = 2T

ε˙ rz =

(3b)

In Eq. (3a), T represents an instantaneous height of the upset ring at any given step during deformation, and therefore T = T0 is set initially, see Fig. 1. A kinematically admissible stream function that satisfies Eqs. (1) and (3) is mathematically found to be

h(0) = 0

ε˙ θθ

(3a)

and Vr (Rn , z) = 0.

  R2n R2n 1 + 2 + 1 + 2 g(r) r r     2 R + r − n g (r) h (z) r


V0 ε˙ rr = 2T

(2b)

where Vr , Vz and Vθ represent the radial, the axial and the circumferential velocity components, respectively. As described before, the kinematic conditions for the velocity field can be mathematically obtained as follows: Vz (r, ±T ) = ∓V0 ,

and (2):

(5b)

According to the relationship described in Avitzur [30], the strain-rate components can be determined from Eqs. (4)

ε˙ θr = ε˙ θz = 0

(6e)

It should be noted that in Eqs. (6), the primed notation denotes the differentiation with respect to z for the function h(z), or the differentiation with respect to r for the function g(r). In order to satisfy the symmetry requirements, i.e. ε˙ rz = 0 for r = Rn or z = 0, Eq. (6d) together with Eq. (5a) provide the following constraints: h (0) = 0

(7)

and g (Rn ) = 0.

(8)

2.3. The upper-bound formulation In absence of surface tractions, the strain hardening effect, and rigid/plastic interfaces, the total forming energy rate, J,

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for the problem of upset rings can be mathematically stated as    4σ0 π T R0 (z) 1 ε˙ ij ε˙ ij r dr dz J= √ 2 3 −T Ri (z)  mσ0 + √ |V | ds (9) 3 Sd √ where σ0 = 3k is the tensile yield strength, m the interfacial friction factor reflecting the magnitude of shear stress at the friction interface and V is the velocity discontinuity at the friction interface whose area is denoted by Sd . The first term on the right hand side of Eq. (9), which will ˙ i , represents the energy rate due to plastic be denoted by W deformation, and it can be easily obtained by direct substitution of Eq. (6) as   πσ0 V0 T R0 (z) ˙ Wi = √ F (r, g, g , g , h, h , h )r dr dz 3T −T Ri (z) (10)

where   2  Rn (1 + g(r)h (T )). H = r− r

where

F = Grr + Gzz + Gθθ + Grz

Since the total forming energy rate, J, calculated by the upper-bound theorem is never less than that required to perform the upset forging of rings, the solution can be determined by minimizing J. Therefore, the method of variational calculus can be applied, since J is expressed as a functional. To this end, while the function, h(z), is considered as an arbitrary function in prior, the other function, g(r), is assumed to be a half-range expansion of Fourier series,     πr 2πr g(r) = α0 + α1 cos + α2 cos , R0 R0

with


 R2 R2 Grr = 2 1 + 2n + 1 + 2n g(r) r r    2 R2n   + r− g (r) h (z) r


Gzz

  2  R2n   = 2 −2 − r − g (r) + 2g(r) h (z) r 

Gθθ = 2 1 −

2 R2n 2 {1 + g(r)h (z)} r2




   R2n R2n  Grz = r− g(r)h (z) − 3 + 2 g (r) r r     2 R2 + r − n g (r) h(z) . r The second term on the right hand side of Eq. (9), which ˙ f , represents the energy loss due to fricwill be denoted by W tion at the interface. By referring to Fig. 1, since the direction of metal-flow along the interface depends on where the neutral surface, r = Rn , will locate during upsetting process, there ˙ f. are two cases that may possibly exist for W Case (i) Rn < Ri . In this case, all the particles on the ˙ f , can be interface flow outward. Hence, the energy loss, W obtained using Eqs. (4) and (2a) as  R0 (T ) 0 V0 m ˙ f = 2πσ √ H(r, g, h )r dr (11a) W 3T Ri (T )

