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A constitutive model for anisotropic tubes incorporating the variation of contractile strain ratio with deformation
International Journal of Mechanical Sciences 90 (2015) 33–43
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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
A constitutive model for anisotropic tubes incorporating the variation of contractile strain ratio with deformation T. Huang, M. Zhan n, H. Yang, J. Guo, J.Q. Tan State Key Laboratory of Solidification Processing, School of Materials Science and Engineering, Northwestern Polytechnical University, Xi'an 710072, China
art ic l e i nf o
a b s t r a c t
Article history: Received 13 May 2014 Received in revised form 23 September 2014 Accepted 20 October 2014 Available online 28 October 2014
The contractile strain ratio (CSR) is an important index measuring the anisotropy of tubes. The values of CSR vary non-linearly with the amount of plastic deformation. However, up to now, there are no constitutive models which are able to take this variation into account. In order to obtain the actual deformation characteristics and improve the prediction capability for plastic deformation processes of anisotropic tubes, this study developed an elasto-plastic constitutive model based on Hill's 48 anisotropic yield function. In this constitutive model, the variation in CSR with plastic deformation was considered. A user material subroutine VUMAT incorporating the constitutive model was developed based on ABAQUS/Explicit platform. The comparisons between the finite element (FE) simulations and experiments for uniaxial tension, uniaxial compression and numerical control (NC) bending process of Ti–3Al–2.5 V titanium alloy anisotropic tube, show that this constitutive model is reliable. The simulation accuracy was improved and the plastic deformation characteristics could be better described for the anisotropic tubes by using this constitutive model. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Anisotropic tubes Constitutive model Contractile strain ratio Plastic deformation Ti–3Al–2.5V
1. Introduction As one kind of key lightweight components for liquid transforming and loading carrying with enormous quantities, lightweight and high-strength bent metal tubes have been widely applied in many industries such as aerospace, aviation, oil, and related high-technology industries [1–3]. However, due to the complex metallurgical process of cold-working, the mechanical performances of these tubes are significantly influenced by the mechanical anisotropy [4]. Such mechanical anisotropy is introduced by the development of preferred crystallographic texture during tube processing. The tubes' anisotropy is usually described by the contractile strain ratio (CSR), which is a very important macroscopic deformation index. A number of methods have been used to measure the CSR of tubes, including the post-test manual measure method [5], the continuously loading–unloading measure method by utilizing the strain gages [6] or the method based on digital speckle correlation method [7,8]. In recent years, some research works on the influence of CSR values on tube bending processes have been carried out. Jiang et al. [9,10] revealed that the increase in CSR led to the decrease in springback angle for titanium alloy bent tubes. Hasanpour et al. [11] revealed that a larger CSR led to a higher flattening of the
n
Corresponding author. E-mail address: [email protected] (M. Zhan).
http://dx.doi.org/10.1016/j.ijmecsci.2014.10.014 0020-7403/& 2014 Elsevier Ltd. All rights reserved.
cross-section, and a serious wall thickness variation when the CSR of the tube is less than 1. Among these research works [9–11], the CSR was always considered as a constant in the whole deformation process independent of the plastic deformation degree. That is to say, these studies did not consider the variation in CSR during the plastic deformation process. However, CSR values vary with the amount of deformation during the plastic forming processes of tubes. Therefore, in order to obtain the deformation characteristics of tubes during plastic forming processes more accurately, the dependence of CSR on deformation should be considered in numerical simulation. Up to now, no commercial finite element (FE) codes can provide such an elasto-plastic constitutive model and there followed by a question that is: how to contain the variation in CSR with deformation into FE model of the plastic forming of tubes. This needs a constitutive model which can take the variation in CSR into consideration at first. Up to now, there are some yield functions which can be used to express the deformation behavior of anisotropic materials, including Hill's 48, Hill's 79, Barlat 89, Barlat 91, and Hosford, etc. [12]. Hill's 48 anisotropic yield function was widely used to analyze deformation of metal sheets due to its general stress state, relative ease of formulation, and easily obtained parameters through uniaxial experiments [13]. Lee et al. [14] developed an elastoplastic constitutive model based on Hill's 48 yield function, and proposed a stable integration algorithm using the Newton‐Raphson method to solve the model. The constitutive model was implemented into the explicit FE code named FORMSYS-DE.
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T. Huang et al. / International Journal of Mechanical Sciences 90 (2015) 33–43
The simulations for the thickness strain of an annealed aluminum and mild steel showed a good correlation with experiments. Ni [15] developed an explicit expression of the elasto-plastic constitutive model based on Hill's 48 and incorporated this model into ABAQUS via the user subroutine VUMAT. The 3D simulation for the drawing process of an aluminum square box showed that this model was reliable. Tang et al. [16] established a constitutive model based on Hill's 48 and calculated the stress and strain increment by a backward-Euler return mapping algorithm. The model was implemented into ABAQUS via the user subroutine UMAT. The results showed that this model could better describe the plastic deformation of aluminum alloy sheet. Zang et al. [17,18] have extended Tang's work by considering the variation in Young's modulus with plastic deformation for high strength steels and aluminum alloys. Similarly, Zang et al. concluded that the constitutive model based on Hill's 48 yield function was generally the most accurate for prediction sheets' deformation. However, according to the aforementioned references, it can be found that the studies for constitutive models have focused on metal sheets, while not much reported on tubes. As there is a similarity between sheets and tubes in deformation characteristics, extending Hill's 48 yield function to plastic deformation of anisotropic tubes is still acceptable. In this study, Hill's 48 yield function was extended to research plastic deformation of anisotropic tubes. An elasto-plastic constitutive model was established to describe the variation in CSR with plastic deformation. Then, on the platform of ABAQUS/Explicit, a user material subroutine VUMAT was developed. The single cubic FE models for uniaxial tension, uniaxial compression, and pure shearing were conducted to verify the correctness and reliability of the constitutive model. Finally, FE simulations and experiments for the uniaxial tension, uniaxial compression and numerical control (NC) bending processes of Ti–3Al–2.5V (ASTM Gr. 9) titanium alloy tubes, which are typical anisotropic tubes, were carried out to determine whether this elasto-plastic constitutive model could help to improve the simulation accuracy of tubes deformation.
2. Extension of Hill's 48 yield function to anisotropic tubes 2.1. Original Hill's 48 anisotropic yield function In 1948 Hill proposed the orthogonal anisotropic yield surface as Φ σ ij ; k ¼ Fðσ 22 σ 33 Þ2 þ Gðσ 33 σ 11 Þ2 þ Hðσ 11 σ 22 Þ2 þ2Lσ 223 þ 2M σ 231 þ 2N σ 212 2k 2 ¼ ϕ σ ij 2k ¼ 0
2
ð1Þ
where F, G, H, L, M, and N are anisotropic characteristic parameters, their values can be determined by experiments; 11, 22, and 33 are orthogonal anisotropic principal axes [12]. The anisotropic parameters F, G, H, L, M, and N in Hill's 48 can be written as 8F 1 1 1 ð2 1Þ > 2 ¼ 2 þ 2 2 > σ 22 σ 33 σ 11 >k > > > G 1 1 1 > ð2 2Þ > 2 ¼ 2 þ 2 2 > k σ 33 σ 11 σ 22 > > >H > 1 1 1 > < k2 ¼ σ 2 þ σ 2 σ 2 ð2 3Þ 11 22 33 ð2Þ L 1 > ð2 4Þ 2 ¼ 2 > k > σ 23 > > > M 1 > > ð2 5Þ > k2 ¼ σ 231 > > > > > N2 ¼ 12 : ð2 6Þ σ 12
k
where σ ij are the yield stresses with the only nonzero stress σ ij ; k is a function of plastic work and hardening. The yield stress can be written as Eq. (3) for isotropic material
σ 11 ¼ σ 22 ¼ σ 33 ¼ σ
ð3Þ
The sum of Eqs. (2-1), (2-2), and (2-3) in Eq. (2) can be written as F þGþH 2
k
¼
3
ð4Þ
σ2
Hill yield surface could be derived as Mises yield surface when 2 2 F ¼ G ¼ H ¼ 1=2, L ¼ M ¼ N ¼ 3=2 and σ 2 ¼ 2k . 2k is applicable for both cases of the isotropic and anisotropic [15]. And Hill anisotropic yield surface could be expressed as Eq. (5). In this study, the Swift hardening equation (Eq. (6)) is adopted [19] Φ σ ij ; σ ¼ ϕ σ ij σ 2 ¼ 0 ð5Þ
σ ¼ Kðεp þ bÞn
ð6Þ
According to yield stress ratio, the anisotropy parameters F, G, H, L, M, and N can be expressed as 8 2 > > > F ¼ σ2 12 þ 12 12 ¼ 12 R12 þ R12 R12 > σ 22 σ 33 σ 11 > 22 33 11 > > > > > 2 > σ 1 1 1 1 1 1 > G ¼ 2 2 þ 2 2 ¼ 2 R2 þ R2 R12 > > σ σ σ > 33 11 22 33 11 22 > > > > 2 > σ 1 1 1 1 1 1 > < H ¼ 2 2 þ 2 2 ¼ 2 2 þ 2 12 R11 R22 R33 σ 11 σ 22 σ 33 ð7Þ 2 > > > 3 τ 3 > ¼ L ¼ > 2 σ 23 > 2R223 > > > 2 > > 3 τ 3 > > > M ¼ 2 σ 13 ¼ 2R213 > > > 2 > > > 3 τ 3 > : N ¼ 2 σ 12 ¼ 2R2 12
Fig. 1. FE models for single cubic element under different loading paths: (a) uniaxial tension, (b) uniaxial compression and (c) pure shearing.
