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An optimum-curved die profile for the hot forward rod extrusion process
Journal of Materials Processing Technology 73 (1998) 97 – 107
An optimum-curved die profile for the hot forward rod extrusion process A.S. Wifi a,*, M.N. Shatla b, A. Abdel-Hamid b,c a
Department of Mechanical Design and Production, Faculty of Engineering, Cairo Uni6ersity, Giza 12 316, Egypt b Department of Engineering and c Department of Science, The American Uni6ersity in Cairo, Cairo, Egypt Received 12 April 1996
Abstract An incremental slab method is implemented to obtain the extrusion pressure of the hot forward rod extrusion process for arbitrarily-curved dies. Two different algorithms utilizing that incremental slab method are developed to obtain the optimum curved die profile that minimizes the extrusion load and stresses at the tool – workpiece interface for the hot forward rod extrusion process. Both techniques are based on the optimization of the effects of friction at the tool – workpiece interface, strain rate, and redundant deformation. The optimum curved die profiles generated by the two different algorithms are found to be very similar. For the material used in the present study which is carbon steel, the shape of the optimum curved die profile is found to depend on the extrusion ratio and Coulomb friction coefficient and not to be affected by the extrusion velocity. The finite element method revealed that the optimum curved die profile developed in the present work produces lower stress levels than those produced using the optimum conical die profile found in the literature. © 1998 Elsevier Science S.A. Keywords: Optimum curved die profile; Tool–workpiece; Friction; Strain rate; Extrusion ratio; Finite element method
1. Introduction The hot forward rod extrusion process can be described as reducing the cross sectional area of a heated rod placed in a container by pushing the rod by a ram and squeezing it through a die. The factors affecting the process are the material properties and the forming conditions, the latter of which are the rod temperature, friction at the tool– workpiece interface, the extrusion ratio and the die profile. Optimization of these parameters has been one of the most important tasks that has attracted the attention of many researchers and engineers using analytical, computer-aided design (CAD) and experimental techniques. In metal forming, the flow stress of the material is a function of strain, strain rate and temperature. If the forming operation takes place above or close to the recrystalization temperature of the processed material, the effect of strain on the flow stress is negligibly small and the flow stress becomes dependent on the strain * Corresponding author. Fax: + 20 2 5723486; e-mail: [email protected] 0924-0136/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved. PII S 0 9 2 4 - 0 1 3 6 ( 9 7 ) 0 0 2 1 8 - 5
rate and temperature only [1]. Different empirical relationships with constants obtained experimentally have been used to express the flow stress of the hot formed material as a function of strain rate and temperature. Some of these relationships have temperature dependent material constants with no temperature terms [1]. Another more convenient expressions of the flow stress as a function of strain rate and temperature utilize the Zener–Holloman parameter in conjunction with temperature independent material constants [2,3]. A slab element can be used to obtain an expression of the strain rate as a function of the axial location in the deformation zone for the forward rod extrusion process, such expression being used by Farag et al. [3] to study the flow stress in the hot extrusion of commercial purity aluminum. Srinivasan et al. [4] also used such expression to study the stain rate distribution in the deformation zone for different die profiles and to generate a controlled strain rate die profile, as will be mentioned later. The expression of the local strain rate as a function of the axial location in the deformation zone can be used in conjunction with the expression of the flow stress as a function of strain rate to determine
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Fig. 1. Incremental slab method for curved dies.
the local flow stress in the deformation zone for the forward hot rod extrusion of metals. The total strain of a workpiece undergoing any plastic deformation consists of a homogeneous component and a redundant component. The homogeneous strain is responsible for the change of dimensions of the workpiece to acquire the required final shape whereas the redundant strain occurs internally and does not contribute anything to the final shape of the product [5]. Thus, the total strain can be expressed as the homogeneous strain multiplied by a factor called the strain redundancy factor. The strain redundancy factor can not have a value of less than unity at which value the total deformation equals the homogeneous deformation. This is an ideal case, in which there is no redundant deformation that adds to the strain hardening of the material causing higher stress levels during the deformation process [5]. For forward rod extrusion, it has been found that the redundant deformation increases with increasing of the die angle, which leads to more strain hardening and consequently higher extrusion load. However, increasing the die angle decreases the tool – workpiece interface length which leads to lower frictional stresses at this interface and consequently to lower extrusion load [5,6]. The optimum die angle that minimizes the extru-
Fig. 2. Description of Bezier curves.
sion load for the forward rod extrusion process achieves a balance between the effects of redundancy and friction [5,6]. This optimum die angle was obtained analytically by Avitzur [6] through the minimization of an upper bound expression of the extrusion load as a function of the die angle with respect to the die angle. The optimum die angle can be obtained by a similar technique using an expression, developed by the slab method, of the extrusion load as a function of the die angle [5]. In the past, conical dies were preferred to curved dies because of their ease of manufacture. However, nowadays, the manufacture of curved dies has become an easy task using computer numerical control machines. Thus, many researchers have become motivated to develop techniques to design curved die profiles for the forward rod extrusion process. Srinivasan et al. [4], as discussed above, used a slab element to develop an expression of the local strain rate as a function of the axial location in the deformation zone, using the expression to study the strain-rate distribution in the deformation zone for different die profiles such as the conical and the cubic die profiles. Moreover, Srinivasan et al. used the expression of the local strain rate as a function of the axial location in the deformation zone
Fig. 3. Utilization of Bezier curves to generate forward extrusion die profiles.
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Fig. 4. Optimum curved die as obtained by the incremental approach for an extrusion ratio ER = 34.6 and ram velocity 6 =3 m s − 1 for 0.25% carbon steel.
to generate the controlled strain rate die profile. A special case of the controlled strain rate die profile is the constant strain rate die profile that has a constant strain rate distribution throughout the deformation zone. These authors used the finite element code ALPID to establish that the strain rate distribution in the deformation zone of the constant strain rate die profile is much more uniform than the distribution in the deformation zone of other die profiles such as the conical and the cubic die profiles.
Blazynski [5] used an iterative algorithm to create a curved die profile based on the criterion of the constancy of the ratios of successive generalized homogeneous strain increments (CRHS). Shatla [7] used mathematical manipulations to obtain a closed form solution for the CRHS die profile. Blazynski [5] designed another curved die profile based on the concept of the constancy of mean strain rate (CMSR) also his own suggestion. Blazynski [5] then used both concepts to generate the CRHS–CMSR die profile that has both criteria.
Fig. 5. Effect of the Coulomb friction coefficient (m), the extrusion ratio (ER) and the ram velocity (6) on the coefficients of the quadratic polynomial used to curve fit the optimum curved die profile.
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tion zone is divided into a finite number of slabs each of which is considered as an infinitesimal conical die, with local die semi-angle, a, and local Coulomb friction coefficient, m, for which the exact solution of the differential equation of the force equilibrium is [5]: bsx − y(1+ b)= C(D)2b
(1)
where y is the current flow stress in the slab, C is the integration constant and: Fig. 6. Effect of the coulomb friction coefficient on the shape of the optimum curved die profile, using a ram velocity of 3 m s − 1 and an extrusion ratio of 16.
