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The strain gradient approach to predict necking in tube hydroforming
Journal of Manufacturing Processes 15 (2013) 51–55
Contents lists available at SciVerse ScienceDirect
Journal of Manufacturing Processes journal homepage: www.elsevier.com/locate/manpro
Technical Paper
The strain gradient approach to predict necking in tube hydroforming Ramin Hashemi a,∗ , Karen Abrinia a , Ahmad Assempour b a b
Mechanical Engineering Department, College of Engineering, University of Tehran, Tehran, Iran Center of Excellence in Design, Robotics and Automation, Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 2 February 2012 Received in revised form 30 October 2012 Accepted 1 November 2012 Available online 21 December 2012 Keywords: Numerical Model Forming
a b s t r a c t A stress-based forming limit diagram for necking prediction which is based on the strain gradient theory of plasticity in conjunction with the M–K model has been represented and used in tube hydroforming. In this study, the finite element model for bulge forming of straight tube has been constructed and verified with published experimental data. The adaptive simulation technique is based on the ability to detect the onset and growth of defects (e.g., wrinkling, and bursting) and to promptly readjust the loading paths. Thus, a suitable load path has been obtained by applying Adaptive Simulation Method in ANSYS Parametric Design Language (APDL). © 2012 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.
1. Introduction The Tube HydroForming (THF) process is a relatively complex manufacturing process. The THF technique offers some advantages as compared to conventional methods in production of complex parts. It could produce lightweight parts and reduce the manufacturing costs [1–3]. However, failures such as buckling, necking could occur in the tube hydroforming process. The causes of such defects are mainly due to wrong loading conditions in applying the fluid pressure and axial feeding simultaneously. Since bursting in THF processes is the consequence of necking, prediction of necking initiation is an important issue before designing the details of the processes [4]. In Ref. [5], the adaptive simulation concept is presented as an effective finite element method (FEM) approach, able to select a feasible THF loading path within a minimum number of simulation runs or even within a single run. The adaptive simulation technique is based on the ability to detect the onset and growth of defects (wrinkling, buckling, bursting) and to promptly readjust the loading paths. The detection and quantitative evaluation of wrinkles play a key role in the implementation of the adaptive simulation. A new geometrical wrinkle indicator is introduced and evaluated with numerical and experimental evidences. The proposed wrinkle indicator is computationally inexpensive, suitable for many die shapes and sensitive to different kinds of wrinkles.
∗ Corresponding author. E-mail addresses: [email protected], [email protected] (R. Hashemi), [email protected] (K. Abrinia), [email protected] (A. Assempour).
A classification of THF processes based on sensitivity to loading parameters has been suggested in Ref. [6]. Gao et al. [6] analyzed the characteristics of the classification in terms of failure mode, dominant loading parameters and their working windows. It is considered that the so-called pressure dominant THF process is the most difficult process for both simulation in FEM analysis and practical operation in real manufacturing situation. Adaptive FEM simulation strategies are mostly needed to effectively determine out the optimum loading path for pressure dominant THF process. It is found in the literature that path-independent stress-based Forming Limit Diagram (FLSD) can be used well for all forming processes without concerning for strain path effects [7]. In addition, it is reported that the path-dependent nature of the strain-based FLD causes the formability prediction method to become ineffective in the analysis of complex forming process, especially THF [8]. There are several works [9–11] which used the conventional M–K model [12] for prediction of sheet metal forming limits. Detailed studies on these approaches using both deformation and flow theory of plasticity reveal a deficiency of the localized necking analysis associated with imperfection-sensitivity. Moreover, they cannot give a detailed description of the profile, especially after initiation of localized necking. In order to obtain more accurate predictions for the limit strain and stress, a more suitable constitutive relation is needed. Such a constitutive relation was advanced by Aifantis and co-workers [13–15] by incorporating higher order strain gradient terms into the flow stress or the yield condition. The higher order strain gradient terms have a significant effect in the localized deformation region, but they are unimportant in the uniform deformation regime. This allows for the use of a unified constitutive equation to describe the different material behavior inside and outside the localization region.
