Mechanism of increasing spinnability by multi-pass spinning forming – Analysis of damage evolution using a modified GTN model

International Journal of Mechanical Sciences 159 (2019) 1–19

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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci

Mechanism of increasing spinnability by multi-pass spinning forming – Analysis of damage evolution using a modified GTN model He Wu a, Wenchen Xu a,∗, Debin Shan a, Bo Cheng Jin b a

School of Materials Science and Engineering and National Key Laboratory for Precision Hot Processing of Metals, Harbin Institute of Technology, Harbin 150001, PR China b Department of Aerospace and Mechanical Engineering, Viterbi School of Engineering, University of Southern California, Los Angeles, CA 90007, United States

a r t i c l e Keywords: Multi-pass spin forming Damage Ductile fracture Finite element method

i n f o

a b s t r a c t Multi-pass spinning forming is one of the major approaches to manufacture rotationally symmetric thin-walled components, which is strongly influenced by the spinnability of materials. But it remains an open question regarding the formability limit in the multi-pass spinning process. According to the spinning experiment phenomena under different spinning passes and thinning rates, a modified GTN model was incorporated into the FE simulation to analyze the damage evolution and cracking mechanism during single-pass tube spinning and multi-pass spinning. The results show that the thinning rate and spinning pass have an important influence on the forming limit of 2024 aluminum alloy. Smaller thinning rates in multi-pass tube spinning led to earlier crack compared to single-pass tube spinning, which was induced mainly by void damage, while the cracking under larger thinning rates was primarily caused by shear damage. When the pass thinning rate reached ∼21%, no crack occurred at the total thinning rate of 38% in multi-pass spinning, which was much higher than the forming limit (no more than 29.85%) in single-pass spinning. In this case, both the insufficient plastic deformation and low stress triaxiality suppress the damage accumulation, contributing to the increase of the spinnability of materials in multi-pass spin forming. This study provides an effective guidance for establishing the thinning strategy for spin forming practice.

1. Introduction Spin forming (or flow forming) is an advanced incremental forming process widely used to produce axisymmetrical or even nonaxisymmetrical thin-walled components, such as tubular, conical and curvilinear workpieces [1–3]. In the spinning process, the rollers advance axially and/or radially over the tube blank/sheet blank revolving together with the mandrel, which induces the progressive change in its geometry and thickness according to the profile of the mandrel [1]. Compared with traditional press forming process using male and female die, spinning forming exhibits its inherent advantages and flexibility such as simple tooling, low load short lead time, high material utilization and improved mechanical properties, so it is increasingly utilized in the aviation, automotive, defense, energy, and electronics industry [1,2]. In consideration of high precision, productivity, and low cost, this process is often performed in multiple passes at room temperature [4]. Nowadays, multi-pass spin forming is employed as one of the major methods to manufacture axisymmetric parts with the diameters of 3 mm–10 m and the thicknesses of 0.3 mm–25 mm [2]. Although spin forming has gained extensive application in different industry fields, the theoretical study of multi-pass spinning forming ∗

is quite lacking because the spinning process is very complex. There are highly non-linear tooling/workpiece interaction, strongly inhomogeneous material flow and complex local stress state existing throughout the entire flow forming process. Actually, cracking often takes place during tube forming of low-ductility materials [5,6], which is possibly because plastic deformation surpasses the general limit of flow formability (or spinnability) characterized by the maximum thinning rate the material can undertake in single or multiple passes. However, it is not clear what is the exact spinnability limit of materials in the spinning process. Since the spinnability “cone” test proposed by Kalpakcioglu et al. [7] underestimates the forming limits of tubes, Bilya et al. put forward a spinnability “step” test, which could present the upper bound value of forming limit through multi-pass tube spinning. However, the thickness reduction in each pass should be optimized for increasing the spinnability, which is also very helpful to establish the thinning strategy for spin forming practice [8]. To this end, it is necessary to analyze and predict the forming limit of materials during tube spinning through robust theoretical methods, which is the prerequisite for determining the proper multi-pass spinning process. Compared with the conventional forming limit diagram (FLD) based on necking models, the FE simulation combined with ductile fracture

Corresponding author. E-mail addresses: [email protected] (W. Xu), [email protected] (D. Shan).

https://doi.org/10.1016/j.ijmecsci.2019.05.030 Received 2 January 2019; Received in revised form 29 April 2019; Accepted 22 May 2019 Available online 23 May 2019 0020-7403/© 2019 Elsevier Ltd. All rights reserved.

H. Wu, W. Xu and D. Shan et al.

International Journal of Mechanical Sciences 159 (2019) 1–19

Table 1 The chemical compositions of 2024-T351 (% in weight).

