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An upper bound analysis of a process-induced side-surface defect in forgings
Journal of Materials Processing Technology 99 (2000) 179±184
An upper bound analysis of a process-induced side-surface defect in forgings Part 2: Characteristics and criteria curves Y.H. Moona,*, C.J. Van Tyneb, W.A. Gordonc a
Engineering Research Center for Net Shape and Die Manufacturing, Pusan National University, San 30 Jangjeondong, Pusan, South Korea b Department of Metallurgical and Materials Engineering, Colorado School of Mines, Golden, CO, USA c Advanced Technology Center, Torrington Co., 59 Field Street, Torrington, CT, USA Received 13 August 1998
Abstract An upper bound analysis for an axisymmetric forging with a double ram action has been developed. The results of the analysis provide manufacturing conditions that can be used without the development of unacceptable side-surface cracks. There are processing conditions that cause the defective ¯ow pattern to be most favorable. The transition from an energetically-favorable sound ¯ow to an energeticallyfavorable defect ¯ow is used to determine the criteria curves for the prevention of this side-surface defect. It is found that the formation of the side-surface cracks is promoted by smaller diameter rams, smaller workpiece thicknesses and lower friction. The ®rst paper in this two-part series presents the velocity ®elds and the individual power terms that were derived based upon these velocity ®elds, whilst this second paper shows the characteristics of the upper bound solution for the double action forging process. This second paper also presents the criteria for the prevention of the side-surface defects that were determined based upon the upper bound solution. # 2000 Elsevier Science S.A. All rights reserved. Keywords: Upper bound analysis; Side-surface defect; Double action forging process; Criteria curve; Constant friction factor
Nomenclature H m mf pave R Ri Ro S T U_ V vf vi WI
height of the outer portion of the workpiece constant friction factor measure of bonding/adhesion within the workpiece average forging pressure radial coordinate in cylindrical coordinate system ram radius chamber radius surface thickness of the center portion of the workpiece ram velocity volume upwards workpiece velocity velocity component internal power of deformation
* Corresponding author. Tel.: 82-51-510-2472; fax: 82-51-512-1722. E-mail address: [email protected] (Y.H. Moon).
WG Wf WCF G d dopt e eopt y so
shear power loss along internal surface frictional power loss power for crack formation surface of velocity discontinuity crack length (a pseudo-independent parameter) crack length that requires lowest ram pressure position variable for internal surface (a pseudoindependent parameter) value of e that requires lowest ram pressure angular coordinate in cylindrical or spherical coordinate systems flow strength of the workpiece
1. Introduction Fig. 1 is a schematic diagram of the double ram action forging process that has been analyzed via the upper bound approach [1]. The workpiece begins as an axisymmetric disk of radius Ro. The upper ram of radius Ri moves downward
0924-0136/00/$ ± see front matter # 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 9 9 ) 0 0 4 1 8 - 5
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Fig. 1. Schematic diagram of the axisymmetric forging process.
_ as the lower ram moves upwards with with a velocity of U/2 _ a velocity of U/2. The material in the inner part of the disk ¯ows outwards and then moves through the gap between the ram and the chamber. During this process cracking can occur on the outside surface of the workpiece at about mid-height. The analysis presented in the previous paper of this series showed the velocity ®elds as well as the power terms for each ¯ow ®elds that are used to analyze the side-surface defect. (See [1] for a full description and drawings of the ¯ow ®elds.) This paper describes the results that have been obtained for the upper bound analysis of the double action forging process. Results from both the sound and the defective ¯ow patterns that were assumed for the process are presented. A comparison of these results for each ¯ow pattern is made and the pattern that requires the least power for a given set of process parameters will be assumed to be that which is operative at that instant in time. The characteristics of sound ¯ow patterns will be described and then the characteristics of the defect ¯ow patterns will be presented. A comparison of the power requirements for the sound ¯ow to the requirements for the defect ¯ow will be used to generate the criteria curves for the prevention of the process-induced side-surface cracking that is of primary interest in this investigation.
Fig. 2. Relative averaged ram pressure as a function of e/Ri for sound ¯ow `C'.
with the defect ¯ow pattern, the optimal value of e will be utilized for the determination of the averaged ram pressure. As shown in Fig. 2, the total power is comprised of contributions from the internal power of deformation, internal shear power losses and frictional power losses.
