- Email: [email protected]
The system for sheet metal forming design of complex parts
Journal of Materials Processing Technology 157–158 (2004) 502–507
The system for sheet metal forming design of complex parts J. Gronostajski∗ , A. Matuszak, A. Niechajowicz, Z. Zimniak Institute of Production Engineering and Automation, Wrocław University of Technology, ul. Łukasiewicza 3/5, 50-371 Wrocław, Poland
Abstract In the paper the system for sheet metal forming design is described. The system incorporates into finite element method the forming limit stress diagrams and the wrinkling criterion, as the limit conditions of forming and constitutive equations and boundary conditions describing the materials reaction on the complex deformation conditions. Application of the system for sheet metal forming design creates the possibility to design the sheet metal forming processes without expensive and time consuming trail and error techniques, so that the necessity of investigation by using real tools may be reduced or eliminated. The system will able to predict the forming loads, to create the geometry of the tools, to calculate the distribution of strain and stress and to determine the process condition. © 2004 Elsevier B.V. All rights reserved. Keywords: Design system; Sheet metals; Forming operations
1. Introduction Sheet metal forming operations are standards manufacturing processes, which enable to obtain different types of drawpieces. The complexity of these processes leads to numerous techniques to predict or evaluate the formability of the raw materials. The challenge in the last years to the use of the more light, corrosion resistance and safety vehicles has raised the application of new generation of materials. The formability problems encountered in press-forming of sheets of enhanced strength and reduced thickness impose important constraints in manufacturing process. Moreover, the requirements for higher standards of corrosion protection of thinner steel components are leading to an increase in application of pre-coated steel sheets, which impose additional constrains in manufacture. In the last years an extensive effort has been devoted toward the development of analytical tools and computer models capable of simulating sheet material forming processes. The purpose of that is to provide the design engineer with analytical tools for designing at the computer terminal so that ∗
Corresponding author. Tel.: +48 713202173; fax: +48 713203422. E-mail address: [email protected] (J. Gronostajski).
0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2004.07.110
the expensive trail and error process with the real tools may be reduced or eliminated. The system should be capable to predict during process design main factors that limit forming of the sheet as a necking and wrinkling. However, sheet metal forming is an industrial process strongly dependent on numerous interactive variables: material behaviour, lubrication, forming equipment, strain rate, etc. The correct choice of these parameters has appeared as one of the main aims of the automobile industry. Formability analysis of sheet materials is usually evaluated through the concept of forming limit diagrams. The construction of forming limit diagrams for particular materials is basically experimental but some analytical models were also developed [1–9]. The forming limit diagram is difficult to use for sheet metal forming process design because it is strongly dependent on the strain path [10–12]. Considering this fact the forming limit stress diagram instead of forming limit diagram was proposed [13–15]. The forming limit stress diagram is almost independent on the strain path therefore the only one forming limit stress diagram can be used for calculation of different forming limit diagrams at various strain paths. In the numerical simulation of deep drawing operations by using finite element method the forming limit stress diagram
J. Gronostajski et al. / Journal of Materials Processing Technology 157–158 (2004) 502–507
503
can be applied directly as a stop test [16]. The onset of necking occurs when the stress states in any element reaches or oversteps the limit stress curve. The other form of instability is wrinkling. In the last years the danger of wrinkling in forming operations constantly increases by trend to use of thinner sheet metals made from high strength materials. Therefore, the prediction of the onset of wrinkling and implementation of wrinkling criterion into design system is so important. The aim of the work is to develop the modern sheet metal forming processes design technique and diagnostic system.
2. Description of the system The proposed method incorporates into finite element method the forming limit stress diagrams and the wrinkling criterion, as the limit conditions of forming. In the programme constitutive equations and boundary conditions, good describing the materials reaction on the complex deformation conditions are also used. The procedure involves tests for evaluating strain hardening and strains softening behaviour, friction tests, plastic anisotropy and instability in necking and wrinkling. The elaborated system using commercial finite element package MARC becomes available to make realistic analysis, simulation and designing of sheet metal forming processes. That way engineers involved in process design will receive a valuable tool, in the form of computer procedure. The algorithm of the system is shown in Fig. 1. The realisation of the projected work contains the following tasks: • development of the theoretical calculation and experimental verification of forming limit stress diagram, • elaboration of the wrinkling criterion, • elaboration of the method for description of contact conditions, • determination of the constitutive equations, • creation of the mathematical model of selected deep drawing processes for laboratory and industrial application. Forming limit stress diagram was chosen because one curve can determine critical conditions for necking independently on strain path. This is very convenient for analysis of the complex shape parts and for multistage as well as one stage forming operations. The onset of necking is checked when the stress reaches or oversteps the stress curve. The forming limit stress diagram is created on the base of theoretical calculation and experimental research [13–15]. Original and modified M–K theories [1] and perturbation analysis [7] were used for the theoretical calculation of the forming limit stress diagram. In the calculation of the forming limit stress diagram, the following assumptions are taken: • the material is orthotropic, • yielding is described by Hill’s anisotropic plasticity theory,
Fig. 1. The algorithm of the system for sheet metal forming design.