1526-6125/$ – see front matter © 2012 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jmapro.2012.11.001
52
R. Hashemi et al. / Journal of Manufacturing Processes 15 (2013) 51–55
Based on Aifantis’s strain gradient theory [13], Wang et al. [16] and Safikhani et al. [17] incorporated the strain gradient theory into the M–K model to analyze the deformation localization and to predict the corresponding FLDs and FLSDs. The higher order strain gradient (second order strain gradient) was used in the power law hardening equation to describe the different material properties inside and outside the localized region. They followed Hill’s criterion [18] and simplified this problem by assuming that the localized necking initiates along the zero extension direction for the left hand side of the FLD and FLSD. For obtaining one point of FLD, it is required to consider all groove angles between 0 and 90◦ . The limit strain as a candidate point on FLD is the minimum of all calculated limit strains for all groove angles. As shown in Ref. [19], on right hand side of FLD, the candidate limit strains occur at zero groove angles. It means for computation of limit strain at right hand side of FLD, it is enough to accomplish the calculation only for zero groove angles. In this case the groove direction is normal to the rolling direction. But on the left hand side of FLD, depends on the material constants, local neck occurs at zero extension. At this direction, one of the non-shear strain components becomes zero. The FLD calculation procedure for left and right hand sides is simplified. On the right hand side of FLD, limit strains are obtained for zero groove angles. On the left hand side of FLD, first the groove angle is calculated and then the limit strain for this angle is obtained. The localized band was assumed to initiate along the minor strain direction for positive strain ratios as proposed by Marciniak and Kuczynski. This approach introduces an internal length scale into the constitutive equations and takes into account the effects of deformation inhomogeneity. It overcomes the imperfection sensitivity encountered in the conventional M–K method. The nonlinear differential equation of the thickness variation is the second order ordinary type and it has been solved by collocation method [17].In this work, the stress-based hydroforming limits are used to predict necking. The methodology for computation of FLSD is based on the strain gradient theory of plasticity in conjunction with the M–K model. This approach introduces an internal length scale into conventional constitutive equations and takes into account the effects of deformation inhomogeneity and material softening. The nonlinear second order ordinary differential equation of the thickness of tube has been solved by the shooting method. The shooting method was first presented by Wang et al. [16] and later developed by Safikhani et al. [17]. The verification of the theoretical results has been done. Ultimately, the suitable load path for bulge forming of straight tube has been determined by applying adaptive simulation method in ANSYS Parametric Design Language (APDL). 2. Review of the sheet metal computations In Marciniak and Kuczynski (M–K) model [12], it is assumed that there is a shallow groove on material surface which causes the localized necking. The initial imperfection factor of the groove f0 is defined as the thickness ratio f0 = hB /hA , where h denotes the material thickness and subscript “0” denotes the initial state. Groove region is called “B” and safe zone is named “A”. A major problem to the conventional M–K model is that the calculated results are so sensitive to the initial value of imperfection factor and the conventional M–K method cannot explain post-localization behavior. In this method to analyze the deformation throughout the material, the constitutive relation for the effective or flow stress should account for both homogeneous and non-homogeneous deformation states. To account for the strain gradient effect corresponding to the initiation of the strain localization, a gradient dependent flow stress has been used which was proposed by Aifantis [13]. In this study, for simplicity and mathematical reasons, the strain gradient plasticity presented by Aifantis and co workers [13–15]
has been used. So, the second order terms are incorporated into the power law isotropic hardening constitutive equation to describe the material behavior during the post-localization process. The constitutive equation used is [20,21]:
2
¯ = 0 (¯ε0 + ε¯ )n ε¯˙ − c1 ∇ ε¯ − c2 ∇ 2 ε¯ m
(1)
where c1 and c2 are the strain gradient coefficients. As shown in Refs. [16,17], the effect of the strain gradient increases when coefficient c2 is increased. This coefficient represents a property of material and can be determined by experiment. If in this equation, it is supposed that c1 and c2 are equal to zero, it reduces to the conventional relationship between effective strain and effective stress (Swift relation): m
¯ = 0 (¯ε0 + ε¯ )n ε¯˙
(2)
Considering a material to plane stress and having a deformation in the form of a neck which is initially inclined at an angle . For consideration of strain gradient effects and triggering the evolution of necking, the authors suppose the continuous function for sheet thickness as a function of sheet length in the neck direction [16]. The initial thickness of the sheet varies along the x-direction according to the relation: hB0[i] =
hA0 + hB[0] 2
+
hA0 − hB[0] 2
cos
1+
xi w0
(3)
where hB[0] , hB and hA0 are the initial sheet thickness in the imperfection center, at position xi and in the uniform region respectively, as shown in the following figure (e.g., Fig. 1b). w0 represents the initial half width of neck. Assuming the load in the direction perpendicular to the neck (y-direction) is Fx . According to the equilibrium condition in the direction perpendicular to the neck, load in the uniform region (A) should be equal to the load at any cross-section inside the neck (B), i.e. FxA = FxB . By using of Hill’s yield criterion for anisotropic material and Levy–Mises flow rule, and the definition of effective stress and strain, the conditions for calculation of neck evolution and limit strain are as follows [17]: hA A
0 (¯ε0 + ε¯ A )
n
A
−1 1 + ˇA
h˙ A hA
m
⎧ m ⎫ ⎪ ⎪ ˙ ⎪ ⎪ h −1 ⎨ 0 (¯ε0 + ε¯ B )n B
⎬ B hB h B 1 + ˇB = B ⎪ ⎪ ⎪ ⎪ ⎩ −c ∇ ε¯ 3 − c ∇ 2 ε¯ ⎭ B B 1 2
(4)
where is ratio of effective strain rate to strain rate in strain path, is ratio of effective stress to stress in strain path and 0 is assumed the same inside and outside the imperfection. For each element, the neck thickness is calculated based on (4). The critical neck shape obtained under different loading conditions is used as the criterion to determine the forming limit diagram. This is in contrast to the use of a limit strain value employed by conventional M–K methods. In connection with the study of neck evolution, the effects of initial imperfection, both with and without considering the strain gradient effects, are discussed in a previous work [17]. The detailed description about strain gradient theory of plasticity, analysis of neck evolution, consideration of the strain gradient effects on the left and right hand sides of FLSD and the shooting solution of neck evolution in this work can be found in Ref. [17]. In a previous work [17] the necking strain value was defined as follows:for the right hand side of FLD: ε1 = ln
(w − w) t (wt − w0 )
,
(5)
R. Hashemi et al. / Journal of Manufacturing Processes 15 (2013) 51–55
53
FLSDs , STKM-11A tube , Tube hydroforming , Hill 1948 , f =0.96 0
1800 1600
Major Stress σ 1 (MPa)
1400 1200 1000 800 600
experimental data [22]
400
conventional M-K method (present work)
200
diffuse neck ing criterion [22]
0 -600
-400
-200
0 200 400 600 Minor Stress σ2 (MPa)
800
1000
1200
Fig. 2. Hydroforming stress limit diagrams.