Nomenclature p q 𝜃 𝜂 f∗ f q1 , q2 𝜀𝑚 𝑞 𝜺p 𝜎s fN 𝜀N SN fc ff Ds D0 Dc Df ks , kv 𝜂 cut

hydrostatic pressure second stress invariant lode angle parameter (− 1 ≤ 𝜃 ≤ 1) stress triaxiality effective porosity void volume fraction adjustment parameters for the GTN model matrix equivalent plastic strain the macroscopic plastic strain tensor flow stress of the fully dense matrix potential nucleation rate mean value of plastic strain for void nucleation standard deviation of 𝜀N critical void volume fraction for initial coalescence void volume fraction at final failure shear damage initial shear damage critical shear damage shear damage at final failure adjustment parameter associated with shear damage cut-off value for stress triaxiality

Si

Fe

Cu

Mn

Mg

Cr

Zn

Ti

0.26

0.29

4.5

0.52

1.43

0.05

0.10

0.06

Fig. 1. The microstructure of 2024-T351 alloy.

ductile fracture during single-pass backward tube spinning of Ti-15-3 alloy and the tube spinnability test of 2024-T351 alloy was successfully predicted. However, the formability limit in the multi-pass tube spinning is still an open question and the influence of spinning pass on the damage evolution is not clear until now due to the complexity of tube spinning, which makes it difficult to control forming defects in the multi-pass tube spinning process. In this work, our research is focused on the influence of thinning rates to the damage evolution under the multi-pass spinning process. The tube spinning process of 2024-T351 aluminum alloy was modeled using a three-dimensional finite element method (FEM). And the two-pass spinning experiments were conducted using different combinations of thinning rate in each pass. To explore the crack initiation mechanism and evaluate the material spinnability during the multi-pass forming process, a shear modified GTN model was proposed and incorporated into finite element software (Abaqus) via user-defined material subroutine (VUMAT) to simulate the damage evolution in different spinning passes. This study can provide an effective guidance for improving the tube spinnability and optimizing the process parameters in the multi-pass spinning process.

criteria (DFC) is generally recognized as a more efficient way to analyze the ductile fracture of metal forming. However, the study on the cracking mechanisms in the multi-pass spinning forming process is still scarce because the forming process is highly non-linear and very complex. Currently, the ductile fracture criteria can be divided into uncoupled and coupled models for fracture prediction by defining whether the damage variable is included in the material constitutive model. The widely used uncoupled fracture criteria include the Rice-Tracey, McClintock, Oh, Cockcroft-Latham, LeRoy and Ayada DFCs. Ma et al. [9] investigated some typical uncoupled DFCs in the tube spinnability test of pure titanium, indicating that the C-L criterion provides higher prediction accuracy regarding the spinnability of TA2 titanium tube through comparing their results with other uncoupled DFCs. On this basis, the damage evolution and cracking mechanism in tube spinning of titanium alloy were investigated by Xu et al. [10]. But, unfortunately, the simulation results are highly dependent on the mesh size. The other category i.e., coupled DFCs, couples the influence of plastic damage into the material constitutive model, such as the Lemaitre model and GTN model. Zhan et al. [11] investigated the damage evolution mechanism in the splitting spinning process based on the Lemaitre model, and the results indicate that the FE simulations using the original Lemaitre model could not accurately predict fracture during splitting spinning. In order to solve these problems, they proposed a modified Lemaitre DFC to investigate the processing window of 5A02-O aluminum alloy in splitting spinning [12]. Another popular coupled DFC is the Gurson-Tvergaard-Needleman (GTN) model deduced by means of the micromechanical method. Although the GTN model has been widely used in predicting ductile failure, it does not perform well under low stress triaxiality condition. To overcome this drawback, several modified Gurson-type models have been proposed by introducing the shear damage to the original model [13–17]. But unfortunately, those modified GTN models predict much earlier failure prediction during the spinning process since shear damage accumulation is not suppressed under negative stress triaxiality [18]. Accordingly, Li et al. [19] and Wang et al. [20] stated that the original GTN model is not appropriate for predicting damage evolution in deformation area because of the negative stress triaxiality. Motivated by the limitation, Wu et al. [18] improved the GTN model through redefining the failure strain as a function of stress triaxiality based on the work of Zhou et al. [14], by which

2. Materials The 2024-T351 aluminium alloy was used in the backward tube spinning experiment. The chemical composition of 2024-T351 aluminium alloy was listed in Table 1. The microstructure of 2024-T351 alloy was examined using a Quanta 200 FEG scanning electron microscope (SEM) with a voltage of 20 kV, as shown in Fig. 1. The basic material properties of the 2024-T351 alloy were obtained through a tensile experiment carried on an AG-X Plus 20 kN/5 kN electronic universal testing machine manufactured by SHIMADZU Corporation in Japan, which were listed as follows: Young’s modulus E = 71.0 GPa, Poisson ratio 𝜈 = 0.33, and initial yield strength 𝜎 s = 366.6 MPa. 3. Constitutive model 3.1. Damage model The Gurson-Tvergaard-Needleman (GTN) [21–23] model is a ductile fracture model, which is established based on the micromechanical 2

