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Experimental and numerical study of an AlMgSc sheet formed by an incremental process
Journal of Materials Processing Technology 211 (2011) 1684–1693
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Experimental and numerical study of an AlMgSc sheet formed by an incremental process C. Bouffioux a,∗ , C. Lequesne a , H. Vanhove b , J.R. Duflou b , P. Pouteau c , L. Duchêne a , A.M. Habraken a a b c
University of Liège, Dept. ArGEnCo, Chemin des Chevreuils 1, B-4000 Liège, Belgium University of Leuven, Dept. of Mechanical Engineering, Celestijnenlaan 300b, B-3001 Heverlee, Belgium CRM Gent, Technologiepark 903c, B-9052 Zwijnaarde, Belgium
a r t i c l e
i n f o
Article history: Received 22 February 2011 Received in revised form 10 May 2011 Accepted 15 May 2011 Available online 20 May 2011 Keywords: Aluminium alloy AlMgSc Material parameter identification Finite element simulation Single Point Incremental Forming
a b s t r a c t A recently developed AlMgSc alloy is studied since this material, which is well adapted to the aeronautic domain, is poorly known. The first objective is to reach a better knowledge of this alloy to provide the missing useful information to the aeronautic industry and to help research institutes who want to simulate sheet forming processes by Finite Element (FE) simulations. A set of experimental tests has been performed on the as-received sheets, material laws have been chosen and the corresponding material parameters have been adjusted to correctly describe the material behaviour. The second objective is to study the applicability of the Single Point Incremental Forming process (SPIF) on this material. Truncated cones with different geometries were formed and the maximum forming angle was determined. A numerical model was developed and proved to be able to predict both the force evolution during the process and the final geometrical shape. Moreover, the model helps reaching a better understanding of the process. The characterisation method described in this research and applied on the AlMgSc alloy can be extended to other alloys. In addition, the numerical simplified model, able to accurately describe the SPIF process with a reduced computation time, can be used to study more complex geometries. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The aeronautic industry is constantly looking for ways to reduce the weight of aeronautic parts and structures. Therefore, the availability of a new material of the AlMgSc alloy family, combining a low specific weight, a high corrosion resistance, a high toughness and an excellent weldability is of major interest in this domain. This recently developed alloy, with compositions ranging from 3.1 to 5.1 at.% for Mg, 0.10 to 0.40 at.% for Sc and 0.05 to 0.20 at.% for Zr, is not commonly used. The first investigations of the material provider showed promising results. However, a detailed study was required to improve the knowledge of the material behaviour, to extend its applications and to study more complex processes both by experiments and by numerical investigations.
∗ Corresponding author. Tel.: +32 4 366 9219; fax: +32 4 366 9192. E-mail addresses: chantal.bouffi[email protected] (C. Bouffioux), [email protected] (C. Lequesne), [email protected] (H. Vanhove), joost.dufl[email protected] (J.R. Duflou), [email protected] (P. Pouteau), [email protected] (L. Duchêne), [email protected] (A.M. Habraken). 0924-0136/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2011.05.010
The Single Point Incremental Forming (SPIF) process is a flexible sheet metal forming method adapted to form various complex shapes using a milling machine, a multi-axis robot or a dedicated machine without the need of specific and costly tools, such as a punch and die, as described by Allwood and Shouler (2009). The sheet, of which the edges are clamped, is locally deformed by a spherical tipped tool following a succession of contours leading to the final shape of the sheet. Depending on the length of the tool path and speed of the tool, the forming process can take up to a few hours for large parts. It is, in consequence, adapted to small batch production and rapid prototyping. Considering the maximum part size is machine dependent, SPIF can be used to form small or large pieces, such as aeronautical components. Furthermore, the SPIF process, which is known to shift Forming Limit Diagrams (FLD) to higher formability (Filice et al., 2002) compared to conventional forming processes can be adapted to materials such as the AlMgSc alloy. Because SPIF has a high industrial interest, the applicability of this technique on the new material studied must be experimentally verified. The problem of shape inaccuracy which is usually an important limiting factor for SPIF applications, as underlined by Micari et al. (2007), must be solved. Moreover, the forming force prediction provides crucial data to choose the forming tool, to verify
C. Bouffioux et al. / Journal of Materials Processing Technology 211 (2011) 1684–1693
2. Experimental material characterisation As underlined by Aerens et al. (2009), before performing any forming process, it is useful to ensure the preservation of the tooling and the machinery and to verify that the platform used is stiff enough to avoid significant deviation resulting in errors in the geometry of the achieved parts. A first evaluation of the required force to form the sheet by the available SPIF equipment indicated that the as-received sheets from the manufacturer (3.2 mm in thickness) were too thick and had to be rolled down to 0.5 mm. A conventional symmetrical cold rolling was applied and followed by an additional annealing to recover the initial material properties. Different annealing temperatures between 100 and 475 ◦ C and two holding times of 30 and 60 min were investigated to recover the initial hardness of 110 HV2 and the mechanical properties. It turned out that an annealing at 450 ◦ C during 30 min was adequate.
Simple tensile tests 500
True stress (MPa).
the stiffness of the set-up and to avoid deformation and risk for the equipment. Aerens et al. (2009) established a useful regression formula to predict an approximate value of the axial force for any material when forming a cone. It is a function of the shape geometry, the work conditions and the material tensile strength only. However, even if the force is usually well predicted by Aerens’ formula, its inaccuracy can exceed 20% when a new material is used. Furthermore, its application is limited to cones. The SPIF process can be optimized by experimental trials and errors and/or by a numerical analysis. The numerical approach predicts the forming forces and the final shape according to the tool path. However such information is accurate only if the simulation has been validated by experiments. Previous researches by Bouffioux et al. (2008, 2010) demonstrated the effect of the material parameter identification method, the material model itself (Flores et al., 2007), the Finite Element Method (FEM) strategy such as the explicit or implicit choice (Henrard, 2009), the finite element type: solid or shell (Eyckens et al., 2010a; Henrard et al., 2010), the mesh density and the contact model (Henrard, 2009). Eyckens (2010b) incorporated through-thickness shear (TTS), also known as out-ofplane shear, into a Marciniak–Kuczynski model and observed that the through-thickness shear has a significant effect on the increasing forming limit curves of the process. Moreover, Emmens and van den Boogaard (2009) presented several mechanisms that can explain the enhanced formability of the SPIF process. The numerical inaccuracy of finite element models is one of the main problems often encountered by scientists. Furthermore, the simulations usually require a very high computation time because the tool path is long and can be complex. The combination of different strategies described by Bouffioux et al. (2008, 2010), Lequesne et al. (2008) to improve the material and numerical models and applied on this study aimes to be able to accurately simulate the process with a short computation time. After this introduction showing the interest of this study and the complexity of reaching a good numerical model of the SPIF process, Section 2 presents all the experimental tests performed on the AlMgSc sheets. The constitutive laws used to describe the material behaviour, the data fitted by the inverse method and the validation on a line test are presented in Section 3. Then, the experimental cone tests, which were performed on this material to evaluate its forming limits, are described in Section 4. A numerical model of a truncated cone was chosen and simulated. The results were compared with the experimental forces and shape to validate the model in Section 5. Finally, Section 6 presents the wall angle sensitivity which was compared with the experimental results and analytical predictions. The last section summarizes and discusses the results.
1685
400 300 200 100 0 0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
True strain Tens_0°
Tens_45°
Tens_90°
Hooke 0.2%
Fig. 1. Effect of the anisotropy on tensile tests.