Step 3: Calculate the effective stress ¯ by using the constitutive equation which is expressed by Eq. (1). In this equation it is supposed that c1 and c2 are equal to zero. Step 4: Calculate stress ratio (˛) according to respective strain ratio (ˇ). The relation between ˛ and ˇ is expressed as follows: ˛=
(1 + R)ˇ + R (1 + R) + Rˇ
(8)
where R is anisotropy coefficient. Step 5: Calculate , ratio of effective stress to stress in strain path, by using the following equation: √ = (/ ¯ 1 ) = (1/ 2) 4 3/2R + 1 (1 + R) ˛2 − 2R˛ + (1 + R) (9) Fig. 1. Schematic diagram of the imperfection model b. Schematic diagram of the neck cross-section in x–z [17].
Step 6: The major and minor stresses can be expressed as follows: 1 =
2 = ˛1
for the left hand side of FLD: ε1 =
ln((wt − w)/(wt − w0 )) , (1 + ˇ12 )
(6)
where w0 is the initial half neck width, w is half neck width after deformation, wt is total half neck width after deformation and ˇ12 = ε˙ 2 /ε˙ 1 is the proportional strain ratio applied to the uniform region. These equations should be used as a necking strain to construct the forming limit diagram (FLD) [19]. By using Eqs. (5) and (6), the major strain, here (ε1 ), has been calculated for right hand side (RHS) and left hand side (LHS) of the FLD. And the amount of strain ratio ˇ12 (proportional loading) is known. Thus, the minor strain, here (ε2 ), can be calculated. To obtain FLSD, the calculations must be done to convert strains to stresses. The steps of computations in this procedure are summarized as the following [17]: Step 1: Obtain theoretical forming limits in strain form (ε1 , ε2 ) from Eqs. (5) and (6). Step 2: Calculate the effective strain ε¯ by using the following equation: ε¯ =
ij ε˙ ij ¯
¯
(7)
Reader could see the detailed explanations for this equation in Ref. [17].
(10) (11)
When the strain gradient effect is included, this approach overcomes the imperfection sensitivity encountered in the conventional M–K method and instability analysis is performed by considering the neck behavior in both pre- and post-localization regimes. 3. Results and discussions In this paper, to verify the stress-based forming limit diagram predicted by the proposed approach, the predicted stress-based FLD has been compared with hydroforming stress limit diagram determined experimentally in Ref. [22]. In this calculation properties of AK steel alloy [17] have been used. The tube material is assumed to obey the stress–strain relationship ¯ Y = 1400¯ε0.17 (MPa) and with normal anisotropy value r = 2.14. This verification has been shown in Fig. 2 and there is good agreement between theoretical results and experimental data. The adaptive simulation is an approach to obtain the load path with minimum defects concerning the wrinkle and neck formations in tube hydroforming process [23,24]. Applying ANSYS LS-DYNA and ANSYS Parametric Design Language (APDL), an adaptive simulation was carried out. The general algorithm of adaptive simulation method can be found in Ref. [10]. A wrinkle predictor and a necking indicator are two essential aspects in any adaptive simulation design system.
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R. Hashemi et al. / Journal of Manufacturing Processes 15 (2013) 51–55
Table 1 Die geometry of models. Materials
OD1 (mm)
OD2 (mm)
L1 (mm)
L2 (mm)
L3 (mm)
R1 (mm)
R2 (mm)
AISI 304 [27] AK steel [17]
48.6 60
64 145
120 50
25 0
150 95
3.4 12.5
7.1 10
End Feed (mm)
20 15 10 Load path… 5 0 0
100
200
300
Pressure (MPa) Fig. 4. The load path [10,27] used for verification of the current FE model. Fig. 3. The geometry of die and tube in BULGE HYDROFORMING [10].
AISI 304 [27] AK steel [17]
n 0.615 0.245
10 8 6 4 2
Current Work data
0
Experimental Data
-2 0
0.05
0.1
0.15
0.2
0.25
Longitudinal length (m) Fig. 5. Comparison of tube wall thinning between the current simulation and experimental data [27] for AISI 304.