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International Journal of Mechanical Sciences 159 (2019) 1–19

theory and can be expressed as follows: ( )2 ( ) ( ( ) ( )2 ) 3𝑞 𝑝 𝑞 𝜙 𝝈, 𝑓 ∗ , 𝜀𝑚 + 2𝑞1 𝑓 ∗ 𝑐𝑜𝑠ℎ − 2 − 1 − 𝑞1 𝑓 ∗ = 𝑞 𝜎𝑠 2 𝜎𝑠

Fig. 1 shows that many second phase particles are distributed in the Al matrix, indicating that the nucleation of voids could be promoted at the second phase particles and inclusions by particle cracking or interface decohesion when the plastic strain level was high [24]. Accordingly, Brünig et al. [25] stated that a cut-off value of stress triaxiality should be defined below which fracture will never occur:

(1)

𝑚 where 𝜎𝑠 = 𝜎𝑠 (𝜀𝑚 𝑞 ) is a function of the equivalent plastic strain 𝜀𝑞 . The void volume fraction accumulation consists of void nucleation and growth:

𝑓̇ = 𝑓̇𝑛 + 𝑓̇𝑔

𝜂𝑐𝑢𝑡 = −0.6 + 0.27 ∗ 𝜃̄ for 𝜃̄ ∈ [0, 1]

(2)

(10)

where 𝑓̇𝑛 is the void volume rate due to nucleation of new voids. It can be defined as:

Thus, a new void nucleation criterion is proposed here, which can be expressed as:

𝑓̇𝑛 = 𝐴 𝜀̇ 𝑚 𝑞

( ) 𝑓̇𝑛 = 𝜑𝑣 𝜂, 𝜃̄ 𝐴 𝜀̇ 𝑚 𝑞

𝐴=

𝑓𝑁 √

𝑆𝑁

(3) ⎡ 1 𝑒𝑥𝑝⎢− ⎢ 2 2𝜋 ⎣

(

𝜀𝑚 𝑞 − 𝜀𝑁 𝑆𝑁

)2

⎤ ⎥ ⎥ ⎦

And the evolution law of shear damage (Ds ) could be modified and expressed as:

(4)

( ) 𝐷̇ 𝑠 = 𝜑𝑠 𝜂, 𝜃̄ 𝑔0 𝑘𝑤 𝐷𝑠 𝜀̇ 𝑚 𝑞

In Eq. (2), 𝑓̇𝑔 is void volume rate due to the growth of existing voids: 𝑓̇𝑔 = (1 − 𝑓 )𝑇 𝑟(𝜺̇ 𝑝 )

In order to incorporate coalescence into the Gurson model, Tvergaard and Needleman [21] identified the porosity evolution as an addition of porosity nucleation and growth mechanism using a specific coalescence function f∗ , which replaces the porosity in the following way:

⎧1 ⎪ 1 ( ) ⎪[ ( 𝜑 𝜂, 𝜃̄ = ⎨ 1 − 𝑞0 ln 1 − ⎪ ⎪ ⎩0

𝑖𝑓 𝑓 ≤ 𝑓𝑐 𝑖𝑓 𝑓 ≥ 𝑓𝑐

( ) 3𝑞2 𝑝 ∗ + 2 𝑞 𝑓 cos ℎ − ( ) 1 )2 2 1 − 𝐷𝑠∗ 𝜎𝑦 1 − 𝐷𝑠∗ 𝜎𝑦2 ( ( )2 ) − 1 + 𝑞1 𝑓 ∗ 𝑞2

(7)

where the evolution equation for f∗ is the same as in the original GTN model (see Eqs. (2)–(6)). The present model adopts two separate scalar damage variables, namely the volume void fraction f and the shear damage variable Ds . Therefore, the model includes two distinct critical values: the critical volume void fraction fc , which is the critical volume of void fraction in the GTN original model, and the critical shear damage Dc , which is regarded as a material constant that needs to be calculated. In addition, similar to the f∗ function, a new 𝐷𝑠∗ function is used to simulate the coalescence of micro-shear bands once the critical condition is reached [13,15]. In this stage, the damage increases rapidly and the softening process accelerates so that material abruptly loses its load carrying capacity. ⎧𝐷 𝑠 ⎪ 𝐷𝑠∗ = ⎨ ) 1 − 𝐷𝑐 ( ⎪𝐷 𝑐 + 𝐷 − 𝐷 𝐷 𝑠 − 𝐷 𝑐 ⎩ 𝑓 𝑐

𝜂>0 𝜂 𝜂𝑐𝑢𝑡

)]