• strain path has significant influence on the formability of materials, • sheet metals is strain hardening and strain rate sensitive. The choice of yield criterion has a large effect on the predicted strain distribution and strain limit [17]. The wrinkling of the sheet metal during forming takes place in the area of the sheet where compressive stress in sheet plane is excessive. For deep drawing process two types of wrinkling may occur. First one is wrinkling of a flat blank and the second one is wrinkling of walls unsupported by tools. The flat blank wrinkling is relatively easy to predict and suppress by blankholder force. Preventing of unsupported by tools wall wrinkling is only possible by change of stress state in the wall in such way that critical condition for wrinkling will not be fulfilled. The knowledge of critical conditions of wrinkling enable designers to optimise sheet metal forming processes. The wrinkling criterion was built on the base of shell theory. The sheet metal buckling during forming operation was considered as the bifurcation type buckling of the shell in elastic–plastic range. As a result the following equation for critical wrinkling stress during sheet forming was obtained 1 h 1/2 σ1cr = √ (L11 L22 − L212 ) R 3 2
(1)
where h is the local thickness, R2 the main curvature radius and Lij the incremental stiffness module.
504
J. Gronostajski et al. / Journal of Materials Processing Technology 157–158 (2004) 502–507
All data needed for calculation of critical stress (1) are determined during FEM analysis of deep drawing process. These data are geometric data (local thickness and local radius of curvature), local effective strain, local effective stress and principal stresses [18]. As a wrinkling criterion a wrinkling factor is used that is defined as: σ1 Wf = (2) σ1cr For Wf < 1 the process should be safe, without wrinkling and for Wf ≥ 1 there is a risk of the folds appearance. The correctness of the simulation of sheet metal forming operations can be considerably restricted by the constitutive equations. Sheet metal forming processes usually are performed at ambient temperature, where the structural softening processes do not take place, so the effect of deformation history can be neglected. In that situation it can be assumed that mathematical models describing the strain hardening curves depend on the strain and strain rate only. For these assumptions the following equations were applied for steel, aluminium and titanium sheets [19,20]: n1
σp = C(ε + ε0 )
ε˙ exp(n2 ε) ˙ε0
(aε+aε2 ) (3)
and for steel sheets [21] σp = C(ε + ε0 )n1
ε˙ ε˙ 0
(aε+aε2 ) (4)
where C, ai are material constants, ε0 the pre-strain, ni the coefficient of strain hardening. These functions give the good fitting both for small and large strain range. In the case of simple parts produced in one operations from the steel sheets also was used Swift equation [22]. The friction greatly influences the material flow and strain distribution therefore it is necessary to use the friction model as close as possible to real conditions for the reliable process simulation. On the base of the theoretical study and analysis of the friction phenomena in sheet metal forming operations the test, based on the single strip pass drawing, has been adopted to simulate friction phenomena. The device can be used to simulate either sliding and concurrent thickness reduction or bending and unbending under tension type strip drawing tests [23]. The simplest configuration of the beads used in friction coefficient test calculation is shown in Fig. 2. The sheet strip was drew with the force FT between two beads clamped together with the force FN . During the test, the strip was elongated to the different degree. The friction coefficient µ was calculated by using following formula [24–26]: µ=
FT − 2FN tan(α/2) 2FN + 1FT tan(α/2)
(5)
where α is the angle of die surface contact with the sheet.
Fig. 2. Scheme for friction coefficient determination.
It has been found that the most influencing factors are rolling direction, sliding velocity, plastic strain, tool radius and lubricant. Conducted experiments show that friction is very complex phenomenon and depends on many factors, so it is difficult to create universal mathematical model that could be used for friction force description. It was assumed that future users of the system for sheet metal forming design should determine the friction coefficients using elaborated testing device for conditions anticipated in the designed process. The elaborated problems was enclosed to FEM program by wrote users-subroutines. The modified post-processor was prepared and that way the forming limit stress diagram and principal stresses are displayed simultaneously with standard FEM results. It makes easier to analyse results of the process modelling. As a result a complete tool for sheet metal forming process analysis was obtained. The FEM analysis was applied to simulate experimentally tested sheet metal forming processes. The forming analysis was performed with MARC K7.3 finite element software package using rigid-plastic flow method. The rigid-plastic method was chosen because this method reduces the total computation time in comparison with the elastic–plastic explicit method. The tooling shape for analysis was converted from Pro/Engineer system into IGES format and meshed preprocessing MENTAT 3.3 program. The detail of performed FEM analysis was presented in the papers [27–33]. The mathematical model of forming processes were elaborated for different steels, aluminium alloys and titanium sheets. All the above-mentioned processes was analysed for different conditions it means: different punch velocities, initial shapes of blank, different blankholder forces and different friction conditions. The calculated results were compared with experimental strain level at necking and the experimental punch travel [34]. The results of simulation for SOLDUR 340 steel for cylindrical cup is shown in Fig. 3 and for L-shape in Fig. 4. Cylindrical cup, the FEM simulations were performed on the 1/4 symmetric area, the initial blank geometry was round shape of 300 mm in diameter. For L-shape, the initial blank geometry was rectangular shape of 255 mm × 200 mm. In both cases, a maximum of 300 increments was carried out in rigid-plastic
J. Gronostajski et al. / Journal of Materials Processing Technology 157–158 (2004) 502–507
505
Fig. 3. Distribution of effective strain in cylindrical cup at punch travel to 43.7 mm and principal stress and components with respect to the FLSD and FLD.