path is decreased and more material is fed into the bulging area. This procedure continues until reaching the necking condition. The distribution of Von-Mises stress in deformed tube and thickness distribution along the tube are shown in Figs. 7 and 8. As could be seen, the minimum thickness at bulging zone is not very critical and no wrinkling occurred in the deformed tube and therefore adaptive simulation method could provide the correct load path successfully. In this study, the strain gradient coefficients (c1 and c2) were obtained experimentally. The same experimental data as in Refs. [17,20,28,29] has been used when the strain gradient coefficients 35 30 25 20 15 Load path
10 5
Table 2 Mechanical properties of tube. Materials
12
Pressure (MPa)
An indicator is needed for optimization of the process concerning wrinkle formation [25,26]. In the present paper, based on die geometry, the slope wrinkling indicator has been used. Considering an axial line on the tube, once the line slope at a particular point changes its sign, there is a wrinkle creation potential at that point. More explanation can be found about this phenomenon in Ref. [10]. The proposed stress-based forming limit approach has been adopted for evaluation of forming severity and occurrence of the necking phenomena. The stress-based limit used in this study reduces the sensitivity of the limit curve to the strain path in the bulging process. While the tube has not reached the tool surface, the tube is under plane stress condition and the present stress-based forming limit diagram can be used as a criterion for bursting failure. As a case study, an adaptive simulation method was implemented in APDL to determine suitable load path for tube hydroforming. Using this load path, suitable formability without wrinkling and bursting failures has been obtained. Theoretically FLSD has been utilized as a criterion for necking prediction. Tool and tube model were made in ANSYS LS-DYNA. A view of tube and die is shown in Fig. 3. The tool is modeled as a rigid body. The tube material is transversely anisotropic elastic-plastic. Tables 1 and 2 display the geometrical parameters and mechanical properties used in this case study. The coulomb friction coefficient of 0.05 was considered in the simulation. The experimental load path in Ref. [27], shown in Fig. 4, is used to verify the present model for material model AISI 304. The result is shown in Fig. 5. As the results show, good agreement with experimental data has been achieved. Therefore, the finite element model can be used with adaptive simulation to obtain suitable load path. Fig. 6 displays the load path obtained from adaptive simulation for AK steel alloys [17]. In this system, at first the pressure and axial end feed are increased by small amounts. Then, the pressure remains constant and only the axial feed is increased. This process continues until wrinkling occurs. To prevent the wrinkling, the pressure and axial end feed are increased simultaneously in a linear path. As thinning occurs in the bulging zone, the slope of the load
Tube wall thining (%)
14
0
ε0
0 (MPa)
R
0.03 0.01
1555 186.525
1 1.2
Thickness (mm) 2.2 1.0
c2 0.0 0.1
0
10
20
30
Axial end feed (mm) Fig. 6. Load path determined by adaptive simulation method (present work).
R. Hashemi et al. / Journal of Manufacturing Processes 15 (2013) 51–55
Tube thickness (mm)
Fig. 7. Von-Mises stress distribution in tube after free bulging (present work).
1.6 1.2 0.8 Thickness along axial line
0.4 0 0
100
200
Distance along one axial line (mm) Fig. 8. Thickness distribution along axial line (present work).
have been considered. Verification of the present approach by performing experiments with different materials for tube still needs to be done, which is an issue to consider in the future. Nevertheless, the present results (for hydroforming limit strain and stress diagrams) show a qualitative agreement with experiences in practice, see for example (e.g., see Refs. [10,22,24,25]). 4. Conclusions A methodology already developed for prediction of the sheet metal forming limits [17] has been extended and utilized to determine the hydroforming stress limit diagram. When the strain gradient effect has been included, this approach reduces the imperfection sensitivity encountered in the conventional M–K methods. Also the post-localization behavior of material can be presented. An adaptive simulation can be used to obtain suitable load path with high formability in tube hydroforming. Acknowledgments The author wishes to thank Dr. Alireza Safikhani for his important role and help in numerical computation, and also to Mr. Ehsan Masoumi Khalil Abad for his help in finite element simulation. References [1] Ray P, Mac Donald BJ. Experimental study and finite element analysis of simple X-shape and T-branch tube hydroforming processes. International Journal of Mechanical Sciences 2005;47:1498–518. [2] Hartl Ch. Research and advances in fundamentals and industrial applications of hydroforming. Journal of Materials Processing Technology 2005; 167(2–3):383–92.
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[3] Ahmetoglu M, Altan T. Tube hydroforming: state-of-the-art and future trends. Journal of Materials Processing Technology 2000;98(1):25–33. [4] Kim J, Kim YW, Kang BS, Hwan SM. Finite element analysis for bursting failure prediction in bulge forming of a seamed tube. Finite Elements in Analysis and Design 2004;40:953–66. [5] Strano M, Jirathearanat S, Altan T. Adaptive FEM simulation for tube hydroforming: a geometry-based approach for wrinkle detection. CIRP Annals – Manufacturing Technology 2001;50(1):185–90. [6] Gao L, Motsch S, Strano M. Classification and analysis of tube hydroforming processes with respect to adaptive FEM simulations. Journal of Materials Processing Technology 2002;129(1–3):261–7. [7] Stoughton TB. A general forming limit criterion for sheet metal forming. International Journal of Mechanical Sciences 2000;42(1):1–27. [8] Stoughton TB, Zhu X. Review of theoretical models of the strain-based FLD and their relevance to the stress based FLD. International Journal of Plasticity 2004;20(8–9):1463–86. [9] Xu G, Weinmann KJ. A new approach to predicting forming limits of steel sheet. Journal of Manufacturing Processes 2000;2(3):158–66. [10] Hashemi R, Assempour A, Masoumi Khali Abad E. Implementation of the forming limit stress diagram to obtain suitable load path in tube hydroforming. Journal of Materials and Design 2009;30:3545–53. [11] Assempour