𝜂𝑐𝑢𝑡 < 𝜂 ≤ 0 𝜂 ≤ 𝜂𝑐𝑢𝑡

𝑖𝑓 𝐷𝑠 ≤ 𝐷𝑐 𝑖𝑓 𝐷𝑠 ≥ 𝐷𝑐

(8)

where Df denotes the shear damage at complete shear failure. The evolution law of shear damage (Ds ) is expressed as [13] ∗ 𝑚 𝐷̇ 𝑠 = 𝑔0 𝑘𝑠 𝐷𝑠 𝜀̇ 𝑚 𝑞 + 𝑔0 𝑘𝑣 𝑓 𝜀̇ 𝑞

(13)

where q0 is the sensitivity coefficient for damage evolution in the range of 𝜂 cut < 𝜂 ≤ 0. Considering the influence of the parameter q0 on damage evolution ̄ with different q0 under negative stress triaxiality, the function 𝜑(𝜂, 𝜃) ̄ invalues is illustrated in Fig. 2. Clearly, the rate of decline of 𝜑(𝜂, 𝜃) creases for larger q0 values. Especially, there is a steep decline in the value of this function when the stress triaxiality is close to the cut-off value 𝜂 cut . Actually, the values of q0 are different for the function 𝜙v in Eq. (11) and 𝜙s in Eq. (12) because the void nucleation and shear damage showed different sensitivity for the negative stress triaxiality. Therefore, the parameter q0 was re-written as qv and qs in the function 𝜙v and 𝜙s , respectively. The element removal technique is applied in the simulation to observe the initial cracks clearly. This technique removes an element when the value of the corresponding solution-dependent state variable (SDV) reaches a specific value [26]. In the current study, we define SDV = 0 when f ≥ ff or Ds ≥ Df , i.e., the element will be removed.

(6)

where ff denotes the shear damage at complete shear failure. In this work, we modified the model proposed by Jiang et al. [13] to describe damage evolution during tube spinning of 2024 aluminum alloy: ( ) ∅ 𝝈, 𝑓 ∗ , 𝜀𝑚 𝑞 , 𝐷𝑠 = (

(12)

In this study, 𝜙v and 𝜙s take the same functional form, i.e., a stress-state dependent function controlling damage evolution in different stress triaxialities, which can be written as:

(5)

⎧𝑓 ⎪ 𝑓∗ = ⎨ ) 1∕𝑞1 − 𝑓𝑐 ( ⎪𝑓 𝑐 + 𝑓 − 𝑓 𝑓 − 𝑓 𝑐 ⎩ 𝑓 𝑐

(11)

(9)

̄ where 𝑔0 = 1 − |𝜃|.

Fig. 2. Effect of q0 on damage evolution in the range of 𝜂 cut < 𝜂 ≤ 0. 3

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International Journal of Mechanical Sciences 159 (2019) 1–19

Fig. 3. Sample geometries and loading types for the specimens.

program was developed based on the program proposed by Afshan [29], in which the Genetic Algorithm (GA) was used for parameter optimization. The values of q1 and q2 suggested by Tvergaard [30] as 1.5 and 1 were adopted in current research. Because the shear damage was very small during the tensile test and the void damage was the major factor to induce the final fracture, the parameters fN , SN , 𝜀N , fc and ff were identified using the tensile test. The parameters kws , kwv , Dc and Df were calibrated from the shear test and compression test. The parameter qs and qv were set as 0.025 and 0.1, respectively, to ensure consistency between experiments and simulations during multi-pass tube spinning. All the parameters obtained for the modified GTN model are presented in Table 3. Fig. 4 illustrates the meshes adopted in the simulations. A sensitivity analysis was carried to out to determine the influence of the mesh on the results. Three FE meshes were analyzed. The results of this mesh sensitivity analysis are shown in Fig. 5(b)–(d). The load and displacement results shows that there was no evident influence of the three mesh models on the results, so mesh I will be used for calibration process. The FE predicted force vs. displacement curves for tensile, shear, and compression tests were used to verify the reliability of the GTN model parameters through comparing the numerical results with experimental data, as shown in Fig. 5(b)–(d). The FE predictions agree well with the experimental curves. Fig. 6 shows the comparison between the simulated and experimental results, which indicates that the simulation results were basically consistent with the experimental observations. Under tensile condition, the load-displacement results show that there was no effect of shear when comparing the experimental results with standard GTN model and shear modified GTN model (see Fig. 6(b)). Fig. 7 shows the comparison of the damage evolution in the fracture zone between the standard GTN model and the modified GTN model under compression condition. It can be seen that the void fraction of standard GTN model is nearly zero in the whole deformation process. In this case, cracking will never take place in the compression process, which is inconsistent with the experiment result. Therefore, the modified GTN model is more able to predict fracture under low stress triaxiality condition.