2. Process analysis and characteristics 2.1. Sound ¯ow `C' Fig. 2 shows the relative averaged ram pressure as a function of e/Ri (e is a pseudo-independent parameter). Eq. (19) in [1] is used to generate this ®gure. The optimum value of e is that at which the ram pressure is a minimum for a given set of process conditions. In Fig. 2 this optimal value of e/Ri is 0.13. For Figs. 3 and 4 discussed in this section and when this ¯ow pattern is used in comparison with the other sound ¯ow patterns as well as when it is used for comparison
Fig. 3. Effect of chamber-to-ram radius ratio (Ro/Ri) and relative workpiece center thickness (T/Ri) on the optimal relative averaged ram pressure for sound ¯ow `C' with no friction (m 0.0).
Y.H. Moon et al. / Journal of Materials Processing Technology 99 (2000) 179±184
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dominate the process rather than the internal power of deformation and the internal shear losses. The internal power of deformation is dependent strongly upon the volume of material being deformed and the internal shear losses depend on the area of the internal surface of velocity discontinuities, which become smaller as the volume of the workpiece decreases. It should be noted that the material being deformed in this analysis is a Mises material, hence the trends that are shown in Figs. 3 and 4 are due to the process and not the strain-hardening of the workpiece. When friction is present, the frictional power losses contribute signi®cantly to the total power and they are particularly large at low values of Ro/Ri, where the contact area between workpiece and tooling is large. 2.2. Defect ¯ow `C'
Fig. 4. Effect of chamber-to-ram radius ratio (Ro/Ri) and relative workpiece center thickness (T/Ri) on the optimal relative averaged ram pressure for sound ¯ow `C' with friction (m 0.2).
Fig. 2 shows that there is a minimum point, which occurs primarily due to the variation of the internal power of zone II and the shear power losses along G2 as e varies. The internal power of zone I as well as the frictional power losses between the workpiece and the tooling remain essentially constant as e changes. As e increases the volume of zone II increases and the internal power of deformation in this zone rises, whereas the shear power losses along G2 decrease with increasing e due to the decreasing surface area of this velocity discontinuity as well as the decreasing difference in tangential velocities along G2. The effect friction on relative averaged ram pressure with varying geometries is shown by comparing Figs. 3 and 4. As would be expected, the required pressure when friction is present is always greater than when friction is absent. These two ®gures also illustrate how the pressure would change as the workpiece becomes thinner during forging. For a ®xed value of Ro/Ri, as dictated by the tooling, the value of T/Ri decreases continually as the process continues. The required pressure as illustrated in Figs. 3 and 4 show a decreasing trend until some critical thickness value. Below that critical thickness there is a dramatic rise in the required pressure, even for low values of friction. In the open-die forging of cylindrical billets a similar trend is observed. This dramatic rise in required forging pressure is often termed the ``thin disk effect''. A similar trend is observed at the end of the stroke in both direct and indirect extrusion processes. This effect occurs because the ratio of the surface area for the tool/workpiece interface to the volume of the material has increased greatly. Hence the surface frictional effects tend to
In this part the characteristics of the defect ¯ow patterns are discussed. The defect ¯ow `C' pattern is compared to the equivalent sound ¯ow `C'. The ¯ow pattern that requires the least amount of total power in these comparisons will that which will be assumed to be operative. The comparison of these two ¯ow patterns will be used to develop criteria curves for the prevention of the process-induced side-surface cracks. Fig. 5 illustrates the relative ram pressure for both sound ¯ow `C' and defect ¯ow `C'. The sound ¯ow equation, Eq. (19) in [1] is used to calculate the right side of this plot, whilst the defect ¯ow equation, Eq. (31) in [1] is used to generate the left side of this ®gure. The geometry for this
Fig. 5. Relative averaged ram pressure as a function of e/Ri for defective ¯ow `C'.