Case (ii) Ri < Rn < R0 . In this case, the particle defined by r < Rn flows inward and that defined by Rn < r < R0 flows ˙ f , becomes to be outward. Hence, the energy loss, W   Rn 0 V0 m ˙ f = 2πσ √ W − H(r, g, h )r dr 3T Ri (T )   R0 (T )  H(r, g, h )r dr + (11b) Rn

It is clearly seen that with the results shown in Eqs. (9)–(11) the total forming energy rate, J, becomes a functional when either of g(r) and h(z) is considered as an arbitrary function. 2.4. Variational approach

Ri ≤ r ≤ R0 ,

(12)

where α0 , α1 and α2 are parameters to be determined. We do consider in this manner that Eq. (12) has been introduced for g(r) before the variational process is performed, because that, as shown in Eqs. (4) and (2a), the radial velocity component, Vr , which characterizes the bulged profiles of the upset ring at the lateral surface, is primarily dominated by the function h(z). Consequently, to derive natural boundary condition for h(z) is much more significant than for g(r), which also simplifies the variational process. Let δ be the first variational operator, then the necessary condition for h(z) to minimize J requires δJ to be zero. Thus, in a very similar manner as presented by Yeh and Yang [15], an equilibrium equation along with two sets of boundary conditions, namely the kinematic and the natural boundary conditions, we obtain for h(z) can be summarized as follows: 2.4.1.  Equilibrium   of h(z) equation ∂2 ∂F ∂ ∂F ∂F − + =0 ∂z2 ∂h ∂z ∂h ∂h

(13)

where F is as given in Eq. (10). It is noted that although the total forming energy rate, J, as shown in Eqs. (9)–(11), has two different forms, depending

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on the interface friction condition, Eq. (13) is the only one derived for governing the function, h(z). 2.4.2. The boundary conditions of h(z) Since J has two different forms depending on the direction of metal flow at the friction interface, there are two cases that may exist for the boundary conditions of h(z). Case (i) Rn < Ri . On the boundary z = ±T,    R0  ∂F ∂ ∂F either r dr = 0, − ∂h ∂z ∂h Ri h(z) is prescribed; and   R0  ∂F 2 2 either r + m(r − Rn )g(r) dr = 0, ∂h Ri or

or

h (z) is prescribed.

Case (ii) Ri < Rn < R0 . On the boundary z = ±T,    R0  ∂F ∂F ∂ r dr = 0, − either ∂h ∂z ∂h Ri or

h(z) is prescribed; and Rn

(14a) (14b) (15a) (15b)

(16a) (16b)

either

or

h (z) is prescribed.

In this investigation, the upper-bound solution was determined numerically as an approximation using the Ritz method [31]. Due to the symmetry requirement that is ψ(r, z) = −ψ(r, −z), and also due to the constraints as given by Eqs. (5) and (7), it is mathematically legal to choose the function h(z) as a polynomial of fifth order, i.e. h(z) = β0 z(z2 − T 2 ) + β1 z(z4 − T 4 ),



 ∂F 2 2 r − m(r − R )g(r) dr n ∂h Ri   R0  ∂F 2 2 + r + m(r − R )g(r) dr = 0, n ∂h Rn 

2.5. The upper-bound solution

(17a)

(18)

where β0 and β1 are parameters to be determined and T is the instantaneous height of the ring at each step of the upsetting process, considered in small steps for computational purpose. A program written in FORTRAN language was developed to implement theoretical formulations. In numerical computations, the energy functional, defined by Eqs. (9)–(11), was algebraically expanded using the Simpson’s integration rule. With the functions given in Eqs. (12) and (18), the VUB solution can be determined by minimizing the energy functional subjected to the symmetry requirement (Eqs. (7) and (8)) and the natural boundary condition (Eq. (15a) or (17a), depending on the interface friction condition). However, in obtaining the traditional UB solution, the natural boundary condition was ignored at all. Accordingly, the number of free parameters left for minimization in the VUB analysis is four, and it is always less than that by one in the traditional UB analysis. Upon derterming the solution, the bulged profile of upset rings is predicted following the technique described in Hosford and Cadell [32], and the total forming energy rate is calculated using Eqs. (9)–(12) and (18).