3.2. Identification of GTN model parameters A series of tests under various stress states were performed to determine the model parameters. Fig. 3 shows the sizes of the specimens and loading types in the tests. The specimens were meshed with C3D8R elements in the FE models. The true stress-strain relationship from the tensile test was described using the mixed isotropic hardening law, which was modeled through a linear combination of the Swift [27] and the Voce [28] law,

(14) where the weighting factor 𝛼 ∈ [0,1]; 𝜀0 , N and K were material parameters defined in the Swift law; 𝜎 0 , b and 𝜎 sat were material parameters defined in the Voce law. The Swift parameters {𝜀0 , N, K} and Voce parameters {𝜎 0 ,b, 𝜎 sat } were determined by fitting the Swift and the Voce law to the experimental true stress-plastic strain data under uniaxial tensile condition. The weighting factor 𝛼 was found to be 0.3, as shown in Fig. 5(a). A summary of all the strain hardening behavior parameters for the 2024-T351 aluminium alloy is listed in Table 2. In this work, the inverse method based on the optimization techniques was used to calibrate the parameters in the modified GTN model. Regarding the identification of the GTN model parameters, a MATLAB Table 2 Strain hardening behavior parameters for 2024-T351 aluminium alloy. Swift (1 − 𝛼)

Voce (𝛼)

Weighting factor

K(MPa)

𝜀0

N

𝛼

𝜎0

b

𝜎 sat

946.17

0.01334

0.2211

0.3

366.368

14.89

285.97

4

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International Journal of Mechanical Sciences 159 (2019) 1–19

Table 3 Parameters for GTN model. q1

q2

qs

qv

𝜀N

SN

fN

fc

ff

ks

kv

D0

Dc

Df

1.5

1.0

0.025

0.1

0.15

0.075

0.025

0.02

0.04

4.0

1.0

0.01

0.06

0.07

Fig. 4. Finite element meshes of the specimens.

3.3. Backward multi-pass tube spinning experiment

since the tube blanks were cooled by emulsion liquid in the spinning experiments. The spinning experiments were carried out on a CNC spin forming machine, as shown in Fig. 9. During the whole spinning process, the tube workpiece was fixed around the mandrel with high rotation speed (100 r/min), as shown in Fig. 10. Although the initial crack occurred at t0 (corresponding to the axial cracking position l0 ), it could not be observed immediately since the tube workpiece rotated around the mandrel. The initial crack could only be observed when the spin process

The tubular blank was mounted on a rotating mandrel and the metal was forced to flow reversely along the axial direction of the mandrel by two rollers during backward multi-pass tube spinning. As a result, the wall thickness of as-spun tubular workpiece was uniformly reduced and the length was gradually increased. In this work, this spinning process was conducted in one pass or two passes to evaluate the spinnability limit, as shown in Fig. 8. The influence of temperature was ignored

5

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International Journal of Mechanical Sciences 159 (2019) 1–19

Fig. 5. Inter- and extrapolation of the isotropic hardening function (a) and comparison of the simulation and experiment force-displacement curves: (b) tensile; (c) shear; (d) compression.

Fig. 6. Comparison of the simulation results with the experimental results.

6

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International Journal of Mechanical Sciences 159 (2019) 1–19

Fig. 7. Comparison of the damage evolution in the fracture zone between the standard GTN model and the modified GTN model under compression condition.

Fig. 9. CNC spin forming machine. Table 4 Fundamental parameters for tube spinning. Parameters

Value

The initial thickness of tube blank (mm) The diameter of the mandrel (mm) The diameter of the roller (mm) Front angle (°) Fillet radius of the roller (mm) Rotational speed (r/min) Feed radio (mm/r)

5 65 200 20 5 100 1.0

The die and process parameters of the spinnability test are shown in Table 4 and the coolant was used to keep tube blanks to room temperature in the spinning experiments. The design scheme of the thinning rate in various spinning experiments is listed in Table 5. The experimental photos of as-spun workpieces are illustrated in Fig. 11 (See Fig. 19 for more details). According to the experiment results, an interesting experimental phenomenon can be found in the spinning process. Cracking did not occur during single-pass spinning when the thinning rates were 12.35%, 21.20% and 26.75%, respectively (Experiment 1, 2, 3).

Fig. 8. Schematic of backward multi-pass tube spinning.

stopped due to the appearance of large crack induced surface fracture at t1 (corresponding to the axial fracture position l1 ). In this study, the value of l0 was used as an index to evaluate the accuracy of the initiation cracking time through comparing the simulation and experimental results.

Fig. 10. Schematic diagram showing the cracks evolution during the spinning process.