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Fig. 6. Geometrical relationship between d and e for: (a) sound ¯ow; and (b) defective ¯ow.
example has Ro/Ri 2, T/(2Ri) 0.25, and H/(2Ri) 0.75 with friction, m 0.2. Each of the relative power terms contributes to the total power and determines the shape of this curve. The continuity from the sound ¯ow region to the defect ¯ow region for each one of the power terms as well as the total power term is seen in Fig. 5 by the continuity of each line through the value of e/Ri 0.0. The left side of this plot incorporates the power contribution due to crack formation. In the x-axis of Fig. 5, a negative value of e/Ri (which is a pseudo-independent parameter for sound ¯ow) has been used in representing the defect ¯ow instead of d/Ri. As d can be represented geometrically by a negative e value, as shown in Fig. 6, the same parameter e has been used for both sound ¯ow and defect ¯ow. To determine whether sound ¯ow or defect ¯ow will prevail under these process conditions the optimal value of e/Ri needs to be determined. As shown in Fig. 5, minima for the total power curve exist both in the sound ¯ow region (where e/Ri is positive) and in the defective ¯ow region (where e/Ri is negative). Under the particular set of processing conditions that were used to generate Fig. 5, the negative e/Ri minimum has a slightly lower ram pressure. Therefore under these processing conditions, there is a danger of surface crack formation, since this is energetically favorable as compared to sound ¯ow. With a standard optimizing routine [2], the minimum values of the relative ram pressure were computed for both, the positive (sound) and negative (defective) e/Ri domains. The two minimum ram pressures that were determined from these calculations are compared in Fig. 7 as a function of T/Ri for three different radius ratios. As shown in Fig. 7, there is a separating point at some critical T/Ri value where the defective ¯ow pattern becomes energetically favorable when compared to the sound ¯ow pattern. All values of T/Ri less than this critical value have their lowest ram pressure in
Fig. 7. Comparison of optimal relative averaged ram pressure for sound ¯ow `C' with defective ¯ow `C' as a function of relative workpiece thickness (T/Ri) with no friction (m 0.0).
the negative e/Ri domain. This indicates that for the given process conditions a workpiece with a T/Ri of less than the critical value a surface crack may be produced, since this is energetically favorable. Thus there is a larger danger of surface crack formation in forgings with a small center thicknesses (i.e. a low values of T/Ri). Fig. 8 shows the change in minima pressures in both e/Ri domains with an increased friction value as compared to Fig. 6 (i.e. m 0.2 in Fig. 8 and m 0.0 in Fig. 7). It is seen that for low friction, the danger of the surface crack is present but for higher friction this danger decreases, since the domain when the defective pattern is favorable is less. This is not the ®rst time that increased friction tends to hinder the formation of defects. Avitzur [3] showed that higher friction hindered the formation of a central burst in direct extrusions through
Fig. 8. Comparison of optimal relative averaged ram pressure for sound ¯ow `C' with defective ¯ow `C' as a function of relative workpiece thickness (T/Ri) with friction (m 0.2).
Y.H. Moon et al. / Journal of Materials Processing Technology 99 (2000) 179±184
183
conical converging dies. (Note that high friction promotes a central burst in wire drawing through conical converging dies.) Gordon and Van Tyne [4] also have shown that internal fractures in double hub forgings can be prevented by the use of higher frictional conditions on the tool/workpiece interface. 3. Criteria curves for defect prevention Based upon the analysis presented in the previous section with regard to the comparison of defective ¯ow `C' with sound ¯ow `C', criteria curves for the formation of processinduced surface cracks can be developed. Fig. 9 is one such curve. It is a summary plot of all of the transition points from sound ¯ow to defective ¯ow as a function of processing parameters. The relative tool radii ratio is plotted on the abscissa and the relative center thickness of the workpiece is plotted on the ordinate. Curves for various friction constants are presented. A value of mf 1.0 (i.e. complete adhesion within the material) is assumed for this ®gure. When the processing conditions are above or to the left of the curve, sound ¯ow is energetically favorable, whilst when the processing conditions are to the right or below the curve, defect ¯ow is favorable and the formation of a surface crack is possible. This cracking region is labeled as the DANGER region on the ®gure and should be avoided when forging these types of components. Fig. 9 indicates that increasing friction decreases the danger zone. In other words, at higher friction conditions with constant tooling (i.e. a ®xed value of Ro/Ri) the workpiece can be forged to a smaller center thicknesses (i.e.
smaller T/Ri values) without the danger of surface cracking. If the ®nal center thickness (i.e. the value of T/Ri) of the workpiece is ®xed by going to a larger ram radius in a ®xed chamber size (i.e. a smaller Ro/Ri value), the process could move from the danger zone to the safe region and the potential for a process-induced surface crack is avoided. Fig. 9 can be used to determine what changes in the process conditions should be made, so that a forging can be produced without the danger of surface cracking.