(17b) 3. Results and discussions

The boundary conditions as given in Eqs. (14)–(17) are discussed subsequently. For case (i) that is Rn < Ri , since Eq. (5), which has shown h(T) = h(−T) = 0, indicates that h(z) is prescribed at z = ±T, the derived Eq. (14b) is of kinematic boundary, and therefore Eq. (14a) can be disregarded. However, since there is no restriction on h (z) at z = ±T, Eq. (15b) cannot be satisfied forever. Instead, the natural boundary condition as shown in Eq. (15a) must be required. Similarly, for case (ii) that is Ri < Rn < R0 , the above discussion can also be applied to conclude that, in addition to the kinematic boundary condition, as shown in Eq. (16b), Eq. (17a) is the derived natural condition available. Meanwhile, we should emphasize that it is impossible to obtain the natural boundary condition so perceptibly as to obtain the kinematic boundary conditions, unless the method of variational calculus is involved theoretically. In several past investigations, such as in Refs. [14,18–21,26], solutions were determined without taking account for such a natural boundary condition. However, the advantage of the natural boundary condition can be in better understanding with the results to be discussed in a subsequent section.

In this section, capability and validity of the VUB method are investigated. Applicability of the proposed model in analysis of the upset forgings of ring also receives attention. Comparisons and discussion will be made among the results, obtained by the VUB method, the UB method and the FEM, respectively. Experimental results of the calibration curves obtained by Lee and Altan [18], Male and Depierre [25] and Moon and Van Tyne [22], respectively, from ring tests were used for quantitative verification of the proposed model in association with the upper-bound theorem. Of major concern is the improvement of the VUB method on the UB method in predicting the bulged profiles of the upset forging of rings and cylinders as well as in calculating the upper-bound energy. As a result, some related experimental data, obtained by Lee and Altan [18], Nagpal et al. [20] and Hsiang and Huang [26], repectively, were compared with theoretical results in order that the advantage of the VUB analysis, which satisfies the natural boundary condition derived in this investigation, can be signified. As shown in the figures to be discussed, the symbol, m, again, indicates the interfacial friction factor, and RH repre-

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397

sents the percentage of reduction in height of the upset ring or disks defined to be RH =

T0 − T × 100%. T0

(19)

As shown in Fig. 2 are the calibration curves determined using the VUB and the traditional UB methods, respectively; the experimental data obtained by Lee and Altan [18] (denoted by circular symbols) and Male and Depierre [25] (denoted by squared symbols) from ring tests are also presented for comparison. In this figure, the change in minimum internal diameter of the ring, expressed as a percentage of the original internal diameter, is expressed as a function of reduction in height on deformation for various values of the interfacial friction factor, m. In the work of Lee and Altan, the specimen sizes used were 3.0 in. o.d. × 1.5 in. i.d. × 1.0 in. height (6:3:2). The m values, they determined from the experimental data is 1 for the test under dry condition (Aceton cleaned) and 0.25 for the test lubricated with MoS2 . The specimen of the same proportion (6:3:2) were also used by Male and Depierre, who obtained m = 0.14 from the test lubricated with Lanolin, and m = 0.32 from the test lubricated with Graphite, respectively. These m values were also used in both VUB and UBs’ analyses for determing the calibration curves. It is clearly seen from Fig. 2 that there is satisfactory agreement between the theoretical and experimental results. A further verification from the calibration curves of Moon and Van Tyne [22] is shown in Fig. 3, where both of the experimental data (denoted by various symbols) and theoretical curves are presented. The m values were experimentally found by the authors to strongly depend on the type of the lubricants as indicated in the figure. The m values given by these curves are compared with the m values obtained from the experiments. This comparison shows that the m values from the theories, the VUB and the UB methods, are generally less than those obtained from the experiments. The factor between the theoretical and experimental m values is between 0.6 and 0.8, approximately. However, for the case with no lubricant (denoted by a solid circular symbol) and

Fig. 2. Comparison of the calibration curves between the theoretical and experimental results.

