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Development of simulation code for calculating residual stress distribution in D-I cans produced by both-sided ironing process
Journal of Materials Processing Technology 140 (2003) 13–18
Development of simulation code for calculating residual stress distribution in D-I cans produced by both-sided ironing process S. Kuriyama a,∗ , Y. Yoshida a , T. Takahashi b , S. Kumagaya b , T. Aoki c , K. Miyauchi a a c
Institute of Physical and Chemical Research, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan b Saitama Institute of Technology, 1690 Fuzaiji, Okabe, Saitama 369-0293, Japan Gunma Industrial Research Laboratories, 190 Toriba-cho, Maebashi, Gunma 371-0845, Japan
Abstract A code of FEM for an elastic and plastic material has been developed to calculate a residual stress distribution in D-I can by both-sided ironing. It is possible for the code to simulate a forming process of deep drawing from a sheet to a can, an ironing process of both-sided of the can wall, and a deformation of the whole of the can under perfect unloading when the can is removed from an equipment of deep drawing and ironing. From these simulations, quite different and contrary distributions of residual stress are obtained according to mutual positions of dies for the ironing, which are considered to expand the wall of can as an ellipsoidal shape or shrink it to a hyperbolic shape. © 2003 Elsevier B.V. All rights reserved. Keywords: Finite element method; D-I can; Simulation code; Deep drawing; Ironing; Elastic–plastic material
1. Introduction In the conventional ironing process, only an outer wall surface of deep drawn cans is ironed to be very smooth. An inner wall surface, however, is requested to be very smooth in some cases such as shock-absorbing tubes. Two of the authors [1] developed a new process for ironing both the sides of the can wall at the same time. The residual stress distribution in the ironed wall is considerably different between the conventional ironing and new both-sided ironing. The distribution is affected by the combination of die profile radii for inner and outer ironing, matching condition of the two dies and the starting point of ironing by the inner die. Especially a mutual position of the inner and outer die makes the wall expand like a barrel or shrink like a saddle. A simulation code of finite element method (FEM) has been developed for a deformation of elastic and plastic material by the authors [2–4], which is a three-dimensional static implicit code and called as FDSolid-S. A can is produced from a sheet by a deep drawing process, and the residual stress remains under a process of perfect unloading when the can is removed from an equipment of the deep drawing. Next, the can being set on another equipment of ironing, a new process of both-sided ironing of the can wall is conducted. The residual stress remains again by removing the
∗ Corresponding author. Tel.: +81-48-467-9317; fax: +81-48-462-4657. E-mail address: [email protected] (S. Kuriyama).
0924-0136/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0924-0136(03)00694-0
can from the equipment of ironing. These processes of sheet metal forming are simulated numerically by using FEM, and distributions of residual stress in the can wall are calculated according to mutual positions of the dies for ironing.
2. Finite element code developed The authors have developed a simulation code of FEM, which is composed of three kinds of solution, that is a static explicit, a static implicit, and a dynamic explicit solution as shown in Fig. 1. The static explicit solution is applied to a formation and deformation of stereolithography of resin by ultra-violet ray [5]. The static implicit solution is applied to a deformation of metal in plastic working produced by a tool [2]. The dynamic explicit solution is applied to a deformation of plastic working and is being applied to a fracture problem in plastic working [6]. These codes are called as FDSolid-R, FDSolid-S, and FDSolid-D, respectively. A both-sided ironing treated in this paper is simulated by a code of the static implicit solution of FDSolid-S. Our treatment of a specimen, which contacts with or releases from a tool, is controlled numerically by a truss element of elastic bar [2], which is set between a surface of the specimen and a center of curvature of the tool as shown in Fig. 2. We consider a case where the specimen enters into the tool inside from time i − 1 to i shown in Fig. 2(a). An elastic bar Pi OQ is set and a displacement QO (=Pi Hi ) is given at node Q of the bar in order to move the specimen Pi
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Fig. 1. Simulation cord developed for formation and deformation of solid (FDSolid).