A, Hashemi R, Abrinia K, Ganjiani M, Masoumi E. A methodology for prediction of forming limit stress diagrams considering the strain path effect. Computation Materials Science 2009;45(2):195–204, http://dx.doi.org/10.1016/j.commatsci.2008.09.025. [12] Marciniak Z, Kuczynski K. Limit strains in the process of stretch-forming sheet metal. International Journal of Mechanical Sciences 1967;9:609–20. [13] Aifantis EC. On the microstructural origin of certain inelastic models. Transactions of ASME Journal of Materials Engineering and Technology 1984;106:326–30. [14] Aifantis EC. The physics of plastic deformation. International Journal of Plasticity 1987;3:211–47. [15] Zbib HM, Aifantis EC. A Gradient-dependent flow theory of plasticity: application to metal and soil instabilities. Applied Mechanics Review 1989;42:295–304. [16] Wang YW, Majlessi SA, Ning I, Aifantis EC. The strain gradient approach for deformation localization and forming limit diagrams. Journal of Mechanical Behavior of Materials 1996;7:265–92. [17] Safikhani AR, Hashemi R, Assempour A. Some numerical aspects of necking solution in prediction of sheet metal forming limits by strain gradient plasticity. Materials and Design 2009;30(3):727–40. [18] Hill R. On discontinuous plastic states, with special reference to localized necking thin sheets. Journal of Mechanics and Physics of Solids 1952;I:19–30. [19] Ganjiani M, Assempour A. Implementation of a robust algorithm for prediction of forming limit diagrams. Journal of Materials Engineering and Performance 2007;17(1):1–6. [20] Wang YW. The effect of VBHF on formability and instability analysis, incorporating strain gradient effect on neck evolution. Ph.D. Dissertation, Michigan Technological University; 1995. [21] Vardoulakis I, Aifantis EC. A gradient flow theory of plasticity for granular materials. Acta Mechanica 1991;87:197–217. [22] Kim J, Kim SW, Song WJ, Kang BS. Analytical approach to bursting in tube hydroforming using diffuse plastic instability. International Journal of Mechanical Sciences 2004;46(10):1535–47. [23] Sillekens WH, Werkhoven RJ. Hydroforming processes for tubular parts: optimisation by means of adaptive and iterative FEM simulation. International Journal of Forming Process 2001;4:377–93. [24] Hashemi R, Faraji G, Abrinia K, Dizaji AF. Application of the hydroforming strain- and stress-limit diagrams to predict necking in metal bellows forming process. The International Journal of Advanced Manufacturing Technology 2010;46(5–8):551–61, http://dx.doi.org/10.1007/s00170-009-2121-9. [25] Aydemir A, de Vree JHP, Brekelmans WAM, Geers MGD, Sillekens WH, Werkhoven RJ. An adaptive simulation approach designed for tube hydroforming processes. Journal of Materials Processing Technology 2005;159(3): 303–10. [26] Strano M, Jirathearanat S, Shr S-G, Altan T. Virtual process development in tube hydroforming. Journal of Materials Processing Technology 2004;146(1): 130–6. [27] Aue-U-Lan Y, Ngaile G, Altan T. Optimizing tube hydroforming using process simulation and experimental verification. Journal of Materials Processing Technology 2004;146(1):137–43. [28] Safikhani AR, Hashemi R, Assempour A. The strain gradient approach for determination of forming limit stress and strain diagrams. Proceedings of IMechE, Part B, Journal of Engineering Manufacture 2008;222(4):467–83. [29] Assempour A, Safikhani AR, Hashemi R. An improved strain gradient approach for determination of deformation localization and forming limit diagrams. Journal of Materials Processing Technology 2009;209(4):1758–69.
Contents lists available at SciVerse ScienceDirect
Journal of Manufacturing Processes journal homepage: www.elsevier.com/locate/manpro
Technical Paper
The strain gradient approach to predict necking in tube hydroforming Ramin Hashemi a,∗ , Karen Abrinia a , Ahmad Assempour b a b
Mechanical Engineering Department, College of Engineering, University of Tehran, Tehran, Iran Center of Excellence in Design, Robotics and Automation, Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 2 February 2012 Received in revised form 30 October 2012 Accepted 1 November 2012 Available online 21 December 2012 Keywords: Numerical Model Forming
a b s t r a c t A stress-based forming limit diagram for necking prediction which is based on the strain gradient theory of plasticity in conjunction with the M–K model has been represented and used in tube hydroforming. In this study, the finite element model for bulge forming of straight tube has been constructed and verified with published experimental data. The adaptive simulation technique is based on the ability to detect the onset and growth of defects (e.g., wrinkling, and bursting) and to promptly readjust the loading paths. Thus, a suitable load path has been obtained by applying Adaptive Simulation Method in ANSYS Parametric Design Language (APDL). © 2012 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.
1. Introduction The Tube HydroForming (THF) process is a relatively complex manufacturing process. The THF technique offers some advantages as compared to conventional methods in production of complex parts. It could produce lightweight parts and reduce the manufacturing costs [1–3]. However, failures such as buckling, necking could occur in the tube hydroforming process. The causes of such defects are mainly due to wrong loading conditions in applying the fluid pressure and axial feeding simultaneously. Since bursting in THF processes is the consequence of necking, prediction of necking initiation is an important issue before designing the details of the processes [4]. In Ref. [5], the adaptive simulation concept is presented as an effective finite element method (FEM) approach, able to select a feasible THF loading path within a minimum number of simulation runs or even within a single run. The adaptive simulation technique is based on the ability to detect the onset and growth of defects (wrinkling, buckling, bursting) and to promptly readjust the loading paths. The detection and quantitative evaluation of wrinkles play a key role in the implementation of the adaptive simulation. A new geometrical wrinkle indicator is introduced and evaluated with numerical and experimental evidences. The proposed wrinkle indicator is computationally inexpensive, suitable for many die shapes and sensitive to different kinds of wrinkles.
∗ Corresponding author. E-mail addresses: [email protected], [email protected] (R. Hashemi), [email protected] (K. Abrinia), [email protected] (A. Assempour).