Fig. 3. Comparison of the calibration curves between the theoretical and experimental results.

also for the case with glycerin lubricant (denoted by a solid squared symbol) at the smallest reduction, the factor appears to be 1.1 or a little higher. Although deviations between the theories and experiments are observed, it is fair to say that the theoretical predictions, employing the proposed model, quanlitatively agrees with the experiments. Also, it is apparently observed from this figure that the calibration curves, determined by these two methods, are almost coincident correspondingly for the cases of lower values of m. However, with increasing values of m, the effect of the natural boundary condition on determining the calibration curve becomes noticeable. As shown in Figs. 4 and 5 are the bulged profiles of the upset forgings of disks compared between the theories and experiments. Also presented in the figures are the result of FEM employing the commercial software package MARC [27] for comparison. In obtaining the results from MARC, the rigid/plastic model with assuming a constant interfacial friction stress was employed, as was used in the approximate methods. However, the detail in operating the MARC is omitted because it is not a primary subject of this investigation, but the reader can be referred to [27] for a guidance. In Fig. 4, the experimental result of Hsiang and Huang [26] was obtained from the specimen made of AISI 1050 steel. The upper-bound solution, they obtained on the basis of a different model of the velocity field that contains two free parameters, is also presented for comparison. The initial size of the specimen is 18 mm (o.d.) × 10 mm (H) and was upset at 46% reduction in height approximately. The value of m = 0.3, they experimentally determined, was used in all theroretical analyses. It can be seen that, by employing the proposed model as shown in Eqs. (4), (12) and (18), the predicted bulged profile is significantly improved if compared with those obtained by Hsiang and Huang [26], and by the FEM, respectively. In addition, one can attend that the results obtained by the UB and the VUB methods are altered slightly, attributed to the effect of the natural boundary condition. This can be further evident from Table 1, where both magnitudes of statistical mean absolute deviation (MAD) and standard

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Table 1 Comparison of MAD and AD among the results obtained by FEM, UB and VUB methods, respectively FEM

VUB

UB

Comparison of MAD and AD from the bulged profile of the upset disk (Hsiang and Huang [26]) MAD 0.0896 0.0826 S.D. 0.1065 0.0912

0.0224 0.0264

0.0192 0.0238

Comparison of MAD and AD from the bulged profile of the upset disk (Lee and Altan [18]) MAD 0.0115 0.0074 S.D. 0.0121 0.0087

0.0057 0.0074

0.0063 0.0086

Comparison of MAD and AD from the bulged inner surface of the upset ring (Nagpal et al. [20]) MAD 0.2580 0.1720 S.D. 0.3102 0.2425

0.2288 0.2513

0.2313 0.2428

Comparison of MAD and AD from the bulged outer surface of the upset ring (Nagpal et al. [20]) MAD 0.2828 0.1324 S.D. 0.4127 0.1663

0.2495 0.3442

0.3482 0.4591

deviation (S.D.) of the predicted bulged profiles from the corresponding experimental data are listed for comparison. The greater the magnitudes of MAD and S.D., the more the predicted profile deviated from the measured profile becomes. It is noted that for analysis of the upset forgings of disk, the radii, Ri and Rn , as appear in Eqs. (9)–(11), were taken to be zero, and thus the constrained Eq. (8) becomes redundant. In Fig. 5, the experimental result of Lee and Altan [18] was obtained from the specimen made of Al 1100O aluminum. The upper-bound solution provided by the authors is based on a simple velocity field, assuming that the axial velocity component is only a function of the axial coordinate, z. The initial size of the specimen is 1.5 in. (o.d.) × 2.25 in. (H) and was upset at 53% reduction in height approximately. The

value of m was experimentally determined to be 0.25 and was used theoretically. From the results shown in this figure, slight deviation between the bulged profiles predicted by the VUB and the UB methods is also observed. However, the best prediction is obtained by the proposed model, independent whether or not the natural boundary condition was imposed theoretically. The magnitudes of MAD and S.D. are also listed in Table 1 for reference. As for the prediction of the velocity fields as well as the equivalent plastic strain curves, Figs. 6 and 7 show the results obtained using the methods as indicated. In these figures, the processing parameters used are the same as those used in Fig. 5, and only the results corresponding to the

Fig. 4. Comparison of the bulged profile of the disk between the theoretical and experimental results.