T. Huang et al. / International Journal of Mechanical Sciences 90 (2015) 33–43
Eq. (12) can be obtained
where Rij is yield stress ratio and the subscript denotes the direction: 11 is the axial direction, 22 is the transverse direction and 33 is the thickness direction; τ ¼ σ =3. Since r of sheets is the ratio of width true strain to thickness true strain, it is necessary to transform the strain ratio into the stress ratio [15]. Eq. (8) can be obtained when the uniaxial tensile deformation is taken in 11 direction for sheets: dε11 : dε22 : dε33 ¼ G þ H : H : G
8 R11 ¼ 1 > > < R22 ¼ 1 1=2 > > : R33 ¼ r0 þ 1 2
H : dε33 ¼ G
2.2. Incorporating CSR's variation with deformation into Hills' 48 yield function
ð9Þ
The tubes' anisotropy is usually described by the CSR. The CSR is defined as the ratio of the true circumferential plastic strain to the true radial (i.e. thickness) plastic strain of a tube [5]. Moreover, the CSR is a function about plastic strain, and varies with plastic deformation. In this study, the relation between the yield stress ratio Rij and CSR can be written as
Eq. (10) can be obtained when the uniaxial tensile deformation is taken in 22 direction for sheets: dε11 : dε22 : dε33 ¼ H : F þ H : F
ð10Þ
The ratio of strain in axial direction to that in thickness direction can be written as r 90 ¼ dε11 : dε33 ¼
ð12Þ
ð8Þ
The ratio of strain in width direction to that in thickness direction can be written as r 0 ¼ dε22
35
H F
8 R ¼1 > > < 11 R22 ¼ 1 > > : R ¼ CSR þ 1 1=2
ð11Þ
r 0 is equal to r 90 and σ is equal to σ 11 for sheets or tubes only considering the anisotropy in thickness direction. Therefore,
33
ð13Þ
2
Table 1 Mechanical constants of Ti–3Al–2.5V titanium alloy anisotropic tubes. Parameters
Poisson's ratio v
Yield stress σ 0:2 (MPa)
Ultimate tensile stress σ b (MPa)
Strength coefficient K (MPa)
Hardening exponent n
Φ12 mm t0.9 mm Φ6 mm t0.5 mm Φ8 mm t0.6 mm
0.29 0.25 0.30
812.26 805.43 750.00
1043.94 1039.91 940.00
1267.14 1261.52 1483.24
0.064 0.077 0.123
1400
1400
Simulation results Input data
1000
Plastic phase
800
Elastic phase
600 400 200 0
1000
0.1
0.2
0.3
0.4
0.5
0.6
Plastic phase
800
Elastic phase
600 400 200 0
0.0
Simulation results Input data
1200
Equivalent stress σe (MPa)
Equivalent stress σe (MPa)
1200
0.7
0.0
0.2
0.8
1.0
Simulation results Input data
1200
Equivalent stress σe (MPa)
0.6 p
Equivalent plastic strain ε
1400
0.4
Equivalent plastic strain ε
p
1000
Plastic phase
800
Elastic phase
600 400 200 0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
p
Equivalent plastic strain ε
Fig. 2. Comparison in stress–strain of simulation results and input data during different deformation processes: (a) uniaxial tension, (b) uniaxial compression and (c) pure shearing.
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T. Huang et al. / International Journal of Mechanical Sciences 90 (2015) 33–43
4.0
4.0
3.5
3.5
Simulation results Input data
3.0
2.5
CSR
CSR
3.0
Simulation results Input data
2.5
2.0
2.0
1.5
1.5
1.0
1.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
Equivalent plastic strain
Equivalent plastic strain
4.0 3.5
Simulation results Input data
CSR
3.0 2.5 2.0 1.5 1.0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
Equivalent plastic strain Fig. 3. Comparison in CSR's variation of simulation results and input data during different deformation processes: (a) uniaxial tension, (b) uniaxial compression and (c) pure shearing.
3.5
3.5
3.0
3.0
Experiment Second-order equation
2.5
CSR
CSR
2.5 2.0
2.0
1.5
1.5
1.0
1.0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18
Experiment Second-order equation
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18
Plastic strain
Plastic strain
4.0 3.5
CSR
3.0
Experiment Second-order equation
2.5 2.0 1.5 1.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 Plastic strain
Fig. 4. Fitting curves between CSR and plastic strain for Ti–3Al–2.5V titanium alloy anisotropic tubes: (a) Φ6 mm t 0.5 mm, (b) Φ8 mm t 0.6 mm and (c) Φ12 mm t 0.9 mm.
T. Huang et al. / International Journal of Mechanical Sciences 90 (2015) 33–43
ϕ σ ij in Eq. (1) can be reformulated as Eq. (14) by a symmetric matrix P (Eq. (15)) given in a vector Voigt notation 1 ϕ σ ij ¼ σ T Pσ 2 The symmetric matrix P in Eq. 0 CSR 1 CSR CSR1þ 1 þ1 B CSR B CSR þ 1 1 CSR1þ 1 B B 2 CSR1þ 1 B 1 CSR þ 1 P ¼ 2B CSR þ 1 B 0 0 0 B B @ 0 0 0 0 0 0
0
where C ijkl is elastic matrix, which can be written as C ijkl ¼ 2μδik δjl þ λδij δkl
ð15Þ
3
0
According to the associated flow rule, plastic strain increment is
dεpij ¼ dλ
∂Φ σ ij ; σ ∂σ ij
ð16Þ
0
0
0
μ
ð24Þ
where H is dσ =dεp . Substituting Eqs. (19) and (21) into Eq. (24), thus ∂Φ σ ij ; σ =∂σ ij C ijkl dεkl dλ ¼ ∂Φ σ ij ; σ =∂σ ij C ijkl ∂Φ σ ij ; σ =∂σ kl þ H
where dλ is plastic multiplier. The plastic work increment is ð17Þ
Using Euler's theorem for function Φ σ ij ; σ , we get ∂Φ σ ij ; σ σ ij ¼σ ∂σ ij
0
The consistency condition can be stated as ∂Φ σ ij ; σ dσ ij ¼ Hdεp ∂σ ij
dwp ¼ σ ij dεpij ¼ σ dεp
ð22Þ
where μ and λ are Lame´ constants, δij is Kronecker symbol, μ ¼ E=ð2ð1 þ νÞÞ, λ ¼ νE=ð1 þ νÞð1 2νÞ; E is Young's modulus, and ν is Poisson's ratio. The symmetric matrix C is shown as 0 1 λ λ 0 0 0 2μ þ λ B C 2μ þ λ λ 0 0 0C B λ B C B λ λ 2μ þ λ 0 0 0 C B C Cijkl ¼ B ð23Þ μ 0 0C 0 0 B 0 C B C B 0 C 0 0 0 μ 0A @
ð14Þ
(14) can be written as 1 0 0 0 C 0 0 0C C C 0 0 0C C 3 0 0C C C 0 3 0A 0
37
ð25Þ
ð18Þ 3. Implementation and verification of the constitutive model
The equivalent plastic strain increment can be written as Eq. (19) by combining Eqs. (16)–(18) dε ¼ dλ p
In Section 2, the elasto-plastic constitutive model considering the variation in CSR with plastic deformation was researched based on Hill's 48 function. However, this constitutive model cannot be directly applied to FE analysis of tube plastic forming processes due to that no commercial FE codes can provide such an elasto-plastic constitutive model up to now. Therefore, the numerical implementation process is discussed in Section 3.1 for incorporating this constitutive model considering CSR's variation into ABAQUS/Explicit modulus. Then the reliability of the constitutive model is verified by using some single cubic element models in Section 3.2.
ð19Þ
The strain increment includes elastic component dε plastic component dεpij , therefore: dεij ¼ dεeij þ dεpij
e ij
and ð20Þ
Based on elastic Hooke's law, we have ∂Φ σ ij ; σ dσ ij ¼ C ijkl dεekl ¼ C ijkl dεkl dλ ∂σ kl
ð21Þ
Table 2 Parameters in Eq. (31). a0
a1
t1
a2
t2
Fitting precision
Φ6 mm t0.5 mm Φ8 mm t0.6 mm Φ12 mm t0.9 mm
1.01687 1.08829 1.11687
2.22562 0.98182 1.02701
0.00359 0.0156 0.02799
0.76629 3.38667 3.25907
0.0249 0.00168 0.00409
0.96377 0.99017 0.98814
4.5
4.5
4.0
4.0
3.5 3.0 2.5 2.0 1.5
Without considering CSR 's variation Considering CSR's variation Experiment
1.0 -25 -20 -15 -10
-5
0
5
10
15
20
25
Distance between measure point and middle point /mm
Variation ratio of outside diameter /%
Wall thinning ratio /%
Parameters
3.5 3.0 2.5 2.0 1.5 1.0
Without considering CSR's variation Considering CSR's variation Experiment
-25 -20 -15 -10
-5
0
5
10
15
20
25
Distance between measure point and middle point /mm
Fig. 5. Comparisons between simulation and experiment of uniaxial tension of Φ12 mm t0.9 mm Ti–3Al–2.5V titanium alloy anisotropic tube: (a) wall thinning ratio and (b) variation ratio of outside diameter.