Additional mechanical tests (tensile tests, simple and reverse shear tests) and microstructure verifications showed that the cold rolled and annealed sheets had the same behaviour as the initial ones. A set of mechanical tests was performed to describe the mechanical behaviour and to fit the parameters of the hardening laws. 2.1. Classical tests To investigate the anisotropy of the material, tensile tests were performed at 0◦ , 45◦ and 90◦ from the rolling direction. Fig. 1 shows a lower yield stress and tensile strength at 45◦ from the rolling direction without any significant change on the strain hardening coefficient n. The average Young’s modulus E = 70,500 MPa was computed from these tests. Tensile tests, performed along the rolling direction (RD, 0◦ ) with 3 different strain rates (0.0075s−1 , 0.075 s−1 and 0.1 s−1 ) did not show any significant change in the mechanical response of the material properties. The Lankford coefficients: r0 , r45 and r90 with r␣ = εp22 /εp33 are classically computed from tensile tests at 0◦ , 45◦ and 90◦ from the RD. ˛ is the angle between the longitudinal direction of the test specimen and the rolling direction. εp22 and εp33 are the plastic strain respectively in the transversal and thickness directions of the sample. As explained in Section 3.1, the Lankford coefficients are used to determine the parameters of the Hill yield locus. Due to the Portevin–Le Chatelier effect (serrated stress-strain curves as seen in Fig. 1, a kind of instability classically observed in AlMg alloys), it was neither possible to accurately define the Poisson ratio, nor the Lankford coefficient at 45◦ (r45 ) and at 90◦ (r90 ) from RD. However, the average r0 coefficient could be evaluated at 0.8 for a strain between 6 and 12%. The Poisson ratio was estimated equal to 0.33 as for classical Al alloys. Additional tests, as used to determine the yield locus shape and to fit the material data of the isotropic and the kinematic hardening laws, were performed: a monotonic shear test (Figs. 2a and 3, left), a large tensile test (Figs. 2b and 3, right), two orthogonal tests (a large tensile test followed by a shear test) at two levels of pre-strain: 4% and 8% (Figs. 2c and 4, left) and two Bauschinger shear tests (a shear test followed by a reverse shear test) at three different levels of pre-strain: Gamma = d/b = 8.8%, 13% and 29% (Figs. 2d and 4, right). Each test was performed three times. As the results showed a good reproducibility, an average curve was used and drawn for each loading state. The tensile, shear and large tensile tests were performed in three directions: at 0◦ (RD), 45◦ and 90◦ (TD) from the rolling direction to see the effect of the anisotropy. It was observed that the behaviour was different at 45◦ than in RD or TD.
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a
d
b
d
c
d
b
Fig. 2. Description of the additional classical tests: (a) shear test, (b) large tensile test, (c) orthogonal test and (d) Bauschinger test, definition of b and d values.
Large tensile tests
Shear tests 250
True stress (MPa).
Shear stress (MPa).
300
200 150 100 50 0 0.0
0.2
0.4 Gamma
Sh_RD
0.6
0.8
500 450 400 350 300 250 200 150 100 50 0 0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
Strain
Sh_45
TL_RD
Sh_TD
TL_45
TL_TD
Fig. 3. Experimental shear tests (left) and large tensile tests (right) in 3 directions (0◦ , 45◦ and 90◦ from RD).
Bauschinger tests Abs(shear stress) (MPa).
True stress / Shear Stress (MPa) .
Orthogonal tests
450 400 350 300 250 200 150 100 50 0 0.00
0.05
0.10 0.15 Major strain / gamma
OR_4%
0.20
250 200 150 100 50 0 0.0
0.2 BA_8.8%
OR_8%
0.4 0.6 Sum (Gamma)
0.8
BA_13%
1.0
BA_29%
Fig. 4. Experimental orthogonal tests (4 and 8%) (left) and Bauschinger tests (8.8, 13 and 29%) (right).
All these tests were used to describe the yield locus shape and its evolution. The decrease of the stress in the second part of the Bauschinger tests indicated that a complex hardening law combining both an isotropic and a kinematic part was required.
2.2. Indent and line tests The line test was performed on a square sheet with a thickness of 0.5 mm, clamped along its edges (Fig. 5, left).
Y
100 mm
Z
Y
X tool
1
X
tool 20 mm
100 mm Fig. 5. Description of the line test: geometry (left) and tool displacements (right).
2 4 60 mm
3 20 mm
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The displacement of the spherical tool, having a radius of 5 mm, was composed of five steps with an initial position tangent to the surface of the sheet (Fig. 5, right): a first indent of 3 mm (step 1), a line movement at the same depth along the X axis (step 2), then a second indent up to the depth of 6 mm (step 3) followed by a line at the same depth along the X axis in the opposite direction (step 4) and the unloading. The first step corresponds to what is called hereafter the indent test, while all of the five steps define the line test. The whole line test was performed three times (with the SPIF set-up) and the bolts of the frame were tightened using the same torque to ensure the reproducibility of the results. As explained in Section 3.1, the total tool force during the indent test was combined with all of the classical tests to fit the material data by the inverse identification method. The line test, presenting deformation mechanism close to the SPIF, was used as validation of the material parameter fitting (see Section 3.2).
The elastic coefficients and most of the parameters related to the yield locus shape could be defined by classical tensile tests, but the inverse method was required to fit the hardening behaviour and the shear Hill coefficients. This inverse method was coupled with the Finite Element code: “Lagamine” described by Duchêne and Habraken (2005). The principle of this method is to choose a set of tests, the results of which are sensitive to the material data to adjust. These tests are simulated using an initial set of data, chosen arbitrarily. Then, the numerical results are compared with the experimental measurements. The Levenberg Marquardt minimization algorithm described by Dennis and Schnabel (1983) is used to iteratively adjust the material data until a sufficient accuracy is reached. All the classical tests, described in Section 2.1, and the indent test (Fig. 5) were simultaneously used in the inverse method. Indeed, Bouffioux et al. (2008, 2010) observed that simple classical tests inducing in-plane stresses were not sufficient to describe the material behaviour during the SPIF process. The indent test gave the missing information about the out-of-plane material behaviour since it induces through thickness shear. In addition, the same shell elements were used for these simulations and for the SPIF process. It was decided to impose isotropic in-plane behaviour because neither the experimental force evolution during a whole contour of a cone, nor the cone wall thickness comparison in the different sections (at 0◦ , 45◦ and 90◦ from RD), shows any effect of anisotropy. This simplification induced the replacement, in the inverse method, of classical test results in three directions by their average curve. The elasto plastic constitutive law was to be used on a simplified numerical model (a 90◦ pie) where the force evolution of a contour was replaced by an average force when the tool was in the central third of the pie. 3.1. Material model The elastic range was described by Hooke’s law and the plastic part by the Hill 48 law (Eq. (1)) where ij are the stress tensor components and where F is the yield stress. 1 H(xx − yy )2 + G(xx − zz )2 FHILL () = 2
− F2 = 0
(1)
The parameters F, G, H of Hill’s law (Table 1) could be identified from the relation between the yield stress limit in RD and TD (Eq. (2)):
yield
TD
=
2 yield H + F RD
Table 1 Material data of the AlMgSc alloy (units: N, mm). Tests used
Classical and indent tests
Yield surface Coefficients (Hill’48)
Voce parameters
Back stress data (Ziegler)
F = 1.11 G = 1.11 H = 0.89 N=3
K = 275 0 = 8.5 n = 439
CA = 930 GA = 18.1
However, the in-plane isotropic assumption above-mentioned was used. yield
TD
yield
= RD
The usual additional condition for Hill (1948) was applied (Eq. (3)): H+G =2
3. Material behaviour modelling
2 2 2 +F(yy − zz )2 + 2N xy + xz + yz
1687
(2)
(3)
Finally, the Lankford coefficient formulation allowed to characterize the out-of-plane anisotropy (Eq. (4)): r0 =
H G
(4)
where r0 = 0.8, as computed from the tensile tests. The isotropic hardening was described by Voce’s formulation (Eq. (5)) because this law, which predicts a saturation (Bouffioux et al., 2010; Habraken et al., 2010), was better adapted to describe the material behaviour for the large deformation encountered during the SPIF process. F = 0 + K(1 − exp(−n• εpl ))
(5) εpl
where 0 , K and n are the material parameters and is the equivalent plastic strain. It was noticed that a simple isotropic hardening model (Eq. (5)) was not sufficient to provide an accurate tool force prediction (Flores et al., 2007 and Henrard et al., 2010). Therefore an elasto-plastic law with a mixed isotropic-kinematic hardening was used. In this law, the stress tensor (in Eq. (1)) is replaced by ( − X), where X is the back-stress. The evolution of the back-stress is described by Ziegler’s hardening equation (Eq. (6)): X˙ = CA
1 ( − X) ε˙ pl − GA • X • ε˙ pl F
(6)
where CA is the initial kinematic hardening modulus and GA is the rate at which the kinematic hardening modulus decreases with an increasing plastic deformation. These models use the Green strains and the second formulation of Piola–Kirchhoff for the stresses. As already specified in Section 2.1, the Young’s modulus was defined by the tensile tests (E = 70,500 MPa) and the Poisson ratio was fixed equal to 0.33 as for other aluminium alloys. The shear parameter N of the Hill’s law, which could not be defined accurately by the classical tests was combined to the Voce’s law data and the back stress parameters to be fitted by the inverse method (Table 1). The final data were fitted using the classical tests and the indent test. All these tests as well as the SPIF process were simulated with quadrilateral shell elements with four nodes. As shown in Figs. 6–8, this set of data was able to accurately predict all the experimental curves. 3.2. Validation of the material model parameters The line test, described in Fig. 5, was used to verify the accuracy of the material data on a complex procedure similar to the incremental forming process, performed with the SPIF set-up and inducing bending, and an out-of-plane stress field.
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shear test 300
Shear stress (MPa).
True stress (MPa).