Fig. 5. Comparison of the bulged profile of the disk between the theoretical and experimental results.

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Fig. 7. Distribution of the equivalent plastic strain curves.

Fig. 6. The determined velocity fields: (a) comparison of the velocity fields determined by FEM and VUB method, respectively, and (b) comparison of the velocity fields determined by the UB and VUB methods, respectively.

final step of the upsetting process are presented for paging limitation. Shown in Fig. 6(a) are the corresponding velocity fields determined using the FEM and the VUB method, respectively, and shown in Fig. 6(b) are those determined using the UB and the VUB methods, respectively. Among the velocity fields are not resulted identically, but only slight deviations are observed in the region of right corner. All the determined fields are characterized such that the velocities of axial component, Vz , move completely toward the symmetric plane, whereas those of radial component, Vr , move completely outward from the symmetric axis. However, the distribution of the equivalent plastic strain curves, as shown in Fig. 7, are resulted with more deviation.

In Fig. 7, for the sake of easy comparison, the plot is arranged to consist of four individual sub-plots, each of which occupies a quarter of the disk and is symmetric to both z and r coordinates. It is apparently observed from all solutions that the equivalent plastic strain increases both along the symmetric plane (z = 0) outward from the symmetric axis (r = 0) and along the symmetric axis upward (or downward) from the contact interface. The maximum equivalent plastic strain is calculated to occur at the original point, and the minimum one is calculated to occur at the intersecting point of the contact interface and the symmetric axis. In regard to the gradient of the equivalent plastic strain, it is found to be greater in the region close to the interface, and all the distributed patterns look qualitatively similar in general. The last case we employ to verify the proposed model is shown in Fig. 8, where the bulged inner and outer surfaces of the upset forgings of ring are compared between the experimental result of Nagpal et al. [20] and the theories. The experiment was obtained from the specimen made of Al 1100F aluminum, which follows 6:3:2 proportion (54 mm (o.d.) × 27 mm (i.d.) × 18 mm (H)), and was upset at 29.5% reduction in height approximately. The value of m was experimentally determined by the authors to be 0.52 and was used theoretically. In this prediction, the FEM arrives at the result most precisely, but the proposed model also improves the upper-bound solution to be closer to the experimental finding. The reader may refer to Table 1 for clarity. As mentioned before, the deviation between the results, determined using the VUB and the UB methods, respectively, can also attributed to the effect of the natural boundary condition. We should note that the solutions, previously obtained by Nagpal et al. [20] and by the VUB method were based on different models of stream function, but they were determined in accordance with allowing equal number of free parameters left for the purpose of minimization. We should also recall that both of the UB and the VUB solutions were based on the same stream function, but the number of free parameters allowed for the VUB analysis is less than that for the UB analysis by one.

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Fig. 9. The determined velocity fields: (a) comparison of the velocity fields determined by FEM and VUB method, respectively, and (b) comparison of the velocity fields determined by the UB and VUB methods, respectively.

Fig. 8. Comparison of the bulged profile of the upset ring: (a) the profile of inner surface and (b) the profile of outer surface.