Fig. 9. Criteria curve for side-surface crack formation with complete adhesion within the workpiece (mf 1.0).
Fig. 11. Criteria curve for side-surface crack formation with some adhesion within the workpiece (mf 0.6).
Fig. 10. Criteria curve for side-surface crack formation with some adhesion within the workpiece (mf 0.8).
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based on the ram pressure versus e/Ri curve as presented in Fig. 5, that if the workpiece is in an energy state that corresponds to the minimum pressure on the positive e/Ri curve, then the material must overcome an energy activation barrier (which is related to the difference between the minimum pressure on the positive side of the curve and the maximum pressure at the e/Ri 0.0 point) before the workpiece falls into the energy state minimum that is on the negative e/Ri side of this curve. An analogous argument was used by Gordon and Van Tyne [4] in their attempt to use the upper bound approach to determine the maximum size of an internal pre-crack that can be tolerated in a double hub forging before an internal central burst is generated. Because of this type of argument, the criteria curves presented here are of a conservative nature. They give a de®nite indication of the safe processing region. If the processing parameters happen to be in the danger region, then the chance of surface cracking is present but it is not certain to occur. Fig. 12. Criteria curve for side-surface crack formation with low adhesion within the workpiece (mf 0.4).
Figs. 10±12 show the same type of criteria curve as was presented and discussed in Fig. 9, the difference being that the value of the bonding strength parameter is decreased by 0.2 for each ®gure. As would be expected, as the adhesive strength of the material is decreased, the processing danger zone becomes larger. In some cases the entire region depicted in the plot is in the danger zone. For example in Fig. 12 with a mf 0.4 (i.e. low adhesion), the entire plot zone is in the danger region for an m value of 0.0. Hence this curve do not appear on the plot at all. The curve for m 0.2 on this ®gure also shows a different type of behavior, such that above certain radii ratios (Ri/Ro > 1.5 in this case) the workpiece is in the danger region for all center thicknesses. If it is assumed that a small pre-existing crack or defect on the surface of the billet causes a decrease in the bonding parameter mf, then these criteria curves imply that presence of a small surface crack or small ¯aw in the workpiece before the forging begins would lead to the inducement of a larger crack during the actual forging operation. A note of caution at this point is appropriate. These criteria curves were generated on the basis of the energetically favorable ¯ow pattern being that which is more likely to occur. These curves should be used to indicate the regions of de®nite safe behavior and regions of potential but not de®nite crack formation. This potential (and the possibility) for crack formation is real. Whether the crack is actualized is still open to some debate. For example, it could be argued
4. Conclusions The analysis for the formation of side-surface cracks by the upper bound method provides a prediction for defect formation. This analysis can be used to develop the criteria curves as shown in Figs. 8±11. From an examination of these curves it is found that increases in the chamber-to-ram radius ratio (Ro/Ri), decreases in the relative center thickness of the billet (T/Ri) and decreases in friction (m) promote the formation of a surface crack. The trends that are observed in these criteria curves are matched by the trends for crack formation as determined by a ®nite element analysis and experimental veri®cation using a plasticine modeling material [5]. Hence these criteria curves provide a reliable means for the prevention of this side-surface cracking in forging. References [1] Y.H. Moon, C.J. Van Tyne, W.A. Gordon, An upper bound analysis of a process-induced side-surface defect in forgings. Part 1: The velocity ®eld and power terms, J. Mater. Process. Technol. 99 (2000) 169±178. [2] G.E. Forsythe, M.A. Malcolm, Computer Method for Mathematical Computations, Prentice-Hall, Englewood Cliffs, NJ, 1977. [3] B. Avitzur, Analysis of central bursting defects in drawing and extrusion, J. ASME, J. Eng. Ind. 90 (1968) 79. [4] W.A. Gordon, C.J. Van Tyne, Analysis of central burst in forging, Proceedings of the NAMRC-XI, 1983, p. 238. [5] Y.H. Moon, C.J. Van Tyne, Validation via FEM and plasticine modeling of upper bound criteria of a process-induced side surface defect in forgings, J. Mater. Process. Technol. 99 (2000) 185±196.