Simple tensile and large tensile tests 500 450 400 350 300 250 200 150 100 50 0 0.00
0.02
0.04
0.06
0.08
250 200 150 100 50 0 0.00
0.10
strain
0.10
0.20
0.30
Gamma
SimpleTens exp LargeTens exp
SimpleTens Num. LargeTens Num.
Shear exp
Shear Num.
Orthogonal tests 450 400 350 300 250 200 150 100 50 0 0.00
Bauschinger tests
Shear stress (MPa).
True stress / Shear stress (MPa).
Fig. 6. Comparison between experimental and numerical tensile and large tensile tests (left) and simple shear tests (right).
0.05
0.10
0.15
0.20
-0.2
250 200 150 100 50 0 -50 0.0 -100 -150 -200 -250
-0.1
0.25
Ortho_4% Num.
Ortho_8% exp
0.2
0.3
Gamma
Major strain / gamma Ortho_4% exp
0.1
Ortho_8% Num.
Bau_8.8 exp
Bau_8.8 Num.
Bau_29 exp
Bau_29 Num.
Fig. 7. Comparison between experimental and numerical orthogonal tests (left) and Bauschinger tests (right).
Indent test 400 350
Force (N).
300 250 200 150 100 50 0 0
0.1
0.2
0.3
0.4
0.5
Time (sec) Indent test exp
Indent test Num.
Fig. 8. Comparison between experimental and numerical tool force during the indent test.
In the FEM simulation, the nodes along the edges were fixed. The tool force was computed by the static implicit strategy. The Coulomb’s friction coefficient was unknown. The same coefficient of 0.05 as Henrard (2009) for the alloy 3013 was applied between the tool and the sheet. It was observed that the impact of this coefficient on the total tool force is small (the force computed with a friction of 15% is 1% higher than the force without friction). The mesh was adjusted to limit the number of elements (576 elements) while keeping accuracy (Fig. 9, left), taking into account the symmetry. The same shell elements as in the inverse method were used. Fig. 9, right, shows that the model gives a good evaluation of the evolution of the experimental tool total force.
4. Experimental incremental forming applicability on a cone As illustrated in Fig. 10, a firmly clamped sheet was formed by a contouring operation of the forming tool. The tool moved along a predetermined contour in the horizontal plane, after which the tool incrementally descended and started a contour in the next horizontal plane, building up the workpiece layer by layer. The two platforms used in the framework of this project were a 6-axis robot and a 3-axis rigid milling machine (Fig. 11), both equipped with a dedicated force measuring device. The forming speed (approx. 2000 mm/min), in combination with the long toolpath, resulted in a slow forming process.
C. Bouffioux et al. / Journal of Materials Processing Technology 211 (2011) 1684–1693
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Y Line test 1400
100 mm
X
Force (N).
1200 1000 800 600 400 200
tool
0 0
5
1 00 mm
10
15 Time (sec)
Line test exp
20
25
Line test Num.
Fig. 9. Line test: geometry and meshing (left) and comparison between experiment and numerical tool force evolution (right).
Fig. 10. Description of the SPIF process.
Fig. 11. Platforms description.
Thinning of the workpiece is the dominant failure mode in SPIF and is related to the workpiece drawing angle ␣ (Fig. 12). For a given material and initial thickness, the maximum drawing angle represents the limits of the conventional incremental forming process.
Fig. 12. Workpiece geometry.
Determining the geometrical forming limits defined by the maximum drawing angle ˛ (Fig. 12) in Single Point Incremental Forming was done by means of well-established cone tests. Forming limit tests typically started with the formation of a cup with a 10◦ wall angle. Then, successive cups with an increasing wall angle were made. The forming limit is the angle at which the sheet fractured. The maximum drawing angle is mainly influenced by the initial thickness of the blank sheet. To a lesser degree, the diameter of the spherically tipped tool and the scallop height affect the drawing angle in a negative way. The scallop height is specified as the theoretic height of the ripple between two passes of the forming tool as calculated for a milling process. The maximum value of ˛, for the studied AlMgSc alloy, was found equal to 46◦ with an accuracy of 1◦ (Table 2).
C. Bouffioux et al. / Journal of Materials Processing Technology 211 (2011) 1684–1693
Table 2 Cone parameters and maximum forming angle. Material
Thickness
Ø tool
Scallop height
Step down
˛
AlMgSc
0.5 mm
10 mm
0.005 mm
0.322 mm
46◦
Comparing different materials reveals a fairly low forming limit for AlMgSc. This is a strong limiting factor for the part geometries which could be made in this material. Duflou et al. (2007) showed that performing incremental forming at elevated temperatures can significantly improve forming limits. However, this property was not verified on the AlmgSc. 5. Numerical models of a cone processed by SPIF A truncated cone test was chosen to validate the material model. A circular AlMgSc sheet with a thickness of 0.5 mm was clamped. 30 contours were performed with a tool radius of 5 mm, a depth increment Z of 0.5 mm between successive contours, corresponding to a scallop height of 0.015 mm, and a wall angle of 40◦ (Fig. 13). Strategies had to be found to reduce the computation time. As introduced in Section 3, the use of shell elements instead of e.g. 3D solid elements aimed to considerably reduce the computation time. Indeed, solid elements would require minimum three layers of elements to model bending. It would lead to a significantly larger number of degrees of freedom. The mesh used to simulate the process is presented in Fig. 14. Since the tool always moved in the same tangential direction, it was possible to reduce the computation time by modelling only a quarter of sheet. Rotational boundary conditions (Henrard, 2009; Henrard et al., 2010) were imposed by a link between the displacements of the edges, as schematically presented in Fig. 14, left: the rotation, the tangential and the radial displacements were forced to have the same values on both the horizontal and vertical boundaries of the mesh. This link is related to the six degrees of freedom (3 translations and 3 rotations) of each node of the edges along the X and Y axes. The remeshing method (Lequesne et al., 2008) was used to refine the mesh by division of the elements close to the tool into nine elements. The refinement was automatically removed when the tool went further, except when the elements distortion was high (Fig. 14, centre and right).
Two meshes were tested: a very fine mesh (2692 elements) without the remeshing method and the coarse mesh of Fig. 14 (248 elements) combined with the remeshing method. The level of the force was almost the same for both simulations, but the computation time was divided by a factor of 49 for the second case. The Coulomb friction coefficient , which was not accurately known, was taken equal to 0.10. 5.1. Tool force comparison The radial FR , tangential FT and axial FZ force components, respectively in R, T and Z directions (Fig. 14, left) and the norm of the force Ftot were compared with the experimental ones. To simplify the comparison, the force evolution during each contour was replaced by the average of the force. The numerical test values were computed when the tool was in the central third of each contour to avoid small inaccuracies due to the boundary conditions. Fig. 15 shows that similar results were obtained with a fine mesh without remeshing and a coarse mesh combined with the remeshing method. Similar small differences could be observed for all the force components.
Total force (N).
1690
550 500 450 400 350 300 250 200 150 100 50 0 0
5
10
15
20
25
30
Contour Experimental
Fine mesh
Coarse mesh + remesh
Fig. 15. Comparison of the experimental total tool force with the simulation results for two cases: the fine mesh without remeshing and the coarse mesh with remeshing.
Y
Z X X
tool
40°
a b
c
b a
92 mm ∅: 90 - 92 mm Fig. 13. Description of the cone test (a = unsupported area = 1 mm, b = 17.88 mm and c = 54.25 mm).
Fig. 14. Cone test: Initial mesh, axes, radial (R) and tangential (T) directions, boundary conditions (left) and mesh evolution with the remeshing method (centre and right).
C. Bouffioux et al. / Journal of Materials Processing Technology 211 (2011) 1684–1693
Ftot
FZ
550 500 450 400 350 300 250 200 150 100 50 0
FR & Ftot (N)
FT & FZ (N)
FZ_S
FT
0
1691
5
10
15
20
25
550 500 450 400 350 300 250 200 150 100 50 0
FR
0
30
5
10
15
Experimental
Numerical
20
25
30
Contour
Contour
Experimental
Analytic
Numerical
Fig. 16. Comparison of the numerical (coarse mesh + remeshing), analytical (FZ S ) and experimental total, axial, tangential and radial forces.