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T. Huang et al. / International Journal of Mechanical Sciences 90 (2015) 33–43
(c) Calculate the strain after plastic correction εp according to Eq. (28)
3.1. Numerical implementation The scheme of numerical implementation consists of three steps. Firstly, the elastic judgment: in plane stress space, the trial stress is σ Tn þ 1 when discrete strain increment Δεn þ 1 is initially assumed to be elastic at given (n þ1) step. The process at the step is considered elastic when the yield condition is less than zero. Secondly, plastic adjustment: if the yield condition is greater than or equal to zero, the step is considered as elasto-plastic. Plastic corrections are needed to make the trial stress locate on the yield surface. And the direction of plastic correction should be perpendicular to the yield surface that the stress after correction locating in. Finally, the tolerance is checked by judging whether the yield condition is satisfied [17,20]. An open interface is provided in ABAQUS/Explicit to allow users to develop a material subroutine VUMAT to incorporate Hills' 48 constitutive model with the variation in CSR with plastic deformation [20]. Therefore, by developing a program based on the following procedures using the open interface, the variation in CSR with plastic deformation was embedded into ABAQUS/Explicit module. (a) Calculate anisotropy parameter R33 ðε according to Eq. (13) (in Eq. (13), CSR is a function of εpn ), update elastic matrix C according to Eq. (23) and symmetric matrix Pðεpn Þ according to Eq. (15). (b) Calculate Φ σ ij ; σ according to Eq. (26), when the value of Φ σ ij ; σ is greater than zero, calculate the plastic strain increment Δεp according to Eq. (27). Here, the Newton– Raphson iterative method is used to solve Δλ [14,16] 2 1 Φ σ ij ; σ ¼ σT Pðεpn Þσ σ εpnðþT Þ1 ð26Þ 2 Δεp ¼ Δλ ¼ 2ϕ σ ij Δλ2 ð27Þ p nÞ
εp ¼ εp þ Δεp
ð28Þ
(d) Calculate the stress after plastic correction σ according to Eq. (29), which locates on yield surface after updating when the yield condition is satisfied
σ ¼ σ þ σ_
ð29Þ
where σ_ is the iterative increment in σ . (e) Check updated yield condition by calculating Φ σ ij ; σ according to Eq. (30) and compare the value to the tolerance Tol. If Φ σ ij ; σ oTol, the current stress has converged to its true solution. If not, returns to step (c) Φ σ ij ; σ ¼ ϕ σ ij σ 2 ð30Þ (f) Update stress tensor σ n þ 1 , equivalent plastic strain εpn þ 1 , total plastic strain εpn þ 1 , total elastic strain εen þ 1 and total strain εn þ 1 . In the above steps, the superscript “T” stands for a trial state, the superscript “p” stands for the plastic deformation state and the subscript denotes the process time step. 3.2. Verification of the constitutive model developed The single cubic element model has many advantages such as simple geometric shape, quick calculation time, easy to check and find errors about the model. Therefore, in this study, single cubic element models for uniaxial tension, uniaxial compression and pure shearing were adopted to check the correctness and capability of the constitutive model, as shown in Fig. 1. The cubic length is 1 mm, the loading speed is 1 mm/s, the loading time is
where Δλ2 ¼ Δλ2 þ Δλ_ 2 (Δλ_ 2 is the iterative increment in Δλ2 ).
Fig. 6. Axial distance within the gauge length.
Fig. 8. FE model for the NC bending process.
Fig. 7. The configuration comparison of Φ12 mm t0.9 mm Ti–3Al–2.5V titanium alloy anisotropic tube after uniaxial compression: (a) simulation and (b) experiment.
T. Huang et al. / International Journal of Mechanical Sciences 90 (2015) 33–43
1.5 s for the uniaxial tension model and the pure shearing model, the loading time is 0.75 s for the uniaxial compression model, the element type is C3D8R. The materials in these models are the same and assumed as Ti–3Al–2.5V titanium alloy, which has the same material constants as the Φ12 mm t0.9 mm Ti–3Al–2.5V titanium alloy anisotropic tube, as listed in Table 1, Eq. (31) and Table 2. The mechanical constants of the material were obtained by the tensile testing based on ASTM E8/E8M-11 [21], and the parameters in CSR expression are obtained through a nonlinear curve fitting method based on the tensile experimental data according to the aerospace standard SAE AS4076 [5] and using the Digital Speckle Correlation Method (DSCM) [7]. The stress– strain responses during these processes are investigated, as shown in Fig. 2. The figure shows that the stress–strain curves obtained with embedded subroutine in the whole deformation phase are in good agreement with the input data. This means that the user material subroutine developed in this paper is correct. Moreover, CSR variation during these processes is investigated and shown in Fig. 3. The CSR–strain curves obtained with the embedded subroutine in the whole deformation phase are also in good agreement with the input data. This also means that the constitutive model considering variation in CSR with plastic deformation developed in this paper is correct, and the constitutive model is capable to correctly character the variation in CSR with plastic deformation during various plastic forming processes.
39
dimension of Φ12 mm t0.9 mm is discussed in Section 4.1. Also using the model, the prediction accuracy for NC bending deformation of Ti–3Al–2.5V titanium alloy anisotropic tubes with dimensions of Φ6 mm t0.5 mm, Φ8 mm t0.6 mm and Φ12 mm t0.9 mm is discussed in Section 4.2. The mechanical property parameters of Φ6 mm t0.5 mm, Φ8 mm t0.6 mm tubes are also shown in Table 1. The variations in CSR of these three Ti–3Al–2.5V anisotropic tubes were obtained by tensile tests according to the aerospace standard SAE AS4076 [5] and using the Digital Speckle Correlation Method (DSCM) [7], as shown in Fig. 4. According to the similar variation trend of CSR with deformation of each tube, a secondorder exponential decay equation (Eq. (31)) has been used to reflect this variation trend. The parameters in Eq. (31) can be obtained by using the nonlinear curve fitting method, as shown in Table 2. The equation can well describe the variation trend of the CSR of Ti–3Al–2.5V anisotropic tubes with a fitting precision higher than 96%, as shown in Table 2 CSR ¼ a0 þ a1 expð εp =t 1 Þ þ a2 expð εp =t 2 Þ
ð31Þ
where ε is plastic strain; a0, a1, a2, t1, and t2 are constants, as shown in Table 2. p
4.1. Uniaxial tension and uniaxial compression of anisotropic tube A 3D elastic–plastic FE model for the uniaxial tension of a
Φ12 mm t0.9 mm Ti–3Al–2.5V titanium alloy tube was estab4. Applications and discussion In order to validate the reliability and practicability of the constitutive model considering the variation in CSR with plastic deformation, the prediction accuracy for uniaxial tension, uniaxial compression deformation of Ti–3Al–2.5V titanium alloy anisotropic tubes with
lished based on ABAQUS/Explicit software environment. The tube was defined as a deformable solid body and was meshed by 3D linear reduction integration continuum elements with eight nodes (C3D8R). The length of the tube specimen is 180 mm. One end of tube specimen was loaded, the other end was fixed, and four layers of elements in the thickness direction were adopted. An initial length of 50 mm in the middle of tube specimen with displacement
Table 3 Bending parameters of Ti–3Al–2.5V titanium alloy anisotropic tubes. Parameters
Φ6 mm t0.5 mm
Φ8 mm t0.6 mm
Φ12 mm t0.9 mm
Bending radii (mm) Axial mandrel feed(mm) Ball number Bending speed (rad/s) Assistant pushing speed (mm/s) Pushing ratio (%) Mandrel diameter (mm) Bending angle (deg.)
12 1 0 0.42 5.04 100 4.84 90
24 1 0 0.84 20.16 100 6.60 90
24 1 1 0.80 19.2 100 9.94 90
Table 4 Mesh sizes of Ti–3Al–2.5V titanium alloy anisotropic tubes. Mesh sizes of tubes
Φ6 mm t0.5 mm
Φ8 mm t0.6 mm
Φ12 mm t0.9 mm
Bending part (mm) (axial size circumferential size) Front and trailing parts (mm) (axial size circumferential size) Thickness direction (mm)
0.398 0.327 0.800 0.327 0.125
0.600 0.395 1.000 0.395 0.150
0.800 0.700 1.500 0.700 0.225
Table 5 Friction conditions in various contact interfaces of Ti–3Al–2.5V titanium alloy anisotropic tubes. Contact interface
Friction coefficients
Tube/wiper die Tube/pressure die Tube/clamp die Tube/bend die Tube/mandrel Tube/balls
0.05 0.25 Rough 0.05 0.05 0.05
40
T. Huang et al. / International Journal of Mechanical Sciences 90 (2015) 33–43
the FE model without considering the variation in CSR is 20.7%, while the result from the FE model considering the variation in CSR is 6.9%. The AARE in outside diameter variation ratio from the FE model without considering the variation in CSR is 17.3%, while the result from the FE model considering the variation in CSR is 7.5%. These mean that with considering the variation in CSR, the simulation accuracy in the wall thinning and outside diameter change was improved by 13.8% and 9.8% compared with that without considering the variation in CSR, respectively. Fig. 7 shows a configuration comparison of simulation and experiment results of Φ12 mm t0.9 mm Ti–3Al–2.5V titanium alloy anisotropic tube with a height of 12 mm after uniaxial
of 4.25 mm (uniform plastic deformation) at a speed of 3 mm/min was investigated. Fig. 5 shows the variation in wall thinning and outside diameter within the gauge length (Fig. 6) of the anisotropic tube after the uniaxial tension. In this study, the predictability of the simulation is quantified by employing statistical parameters: average absolute relative error (AARE) [22] as AAREð%Þ ¼
i i 1 N P p P exp ∑ 100 N i ¼ 1 P iexp
ð32Þ
where P ip is the predicted value, P iexp is the experimental value, and N is the total number of data. The AARE in wall thinning ratio from
Fig. 9. The configuration comparison of Ti–3Al–2.5V titanium alloy anisotropic bent tubes: (a) simulation and (b) experiment.
24
14 12
Wall thinning ratio /%
Wall thinning ratio /%
20 16 12 8 Without considering CSR's variation Considering CSR's variation Experiment
4 0
-10
0
10
20
30
40
50
60
70
80
10 8 6 Without considering CSR's variation Considering CSR's variation Experiment
4 2 0
90 100
-10
Angle between measure point and initial bending section / o
0
10
20
30
40
50
60
70
80
90 100
Angle between measure point and initial bending section / o
20
Wall thinning ratio /%
18 16 14 12 10 8 Without considering CSR's variation Considering CSR's variation Experiment
6 4 2 0
-10
0
10
20
30
40
50
60
70
80
90 100
Angle between measure point and initial bending section / o Fig. 10. Comparison in the wall thinning ratio between simulation and experiment for Ti–3Al–2.5V titanium alloy anisotropic tubes: (a) Φ6 mm t0.5 mm, (b) Φ8 mm t0.6 mm and (c) Φ12 mm t0.9 mm.
T. Huang et al. / International Journal of Mechanical Sciences 90 (2015) 33–43
16
compression at a speed of 1 mm/min. However, the drum deformation occurs in the zone close to the end of tube specimen without considering the variation in CSR. The result shows that considering the variation in CSR correlates better with the experiment one. Springback angle / o
12
Wall thinning and the flattening of the cross-section are primary quality issues during a NC tube bending process. The springback is another key factor which affects the forming quality of tube bending. The bent tube dimension cannot satisfy the requirement of industrial application when the value of springback exceeds the error range. Therefore, in this section, a 3D elastic–plastic FE model which incorporates the constitutive model considering the variation in CSR for the NC bending process of Ti–3Al–2.5V titanium alloy anisotropic tubes was developed based on ABAQUS/Explicit, as shown in Fig. 8. The tubes were defined as a deformable solid body and were meshed by C3D8R element. While the dies were defined as discrete rigid bodies and were meshed by 3D bilinear rigid quadrilateral elements with four nodes (R3D4). Tables 3 and 4 show the bending parameters of tubes. The different friction coefficients have been assigned to the different contact interfaces as shown in Table 5. The simulation results were verified by experiments using a GQ W27YPC-63 PLC hydraulic bender with the same forming principle and dies structure. Bent Ti–3Al–2.5V titanium alloy anisotropic tubes under different dimensions are shown in Fig. 9. The figure shows that the configurations of the bent tube obtained from the simulations considering the variation in CSR are similar to those obtained in the experiments. The wall thinning predicted by different FE models and from the experiments for different dimensional tubes is shown in Fig. 10. The AARE of wall thinning for Φ6 mm t0.5 mm, Φ8 mm t0.6 mm
8 6 4
0 Φ6mm× t0.5mm Φ8mm× t0.6mm Φ12mm× t0.9mm
Fig. 12. Comparison in the springback angle between simulation and experiment for Ti–3Al–2.5V titanium alloy anisotropic tubes.
4.0
Simulation results Input data
3.5 3.0
CSR
2.5 2.0 1.5 1.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
Equivalent plastic strain Fig. 13. Comparison in CSR's variation of simulation results and input data during the NC bending process.
2.4
4.0 3.5
Flattening of cross-section /%
Flattening of cross-section /%
10
2
4.5
3.0 2.5 2.0 1.5 1.0
Without considering CSR's variation Considering CSR's variation Experiment
0.5 0.0 -0.5
Without considering CSR's variation Considering CSR's variation Experiment
14
4.2. NC bending of anisotropic tube
41
-10
0
10
20
30
40
50
60
70
80
2.0 1.6 1.2
0.4 0.0
90 100
Without considering CSR's variation Considering CSR's variation Experiment
0.8
Angle between measure section and initial bending section / o
-10
0
10
20
30
40
50
60
70
80
90 100
Angle between measure point and initial bending section / o
4.5
Flattening of cross-section /%
4.0 3.5 3.0 2.5 2.0 1.5 Without considering CSR's variation Considering CSR's variation Experiment
1.0 0.5 0.0
-10
0
10
20
30
40
50
60
70
80
90 100
Angle between measure section and initial bending section / o Fig. 11. Comparison in flattening of the cross-section between simulation and experiment of Ti–3Al–2.5V titanium alloy anisotropic tubes: (a) Φ6 mm t0.5 mm, (b) Φ8 mm t0.6 mm and (c) Φ12 mm t0.9 mm.
42
T. Huang et al. / International Journal of Mechanical Sciences 90 (2015) 33–43
Fig. 14. Distribution of CSR at different bending angles of bent tube: (a) 101, (b) 451 and (c) 901.
and Φ12 mm t0.9 mm are 21.7%, 22.2%, and 20.7% by the FE models without considering the variation in CSR, and 8.6%, 15.1%, and 8.5% by the FE models with considering the variation in CSR, respectively. This means that the simulation accuracy in wall thinning can be improved by 13.1%, 7.1%, and 12.2% for three dimensional tubes, respectively. The flattening of the cross-section predicted by different FE models and from the experiments for different dimensional tubes is shown in Fig. 11. Compared with the experiments, the AARE in flattening of the cross-section for Φ6 mm t0.5 mm, Φ8 mm t0.6 mm and Φ12 mm t0.9 mm are 20.5%, 12.4%, and 25.9%, respectively, by the FE models without considering the CSR's variation, and 12.3%, 11.0%, and 20.0% by the FE models with consideration of the CSR's variation. Considering the variation in CSR, the simulation accuracy in the flattening degree of the cross-section was improved by 8.2%, 1.4%, and 5.9%, respectively, compared with that without considering the variation in CSR. Fig. 12 shows the springback angles predicted by different FE models and from the experiments for different dimensional tubes. With considering the variation in CSR based on constitutive model for FE model, the AARE in springback angle is 4.1%, 12.9% and 5.6% for Φ6 mm t0.5 mm, Φ8 mm t0.6 mm and Φ12 mm t0.9 mm tubes, respectively. However, without considering the variation in CSR, the AARE is 10.4%, 17.5% and 13.8%, respectively. This means that the simulation accuracy in the springback angle was improved by 8.2% mostly. The results show that the wall thinning, the flattening of the cross-section and the springback angle predicted have a better agreement with the experiments when using the constitutive model considering the variation in CSR for different dimensions anisotropic tubes. This is related to the variation and distribution in CSR during the plastic deformation process of tubes. Fig. 13 shows the variation of CSR with plastic strain of a feature element on Φ12 mm t 0.9 mm Ti–3Al–2.5V titanium alloy anisotropic tube. As observed in Fig. 13, the values of the CSR obtained from the simulation are in good agreement with the input data. Fig. 14 shows the distribution of CSR during the bending process of tube. In the initial deformation stage, the CSR shows a non-uniform distribution, and the CSR value in the inside and outside of the bending is less than that on the neutral layer zone. This may be due to the fact that larger strain occurs in the outside and inside of the bent tube, and the CSR decreases with the increase of plastic deformation. In the later bending deformation, CSR value decreases and tends to be stable and the distribution of CSR tends to be uniform within main deformation zone. The uneven distribution zones of CSR are focused on transitional zones between deformation zone and the non-deformation zone. This shows that the variation in CSR in the deformation zone is different when deformation degree of tube is different. However, FE model cannot
accurately describe the plastic deformation of tube without considering variation in CSR in tube bending, and due to it ignoring CSR variation characteristic. Therefore, the simulation accuracies of FE model which incorporate the constitutive model considering the variation in CSR are closer to the experimental values compared with the CSR for constant simulation.
5. Conclusions In this study, an elasto-plastic constitutive model which contains the variation in CSR with plastic deformation was developed and evaluated. The conclusions are as follows. An elasto-plastic constitutive model was developed based on the Hill's 48 anisotropic yield function, where the variation in CSR with plastic deformation was incorporated. The constitutive model was implemented into ABAQUS/Explicit by using the user material subroutine VUMAT. This constitutive model was testified reliable by using the single cubic element FE models for uniaxial tension, uniaxial compression and pure shearing process. Moreover, comparisons between simulation results and experiment results of the uniaxial tension, the uniaxial compression and the NC bending show that the elasto-plastic constitutive model considering the variation in CSR could improve the prediction accuracy for the plastic deformation processes of anisotropic tubes.
Acknowledgments The authors would like to thank the National Science Fund for Excellent Young Scholars of China (51222509), the National Natural Science Foundation of China (51175429), the Foundation of NWPU (JC201136), and Project 111 of China (B08040). References [1] Zhan M, Yang H, Huang L, Gu RJ. Springback analysis of numerical control bending of thin-walled tube using numerical-analytic method. J Mater Process Technol 2006;177:197–201. [2] Jiang ZQ, Yang H, Zhan M, Yue YB, Liu J, Xu XD, et al. Establishment of a 3D FE model for the bending of a titanium alloy tube. Int J Mech Sci 2010;52:1115–24. [3] Yang H, Li H, Zhang ZY, Zhan M, Liu J, Li GJ. Advances and trends on tube bending forming technologies. Chin J Aeronaut 2012;25:1–12. [4] Srbislav A, Milentije S, Dragan A, Vukić L. Variation of normal anisotropy ratio “r” during plastic forming. J Mech Eng 2009;55:392–9. [5] Contractile strain ratio testing of titanium hydraulic tubing. Aerospace standard, SAE AS4076; 2001. [6] Hwang SK, Sabol GP. A new strain gage method for measuring the contractile strain ratio of zircaloy tubing. J Nucl Mater 1988;151:327–33. [7] Peters WH, Ranson WF. Digital image techniques in experimental stress analysis. Opt Eng 1982;21:427–31.
T. Huang et al. / International Journal of Mechanical Sciences 90 (2015) 33–43 [8] Meng LB, Jin GC, Yao XF. Application of iteration and finite element smoothing technique for displacement and strain measurement of digital speckle correlation. Opt Laser Eng 2007;45(1):57–63. [9] Jiang ZQ, Yang H, Zhan M, Xu XD, Li GJ. Coupling effects of material properties and the bending angle on the springback angle of a titanium alloy tube during numerically controlled bending. Mater Des 2010;31:2001–10. [10] Jiang ZQ. ([Dissertation]). Deformation laws of medium-strength titanium alloy thick-walled tubes in NC bending process. Xi'an (SN): Northwestern Polytechnical University; 2010 (Chinese). [11] Poursina M, Amini B, Hasanpour K, Barati M. The effect of normal anisotropy on thin-walled tube bending. Int J Eng Trans A: Basics 2013;26:83–9. [12] Wu XD. Research on the plastic deformation behavior of anisotropic sheet metal under different loading paths. ([Dissertation]). Beijing: Beihang University; 2004 (Chinese). [13] Li HW. Numeralization of macroscopic and microscopic constitutive relations and their applications in Fe simulation. ([Dissertation]). Xi'an (SN): Northwestern Polytechnical University; 2007 (Chinese). [14] Lee SW, Yoon JW, Yang DY. A stress integration algorithm for plane stress elasto plasticity and its applications to explicit finite element analysis of sheet metal forming processes. Comput Struct 1998;66(2–3):301–11.
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[15] Ni XG. The study of mechanical behavior and numerical simulation techniques for sheet forming processes. ([Dissertation]). Hefei (ANH): University of Science and Technology of China; 2000 (Chinese). [16] Tang BT, Zhao GQ, Wang ZQ. A mixed hardening rule coupled with Hill48' yielding function to predict the springback of sheet U-bending. Int J Mater Form 2008;1:169–75. [17] Zang SL, Liang J, Guo C. A constitutive model for spring-back prediction in which the change of Young's modulus with plastic deformation is considered. Int J Mach Tool Manuf 2007;47:1791–7. [18] Zang SL. Research on elasto-plastic constitutive model and sheet springback. ([Dissertation]). Xi'an (SN): Xi'an Jiaotong University; 2007 (Chinese). [19] Swift HW. Plastic instability under plane stress. J Mech Phys Solids 1952;1:1–18. [20] (Version 6.9. Hibbit). ABAQUS analysis user's manual. USA: Karlsson and Sorensen Inc.; 2008. [21] Standard test methods for tension testing of metallic materials. ASTM E8/E8M11; 2011. [22] Cai J, Li FG, Liu TY, Chen B. Microindentation study of Ti–6Al–4V alloy. Mater Des 2011;32:2756–62.
Contents lists available at ScienceDirect
International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
A constitutive model for anisotropic tubes incorporating the variation of contractile strain ratio with deformation T. Huang, M. Zhan n, H. Yang, J. Guo, J.Q. Tan State Key Laboratory of Solidification Processing, School of Materials Science and Engineering, Northwestern Polytechnical University, Xi'an 710072, China
art ic l e i nf o
a b s t r a c t
Article history: Received 13 May 2014 Received in revised form 23 September 2014 Accepted 20 October 2014 Available online 28 October 2014
The contractile strain ratio (CSR) is an important index measuring the anisotropy of tubes. The values of CSR vary non-linearly with the amount of plastic deformation. However, up to now, there are no constitutive models which are able to take this variation into account. In order to obtain the actual deformation characteristics and improve the prediction capability for plastic deformation processes of anisotropic tubes, this study developed an elasto-plastic constitutive model based on Hill's 48 anisotropic yield function. In this constitutive model, the variation in CSR with plastic deformation was considered. A user material subroutine VUMAT incorporating the constitutive model was developed based on ABAQUS/Explicit platform. The comparisons between the finite element (FE) simulations and experiments for uniaxial tension, uniaxial compression and numerical control (NC) bending process of Ti–3Al–2.5 V titanium alloy anisotropic tube, show that this constitutive model is reliable. The simulation accuracy was improved and the plastic deformation characteristics could be better described for the anisotropic tubes by using this constitutive model. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Anisotropic tubes Constitutive model Contractile strain ratio Plastic deformation Ti–3Al–2.5V
1. Introduction As one kind of key lightweight components for liquid transforming and loading carrying with enormous quantities, lightweight and high-strength bent metal tubes have been widely applied in many industries such as aerospace, aviation, oil, and related high-technology industries [1–3]. However, due to the complex metallurgical process of cold-working, the mechanical performances of these tubes are significantly influenced by the mechanical anisotropy [4]. Such mechanical anisotropy is introduced by the development of preferred crystallographic texture during tube processing. The tubes' anisotropy is usually described by the contractile strain ratio (CSR), which is a very important macroscopic deformation index. A number of methods have been used to measure the CSR of tubes, including the post-test manual measure method [5], the continuously loading–unloading measure method by utilizing the strain gages [6] or the method based on digital speckle correlation method [7,8]. In recent years, some research works on the influence of CSR values on tube bending processes have been carried out. Jiang et al. [9,10] revealed that the increase in CSR led to the decrease in springback angle for titanium alloy bent tubes. Hasanpour et al. [11] revealed that a larger CSR led to a higher flattening of the
n
Corresponding author. E-mail address: [email protected] (M. Zhan).
http://dx.doi.org/10.1016/j.ijmecsci.2014.10.014 0020-7403/& 2014 Elsevier Ltd. All rights reserved.
cross-section, and a serious wall thickness variation when the CSR of the tube is less than 1. Among these research works [9–11], the CSR was always considered as a constant in the whole deformation process independent of the plastic deformation degree. That is to say, these studies did not consider the variation in CSR during the plastic deformation process. However, CSR values vary with the amount of deformation during the plastic forming processes of tubes. Therefore, in order to obtain the deformation characteristics of tubes during plastic forming processes more accurately, the dependence of CSR on deformation should be considered in numerical simulation. Up to now, no commercial finite element (FE) codes can provide such an elasto-plastic constitutive model and there followed by a question that is: how to contain the variation in CSR with deformation into FE model of the plastic forming of tubes. This needs a constitutive model which can take the variation in CSR into consideration at first. Up to now, there are some yield functions which can be used to express the deformation behavior of anisotropic materials, including Hill's 48, Hill's 79, Barlat 89, Barlat 91, and Hosford, etc. [12]. Hill's 48 anisotropic yield function was widely used to analyze deformation of metal sheets due to its general stress state, relative ease of formulation, and easily obtained parameters through uniaxial experiments [13]. Lee et al. [14] developed an elastoplastic constitutive model based on Hill's 48 yield function, and proposed a stable integration algorithm using the Newton‐Raphson method to solve the model. The constitutive model was implemented into the explicit FE code named FORMSYS-DE.
34
T. Huang et al. / International Journal of Mechanical Sciences 90 (2015) 33–43
The simulations for the thickness strain of an annealed aluminum and mild steel showed a good correlation with experiments. Ni [15] developed an explicit expression of the elasto-plastic constitutive model based on Hill's 48 and incorporated this model into ABAQUS via the user subroutine VUMAT. The 3D simulation for the drawing process of an aluminum square box showed that this model was reliable. Tang et al. [16] established a constitutive model based on Hill's 48 and calculated the stress and strain increment by a backward-Euler return mapping algorithm. The model was implemented into ABAQUS via the user subroutine UMAT. The results showed that this model could better describe the plastic deformation of aluminum alloy sheet. Zang et al. [17,18] have extended Tang's work by considering the variation in Young's modulus with plastic deformation for high strength steels and aluminum alloys. Similarly, Zang et al. concluded that the constitutive model based on Hill's 48 yield function was generally the most accurate for prediction sheets' deformation. However, according to the aforementioned references, it can be found that the studies for constitutive models have focused on metal sheets, while not much reported on tubes. As there is a similarity between sheets and tubes in deformation characteristics, extending Hill's 48 yield function to plastic deformation of anisotropic tubes is still acceptable. In this study, Hill's 48 yield function was extended to research plastic deformation of anisotropic tubes. An elasto-plastic constitutive model was established to describe the variation in CSR with plastic deformation. Then, on the platform of ABAQUS/Explicit, a user material subroutine VUMAT was developed. The single cubic FE models for uniaxial tension, uniaxial compression, and pure shearing were conducted to verify the correctness and reliability of the constitutive model. Finally, FE simulations and experiments for the uniaxial tension, uniaxial compression and numerical control (NC) bending processes of Ti–3Al–2.5V (ASTM Gr. 9) titanium alloy tubes, which are typical anisotropic tubes, were carried out to determine whether this elasto-plastic constitutive model could help to improve the simulation accuracy of tubes deformation.
2. Extension of Hill's 48 yield function to anisotropic tubes 2.1. Original Hill's 48 anisotropic yield function In 1948 Hill proposed the orthogonal anisotropic yield surface as Φ σ ij ; k ¼ Fðσ 22 σ 33 Þ2 þ Gðσ 33 σ 11 Þ2 þ Hðσ 11 σ 22 Þ2 þ2Lσ 223 þ 2M σ 231 þ 2N σ 212 2k 2 ¼ ϕ σ ij 2k ¼ 0
2
ð1Þ
where F, G, H, L, M, and N are anisotropic characteristic parameters, their values can be determined by experiments; 11, 22, and 33 are orthogonal anisotropic principal axes [12]. The anisotropic parameters F, G, H, L, M, and N in Hill's 48 can be written as 8F 1 1 1 ð2 1Þ > 2 ¼ 2 þ 2 2 > σ 22 σ 33 σ 11 >k > > > G 1 1 1 > ð2 2Þ > 2 ¼ 2 þ 2 2 > k σ 33 σ 11 σ 22 > > >H > 1 1 1 > < k2 ¼ σ 2 þ σ 2 σ 2 ð2 3Þ 11 22 33 ð2Þ L 1 > ð2 4Þ 2 ¼ 2 > k > σ 23 > > > M 1 > > ð2 5Þ > k2 ¼ σ 231 > > > > > N2 ¼ 12 : ð2 6Þ σ 12
k
where σ ij are the yield stresses with the only nonzero stress σ ij ; k is a function of plastic work and hardening. The yield stress can be written as Eq. (3) for isotropic material
σ 11 ¼ σ 22 ¼ σ 33 ¼ σ
ð3Þ
The sum of Eqs. (2-1), (2-2), and (2-3) in Eq. (2) can be written as F þGþH 2
k
¼
3
ð4Þ
σ2
Hill yield surface could be derived as Mises yield surface when 2 2 F ¼ G ¼ H ¼ 1=2, L ¼ M ¼ N ¼ 3=2 and σ 2 ¼ 2k . 2k is applicable for both cases of the isotropic and anisotropic [15]. And Hill anisotropic yield surface could be expressed as Eq. (5). In this study, the Swift hardening equation (Eq. (6)) is adopted [19] Φ σ ij ; σ ¼ ϕ σ ij σ 2 ¼ 0 ð5Þ
σ ¼ Kðεp þ bÞn
ð6Þ
According to yield stress ratio, the anisotropy parameters F, G, H, L, M, and N can be expressed as 8 2 > > > F ¼ σ2 12 þ 12 12 ¼ 12 R12 þ R12 R12 > σ 22 σ 33 σ 11 > 22 33 11 > > > > > 2 > σ 1 1 1 1 1 1 > G ¼ 2 2 þ 2 2 ¼ 2 R2 þ R2 R12 > > σ σ σ > 33 11 22 33 11 22 > > > > 2 > σ 1 1 1 1 1 1 > < H ¼ 2 2 þ 2 2 ¼ 2 2 þ 2 12 R11 R22 R33 σ 11 σ 22 σ 33 ð7Þ 2 > > > 3 τ 3 > ¼ L ¼ > 2 σ 23 > 2R223 > > > 2 > > 3 τ 3 > > > M ¼ 2 σ 13 ¼ 2R213 > > > 2 > > > 3 τ 3 > : N ¼ 2 σ 12 ¼ 2R2 12
Fig. 1. FE models for single cubic element under different loading paths: (a) uniaxial tension, (b) uniaxial compression and (c) pure shearing.
T. Huang et al. / International Journal of Mechanical Sciences 90 (2015) 33–43
Eq. (12) can be obtained
where Rij is yield stress ratio and the subscript denotes the direction: 11 is the axial direction, 22 is the transverse direction and 33 is the thickness direction; τ ¼ σ =3. Since r of sheets is the ratio of width true strain to thickness true strain, it is necessary to transform the strain ratio into the stress ratio [15]. Eq. (8) can be obtained when the uniaxial tensile deformation is taken in 11 direction for sheets: dε11 : dε22 : dε33 ¼ G þ H : H : G
8 R11 ¼ 1 > > < R22 ¼ 1 1=2 > > : R33 ¼ r0 þ 1 2
H : dε33 ¼ G
2.2. Incorporating CSR's variation with deformation into Hills' 48 yield function
ð9Þ
The tubes' anisotropy is usually described by the CSR. The CSR is defined as the ratio of the true circumferential plastic strain to the true radial (i.e. thickness) plastic strain of a tube [5]. Moreover, the CSR is a function about plastic strain, and varies with plastic deformation. In this study, the relation between the yield stress ratio Rij and CSR can be written as
Eq. (10) can be obtained when the uniaxial tensile deformation is taken in 22 direction for sheets: dε11 : dε22 : dε33 ¼ H : F þ H : F
ð10Þ
The ratio of strain in axial direction to that in thickness direction can be written as r 90 ¼ dε11 : dε33 ¼
ð12Þ
ð8Þ
The ratio of strain in width direction to that in thickness direction can be written as r 0 ¼ dε22
35
H F
8 R ¼1 > > < 11 R22 ¼ 1 > > : R ¼ CSR þ 1 1=2
ð11Þ
r 0 is equal to r 90 and σ is equal to σ 11 for sheets or tubes only considering the anisotropy in thickness direction. Therefore,
33
ð13Þ
2
Table 1 Mechanical constants of Ti–3Al–2.5V titanium alloy anisotropic tubes. Parameters
Poisson's ratio v
Yield stress σ 0:2 (MPa)
Ultimate tensile stress σ b (MPa)
Strength coefficient K (MPa)
Hardening exponent n
Φ12 mm t0.9 mm Φ6 mm t0.5 mm Φ8 mm t0.6 mm
0.29 0.25 0.30
812.26 805.43 750.00
1043.94 1039.91 940.00
1267.14 1261.52 1483.24
0.064 0.077 0.123
1400
1400
Simulation results Input data
1000
Plastic phase
800
Elastic phase
600 400 200 0
1000
0.1
0.2
0.3
0.4
0.5
0.6
Plastic phase
800
Elastic phase
600 400 200 0
0.0
Simulation results Input data
1200
Equivalent stress σe (MPa)
Equivalent stress σe (MPa)
1200
0.7
0.0
0.2
0.8
1.0
Simulation results Input data
1200
Equivalent stress σe (MPa)
0.6 p
Equivalent plastic strain ε
1400
0.4
Equivalent plastic strain ε
p
1000
Plastic phase
800
Elastic phase
600 400 200 0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
p
Equivalent plastic strain ε
Fig. 2. Comparison in stress–strain of simulation results and input data during different deformation processes: (a) uniaxial tension, (b) uniaxial compression and (c) pure shearing.
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T. Huang et al. / International Journal of Mechanical Sciences 90 (2015) 33–43
4.0
4.0
3.5
3.5
Simulation results Input data
3.0
2.5
CSR
CSR
3.0
Simulation results Input data
2.5
2.0
2.0
1.5
1.5
1.0
1.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
Equivalent plastic strain
Equivalent plastic strain
4.0 3.5
Simulation results Input data
CSR
3.0 2.5 2.0 1.5 1.0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
Equivalent plastic strain Fig. 3. Comparison in CSR's variation of simulation results and input data during different deformation processes: (a) uniaxial tension, (b) uniaxial compression and (c) pure shearing.
3.5
3.5
3.0
3.0
Experiment Second-order equation
2.5
CSR
CSR
2.5 2.0
2.0
1.5
1.5
1.0
1.0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18
Experiment Second-order equation
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18
Plastic strain
Plastic strain
4.0 3.5
CSR
3.0
Experiment Second-order equation
2.5 2.0 1.5 1.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 Plastic strain
Fig. 4. Fitting curves between CSR and plastic strain for Ti–3Al–2.5V titanium alloy anisotropic tubes: (a) Φ6 mm t 0.5 mm, (b) Φ8 mm t 0.6 mm and (c) Φ12 mm t 0.9 mm.
T. Huang et al. / International Journal of Mechanical Sciences 90 (2015) 33–43
ϕ σ ij in Eq. (1) can be reformulated as Eq. (14) by a symmetric matrix P (Eq. (15)) given in a vector Voigt notation 1 ϕ σ ij ¼ σ T Pσ 2 The symmetric matrix P in Eq. 0 CSR 1 CSR CSR1þ 1 þ1 B CSR B CSR þ 1 1 CSR1þ 1 B B 2 CSR1þ 1 B 1 CSR þ 1 P ¼ 2B CSR þ 1 B 0 0 0 B B @ 0 0 0 0 0 0
0
where C ijkl is elastic matrix, which can be written as C ijkl ¼ 2μδik δjl þ λδij δkl
ð15Þ
3
0
According to the associated flow rule, plastic strain increment is
dεpij ¼ dλ
∂Φ σ ij ; σ ∂σ ij
ð16Þ
0
0
0
μ
ð24Þ
where H is dσ =dεp . Substituting Eqs. (19) and (21) into Eq. (24), thus ∂Φ σ ij ; σ =∂σ ij C ijkl dεkl dλ ¼ ∂Φ σ ij ; σ =∂σ ij C ijkl ∂Φ σ ij ; σ =∂σ kl þ H
where dλ is plastic multiplier. The plastic work increment is ð17Þ
Using Euler's theorem for function Φ σ ij ; σ , we get ∂Φ σ ij ; σ σ ij ¼σ ∂σ ij
0
The consistency condition can be stated as ∂Φ σ ij ; σ dσ ij ¼ Hdεp ∂σ ij
dwp ¼ σ ij dεpij ¼ σ dεp
ð22Þ
where μ and λ are Lame´ constants, δij is Kronecker symbol, μ ¼ E=ð2ð1 þ νÞÞ, λ ¼ νE=ð1 þ νÞð1 2νÞ; E is Young's modulus, and ν is Poisson's ratio. The symmetric matrix C is shown as 0 1 λ λ 0 0 0 2μ þ λ B C 2μ þ λ λ 0 0 0C B λ B C B λ λ 2μ þ λ 0 0 0 C B C Cijkl ¼ B ð23Þ μ 0 0C 0 0 B 0 C B C B 0 C 0 0 0 μ 0A @
ð14Þ
(14) can be written as 1 0 0 0 C 0 0 0C C C 0 0 0C C 3 0 0C C C 0 3 0A 0
37
ð25Þ
ð18Þ 3. Implementation and verification of the constitutive model
The equivalent plastic strain increment can be written as Eq. (19) by combining Eqs. (16)–(18) dε ¼ dλ p
In Section 2, the elasto-plastic constitutive model considering the variation in CSR with plastic deformation was researched based on Hill's 48 function. However, this constitutive model cannot be directly applied to FE analysis of tube plastic forming processes due to that no commercial FE codes can provide such an elasto-plastic constitutive model up to now. Therefore, the numerical implementation process is discussed in Section 3.1 for incorporating this constitutive model considering CSR's variation into ABAQUS/Explicit modulus. Then the reliability of the constitutive model is verified by using some single cubic element models in Section 3.2.
ð19Þ
The strain increment includes elastic component dε plastic component dεpij , therefore: dεij ¼ dεeij þ dεpij
e ij
and ð20Þ
Based on elastic Hooke's law, we have ∂Φ σ ij ; σ dσ ij ¼ C ijkl dεekl ¼ C ijkl dεkl dλ ∂σ kl
ð21Þ
Table 2 Parameters in Eq. (31). a0
a1
t1
a2
t2
Fitting precision
Φ6 mm t0.5 mm Φ8 mm t0.6 mm Φ12 mm t0.9 mm
1.01687 1.08829 1.11687
2.22562 0.98182 1.02701
0.00359 0.0156 0.02799
0.76629 3.38667 3.25907
0.0249 0.00168 0.00409
0.96377 0.99017 0.98814
4.5
4.5
4.0
4.0
3.5 3.0 2.5 2.0 1.5
Without considering CSR 's variation Considering CSR's variation Experiment
1.0 -25 -20 -15 -10
-5
0
5
10
15
20
25
Distance between measure point and middle point /mm
Variation ratio of outside diameter /%
Wall thinning ratio /%
Parameters
3.5 3.0 2.5 2.0 1.5 1.0
Without considering CSR's variation Considering CSR's variation Experiment
-25 -20 -15 -10
-5
0
5
10
15
20
25
Distance between measure point and middle point /mm
Fig. 5. Comparisons between simulation and experiment of uniaxial tension of Φ12 mm t0.9 mm Ti–3Al–2.5V titanium alloy anisotropic tube: (a) wall thinning ratio and (b) variation ratio of outside diameter.
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T. Huang et al. / International Journal of Mechanical Sciences 90 (2015) 33–43
(c) Calculate the strain after plastic correction εp according to Eq. (28)
3.1. Numerical implementation The scheme of numerical implementation consists of three steps. Firstly, the elastic judgment: in plane stress space, the trial stress is σ Tn þ 1 when discrete strain increment Δεn þ 1 is initially assumed to be elastic at given (n þ1) step. The process at the step is considered elastic when the yield condition is less than zero. Secondly, plastic adjustment: if the yield condition is greater than or equal to zero, the step is considered as elasto-plastic. Plastic corrections are needed to make the trial stress locate on the yield surface. And the direction of plastic correction should be perpendicular to the yield surface that the stress after correction locating in. Finally, the tolerance is checked by judging whether the yield condition is satisfied [17,20]. An open interface is provided in ABAQUS/Explicit to allow users to develop a material subroutine VUMAT to incorporate Hills' 48 constitutive model with the variation in CSR with plastic deformation [20]. Therefore, by developing a program based on the following procedures using the open interface, the variation in CSR with plastic deformation was embedded into ABAQUS/Explicit module. (a) Calculate anisotropy parameter R33 ðε according to Eq. (13) (in Eq. (13), CSR is a function of εpn ), update elastic matrix C according to Eq. (23) and symmetric matrix Pðεpn Þ according to Eq. (15). (b) Calculate Φ σ ij ; σ according to Eq. (26), when the value of Φ σ ij ; σ is greater than zero, calculate the plastic strain increment Δεp according to Eq. (27). Here, the Newton– Raphson iterative method is used to solve Δλ [14,16] 2 1 Φ σ ij ; σ ¼ σT Pðεpn Þσ σ εpnðþT Þ1 ð26Þ 2 Δεp ¼ Δλ ¼ 2ϕ σ ij Δλ2 ð27Þ p nÞ
εp ¼ εp þ Δεp
ð28Þ
(d) Calculate the stress after plastic correction σ according to Eq. (29), which locates on yield surface after updating when the yield condition is satisfied
σ ¼ σ þ σ_
ð29Þ
where σ_ is the iterative increment in σ . (e) Check updated yield condition by calculating Φ σ ij ; σ according to Eq. (30) and compare the value to the tolerance Tol. If Φ σ ij ; σ oTol, the current stress has converged to its true solution. If not, returns to step (c) Φ σ ij ; σ ¼ ϕ σ ij σ 2 ð30Þ (f) Update stress tensor σ n þ 1 , equivalent plastic strain εpn þ 1 , total plastic strain εpn þ 1 , total elastic strain εen þ 1 and total strain εn þ 1 . In the above steps, the superscript “T” stands for a trial state, the superscript “p” stands for the plastic deformation state and the subscript denotes the process time step. 3.2. Verification of the constitutive model developed The single cubic element model has many advantages such as simple geometric shape, quick calculation time, easy to check and find errors about the model. Therefore, in this study, single cubic element models for uniaxial tension, uniaxial compression and pure shearing were adopted to check the correctness and capability of the constitutive model, as shown in Fig. 1. The cubic length is 1 mm, the loading speed is 1 mm/s, the loading time is
where Δλ2 ¼ Δλ2 þ Δλ_ 2 (Δλ_ 2 is the iterative increment in Δλ2 ).
Fig. 6. Axial distance within the gauge length.
Fig. 8. FE model for the NC bending process.
Fig. 7. The configuration comparison of Φ12 mm t0.9 mm Ti–3Al–2.5V titanium alloy anisotropic tube after uniaxial compression: (a) simulation and (b) experiment.
T. Huang et al. / International Journal of Mechanical Sciences 90 (2015) 33–43
1.5 s for the uniaxial tension model and the pure shearing model, the loading time is 0.75 s for the uniaxial compression model, the element type is C3D8R. The materials in these models are the same and assumed as Ti–3Al–2.5V titanium alloy, which has the same material constants as the Φ12 mm t0.9 mm Ti–3Al–2.5V titanium alloy anisotropic tube, as listed in Table 1, Eq. (31) and Table 2. The mechanical constants of the material were obtained by the tensile testing based on ASTM E8/E8M-11 [21], and the parameters in CSR expression are obtained through a nonlinear curve fitting method based on the tensile experimental data according to the aerospace standard SAE AS4076 [5] and using the Digital Speckle Correlation Method (DSCM) [7]. The stress– strain responses during these processes are investigated, as shown in Fig. 2. The figure shows that the stress–strain curves obtained with embedded subroutine in the whole deformation phase are in good agreement with the input data. This means that the user material subroutine developed in this paper is correct. Moreover, CSR variation during these processes is investigated and shown in Fig. 3. The CSR–strain curves obtained with the embedded subroutine in the whole deformation phase are also in good agreement with the input data. This also means that the constitutive model considering variation in CSR with plastic deformation developed in this paper is correct, and the constitutive model is capable to correctly character the variation in CSR with plastic deformation during various plastic forming processes.
39
dimension of Φ12 mm t0.9 mm is discussed in Section 4.1. Also using the model, the prediction accuracy for NC bending deformation of Ti–3Al–2.5V titanium alloy anisotropic tubes with dimensions of Φ6 mm t0.5 mm, Φ8 mm t0.6 mm and Φ12 mm t0.9 mm is discussed in Section 4.2. The mechanical property parameters of Φ6 mm t0.5 mm, Φ8 mm t0.6 mm tubes are also shown in Table 1. The variations in CSR of these three Ti–3Al–2.5V anisotropic tubes were obtained by tensile tests according to the aerospace standard SAE AS4076 [5] and using the Digital Speckle Correlation Method (DSCM) [7], as shown in Fig. 4. According to the similar variation trend of CSR with deformation of each tube, a secondorder exponential decay equation (Eq. (31)) has been used to reflect this variation trend. The parameters in Eq. (31) can be obtained by using the nonlinear curve fitting method, as shown in Table 2. The equation can well describe the variation trend of the CSR of Ti–3Al–2.5V anisotropic tubes with a fitting precision higher than 96%, as shown in Table 2 CSR ¼ a0 þ a1 expð εp =t 1 Þ þ a2 expð εp =t 2 Þ
ð31Þ
where ε is plastic strain; a0, a1, a2, t1, and t2 are constants, as shown in Table 2. p
4.1. Uniaxial tension and uniaxial compression of anisotropic tube A 3D elastic–plastic FE model for the uniaxial tension of a
Φ12 mm t0.9 mm Ti–3Al–2.5V titanium alloy tube was estab4. Applications and discussion In order to validate the reliability and practicability of the constitutive model considering the variation in CSR with plastic deformation, the prediction accuracy for uniaxial tension, uniaxial compression deformation of Ti–3Al–2.5V titanium alloy anisotropic tubes with
lished based on ABAQUS/Explicit software environment. The tube was defined as a deformable solid body and was meshed by 3D linear reduction integration continuum elements with eight nodes (C3D8R). The length of the tube specimen is 180 mm. One end of tube specimen was loaded, the other end was fixed, and four layers of elements in the thickness direction were adopted. An initial length of 50 mm in the middle of tube specimen with displacement
Table 3 Bending parameters of Ti–3Al–2.5V titanium alloy anisotropic tubes. Parameters
Φ6 mm t0.5 mm
Φ8 mm t0.6 mm
Φ12 mm t0.9 mm
Bending radii (mm) Axial mandrel feed(mm) Ball number Bending speed (rad/s) Assistant pushing speed (mm/s) Pushing ratio (%) Mandrel diameter (mm) Bending angle (deg.)
12 1 0 0.42 5.04 100 4.84 90
24 1 0 0.84 20.16 100 6.60 90
24 1 1 0.80 19.2 100 9.94 90
Table 4 Mesh sizes of Ti–3Al–2.5V titanium alloy anisotropic tubes. Mesh sizes of tubes
Φ6 mm t0.5 mm
Φ8 mm t0.6 mm
Φ12 mm t0.9 mm
Bending part (mm) (axial size circumferential size) Front and trailing parts (mm) (axial size circumferential size) Thickness direction (mm)
0.398 0.327 0.800 0.327 0.125
0.600 0.395 1.000 0.395 0.150
0.800 0.700 1.500 0.700 0.225
Table 5 Friction conditions in various contact interfaces of Ti–3Al–2.5V titanium alloy anisotropic tubes. Contact interface
Friction coefficients
Tube/wiper die Tube/pressure die Tube/clamp die Tube/bend die Tube/mandrel Tube/balls
0.05 0.25 Rough 0.05 0.05 0.05
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T. Huang et al. / International Journal of Mechanical Sciences 90 (2015) 33–43
the FE model without considering the variation in CSR is 20.7%, while the result from the FE model considering the variation in CSR is 6.9%. The AARE in outside diameter variation ratio from the FE model without considering the variation in CSR is 17.3%, while the result from the FE model considering the variation in CSR is 7.5%. These mean that with considering the variation in CSR, the simulation accuracy in the wall thinning and outside diameter change was improved by 13.8% and 9.8% compared with that without considering the variation in CSR, respectively. Fig. 7 shows a configuration comparison of simulation and experiment results of Φ12 mm t0.9 mm Ti–3Al–2.5V titanium alloy anisotropic tube with a height of 12 mm after uniaxial
of 4.25 mm (uniform plastic deformation) at a speed of 3 mm/min was investigated. Fig. 5 shows the variation in wall thinning and outside diameter within the gauge length (Fig. 6) of the anisotropic tube after the uniaxial tension. In this study, the predictability of the simulation is quantified by employing statistical parameters: average absolute relative error (AARE) [22] as AAREð%Þ ¼
i i 1 N P p P exp ∑ 100 N i ¼ 1 P iexp
ð32Þ
where P ip is the predicted value, P iexp is the experimental value, and N is the total number of data. The AARE in wall thinning ratio from
Fig. 9. The configuration comparison of Ti–3Al–2.5V titanium alloy anisotropic bent tubes: (a) simulation and (b) experiment.
24
14 12
Wall thinning ratio /%
Wall thinning ratio /%
20 16 12 8 Without considering CSR's variation Considering CSR's variation Experiment
4 0
-10
0
10
20
30
40
50
60
70
80
10 8 6 Without considering CSR's variation Considering CSR's variation Experiment
4 2 0
90 100
-10
Angle between measure point and initial bending section / o
0
10
20
30
40
50
60
70
80
90 100
Angle between measure point and initial bending section / o
20
Wall thinning ratio /%
18 16 14 12 10 8 Without considering CSR's variation Considering CSR's variation Experiment
6 4 2 0
-10
0
10
20
30
40
50
60
70
80
90 100
Angle between measure point and initial bending section / o Fig. 10. Comparison in the wall thinning ratio between simulation and experiment for Ti–3Al–2.5V titanium alloy anisotropic tubes: (a) Φ6 mm t0.5 mm, (b) Φ8 mm t0.6 mm and (c) Φ12 mm t0.9 mm.
T. Huang et al. / International Journal of Mechanical Sciences 90 (2015) 33–43
16
compression at a speed of 1 mm/min. However, the drum deformation occurs in the zone close to the end of tube specimen without considering the variation in CSR. The result shows that considering the variation in CSR correlates better with the experiment one. Springback angle / o
12
Wall thinning and the flattening of the cross-section are primary quality issues during a NC tube bending process. The springback is another key factor which affects the forming quality of tube bending. The bent tube dimension cannot satisfy the requirement of industrial application when the value of springback exceeds the error range. Therefore, in this section, a 3D elastic–plastic FE model which incorporates the constitutive model considering the variation in CSR for the NC bending process of Ti–3Al–2.5V titanium alloy anisotropic tubes was developed based on ABAQUS/Explicit, as shown in Fig. 8. The tubes were defined as a deformable solid body and were meshed by C3D8R element. While the dies were defined as discrete rigid bodies and were meshed by 3D bilinear rigid quadrilateral elements with four nodes (R3D4). Tables 3 and 4 show the bending parameters of tubes. The different friction coefficients have been assigned to the different contact interfaces as shown in Table 5. The simulation results were verified by experiments using a GQ W27YPC-63 PLC hydraulic bender with the same forming principle and dies structure. Bent Ti–3Al–2.5V titanium alloy anisotropic tubes under different dimensions are shown in Fig. 9. The figure shows that the configurations of the bent tube obtained from the simulations considering the variation in CSR are similar to those obtained in the experiments. The wall thinning predicted by different FE models and from the experiments for different dimensional tubes is shown in Fig. 10. The AARE of wall thinning for Φ6 mm t0.5 mm, Φ8 mm t0.6 mm
8 6 4
0 Φ6mm× t0.5mm Φ8mm× t0.6mm Φ12mm× t0.9mm
Fig. 12. Comparison in the springback angle between simulation and experiment for Ti–3Al–2.5V titanium alloy anisotropic tubes.
4.0
Simulation results Input data
3.5 3.0
CSR
2.5 2.0 1.5 1.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
Equivalent plastic strain Fig. 13. Comparison in CSR's variation of simulation results and input data during the NC bending process.
2.4
4.0 3.5
Flattening of cross-section /%
Flattening of cross-section /%
10
2
4.5
3.0 2.5 2.0 1.5 1.0
Without considering CSR's variation Considering CSR's variation Experiment
0.5 0.0 -0.5
Without considering CSR's variation Considering CSR's variation Experiment
14
4.2. NC bending of anisotropic tube
41
-10
0
10
20
30
40
50
60
70
80
2.0 1.6 1.2
0.4 0.0
90 100
Without considering CSR's variation Considering CSR's variation Experiment
0.8
Angle between measure section and initial bending section / o
-10
0
10
20
30
40
50
60
70
80
90 100
Angle between measure point and initial bending section / o
4.5
Flattening of cross-section /%
4.0 3.5 3.0 2.5 2.0 1.5 Without considering CSR's variation Considering CSR's variation Experiment
1.0 0.5 0.0
-10
0
10
20
30
40
50
60
70
80
90 100
Angle between measure section and initial bending section / o Fig. 11. Comparison in flattening of the cross-section between simulation and experiment of Ti–3Al–2.5V titanium alloy anisotropic tubes: (a) Φ6 mm t0.5 mm, (b) Φ8 mm t0.6 mm and (c) Φ12 mm t0.9 mm.
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T. Huang et al. / International Journal of Mechanical Sciences 90 (2015) 33–43
Fig. 14. Distribution of CSR at different bending angles of bent tube: (a) 101, (b) 451 and (c) 901.
and Φ12 mm t0.9 mm are 21.7%, 22.2%, and 20.7% by the FE models without considering the variation in CSR, and 8.6%, 15.1%, and 8.5% by the FE models with considering the variation in CSR, respectively. This means that the simulation accuracy in wall thinning can be improved by 13.1%, 7.1%, and 12.2% for three dimensional tubes, respectively. The flattening of the cross-section predicted by different FE models and from the experiments for different dimensional tubes is shown in Fig. 11. Compared with the experiments, the AARE in flattening of the cross-section for Φ6 mm t0.5 mm, Φ8 mm t0.6 mm and Φ12 mm t0.9 mm are 20.5%, 12.4%, and 25.9%, respectively, by the FE models without considering the CSR's variation, and 12.3%, 11.0%, and 20.0% by the FE models with consideration of the CSR's variation. Considering the variation in CSR, the simulation accuracy in the flattening degree of the cross-section was improved by 8.2%, 1.4%, and 5.9%, respectively, compared with that without considering the variation in CSR. Fig. 12 shows the springback angles predicted by different FE models and from the experiments for different dimensional tubes. With considering the variation in CSR based on constitutive model for FE model, the AARE in springback angle is 4.1%, 12.9% and 5.6% for Φ6 mm t0.5 mm, Φ8 mm t0.6 mm and Φ12 mm t0.9 mm tubes, respectively. However, without considering the variation in CSR, the AARE is 10.4%, 17.5% and 13.8%, respectively. This means that the simulation accuracy in the springback angle was improved by 8.2% mostly. The results show that the wall thinning, the flattening of the cross-section and the springback angle predicted have a better agreement with the experiments when using the constitutive model considering the variation in CSR for different dimensions anisotropic tubes. This is related to the variation and distribution in CSR during the plastic deformation process of tubes. Fig. 13 shows the variation of CSR with plastic strain of a feature element on Φ12 mm t 0.9 mm Ti–3Al–2.5V titanium alloy anisotropic tube. As observed in Fig. 13, the values of the CSR obtained from the simulation are in good agreement with the input data. Fig. 14 shows the distribution of CSR during the bending process of tube. In the initial deformation stage, the CSR shows a non-uniform distribution, and the CSR value in the inside and outside of the bending is less than that on the neutral layer zone. This may be due to the fact that larger strain occurs in the outside and inside of the bent tube, and the CSR decreases with the increase of plastic deformation. In the later bending deformation, CSR value decreases and tends to be stable and the distribution of CSR tends to be uniform within main deformation zone. The uneven distribution zones of CSR are focused on transitional zones between deformation zone and the non-deformation zone. This shows that the variation in CSR in the deformation zone is different when deformation degree of tube is different. However, FE model cannot
accurately describe the plastic deformation of tube without considering variation in CSR in tube bending, and due to it ignoring CSR variation characteristic. Therefore, the simulation accuracies of FE model which incorporate the constitutive model considering the variation in CSR are closer to the experimental values compared with the CSR for constant simulation.
5. Conclusions In this study, an elasto-plastic constitutive model which contains the variation in CSR with plastic deformation was developed and evaluated. The conclusions are as follows. An elasto-plastic constitutive model was developed based on the Hill's 48 anisotropic yield function, where the variation in CSR with plastic deformation was incorporated. The constitutive model was implemented into ABAQUS/Explicit by using the user material subroutine VUMAT. This constitutive model was testified reliable by using the single cubic element FE models for uniaxial tension, uniaxial compression and pure shearing process. Moreover, comparisons between simulation results and experiment results of the uniaxial tension, the uniaxial compression and the NC bending show that the elasto-plastic constitutive model considering the variation in CSR could improve the prediction accuracy for the plastic deformation processes of anisotropic tubes.
Acknowledgments The authors would like to thank the National Science Fund for Excellent Young Scholars of China (51222509), the National Natural Science Foundation of China (51175429), the Foundation of NWPU (JC201136), and Project 111 of China (B08040). References [1] Zhan M, Yang H, Huang L, Gu RJ. Springback analysis of numerical control bending of thin-walled tube using numerical-analytic method. J Mater Process Technol 2006;177:197–201. [2] Jiang ZQ, Yang H, Zhan M, Yue YB, Liu J, Xu XD, et al. Establishment of a 3D FE model for the bending of a titanium alloy tube. Int J Mech Sci 2010;52:1115–24. [3] Yang H, Li H, Zhang ZY, Zhan M, Liu J, Li GJ. Advances and trends on tube bending forming technologies. Chin J Aeronaut 2012;25:1–12. [4] Srbislav A, Milentije S, Dragan A, Vukić L. Variation of normal anisotropy ratio “r” during plastic forming. J Mech Eng 2009;55:392–9. [5] Contractile strain ratio testing of titanium hydraulic tubing. Aerospace standard, SAE AS4076; 2001. [6] Hwang SK, Sabol GP. A new strain gage method for measuring the contractile strain ratio of zircaloy tubing. J Nucl Mater 1988;151:327–33. [7] Peters WH, Ranson WF. Digital image techniques in experimental stress analysis. Opt Eng 1982;21:427–31.
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