In connection with the case of the upset forgings of the ring just discussed, the determined velocity fields as well as the neutral surfaces at the final step of the upsetting process are shown in Fig. 9, where the employed methods are also indicated. As found in Fig. 9 for the case of the upset ring, among the determined velocity fields are observed to have slight deviations in the region close to the lateral surfaces. Specifically, all the determined velocity fields are characterized such that Vz moves completely toward the symmetric plane, whereas Vr moves toward the lateral surfaces from the neutral surface. However, the neutral surface resulted from the FEM looks like a barrelling shape, unlike the ones

assumed by the upper-bound methods in prior to be cylindrical. Although the velocity field determined by the UB and the VUB methods are different, yet the neutral surfaces they predict are almost coincident. Also, the distribution of the equivalent plastic strain curves, as shown in Fig. 10, are resulted with more deviation. In order to properly magnify the plot for the sake of comparison, the plots in Fig. 10 are arranged to consist of two sub-plots, each of which occupies a quarter of the ring, and the r, coordinate begins at 10 mm from the symmetric axis. It is observed that the maximum equivalent plastic strain is computed to occur at the intersecting point of the inner surface and the symmetric plane, but the minimum one, occuring on the contact interface, varies with the method used. However, the distributed patterns look qualitatively similar in general. As shown in Fig. 11 is the curve of the ratio of average pressure versus the reduction in height on deformation for various values of the interfacial friction factor, m. The ratio of average pressure is proportional to the total forming energy rate, J, √ given by Eq. (9), by a constant factor 3/(2πσ0 AV0 ), where A is area of the contact interface and σ 0 and V0 are constants as explained before. Since the upper-bound method gives the calculated J closer to the actual total forming energy rate from above, the smaller the calculated J, the better the upper-bound solution is attained. Accordingly, for lower values of m, say m ≤ 0.3, both of the VUB and the UB methods predict the result almost indistinguishable. However, for the value of m greater than 0.3 approximately, the VUB method undoubtedly shows an improvement on the UB method in predicting the ratio of average pressure, and this improvement becomes more apparent with the increasing value of m.

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It has been generally recognized that with an increasing number of parameters in the velocity field, the solution improves while the computation becomes more complex. However, one should recall from Eq. (15a) or (17a), which shows the interfacial friction factor, m, is incorporated in the natural boundary condition. Alternatively, such a condition can be interpreted physically as to constrain plastic flow of the upset ring upon the contact interface during upsetting process. When it is imposed in the VUB analysis, not only the number of free parameters can be reduced, but also the upperbound solution is improved. Clearly, this is the advantage of the VUB method over the traditional UB method.

4. Conclusion

Fig. 10. Distribution of the equivalent plastic strain curves: (a) comparison between the equivalent plastic strain curves obtained by the FE and the VUB methods, respectively, and (b) comparison between the equivalent plastic strain curves obtained by the UB and the VUB methods, respectively.

In this investigation, a mathematical model of stream function was proposed for the upsetting problems. To obtain the upper-bound solution, the method of variational calculus is applied to derive the natural boundary condition. The results obtained in this manner, together with those obtained by the FEM and by the traditional UB method were discussed and compared with related experiments. On the basis of the given theoretical and experimental results, some conclusions can be drawn subsequently. In verification of the proposed model, the theoretical prediction of the calibration curves satisfactorily agrees with the experimental results. Also, the model has shown to improve the upper-bound solution in predicting the bulged profile of the upset disk and ring when compared with a variety of mathematical models previously proposed. The FEM is shown to be better in the prediction of the bulged profile of the upset ring, but the upper-bound method associated with proposed model is better for the upset disk. Since the natural boundary condition, taking account for the interfacial friction condition, is imposed, the VUB method not only improves the solution both in predicting the bulged profiles of the upset ring and in calculating the total forming energy rate, but also makes the computational work simpler. While keeping the same number of free parameters, the VUB method is permitted to use the approximation (Eq. (18)) of higher order when compared with the UB method because of presence of the natural boundary condition. Since the neutral surface of the upsetting ring is found by the FEM to be like a barreling shape, it may considered as a function of z in further works of the upper-bound analysis, assuming it is cylindrical.

Acknowledgments

Fig. 11. Comparison between the ratio of average pressure obtained by the UB and the VUB methods, respectively.

The authors gratefully acknowledge the financial support provided by National Science Council of Taiwan, ROC under grant numbers NSC90-2212-E008-012 and partially under NSC92-2212-E008-023.

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