A classification of THF processes based on sensitivity to loading parameters has been suggested in Ref. [6]. Gao et al. [6] analyzed the characteristics of the classification in terms of failure mode, dominant loading parameters and their working windows. It is considered that the so-called pressure dominant THF process is the most difficult process for both simulation in FEM analysis and practical operation in real manufacturing situation. Adaptive FEM simulation strategies are mostly needed to effectively determine out the optimum loading path for pressure dominant THF process. It is found in the literature that path-independent stress-based Forming Limit Diagram (FLSD) can be used well for all forming processes without concerning for strain path effects [7]. In addition, it is reported that the path-dependent nature of the strain-based FLD causes the formability prediction method to become ineffective in the analysis of complex forming process, especially THF [8]. There are several works [9–11] which used the conventional M–K model [12] for prediction of sheet metal forming limits. Detailed studies on these approaches using both deformation and flow theory of plasticity reveal a deficiency of the localized necking analysis associated with imperfection-sensitivity. Moreover, they cannot give a detailed description of the profile, especially after initiation of localized necking. In order to obtain more accurate predictions for the limit strain and stress, a more suitable constitutive relation is needed. Such a constitutive relation was advanced by Aifantis and co-workers [13–15] by incorporating higher order strain gradient terms into the flow stress or the yield condition. The higher order strain gradient terms have a significant effect in the localized deformation region, but they are unimportant in the uniform deformation regime. This allows for the use of a unified constitutive equation to describe the different material behavior inside and outside the localization region.
1526-6125/$ – see front matter © 2012 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jmapro.2012.11.001
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R. Hashemi et al. / Journal of Manufacturing Processes 15 (2013) 51–55
Based on Aifantis’s strain gradient theory [13], Wang et al. [16] and Safikhani et al. [17] incorporated the strain gradient theory into the M–K model to analyze the deformation localization and to predict the corresponding FLDs and FLSDs. The higher order strain gradient (second order strain gradient) was used in the power law hardening equation to describe the different material properties inside and outside the localized region. They followed Hill’s criterion [18] and simplified this problem by assuming that the localized necking initiates along the zero extension direction for the left hand side of the FLD and FLSD. For obtaining one point of FLD, it is required to consider all groove angles between 0 and 90◦ . The limit strain as a candidate point on FLD is the minimum of all calculated limit strains for all groove angles. As shown in Ref. [19], on right hand side of FLD, the candidate limit strains occur at zero groove angles. It means for computation of limit strain at right hand side of FLD, it is enough to accomplish the calculation only for zero groove angles. In this case the groove direction is normal to the rolling direction. But on the left hand side of FLD, depends on the material constants, local neck occurs at zero extension. At this direction, one of the non-shear strain components becomes zero. The FLD calculation procedure for left and right hand sides is simplified. On the right hand side of FLD, limit strains are obtained for zero groove angles. On the left hand side of FLD, first the groove angle is calculated and then the limit strain for this angle is obtained. The localized band was assumed to initiate along the minor strain direction for positive strain ratios as proposed by Marciniak and Kuczynski. This approach introduces an internal length scale into the constitutive equations and takes into account the effects of deformation inhomogeneity. It overcomes the imperfection sensitivity encountered in the conventional M–K method. The nonlinear differential equation of the thickness variation is the second order ordinary type and it has been solved by collocation method [17].In this work, the stress-based hydroforming limits are used to predict necking. The methodology for computation of FLSD is based on the strain gradient theory of plasticity in conjunction with the M–K model. This approach introduces an internal length scale into conventional constitutive equations and takes into account the effects of deformation inhomogeneity and material softening. The nonlinear second order ordinary differential equation of the thickness of tube has been solved by the shooting method. The shooting method was first presented by Wang et al. [16] and later developed by Safikhani et al. [17]. The verification of the theoretical results has been done. Ultimately, the suitable load path for bulge forming of straight tube has been determined by applying adaptive simulation method in ANSYS Parametric Design Language (APDL). 2. Review of the sheet metal computations In Marciniak and Kuczynski (M–K) model [12], it is assumed that there is a shallow groove on material surface which causes the localized necking. The initial imperfection factor of the groove f0 is defined as the thickness ratio f0 = hB /hA , where h denotes the material thickness and subscript “0” denotes the initial state. Groove region is called “B” and safe zone is named “A”. A major problem to the conventional M–K model is that the calculated results are so sensitive to the initial value of imperfection factor and the conventional M–K method cannot explain post-localization behavior. In this method to analyze the deformation throughout the material, the constitutive relation for the effective or flow stress should account for both homogeneous and non-homogeneous deformation states. To account for the strain gradient effect corresponding to the initiation of the strain localization, a gradient dependent flow stress has been used which was proposed by Aifantis [13]. In this study, for simplicity and mathematical reasons, the strain gradient plasticity presented by Aifantis and co workers [13–15]
has been used. So, the second order terms are incorporated into the power law isotropic hardening constitutive equation to describe the material behavior during the post-localization process. The constitutive equation used is [20,21]:
2
¯ = 0 (¯ε0 + ε¯ )n ε¯˙ − c1 ∇ ε¯ − c2 ∇ 2 ε¯ m
(1)
where c1 and c2 are the strain gradient coefficients. As shown in Refs. [16,17], the effect of the strain gradient increases when coefficient c2 is increased. This coefficient represents a property of material and can be determined by experiment. If in this equation, it is supposed that c1 and c2 are equal to zero, it reduces to the conventional relationship between effective strain and effective stress (Swift relation): m
¯ = 0 (¯ε0 + ε¯ )n ε¯˙
(2)
Considering a material to plane stress and having a deformation in the form of a neck which is initially inclined at an angle . For consideration of strain gradient effects and triggering the evolution of necking, the authors suppose the continuous function for sheet thickness as a function of sheet length in the neck direction [16]. The initial thickness of the sheet varies along the x-direction according to the relation: hB0[i] =
hA0 + hB[0] 2
+
hA0 − hB[0] 2
cos
1+
xi w0
(3)
where hB[0] , hB and hA0 are the initial sheet thickness in the imperfection center, at position xi and in the uniform region respectively, as shown in the following figure (e.g., Fig. 1b). w0 represents the initial half width of neck. Assuming the load in the direction perpendicular to the neck (y-direction) is Fx . According to the equilibrium condition in the direction perpendicular to the neck, load in the uniform region (A) should be equal to the load at any cross-section inside the neck (B), i.e. FxA = FxB . By using of Hill’s yield criterion for anisotropic material and Levy–Mises flow rule, and the definition of effective stress and strain, the conditions for calculation of neck evolution and limit strain are as follows [17]: hA A
0 (¯ε0 + ε¯ A )
n
A
−1 1 + ˇA
h˙ A hA
m
⎧ m ⎫ ⎪ ⎪ ˙ ⎪ ⎪ h −1 ⎨ 0 (¯ε0 + ε¯ B )n B
⎬ B hB h B 1 + ˇB = B ⎪ ⎪ ⎪ ⎪ ⎩ −c ∇ ε¯ 3 − c ∇ 2 ε¯ ⎭ B B 1 2
(4)
where is ratio of effective strain rate to strain rate in strain path, is ratio of effective stress to stress in strain path and 0 is assumed the same inside and outside the imperfection. For each element, the neck thickness is calculated based on (4). The critical neck shape obtained under different loading conditions is used as the criterion to determine the forming limit diagram. This is in contrast to the use of a limit strain value employed by conventional M–K methods. In connection with the study of neck evolution, the effects of initial imperfection, both with and without considering the strain gradient effects, are discussed in a previous work [17]. The detailed description about strain gradient theory of plasticity, analysis of neck evolution, consideration of the strain gradient effects on the left and right hand sides of FLSD and the shooting solution of neck evolution in this work can be found in Ref. [17]. In a previous work [17] the necking strain value was defined as follows:for the right hand side of FLD: ε1 = ln
(w − w) t (wt − w0 )
,
(5)
R. Hashemi et al. / Journal of Manufacturing Processes 15 (2013) 51–55
53
FLSDs , STKM-11A tube , Tube hydroforming , Hill 1948 , f =0.96 0
1800 1600
Major Stress σ 1 (MPa)
1400 1200 1000 800 600
experimental data [22]
400
conventional M-K method (present work)
200
diffuse neck ing criterion [22]
0 -600
-400
-200
0 200 400 600 Minor Stress σ2 (MPa)
800
1000
1200
Fig. 2. Hydroforming stress limit diagrams.
Step 3: Calculate the effective stress ¯ by using the constitutive equation which is expressed by Eq. (1). In this equation it is supposed that c1 and c2 are equal to zero. Step 4: Calculate stress ratio (˛) according to respective strain ratio (ˇ). The relation between ˛ and ˇ is expressed as follows: ˛=
(1 + R)ˇ + R (1 + R) + Rˇ
(8)
where R is anisotropy coefficient. Step 5: Calculate , ratio of effective stress to stress in strain path, by using the following equation: √ = (/ ¯ 1 ) = (1/ 2) 4 3/2R + 1 (1 + R) ˛2 − 2R˛ + (1 + R) (9) Fig. 1. Schematic diagram of the imperfection model b. Schematic diagram of the neck cross-section in x–z [17].
Step 6: The major and minor stresses can be expressed as follows: 1 =
2 = ˛1
for the left hand side of FLD: ε1 =
ln((wt − w)/(wt − w0 )) , (1 + ˇ12 )
(6)
where w0 is the initial half neck width, w is half neck width after deformation, wt is total half neck width after deformation and ˇ12 = ε˙ 2 /ε˙ 1 is the proportional strain ratio applied to the uniform region. These equations should be used as a necking strain to construct the forming limit diagram (FLD) [19]. By using Eqs. (5) and (6), the major strain, here (ε1 ), has been calculated for right hand side (RHS) and left hand side (LHS) of the FLD. And the amount of strain ratio ˇ12 (proportional loading) is known. Thus, the minor strain, here (ε2 ), can be calculated. To obtain FLSD, the calculations must be done to convert strains to stresses. The steps of computations in this procedure are summarized as the following [17]: Step 1: Obtain theoretical forming limits in strain form (ε1 , ε2 ) from Eqs. (5) and (6). Step 2: Calculate the effective strain ε¯ by using the following equation: ε¯ =
ij ε˙ ij ¯
¯
(7)
Reader could see the detailed explanations for this equation in Ref. [17].
(10) (11)
When the strain gradient effect is included, this approach overcomes the imperfection sensitivity encountered in the conventional M–K method and instability analysis is performed by considering the neck behavior in both pre- and post-localization regimes. 3. Results and discussions In this paper, to verify the stress-based forming limit diagram predicted by the proposed approach, the predicted stress-based FLD has been compared with hydroforming stress limit diagram determined experimentally in Ref. [22]. In this calculation properties of AK steel alloy [17] have been used. The tube material is assumed to obey the stress–strain relationship ¯ Y = 1400¯ε0.17 (MPa) and with normal anisotropy value r = 2.14. This verification has been shown in Fig. 2 and there is good agreement between theoretical results and experimental data. The adaptive simulation is an approach to obtain the load path with minimum defects concerning the wrinkle and neck formations in tube hydroforming process [23,24]. Applying ANSYS LS-DYNA and ANSYS Parametric Design Language (APDL), an adaptive simulation was carried out. The general algorithm of adaptive simulation method can be found in Ref. [10]. A wrinkle predictor and a necking indicator are two essential aspects in any adaptive simulation design system.
