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Numerical simulation of the cross wedge rolling process including upsetting
Journal of Materials Processing Technology 92±93 (1999) 468±473
Numerical simulation of the cross wedge rolling process including upsetting Zb. Pater* Technical University of Lublin, Mechanical Department, Nadbystrzycka 36, Lublin, Poland
Abstract A new method developed by the author for the numerical simulation of the cross wedge rolling (CWR) process including upsetting is described in the present study. During the calculation sequence the strain area has been divided into several layers to be analysed successively for a plane state of strain as typical for rotary compression processes. The results of calculations based upon the upper-bound method have enabled distribution diagrams to be obtained for: the rolling forces; the contact surface between the material and the tool; and the rolling radius within the total range of the forming process. Furthermore, phenomena have been predicted (slipping and buckling) impairing process stability as well as the dimensions of the product obtained as a result of the CWR process. # 1999 Elsevier Science S.A. All rights reserved. Keywords: Metal forming; Cross wedge rolling; Mathematical modelling; Upper-bound method
1. General The cross wedge rolling process (CWR) is a process forming a forged part or a billet by means of wedge segments that are ®xed on to cross-rolling mills. The CWR process belongs to the class of rotary forming methods. During industrial cross-rolling processes, Fig. 1(a), the tool, being a wedge segment cuts into the material to the desired depth, forming a wedge-shaped groove on its circumference. Then this groove is enlarged by the forming (side) surfaces of the wedges from the centreline towards the product faces to obtain the required width. The forging area reduction of the cross-section is accompanied by free axial elongation. The potential capabilities of the cross-rolling process can be increased by an inverted direction of wedges action on the material being formed as shown in Fig. 1(b). Then the angular wedge groove formed in the ®rst stage of the rolling process is enlarged from the product faces towards the centreline of the product being formed. Such an inverted position of the tool segments results in the axial compression of the central part of the forged part. When the effective stress caused by these forces is greater than the yield stress of the material being formed, the central part of the product is *Tel.: +48-81-525-9061 extn. 396; fax: +48-81-525-0808.
subjected of upsetting. Therefore, the obtaining of a forged part with a diameter greater than the diameter of the billet seems to be practicable. If the axial stresses are insuf®cient, local strains and local diameter increase will occur, with material being concentrated at tools/material contact areas as in the case of the conventional cross/rolling processes. It should be mentioned that the rolling process with upsetting by means of wedge tool segments has not yet been studied suf®ciently. This is a serious disadvantage to potential industrial applications of this process. Therefore, theoretical and experimental studies have been initiated in order to determine the principles affecting this process. The CWR process-modelling concept on the grounds of upperbound method is discussed in the present paper. Due to the wide range of these problems, only particular solutions or problems have been mentioned. The computation capabilities of the present method have been shown by means of an example of the CWR process including upsetting. 2. Mathematical model of CWR process Aiming at solutions for a possibly wide range of crosswedge rolling process parameters, a modelling concept for this forming process has been developed using the upperbound method on the basis of the following assumptions: (i) individual layers of strain area are simulated; (ii) the state of
0924-0136/99/$ ± see front matter # 1999 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 9 9 ) 0 0 2 3 1 - 9
Zb. Pater / Journal of Materials Processing Technology 92±93 (1999) 468±473
469
The rotation angle ' is the main variable in the calculation process, being the basis to determine displacement y (Fig. 2) and the stage of process. The calculation sequence for step i is described below. In the ®rst stage of the calculations the value of displacement yi is determined corresponding to the actual forging rotation angle (' i'). Therefore, the following equation is used Fig. 1. Diagram of the cross wedge rolling process: (a) conventional CWR process; (b) the CWR process including upsetting.