Fig. 4. Distribution of effective strain in L-shape at punch travel of 18 mm and principal stress and strain components with respect to the FLSD and FLD.
506
J. Gronostajski et al. / Journal of Materials Processing Technology 157–158 (2004) 502–507
calculate the distribution of strain and stress and determine the critical process condition.
References
Fig. 5. Wrinkling factor, Wf calculated for different hemispherical punch strokes, h, punch diameter 192 mm, blank diameter 270 mm, blankholder pressure 7.95 MPa.
flow method. During the analysis the punch moves with velocity of 3 mm/s. The figures show the distribution of the equivalent plastic strain in the drawpieces and the principal stress and strain components with respect to the forming limit stress diagram and forming limit diagram, respectively. The benefit of using forming limit stress diagram instead of forming limit diagram is clearly visible by comparison of both the diagrams. On the forming limit diagram strain components show that limit strain is exceeded but in forming limit stress diagram all stress components lie below limit stress. That last result is in good agreement with experimental date. This shows that the forming limit strain for good simulation of the forming processes must be taken just for defined strain path in considered area. But forming limit stress diagram describes generally all the strain paths. Wrinkling factor calculated during simulation of hemispherical part forming is shown in Fig. 4. The wrinkling factor for initial stages of forming is greater than 1 and pointed out the risk of wrinkling. With increasing of the punch travel the factor decreases to low value and then increases again. In experimental observation at the beginning of forming small folds were observed that disappeared with punch travel or unloading. This point out that wrinkling in elastic state take place, but it does not take place during further punch travel in plastic range for conditions described in the caption of Fig. 5. Comparison of results calculated according to elaborated system with experimental results shows satisfactory agreement between them.
3. Conclusion Application of the system for sheet metal forming design creates the possibility to design the sheet metal forming processes without expensive and time consuming trail and error techniques, so that the necessity of investigation by using real tools may be reduced or eliminated. The system will be able to predict the forming loads, create the geometry of the tools,
[1] Z. Marciniak, K. Kuczy´nski, Limit strain in the process of stretch forming sheet, J. Mech. Sci. 9 (1967) 609. [2] J. Gronostajski, Z. Zimniak, Theoretical simulation of sheet behaviour in forming processes, J. Mater. Process Technol. 31 (1992) 57. [3] J. Gronostajski, Z. Zimniak, Prediction of forming limit diagrams with experimentally determined heterogeneity of anisotropic sheets, J. Mater. Process. Technol. 32 (1992) 263. [4] J. Gronostajski, Z. Zimniak, A few methods of analytical calculation of forming-limit curves, J. Mater. Process. Technol. 55 (1996) 213. [5] Z. Zimniak, The system to predict the onset of localised necking in deep drawing processes, in: Proceedings of the International Conference on Advances in Materials and Processing Technologies, AMPT’99, vol. 1, Dublin, 1999, p. 257. [6] Z. Zimniak, Forming limit predictions with the perturbation method and approach in FEM simulations, in: Proceedings of the 8th International Conference on Metal Forming 2000, Krak´ow, 2000, p. 385. [7] D. Dudzi´nski, A. Molinari, Perturbation analysis of thermoviscoplastic instabilities in biaxial loading, Int. J. Solids Struct. 