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International Journal of Mechanical Sciences 159 (2019) 1–19

Table 5 Experimental results during multi-pass tube spinning of 2024-T351 aluminium alloy (where l0 represent the axial cracking position of the initial crack). Total Experiment reduction (%) 1 2 3 4 5 6 7 8

12.35 21.20 26.75 29.85 24.45 30.35 38.00 39.77

1st pass

2nd pass

Reduction (%) Result Reduction (%) Result 12.35 21.20 26.75 29.85 12.35 21.70 21.40 21.50

Good Good Good Crack Good Good Good Good

– – – – 13.80 11.05 21.12 23.27

– – – –– Crack Crack Good Crack

l0 (mm) / / / 37.5 29.5 15.3 / 39.1

Fig. 12. 3D elastic-plastic FE model for tube spinning.

no cracking appeared (Experiment 3). Interestingly, the crack was eliminated if the pass thinning rate increased to a certain extent relative to Experiment 5, as shown in Fig. 11 (Experiment 7). However, when the thinning rate in the first pass was fixed nearly at the same value, followed by increasing or decreasing the second-pass thinning rate, the cracks emerged again (Experiment 6 and 8). The experiment results show that the thinning rate in each pass exerted a complicated influence on the formability limit of 2024 aluminium alloy in the multi-pass spinning process, which needs further analysis on the damage evolution mechanism by means of FE simulation. 3.4. Development of the FE model The FE models were established with the ABAQUS/Explicit, as shown in Fig. 12. In this model, the tube blank was defined as a deformable body, and the material of the deformed body was assumed to be homogeneous and isotropic. To improve computational efficiency, the rollers and the mandrel were set as rigid bodies. The Arbitrary Lagrangian Eulerian (ALE) adaptive meshing technique was adopted to automatically regenerate the finite-element mesh and control element distortion at each time increment during tube spinning [10]. Hence the tube blank was meshed by 3D 8-node brick elements with reduced integration (C3D8R). The penalty contact method and the Coulomb’s friction law were used to describe the contact behavior between the material and tools. The contact pairs between the tooling and blank were assigned two different frictional coefficients: roller-blank 0.02, mandrel-blank 0.2 [31]. Theoretically, the mesh size should influence the prediction accuracy. To reasonably determine the mesh size, three FE models with the different mesh sizes were analyzed, as shown in Fig. 13. The matrix

Fig. 11. Photographs of 2024-T351 aluminium alloy spin formed tubes.

And cracks could be found when the thinning rate increased to 29.85% (Experiment 4). However, the combination of two small thinning rates (12.35% First pass + 13.80% Second pass = 24.45% Total ) could induce the cracks on the tube surface (Experiment 5). It was worth noting that the total thinning rate of the multi-pass spinning process (Experiment 5) was even smaller than that in the one-pass spinning process, in which

Fig. 13. Three mesh models with different mesh sizes. 8

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International Journal of Mechanical Sciences 159 (2019) 1–19

equivalent plastic strain, void fraction and shear damage distribution were compared, as shown in Fig. 14. It is observed that the values of matrix equivalent plastic strain, void fraction and shear damage presented little difference among the three mesh models. Therefore, the mesh size showed no significant influence on the results in the FE models, so model I was used in subsequent simulations of tube spinning to reduce computation time in this study.

4. FE simulation results 4.1. Single-pass backward tube spinning The damage distribution predicted with the modified GTN model during backward tube spinning process was discussed here. Since tube spinning was a continuously local plastic forming process, a material point should experience the uplift deformation stage and the thinning deformation stage according to the analysis by Ma et al. [9]. Fig. 15 shows the un-deformed area (I), uplift area (II), deforming area (III) and deformed area (IV) during backward tube spinning. Fig. 16 shows the damage distribution during the tube spinning process using the standard GTN model. As the thinning rate increased to 40%, the maximum void damage predicted by the standard GTN model was smaller than the critical value due to the negative stress triaxiality. However, the crack took place only when the thinning rate reached 29.85% in the spinning experiment (see Fig. 11 #4), which indicates that the standard GTN model could not predict material failure during tube spinning. Therefore, the modified GTN model was utilized to simulate the damage evolution in the spinning process in this study. As the material failure during plastic forming mainly depended on shear damage and void volume fraction in the current model, the shear damage distribution and void volume fraction distribution under various thinning rates are shown in Fig. 17. It can be found that although shear damage fluctuation occurred in the thinning rates ranging from 12% to 16%, the value of shear damage presented an overall upward tendency with increasing thinning rate, as shown in Fig. 18. Besides, it is should be noted that the maximum shear damage was located on the outer layers when the thinning rate was less than 12%, and then the location corresponding to the maximum shear damage moved to the inner layer when the thinning rate increased to 21.5%, as shown in Fig. 17 (a and b). Regarding void volume fraction, the maximum value was always located on the inner layer of the deforming area. The void volume fraction at a 16% thinning rate was larger than that under other thinning rates. This could be explained by means of the void volume fraction evolution law (See Eqs. (3)–(5)). Plastic deformation was small

Fig. 14. Matrix equivalent plastic strain, void fraction and shear damage distribution at thinning rate 33%.

Fig. 15. Schematic of area division of the tube in backward tube spinning (Thickness variation in red marked section). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 9

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International Journal of Mechanical Sciences 159 (2019) 1–19

Fig. 16. Damage distribution using standard GTN model: (a) thinning rate 33%; (b) thinning rate 40%.

Fig. 17. Damage distribution under the different thinning rates.