In the present work, an incremental slab method to calculate the extrusion pressure for arbitrarily-curved dies is implemented. Two different techniques utilizing that incremental slab method are developed to generate the optimum curved die profile that minimizes the extrusion load and stresses at the tool – workpiece interface for the hot forward rod extrusion process. The first technique is incremental and yields a set of points that can be curve fitted to obtain the optimum curved die profile. The second technique utilizes Bezier curves to scan all the possible die profiles to locate the optimum curved die profile. The effect of the hot forward extrusion parameters such as the ram velocity, the extrusion ratio, and friction at the tool – workpiece interface on the shape of the optimum curved die profile are investigated. Moreover, the finite element code MARC [8] is used to compare the stresses developed at the tool– workpiece interface when using the optimum conical die profile found in the literature and the optimum curved die profile developed in the present work.
b= m cot a
Considering the inlet and exit diameters for each slab to be Di + 1 and Di, respectively, where i is from 1 to n−1 and n− 1 is the number of slabs, the homogeneous strain is calculated from: oh = 2 ln
The slab method can be used to obtain the extrusion pressure for curved dies as well as for conical dies. To utilize the slab method to calculate the extrusion pressure for curved dies, as shown in Fig. 1, the deforma-
Di + 1 Di
(3)
and the total strain from: ot = foh
(4)
The redundancy factor, f, for steel is calculated for all of the slabs from the deformation zone geometry parameter D2 using the relationship [5]: f= 1,
D2 5 0.867
f= 0.89+ 0.092D2,
D2 \ 0.867
(5)
D2 is obtained by calculating the tool–workpiece contact length numerically and dividing the die diameter at the mid-point of the face by that length [5]. The axial stress sx1 of the first slab that is located at the exit of the die is determined by: sx = y
2. An incremental slab method for curved dies
(2)
1+ b [1− exp(bsoh)] b
(6)
The boundary condition that is at D= D1 = Df, sx =0 is used to evaluate the constant of integration C in Eq. (1). To obtain sx2 of the second slab between D2 and D3, Eq. (1) is used and the constant of integration C is obtained by considering the boundary condition which is at D= D2, sx = sx1 to yield: 1+ b2 b2sx1 − y2(1+b2) + exp(b2foh2) b2 b2
sx2 = y2
(7)
The general equation of sxi for the ith slab between Di + 1 and Di is: sxi = yi
1+ bi bisxi − 1 − yi (1+bi ) + exp(bifohi ) bi bi
(8)
In hot forming the flow stress y is a function of strain rate and temperature, the following equation being used [2]: Fig. 7. Effect of the extrusion ratio on the shape of the optimum curved die profile, using 6 =3 m s − 1 and Coulomb friction coefficient of 0.04.
n
1 o; Q/RT y = sinh − 1 e a¯ A
1/n
(9)
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Fig. 8. Effect of the extrusion ratio, using a ram velocity of 3 m s − 1 on: (a) the die length of the optimum curved die profile; (b) the redundancy factor of the optimum curved die profile.
where Q (kJ kmol − 1) is the derived activation energy, R is the universal gas constant (R =8.315 kJ (kmol C) − 1) and T is the temperature (°C). Quantities A, a¯ , and n are temperature independent material parameters. The current strain rate o; in the slab needed in Eq. (9) is calculated for each slab from the following equation [3,4]: o; =
2nr 20 Dr r 3 Dx
(10)
where 6 is the ram velocity, Dr and Dx are increments along the radial and axial directions respectively, r0 is the inlet radius and r is the current radius at a particular axial position x.
calculate the deformation zone geometry parameter, D2, which is used to obtain f for the second iteration and so on until f converges. The algorithm to obtain the optimum curved die profile using this technique is given in Appendix A. The die profile obtained by this algorithm is not smooth and if left as it is, it will cause surface defects in the final product in addition to die wear. To overcome this problem, the points obtained by the above algorithm are curve fitted using a second order polynomial, in the form r= a2x 2 + a1x+ a0, since using higher order polynomials does not improve the accuracy of the curve fitting. Quantity a0 always takes the value of the initial radius of the billet. Increasing a2 increases the curvature of the profile, whereas increasing a1 increases its steepness.
3. An incremental technique to obtain the optimum curved die profile To obtain the optimum curved die profile between a particular inlet radius, r0, and exit radius, rf, the deformation zone is divided into a finite number of slabs, n, where the length of each slab, Dx, is unknown whereas the vertical distance Dr covered in each slab is (r0 − rf)/ n. Taking slab 1 at the exit of the die, as shown in Fig. 1, where x = 0, the angle of the cone of the slab is varied from 89 to 1 to obtain the angle that produces the minimum stress in the slab by optimizing the effects of friction, redundancy and local strain rate. Dx is calculated from that angle from the relationship Dx = Dr/tan a. The ith slab will start at xi =xi − 1 + Dxi where i is from 1 to n. This is repeated until i= n where r= r0 and x= L. At the beginning, the strain redundancy factor, f, is unknown because it is calculated from the deformation zone geometry parameter D2, that can not be obtained unless the tool profile is known. To overcome this problem, an iterative technique is used where in the first iteration the die profile is obtained by assuming f= 1. The die profile obtained is used to
4. Utilization of Bezier curves to generate an optimum curved die profile for the hot forward rod extrusion process This technique is based on the utilization of Bezier curves to generate all the possible die profiles for a given extrusion ratio and initial radius of the billet for the hot forward rod extrusion process. The extrusion load for each generated die profile is calculated using the incremental slab method for curved dies developed in the present work. Thus, the optimum curved die profile that has the minimum extrusion load can be obtained. The Bezier-curve technique can be used to generate the optimum curved die profile at high extrusion ratios where the incremental slab technique gives numerical errors. As shown in Fig. 2, a Bezier curve is described by a set of guiding points P0, P1, P2, P3,…, Pm − 1, Pm [9]. The control polygon of the Bezier curve is made of line segments that connect Pi, Pi + 1 (i= 1 to m− 1). The Bezier curve is tangential to the first and last line
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Fig. 9. Effect of Coulomb friction coefficient and extrusion ratio on the axial stress at the die inlet of the optimum curved die and optimum conical die profiles using different Coulomb friction coefficients and extrusion ratios for 6= 3 m s − 1.
segments of the control polygon that connect P0 and P1 and Pm − 1 and Pm receptively. As shown in Fig. 3, the utilization of Bezier curves to create the die profile is done by using two Bezier curves, each of which has three control points. The two Bezier curves L1, L2, determined by the control points P1, P2 and P3 and P3, P4 and P5 respectively, must be tangential to each other at P3, which imposes the first constraint on the control points, i.e. P2, P3 and P4 must lie on a straight line. Points P1 and P5 are at the die inlet and die exit respectively, which imposes the second constraint on the control points. The third constraint on the control points results from the requirement that L1 must be tangential to the container at P1 and L2 must be tangential to the die land at P5. To achieve this
requirement, points P2 and P4 must lie on the same horizontal level as points P1 and P5, respectively. The last constraint must be that the axial distance from P1 to P2 must be less than the axial distance from P1 to P4, otherwise the die profile will be unrealistic. The way to create all the possible die profiles that satisfy the requirements discussed above is by moving the control points P2, P3, P4 and P5 to all possible locations such a way that the above four constraints are satisfied. The conical die profile can also be created by this technique if point P2 coincides with point P1 and Point P4 coincides with point P5. The algorithm for generating a Bezier polynomial determined by a set of control points is given in [9]. An algorithm to obtain the optimum curved die profile for the hot forward rod
Fig. 10. Strain-rate and axial-stress distributions as calculated by the slab method for ER =16, 6= 4 m s − 1 and m =0.06 for the optimum conical and optimum curved die profiles.
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Fig. 11. Deformed mesh at different increments for the optimum conical die profile.
extrusion process by scanning all the possible die profiles for a given extrusion ratio and initial radius via moving the control points P2 through P5 to all possible locations that satisfy the constraints discussed above is given in Appendix B.
5. Results and discussion
5.1. Comparison of the optimum-cur6ed die profiles generated by both algorithms The optimum curved die profile generated by the Bezier-curve technique is compared in Fig. 4 with the optimum curved die profile generated by the incremental technique for different extrusion conditions. It is clear from the figure that both techniques generate nearly the same optimum curved die profile for the same extrusion conditions. The slight difference between the two profiles generated by the two techniques can even be minimized by minimizing the increments of the Bezier points, Dx, which will increase the number of
die profiles scanned and give better accuracy. Although the Bezier-curve approach requires much more calculation time by the computer, about 15 times that required by the incremental slab technique, the Bezier curve approach can produce optimally-curved die profiles that have continuous first derivatives at the die inlet and exit, causing the profile to be smoothly merged with the container at the inlet and with the die land at the exit.
5.2. Some factors affecting the optimum cur6ed die profile The first algorithm, which is the incremental slab technique, will be used in this study to generate the optimum curved die profile because it converges much faster and requires much less calculation time by the computer than the second algorithm, which is the Bezier-curve technique. The material used is carbon steel (0.25% C) for which the material constants required in Eq. (9) to calculate the flow stress are Q=306 kJ mol − 1, n=4.55, 1/a¯ =63.8 N mm − 2 and A=1.5×
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Fig. 12. Deformed mesh at different increments for the optimum curved die profile.
1011 s − 1 [2]. The extrusion temperature for this hard-toextrude material must be 1100°C [2,10]. To prevent the billet from cooling, the extrusion velocity must be high in the range from 3 to 6 m s − 1 and the maximum extrusion ratio that can be attained is 100 [2,10]. Algorithm A.1. is used to generate the optimum-curved die profiles that can be used to extrude carbon steel (0.25% C) for extrusion ratios, ER, of 4, 8.16, 16, 25, and 34.6, the latter being the maximum value of extrusion ratio above which the incremental slab technique produces numerical errors, with the initial diameter and length of the billet fixed to 200 and 500 mm, respectively. The velocity is changed amongst the values 3, 4, 5, and 6 m s − 1 for all of the extrusion ratios to study the effect of extrusion velocity. For each combination of extrusion ratio and extrusion velocity, the Coulomb friction coefficient is varied amongst the values 0.02, 0.04, 0.06, 0.08, 0.1, and 0.12. As shown in Fig. 5, the absolute values of a1 and a2 of the second order polynomial r =a2x 2 +a1x +a0 representing the die profile are directly proportional to both the values of friction coefficient and extrusion ratio while they are insignificantly affected by the ram velocity for the material investigated. The insignificant effect of ram velocity on the shape of the optimumcurved die profile for the material investigated may be attributed to the low sensitivity of the flow stress of this material to strain rate. Figs. 6 and 7 illustrate the effect of Coulomb friction coefficient and extrusion ratio,
respectively, on the shape of the optimum curved die profile. Fig. 6 shows the decrease of die length with friction to minimize the stresses developed along the tool–workpiece interface. As shown in Fig. 7, the die length increases with an increase in the extrusion ratio until the extrusion ratio attains the value of 8.16, above which the die length starts to decrease. Figs. 6 and 7 also illustrate the increase of the curvature and steepness of the profile with increasing the Coulomb friction coefficient and extrusion ratio. The behavior of the change of die length with extrusion ratio discussed above is illustrated in Fig. 8. The cause of this behavior is, as shown in Fig. 8, due to the decrease of the redundancy factor with extrusion ratio. For small extrusion ratios, the redundancy factor is high, causing the die length to increase with extrusion ratio to minimize the redundancy effects. At high extrusion ratios, the redundancy factor approaches unity and the effect of redundancy starts to disappear, whilst the frictional effect becomes dominant causing the deformation zone length to decrease with extrusion ratio to minimize this frictional effect. Fig. 9 illustrates a comparison between the optimum curved and optimum conical die profiles regarding the effect of extrusion ratio and Coulomb friction coefficient on the axial stress at the inlet to the die as calculated by the slab method. In this figure the stressvs.-friction and stress-vs.-extrusion ratio curves of the optimum conical die profile are steep in comparison
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Fig. 13. Axial stress vs. axial distance in the deformation zone as calculated by the finite-element method on the surface of the tool profile (FE(s) and at the center line of the workpiece (FE(c)) and as calculated by the slab method (slab), using a ram velocity of 4 m s − 1, an extrusion ratio of 16 and a Coulomb friction coefficient of 0.06 for: (a) the optimum conical die profile; (b) the optimum curved die profile.
with those of the optimum curved die profile being nearly horizontal and showing the capability of the optimum curved die profile to accommodate the increase of friction coefficient and extrusion ratio. The success in decreasing the axial stress achieved by the optimum curved die profile is reflected in the stresses produced on the tooling, as will be discussed later. Fig. 10 illustrates a comparison between the optimum curved and optimum conical die profiles regarding the axial stress and strain rate distributions as functions of axial coordinate (x), as calculated by the slab method. Fig. 10(a) shows that the axial stress distribution for the optimum curved die profile is much less than that for the optimum conical die profile, which has a more curved axial stress distribution. Fig. 10(b) illustrates the higher value of the strain-rate distribution depicted in the optimum curved die profile, which is attributed to the shorter die length for the material investigated. If the value of the flow stress of the material is more sensitive to strain rate, the length of the optimum curved die profile is expected to be longer to decrease the value of the strain rate since the profile is based on the optimization of strain rate, redundancy and friction to minimize the extrusion pressure, as discussed earlier. Thus, the die length of the optimum curved die profile for the material investigated came to be less than that of the optimum conical die profile, causing the die cost to decrease, since less amount of expensive tool steel will be used to manufacture the optimum curved die profile.
5.3. A finite-element comparison of the stress de6eloped in the optimum cur6ed and optimum conical die An axisymmetric large strain, updated Lagrangian elastic-plastic finite element model using the MARC program [8] is utilized to compare the different compo-
nents of stress developed by the optimum curved and the optimum conical die profiles during the forward hot rod extrusion of 0.25% carbon steel. The deformed meshes at different increments, produced by this finite element model, for the optimum conical and optimum curved die profiles are illustrated in Figs. 11 and 12, respectively. In Fig. 13, the axial stress distribution along the die length, calculated by the slab method, is matched against that calculated by the finite element method for both profiles using an extrusion ratio of 16, a ram velocity of 4 m s − 1 and a Coulomb friction coefficient of 0.06: Good correspondence is found which may be attributed to the assumption used for the solution by the slab method that the stresses on a slab surface, perpendicular to the flow direction, are not permitted to vary on this surface. This assumption can be closer to reality for longer die lengths such as those for the optimum conical die profile where the stress distribution perpendicular to the flow direction is more uniform, which is not the case for shorter die lengths such as those for the optimum curved die profile obtained in the present work. For the same extrusion conditions of Fig. 13, Fig. 14 demonstrates a comparison between the optimum conical and optimum curved die profiles regarding the different components of stress, calculated by the finite element method, acting on the workpiece along the tool–workpiece interface, at steady state. It is clear from this figure that all of the components of stress developed when using the optimum curved die profile are less than those developed when using the optimum conical die profile leading to a greater tool life of the former. Irregularities in the stress distribution near to the die exit may be attributed to geometrical discontinuity in the finite element model used.
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Fig. 14. Different components of stress along the tool–workpiece interface for the optimum conical and optimum curved die profiles, using a ram velocity of 4 m s − 1, an extrusion ratio of 16 and a coulomb friction coefficient of 0.06.
6. Concluding remarks An incremental slab method has been developed to calculate the extrusion pressure for arbitrarily-curved dies for the hot forward rod extrusion process. Two numerical techniques, the incremental slab technique and the Bezier-curve technique, which are based on the incremental slab method, have been developed to generate the optimum curved die profile for the forward hot rod extrusion process. Both techniques have been found to yield nearly the same optimum curved die profiles for the same extrusion conditions. Although, the incremental slab technique has been found to converge much faster than the Bezier-curve technique, the incremental slab technique can not generate the optimumcurved die profile at high extrusion ratios because of round-off numerical errors. Thus, the incremental slab technique is recommended to be used to generate the optimum curved die profile for relatively small extrusion ratios. For high extrusion ratios, above 34.6, at which the incremental slab technique can not converge, the Bezier-curve technique should be used. For the material investigated in this study, the shape of the optimum curved die profile has been found to be affected by both the extrusion ratio and Coulomb friction coefficient. However, the ram velocity seems to have no significant influence on the shape of the optimum
curved die profile because of the low strain rate sensitivity of the material investigated. The slab method has revealed that the extrusion pressure of the optimum curved die profile is less than that of the optimum conical die profile, leading to less energy requirements, especially when forming difficult-to-press metals such as the high alloy and carbon steels. Moreover, the finite element analysis has revealed that all of the components of stress developed at the tool–workpiece interface of the optimum curved die profile are less than those of the optimum conical die profile. Thus, the tool life of the optimum curved die profile is expected to be longer than that of the optimum conical die profile. For the material investigated in this study, the length of the optimum curved die profile has been found to be less than that of the optimum conical die profile, therefore the optimum curved die profile is expected to be cheaper because it requires a lesser amount of expensive tool steel.
Appendix A. Algorithm to generate the optimum curved die profile based on the incremental slab technique 1. Read and store data. 2. Calculate the vertical distance covered by the profile ly = r0 − rf.
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3. Get Dly = ly /n is the number of divisions from slab 1 to slab n where slab 1 is at the exit from the die and slab n is at the inlet to the die}. 4. Assume the strain redundancy factor f =1. Repeat 5. For each value of ri (i is from 1 to n) from rf to r0 by Dly do: Begin 6. Get ri + 1. 7. Get oh from Eq. (3). 8. Get ot from Eq. (4) 9. For each value of the die angle a from 1 to 89 by Da and coefficient of friction m of a slab bounded by ri, ri + 1 do: Begin. 10. Calculate b =m cot a. 11. Calculate the current strain rate using Eq. (10) 12. Calculate the current flow stress in the slab from Eq. (9). 13. Calculate sxi for slab i. {for i= 1, use Eq. (6) and for i= 2 to i= n −1, use Eq. (8)} End. 14. Get a that produces the minimum sxi in the slab. 15. Get xi + 1 =xi +Dly /tan a. {x1 =0 at the exit where r1 = rf} End. 16. Calculate the diameter at the mid-point of the face. 17. Calculate the contact length of the die. 18. Calculate the deformation zone geometry parameter D2. 19. Calculate the strain redundancy factor f. Until f converges. 20. End of Algorithm.
Appendix B Algorithm to utilize the Bezier polynomials to scan all the possible die profiles for a given initial radius and extrusion ratio to obtain the optimum-curved die profile for the hot forward rod extrusion process 1. Read and store data. 2. P1.r = r0 . {where P.r is the radial position of point P} 3. P1.x =0. {where P.x is the axial position of point P} 4. P2.r =r0. 5. P4.r = rf. 6. P5.r =rf.
.
107
7. For each value of P5.x from Lmin to Lmax by DL do: Begin. 8. P2.x = 0. 9. P4.x = L. 10. For each value of P4.x from P5.x down to Dx1 by Dx1 do: begin. 11. P2.x = 0. 12. For each value of P2.x from 0 to (P4.x − Dx1) do: begin. 13. For each value of P3.x from (P2.x +Dx2) to (P4.x −Dx2) by Dx2 do: begin. P4.x −P3 · x 14. Calculate P3.r =rf + (r0 − rf) P4.x −P2.x 15. Get the Profile using P1, P2, P3, P4, and P5. 16. Calculate the extrusion pressure by the slab method and compare with the minimum End. End. End. End. 17. Save the die profile with minimum extrusion pressure. 18. End of Algorithm.
References [1] T. Altan, F.W. Boulger, Flow stress of metals and its application in metal forming analysis, J. Eng. Ind. 95 (1973) 1009 –1019. [2] K. Lange, Metal Forming Handbook, English edn., Ch. 16, McGraw-Hill, 1985. [3] M.M. Farag, C.M. Sellars, Flow stress in hot extrusion of commercial purity aluminum, J. Inst. Metals 101 (1973) 137– 145. [4] R. Srinivasan, J.S. Gunasekera, H.L. Gegel, S.M. Doraivelu, Extrusion through controlled strain rate dies. J. Mater. Shaping Technol. 8 (2) (1990) 133 – 141. [5] T.Z. Blazynski, Metal Forming-Tool Profiles and Flow, Macmillan Press, New York, 1976. [6] B. Avitzur, Metal Forming Processes and Analysis, McGrawHill, New York, 1968. [7] M.N. Shatla, A Finite Element Analysis of the Forward Rod Extrusion Process With an Emphasis on Die Profile Optimization, A Master Thesis Submitted to the Engineering Department, The American University in Cairo, 1995. [8] Corporate Headquarters, MARC Analysis Research Corporation, 260 Sheridan Avenue Palo Alto, CA 94 306, 1994. [9] T. Pavlidis, Algorithms for Graphs and Image Processing, Computer Science Press, 1982. [10] K. Laue, H. Stenger, G. Lang, Extrusion-Process, Machinery, Tooling, American Society for Metals, Ohio, 1981.
An optimum-curved die profile for the hot forward rod extrusion process A.S. Wifi a,*, M.N. Shatla b, A. Abdel-Hamid b,c a
Department of Mechanical Design and Production, Faculty of Engineering, Cairo Uni6ersity, Giza 12 316, Egypt b Department of Engineering and c Department of Science, The American Uni6ersity in Cairo, Cairo, Egypt Received 12 April 1996
Abstract An incremental slab method is implemented to obtain the extrusion pressure of the hot forward rod extrusion process for arbitrarily-curved dies. Two different algorithms utilizing that incremental slab method are developed to obtain the optimum curved die profile that minimizes the extrusion load and stresses at the tool – workpiece interface for the hot forward rod extrusion process. Both techniques are based on the optimization of the effects of friction at the tool – workpiece interface, strain rate, and redundant deformation. The optimum curved die profiles generated by the two different algorithms are found to be very similar. For the material used in the present study which is carbon steel, the shape of the optimum curved die profile is found to depend on the extrusion ratio and Coulomb friction coefficient and not to be affected by the extrusion velocity. The finite element method revealed that the optimum curved die profile developed in the present work produces lower stress levels than those produced using the optimum conical die profile found in the literature. © 1998 Elsevier Science S.A. Keywords: Optimum curved die profile; Tool–workpiece; Friction; Strain rate; Extrusion ratio; Finite element method
1. Introduction The hot forward rod extrusion process can be described as reducing the cross sectional area of a heated rod placed in a container by pushing the rod by a ram and squeezing it through a die. The factors affecting the process are the material properties and the forming conditions, the latter of which are the rod temperature, friction at the tool– workpiece interface, the extrusion ratio and the die profile. Optimization of these parameters has been one of the most important tasks that has attracted the attention of many researchers and engineers using analytical, computer-aided design (CAD) and experimental techniques. In metal forming, the flow stress of the material is a function of strain, strain rate and temperature. If the forming operation takes place above or close to the recrystalization temperature of the processed material, the effect of strain on the flow stress is negligibly small and the flow stress becomes dependent on the strain * Corresponding author. Fax: + 20 2 5723486; e-mail: [email protected] 0924-0136/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved. PII S 0 9 2 4 - 0 1 3 6 ( 9 7 ) 0 0 2 1 8 - 5
rate and temperature only [1]. Different empirical relationships with constants obtained experimentally have been used to express the flow stress of the hot formed material as a function of strain rate and temperature. Some of these relationships have temperature dependent material constants with no temperature terms [1]. Another more convenient expressions of the flow stress as a function of strain rate and temperature utilize the Zener–Holloman parameter in conjunction with temperature independent material constants [2,3]. A slab element can be used to obtain an expression of the strain rate as a function of the axial location in the deformation zone for the forward rod extrusion process, such expression being used by Farag et al. [3] to study the flow stress in the hot extrusion of commercial purity aluminum. Srinivasan et al. [4] also used such expression to study the stain rate distribution in the deformation zone for different die profiles and to generate a controlled strain rate die profile, as will be mentioned later. The expression of the local strain rate as a function of the axial location in the deformation zone can be used in conjunction with the expression of the flow stress as a function of strain rate to determine
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Fig. 1. Incremental slab method for curved dies.
the local flow stress in the deformation zone for the forward hot rod extrusion of metals. The total strain of a workpiece undergoing any plastic deformation consists of a homogeneous component and a redundant component. The homogeneous strain is responsible for the change of dimensions of the workpiece to acquire the required final shape whereas the redundant strain occurs internally and does not contribute anything to the final shape of the product [5]. Thus, the total strain can be expressed as the homogeneous strain multiplied by a factor called the strain redundancy factor. The strain redundancy factor can not have a value of less than unity at which value the total deformation equals the homogeneous deformation. This is an ideal case, in which there is no redundant deformation that adds to the strain hardening of the material causing higher stress levels during the deformation process [5]. For forward rod extrusion, it has been found that the redundant deformation increases with increasing of the die angle, which leads to more strain hardening and consequently higher extrusion load. However, increasing the die angle decreases the tool – workpiece interface length which leads to lower frictional stresses at this interface and consequently to lower extrusion load [5,6]. The optimum die angle that minimizes the extru-
Fig. 2. Description of Bezier curves.
sion load for the forward rod extrusion process achieves a balance between the effects of redundancy and friction [5,6]. This optimum die angle was obtained analytically by Avitzur [6] through the minimization of an upper bound expression of the extrusion load as a function of the die angle with respect to the die angle. The optimum die angle can be obtained by a similar technique using an expression, developed by the slab method, of the extrusion load as a function of the die angle [5]. In the past, conical dies were preferred to curved dies because of their ease of manufacture. However, nowadays, the manufacture of curved dies has become an easy task using computer numerical control machines. Thus, many researchers have become motivated to develop techniques to design curved die profiles for the forward rod extrusion process. Srinivasan et al. [4], as discussed above, used a slab element to develop an expression of the local strain rate as a function of the axial location in the deformation zone, using the expression to study the strain-rate distribution in the deformation zone for different die profiles such as the conical and the cubic die profiles. Moreover, Srinivasan et al. used the expression of the local strain rate as a function of the axial location in the deformation zone
Fig. 3. Utilization of Bezier curves to generate forward extrusion die profiles.
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Fig. 4. Optimum curved die as obtained by the incremental approach for an extrusion ratio ER = 34.6 and ram velocity 6 =3 m s − 1 for 0.25% carbon steel.
to generate the controlled strain rate die profile. A special case of the controlled strain rate die profile is the constant strain rate die profile that has a constant strain rate distribution throughout the deformation zone. These authors used the finite element code ALPID to establish that the strain rate distribution in the deformation zone of the constant strain rate die profile is much more uniform than the distribution in the deformation zone of other die profiles such as the conical and the cubic die profiles.
Blazynski [5] used an iterative algorithm to create a curved die profile based on the criterion of the constancy of the ratios of successive generalized homogeneous strain increments (CRHS). Shatla [7] used mathematical manipulations to obtain a closed form solution for the CRHS die profile. Blazynski [5] designed another curved die profile based on the concept of the constancy of mean strain rate (CMSR) also his own suggestion. Blazynski [5] then used both concepts to generate the CRHS–CMSR die profile that has both criteria.
Fig. 5. Effect of the Coulomb friction coefficient (m), the extrusion ratio (ER) and the ram velocity (6) on the coefficients of the quadratic polynomial used to curve fit the optimum curved die profile.
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tion zone is divided into a finite number of slabs each of which is considered as an infinitesimal conical die, with local die semi-angle, a, and local Coulomb friction coefficient, m, for which the exact solution of the differential equation of the force equilibrium is [5]: bsx − y(1+ b)= C(D)2b
(1)
where y is the current flow stress in the slab, C is the integration constant and: Fig. 6. Effect of the coulomb friction coefficient on the shape of the optimum curved die profile, using a ram velocity of 3 m s − 1 and an extrusion ratio of 16.
In the present work, an incremental slab method to calculate the extrusion pressure for arbitrarily-curved dies is implemented. Two different techniques utilizing that incremental slab method are developed to generate the optimum curved die profile that minimizes the extrusion load and stresses at the tool – workpiece interface for the hot forward rod extrusion process. The first technique is incremental and yields a set of points that can be curve fitted to obtain the optimum curved die profile. The second technique utilizes Bezier curves to scan all the possible die profiles to locate the optimum curved die profile. The effect of the hot forward extrusion parameters such as the ram velocity, the extrusion ratio, and friction at the tool – workpiece interface on the shape of the optimum curved die profile are investigated. Moreover, the finite element code MARC [8] is used to compare the stresses developed at the tool– workpiece interface when using the optimum conical die profile found in the literature and the optimum curved die profile developed in the present work.
b= m cot a
Considering the inlet and exit diameters for each slab to be Di + 1 and Di, respectively, where i is from 1 to n−1 and n− 1 is the number of slabs, the homogeneous strain is calculated from: oh = 2 ln
The slab method can be used to obtain the extrusion pressure for curved dies as well as for conical dies. To utilize the slab method to calculate the extrusion pressure for curved dies, as shown in Fig. 1, the deforma-
Di + 1 Di
(3)
and the total strain from: ot = foh
(4)
The redundancy factor, f, for steel is calculated for all of the slabs from the deformation zone geometry parameter D2 using the relationship [5]: f= 1,
D2 5 0.867
f= 0.89+ 0.092D2,
D2 \ 0.867
(5)
D2 is obtained by calculating the tool–workpiece contact length numerically and dividing the die diameter at the mid-point of the face by that length [5]. The axial stress sx1 of the first slab that is located at the exit of the die is determined by: sx = y
2. An incremental slab method for curved dies
(2)
1+ b [1− exp(bsoh)] b
(6)
The boundary condition that is at D= D1 = Df, sx =0 is used to evaluate the constant of integration C in Eq. (1). To obtain sx2 of the second slab between D2 and D3, Eq. (1) is used and the constant of integration C is obtained by considering the boundary condition which is at D= D2, sx = sx1 to yield: 1+ b2 b2sx1 − y2(1+b2) + exp(b2foh2) b2 b2
sx2 = y2
(7)
The general equation of sxi for the ith slab between Di + 1 and Di is: sxi = yi
1+ bi bisxi − 1 − yi (1+bi ) + exp(bifohi ) bi bi
(8)
In hot forming the flow stress y is a function of strain rate and temperature, the following equation being used [2]: Fig. 7. Effect of the extrusion ratio on the shape of the optimum curved die profile, using 6 =3 m s − 1 and Coulomb friction coefficient of 0.04.
n
1 o; Q/RT y = sinh − 1 e a¯ A
1/n
(9)
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Fig. 8. Effect of the extrusion ratio, using a ram velocity of 3 m s − 1 on: (a) the die length of the optimum curved die profile; (b) the redundancy factor of the optimum curved die profile.
where Q (kJ kmol − 1) is the derived activation energy, R is the universal gas constant (R =8.315 kJ (kmol C) − 1) and T is the temperature (°C). Quantities A, a¯ , and n are temperature independent material parameters. The current strain rate o; in the slab needed in Eq. (9) is calculated for each slab from the following equation [3,4]: o; =
2nr 20 Dr r 3 Dx
(10)
where 6 is the ram velocity, Dr and Dx are increments along the radial and axial directions respectively, r0 is the inlet radius and r is the current radius at a particular axial position x.
calculate the deformation zone geometry parameter, D2, which is used to obtain f for the second iteration and so on until f converges. The algorithm to obtain the optimum curved die profile using this technique is given in Appendix A. The die profile obtained by this algorithm is not smooth and if left as it is, it will cause surface defects in the final product in addition to die wear. To overcome this problem, the points obtained by the above algorithm are curve fitted using a second order polynomial, in the form r= a2x 2 + a1x+ a0, since using higher order polynomials does not improve the accuracy of the curve fitting. Quantity a0 always takes the value of the initial radius of the billet. Increasing a2 increases the curvature of the profile, whereas increasing a1 increases its steepness.
3. An incremental technique to obtain the optimum curved die profile To obtain the optimum curved die profile between a particular inlet radius, r0, and exit radius, rf, the deformation zone is divided into a finite number of slabs, n, where the length of each slab, Dx, is unknown whereas the vertical distance Dr covered in each slab is (r0 − rf)/ n. Taking slab 1 at the exit of the die, as shown in Fig. 1, where x = 0, the angle of the cone of the slab is varied from 89 to 1 to obtain the angle that produces the minimum stress in the slab by optimizing the effects of friction, redundancy and local strain rate. Dx is calculated from that angle from the relationship Dx = Dr/tan a. The ith slab will start at xi =xi − 1 + Dxi where i is from 1 to n. This is repeated until i= n where r= r0 and x= L. At the beginning, the strain redundancy factor, f, is unknown because it is calculated from the deformation zone geometry parameter D2, that can not be obtained unless the tool profile is known. To overcome this problem, an iterative technique is used where in the first iteration the die profile is obtained by assuming f= 1. The die profile obtained is used to
4. Utilization of Bezier curves to generate an optimum curved die profile for the hot forward rod extrusion process This technique is based on the utilization of Bezier curves to generate all the possible die profiles for a given extrusion ratio and initial radius of the billet for the hot forward rod extrusion process. The extrusion load for each generated die profile is calculated using the incremental slab method for curved dies developed in the present work. Thus, the optimum curved die profile that has the minimum extrusion load can be obtained. The Bezier-curve technique can be used to generate the optimum curved die profile at high extrusion ratios where the incremental slab technique gives numerical errors. As shown in Fig. 2, a Bezier curve is described by a set of guiding points P0, P1, P2, P3,…, Pm − 1, Pm [9]. The control polygon of the Bezier curve is made of line segments that connect Pi, Pi + 1 (i= 1 to m− 1). The Bezier curve is tangential to the first and last line
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Fig. 9. Effect of Coulomb friction coefficient and extrusion ratio on the axial stress at the die inlet of the optimum curved die and optimum conical die profiles using different Coulomb friction coefficients and extrusion ratios for 6= 3 m s − 1.
segments of the control polygon that connect P0 and P1 and Pm − 1 and Pm receptively. As shown in Fig. 3, the utilization of Bezier curves to create the die profile is done by using two Bezier curves, each of which has three control points. The two Bezier curves L1, L2, determined by the control points P1, P2 and P3 and P3, P4 and P5 respectively, must be tangential to each other at P3, which imposes the first constraint on the control points, i.e. P2, P3 and P4 must lie on a straight line. Points P1 and P5 are at the die inlet and die exit respectively, which imposes the second constraint on the control points. The third constraint on the control points results from the requirement that L1 must be tangential to the container at P1 and L2 must be tangential to the die land at P5. To achieve this
requirement, points P2 and P4 must lie on the same horizontal level as points P1 and P5, respectively. The last constraint must be that the axial distance from P1 to P2 must be less than the axial distance from P1 to P4, otherwise the die profile will be unrealistic. The way to create all the possible die profiles that satisfy the requirements discussed above is by moving the control points P2, P3, P4 and P5 to all possible locations such a way that the above four constraints are satisfied. The conical die profile can also be created by this technique if point P2 coincides with point P1 and Point P4 coincides with point P5. The algorithm for generating a Bezier polynomial determined by a set of control points is given in [9]. An algorithm to obtain the optimum curved die profile for the hot forward rod
Fig. 10. Strain-rate and axial-stress distributions as calculated by the slab method for ER =16, 6= 4 m s − 1 and m =0.06 for the optimum conical and optimum curved die profiles.
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Fig. 11. Deformed mesh at different increments for the optimum conical die profile.
extrusion process by scanning all the possible die profiles for a given extrusion ratio and initial radius via moving the control points P2 through P5 to all possible locations that satisfy the constraints discussed above is given in Appendix B.
5. Results and discussion
5.1. Comparison of the optimum-cur6ed die profiles generated by both algorithms The optimum curved die profile generated by the Bezier-curve technique is compared in Fig. 4 with the optimum curved die profile generated by the incremental technique for different extrusion conditions. It is clear from the figure that both techniques generate nearly the same optimum curved die profile for the same extrusion conditions. The slight difference between the two profiles generated by the two techniques can even be minimized by minimizing the increments of the Bezier points, Dx, which will increase the number of
die profiles scanned and give better accuracy. Although the Bezier-curve approach requires much more calculation time by the computer, about 15 times that required by the incremental slab technique, the Bezier curve approach can produce optimally-curved die profiles that have continuous first derivatives at the die inlet and exit, causing the profile to be smoothly merged with the container at the inlet and with the die land at the exit.
5.2. Some factors affecting the optimum cur6ed die profile The first algorithm, which is the incremental slab technique, will be used in this study to generate the optimum curved die profile because it converges much faster and requires much less calculation time by the computer than the second algorithm, which is the Bezier-curve technique. The material used is carbon steel (0.25% C) for which the material constants required in Eq. (9) to calculate the flow stress are Q=306 kJ mol − 1, n=4.55, 1/a¯ =63.8 N mm − 2 and A=1.5×
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Fig. 12. Deformed mesh at different increments for the optimum curved die profile.
1011 s − 1 [2]. The extrusion temperature for this hard-toextrude material must be 1100°C [2,10]. To prevent the billet from cooling, the extrusion velocity must be high in the range from 3 to 6 m s − 1 and the maximum extrusion ratio that can be attained is 100 [2,10]. Algorithm A.1. is used to generate the optimum-curved die profiles that can be used to extrude carbon steel (0.25% C) for extrusion ratios, ER, of 4, 8.16, 16, 25, and 34.6, the latter being the maximum value of extrusion ratio above which the incremental slab technique produces numerical errors, with the initial diameter and length of the billet fixed to 200 and 500 mm, respectively. The velocity is changed amongst the values 3, 4, 5, and 6 m s − 1 for all of the extrusion ratios to study the effect of extrusion velocity. For each combination of extrusion ratio and extrusion velocity, the Coulomb friction coefficient is varied amongst the values 0.02, 0.04, 0.06, 0.08, 0.1, and 0.12. As shown in Fig. 5, the absolute values of a1 and a2 of the second order polynomial r =a2x 2 +a1x +a0 representing the die profile are directly proportional to both the values of friction coefficient and extrusion ratio while they are insignificantly affected by the ram velocity for the material investigated. The insignificant effect of ram velocity on the shape of the optimumcurved die profile for the material investigated may be attributed to the low sensitivity of the flow stress of this material to strain rate. Figs. 6 and 7 illustrate the effect of Coulomb friction coefficient and extrusion ratio,
respectively, on the shape of the optimum curved die profile. Fig. 6 shows the decrease of die length with friction to minimize the stresses developed along the tool–workpiece interface. As shown in Fig. 7, the die length increases with an increase in the extrusion ratio until the extrusion ratio attains the value of 8.16, above which the die length starts to decrease. Figs. 6 and 7 also illustrate the increase of the curvature and steepness of the profile with increasing the Coulomb friction coefficient and extrusion ratio. The behavior of the change of die length with extrusion ratio discussed above is illustrated in Fig. 8. The cause of this behavior is, as shown in Fig. 8, due to the decrease of the redundancy factor with extrusion ratio. For small extrusion ratios, the redundancy factor is high, causing the die length to increase with extrusion ratio to minimize the redundancy effects. At high extrusion ratios, the redundancy factor approaches unity and the effect of redundancy starts to disappear, whilst the frictional effect becomes dominant causing the deformation zone length to decrease with extrusion ratio to minimize this frictional effect. Fig. 9 illustrates a comparison between the optimum curved and optimum conical die profiles regarding the effect of extrusion ratio and Coulomb friction coefficient on the axial stress at the inlet to the die as calculated by the slab method. In this figure the stressvs.-friction and stress-vs.-extrusion ratio curves of the optimum conical die profile are steep in comparison
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Fig. 13. Axial stress vs. axial distance in the deformation zone as calculated by the finite-element method on the surface of the tool profile (FE(s) and at the center line of the workpiece (FE(c)) and as calculated by the slab method (slab), using a ram velocity of 4 m s − 1, an extrusion ratio of 16 and a Coulomb friction coefficient of 0.06 for: (a) the optimum conical die profile; (b) the optimum curved die profile.
with those of the optimum curved die profile being nearly horizontal and showing the capability of the optimum curved die profile to accommodate the increase of friction coefficient and extrusion ratio. The success in decreasing the axial stress achieved by the optimum curved die profile is reflected in the stresses produced on the tooling, as will be discussed later. Fig. 10 illustrates a comparison between the optimum curved and optimum conical die profiles regarding the axial stress and strain rate distributions as functions of axial coordinate (x), as calculated by the slab method. Fig. 10(a) shows that the axial stress distribution for the optimum curved die profile is much less than that for the optimum conical die profile, which has a more curved axial stress distribution. Fig. 10(b) illustrates the higher value of the strain-rate distribution depicted in the optimum curved die profile, which is attributed to the shorter die length for the material investigated. If the value of the flow stress of the material is more sensitive to strain rate, the length of the optimum curved die profile is expected to be longer to decrease the value of the strain rate since the profile is based on the optimization of strain rate, redundancy and friction to minimize the extrusion pressure, as discussed earlier. Thus, the die length of the optimum curved die profile for the material investigated came to be less than that of the optimum conical die profile, causing the die cost to decrease, since less amount of expensive tool steel will be used to manufacture the optimum curved die profile.
5.3. A finite-element comparison of the stress de6eloped in the optimum cur6ed and optimum conical die An axisymmetric large strain, updated Lagrangian elastic-plastic finite element model using the MARC program [8] is utilized to compare the different compo-
nents of stress developed by the optimum curved and the optimum conical die profiles during the forward hot rod extrusion of 0.25% carbon steel. The deformed meshes at different increments, produced by this finite element model, for the optimum conical and optimum curved die profiles are illustrated in Figs. 11 and 12, respectively. In Fig. 13, the axial stress distribution along the die length, calculated by the slab method, is matched against that calculated by the finite element method for both profiles using an extrusion ratio of 16, a ram velocity of 4 m s − 1 and a Coulomb friction coefficient of 0.06: Good correspondence is found which may be attributed to the assumption used for the solution by the slab method that the stresses on a slab surface, perpendicular to the flow direction, are not permitted to vary on this surface. This assumption can be closer to reality for longer die lengths such as those for the optimum conical die profile where the stress distribution perpendicular to the flow direction is more uniform, which is not the case for shorter die lengths such as those for the optimum curved die profile obtained in the present work. For the same extrusion conditions of Fig. 13, Fig. 14 demonstrates a comparison between the optimum conical and optimum curved die profiles regarding the different components of stress, calculated by the finite element method, acting on the workpiece along the tool–workpiece interface, at steady state. It is clear from this figure that all of the components of stress developed when using the optimum curved die profile are less than those developed when using the optimum conical die profile leading to a greater tool life of the former. Irregularities in the stress distribution near to the die exit may be attributed to geometrical discontinuity in the finite element model used.
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Fig. 14. Different components of stress along the tool–workpiece interface for the optimum conical and optimum curved die profiles, using a ram velocity of 4 m s − 1, an extrusion ratio of 16 and a coulomb friction coefficient of 0.06.
6. Concluding remarks An incremental slab method has been developed to calculate the extrusion pressure for arbitrarily-curved dies for the hot forward rod extrusion process. Two numerical techniques, the incremental slab technique and the Bezier-curve technique, which are based on the incremental slab method, have been developed to generate the optimum curved die profile for the forward hot rod extrusion process. Both techniques have been found to yield nearly the same optimum curved die profiles for the same extrusion conditions. Although, the incremental slab technique has been found to converge much faster than the Bezier-curve technique, the incremental slab technique can not generate the optimumcurved die profile at high extrusion ratios because of round-off numerical errors. Thus, the incremental slab technique is recommended to be used to generate the optimum curved die profile for relatively small extrusion ratios. For high extrusion ratios, above 34.6, at which the incremental slab technique can not converge, the Bezier-curve technique should be used. For the material investigated in this study, the shape of the optimum curved die profile has been found to be affected by both the extrusion ratio and Coulomb friction coefficient. However, the ram velocity seems to have no significant influence on the shape of the optimum
curved die profile because of the low strain rate sensitivity of the material investigated. The slab method has revealed that the extrusion pressure of the optimum curved die profile is less than that of the optimum conical die profile, leading to less energy requirements, especially when forming difficult-to-press metals such as the high alloy and carbon steels. Moreover, the finite element analysis has revealed that all of the components of stress developed at the tool–workpiece interface of the optimum curved die profile are less than those of the optimum conical die profile. Thus, the tool life of the optimum curved die profile is expected to be longer than that of the optimum conical die profile. For the material investigated in this study, the length of the optimum curved die profile has been found to be less than that of the optimum conical die profile, therefore the optimum curved die profile is expected to be cheaper because it requires a lesser amount of expensive tool steel.
Appendix A. Algorithm to generate the optimum curved die profile based on the incremental slab technique 1. Read and store data. 2. Calculate the vertical distance covered by the profile ly = r0 − rf.
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3. Get Dly = ly /n is the number of divisions from slab 1 to slab n where slab 1 is at the exit from the die and slab n is at the inlet to the die}. 4. Assume the strain redundancy factor f =1. Repeat 5. For each value of ri (i is from 1 to n) from rf to r0 by Dly do: Begin 6. Get ri + 1. 7. Get oh from Eq. (3). 8. Get ot from Eq. (4) 9. For each value of the die angle a from 1 to 89 by Da and coefficient of friction m of a slab bounded by ri, ri + 1 do: Begin. 10. Calculate b =m cot a. 11. Calculate the current strain rate using Eq. (10) 12. Calculate the current flow stress in the slab from Eq. (9). 13. Calculate sxi for slab i. {for i= 1, use Eq. (6) and for i= 2 to i= n −1, use Eq. (8)} End. 14. Get a that produces the minimum sxi in the slab. 15. Get xi + 1 =xi +Dly /tan a. {x1 =0 at the exit where r1 = rf} End. 16. Calculate the diameter at the mid-point of the face. 17. Calculate the contact length of the die. 18. Calculate the deformation zone geometry parameter D2. 19. Calculate the strain redundancy factor f. Until f converges. 20. End of Algorithm.
Appendix B Algorithm to utilize the Bezier polynomials to scan all the possible die profiles for a given initial radius and extrusion ratio to obtain the optimum-curved die profile for the hot forward rod extrusion process 1. Read and store data. 2. P1.r = r0 . {where P.r is the radial position of point P} 3. P1.x =0. {where P.x is the axial position of point P} 4. P2.r =r0. 5. P4.r = rf. 6. P5.r =rf.
.
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7. For each value of P5.x from Lmin to Lmax by DL do: Begin. 8. P2.x = 0. 9. P4.x = L. 10. For each value of P4.x from P5.x down to Dx1 by Dx1 do: begin. 11. P2.x = 0. 12. For each value of P2.x from 0 to (P4.x − Dx1) do: begin. 13. For each value of P3.x from (P2.x +Dx2) to (P4.x −Dx2) by Dx2 do: begin. P4.x −P3 · x 14. Calculate P3.r =rf + (r0 − rf) P4.x −P2.x 15. Get the Profile using P1, P2, P3, P4, and P5. 16. Calculate the extrusion pressure by the slab method and compare with the minimum End. End. End. End. 17. Save the die profile with minimum extrusion pressure. 18. End of Algorithm.
References [1] T. Altan, F.W. Boulger, Flow stress of metals and its application in metal forming analysis, J. Eng. Ind. 95 (1973) 1009 –1019. [2] K. Lange, Metal Forming Handbook, English edn., Ch. 16, McGraw-Hill, 1985. [3] M.M. Farag, C.M. Sellars, Flow stress in hot extrusion of commercial purity aluminum, J. Inst. Metals 101 (1973) 137– 145. [4] R. Srinivasan, J.S. Gunasekera, H.L. Gegel, S.M. Doraivelu, Extrusion through controlled strain rate dies. J. Mater. Shaping Technol. 8 (2) (1990) 133 – 141. [5] T.Z. Blazynski, Metal Forming-Tool Profiles and Flow, Macmillan Press, New York, 1976. [6] B. Avitzur, Metal Forming Processes and Analysis, McGrawHill, New York, 1968. [7] M.N. Shatla, A Finite Element Analysis of the Forward Rod Extrusion Process With an Emphasis on Die Profile Optimization, A Master Thesis Submitted to the Engineering Department, The American University in Cairo, 1995. [8] Corporate Headquarters, MARC Analysis Research Corporation, 260 Sheridan Avenue Palo Alto, CA 94 306, 1994. [9] T. Pavlidis, Algorithms for Graphs and Image Processing, Computer Science Press, 1982. [10] K. Laue, H. Stenger, G. Lang, Extrusion-Process, Machinery, Tooling, American Society for Metals, Ohio, 1981.