Fig. 16, left and right show that the numerical model with a coarse mesh combined with the remeshing method can predict all the force components: FR , FT , FZ and Ftot with accuracy. In addition, it would be possible to improve the level of the FT component by adapting the friction coefficient. Henrard (2009) observed that friction has almost no impact on FR and FZ while FT is linearly affected by the friction coefficient. The ratio FT /FZ reached by two simulations with = 0.05 and = 0.10 was respectively: 0.157 and 0.212. By extrapolation, the value: = 0.14 should lead to the experimentally measured ratio FT /FZ of 0.256. The axial force when a steady state is reached: FZ S could also be predicted by an analytic generalized formula (Aerens et al., 2009) based on the tensile strength of the material Rm , the sheet thickness t, the tool diameter dt , the scallop height h and the wall angle ˛ (Eq. (7)): FZ
S
=
0.41 • 0.0716• Rm • t 1.57• dt h0.09• ˛ cos ˛
where FZ S is expressed in N, Rm in t, dt and h in mm and ˛ in degrees. As before-mentioned, the scallop height h is the theoretic height of the ripple between two passes of the forming tool. It is linked to the depth increment Z, the tool diameter and the wall angle by the relation (Eq. (8)):
h(dt − h) ≈ 2 sin ˛
h• dt
(8)
The tensile strength determined by the tensile tests was 352 MPa, the scallop height was 0.0151 mm from Eq. (8) and thus the force prediction FZ S was equal to 458.5 N. This level, represented in Fig. 16, left, gave a good prediction for this alloy and geometry. The radial force FR could be linked to the axial force FZ S (Aerens et al., 2009) by the following relation (Eq. (9)):
FR = FZ
s
• tan
-30
-10
10
30
50
-4 -8 -12 -16 -20
Numerical shape
Robot mid-thickness, X=0
Robot mid-thickness, Y=0
Fig. 17. Comparison of the mid-thickness shapes.
Only a small difference was observed between the two sections: X = 0 and Y = 0 and between the cones formed by the robot and the milling machine used for the experiments. The accuracy measurements were obtained on the tool side; it is the inner shape. The local thickness was found by comparing the scan data from the outside of the part with respect to the inside. The mid-thickness layer was computed (Fig. 17) point by point, taking into account the local wall angle and the thickness. This correction had a small impact on the shape due to the low sheet thickness. The numerical test shape was extracted out of the section: X = Y to avoid small imprecision due to the boundary conditions. It was the mid-thickness layer obtained by the nodal coordinates. Fig. 17 shows that the final shape was well predicted by the model. The higher discrepancy was observed at the centre of the cone with an error of 0.9 mm.
−c
˛ + ˇ − 17.2• dt /10 2
0 -50
(7)
N/mm2 ,
Z = 2 sin ˛
Experimental and numerical shapes
(9)
where ˇ = arccos (1 − 2·h/dt ) (in degrees) and c = 2.54 for aluminium alloys. This equation provided a first approximation of FR = 111.2 N which was too low in the current case. However, as mentioned by Aerens et al. (2009), a correction term should be used to improve this equation, but its determination would require additional experiments that were not performed on the AlMgSc material. 5.2. Shape comparison The experimental shapes were measured by means of laser line scanning at the end of the process when the tool was removed.
6. Wall angle sensitivity Earlier research on forces in SPIF revealed the important and complex non-linear influence of the draw angle on the forming forces. Because of this irregular behaviour a more in-depth study was performed towards this influence. A set of cones similar to the cones used for the forming limit analysis was experimentally formed to see the angle sensitivity. Once again, a circular AlMgSc metal sheet with a thickness of 0.5 mm was clamped on a backing plate with a diameter of 90 mm. The contours were performed with a tool diameter of 10 mm and the same scallop height of 0.005 mm for all the cones. While forming the cones with draw angles between 10◦ and 46◦ and the step down indicated in Table 3, forces were measured when a steady state was reached.
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Fig. 18. Effect of the wall angle on the force components and comparison between experimental and numerically simulated forces (left) and comparison between experimental, analytical and simulated axial (FZ ) and radial (FR ) forces (right). Table 3 Step down with respect to wall angles for a tool diameter of 10 mm and a scallop height of 0.005 mm. Wall angle (◦ )
10
20
30
40
45
46
Step down (mm)
0.078
0.153
0.224
0.287
0.316
0.322
Three cones with the wall angles of 20, 30 and 40◦ were also simulated using a numerical model with a coarse mesh and the remeshing method, the same tool path as in the experiment and a friction coefficient of 0.14. 30 contours were performed as it was observed that the forces reached a steady state from the 12th contour. Then, the average values of the tool forces were computed when the tool was in the central third of each contour. As a result, these force components with respect to the wall angle were compared with the experimental forces (Fig. 18, left and right). It is clearly visible in Fig. 18, right, that the axial force: FZ was the dominating force component in the SPIF process. Also, the nonlinearity between the draw angle and the force components is visible. The numerical model was able to give a good approximation of the force components. The radial FR and axial FZ tool forces were also computed by Eqs. (7) and (9) (Aerens et al., 2009). These equations which were established for five different materials gave also a good estimation of these two components (Fig. 18, right) even if the AlMgSc alloy was not considered in that study.
tion time. It was possible to simulate the 30 contours of a cone test with a simple personal computer within 5 h and last but not least, with a good precision. The effect of the wall angle on the forming forces was also studied experimentally, analytically and by numerical simulation. Once again, the numerical simulation model was validated. In future work, the results could be further improved by using more complex elements, such as solid-shell type elements (Alves de Sousa et al., 2007) which would take into account the throughthickness shear. However, more complex elements will affect the computation time. As conclusions, it is clear that this study gives a new contribution in AlMgSc characterisation and in incremental forming application and simulations. Acknowledgments The authors would like to thank the Belgian Federal Science Policy Office (IAP: project P6/24 and ALECASPIF: Pat2 project P2/00/01) for its financial support and Aleris for the material supply. A.M. Habraken and L. Duchêne would like to thank the Fund for Scientific Research (FNRS, Belgium) for its support. J. Duflou and H. Vanhove recognize the support by FWOvlaanderen for the project. References
7. Conclusions The aims of this research were to improve the knowledge of the AlMgSc sheets, to study the applicability of the SPIF process and to simulate it on this material. Classical tests gave information about in-plane material behaviour. An indent test performed with the SPIF setup, inducing heterogeneous stress and strain fields similar to those present in the SPIF process, provided information for both in-plane and out-of-plane strain states. A mixed isotropic-kinematic hardening, described by both a saturating hardening law such as Voce’s law and the kinematic Ziegler’s law, accurately models the material behaviour as observed during all the simulated tests. To quantify the formability of the AlMgSc alloy in the SPIF process, experimental investigations on cup tests showed that for a spherically tipped tool of Ø10 mm and a scallop height of 0.005 mm, a maximum wall angle of 46◦ was found. To simulate the SPIF process, the use of shell elements, a 90◦ pie coarse mesh combined with the remeshing method and rotational boundary conditions allowed to considerably reduce the computa-
Aerens, R., Eyckens, P., Van Bael, A., Duflou, J.R., 2009. Force prediction for single point incremental forming deduced from experimental and FEM observations. International Journal of Advanced Manufacturing Technology, 969–982, doi:10.1007/s00170-009-2160-2. Allwood, J.M., Shouler, D.R., 2009. Generalised forming limit diagrams showing increased forming limits with non-planar stress states. International Journal of Plasticity 25 (7), 1207–1230, doi:10.1016/j.ijplas.2008.11.001. Alves de Sousa, R.J., Yoon, J.W., Cardoso, R.P.R., Fontes Valente, R.A., Grácio, J.J., 2007. On the use of a reduced enhanced solid-shell (RESS) element for sheet forming simulations. International Journal of Plasticity 23 (3), 490–515. Bouffioux, C., Henrard, C., Eyckens, P., Aerens, R., Van Bael, A., Sol, H., Duflou, J.R., Habraken, A.M., 2008. Comparison of the tests chosen for material parameter identification to predict single point incremental forming forces. In: Proceedings of IDDRG 2008 Conference , Olofström, Sweden, pp. 133–144. Bouffioux, C., Pouteau, P., Duchêne, L., Vanhove, H., Duflou, J.R., Habraken, A.M., 2010. Material data identification to model the single point incremental forming process. International Journal of Material Forming 3 (Suppl. 1), 979–982, doi:10.1007/s12289-010-0933-7. Dennis, J.E., Schnabel, R.B., 1983. Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice, Hall, Englewood Cli s, NJ, USA. Duchêne, L., Habraken, A.M., 2005. Analysis of the sensitivity of FEM predictions to numerical parameters in deep drawing simulations. European Journal of Mechanics - A/Solids 24, 614–629. Duflou, J.R., Callebaut, B., Verbert, J., Debaerdemaeker, H., 2007. Laser assisted incremental forming: formability and accuracy improvement. CIRP Annals—Manufacturing Technology 56 (1), 273–276.