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Fig. 18. Shear damage/void volume fraction for single pass tube spinning with different thinning rate.

at a low thinning rate, thus void nucleation was insufficient. However, negative stress triaxiality could suppress the void nucleation in the uplift deformation stage and close voids within the deforming area during in the thinning deformation stage when the thinning rate was larger. This should be the reason that the void volume fraction value in the steady deformed area was less than that in the deforming area, as shown in Fig. 18. Finally, fracture occurred when the thinning rate reached 33% according to the FE simulation result, which was only slightly greater to the experimental fracture thinning rate of 29.85%. The simulations show that the shear damage was greater than the critical shear damage value (Dc = 0.06) and the void volume fraction was less than the critical void volume fraction (fc = 0.02) at the thinning rate of 33%, indicating that the shear damage actually dominated the initiation of ductile fracture on the inner surface of as-spun tube workpiece. Similar experimental results during the tube spinning were also reported by Bylya et al. [8] and Wu et al. [18]. Also, void volume fraction would reduce with increasing thinning rate when the thinning rate exceeded a certain value. 4.2. Multi-pass backward tube spinning Fig. 19 shows the experimental photos of multipass spin formed tube, and Fig. 20 shows the shear and void volume fraction distribution at different thinning rates during multi-pass tube spinning. Table 6 presents the comparisons between the numerical predictions and the experimental results of tube spinning. The experimental results show that the damage evolution exhibited the strong dependence on the thinning rate during multi-pass spinning. As seen in Table. 6, most of the predicted thinning rates show a good correlation with the experiments, whereas a small difference can be found in the prediction of the fracture time at the total thinning rates of 24.45% (First pass: 12.35% + Second pass: 13.80%). Basically, the predicted thinning rates and fracture time coincide well with the experimental results. In other words, the simulation results confirm that the shear modified GTN model is able to predict the material failure in multi-pass tube spinning. It can be observed that crack initiation always occurred on the inner layer, as shown in Fig. 20(a), (b) and (d). The simulation results show that the cracking mechanism for crack formation changed under different thinning rates. Fig. 20(a) and (b) show that the maximum shear damage was less than the critical shear damage value (Dc = 0.06), thus void coalescence should be the primary cause of material failure, where the maximum void volume fraction reached the final void volume fraction (ff = 0.04) at the total thinning rates of 24.45% (First pass: 12.35% + Second pass: 13.80%) and 30.35% (First pass:

Fig. 19. Photo of the multi-pass backward spin formed tubes. 11

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Fig. 20. Damage distribution under the different thinning rates for multi-pass spinning and the image of the fractured tube.

5. Discussions

21.70% + Second pass: 11.05%). When the total thinning rate increased to 38.0%, both the shear damage and void volume fraction were suppressed, thus no crack was observed. However, when the total thinning rate increased to 39.77% (First pass: 21.50% + Second pass: 23.27%), shear damage became the primary failure mechanism.

As indicated in Ma et al. [9], the evolution of the internal state variables for a material point in tube blank generally follows a periodic pattern during tube spinning since metal spinning is a continuously 12

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Table 6 Comparisons between the numerical predictions and the experimental results (where l0 represents the axial cracking position of the initial crack). Simulation

Experiment

Total reduction

1st pass reduction

Result

2nd pass reduction

Result

l0 (mm)

Total reduction

1st pass reduction

Result

2nd pass reduction

Result

l0 (mm)

25.00% 30.00% 38.00% 39.50%

12.00% 21.50% 21.50% 21.50%

Good Good Good Good

14.80% 10.80% 21.02% 22.93%

Crack Crack Good Crack

20.8 13.6 / 36.3

24.45% 30.35% 38.00% 39.77%

12.35% 21.70% 21.40% 21.50%

Good Good Good Good

13.80% 11.05% 21.12% 23.27%

Crack Crack Good Crack

29.5 15.3 / 39.1

Fig. 21. Definition of one spinning period.

local forming process. To analyze the damage evolution conveniently, one spinning period in the spinning process was defined, as shown in Fig. 21. 5.1. Mechanism of increasing spinnability by multi-pass spinning forming In terms of the damage evolution laws Eq. (3), ((5) and (9)) during plastic forming, the equivalent plastic strain was an important factor that led to material failure. Fig. 22(a) (Left) shows the equivalent plastic strain distribution along the radial direction in single-pass tube spinning with different thinning rates. It can be seen that the equivalent plastic strain of all the layers increased monotonously with increasing thinning rate. Besides, it is worth noting that the equivalent plastic strain value exhibited a monotonic decrease from the outer to inner layers in the radial direction. The normalized equivalent plastic strain is plotted in Fig. 22(a) (Right) to analyze the influence of thinning rate on the equivalent plastic strain. In this study, the normalized equivalent plastic strain denoted the ratio of the maximum equivalent plastic strain of each layer to the maximum equivalent plastic strain of all the layers under various thinning rates, which indicates the deformation nonuniformity along the thickness of as-spun tube in the spinning process. The homogeneity of the equivalent plastic strain along the radial direction was increased with the increase of the thinning rate. This was ascribed to the influence of local compression stress of the contact area between the roller and workpiece, which reduced gradually from the outer layer to the inner layer of the tube, as shown in Fig. 22(b). In the case of a small reduction rate, the effect of local compression on the inner surface was weakened remarkably, resulting in a significant difference in the equivalent plastic strain distribution along the thickness of the tube. The similar effect was found in the multi-pass spinning process. For the comparative purpose, the matrix equivalent plastic strain distribution on the axial section of the as-spun tube during two-pass spinning