strain in any layer of the forming zone can be simulated by rotary compression analysis for a plane state of strain; (iii) behaviour of the material being formed is similar of that for rigid/perfectly plastic material; (iv) no friction forces are caused by the guide strips/rolls; and (v) there is a constant value of the friction coef®cient along the whole of the material/tool contact surface. The following parameters: forming angle ; spreading angle ; spacing of wedges l; length of sizing zone lS; billet diameter d0 and relative reduction ( d1/d); forging rotation angle increase per single step '; friction factor on the wedge side m; friction factor on the sizing surface mS; and a group of parameters determining the course, time and accuracy of solution; are the input data for calculations.
yi yiÿ1
RT i RTiÿ1 : 2
(1)
The rolling radius being also simulated for the actual step, RT[i], is required for the carrying out of the calculations. Therefore, RT[i] RT[iÿ1] is assumed for the ®rst iteration. Then the value of the rolling stroke is calculated to determine the travel of the wedge side in direction x occurring during the last half of forging rotation i.e. after last tool contact s tan yi ÿ yiÿy
(2)
where j is the number of steps per 1/2 rotation made by the forging. Knowing the rolling stroke and the geometry parameters , d1 and d, one can determine the shape of the forming zone shown in Fig. 3. Then forming zone is divided into n layers and the surface pressure p is calculated for each layer successively. For instance, for a layer with its position determined by abscissa of x (Fig. 3), the following are calculated on the grounds of geometrical conditions: initial radius rx; rolling depth rx; and the height after rolling hx rx ÿ rx . Then using Eq. [1] bx
1 p 3rx rx ; cos
(3)
the contact surface width bx is determined. In order to determine the width fx (Fig. 3(b)) experimental results were
Fig. 2. Wedge element geometry and product shape development during the CWR process including upsetting.
Fig. 3. Layout of the layer simulation for the strain area in the cross wedge rolling process: (a) axial sectional view; (b) radial shape of the x-layer separated.
470
Zb. Pater / Journal of Materials Processing Technology 92±93 (1999) 468±473
Fig. 4. Rotary compression in the case of strain propagation up to centreline of the product being formed: (a) mechanism of plastic flow; (b) rate hodograph.
(i) in the case of the strain reaching the axis of the product
used [2±4] with the following obtained therefrom 1 rx ÿ 1:00106 : fx rx 1:01257 rx ÿ rx 2
(4)
On the grounds of bx and fx using the geometrical relationships resulting from Fig. 3(b), the remaining parameters are found, i.e. the free angle x and radius r0x explicitly describing the shape of the layer separated. Then the average unit pressure is determined for the contact surface using a method developed for rotary compression. A solution of Hayama [4] obtained by means of the upper-bound method and modi®ed by the present author has been used to determine the surface pressure. A mechanism of plastic ¯ow has been assumed in the form of sliding rigid blocks created as a result of dividing the area of the crosssection into the sub-sections. (Refer to Fig. 4 for a kinetically permissible strain mechanism in the case of the strain reaching the axis of the product being formed, and to Fig. 5 for the case of surface strain. Additionally, speed hodographs have been shown in these ®gures corresponding to the strain mechanism being presented). Non-dimensional relationships for the average surface pressure on the grounds of the balance of consumed and available power have been determined:
p 1 ÿ l 1;2 1;2 l 2;4 2;4 mb ; 2k 2b
(5)
(ii) in the case of surface strain p 1 ÿ l 1;2 1;2 l 1;3 1;3 2k 2b l 2;4 2;4 l 3;4 3;4 mb :
(6)
The lengths of the slipping surface are indicated, in Eqs. (5) and (6) by l (. . .) with the corresponding slipping speed values by (. . .). The material yield point for the pure shear condition is indicated as k is well-known. As, several simulation calculations are required to calculate the surface pressure in the rotary compression process by means of the upper ± bound method, with simultaneous change of parameters describing the strain mechanism. The parameters providing minimum value of pressure are used for the ®nal determination of p/2k. The Monte±Carlo optimisation method has been used in the present study to ®nd the angular parameters ensuring the minimum value of p/2k. Using the a/m procedure, values of p/2k are determined for all layers separated along the length of the forming zone. Then the rolling radius RT is calculated. The rotation movement rule is then used i.e. the total moment resulting from
Fig. 5. Rotary compression in the case of strain distributed on the surface: (a) mechanism of plastic flow; (b) rate hodograph.