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R. Hashemi et al. / Journal of Manufacturing Processes 15 (2013) 51–55
Table 1 Die geometry of models. Materials
OD1 (mm)
OD2 (mm)
L1 (mm)
L2 (mm)
L3 (mm)
R1 (mm)
R2 (mm)
AISI 304 [27] AK steel [17]
48.6 60
64 145
120 50
25 0
150 95
3.4 12.5
7.1 10
End Feed (mm)
20 15 10 Load path… 5 0 0
100
200
300
Pressure (MPa) Fig. 4. The load path [10,27] used for verification of the current FE model. Fig. 3. The geometry of die and tube in BULGE HYDROFORMING [10].
AISI 304 [27] AK steel [17]
n 0.615 0.245
10 8 6 4 2
Current Work data
0
Experimental Data
-2 0
0.05
0.1
0.15
0.2
0.25
Longitudinal length (m) Fig. 5. Comparison of tube wall thinning between the current simulation and experimental data [27] for AISI 304.
path is decreased and more material is fed into the bulging area. This procedure continues until reaching the necking condition. The distribution of Von-Mises stress in deformed tube and thickness distribution along the tube are shown in Figs. 7 and 8. As could be seen, the minimum thickness at bulging zone is not very critical and no wrinkling occurred in the deformed tube and therefore adaptive simulation method could provide the correct load path successfully. In this study, the strain gradient coefficients (c1 and c2) were obtained experimentally. The same experimental data as in Refs. [17,20,28,29] has been used when the strain gradient coefficients 35 30 25 20 15 Load path
10 5
Table 2 Mechanical properties of tube. Materials
12
Pressure (MPa)
An indicator is needed for optimization of the process concerning wrinkle formation [25,26]. In the present paper, based on die geometry, the slope wrinkling indicator has been used. Considering an axial line on the tube, once the line slope at a particular point changes its sign, there is a wrinkle creation potential at that point. More explanation can be found about this phenomenon in Ref. [10]. The proposed stress-based forming limit approach has been adopted for evaluation of forming severity and occurrence of the necking phenomena. The stress-based limit used in this study reduces the sensitivity of the limit curve to the strain path in the bulging process. While the tube has not reached the tool surface, the tube is under plane stress condition and the present stress-based forming limit diagram can be used as a criterion for bursting failure. As a case study, an adaptive simulation method was implemented in APDL to determine suitable load path for tube hydroforming. Using this load path, suitable formability without wrinkling and bursting failures has been obtained. Theoretically FLSD has been utilized as a criterion for necking prediction. Tool and tube model were made in ANSYS LS-DYNA. A view of tube and die is shown in Fig. 3. The tool is modeled as a rigid body. The tube material is transversely anisotropic elastic-plastic. Tables 1 and 2 display the geometrical parameters and mechanical properties used in this case study. The coulomb friction coefficient of 0.05 was considered in the simulation. The experimental load path in Ref. [27], shown in Fig. 4, is used to verify the present model for material model AISI 304. The result is shown in Fig. 5. As the results show, good agreement with experimental data has been achieved. Therefore, the finite element model can be used with adaptive simulation to obtain suitable load path. Fig. 6 displays the load path obtained from adaptive simulation for AK steel alloys [17]. In this system, at first the pressure and axial end feed are increased by small amounts. Then, the pressure remains constant and only the axial feed is increased. This process continues until wrinkling occurs. To prevent the wrinkling, the pressure and axial end feed are increased simultaneously in a linear path. As thinning occurs in the bulging zone, the slope of the load
Tube wall thining (%)
14
0
ε0
0 (MPa)
R
0.03 0.01
1555 186.525
1 1.2
Thickness (mm) 2.2 1.0
c2 0.0 0.1
0
10
20
30
Axial end feed (mm) Fig. 6. Load path determined by adaptive simulation method (present work).
R. Hashemi et al. / Journal of Manufacturing Processes 15 (2013) 51–55
Tube thickness (mm)
Fig. 7. Von-Mises stress distribution in tube after free bulging (present work).
1.6 1.2 0.8 Thickness along axial line
0.4 0 0
100
200
Distance along one axial line (mm) Fig. 8. Thickness distribution along axial line (present work).
have been considered. Verification of the present approach by performing experiments with different materials for tube still needs to be done, which is an issue to consider in the future. Nevertheless, the present results (for hydroforming limit strain and stress diagrams) show a qualitative agreement with experiences in practice, see for example (e.g., see Refs. [10,22,24,25]). 4. Conclusions A methodology already developed for prediction of the sheet metal forming limits [17] has been extended and utilized to determine the hydroforming stress limit diagram. When the strain gradient effect has been included, this approach reduces the imperfection sensitivity encountered in the conventional M–K methods. Also the post-localization behavior of material can be presented. An adaptive simulation can be used to obtain suitable load path with high formability in tube hydroforming. Acknowledgments The author wishes to thank Dr. Alireza Safikhani for his important role and help in numerical computation, and also to Mr. Ehsan Masoumi Khalil Abad for his help in finite element simulation. References [1] Ray P, Mac Donald BJ. Experimental study and finite element analysis of simple X-shape and T-branch tube hydroforming processes. International Journal of Mechanical Sciences 2005;47:1498–518. [2] Hartl Ch. Research and advances in fundamentals and industrial applications of hydroforming. Journal of Materials Processing Technology 2005; 167(2–3):383–92.
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