27 (1991) 601. [8] A. Barata da Rocha, E. Barlat, J.M. Jalinier, Prediction of the forming limit diagrams of anisotropic sheet in linear and nonlinear loading, Mater. Sci. Eng. 68 (1985) 151. [9] A. Barata da Rocha, J.M. Jalinier, Development of strain gradient in sheet metal forming processes, in: Proceedings of the International Symposium on Plastic Institute, Ecole Nationale des Ponts et Chauss´ees, Paris, 1985, p. 35. [10] J. Gronostajski, A. Dolny, The influence of strain path on sheet metal forming limits, in: Development in the Drawing of Metals, The Metals Society, London, 1983. [11] J. Gronostajski, Use of new yield criteria determined the effect of strain paths on forming limit diagrams, Adv. Technol. Plasticity 1 (1984) 723. [12] J. Gronostajski, Effect of changing of anisotropy during stamping of sheet steel on forming limit diagram, Adv. Technol. Plasticity 3 (1990) 1173. [13] R. Arrieux, C. Bedrin, M. Boivin, Determination of an intrinsic forming limit stress diagram for isotropic metal sheets, in: Proceedings of the 12th Biennial Congress International of Deep Drawing Research Group, S. Marherita Ligure, 1982, p. 61. [14] J. Gronostajski, Sheet metal forming-limit for complex strain paths, J. Mech. Work. Technol. 10 (1984) 349. [15] J. Gronostajski, Application of limit stresses to determine limit strains at complex strains paths, Archiv. Metall. 30 (1985) 41. [16] Z. Gronostajski, Application of forming limit stress diagram in analysis of sheet metals forming processes, in: Proceedings of the International Conference on Material Formability, Forming’2000, Ustro´n, 2000, p. 89 (in Polish). [17] Z. Gronostajski, The meaning of constitutive equations in construction of forming limit stress diagrams, Obr´obka Plastyczna Metali 11 (2000) 29 (in Polish). [18] A. Niechajowicz, Analysis of wrinkling during deep drawing of hemispherical parts, in: Proceedings of the International Conference on Material Formability, Forming’2000, Ustro´n, 2000, p. 83. [19] Z. Gronostajski, Constitutive equations applied in sheet metal forming processes, in: Proceedings of the International Conference on Advances in Materials and Processing Technologies, AMPT’95, Guimar¨aes, Portugal, 1997, p. 919. [20] Z. Gronostajski, Constitutive equations for sheet metals cold forming, Obr´obka Plastyczna Metali 8 (1997) 19 (in Polish).
J. Gronostajski et al. / Journal of Materials Processing Technology 157–158 (2004) 502–507 [21] Z. Gronostajski, W. Weiler, Material models for analysis of sheet metal forming, Obr´obka Plastyczna Metali 12 (2001) 16 (in Polish). [22] H.W. Swift, Length changes in metals under torsional overstrain, Engineering 163 (1947) 253. [23] A. Matuszak, Determination of the frictional properties of coated steel sheets, J. Mater. Process. Technol. 106 (2000) 107. [24] A. Matuszak, Factors influencing friction in steel sheet forming, J. Mater. Process. Technol. 106 (2000) 250. [25] A. Matuszak, K. Gładysz, Description of friction conditions for computer simulation of sheet metal forming processes, Przegl˛ad Mechaniczny 60 (2001) 31 (in Polish). [26] A. Matuszak, K. Gładysz, Friction problem in sheet metal forming, in: Proceedings of the 8th International Conference on Metal Forming, Krak´ow, 2000, p. 87. [27] J. Gronostajski, Z. Gronostajski, A. Matuszak, A. Niechajowicz, Z. Zimniak, System for sheet metal forming design, Rudy i Metale Nie˙zelazne 42 (1997) 499 (in Polish). [28] J. Gronostajski, Z. Gronostajski, A. Matuszak, A. Niechajowicz, Z. Zimniak, The system for the sheet metal forming design, In˙zynieria Materiałowa 19 (1998) 268.
507
[29] J. Gronostajski, Z. Gronostajski, A. Matuszak, A. Niechajowicz, Z. Zimniak, The system for the sheet metal forming design, in: Proceedings of the International Conference on Material Formability, Forming’2000, Ustro´n, 2000, p. 95. [30] J. Gronostajski, Z. Gronostajski, A. Matuszak, A. Niechajowicz, Z. Zimniak, The system for sheet metal forming design of complex parts, in: Proceedings of the 9th International Science Conference on Achievements in Mechanical and Materials Engineering, AMME’2000, Sopot, 2000, p. 231. [31] J. Gronostajski, Z. Gronostajski, A. Matuszak, A. Niechajowicz, Z. Zimniak, System for sheet metals forming design, Obr´obka Plastyczna Metali 11 (2000) 23 (in Polish). [32] J. Gronostajski, Z. Gronostajski, A. Matuszak, A. Niechajowicz, Z. Zimniak, The system for the sheet metal forming design, Archive Metall. 46 (2001) 87. [33] R. Arieux, J. Gronostajski, Z. Gronostajski, A. Matuszak, A. Niechajowicz, Z. Zimniak, Design methods of sheet metal forming processes, Obr´obka Plastyczna Metali 12 (2001) 12. [34] Z. Zimniak, System for the sheet metal forming design, in: Proceedings of the International Conference on Sheet Metal’99, Erlangen, 1999, p. 99.
The system for sheet metal forming design of complex parts J. Gronostajski∗ , A. Matuszak, A. Niechajowicz, Z. Zimniak Institute of Production Engineering and Automation, Wrocław University of Technology, ul. Łukasiewicza 3/5, 50-371 Wrocław, Poland
Abstract In the paper the system for sheet metal forming design is described. The system incorporates into finite element method the forming limit stress diagrams and the wrinkling criterion, as the limit conditions of forming and constitutive equations and boundary conditions describing the materials reaction on the complex deformation conditions. Application of the system for sheet metal forming design creates the possibility to design the sheet metal forming processes without expensive and time consuming trail and error techniques, so that the necessity of investigation by using real tools may be reduced or eliminated. The system will able to predict the forming loads, to create the geometry of the tools, to calculate the distribution of strain and stress and to determine the process condition. © 2004 Elsevier B.V. All rights reserved. Keywords: Design system; Sheet metals; Forming operations
1. Introduction Sheet metal forming operations are standards manufacturing processes, which enable to obtain different types of drawpieces. The complexity of these processes leads to numerous techniques to predict or evaluate the formability of the raw materials. The challenge in the last years to the use of the more light, corrosion resistance and safety vehicles has raised the application of new generation of materials. The formability problems encountered in press-forming of sheets of enhanced strength and reduced thickness impose important constraints in manufacturing process. Moreover, the requirements for higher standards of corrosion protection of thinner steel components are leading to an increase in application of pre-coated steel sheets, which impose additional constrains in manufacture. In the last years an extensive effort has been devoted toward the development of analytical tools and computer models capable of simulating sheet material forming processes. The purpose of that is to provide the design engineer with analytical tools for designing at the computer terminal so that ∗
Corresponding author. Tel.: +48 713202173; fax: +48 713203422. E-mail address: [email protected] (J. Gronostajski).
0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2004.07.110
the expensive trail and error process with the real tools may be reduced or eliminated. The system should be capable to predict during process design main factors that limit forming of the sheet as a necking and wrinkling. However, sheet metal forming is an industrial process strongly dependent on numerous interactive variables: material behaviour, lubrication, forming equipment, strain rate, etc. The correct choice of these parameters has appeared as one of the main aims of the automobile industry. Formability analysis of sheet materials is usually evaluated through the concept of forming limit diagrams. The construction of forming limit diagrams for particular materials is basically experimental but some analytical models were also developed [1–9]. The forming limit diagram is difficult to use for sheet metal forming process design because it is strongly dependent on the strain path [10–12]. Considering this fact the forming limit stress diagram instead of forming limit diagram was proposed [13–15]. The forming limit stress diagram is almost independent on the strain path therefore the only one forming limit stress diagram can be used for calculation of different forming limit diagrams at various strain paths. In the numerical simulation of deep drawing operations by using finite element method the forming limit stress diagram
J. Gronostajski et al. / Journal of Materials Processing Technology 157–158 (2004) 502–507
503
can be applied directly as a stop test [16]. The onset of necking occurs when the stress states in any element reaches or oversteps the limit stress curve. The other form of instability is wrinkling. In the last years the danger of wrinkling in forming operations constantly increases by trend to use of thinner sheet metals made from high strength materials. Therefore, the prediction of the onset of wrinkling and implementation of wrinkling criterion into design system is so important. The aim of the work is to develop the modern sheet metal forming processes design technique and diagnostic system.
2. Description of the system The proposed method incorporates into finite element method the forming limit stress diagrams and the wrinkling criterion, as the limit conditions of forming. In the programme constitutive equations and boundary conditions, good describing the materials reaction on the complex deformation conditions are also used. The procedure involves tests for evaluating strain hardening and strains softening behaviour, friction tests, plastic anisotropy and instability in necking and wrinkling. The elaborated system using commercial finite element package MARC becomes available to make realistic analysis, simulation and designing of sheet metal forming processes. That way engineers involved in process design will receive a valuable tool, in the form of computer procedure. The algorithm of the system is shown in Fig. 1. The realisation of the projected work contains the following tasks: • development of the theoretical calculation and experimental verification of forming limit stress diagram, • elaboration of the wrinkling criterion, • elaboration of the method for description of contact conditions, • determination of the constitutive equations, • creation of the mathematical model of selected deep drawing processes for laboratory and industrial application. Forming limit stress diagram was chosen because one curve can determine critical conditions for necking independently on strain path. This is very convenient for analysis of the complex shape parts and for multistage as well as one stage forming operations. The onset of necking is checked when the stress reaches or oversteps the stress curve. The forming limit stress diagram is created on the base of theoretical calculation and experimental research [13–15]. Original and modified M–K theories [1] and perturbation analysis [7] were used for the theoretical calculation of the forming limit stress diagram. In the calculation of the forming limit stress diagram, the following assumptions are taken: • the material is orthotropic, • yielding is described by Hill’s anisotropic plasticity theory,
Fig. 1. The algorithm of the system for sheet metal forming design.
• strain path has significant influence on the formability of materials, • sheet metals is strain hardening and strain rate sensitive. The choice of yield criterion has a large effect on the predicted strain distribution and strain limit [17]. The wrinkling of the sheet metal during forming takes place in the area of the sheet where compressive stress in sheet plane is excessive. For deep drawing process two types of wrinkling may occur. First one is wrinkling of a flat blank and the second one is wrinkling of walls unsupported by tools. The flat blank wrinkling is relatively easy to predict and suppress by blankholder force. Preventing of unsupported by tools wall wrinkling is only possible by change of stress state in the wall in such way that critical condition for wrinkling will not be fulfilled. The knowledge of critical conditions of wrinkling enable designers to optimise sheet metal forming processes. The wrinkling criterion was built on the base of shell theory. The sheet metal buckling during forming operation was considered as the bifurcation type buckling of the shell in elastic–plastic range. As a result the following equation for critical wrinkling stress during sheet forming was obtained 1 h 1/2 σ1cr = √ (L11 L22 − L212 ) R 3 2
(1)
where h is the local thickness, R2 the main curvature radius and Lij the incremental stiffness module.
504
J. Gronostajski et al. / Journal of Materials Processing Technology 157–158 (2004) 502–507
All data needed for calculation of critical stress (1) are determined during FEM analysis of deep drawing process. These data are geometric data (local thickness and local radius of curvature), local effective strain, local effective stress and principal stresses [18]. As a wrinkling criterion a wrinkling factor is used that is defined as: σ1 Wf = (2) σ1cr For Wf < 1 the process should be safe, without wrinkling and for Wf ≥ 1 there is a risk of the folds appearance. The correctness of the simulation of sheet metal forming operations can be considerably restricted by the constitutive equations. Sheet metal forming processes usually are performed at ambient temperature, where the structural softening processes do not take place, so the effect of deformation history can be neglected. In that situation it can be assumed that mathematical models describing the strain hardening curves depend on the strain and strain rate only. For these assumptions the following equations were applied for steel, aluminium and titanium sheets [19,20]: n1
σp = C(ε + ε0 )
ε˙ exp(n2 ε) ˙ε0
(aε+aε2 ) (3)
and for steel sheets [21] σp = C(ε + ε0 )n1
ε˙ ε˙ 0
(aε+aε2 ) (4)
where C, ai are material constants, ε0 the pre-strain, ni the coefficient of strain hardening. These functions give the good fitting both for small and large strain range. In the case of simple parts produced in one operations from the steel sheets also was used Swift equation [22]. The friction greatly influences the material flow and strain distribution therefore it is necessary to use the friction model as close as possible to real conditions for the reliable process simulation. On the base of the theoretical study and analysis of the friction phenomena in sheet metal forming operations the test, based on the single strip pass drawing, has been adopted to simulate friction phenomena. The device can be used to simulate either sliding and concurrent thickness reduction or bending and unbending under tension type strip drawing tests [23]. The simplest configuration of the beads used in friction coefficient test calculation is shown in Fig. 2. The sheet strip was drew with the force FT between two beads clamped together with the force FN . During the test, the strip was elongated to the different degree. The friction coefficient µ was calculated by using following formula [24–26]: µ=
FT − 2FN tan(α/2) 2FN + 1FT tan(α/2)
(5)
where α is the angle of die surface contact with the sheet.
Fig. 2. Scheme for friction coefficient determination.
It has been found that the most influencing factors are rolling direction, sliding velocity, plastic strain, tool radius and lubricant. Conducted experiments show that friction is very complex phenomenon and depends on many factors, so it is difficult to create universal mathematical model that could be used for friction force description. It was assumed that future users of the system for sheet metal forming design should determine the friction coefficients using elaborated testing device for conditions anticipated in the designed process. The elaborated problems was enclosed to FEM program by wrote users-subroutines. The modified post-processor was prepared and that way the forming limit stress diagram and principal stresses are displayed simultaneously with standard FEM results. It makes easier to analyse results of the process modelling. As a result a complete tool for sheet metal forming process analysis was obtained. The FEM analysis was applied to simulate experimentally tested sheet metal forming processes. The forming analysis was performed with MARC K7.3 finite element software package using rigid-plastic flow method. The rigid-plastic method was chosen because this method reduces the total computation time in comparison with the elastic–plastic explicit method. The tooling shape for analysis was converted from Pro/Engineer system into IGES format and meshed preprocessing MENTAT 3.3 program. The detail of performed FEM analysis was presented in the papers [27–33]. The mathematical model of forming processes were elaborated for different steels, aluminium alloys and titanium sheets. All the above-mentioned processes was analysed for different conditions it means: different punch velocities, initial shapes of blank, different blankholder forces and different friction conditions. The calculated results were compared with experimental strain level at necking and the experimental punch travel [34]. The results of simulation for SOLDUR 340 steel for cylindrical cup is shown in Fig. 3 and for L-shape in Fig. 4. Cylindrical cup, the FEM simulations were performed on the 1/4 symmetric area, the initial blank geometry was round shape of 300 mm in diameter. For L-shape, the initial blank geometry was rectangular shape of 255 mm × 200 mm. In both cases, a maximum of 300 increments was carried out in rigid-plastic
J. Gronostajski et al. / Journal of Materials Processing Technology 157–158 (2004) 502–507
505
Fig. 3. Distribution of effective strain in cylindrical cup at punch travel to 43.7 mm and principal stress and components with respect to the FLSD and FLD.
Fig. 4. Distribution of effective strain in L-shape at punch travel of 18 mm and principal stress and strain components with respect to the FLSD and FLD.
506
J. Gronostajski et al. / Journal of Materials Processing Technology 157–158 (2004) 502–507
calculate the distribution of strain and stress and determine the critical process condition.
References
Fig. 5. Wrinkling factor, Wf calculated for different hemispherical punch strokes, h, punch diameter 192 mm, blank diameter 270 mm, blankholder pressure 7.95 MPa.
flow method. During the analysis the punch moves with velocity of 3 mm/s. The figures show the distribution of the equivalent plastic strain in the drawpieces and the principal stress and strain components with respect to the forming limit stress diagram and forming limit diagram, respectively. The benefit of using forming limit stress diagram instead of forming limit diagram is clearly visible by comparison of both the diagrams. On the forming limit diagram strain components show that limit strain is exceeded but in forming limit stress diagram all stress components lie below limit stress. That last result is in good agreement with experimental date. This shows that the forming limit strain for good simulation of the forming processes must be taken just for defined strain path in considered area. But forming limit stress diagram describes generally all the strain paths. Wrinkling factor calculated during simulation of hemispherical part forming is shown in Fig. 4. The wrinkling factor for initial stages of forming is greater than 1 and pointed out the risk of wrinkling. With increasing of the punch travel the factor decreases to low value and then increases again. In experimental observation at the beginning of forming small folds were observed that disappeared with punch travel or unloading. This point out that wrinkling in elastic state take place, but it does not take place during further punch travel in plastic range for conditions described in the caption of Fig. 5. Comparison of results calculated according to elaborated system with experimental results shows satisfactory agreement between them.
3. Conclusion Application of the system for sheet metal forming design creates the possibility to design the sheet metal forming processes without expensive and time consuming trail and error techniques, so that the necessity of investigation by using real tools may be reduced or eliminated. The system will be able to predict the forming loads, create the geometry of the tools,
[1] Z. Marciniak, K. Kuczy´nski, Limit strain in the process of stretch forming sheet, J. Mech. Sci. 9 (1967) 609. [2] J. Gronostajski, Z. Zimniak, Theoretical simulation of sheet behaviour in forming processes, J. Mater. Process Technol. 31 (1992) 57. [3] J. Gronostajski, Z. Zimniak, Prediction of forming limit diagrams with experimentally determined heterogeneity of anisotropic sheets, J. Mater. Process. Technol. 32 (1992) 263. [4] J. Gronostajski, Z. Zimniak, A few methods of analytical calculation of forming-limit curves, J. Mater. Process. Technol. 55 (1996) 213. [5] Z. Zimniak, The system to predict the onset of localised necking in deep drawing processes, in: Proceedings of the International Conference on Advances in Materials and Processing Technologies, AMPT’99, vol. 1, Dublin, 1999, p. 257. [6] Z. Zimniak, Forming limit predictions with the perturbation method and approach in FEM simulations, in: Proceedings of the 8th International Conference on Metal Forming 2000, Krak´ow, 2000, p. 385. [7] D. Dudzi´nski, A. Molinari, Perturbation analysis of thermoviscoplastic instabilities in biaxial loading, Int. J. Solids Struct. 