Fig. 22. Distributions of equivalent plastic strain/ Von Mises effective stress and normalized equivalent plastic strain/ normalized Von Mises effective stress for single pass tube spinning with different thinning rates in radial direction.

with total 38% thinning rate and single-pass spinning with 33% thinning rate is shown in Fig. 23. And the change of matrix equivalent plastic strain and Von Mises effective stress along the thickness direction is shown in Fig. 23. The matrix equivalent plastic strain at the thinning rate of 21.50% (first pass reduction) was less than the value in single-pass spinning with 33% thinning rate for all the layers since the corresponding Mises stress under the thinning rate of 21.50% (first pass reduction) was less than that during single-pass spinning with 33% thinning rate. However, the situation changed when the second pass spinning was finished. The simulation results show that the matrix equivalent plastic strain in two-pass spinning with the total thinning rate of 38% was not always greater than the value in single-pass spinning under the thinning rate of 33% for each layer. Because the second-pass thinning rate was only 21.02% relative to the thickness of spin formed tube after the first spinning pass was finished, the effect of local compression exerted by the rollers on the inner-layer material was not so dramatic. Fig. 24 also shows that in the second pass of two-pass spinning, the stress of the outer and second outer layers was larger than that in single-pass spinning, but 13

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International Journal of Mechanical Sciences 159 (2019) 1–19

Fig. 23. Matrix equivalent plastic strain distribution along thickness direction in the tube spinning process.

the maximum value close to the critical void volume fraction (fc = 0.02), beyond which it reduced gradually in the thinning deformation stage, while the void volume fraction for the outer layer was far below the critical value. However, the shear damage for inner layer grew progressively in the whole spinning process, which exceeded the critical shear damage value (Dc = 0.06) in the thinning deformation stage. To further investigate the damage evolution mechanism, the changes of the matrix equivalent plastic strain, shear damage, void volume fraction, stress triaxiality and the cut-off value for stress triaxiality in one period of the thinning deformation stage were analyzed, as shown in Fig. 26(a). As per the definition of the spinning period, plastic deformation took place primarily between the moment b and d (see Fig. 21). For the outer-layer element, the stress triaxiality was less than the cut-off value between the moment b and d, as shown in Fig. 26(a1), which caused the void volume fraction to reduce close to zero and suppressed the accumulation of shear damage in the thinning deformation stage. As shown in Fig. 26(a2), there was a zone, i.e. nearly the first half of the period b–d, where the stress triaxiality for the inner-layer element was higher than the cut-off value, suggesting that the accumulation of void volume fraction and shear damage would not be suppressed in the period b–d. And then the void volume fraction decreased and shear damage remained unchanged since the stress triaxiality dropped below the cut-off value of stress triaxiality. Therefore, the shear damage in the inner layer could more easily surpass the critical damage value Dc and thus caused the material failure in single-pass spinning with larger thinning rate (see Fig. 11). Fig. 25(b) shows the damage evolution in multi-pass spinning with the total thinning rate of 38%. In the first pass under 21.5% thinning rate, the void volume fraction of inner layer increased because the stress triaxiality was higher than the cut-off value between the moment b and d (see Fig. 26(b2)). In the second pass under 21.02% thinning rate, the void volume fraction decreased in the thinning deformation stage, wherein the stress triaxiality was basically less than the zero between the moment b and d (see Fig. 26(c2)). However, the void volume

Fig. 24. Change of matrix equivalent plastic strain and Von Mises effective stress with thickness direction.

the inner layers (the inner and second inner layers) present the opposite results. Therefore, the matrix equivalent plastic strain in the inner layer and second inner layer during multi-pass spinning was smaller than that during single-pass spinning even though the total thinning rate in multipass spinning was greater than that in single-pass spinning. According to the analysis, it can be inferred that the matrix equivalent plastic strain on the inner layer under multi-pass spinning should be less than the value during single-pass spinning when the total thickness reduction was equal, which may cause different damage evolution behaviors in the spinning process. Fig. 25(a) shows the damage evolution in single-pass tube spinning under the thinning rate of 33%. Both the void volume fraction for outer and inner layers increased at first and then decreased in the spinning process. Clearly, the void volume fraction for inner layer increased to 14

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International Journal of Mechanical Sciences 159 (2019) 1–19

Fig. 25. