A new approach to visioplasticity in dynamic plane upsetting

Comput. Methods Appl. Mech. Engrg. 190 (2000) 1689±1700

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A new approach to visioplasticity in dynamic plane upsetting J.P. Wang Department of Marine Engineering and Technology, National Taiwan Ocean University, 2 Pei Ning Road, Keelung 20224, Taiwan, ROC Received 15 July 1998; received in revised form 19 July 1999

Abstract A new numerical approach to analyze the stress distributions in the plane-strain upsetting process is developed. In this procedure, the generalizing stream function with the velocity parameters used to ®nd the optimal ¯ow pattern is produced ®rst, and then the velocity and strain-rate ®elds can be obtained. From the strain-rate ®elds calculated, the stress ®elds can be found by the constructed slip-lines from Cauchy±Riemann equations. Along these slip-lines, the stress ®elds can be calculated easily from Hencky equations. The results show that this method presents a good and reasonable solution compared with the slab method and traditional visioplasticity. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: Gerneralizing stream function; Slip-line functions; Cauchy-Riemann equations

Nomenclature v0 u; v T E e_x ; e_y ; e_xy e_v e_ r rx ; ry ; sxy rm / h k a…x; y†; b…x; y†

unit velocity velocity in the x- and y-directions half thickness of the strip error norm strain-rate component volumetric strain-rate e€ective strain-rate e€ective stress stress component mean stress ¯ow function the ®rst shear direction angle constant ratio slip-line functions

1. Introduction The knowledge of stress distributions in the deforming body play an important role in the design of machine tools. The concentrated stress will cause the fracture of work piece and the wear on the tool surface. For the purpose of ®nding stresses in forming region, many numerical methods have been proposed, such as: ®nite element method, ®nite di€erent method, and weighted residual method. In these methods, there is the same question to be proposed. This question is which of the results is the best, and the closest to the true solutions. To consider this problem, solutions obtained from experimental mechanics can be proposed as a reference data. The visioplasticity [1±3] is one of the fundamental works of experimental mechanics. In this 0045-7825/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 5 - 7 8 2 5 ( 0 0 ) 0 0 1 8 3 - 3

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method, there exist many diculties: ®rstly, the ¯ow lines obtained from experimental data are not smooth; secondly, the velocity ®elds calculated from these ¯ow lines may not satisfy the condition of continuity (incompressibility); and ®nally, the error of the stress ®elds calculated by using integral equation are inevitable. Hence, it is seldom used to analyze the metal forming processes recently. For the purpose of smoothing the ¯ow lines, many numerical methods have been developed. The smoothing procedure mentioned by Shabaik and Kobayashi [4] is based on a simple averaging of the points and causes diculties when treating data that are ill-de®ned. Ishikawa et al. [5] have used the hyperbolic tangent curve to ®t the ¯ow lines, but there was great discrepancy in the case of non-hyperbolic tangent curve dies. Farmer et al. [6,7] and Dwivedi [8] have used high-order polynomials to smooth the ¯ow lines and their results have shown good results, but the numerical errors are unavoidable. Pehle and Kopp [9] have used marked lines at spacing to process the error analysis for visioplasticity data. In this investigation, it was shown that the actual values of the shear stress on the tool boundary could not be calculated with suciently high accuracy to provide reasonable solutions. In this paper, a numerical approach is proposed to establish a new procedure for the visioplasticity. In this procedure, a generalizing stream function with optimizing parameters is used to ®nd the ¯ow patterns. Hence, the condition of continuity and the procedures of smoothing the ¯ow lines can be automatically ful®lled. When the optimal ¯ow pattern is produced, the velocity and strain-rate ®elds can be obtained. From the calculated strain-rate ®elds, the stress ®elds can be found by the constructed slip-lines from Cauchy±Riemann equations. Along these slip-lines, the stress ®elds can be calculated easily from Hencky equations [10,11]. In this procedure, the slip-line method is adopted to ®nd the stess ®elds to improve the error from integral equation using in the traditional technique. In this paper, the plane-strain upsetting process is used to illustrate this procedure. The computed solutions will be compared with slab method [12] and traditional visioplasticity [13]. 2. The model of stream function 2.1. Velocity ®eld from stream function For incompressible material, the velocity ®eld for plane strain can be expressed by the stream function /…x; y† as follows: uˆ

o/ ; oy

vˆÿ

o/ ; ox

where u and v are the velocity components in the x- and y-directions. The strain-rates are:     ou o2 / ov o2 / 1 ou ov 1 o2 / o2 / ˆ ; e_y ˆ ˆÿ ; e_xy ˆ ÿ : ‡ ˆ e_x ˆ ox oxy oy oxy 2 oy ox 2 o2 y o 2 x The e€ective strain-rate is calculated from its de®nition 1=2 2  : e_ ˆ p e_2x ‡ e_2xy 3

…1†

…2†

…3†

The e€ective strain e can be evaluated from the integration of the e€ective strain-rate e_ along the ¯ow lines with respect to time Z t e_ dt: …4† eˆ 0

The volumetric strain-rate e_v is e_v ˆ e_x ‡ e_y ˆ

ou ov o2 / o2 / ‡ ˆ ÿ ˆ 0: ox oy ox oy ox oy

It is to be noted from Eq. (5) that the condition of incompressibility is satis®ed automatically.

…5†

J.P. Wang / Comput. Methods Appl. Mech. Engrg. 190 (2000) 1689±1700

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2.2. Experimental determination of the ¯ow ®eld An experimental setup was manufactured and installed in a 50 ton hydraulic press. For the purpose to con®ne the material ¯ow as plane strain upsetting, two plates were placed on the two end surfaces of the specimen. As working material, pure lead was chosen. The test specimens were rectangular blocks (24 mm  24 mm  25 mm) with 2 mm square grids marked on their surfaces. The die was designed so that the punches were moved at the same velocity but in the opposed direction (Fig. 1). To decrease the frictional drag on the surfaces, the specimens (except for the contacted surfaces between the punch and specimen) were sprayed with Te¯on ®lm before the test, the latter being carried out at a punch speed of 0.1 mm/s. A photograph of this process is shown in Fig. 2. Using a digital-scanning machine, the coordinate of the cross-point on the surfaces of specimens can be calculated. 2.3. The generalizing stream-function used for ®tting In this paper, a generalizing stream function with polynomial expression is chosen as u…x; y† ˆ

1 X 1 X iˆ0

aij xi y j :

…6†

jˆ0

From Eq. (1), the velocities u and v can be expressed as u…x; y† ˆ

1 X 1 X iˆ0

jaij xi y jÿ1 ;

v…x; y† ˆ

jˆ0

1 X 1 X iˆ0

iaij xiÿ1 y j :

…7†

jˆ0

To consider Eq. (7) and the velocity boundary condition (Fig. 1) at y ˆ 0; v ˆ 0 and y ˆ T ; v ˆ v0 ; the velocity of v…x; y† can be expressed as v…x; 0† ˆ

1 X iˆ0

iai0 xiÿ1 ˆ 0;

v…x; T † ˆ

1 X 1 X iˆ0

iaij xiÿ1 T j ˆ v0 :

jˆ0

Fig. 1. The model of plane upsetting.

…8†

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Fig. 2. The deformation of the grid inscribed on strip.

Equating coecients in Eq. (8), it is found that all the coecients become zeros on the sides of variable x, and a trivial velocity ®eld will be obtained. Hence, the velocity of v…x; y† can be assumed to be independent with coordinate x. That means v ˆ v…y† and the stream function can be described as u…x; y† ˆ ÿxv…y† ‡ f …y†:

…9†

The center line …x ˆ 0† is a stream line because no metal ¯ows across it, i.e., u ˆ 0; /…0; y† ˆ 0, and f …y† ˆ 0. Hence, Eq. (9) can be expressed as u ˆ ÿxv…y†:

…10†

J.P. Wang / Comput. Methods Appl. Mech. Engrg. 190 (2000) 1689±1700

From Eqs. (1) and (2), the velocity and strain-rate ®elds can be described as   dv…y† dv…y† 1 d2 v…y† ; v ˆ v…y†; e_x ˆ ÿe_y ˆ ÿ ; e_xy ˆ ÿx u ˆ ÿx : dy dy 2 dy 2

1693

…11†

It is interesting to note that the ¯ow patterns of plane-strain upsetting are dependent on the chosen v…y†. In this paper, the velocity of v…y† is chosen as cubic spline ®tting as shown in Fig. 3. The ends of the cubic spline line are set as v…0† ˆ 0; v…T † ˆ v0 , and the unknown velocities v1 ; v2 ; . . . ; vn ; . . . are chosen as optimizing parameters. For the purpose of smoothing, the second derivatives are speci®ed as the same at each internal point between the two cubic lines in each interval. In this method, the end conditions of second derivatives d2 v…y†=dy 2 are chosen from the conditions of shear stresses as: (a) At y ˆ 0, shear stress sxy is 0, that means   1 d2 v…0† d2 v…0† ˆ 0 ˆ s0 : ÿx ˆ 0 and e_xy …x; 0† ˆ 2 2 dy dy 2 (b) At y ˆ T , shear stress sxy is unknown that means d2 v…T † ˆ sT dy 2

and the unknown sT is chosen as an optimizing parameter:

In this method, six parameters (®ve velocity parameters and one unknown parameter of sT ) are chosen to ®nd the optimal ¯ow pattern. 2.4. Optimization with velocity parameters Since the metal ¯ow in the plane upsetting is symmetrical to the x and y axes, the upper right quarter of the workpiece is considered (Fig. 1). The velocity at cross-points on the specimen can be calculated as ui‡1 ˆ

xi‡1 ÿ xi ; Dt

vi‡1 ˆ

yi‡1 ÿ yi ; Dt

…12†

where ui‡1 ; vi‡1 are the velocities of the coordinate in the x- and y-directions for the i ‡ 1 step, and Dt is the time increment from the i to the i ‡ 1 steps. The stream function is chosen as the object value for the minimizing formulation. The unknown velocity parameter and sT can be determined from the following minimization:

Fig. 3. The cubic spline ®tting for v…y†.

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minimize

v uX u m 2 2 E ˆ t ‰…u…v0 ; v1 ; . . . ; v5 ; sT † ÿ uj † ‡ …v…v0 ; . . . ; v5 ; sT † ÿ vj † Š

…13†

jˆ1

subject to v0 6 vi 6 0;

i ˆ 1; 2; 3; 4; 5;

where E is the error norm (which is the summation for the velocity differences between the experimental and the numerical points), m is the number of experimental points. It is interesting to note that six parameters are needed to minimize Eq. (13), and the ¯exible-tolerance method [14] is adopted for the minimizing procedure. 3. The model of the slip-line 3.1. The constructed model of slip-line ®eld by Cauchy±Riemann equations The purpose of constructing the slip-line ®eld is to ®nd the curvilinear coordinates of a- and b-lines. The a-line, being the ®rst maximum shear direction, is taken 45° clockwise from the ®rst principal direction. The

Fig. 4. The relation between slip-lines and Mohr circle on the point p.

Fig. 5. The calculation procedure for constants C1 and C2 on slip-lines.

J.P. Wang / Comput. Methods Appl. Mech. Engrg. 190 (2000) 1689±1700

Fig. 6. The deforming of grids (left) and smoothed velocity ®eld (right) at reductions: (a) R ˆ 24%, and (b) R ˆ 30%.

Fig. 7. The contours of shear stress sxy at 30% reduction of height.

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b-line is the second maximum shear direction, 90° counterclockwise from the ®rst maximum shear direction. That means the curvilinear coordinates of a and b are orthogonal set. De®nite a complex function as w…x; y† ˆ a…x; y† ‡ ib…x; y†:

…14†

And the a…x; y†; b…x; y† are the slip-line functions. If w…x; y† is analytic in deforming region, from Cauchy±Riemann law oa ob ˆ ; ox oy

oa ob ˆÿ : oy ox

…15†

And the lines of a…x; y† ˆ c1 ; b…x; y† ˆ c2 will be mutually orthogonal. As shown in Fig. 4, along the a-line

ob=ox dy k sin h ˆ ˆ tan h ˆ : ob=oy dx k cos h

…16†

The h is the angle (®rst shear directions) p that  a lines make with the x-axis (measured counterclockwise), and k is constant ratio that can be set as 1= 3. From Eqs. (15) and (16) k sin h ˆ

ob ; ox

k cos h ˆ

ob ; oy

k sin h ˆ ÿ

oa ; oy

k cos h ˆ

oa : ox

…17†

The numerical construction of slip-line ®eld can be derived from Eq. (17) da ˆ

oa oa dx ‡ dy ˆ k cos h dx ÿ k sin h dy; ox oy

db ˆ

ob ob dx ‡ dy ˆ k sin h dx ‡ k cos h dy: ox oy

…18†

From Eq. (18), the slip-lines can be constructed when the ®rst shear directions (h) are known at every point for all deforming body. The origin of the curvilinear coordinates can be set to any point of deforming region (the case of this paper is set at a-point in Fig. 5). 3.2. Stress calculation By combining the calculated strain-rate ®eld with Eq. (17), the a- and b-lines are constructed. The stress ®eld can be carried out by Hencky equations:

Fig. 8. The contours of slip-lines at reduction of height (left: 24%, right: 30%).

J.P. Wang / Comput. Methods Appl. Mech. Engrg. 190 (2000) 1689±1700

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Fig. 9. The contours of stress ry =k compared with traditional visioplasticity method at 30% reduction of height: (a) normal presssure, (b) this method, and (c) traditional method.

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rm ‡ 2kh ˆ C1

along the first slip-line;

rm ÿ 2kh ˆ C2

along the second slip-line;

…19†

where rm is the mean stress. The constants C1 and C2 can be obtained from the conditions of free boundary. The components of stresses in terms of rm and h can be expressed as: rx ˆ rm ÿ k sin 2h;

ry ˆ rm ‡ k sin 2h;

s ˆ k cos 2h:

…20†

Using Eqs. (19) and (20), the stress components at any point in the deformation regions can be calculated. A computer program for the calculation of the state of stress at any point is developed. As shown in Fig. 5, the region-A is ®rst calculated using the conditions of free boundary by b-line …rx ˆ 0; rm ˆ p13 at a-point) and then, the region-B is obtained by a-line according the calculated data in region-A. The region-C is also calculated by the same procedure using b- and a-lines.

Fig. 10. The contours of mean stress rm compared with traditional visioplasticity method at 30% reduction of height: (a) this method, and (b) traditional method.

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4. Results and discussion The deforming grid and smoothed velocity ®eld at 24% and 30% reductions of height are shown in Fig. 6. The result shows that the ¯ow lines ¯ow from the top of strip to the free surface, and greater deforming of grids can be observed at the location of symmetrical surfaces (y ˆ 0) near the free boundary. The contour of shear stress ®eld at 30% reduction of height is shown in Fig. 7. It can be seen that the shear stress along the friction boundary increases parabolically, and approaches the level of sticking at the end edge. This result shows that the solution obtained from this paper has a good coincident with the experiment data proposed by Shabaik [15]. It is interesting to note that this shear stress is zero along the surface of symmetry. The slip-line ®elds at 24% and 30% reductions of height are shown in Fig. 8. It can be seen that the sliplines are tangential to the symmetrical surface (y ˆ 0) and center line (x ˆ 0) at 45°, and show a good agreement with the shear boundary condition (sxy ˆ 0). It is also to be shown that the slip-lines tangential on the friction boundary are decreased owing to the increase of the friction shear stress. The contour of stress ry =k stress at 30% reduction of height is shown in Fig. 9, the normal pressure obtained by this method, traditional visioplasticity [13], the slab [12], and traditional slip-line method proposed by Shabaik [15] also are given in this ®gure. It is seen that all of them have the same trend, and this paper presents a good agreement with traditional slip-line method. It is interesting to note that there exists a great discrepancy between this method and traditional visioplasticity. The discrepancy can be explained as the numerical errors by the procedure of integration. The contour of mean stress rm compared with traditional method at 30% reduction of height is shown in Fig. 10. It is found that the trend of means stress described in this method increases from top to the bottom and is contrary to traditional method. That means the highest pressure will be presented on the bottom. This phenomenon shows a reasonable result compared with traditional method. 5. Conclusions This paper presents a new way to rebuild the procedure of visioplasticity for plane-strain upsetting. There are two models expressed in this paper. One is used to ®nd the optimal ¯ow that satis®es the condition of continuity and the smoothing of ¯ow lines. The other is used to calculate the stress ®elds by the slip-line method combined with the theory of Cauchy±Riemann equations. The plane-strain upsetting process is used to illustrate this procedure. From the above discussion, the following conclusions can be drawn: (1) The shear stress along the friction boundary increases parabolically. It is shown that this result is well coincident with the experiment data proposed by Shabaik. (2) The normal pressure along the friction boundary has a good agreement with traditional slip-line method, and indicates that this method has a good approach compared with the solutions of traditional visioplasticity method. (3) The distribution of mean stress is found to increase from top toward bottom, contrary to traditional method. This phenomenon shows that a reasonable result is obtained by this method. (4) The present method proposes new procedures to visioplasticity for dynamic plane upsetting. From this way, an optimal ¯ow pattern and reasonable solution can be obtained. References [1] [2] [3] [4] [5] [6] [7]

E.G. Thomsen, A new approach to metal-forming problems, Trans. ASME 77 (1955) 515. E.G. Thomsen, Visioplasticity, Ann. CIRP 11 (1964) 127. L.E. Farmer, R.F., An experimental procedure for studying the ¯ow in the plane extrusion, Int. J. Mech. Sci. 21 (1979) 599. A.H. Shabaik, S. Kobayashi, Computer application to the visioplasticity method, J. Eng. Ind. 89 (1967) 339. H. Ishikawa, K.I. Hata, M. Goto, On one analysis of axisymmetrical extrusion by the use of ¯ow lines, J. Eng. Ind. 99 (1977) 419. L.E. Farmer, S.W. Conning, Numerical smoothing of ¯ow patterns, Int. J. Mech. Sci. 21 (1979) 577. L.E. Farmer, P.L.B. Oxley, A computer-aid methods for the construction and analysis of hardening slip-line ®elds for experimental ¯ow ®elds, in: Proceedings of the 25th International Mach. Tool Design and Res. Conference, 1985, p. 427.

1700 [8] [9] [10] [11] [12] [13] [14] [15]

J.P. Wang / Comput. Methods Appl. Mech. Engrg. 190 (2000) 1689±1700

S.N. Dwivedi, Plane strain visioplasticity for dynamic and quasi-static deformation processes, J. Eng. Ind. 105 (1983) 197. H.J. Pehle, R. Kopp, An error analysis for visioplasticity data processing, Steel Res. 57 (1986) 318. S.K. Samanta, Slip-line ®eld for extrusion through cosine-shaped dies, J. Mech. Phys. Solids 18 (1970) 311. L.E. Farmer, P.L.B. Oxley, A slip-line ®eld for plane-strain extrusion of a strain hardening material, J. Mech. Phys. Solids 19 (1971) 369. R.H. Wagomer, Fundamentals of Metal Forming, Wiley, New York, 1996. J.P. Wang, Y.H. Tsai, J.J. Wang, The dynamic analysis of visioplasticity for the plane upsetting process by ¯ow-function element technique, J. Mater. Process. Tech. 63 (1997) 736. D.M. Himmelblau, Applied Nonlinear Programming, McGraw-Hill, Maidenhead, UK, 1972, p. 148. A.H. Shabaik, Prediction of the geometry changes of the free boundary during upsetting by slip-line theory, J. Eng. Ind. 93 (1971) 586.