C. Bouffioux et al. / Journal of Materials Processing Technology 211 (2011) 1684–1693 Emmens, W.C., van den Boogaard, A.H., 2009. An overview of stabilizing deformation mechanisms in incremental sheet forming. Journal of Materials Processing Technology 209, 3688–3695. Eyckens, P., Belkassem, B., Henrard, C., Gu, J., Sol, H., Habraken, A.M., Duflou, J.R., Van Bael, A., Van Houtte, P., 2010a. Strain evolution in the single point incremental forming process: digital image correlation measurement and finite element prediction. International Journal of Material Forming, doi:10.1007/s12289-0100995-6. Eyckens, P., 2010. Formability in Incremental Sheet Forming: Generalization of the Marciniak–Kuczynski model. PhD thesis. ISBN 978-94-6018-210-5. Filice, L., Fratini, L., Micari, F., 2002. Analysis of material formability in incremental forming. CIRP annals—Manufacturing Technology 51 (1), 199–202. Flores, P., Duchêne, L., Bouffioux, C., Lelotte, T., Henrard, C., Pernin, N., Van Bael, A., He, S., Duflou, J., Habraken, A.M., 2007. Model identification and FE simulations: effect of different yield loci and hardening laws in sheet forming. International Journal of Plasticity 23 (3), 420–449. Habraken, A.M., Bouffioux, C., Lequesne, C., Rosochowski, A., Rosochowska, M., Verlinden, B., Vidal, V., Ailinca, G., Duflou, J.R., Vanhove, H., Schmitz, A., Pouteau,
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P., 2010. An Aluminium alloy improved by ECAP and formed by SPIF process, Alecaspif, Final Report, Belgian Science Policy (Programme to stimulate knowledge transfer in areas of strategic importance–TAP2), 112 pages, Available from: http://www.belspo.be/belspo/home/publ/rappP2 en.stm. Henrard, C., 2009. Numerical simulations of the Single Point Incremental Forming Process. PhD thesis. Available from: http://bictel.ulg.ac.be/ETDdb/collection/available/ULgetd-04182009-184505/. Henrard, C., Bouffioux, C., Eyckens, P., Sol, H., Duflou, J.R., Van Houtte, P., Van Bael, A., Duchêne, L., Habraken, A.M., 2010. Forming forces in single point incremental forming. Prediction by finite element simulations, validation and sensitivity. Computational Mechanics, doi:10.1007/s00466-0100563-4. Lequesne, C., Henrard, C., Bouffioux, C., Duflou, J.R., Habraken, A.M., 2008. Adaptive remeshing for incremental forming simulation. In: Proceedings of Numisheet 2008 Conference , Interlaken, Switzerland, pp. 399–403. Micari, F., Ambrogio, G., Filice, L., 2007. Shape and dimensional accuracy in single point incremental forming: state of the art and future trends. Journal of Materials Processing Technology 191, 305–390.
Contents lists available at ScienceDirect
Journal of Materials Processing Technology journal homepage: www.elsevier.com/locate/jmatprotec
Experimental and numerical study of an AlMgSc sheet formed by an incremental process C. Bouffioux a,∗ , C. Lequesne a , H. Vanhove b , J.R. Duflou b , P. Pouteau c , L. Duchêne a , A.M. Habraken a a b c
University of Liège, Dept. ArGEnCo, Chemin des Chevreuils 1, B-4000 Liège, Belgium University of Leuven, Dept. of Mechanical Engineering, Celestijnenlaan 300b, B-3001 Heverlee, Belgium CRM Gent, Technologiepark 903c, B-9052 Zwijnaarde, Belgium
a r t i c l e
i n f o
Article history: Received 22 February 2011 Received in revised form 10 May 2011 Accepted 15 May 2011 Available online 20 May 2011 Keywords: Aluminium alloy AlMgSc Material parameter identification Finite element simulation Single Point Incremental Forming
a b s t r a c t A recently developed AlMgSc alloy is studied since this material, which is well adapted to the aeronautic domain, is poorly known. The first objective is to reach a better knowledge of this alloy to provide the missing useful information to the aeronautic industry and to help research institutes who want to simulate sheet forming processes by Finite Element (FE) simulations. A set of experimental tests has been performed on the as-received sheets, material laws have been chosen and the corresponding material parameters have been adjusted to correctly describe the material behaviour. The second objective is to study the applicability of the Single Point Incremental Forming process (SPIF) on this material. Truncated cones with different geometries were formed and the maximum forming angle was determined. A numerical model was developed and proved to be able to predict both the force evolution during the process and the final geometrical shape. Moreover, the model helps reaching a better understanding of the process. The characterisation method described in this research and applied on the AlMgSc alloy can be extended to other alloys. In addition, the numerical simplified model, able to accurately describe the SPIF process with a reduced computation time, can be used to study more complex geometries. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The aeronautic industry is constantly looking for ways to reduce the weight of aeronautic parts and structures. Therefore, the availability of a new material of the AlMgSc alloy family, combining a low specific weight, a high corrosion resistance, a high toughness and an excellent weldability is of major interest in this domain. This recently developed alloy, with compositions ranging from 3.1 to 5.1 at.% for Mg, 0.10 to 0.40 at.% for Sc and 0.05 to 0.20 at.% for Zr, is not commonly used. The first investigations of the material provider showed promising results. However, a detailed study was required to improve the knowledge of the material behaviour, to extend its applications and to study more complex processes both by experiments and by numerical investigations.
∗ Corresponding author. Tel.: +32 4 366 9219; fax: +32 4 366 9192. E-mail addresses: chantal.bouffi[email protected] (C. Bouffioux), [email protected] (C. Lequesne), [email protected] (H. Vanhove), joost.dufl[email protected] (J.R. Duflou), [email protected] (P. Pouteau), [email protected] (L. Duchêne), [email protected] (A.M. Habraken). 0924-0136/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2011.05.010
The Single Point Incremental Forming (SPIF) process is a flexible sheet metal forming method adapted to form various complex shapes using a milling machine, a multi-axis robot or a dedicated machine without the need of specific and costly tools, such as a punch and die, as described by Allwood and Shouler (2009). The sheet, of which the edges are clamped, is locally deformed by a spherical tipped tool following a succession of contours leading to the final shape of the sheet. Depending on the length of the tool path and speed of the tool, the forming process can take up to a few hours for large parts. It is, in consequence, adapted to small batch production and rapid prototyping. Considering the maximum part size is machine dependent, SPIF can be used to form small or large pieces, such as aeronautical components. Furthermore, the SPIF process, which is known to shift Forming Limit Diagrams (FLD) to higher formability (Filice et al., 2002) compared to conventional forming processes can be adapted to materials such as the AlMgSc alloy. Because SPIF has a high industrial interest, the applicability of this technique on the new material studied must be experimentally verified. The problem of shape inaccuracy which is usually an important limiting factor for SPIF applications, as underlined by Micari et al. (2007), must be solved. Moreover, the forming force prediction provides crucial data to choose the forming tool, to verify
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2. Experimental material characterisation As underlined by Aerens et al. (2009), before performing any forming process, it is useful to ensure the preservation of the tooling and the machinery and to verify that the platform used is stiff enough to avoid significant deviation resulting in errors in the geometry of the achieved parts. A first evaluation of the required force to form the sheet by the available SPIF equipment indicated that the as-received sheets from the manufacturer (3.2 mm in thickness) were too thick and had to be rolled down to 0.5 mm. A conventional symmetrical cold rolling was applied and followed by an additional annealing to recover the initial material properties. Different annealing temperatures between 100 and 475 ◦ C and two holding times of 30 and 60 min were investigated to recover the initial hardness of 110 HV2 and the mechanical properties. It turned out that an annealing at 450 ◦ C during 30 min was adequate.
Simple tensile tests 500
True stress (MPa).
the stiffness of the set-up and to avoid deformation and risk for the equipment. Aerens et al. (2009) established a useful regression formula to predict an approximate value of the axial force for any material when forming a cone. It is a function of the shape geometry, the work conditions and the material tensile strength only. However, even if the force is usually well predicted by Aerens’ formula, its inaccuracy can exceed 20% when a new material is used. Furthermore, its application is limited to cones. The SPIF process can be optimized by experimental trials and errors and/or by a numerical analysis. The numerical approach predicts the forming forces and the final shape according to the tool path. However such information is accurate only if the simulation has been validated by experiments. Previous researches by Bouffioux et al. (2008, 2010) demonstrated the effect of the material parameter identification method, the material model itself (Flores et al., 2007), the Finite Element Method (FEM) strategy such as the explicit or implicit choice (Henrard, 2009), the finite element type: solid or shell (Eyckens et al., 2010a; Henrard et al., 2010), the mesh density and the contact model (Henrard, 2009). Eyckens (2010b) incorporated through-thickness shear (TTS), also known as out-ofplane shear, into a Marciniak–Kuczynski model and observed that the through-thickness shear has a significant effect on the increasing forming limit curves of the process. Moreover, Emmens and van den Boogaard (2009) presented several mechanisms that can explain the enhanced formability of the SPIF process. The numerical inaccuracy of finite element models is one of the main problems often encountered by scientists. Furthermore, the simulations usually require a very high computation time because the tool path is long and can be complex. The combination of different strategies described by Bouffioux et al. (2008, 2010), Lequesne et al. (2008) to improve the material and numerical models and applied on this study aimes to be able to accurately simulate the process with a short computation time. After this introduction showing the interest of this study and the complexity of reaching a good numerical model of the SPIF process, Section 2 presents all the experimental tests performed on the AlMgSc sheets. The constitutive laws used to describe the material behaviour, the data fitted by the inverse method and the validation on a line test are presented in Section 3. Then, the experimental cone tests, which were performed on this material to evaluate its forming limits, are described in Section 4. A numerical model of a truncated cone was chosen and simulated. The results were compared with the experimental forces and shape to validate the model in Section 5. Finally, Section 6 presents the wall angle sensitivity which was compared with the experimental results and analytical predictions. The last section summarizes and discusses the results.