to a tool surface Pi+1 as shown in Fig. 2(b). The elastic bar does not permit the specimen to enter into the tool inside but makes it rotate along the tool surface. The rigidity of the elastic bar is not high and variable. From this treatment of elastic rigidity, not only the specimen but also the tool deforms so that divergence of solution is suppressed slightly. When divergence occurs in our implicit method, a solving technique is changed from the implicit method to the explicit method to continue calculations. After the explicit method being used during about 20 steps, the solving technique is returned to the implicit method again. A surface of tool is divided into five kinds of curved surface, namely plane, cylindrical, spherical, conical, and doughnut surface. Geometrical information about the each element is obtained from a relationship among nine nodes. The inside or outside of each surface is distinguished by four walls installed at boundaries of the element as shown in Fig. 3. When a can is removed from an equipment of deep drawing and both-sided ironing, it is necessary to simulate a deformation of the can under a perfect unloading process to calculate a distribution of residual stress. Two sets of triangle defined by three nodes are prepared in the specimen shown in Fig. 4 for removing an external applied force perfectly. Deformation of the specimen under perfect unloading is obtained by solving several times on alternate condition
Fig. 2. Numerical control in contact with tool by an elastic truss element.
Fig. 3. Tool surfaces divided are expressed by five kinds of curved surface and four walls installed.
Fig. 4. Position of two triangles for perfect unloading.
Fig. 5. Schematic diagram of varying displacement under loading and unloading processes.
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of fixed and free nodes of triangles 1 and 2. A schematic diagram of varying displacement at a certain node is shown in Fig. 5 under two loading processes of an elastic and a plastic deformation, an unloading process of tool movement in reverse, and a perfect unloading process of release from a tool.
3. Simulation by FEM A code of FEM developed is on a base of the finite deformation theory [3]. Three-dimensional solid element is used and a shape function of element is of ‘Serendipity’ family expressed by 20 nodes. BFGS method is used to solve nonlinear problem of deformation and number of integrating points for Gauss integration is 3 × 3 × 2 = 18. It is enough to solve a deformation of a quarter of specimen, because it is formed by a symmetrical tool. Material of the specimen is assumed to be steel and to be expressed by linear work-hardening. 3.1. Forming process of can by deep drawing A steel can is formed from a sheet through a process of deep drawing shown in Fig. 6. Dimension of a specimen is a quarter of sheet metal shown in Fig. 7, whose diameter and thickness is φ 70 and 0.8 mm, respectively. A blank of the sheet is divided into 342 elements of three-dimensional solid with two layers. Numbers of elastic truss elements in contact with tool are 1140, and total number of nodes is 5187. It is assumed for material properties of specimen that Young’s modulus is E = 21 000 kg/mm2 , yield stress is σy = 21.0 kg/mm2 , rate of linear work-hardening is Hp = 210.0 kg/mm2 , and Poisson’s ratio is ν = 0.3. A surface division of tool for deep drawing is shown in Fig. 8, where a circular flat punch and a die is composed of three surfaces, namely one plane, one doughnut and one
Fig. 8. Division and expression of tool by curved surface for deep drawing.
cylindrical surface, and a blank-holder is composed of two surfaces, namely one plane and one cylindrical surface. A diameter of the punch and the die are φ 50 and 51.6 mm, radius of a punch and a die shoulder are 7 and 3 mm, respectively. Numerical results of formed shape are shown in Fig. 9(a) and (b). They are formed from a sheet of Fig. 7 whose material is annealed, and a can shown in Fig. 9(b) is released from an equipment of deep drawing. Their distance of punch stroke are 25 mm at step 187 and 50 mm at step 324, respectively. Computing time is about 4 h. 3.2. Both-sided ironing of can wall Ironing of both sides of a can wall is conducted by an equipment constituted of three parts of an inner die, an outer
Fig. 6. Process of deep drawing.
Fig. 7. Mesh division of sheet metal for deep drawing process.
Fig. 9. Formed shape of a can from sheet by deep drawing.
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Fig. 13. Division and expression of tool by curved surface for both-sided ironing. Fig. 10. Process of both-sided ironing.