Zb. Pater / Journal of Materials Processing Technology 92±93 (1999) 468±473
the forces provoking the forging rotation equals to total moment resulting from the forces preventing this rotation. Special iterative procedure has been used in this program to ®nd such value of rolling radius RT (separating the areas of forward and backward slipping) where the total moment resulting from the forces acting on the forging is equal to zero. When the obtained values of rolling radius by this method exceeded the billet radius by more than 20%, it was assumed that the forming process stability was impaired by excessive slipping and the calculations were discontinued. However, in the remaining cases the values of the rolling radius assumed preliminary and obtained from calculations were compared. When the difference of the two values was greater than the error assumed, it was assumed that the initial radius value equals that determined latterly and the calculations were repeated starting from coordinate determination yi , Eq. (1). For errors less than permissible value, the material/tool contact surface and rolling force components were calculated i.e., tangent component Py, axial component Px and radial component Pr (Fig. 6). Checking for the upsetting condition of the forging section being compressed between the wedges was the last step of the calculations made in step i. This condition can be described using the following equation 8Px 0 2 di
471
3. Calculation example In order to present the computation capabilities of the cross wedge rolling process developed for simulation, an example of a forming process with : 308; : 58; : 2.0; lF: 300 mm; lS: 50 mm; l: 80 mm; m: 1.0; mS: 0.5; and 0: 17N/ mm2 has been used. Such selection of process parameters ensured a proper value of the axial component of rolling force causing material upsetting. During the initial stage of the CWR process, the wedges cut into the material to the desired depth, pressing it outside. Therefore, the diameter (d1) of the material between the wedges is not changed, Fig. 7. Then, after reaching the proper value of the component Px, gradual upsetting of the material occurs and continues until the sizing stage is reached. Faster increase of the upset section diameter in the ®nal forming stage occurs as a result of the decreasing material length between the edges. Whilst analysing the rolling radius distribution RT, Fig. 8, during the CWR process, initially, its reduction can be noted
(7)
If the inequality (7) was not satis®ed, a new diameter of the upset section of the forging was determined under the assumption that whole volume of material being displaced during this step is transferred to the product section between the wedges. Then the next calculation step was initiated and whole computation cycle was repeated until the value of displacement y was not greater than the assumed length of the tool segment.
Fig. 7. Diameter distribution for the upset product section versus wedge displacement, determined for the CWR process under analysis.
Fig. 6. Load distribution on the tool segment in the CWR process.
Fig. 8. Rolling radius distribution versus wedge displacement, determined for the CWR process under analysis.
472
Zb. Pater / Journal of Materials Processing Technology 92±93 (1999) 468±473
4. Conclusions
Fig. 9. Distribution of projections for the material/tool contact surface versus wedge displacement, determined for the CWR process under analysis.
in result in increasing rolling depth. Then increase of material diameterd1 is accompanied by gradual increase of the RT value. In the ®nal stageoftheforming process,violent reduction ofthe rolling radius occurs. Decreasing resistance values during the sizing stage of the part being formed can be considered as a possible cause of this phenomenon. Refer to Fig. 9 for the distribution of the contact surface projection in the radial direction Sr, axial direction Sx and tangential direction Sy, obtained for the CWR process being analysed. Obviously, increase of the material diameter d1 is accompanied by increased material/tool contact surface. The distribution of the rolling forces during the forming process is affected by the size of material/tool contact surface. As shown in Fig. 10, the variations of rolling forces during the CWR process are appropriate for the corresponding distributions of the material/tool contact surface projection. An interesting fact can be mentioned, i.e. the reduction of the tangential component of the rolling force Py during the forming process in the location corresponding to the moment when the required cutting depth of rolling is achieved. Therefore, it can be supposed that for propagation of bar crosssection reduction, a signi®cantly lower force is required than for tool cutting performed in the ®rst process stage
Fig. 10. Distribution of the rolling force components versus wedge displacement, determined for the CWR process under analysis.
The simulation method of the cross wedge rolling CWR process (including upsetting) based upon dividing the forming zone into layers has been described in the present study. The proposed solution can also be used for typical CWR processes without material upsetting. At the present stage it is possible to determine the rolling force distributions, the material/tool contact surface, and the rolling radius during the forming process to calculate the forging dimensions after rolling and to check the process stability conditions. A series of experimental tests will be carried out soon by the present author to verify the accepted mathematical model of the CWR process. After that, an extension of the computation program is expected by including the strain-hardening problem, and possibility the selection of an optimisation method, etc. 5. Notations P Pr,
x, y
RT Sr, x, T b d d0 d1 f h i k l lF lS m mS n p r r0 s y r '
y
pressure force radial, axial and tangent component of the rolling load rolling radius contact surface projection in the radial, axial or tangential direction friction force contact width forging diameter after rolling billet diameter diameter of the central section of the forging part of contact width, Fig. 3 rolling height step of calculation yield shear stress spacing of the wedges length of the forming zone length of the sizing zone friction factor on the wedge side friction factor on the sizing surface number of layers selected in the forming zone contact pressure initial radius free radius rolling pitch velocity of subsection displacement of rolling tool rolling depth forming angle spreading angle relative reduction; d1/d forging rotation angle
Zb. Pater / Journal of Materials Processing Technology 92±93 (1999) 468±473
0 , , , , '1, !,
yield stress angular parameters
Acknowledgements The present study has been completed under Grant KBN No. 7 T08B 038 10 Analysis of rolling process including upsetting by means of wedge tool segments.
473
References [1] Zb. Pater, W. WeronÂski, Theoretical basis of cross wedge rolling process, LTN, Lublin, 1995. [2] H. Kasuga, Japan Soc. Mech. Eng (1971) 73. [3] G.W. Andrejew et al., Cross Wedge Rolling, Science and Technology, Minsk (1974). [4] M. Hayama, Bulletin of Fac. Eng. Yokohama Nat. Univ. 23 (1974) 83.
Numerical simulation of the cross wedge rolling process including upsetting Zb. Pater* Technical University of Lublin, Mechanical Department, Nadbystrzycka 36, Lublin, Poland
Abstract A new method developed by the author for the numerical simulation of the cross wedge rolling (CWR) process including upsetting is described in the present study. During the calculation sequence the strain area has been divided into several layers to be analysed successively for a plane state of strain as typical for rotary compression processes. The results of calculations based upon the upper-bound method have enabled distribution diagrams to be obtained for: the rolling forces; the contact surface between the material and the tool; and the rolling radius within the total range of the forming process. Furthermore, phenomena have been predicted (slipping and buckling) impairing process stability as well as the dimensions of the product obtained as a result of the CWR process. # 1999 Elsevier Science S.A. All rights reserved. Keywords: Metal forming; Cross wedge rolling; Mathematical modelling; Upper-bound method
1. General The cross wedge rolling process (CWR) is a process forming a forged part or a billet by means of wedge segments that are ®xed on to cross-rolling mills. The CWR process belongs to the class of rotary forming methods. During industrial cross-rolling processes, Fig. 1(a), the tool, being a wedge segment cuts into the material to the desired depth, forming a wedge-shaped groove on its circumference. Then this groove is enlarged by the forming (side) surfaces of the wedges from the centreline towards the product faces to obtain the required width. The forging area reduction of the cross-section is accompanied by free axial elongation. The potential capabilities of the cross-rolling process can be increased by an inverted direction of wedges action on the material being formed as shown in Fig. 1(b). Then the angular wedge groove formed in the ®rst stage of the rolling process is enlarged from the product faces towards the centreline of the product being formed. Such an inverted position of the tool segments results in the axial compression of the central part of the forged part. When the effective stress caused by these forces is greater than the yield stress of the material being formed, the central part of the product is *Tel.: +48-81-525-9061 extn. 396; fax: +48-81-525-0808.
subjected of upsetting. Therefore, the obtaining of a forged part with a diameter greater than the diameter of the billet seems to be practicable. If the axial stresses are insuf®cient, local strains and local diameter increase will occur, with material being concentrated at tools/material contact areas as in the case of the conventional cross/rolling processes. It should be mentioned that the rolling process with upsetting by means of wedge tool segments has not yet been studied suf®ciently. This is a serious disadvantage to potential industrial applications of this process. Therefore, theoretical and experimental studies have been initiated in order to determine the principles affecting this process. The CWR process-modelling concept on the grounds of upperbound method is discussed in the present paper. Due to the wide range of these problems, only particular solutions or problems have been mentioned. The computation capabilities of the present method have been shown by means of an example of the CWR process including upsetting. 2. Mathematical model of CWR process Aiming at solutions for a possibly wide range of crosswedge rolling process parameters, a modelling concept for this forming process has been developed using the upperbound method on the basis of the following assumptions: (i) individual layers of strain area are simulated; (ii) the state of
0924-0136/99/$ ± see front matter # 1999 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 9 9 ) 0 0 2 3 1 - 9
Zb. Pater / Journal of Materials Processing Technology 92±93 (1999) 468±473
469
The rotation angle ' is the main variable in the calculation process, being the basis to determine displacement y (Fig. 2) and the stage of process. The calculation sequence for step i is described below. In the ®rst stage of the calculations the value of displacement yi is determined corresponding to the actual forging rotation angle (' i'). Therefore, the following equation is used Fig. 1. Diagram of the cross wedge rolling process: (a) conventional CWR process; (b) the CWR process including upsetting.
strain in any layer of the forming zone can be simulated by rotary compression analysis for a plane state of strain; (iii) behaviour of the material being formed is similar of that for rigid/perfectly plastic material; (iv) no friction forces are caused by the guide strips/rolls; and (v) there is a constant value of the friction coef®cient along the whole of the material/tool contact surface. The following parameters: forming angle ; spreading angle ; spacing of wedges l; length of sizing zone lS; billet diameter d0 and relative reduction ( d1/d); forging rotation angle increase per single step '; friction factor on the wedge side m; friction factor on the sizing surface mS; and a group of parameters determining the course, time and accuracy of solution; are the input data for calculations.
yi yiÿ1
RT i RTiÿ1 : 2
(1)
The rolling radius being also simulated for the actual step, RT[i], is required for the carrying out of the calculations. Therefore, RT[i] RT[iÿ1] is assumed for the ®rst iteration. Then the value of the rolling stroke is calculated to determine the travel of the wedge side in direction x occurring during the last half of forging rotation i.e. after last tool contact s tan yi ÿ yiÿy
(2)
where j is the number of steps per 1/2 rotation made by the forging. Knowing the rolling stroke and the geometry parameters , d1 and d, one can determine the shape of the forming zone shown in Fig. 3. Then forming zone is divided into n layers and the surface pressure p is calculated for each layer successively. For instance, for a layer with its position determined by abscissa of x (Fig. 3), the following are calculated on the grounds of geometrical conditions: initial radius rx; rolling depth rx; and the height after rolling hx rx ÿ rx . Then using Eq. [1] bx
1 p 3rx rx ; cos
(3)
the contact surface width bx is determined. In order to determine the width fx (Fig. 3(b)) experimental results were
Fig. 2. Wedge element geometry and product shape development during the CWR process including upsetting.
Fig. 3. Layout of the layer simulation for the strain area in the cross wedge rolling process: (a) axial sectional view; (b) radial shape of the x-layer separated.
470
Zb. Pater / Journal of Materials Processing Technology 92±93 (1999) 468±473
Fig. 4. Rotary compression in the case of strain propagation up to centreline of the product being formed: (a) mechanism of plastic flow; (b) rate hodograph.
(i) in the case of the strain reaching the axis of the product
used [2±4] with the following obtained therefrom 1 rx ÿ 1:00106 : fx rx 1:01257 rx ÿ rx 2
(4)
On the grounds of bx and fx using the geometrical relationships resulting from Fig. 3(b), the remaining parameters are found, i.e. the free angle x and radius r0x explicitly describing the shape of the layer separated. Then the average unit pressure is determined for the contact surface using a method developed for rotary compression. A solution of Hayama [4] obtained by means of the upper-bound method and modi®ed by the present author has been used to determine the surface pressure. A mechanism of plastic ¯ow has been assumed in the form of sliding rigid blocks created as a result of dividing the area of the crosssection into the sub-sections. (Refer to Fig. 4 for a kinetically permissible strain mechanism in the case of the strain reaching the axis of the product being formed, and to Fig. 5 for the case of surface strain. Additionally, speed hodographs have been shown in these ®gures corresponding to the strain mechanism being presented). Non-dimensional relationships for the average surface pressure on the grounds of the balance of consumed and available power have been determined:
p 1 ÿ l 1;2 1;2 l 2;4 2;4 mb ; 2k 2b
(5)
(ii) in the case of surface strain p 1 ÿ l 1;2 1;2 l 1;3 1;3 2k 2b l 2;4 2;4 l 3;4 3;4 mb :
(6)
The lengths of the slipping surface are indicated, in Eqs. (5) and (6) by l (. . .) with the corresponding slipping speed values by (. . .). The material yield point for the pure shear condition is indicated as k is well-known. As, several simulation calculations are required to calculate the surface pressure in the rotary compression process by means of the upper ± bound method, with simultaneous change of parameters describing the strain mechanism. The parameters providing minimum value of pressure are used for the ®nal determination of p/2k. The Monte±Carlo optimisation method has been used in the present study to ®nd the angular parameters ensuring the minimum value of p/2k. Using the a/m procedure, values of p/2k are determined for all layers separated along the length of the forming zone. Then the rolling radius RT is calculated. The rotation movement rule is then used i.e. the total moment resulting from
Fig. 5. Rotary compression in the case of strain distributed on the surface: (a) mechanism of plastic flow; (b) rate hodograph.
Zb. Pater / Journal of Materials Processing Technology 92±93 (1999) 468±473
the forces provoking the forging rotation equals to total moment resulting from the forces preventing this rotation. Special iterative procedure has been used in this program to ®nd such value of rolling radius RT (separating the areas of forward and backward slipping) where the total moment resulting from the forces acting on the forging is equal to zero. When the obtained values of rolling radius by this method exceeded the billet radius by more than 20%, it was assumed that the forming process stability was impaired by excessive slipping and the calculations were discontinued. However, in the remaining cases the values of the rolling radius assumed preliminary and obtained from calculations were compared. When the difference of the two values was greater than the error assumed, it was assumed that the initial radius value equals that determined latterly and the calculations were repeated starting from coordinate determination yi , Eq. (1). For errors less than permissible value, the material/tool contact surface and rolling force components were calculated i.e., tangent component Py, axial component Px and radial component Pr (Fig. 6). Checking for the upsetting condition of the forging section being compressed between the wedges was the last step of the calculations made in step i. This condition can be described using the following equation 8Px 0 2 di
471
3. Calculation example In order to present the computation capabilities of the cross wedge rolling process developed for simulation, an example of a forming process with : 308; : 58; : 2.0; lF: 300 mm; lS: 50 mm; l: 80 mm; m: 1.0; mS: 0.5; and 0: 17N/ mm2 has been used. Such selection of process parameters ensured a proper value of the axial component of rolling force causing material upsetting. During the initial stage of the CWR process, the wedges cut into the material to the desired depth, pressing it outside. Therefore, the diameter (d1) of the material between the wedges is not changed, Fig. 7. Then, after reaching the proper value of the component Px, gradual upsetting of the material occurs and continues until the sizing stage is reached. Faster increase of the upset section diameter in the ®nal forming stage occurs as a result of the decreasing material length between the edges. Whilst analysing the rolling radius distribution RT, Fig. 8, during the CWR process, initially, its reduction can be noted
(7)
If the inequality (7) was not satis®ed, a new diameter of the upset section of the forging was determined under the assumption that whole volume of material being displaced during this step is transferred to the product section between the wedges. Then the next calculation step was initiated and whole computation cycle was repeated until the value of displacement y was not greater than the assumed length of the tool segment.
Fig. 7. Diameter distribution for the upset product section versus wedge displacement, determined for the CWR process under analysis.
Fig. 6. Load distribution on the tool segment in the CWR process.
Fig. 8. Rolling radius distribution versus wedge displacement, determined for the CWR process under analysis.
472
Zb. Pater / Journal of Materials Processing Technology 92±93 (1999) 468±473
4. Conclusions
Fig. 9. Distribution of projections for the material/tool contact surface versus wedge displacement, determined for the CWR process under analysis.
in result in increasing rolling depth. Then increase of material diameterd1 is accompanied by gradual increase of the RT value. In the ®nal stageoftheforming process,violent reduction ofthe rolling radius occurs. Decreasing resistance values during the sizing stage of the part being formed can be considered as a possible cause of this phenomenon. Refer to Fig. 9 for the distribution of the contact surface projection in the radial direction Sr, axial direction Sx and tangential direction Sy, obtained for the CWR process being analysed. Obviously, increase of the material diameter d1 is accompanied by increased material/tool contact surface. The distribution of the rolling forces during the forming process is affected by the size of material/tool contact surface. As shown in Fig. 10, the variations of rolling forces during the CWR process are appropriate for the corresponding distributions of the material/tool contact surface projection. An interesting fact can be mentioned, i.e. the reduction of the tangential component of the rolling force Py during the forming process in the location corresponding to the moment when the required cutting depth of rolling is achieved. Therefore, it can be supposed that for propagation of bar crosssection reduction, a signi®cantly lower force is required than for tool cutting performed in the ®rst process stage
Fig. 10. Distribution of the rolling force components versus wedge displacement, determined for the CWR process under analysis.
The simulation method of the cross wedge rolling CWR process (including upsetting) based upon dividing the forming zone into layers has been described in the present study. The proposed solution can also be used for typical CWR processes without material upsetting. At the present stage it is possible to determine the rolling force distributions, the material/tool contact surface, and the rolling radius during the forming process to calculate the forging dimensions after rolling and to check the process stability conditions. A series of experimental tests will be carried out soon by the present author to verify the accepted mathematical model of the CWR process. After that, an extension of the computation program is expected by including the strain-hardening problem, and possibility the selection of an optimisation method, etc. 5. Notations P Pr,
x, y
RT Sr, x, T b d d0 d1 f h i k l lF lS m mS n p r r0 s y r '
y
pressure force radial, axial and tangent component of the rolling load rolling radius contact surface projection in the radial, axial or tangential direction friction force contact width forging diameter after rolling billet diameter diameter of the central section of the forging part of contact width, Fig. 3 rolling height step of calculation yield shear stress spacing of the wedges length of the forming zone length of the sizing zone friction factor on the wedge side friction factor on the sizing surface number of layers selected in the forming zone contact pressure initial radius free radius rolling pitch velocity of subsection displacement of rolling tool rolling depth forming angle spreading angle relative reduction; d1/d forging rotation angle
Zb. Pater / Journal of Materials Processing Technology 92±93 (1999) 468±473
0 , , , , '1, !,
yield stress angular parameters
Acknowledgements The present study has been completed under Grant KBN No. 7 T08B 038 10 Analysis of rolling process including upsetting by means of wedge tool segments.
473
References [1] Zb. Pater, W. WeronÂski, Theoretical basis of cross wedge rolling process, LTN, Lublin, 1995. [2] H. Kasuga, Japan Soc. Mech. Eng (1971) 73. [3] G.W. Andrejew et al., Cross Wedge Rolling, Science and Technology, Minsk (1974). [4] M. Hayama, Bulletin of Fac. Eng. Yokohama Nat. Univ. 23 (1974) 83.