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400 300 200 100 0 0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
True strain Tens_0°
Tens_45°
Tens_90°
Hooke 0.2%
Fig. 1. Effect of the anisotropy on tensile tests.
Additional mechanical tests (tensile tests, simple and reverse shear tests) and microstructure verifications showed that the cold rolled and annealed sheets had the same behaviour as the initial ones. A set of mechanical tests was performed to describe the mechanical behaviour and to fit the parameters of the hardening laws. 2.1. Classical tests To investigate the anisotropy of the material, tensile tests were performed at 0◦ , 45◦ and 90◦ from the rolling direction. Fig. 1 shows a lower yield stress and tensile strength at 45◦ from the rolling direction without any significant change on the strain hardening coefficient n. The average Young’s modulus E = 70,500 MPa was computed from these tests. Tensile tests, performed along the rolling direction (RD, 0◦ ) with 3 different strain rates (0.0075s−1 , 0.075 s−1 and 0.1 s−1 ) did not show any significant change in the mechanical response of the material properties. The Lankford coefficients: r0 , r45 and r90 with r␣ = εp22 /εp33 are classically computed from tensile tests at 0◦ , 45◦ and 90◦ from the RD. ˛ is the angle between the longitudinal direction of the test specimen and the rolling direction. εp22 and εp33 are the plastic strain respectively in the transversal and thickness directions of the sample. As explained in Section 3.1, the Lankford coefficients are used to determine the parameters of the Hill yield locus. Due to the Portevin–Le Chatelier effect (serrated stress-strain curves as seen in Fig. 1, a kind of instability classically observed in AlMg alloys), it was neither possible to accurately define the Poisson ratio, nor the Lankford coefficient at 45◦ (r45 ) and at 90◦ (r90 ) from RD. However, the average r0 coefficient could be evaluated at 0.8 for a strain between 6 and 12%. The Poisson ratio was estimated equal to 0.33 as for classical Al alloys. Additional tests, as used to determine the yield locus shape and to fit the material data of the isotropic and the kinematic hardening laws, were performed: a monotonic shear test (Figs. 2a and 3, left), a large tensile test (Figs. 2b and 3, right), two orthogonal tests (a large tensile test followed by a shear test) at two levels of pre-strain: 4% and 8% (Figs. 2c and 4, left) and two Bauschinger shear tests (a shear test followed by a reverse shear test) at three different levels of pre-strain: Gamma = d/b = 8.8%, 13% and 29% (Figs. 2d and 4, right). Each test was performed three times. As the results showed a good reproducibility, an average curve was used and drawn for each loading state. The tensile, shear and large tensile tests were performed in three directions: at 0◦ (RD), 45◦ and 90◦ (TD) from the rolling direction to see the effect of the anisotropy. It was observed that the behaviour was different at 45◦ than in RD or TD.
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a
d
b
d
c
d
b
Fig. 2. Description of the additional classical tests: (a) shear test, (b) large tensile test, (c) orthogonal test and (d) Bauschinger test, definition of b and d values.
Large tensile tests
Shear tests 250
True stress (MPa).
Shear stress (MPa).
300
200 150 100 50 0 0.0
0.2
0.4 Gamma
Sh_RD
0.6
0.8
500 450 400 350 300 250 200 150 100 50 0 0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
Strain
Sh_45
TL_RD
Sh_TD
TL_45
TL_TD
Fig. 3. Experimental shear tests (left) and large tensile tests (right) in 3 directions (0◦ , 45◦ and 90◦ from RD).
Bauschinger tests Abs(shear stress) (MPa).
True stress / Shear Stress (MPa) .
Orthogonal tests
450 400 350 300 250 200 150 100 50 0 0.00
0.05
0.10 0.15 Major strain / gamma
OR_4%
0.20
250 200 150 100 50 0 0.0
0.2 BA_8.8%
OR_8%
0.4 0.6 Sum (Gamma)
0.8
BA_13%
1.0
BA_29%
Fig. 4. Experimental orthogonal tests (4 and 8%) (left) and Bauschinger tests (8.8, 13 and 29%) (right).
All these tests were used to describe the yield locus shape and its evolution. The decrease of the stress in the second part of the Bauschinger tests indicated that a complex hardening law combining both an isotropic and a kinematic part was required.
2.2. Indent and line tests The line test was performed on a square sheet with a thickness of 0.5 mm, clamped along its edges (Fig. 5, left).
Y
100 mm
Z
Y
X tool
1
X
tool 20 mm
100 mm Fig. 5. Description of the line test: geometry (left) and tool displacements (right).
2 4 60 mm
3 20 mm
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The displacement of the spherical tool, having a radius of 5 mm, was composed of five steps with an initial position tangent to the surface of the sheet (Fig. 5, right): a first indent of 3 mm (step 1), a line movement at the same depth along the X axis (step 2), then a second indent up to the depth of 6 mm (step 3) followed by a line at the same depth along the X axis in the opposite direction (step 4) and the unloading. The first step corresponds to what is called hereafter the indent test, while all of the five steps define the line test. The whole line test was performed three times (with the SPIF set-up) and the bolts of the frame were tightened using the same torque to ensure the reproducibility of the results. As explained in Section 3.1, the total tool force during the indent test was combined with all of the classical tests to fit the material data by the inverse identification method. The line test, presenting deformation mechanism close to the SPIF, was used as validation of the material parameter fitting (see Section 3.2).
The elastic coefficients and most of the parameters related to the yield locus shape could be defined by classical tensile tests, but the inverse method was required to fit the hardening behaviour and the shear Hill coefficients. This inverse method was coupled with the Finite Element code: “Lagamine” described by Duchêne and Habraken (2005). The principle of this method is to choose a set of tests, the results of which are sensitive to the material data to adjust. These tests are simulated using an initial set of data, chosen arbitrarily. Then, the numerical results are compared with the experimental measurements. The Levenberg Marquardt minimization algorithm described by Dennis and Schnabel (1983) is used to iteratively adjust the material data until a sufficient accuracy is reached. All the classical tests, described in Section 2.1, and the indent test (Fig. 5) were simultaneously used in the inverse method. Indeed, Bouffioux et al. (2008, 2010) observed that simple classical tests inducing in-plane stresses were not sufficient to describe the material behaviour during the SPIF process. The indent test gave the missing information about the out-of-plane material behaviour since it induces through thickness shear. In addition, the same shell elements were used for these simulations and for the SPIF process. It was decided to impose isotropic in-plane behaviour because neither the experimental force evolution during a whole contour of a cone, nor the cone wall thickness comparison in the different sections (at 0◦ , 45◦ and 90◦ from RD), shows any effect of anisotropy. This simplification induced the replacement, in the inverse method, of classical test results in three directions by their average curve. The elasto plastic constitutive law was to be used on a simplified numerical model (a 90◦ pie) where the force evolution of a contour was replaced by an average force when the tool was in the central third of the pie. 3.1. Material model The elastic range was described by Hooke’s law and the plastic part by the Hill 48 law (Eq. (1)) where ij are the stress tensor components and where F is the yield stress. 1 H(xx − yy )2 + G(xx − zz )2 FHILL () = 2
− F2 = 0
(1)
The parameters F, G, H of Hill’s law (Table 1) could be identified from the relation between the yield stress limit in RD and TD (Eq. (2)):
yield
TD
=
2 yield H + F RD
Table 1 Material data of the AlMgSc alloy (units: N, mm). Tests used
Classical and indent tests
Yield surface Coefficients (Hill’48)
Voce parameters
Back stress data (Ziegler)
F = 1.11 G = 1.11 H = 0.89 N=3
K = 275 0 = 8.5 n = 439
CA = 930 GA = 18.1
However, the in-plane isotropic assumption above-mentioned was used. yield
TD
yield
= RD
The usual additional condition for Hill (1948) was applied (Eq. (3)): H+G =2
3. Material behaviour modelling
2 2 2 +F(yy − zz )2 + 2N xy + xz + yz
1687
(2)
(3)
Finally, the Lankford coefficient formulation allowed to characterize the out-of-plane anisotropy (Eq. (4)): r0 =
H G
(4)
where r0 = 0.8, as computed from the tensile tests. The isotropic hardening was described by Voce’s formulation (Eq. (5)) because this law, which predicts a saturation (Bouffioux et al., 2010; Habraken et al., 2010), was better adapted to describe the material behaviour for the large deformation encountered during the SPIF process. F = 0 + K(1 − exp(−n• εpl ))
(5) εpl
where 0 , K and n are the material parameters and is the equivalent plastic strain. It was noticed that a simple isotropic hardening model (Eq. (5)) was not sufficient to provide an accurate tool force prediction (Flores et al., 2007 and Henrard et al., 2010). Therefore an elasto-plastic law with a mixed isotropic-kinematic hardening was used. In this law, the stress tensor (in Eq. (1)) is replaced by ( − X), where X is the back-stress. The evolution of the back-stress is described by Ziegler’s hardening equation (Eq. (6)): X˙ = CA
1 ( − X) ε˙ pl − GA • X • ε˙ pl F
(6)
where CA is the initial kinematic hardening modulus and GA is the rate at which the kinematic hardening modulus decreases with an increasing plastic deformation. These models use the Green strains and the second formulation of Piola–Kirchhoff for the stresses. As already specified in Section 2.1, the Young’s modulus was defined by the tensile tests (E = 70,500 MPa) and the Poisson ratio was fixed equal to 0.33 as for other aluminium alloys. The shear parameter N of the Hill’s law, which could not be defined accurately by the classical tests was combined to the Voce’s law data and the back stress parameters to be fitted by the inverse method (Table 1). The final data were fitted using the classical tests and the indent test. All these tests as well as the SPIF process were simulated with quadrilateral shell elements with four nodes. As shown in Figs. 6–8, this set of data was able to accurately predict all the experimental curves. 3.2. Validation of the material model parameters The line test, described in Fig. 5, was used to verify the accuracy of the material data on a complex procedure similar to the incremental forming process, performed with the SPIF set-up and inducing bending, and an out-of-plane stress field.
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shear test 300
Shear stress (MPa).
True stress (MPa).
Simple tensile and large tensile tests 500 450 400 350 300 250 200 150 100 50 0 0.00
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strain
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Gamma
SimpleTens exp LargeTens exp
SimpleTens Num. LargeTens Num.
Shear exp
Shear Num.
Orthogonal tests 450 400 350 300 250 200 150 100 50 0 0.00
Bauschinger tests
Shear stress (MPa).
True stress / Shear stress (MPa).
Fig. 6. Comparison between experimental and numerical tensile and large tensile tests (left) and simple shear tests (right).
0.05
0.10
0.15
0.20
-0.2
250 200 150 100 50 0 -50 0.0 -100 -150 -200 -250
-0.1
0.25
Ortho_4% Num.
Ortho_8% exp
0.2
0.3
Gamma
Major strain / gamma Ortho_4% exp
0.1
Ortho_8% Num.
Bau_8.8 exp
Bau_8.8 Num.
Bau_29 exp
Bau_29 Num.
Fig. 7. Comparison between experimental and numerical orthogonal tests (left) and Bauschinger tests (right).
Indent test 400 350
Force (N).
300 250 200 150 100 50 0 0
0.1
0.2
0.3
0.4
0.5
Time (sec) Indent test exp
Indent test Num.
Fig. 8. Comparison between experimental and numerical tool force during the indent test.
In the FEM simulation, the nodes along the edges were fixed. The tool force was computed by the static implicit strategy. The Coulomb’s friction coefficient was unknown. The same coefficient of 0.05 as Henrard (2009) for the alloy 3013 was applied between the tool and the sheet. It was observed that the impact of this coefficient on the total tool force is small (the force computed with a friction of 15% is 1% higher than the force without friction). The mesh was adjusted to limit the number of elements (576 elements) while keeping accuracy (Fig. 9, left), taking into account the symmetry. The same shell elements as in the inverse method were used. Fig. 9, right, shows that the model gives a good evaluation of the evolution of the experimental tool total force.
4. Experimental incremental forming applicability on a cone As illustrated in Fig. 10, a firmly clamped sheet was formed by a contouring operation of the forming tool. The tool moved along a predetermined contour in the horizontal plane, after which the tool incrementally descended and started a contour in the next horizontal plane, building up the workpiece layer by layer. The two platforms used in the framework of this project were a 6-axis robot and a 3-axis rigid milling machine (Fig. 11), both equipped with a dedicated force measuring device. The forming speed (approx. 2000 mm/min), in combination with the long toolpath, resulted in a slow forming process.
C. Bouffioux et al. / Journal of Materials Processing Technology 211 (2011) 1684–1693
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Y Line test 1400
100 mm
X
Force (N).
1200 1000 800 600 400 200
tool
0 0
5
1 00 mm
10
15 Time (sec)
Line test exp
20
25
Line test Num.
Fig. 9. Line test: geometry and meshing (left) and comparison between experiment and numerical tool force evolution (right).
Fig. 10. Description of the SPIF process.
Fig. 11. Platforms description.
Thinning of the workpiece is the dominant failure mode in SPIF and is related to the workpiece drawing angle ␣ (Fig. 12). For a given material and initial thickness, the maximum drawing angle represents the limits of the conventional incremental forming process.
Fig. 12. Workpiece geometry.
Determining the geometrical forming limits defined by the maximum drawing angle ˛ (Fig. 12) in Single Point Incremental Forming was done by means of well-established cone tests. Forming limit tests typically started with the formation of a cup with a 10◦ wall angle. Then, successive cups with an increasing wall angle were made. The forming limit is the angle at which the sheet fractured. The maximum drawing angle is mainly influenced by the initial thickness of the blank sheet. To a lesser degree, the diameter of the spherically tipped tool and the scallop height affect the drawing angle in a negative way. The scallop height is specified as the theoretic height of the ripple between two passes of the forming tool as calculated for a milling process. The maximum value of ˛, for the studied AlMgSc alloy, was found equal to 46◦ with an accuracy of 1◦ (Table 2).
C. Bouffioux et al. / Journal of Materials Processing Technology 211 (2011) 1684–1693
Table 2 Cone parameters and maximum forming angle. Material
Thickness
Ø tool
Scallop height
Step down
˛
AlMgSc
0.5 mm
10 mm
0.005 mm
0.322 mm
46◦
Comparing different materials reveals a fairly low forming limit for AlMgSc. This is a strong limiting factor for the part geometries which could be made in this material. Duflou et al. (2007) showed that performing incremental forming at elevated temperatures can significantly improve forming limits. However, this property was not verified on the AlmgSc. 5. Numerical models of a cone processed by SPIF A truncated cone test was chosen to validate the material model. A circular AlMgSc sheet with a thickness of 0.5 mm was clamped. 30 contours were performed with a tool radius of 5 mm, a depth increment Z of 0.5 mm between successive contours, corresponding to a scallop height of 0.015 mm, and a wall angle of 40◦ (Fig. 13). Strategies had to be found to reduce the computation time. As introduced in Section 3, the use of shell elements instead of e.g. 3D solid elements aimed to considerably reduce the computation time. Indeed, solid elements would require minimum three layers of elements to model bending. It would lead to a significantly larger number of degrees of freedom. The mesh used to simulate the process is presented in Fig. 14. Since the tool always moved in the same tangential direction, it was possible to reduce the computation time by modelling only a quarter of sheet. Rotational boundary conditions (Henrard, 2009; Henrard et al., 2010) were imposed by a link between the displacements of the edges, as schematically presented in Fig. 14, left: the rotation, the tangential and the radial displacements were forced to have the same values on both the horizontal and vertical boundaries of the mesh. This link is related to the six degrees of freedom (3 translations and 3 rotations) of each node of the edges along the X and Y axes. The remeshing method (Lequesne et al., 2008) was used to refine the mesh by division of the elements close to the tool into nine elements. The refinement was automatically removed when the tool went further, except when the elements distortion was high (Fig. 14, centre and right).
Two meshes were tested: a very fine mesh (2692 elements) without the remeshing method and the coarse mesh of Fig. 14 (248 elements) combined with the remeshing method. The level of the force was almost the same for both simulations, but the computation time was divided by a factor of 49 for the second case. The Coulomb friction coefficient , which was not accurately known, was taken equal to 0.10. 5.1. Tool force comparison The radial FR , tangential FT and axial FZ force components, respectively in R, T and Z directions (Fig. 14, left) and the norm of the force Ftot were compared with the experimental ones. To simplify the comparison, the force evolution during each contour was replaced by the average of the force. The numerical test values were computed when the tool was in the central third of each contour to avoid small inaccuracies due to the boundary conditions. Fig. 15 shows that similar results were obtained with a fine mesh without remeshing and a coarse mesh combined with the remeshing method. Similar small differences could be observed for all the force components.
Total force (N).
1690
550 500 450 400 350 300 250 200 150 100 50 0 0
5
10
15
20
25
30
Contour Experimental
Fine mesh
Coarse mesh + remesh
Fig. 15. Comparison of the experimental total tool force with the simulation results for two cases: the fine mesh without remeshing and the coarse mesh with remeshing.
Y
Z X X
tool
40°
a b
c
b a
92 mm ∅: 90 - 92 mm Fig. 13. Description of the cone test (a = unsupported area = 1 mm, b = 17.88 mm and c = 54.25 mm).
Fig. 14. Cone test: Initial mesh, axes, radial (R) and tangential (T) directions, boundary conditions (left) and mesh evolution with the remeshing method (centre and right).
C. Bouffioux et al. / Journal of Materials Processing Technology 211 (2011) 1684–1693
Ftot
FZ
550 500 450 400 350 300 250 200 150 100 50 0
FR & Ftot (N)
FT & FZ (N)
FZ_S
FT
0
1691
5
10
15
20
25
550 500 450 400 350 300 250 200 150 100 50 0
FR
0
30
5
10
15
Experimental
Numerical
20
25
30
Contour
Contour
Experimental
Analytic
Numerical
Fig. 16. Comparison of the numerical (coarse mesh + remeshing), analytical (FZ S ) and experimental total, axial, tangential and radial forces.
Fig. 16, left and right show that the numerical model with a coarse mesh combined with the remeshing method can predict all the force components: FR , FT , FZ and Ftot with accuracy. In addition, it would be possible to improve the level of the FT component by adapting the friction coefficient. Henrard (2009) observed that friction has almost no impact on FR and FZ while FT is linearly affected by the friction coefficient. The ratio FT /FZ reached by two simulations with = 0.05 and = 0.10 was respectively: 0.157 and 0.212. By extrapolation, the value: = 0.14 should lead to the experimentally measured ratio FT /FZ of 0.256. The axial force when a steady state is reached: FZ S could also be predicted by an analytic generalized formula (Aerens et al., 2009) based on the tensile strength of the material Rm , the sheet thickness t, the tool diameter dt , the scallop height h and the wall angle ˛ (Eq. (7)): FZ
S
=
0.41 • 0.0716• Rm • t 1.57• dt h0.09• ˛ cos ˛
where FZ S is expressed in N, Rm in t, dt and h in mm and ˛ in degrees. As before-mentioned, the scallop height h is the theoretic height of the ripple between two passes of the forming tool. It is linked to the depth increment Z, the tool diameter and the wall angle by the relation (Eq. (8)):
h(dt − h) ≈ 2 sin ˛
h• dt
(8)
The tensile strength determined by the tensile tests was 352 MPa, the scallop height was 0.0151 mm from Eq. (8) and thus the force prediction FZ S was equal to 458.5 N. This level, represented in Fig. 16, left, gave a good prediction for this alloy and geometry. The radial force FR could be linked to the axial force FZ S (Aerens et al., 2009) by the following relation (Eq. (9)):
FR = FZ
s
• tan
-30
-10
10
30
50
-4 -8 -12 -16 -20
Numerical shape
Robot mid-thickness, X=0
Robot mid-thickness, Y=0
Fig. 17. Comparison of the mid-thickness shapes.
Only a small difference was observed between the two sections: X = 0 and Y = 0 and between the cones formed by the robot and the milling machine used for the experiments. The accuracy measurements were obtained on the tool side; it is the inner shape. The local thickness was found by comparing the scan data from the outside of the part with respect to the inside. The mid-thickness layer was computed (Fig. 17) point by point, taking into account the local wall angle and the thickness. This correction had a small impact on the shape due to the low sheet thickness. The numerical test shape was extracted out of the section: X = Y to avoid small imprecision due to the boundary conditions. It was the mid-thickness layer obtained by the nodal coordinates. Fig. 17 shows that the final shape was well predicted by the model. The higher discrepancy was observed at the centre of the cone with an error of 0.9 mm.
−c
˛ + ˇ − 17.2• dt /10 2
0 -50
(7)
N/mm2 ,
Z = 2 sin ˛
Experimental and numerical shapes
(9)
where ˇ = arccos (1 − 2·h/dt ) (in degrees) and c = 2.54 for aluminium alloys. This equation provided a first approximation of FR = 111.2 N which was too low in the current case. However, as mentioned by Aerens et al. (2009), a correction term should be used to improve this equation, but its determination would require additional experiments that were not performed on the AlMgSc material. 5.2. Shape comparison The experimental shapes were measured by means of laser line scanning at the end of the process when the tool was removed.
6. Wall angle sensitivity Earlier research on forces in SPIF revealed the important and complex non-linear influence of the draw angle on the forming forces. Because of this irregular behaviour a more in-depth study was performed towards this influence. A set of cones similar to the cones used for the forming limit analysis was experimentally formed to see the angle sensitivity. Once again, a circular AlMgSc metal sheet with a thickness of 0.5 mm was clamped on a backing plate with a diameter of 90 mm. The contours were performed with a tool diameter of 10 mm and the same scallop height of 0.005 mm for all the cones. While forming the cones with draw angles between 10◦ and 46◦ and the step down indicated in Table 3, forces were measured when a steady state was reached.
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Fig. 18. Effect of the wall angle on the force components and comparison between experimental and numerically simulated forces (left) and comparison between experimental, analytical and simulated axial (FZ ) and radial (FR ) forces (right). Table 3 Step down with respect to wall angles for a tool diameter of 10 mm and a scallop height of 0.005 mm. Wall angle (◦ )
10
20
30
40
45
46
Step down (mm)
0.078
0.153
0.224
0.287
0.316
0.322
Three cones with the wall angles of 20, 30 and 40◦ were also simulated using a numerical model with a coarse mesh and the remeshing method, the same tool path as in the experiment and a friction coefficient of 0.14. 30 contours were performed as it was observed that the forces reached a steady state from the 12th contour. Then, the average values of the tool forces were computed when the tool was in the central third of each contour. As a result, these force components with respect to the wall angle were compared with the experimental forces (Fig. 18, left and right). It is clearly visible in Fig. 18, right, that the axial force: FZ was the dominating force component in the SPIF process. Also, the nonlinearity between the draw angle and the force components is visible. The numerical model was able to give a good approximation of the force components. The radial FR and axial FZ tool forces were also computed by Eqs. (7) and (9) (Aerens et al., 2009). These equations which were established for five different materials gave also a good estimation of these two components (Fig. 18, right) even if the AlMgSc alloy was not considered in that study.
tion time. It was possible to simulate the 30 contours of a cone test with a simple personal computer within 5 h and last but not least, with a good precision. The effect of the wall angle on the forming forces was also studied experimentally, analytically and by numerical simulation. Once again, the numerical simulation model was validated. In future work, the results could be further improved by using more complex elements, such as solid-shell type elements (Alves de Sousa et al., 2007) which would take into account the throughthickness shear. However, more complex elements will affect the computation time. As conclusions, it is clear that this study gives a new contribution in AlMgSc characterisation and in incremental forming application and simulations. Acknowledgments The authors would like to thank the Belgian Federal Science Policy Office (IAP: project P6/24 and ALECASPIF: Pat2 project P2/00/01) for its financial support and Aleris for the material supply. A.M. Habraken and L. Duchêne would like to thank the Fund for Scientific Research (FNRS, Belgium) for its support. J. Duflou and H. Vanhove recognize the support by FWOvlaanderen for the project. References
7. Conclusions The aims of this research were to improve the knowledge of the AlMgSc sheets, to study the applicability of the SPIF process and to simulate it on this material. Classical tests gave information about in-plane material behaviour. An indent test performed with the SPIF setup, inducing heterogeneous stress and strain fields similar to those present in the SPIF process, provided information for both in-plane and out-of-plane strain states. A mixed isotropic-kinematic hardening, described by both a saturating hardening law such as Voce’s law and the kinematic Ziegler’s law, accurately models the material behaviour as observed during all the simulated tests. To quantify the formability of the AlMgSc alloy in the SPIF process, experimental investigations on cup tests showed that for a spherically tipped tool of Ø10 mm and a scallop height of 0.005 mm, a maximum wall angle of 46◦ was found. To simulate the SPIF process, the use of shell elements, a 90◦ pie coarse mesh combined with the remeshing method and rotational boundary conditions allowed to considerably reduce the computa-
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