An upper bound analysis of a process-induced side-surface defect in forgings Part 2: Characteristics and criteria curves Y.H. Moona,*, C.J. Van Tyneb, W.A. Gordonc a
Engineering Research Center for Net Shape and Die Manufacturing, Pusan National University, San 30 Jangjeondong, Pusan, South Korea b Department of Metallurgical and Materials Engineering, Colorado School of Mines, Golden, CO, USA c Advanced Technology Center, Torrington Co., 59 Field Street, Torrington, CT, USA Received 13 August 1998
Abstract An upper bound analysis for an axisymmetric forging with a double ram action has been developed. The results of the analysis provide manufacturing conditions that can be used without the development of unacceptable side-surface cracks. There are processing conditions that cause the defective ¯ow pattern to be most favorable. The transition from an energetically-favorable sound ¯ow to an energeticallyfavorable defect ¯ow is used to determine the criteria curves for the prevention of this side-surface defect. It is found that the formation of the side-surface cracks is promoted by smaller diameter rams, smaller workpiece thicknesses and lower friction. The ®rst paper in this two-part series presents the velocity ®elds and the individual power terms that were derived based upon these velocity ®elds, whilst this second paper shows the characteristics of the upper bound solution for the double action forging process. This second paper also presents the criteria for the prevention of the side-surface defects that were determined based upon the upper bound solution. # 2000 Elsevier Science S.A. All rights reserved. Keywords: Upper bound analysis; Side-surface defect; Double action forging process; Criteria curve; Constant friction factor
Nomenclature H m mf pave R Ri Ro S T U_ V vf vi WI
height of the outer portion of the workpiece constant friction factor measure of bonding/adhesion within the workpiece average forging pressure radial coordinate in cylindrical coordinate system ram radius chamber radius surface thickness of the center portion of the workpiece ram velocity volume upwards workpiece velocity velocity component internal power of deformation
* Corresponding author. Tel.: 82-51-510-2472; fax: 82-51-512-1722. E-mail address: [email protected] (Y.H. Moon).
WG Wf WCF G d dopt e eopt y so
shear power loss along internal surface frictional power loss power for crack formation surface of velocity discontinuity crack length (a pseudo-independent parameter) crack length that requires lowest ram pressure position variable for internal surface (a pseudoindependent parameter) value of e that requires lowest ram pressure angular coordinate in cylindrical or spherical coordinate systems flow strength of the workpiece
1. Introduction Fig. 1 is a schematic diagram of the double ram action forging process that has been analyzed via the upper bound approach [1]. The workpiece begins as an axisymmetric disk of radius Ro. The upper ram of radius Ri moves downward
0924-0136/00/$ ± see front matter # 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 9 9 ) 0 0 4 1 8 - 5
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Y.H. Moon et al. / Journal of Materials Processing Technology 99 (2000) 179±184
Fig. 1. Schematic diagram of the axisymmetric forging process.
_ as the lower ram moves upwards with with a velocity of U/2 _ a velocity of U/2. The material in the inner part of the disk ¯ows outwards and then moves through the gap between the ram and the chamber. During this process cracking can occur on the outside surface of the workpiece at about mid-height. The analysis presented in the previous paper of this series showed the velocity ®elds as well as the power terms for each ¯ow ®elds that are used to analyze the side-surface defect. (See [1] for a full description and drawings of the ¯ow ®elds.) This paper describes the results that have been obtained for the upper bound analysis of the double action forging process. Results from both the sound and the defective ¯ow patterns that were assumed for the process are presented. A comparison of these results for each ¯ow pattern is made and the pattern that requires the least power for a given set of process parameters will be assumed to be that which is operative at that instant in time. The characteristics of sound ¯ow patterns will be described and then the characteristics of the defect ¯ow patterns will be presented. A comparison of the power requirements for the sound ¯ow to the requirements for the defect ¯ow will be used to generate the criteria curves for the prevention of the process-induced side-surface cracking that is of primary interest in this investigation.
Fig. 2. Relative averaged ram pressure as a function of e/Ri for sound ¯ow `C'.
with the defect ¯ow pattern, the optimal value of e will be utilized for the determination of the averaged ram pressure. As shown in Fig. 2, the total power is comprised of contributions from the internal power of deformation, internal shear power losses and frictional power losses.
2. Process analysis and characteristics 2.1. Sound ¯ow `C' Fig. 2 shows the relative averaged ram pressure as a function of e/Ri (e is a pseudo-independent parameter). Eq. (19) in [1] is used to generate this ®gure. The optimum value of e is that at which the ram pressure is a minimum for a given set of process conditions. In Fig. 2 this optimal value of e/Ri is 0.13. For Figs. 3 and 4 discussed in this section and when this ¯ow pattern is used in comparison with the other sound ¯ow patterns as well as when it is used for comparison
Fig. 3. Effect of chamber-to-ram radius ratio (Ro/Ri) and relative workpiece center thickness (T/Ri) on the optimal relative averaged ram pressure for sound ¯ow `C' with no friction (m 0.0).
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181
dominate the process rather than the internal power of deformation and the internal shear losses. The internal power of deformation is dependent strongly upon the volume of material being deformed and the internal shear losses depend on the area of the internal surface of velocity discontinuities, which become smaller as the volume of the workpiece decreases. It should be noted that the material being deformed in this analysis is a Mises material, hence the trends that are shown in Figs. 3 and 4 are due to the process and not the strain-hardening of the workpiece. When friction is present, the frictional power losses contribute signi®cantly to the total power and they are particularly large at low values of Ro/Ri, where the contact area between workpiece and tooling is large. 2.2. Defect ¯ow `C'
Fig. 4. Effect of chamber-to-ram radius ratio (Ro/Ri) and relative workpiece center thickness (T/Ri) on the optimal relative averaged ram pressure for sound ¯ow `C' with friction (m 0.2).
Fig. 2 shows that there is a minimum point, which occurs primarily due to the variation of the internal power of zone II and the shear power losses along G2 as e varies. The internal power of zone I as well as the frictional power losses between the workpiece and the tooling remain essentially constant as e changes. As e increases the volume of zone II increases and the internal power of deformation in this zone rises, whereas the shear power losses along G2 decrease with increasing e due to the decreasing surface area of this velocity discontinuity as well as the decreasing difference in tangential velocities along G2. The effect friction on relative averaged ram pressure with varying geometries is shown by comparing Figs. 3 and 4. As would be expected, the required pressure when friction is present is always greater than when friction is absent. These two ®gures also illustrate how the pressure would change as the workpiece becomes thinner during forging. For a ®xed value of Ro/Ri, as dictated by the tooling, the value of T/Ri decreases continually as the process continues. The required pressure as illustrated in Figs. 3 and 4 show a decreasing trend until some critical thickness value. Below that critical thickness there is a dramatic rise in the required pressure, even for low values of friction. In the open-die forging of cylindrical billets a similar trend is observed. This dramatic rise in required forging pressure is often termed the ``thin disk effect''. A similar trend is observed at the end of the stroke in both direct and indirect extrusion processes. This effect occurs because the ratio of the surface area for the tool/workpiece interface to the volume of the material has increased greatly. Hence the surface frictional effects tend to
In this part the characteristics of the defect ¯ow patterns are discussed. The defect ¯ow `C' pattern is compared to the equivalent sound ¯ow `C'. The ¯ow pattern that requires the least amount of total power in these comparisons will that which will be assumed to be operative. The comparison of these two ¯ow patterns will be used to develop criteria curves for the prevention of the process-induced side-surface cracks. Fig. 5 illustrates the relative ram pressure for both sound ¯ow `C' and defect ¯ow `C'. The sound ¯ow equation, Eq. (19) in [1] is used to calculate the right side of this plot, whilst the defect ¯ow equation, Eq. (31) in [1] is used to generate the left side of this ®gure. The geometry for this
Fig. 5. Relative averaged ram pressure as a function of e/Ri for defective ¯ow `C'.
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Fig. 6. Geometrical relationship between d and e for: (a) sound ¯ow; and (b) defective ¯ow.
example has Ro/Ri 2, T/(2Ri) 0.25, and H/(2Ri) 0.75 with friction, m 0.2. Each of the relative power terms contributes to the total power and determines the shape of this curve. The continuity from the sound ¯ow region to the defect ¯ow region for each one of the power terms as well as the total power term is seen in Fig. 5 by the continuity of each line through the value of e/Ri 0.0. The left side of this plot incorporates the power contribution due to crack formation. In the x-axis of Fig. 5, a negative value of e/Ri (which is a pseudo-independent parameter for sound ¯ow) has been used in representing the defect ¯ow instead of d/Ri. As d can be represented geometrically by a negative e value, as shown in Fig. 6, the same parameter e has been used for both sound ¯ow and defect ¯ow. To determine whether sound ¯ow or defect ¯ow will prevail under these process conditions the optimal value of e/Ri needs to be determined. As shown in Fig. 5, minima for the total power curve exist both in the sound ¯ow region (where e/Ri is positive) and in the defective ¯ow region (where e/Ri is negative). Under the particular set of processing conditions that were used to generate Fig. 5, the negative e/Ri minimum has a slightly lower ram pressure. Therefore under these processing conditions, there is a danger of surface crack formation, since this is energetically favorable as compared to sound ¯ow. With a standard optimizing routine [2], the minimum values of the relative ram pressure were computed for both, the positive (sound) and negative (defective) e/Ri domains. The two minimum ram pressures that were determined from these calculations are compared in Fig. 7 as a function of T/Ri for three different radius ratios. As shown in Fig. 7, there is a separating point at some critical T/Ri value where the defective ¯ow pattern becomes energetically favorable when compared to the sound ¯ow pattern. All values of T/Ri less than this critical value have their lowest ram pressure in
Fig. 7. Comparison of optimal relative averaged ram pressure for sound ¯ow `C' with defective ¯ow `C' as a function of relative workpiece thickness (T/Ri) with no friction (m 0.0).
the negative e/Ri domain. This indicates that for the given process conditions a workpiece with a T/Ri of less than the critical value a surface crack may be produced, since this is energetically favorable. Thus there is a larger danger of surface crack formation in forgings with a small center thicknesses (i.e. a low values of T/Ri). Fig. 8 shows the change in minima pressures in both e/Ri domains with an increased friction value as compared to Fig. 6 (i.e. m 0.2 in Fig. 8 and m 0.0 in Fig. 7). It is seen that for low friction, the danger of the surface crack is present but for higher friction this danger decreases, since the domain when the defective pattern is favorable is less. This is not the ®rst time that increased friction tends to hinder the formation of defects. Avitzur [3] showed that higher friction hindered the formation of a central burst in direct extrusions through
Fig. 8. Comparison of optimal relative averaged ram pressure for sound ¯ow `C' with defective ¯ow `C' as a function of relative workpiece thickness (T/Ri) with friction (m 0.2).
Y.H. Moon et al. / Journal of Materials Processing Technology 99 (2000) 179±184
183
conical converging dies. (Note that high friction promotes a central burst in wire drawing through conical converging dies.) Gordon and Van Tyne [4] also have shown that internal fractures in double hub forgings can be prevented by the use of higher frictional conditions on the tool/workpiece interface. 3. Criteria curves for defect prevention Based upon the analysis presented in the previous section with regard to the comparison of defective ¯ow `C' with sound ¯ow `C', criteria curves for the formation of processinduced surface cracks can be developed. Fig. 9 is one such curve. It is a summary plot of all of the transition points from sound ¯ow to defective ¯ow as a function of processing parameters. The relative tool radii ratio is plotted on the abscissa and the relative center thickness of the workpiece is plotted on the ordinate. Curves for various friction constants are presented. A value of mf 1.0 (i.e. complete adhesion within the material) is assumed for this ®gure. When the processing conditions are above or to the left of the curve, sound ¯ow is energetically favorable, whilst when the processing conditions are to the right or below the curve, defect ¯ow is favorable and the formation of a surface crack is possible. This cracking region is labeled as the DANGER region on the ®gure and should be avoided when forging these types of components. Fig. 9 indicates that increasing friction decreases the danger zone. In other words, at higher friction conditions with constant tooling (i.e. a ®xed value of Ro/Ri) the workpiece can be forged to a smaller center thicknesses (i.e.
smaller T/Ri values) without the danger of surface cracking. If the ®nal center thickness (i.e. the value of T/Ri) of the workpiece is ®xed by going to a larger ram radius in a ®xed chamber size (i.e. a smaller Ro/Ri value), the process could move from the danger zone to the safe region and the potential for a process-induced surface crack is avoided. Fig. 9 can be used to determine what changes in the process conditions should be made, so that a forging can be produced without the danger of surface cracking.