Fig. 11. Bending and unbending process of cup wall by both-sided ironing.
die, and a punch as shown in Fig. 10. Deforming behavior of wall is considered to be composed of two kinds of deformation. One is a deformation generated by a relative movement between the dies and the punch, which produces a tensile stress in a wall. Another is a bending and unbending process, which occurs twice at the inner and outer die as shown in Fig. 11. Dimension of a specimen is a quarter of cup, whose diameter of inside, outside and thickness is φ 50, 51.6 and 0.8 mm, respectively, as shown in Fig. 12. The cup is divided into 240 elements of three-dimensional solid with two layers. Number of elastic truss elements in contact with a tool is 806, and total number of nodes is 3661. It is assumed for material properties of specimen as annealed that Young’s modulus is E = 21 000 kg/mm2 , yield stress is
Fig. 12. Mesh division of cup for both-sided ironing.
σy = 232.0 kg/mm2 , rate of linear work-hardening is Hp = 270.9 kg/mm2 , and Poisson’s ratio is ν = 0.3. A surface division of tool for both-sided ironing is shown in Fig. 13, where a punch is composed of five surfaces, namely two plane, one doughnut, two cylindrical surfaces, an inner die is composed of one doughnut, one cylindrical, one plane surface, and an outer die is composed of one doughnut and one cylindrical surface. A diameter of the punch is φ 49 mm, and a radius of a punch shoulder is 7 mm. An outer diameter of the inner die is φ 50.88 mm, an inner diameter of the outer die is φ 50 mm, and curvatures of both dies are 10 mm. 3.3. Numerical results of wall by both-sided ironing Numerical results of both-sided ironing of can are shown in Figs. 14–16. It is assumed that the wall of can is ironed from an annealed condition. Fig. 14 shows a deformation of can in a case where a mutual distance h between inner and outer dies shown in Fig. 11 is zero and when positions of the both dies are 20 mm at step 200. Figs. 15 and 16 show deformations of cross section of wall in the cases where the mutual distance h of inner and outer dies are 2 mm. Computing time is about 4 h. When the inner die is higher than the outer die, a reduction of wall thickness is shown in Fig. 15(a) along the wall and distributions of hoop tension or
Fig. 14. Both-sided ironing when a position of inner die is same as one of outer die.
S. Kuriyama et al. / Journal of Materials Processing Technology 140 (2003) 13–18
Fig. 15. Both-sided ironing when an inner die is higher than an outer die in 2 mm.
compression σ θ along a wall thickness are shown in Fig. 15(b). On the contrary, when the outer die is higher than the inner die, a reduction of wall thickness and distributions of hoop stress σ θ are shown in Fig. 16(a) and (b), respectively. The distributions of the stress in the wall are quite different from Fig. 15(b) and converse. It means that these distributions of stress generate an expansion or shrinkage of the wall of can.
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Fig. 16. Both-sided ironing when an outer die is higher than an inner die in 2 mm.
die is higher than that of outer die, the residual stress in the wall is of extension and the wall of can expands. Conversely, when a position of outer die is higher than that of inner die, the residual stress in the wall is of compression and the wall shrinks.
4. Experiment A can was formed from a sheet of steel through a process of deep drawing, and a wall of the can was ironed by inner and outer dies [1]. The wall of can was sliced along a circumferential direction as a strip of ring and the ring was cut to open and to measure residual stress as shown in Fig. 17. The residual stress was estimated from Crampton’s method using a distance of ring opened by the cut. The residual stress distribution in experiment is shown in Fig. 18 on an effect of mutual difference h in height between inner and outer dies as shown in Fig. 11. When a position of inner
Fig. 17. Cup sliced to estimate residual stress (a) and Crampton’s method (b).
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This means that a cutting plane along the wall opened when the wall of can was sliced and cut in experiment. However, the authors consider that quite different bending moments generated by different distributions of residual stress make the wall of can expand or shrink. In experiment a specimen is often cut to examine a residual stress. The cup in Fig. 17 was sliced in circumferential direction and cut in wall direction. Then it is necessary to treat and simulate not only a deformation under loading and perfect unloading, but also separating phenomena of material due to slicing and cutting, which is now being developed for a fracture problem by the authors as FDSolid-D.
Acknowledgements The authors wish to thank members of Material Fabrication Laboratory, Institute of Physical and Chemical Research (Riken), and many students of Saitama Institute of Technology, who got training in Riken and have already graduated from their Institute for their useful discussions and assistance in calculation. Fig. 18. Effect of mutual difference h in height between inner and outer dies on residual stress distribution.