27 (1991) 601. [8] A. Barata da Rocha, E. Barlat, J.M. Jalinier, Prediction of the forming limit diagrams of anisotropic sheet in linear and nonlinear loading, Mater. Sci. Eng. 68 (1985) 151. [9] A. Barata da Rocha, J.M. Jalinier, Development of strain gradient in sheet metal forming processes, in: Proceedings of the International Symposium on Plastic Institute, Ecole Nationale des Ponts et Chauss´ees, Paris, 1985, p. 35. [10] J. Gronostajski, A. Dolny, The influence of strain path on sheet metal forming limits, in: Development in the Drawing of Metals, The Metals Society, London, 1983. [11] J. Gronostajski, Use of new yield criteria determined the effect of strain paths on forming limit diagrams, Adv. Technol. Plasticity 1 (1984) 723. [12] J. Gronostajski, Effect of changing of anisotropy during stamping of sheet steel on forming limit diagram, Adv. Technol. Plasticity 3 (1990) 1173. [13] R. Arrieux, C. Bedrin, M. Boivin, Determination of an intrinsic forming limit stress diagram for isotropic metal sheets, in: Proceedings of the 12th Biennial Congress International of Deep Drawing Research Group, S. Marherita Ligure, 1982, p. 61. [14] J. Gronostajski, Sheet metal forming-limit for complex strain paths, J. Mech. Work. Technol. 10 (1984) 349. [15] J. Gronostajski, Application of limit stresses to determine limit strains at complex strains paths, Archiv. Metall. 30 (1985) 41. [16] Z. Gronostajski, Application of forming limit stress diagram in analysis of sheet metals forming processes, in: Proceedings of the International Conference on Material Formability, Forming’2000, Ustro´n, 2000, p. 89 (in Polish). [17] Z. Gronostajski, The meaning of constitutive equations in construction of forming limit stress diagrams, Obr´obka Plastyczna Metali 11 (2000) 29 (in Polish). [18] A. Niechajowicz, Analysis of wrinkling during deep drawing of hemispherical parts, in: Proceedings of the International Conference on Material Formability, Forming’2000, Ustro´n, 2000, p. 83. [19] Z. Gronostajski, Constitutive equations applied in sheet metal forming processes, in: Proceedings of the International Conference on Advances in Materials and Processing Technologies, AMPT’95, Guimar¨aes, Portugal, 1997, p. 919. [20] Z. Gronostajski, Constitutive equations for sheet metals cold forming, Obr´obka Plastyczna Metali 8 (1997) 19 (in Polish).
J. Gronostajski et al. / Journal of Materials Processing Technology 157–158 (2004) 502–507 [21] Z. Gronostajski, W. Weiler, Material models for analysis of sheet metal forming, Obr´obka Plastyczna Metali 12 (2001) 16 (in Polish). [22] H.W. Swift, Length changes in metals under torsional overstrain, Engineering 163 (1947) 253. [23] A. Matuszak, Determination of the frictional properties of coated steel sheets, J. Mater. Process. Technol. 106 (2000) 107. [24] A. Matuszak, Factors influencing friction in steel sheet forming, J. Mater. Process. Technol. 106 (2000) 250. [25] A. Matuszak, K. Gładysz, Description of friction conditions for computer simulation of sheet metal forming processes, Przegl˛ad Mechaniczny 60 (2001) 31 (in Polish). [26] A. Matuszak, K. Gładysz, Friction problem in sheet metal forming, in: Proceedings of the 8th International Conference on Metal Forming, Krak´ow, 2000, p. 87. [27] J. Gronostajski, Z. Gronostajski, A. Matuszak, A. Niechajowicz, Z. Zimniak, System for sheet metal forming design, Rudy i Metale Nie˙zelazne 42 (1997) 499 (in Polish). [28] J. Gronostajski, Z. Gronostajski, A. Matuszak, A. Niechajowicz, Z. Zimniak, The system for the sheet metal forming design, In˙zynieria Materiałowa 19 (1998) 268.
507
[29] J. Gronostajski, Z. Gronostajski, A. Matuszak, A. Niechajowicz, Z. Zimniak, The system for the sheet metal forming design, in: Proceedings of the International Conference on Material Formability, Forming’2000, Ustro´n, 2000, p. 95. [30] J. Gronostajski, Z. Gronostajski, A. Matuszak, A. Niechajowicz, Z. Zimniak, The system for sheet metal forming design of complex parts, in: Proceedings of the 9th International Science Conference on Achievements in Mechanical and Materials Engineering, AMME’2000, Sopot, 2000, p. 231. [31] J. Gronostajski, Z. Gronostajski, A. Matuszak, A. Niechajowicz, Z. Zimniak, System for sheet metals forming design, Obr´obka Plastyczna Metali 11 (2000) 23 (in Polish). [32] J. Gronostajski, Z. Gronostajski, A. Matuszak, A. Niechajowicz, Z. Zimniak, The system for the sheet metal forming design, Archive Metall. 46 (2001) 87. [33] R. Arieux, J. Gronostajski, Z. Gronostajski, A. Matuszak, A. Niechajowicz, Z. Zimniak, Design methods of sheet metal forming processes, Obr´obka Plastyczna Metali 12 (2001) 12. [34] Z. Zimniak, System for the sheet metal forming design, in: Proceedings of the International Conference on Sheet Metal’99, Erlangen, 1999, p. 99.