Fig. 9. Criteria curve for side-surface crack formation with complete adhesion within the workpiece (mf 1.0).
Fig. 11. Criteria curve for side-surface crack formation with some adhesion within the workpiece (mf 0.6).
Fig. 10. Criteria curve for side-surface crack formation with some adhesion within the workpiece (mf 0.8).
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based on the ram pressure versus e/Ri curve as presented in Fig. 5, that if the workpiece is in an energy state that corresponds to the minimum pressure on the positive e/Ri curve, then the material must overcome an energy activation barrier (which is related to the difference between the minimum pressure on the positive side of the curve and the maximum pressure at the e/Ri 0.0 point) before the workpiece falls into the energy state minimum that is on the negative e/Ri side of this curve. An analogous argument was used by Gordon and Van Tyne [4] in their attempt to use the upper bound approach to determine the maximum size of an internal pre-crack that can be tolerated in a double hub forging before an internal central burst is generated. Because of this type of argument, the criteria curves presented here are of a conservative nature. They give a de®nite indication of the safe processing region. If the processing parameters happen to be in the danger region, then the chance of surface cracking is present but it is not certain to occur. Fig. 12. Criteria curve for side-surface crack formation with low adhesion within the workpiece (mf 0.4).
Figs. 10±12 show the same type of criteria curve as was presented and discussed in Fig. 9, the difference being that the value of the bonding strength parameter is decreased by 0.2 for each ®gure. As would be expected, as the adhesive strength of the material is decreased, the processing danger zone becomes larger. In some cases the entire region depicted in the plot is in the danger zone. For example in Fig. 12 with a mf 0.4 (i.e. low adhesion), the entire plot zone is in the danger region for an m value of 0.0. Hence this curve do not appear on the plot at all. The curve for m 0.2 on this ®gure also shows a different type of behavior, such that above certain radii ratios (Ri/Ro > 1.5 in this case) the workpiece is in the danger region for all center thicknesses. If it is assumed that a small pre-existing crack or defect on the surface of the billet causes a decrease in the bonding parameter mf, then these criteria curves imply that presence of a small surface crack or small ¯aw in the workpiece before the forging begins would lead to the inducement of a larger crack during the actual forging operation. A note of caution at this point is appropriate. These criteria curves were generated on the basis of the energetically favorable ¯ow pattern being that which is more likely to occur. These curves should be used to indicate the regions of de®nite safe behavior and regions of potential but not de®nite crack formation. This potential (and the possibility) for crack formation is real. Whether the crack is actualized is still open to some debate. For example, it could be argued
4. Conclusions The analysis for the formation of side-surface cracks by the upper bound method provides a prediction for defect formation. This analysis can be used to develop the criteria curves as shown in Figs. 8±11. From an examination of these curves it is found that increases in the chamber-to-ram radius ratio (Ro/Ri), decreases in the relative center thickness of the billet (T/Ri) and decreases in friction (m) promote the formation of a surface crack. The trends that are observed in these criteria curves are matched by the trends for crack formation as determined by a ®nite element analysis and experimental veri®cation using a plasticine modeling material [5]. Hence these criteria curves provide a reliable means for the prevention of this side-surface cracking in forging. References [1] Y.H. Moon, C.J. Van Tyne, W.A. Gordon, An upper bound analysis of a process-induced side-surface defect in forgings. Part 1: The velocity ®eld and power terms, J. Mater. Process. Technol. 99 (2000) 169±178. [2] G.E. Forsythe, M.A. Malcolm, Computer Method for Mathematical Computations, Prentice-Hall, Englewood Cliffs, NJ, 1977. [3] B. Avitzur, Analysis of central bursting defects in drawing and extrusion, J. ASME, J. Eng. Ind. 90 (1968) 79. [4] W.A. Gordon, C.J. Van Tyne, Analysis of central burst in forging, Proceedings of the NAMRC-XI, 1983, p. 238. [5] Y.H. Moon, C.J. Van Tyne, Validation via FEM and plasticine modeling of upper bound criteria of a process-induced side surface defect in forgings, J. Mater. Process. Technol. 99 (2000) 185±196.