References 5. Conclusion A code of FEM has been developed for calculating the residual stress distribution in D-I can. It is possible for our code to simulate a process of deep drawing from a sheet to a can, a process of both-sided ironing of can wall, and a deformation of the whole of can under perfect unloading when the specimen is released from an equipment of deep drawing and of ironing. The authors simulated a process of deep drawing from a sheet to form a can, and simulated a process of both-sided ironing of a can and obtained distributions of circumferential stress σ θ along a wall thickness on a cross section of can wall. It is found that distributions of the circumferential stress in a case where inner die is higher than outer die is quite different from and contrary to distributions where outer die is higher than inner die. Summation of the hoop stress σ θ along the thickness is tensile in both cases.
[1] T. Aoki, K. Miyauchi, Forming characteristics of ironed cans, Technol. Mater. Test 36 (2) (1991) pp. 36–42. [2] S. Kuriyama, Developing of FEM code for simulating plastic working controlled by rotating elastic bar, Report of Research Summary to Amada Foundation for Metal Work Technology, vol. 11, 1999, pp. 58–63. [3] K.J. Bathe, A Finite Element Program for Automatic Dynamic Incremental Nonlinear Analysis (ADINA), Massachusetts Institute of Technology, Massachusetts, USA, 1975. [4] O.C. Zienkiewicz, The Finite Element Method, McGraw-Hill, New York, 1977. [5] S. Kuriyama, Y. Xu, T. Nakagawa, Development of FE simulation code for stereolithography of resin cured by U.V. ray, in: Proceedings of the Seventh International Conference of NUMIFORM, Toyohashi, Japan, June 2001, pp. 337–342. [6] S. Kuriyama, T. Takahashi, S. Kumagaya, M. Yoshida, Development of FE simulation code for metal forming with and by cracks, first report application of dynamic code to propagation of crack, in: Proceedings of the 2002 Japanese Spring Conference for Technology of Plasticity, 2002, pp. 213–214.
Development of simulation code for calculating residual stress distribution in D-I cans produced by both-sided ironing process S. Kuriyama a,∗ , Y. Yoshida a , T. Takahashi b , S. Kumagaya b , T. Aoki c , K. Miyauchi a a c
Institute of Physical and Chemical Research, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan b Saitama Institute of Technology, 1690 Fuzaiji, Okabe, Saitama 369-0293, Japan Gunma Industrial Research Laboratories, 190 Toriba-cho, Maebashi, Gunma 371-0845, Japan
Abstract A code of FEM for an elastic and plastic material has been developed to calculate a residual stress distribution in D-I can by both-sided ironing. It is possible for the code to simulate a forming process of deep drawing from a sheet to a can, an ironing process of both-sided of the can wall, and a deformation of the whole of the can under perfect unloading when the can is removed from an equipment of deep drawing and ironing. From these simulations, quite different and contrary distributions of residual stress are obtained according to mutual positions of dies for the ironing, which are considered to expand the wall of can as an ellipsoidal shape or shrink it to a hyperbolic shape. © 2003 Elsevier B.V. All rights reserved. Keywords: Finite element method; D-I can; Simulation code; Deep drawing; Ironing; Elastic–plastic material
1. Introduction In the conventional ironing process, only an outer wall surface of deep drawn cans is ironed to be very smooth. An inner wall surface, however, is requested to be very smooth in some cases such as shock-absorbing tubes. Two of the authors [1] developed a new process for ironing both the sides of the can wall at the same time. The residual stress distribution in the ironed wall is considerably different between the conventional ironing and new both-sided ironing. The distribution is affected by the combination of die profile radii for inner and outer ironing, matching condition of the two dies and the starting point of ironing by the inner die. Especially a mutual position of the inner and outer die makes the wall expand like a barrel or shrink like a saddle. A simulation code of finite element method (FEM) has been developed for a deformation of elastic and plastic material by the authors [2–4], which is a three-dimensional static implicit code and called as FDSolid-S. A can is produced from a sheet by a deep drawing process, and the residual stress remains under a process of perfect unloading when the can is removed from an equipment of the deep drawing. Next, the can being set on another equipment of ironing, a new process of both-sided ironing of the can wall is conducted. The residual stress remains again by removing the
∗ Corresponding author. Tel.: +81-48-467-9317; fax: +81-48-462-4657. E-mail address: [email protected] (S. Kuriyama).
0924-0136/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0924-0136(03)00694-0
can from the equipment of ironing. These processes of sheet metal forming are simulated numerically by using FEM, and distributions of residual stress in the can wall are calculated according to mutual positions of the dies for ironing.
2. Finite element code developed The authors have developed a simulation code of FEM, which is composed of three kinds of solution, that is a static explicit, a static implicit, and a dynamic explicit solution as shown in Fig. 1. The static explicit solution is applied to a formation and deformation of stereolithography of resin by ultra-violet ray [5]. The static implicit solution is applied to a deformation of metal in plastic working produced by a tool [2]. The dynamic explicit solution is applied to a deformation of plastic working and is being applied to a fracture problem in plastic working [6]. These codes are called as FDSolid-R, FDSolid-S, and FDSolid-D, respectively. A both-sided ironing treated in this paper is simulated by a code of the static implicit solution of FDSolid-S. Our treatment of a specimen, which contacts with or releases from a tool, is controlled numerically by a truss element of elastic bar [2], which is set between a surface of the specimen and a center of curvature of the tool as shown in Fig. 2. We consider a case where the specimen enters into the tool inside from time i − 1 to i shown in Fig. 2(a). An elastic bar Pi OQ is set and a displacement QO (=Pi Hi ) is given at node Q of the bar in order to move the specimen Pi
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Fig. 1. Simulation cord developed for formation and deformation of solid (FDSolid).
to a tool surface Pi+1 as shown in Fig. 2(b). The elastic bar does not permit the specimen to enter into the tool inside but makes it rotate along the tool surface. The rigidity of the elastic bar is not high and variable. From this treatment of elastic rigidity, not only the specimen but also the tool deforms so that divergence of solution is suppressed slightly. When divergence occurs in our implicit method, a solving technique is changed from the implicit method to the explicit method to continue calculations. After the explicit method being used during about 20 steps, the solving technique is returned to the implicit method again. A surface of tool is divided into five kinds of curved surface, namely plane, cylindrical, spherical, conical, and doughnut surface. Geometrical information about the each element is obtained from a relationship among nine nodes. The inside or outside of each surface is distinguished by four walls installed at boundaries of the element as shown in Fig. 3. When a can is removed from an equipment of deep drawing and both-sided ironing, it is necessary to simulate a deformation of the can under a perfect unloading process to calculate a distribution of residual stress. Two sets of triangle defined by three nodes are prepared in the specimen shown in Fig. 4 for removing an external applied force perfectly. Deformation of the specimen under perfect unloading is obtained by solving several times on alternate condition
Fig. 2. Numerical control in contact with tool by an elastic truss element.
Fig. 3. Tool surfaces divided are expressed by five kinds of curved surface and four walls installed.
Fig. 4. Position of two triangles for perfect unloading.
Fig. 5. Schematic diagram of varying displacement under loading and unloading processes.
S. Kuriyama et al. / Journal of Materials Processing Technology 140 (2003) 13–18
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of fixed and free nodes of triangles 1 and 2. A schematic diagram of varying displacement at a certain node is shown in Fig. 5 under two loading processes of an elastic and a plastic deformation, an unloading process of tool movement in reverse, and a perfect unloading process of release from a tool.
3. Simulation by FEM A code of FEM developed is on a base of the finite deformation theory [3]. Three-dimensional solid element is used and a shape function of element is of ‘Serendipity’ family expressed by 20 nodes. BFGS method is used to solve nonlinear problem of deformation and number of integrating points for Gauss integration is 3 × 3 × 2 = 18. It is enough to solve a deformation of a quarter of specimen, because it is formed by a symmetrical tool. Material of the specimen is assumed to be steel and to be expressed by linear work-hardening. 3.1. Forming process of can by deep drawing A steel can is formed from a sheet through a process of deep drawing shown in Fig. 6. Dimension of a specimen is a quarter of sheet metal shown in Fig. 7, whose diameter and thickness is φ 70 and 0.8 mm, respectively. A blank of the sheet is divided into 342 elements of three-dimensional solid with two layers. Numbers of elastic truss elements in contact with tool are 1140, and total number of nodes is 5187. It is assumed for material properties of specimen that Young’s modulus is E = 21 000 kg/mm2 , yield stress is σy = 21.0 kg/mm2 , rate of linear work-hardening is Hp = 210.0 kg/mm2 , and Poisson’s ratio is ν = 0.3. A surface division of tool for deep drawing is shown in Fig. 8, where a circular flat punch and a die is composed of three surfaces, namely one plane, one doughnut and one
Fig. 8. Division and expression of tool by curved surface for deep drawing.
cylindrical surface, and a blank-holder is composed of two surfaces, namely one plane and one cylindrical surface. A diameter of the punch and the die are φ 50 and 51.6 mm, radius of a punch and a die shoulder are 7 and 3 mm, respectively. Numerical results of formed shape are shown in Fig. 9(a) and (b). They are formed from a sheet of Fig. 7 whose material is annealed, and a can shown in Fig. 9(b) is released from an equipment of deep drawing. Their distance of punch stroke are 25 mm at step 187 and 50 mm at step 324, respectively. Computing time is about 4 h. 3.2. Both-sided ironing of can wall Ironing of both sides of a can wall is conducted by an equipment constituted of three parts of an inner die, an outer
Fig. 6. Process of deep drawing.
Fig. 7. Mesh division of sheet metal for deep drawing process.
Fig. 9. Formed shape of a can from sheet by deep drawing.
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Fig. 13. Division and expression of tool by curved surface for both-sided ironing. Fig. 10. Process of both-sided ironing.
Fig. 11. Bending and unbending process of cup wall by both-sided ironing.
die, and a punch as shown in Fig. 10. Deforming behavior of wall is considered to be composed of two kinds of deformation. One is a deformation generated by a relative movement between the dies and the punch, which produces a tensile stress in a wall. Another is a bending and unbending process, which occurs twice at the inner and outer die as shown in Fig. 11. Dimension of a specimen is a quarter of cup, whose diameter of inside, outside and thickness is φ 50, 51.6 and 0.8 mm, respectively, as shown in Fig. 12. The cup is divided into 240 elements of three-dimensional solid with two layers. Number of elastic truss elements in contact with a tool is 806, and total number of nodes is 3661. It is assumed for material properties of specimen as annealed that Young’s modulus is E = 21 000 kg/mm2 , yield stress is
Fig. 12. Mesh division of cup for both-sided ironing.
σy = 232.0 kg/mm2 , rate of linear work-hardening is Hp = 270.9 kg/mm2 , and Poisson’s ratio is ν = 0.3. A surface division of tool for both-sided ironing is shown in Fig. 13, where a punch is composed of five surfaces, namely two plane, one doughnut, two cylindrical surfaces, an inner die is composed of one doughnut, one cylindrical, one plane surface, and an outer die is composed of one doughnut and one cylindrical surface. A diameter of the punch is φ 49 mm, and a radius of a punch shoulder is 7 mm. An outer diameter of the inner die is φ 50.88 mm, an inner diameter of the outer die is φ 50 mm, and curvatures of both dies are 10 mm. 3.3. Numerical results of wall by both-sided ironing Numerical results of both-sided ironing of can are shown in Figs. 14–16. It is assumed that the wall of can is ironed from an annealed condition. Fig. 14 shows a deformation of can in a case where a mutual distance h between inner and outer dies shown in Fig. 11 is zero and when positions of the both dies are 20 mm at step 200. Figs. 15 and 16 show deformations of cross section of wall in the cases where the mutual distance h of inner and outer dies are 2 mm. Computing time is about 4 h. When the inner die is higher than the outer die, a reduction of wall thickness is shown in Fig. 15(a) along the wall and distributions of hoop tension or
Fig. 14. Both-sided ironing when a position of inner die is same as one of outer die.
S. Kuriyama et al. / Journal of Materials Processing Technology 140 (2003) 13–18
Fig. 15. Both-sided ironing when an inner die is higher than an outer die in 2 mm.
compression σ θ along a wall thickness are shown in Fig. 15(b). On the contrary, when the outer die is higher than the inner die, a reduction of wall thickness and distributions of hoop stress σ θ are shown in Fig. 16(a) and (b), respectively. The distributions of the stress in the wall are quite different from Fig. 15(b) and converse. It means that these distributions of stress generate an expansion or shrinkage of the wall of can.
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Fig. 16. Both-sided ironing when an outer die is higher than an inner die in 2 mm.
die is higher than that of outer die, the residual stress in the wall is of extension and the wall of can expands. Conversely, when a position of outer die is higher than that of inner die, the residual stress in the wall is of compression and the wall shrinks.
4. Experiment A can was formed from a sheet of steel through a process of deep drawing, and a wall of the can was ironed by inner and outer dies [1]. The wall of can was sliced along a circumferential direction as a strip of ring and the ring was cut to open and to measure residual stress as shown in Fig. 17. The residual stress was estimated from Crampton’s method using a distance of ring opened by the cut. The residual stress distribution in experiment is shown in Fig. 18 on an effect of mutual difference h in height between inner and outer dies as shown in Fig. 11. When a position of inner
Fig. 17. Cup sliced to estimate residual stress (a) and Crampton’s method (b).
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This means that a cutting plane along the wall opened when the wall of can was sliced and cut in experiment. However, the authors consider that quite different bending moments generated by different distributions of residual stress make the wall of can expand or shrink. In experiment a specimen is often cut to examine a residual stress. The cup in Fig. 17 was sliced in circumferential direction and cut in wall direction. Then it is necessary to treat and simulate not only a deformation under loading and perfect unloading, but also separating phenomena of material due to slicing and cutting, which is now being developed for a fracture problem by the authors as FDSolid-D.
Acknowledgements The authors wish to thank members of Material Fabrication Laboratory, Institute of Physical and Chemical Research (Riken), and many students of Saitama Institute of Technology, who got training in Riken and have already graduated from their Institute for their useful discussions and assistance in calculation. Fig. 18. Effect of mutual difference h in height between inner and outer dies on residual stress distribution.
References 5. Conclusion A code of FEM has been developed for calculating the residual stress distribution in D-I can. It is possible for our code to simulate a process of deep drawing from a sheet to a can, a process of both-sided ironing of can wall, and a deformation of the whole of can under perfect unloading when the specimen is released from an equipment of deep drawing and of ironing. The authors simulated a process of deep drawing from a sheet to form a can, and simulated a process of both-sided ironing of a can and obtained distributions of circumferential stress σ θ along a wall thickness on a cross section of can wall. It is found that distributions of the circumferential stress in a case where inner die is higher than outer die is quite different from and contrary to distributions where outer die is higher than inner die. Summation of the hoop stress σ θ along the thickness is tensile in both cases.
[1] T. Aoki, K. Miyauchi, Forming characteristics of ironed cans, Technol. Mater. Test 36 (2) (1991) pp. 36–42. [2] S. Kuriyama, Developing of FEM code for simulating plastic working controlled by rotating elastic bar, Report of Research Summary to Amada Foundation for Metal Work Technology, vol. 11, 1999, pp. 58–63. [3] K.J. Bathe, A Finite Element Program for Automatic Dynamic Incremental Nonlinear Analysis (ADINA), Massachusetts Institute of Technology, Massachusetts, USA, 1975. [4] O.C. Zienkiewicz, The Finite Element Method, McGraw-Hill, New York, 1977. [5] S. Kuriyama, Y. Xu, T. Nakagawa, Development of FE simulation code for stereolithography of resin cured by U.V. ray, in: Proceedings of the Seventh International Conference of NUMIFORM, Toyohashi, Japan, June 2001, pp. 337–342. [6] S. Kuriyama, T. Takahashi, S. Kumagaya, M. Yoshida, Development of FE simulation code for metal forming with and by cracks, first report application of dynamic code to propagation of crack, in: Proceedings of the 2002 Japanese Spring Conference for Technology of Plasticity, 2002, pp. 213–214.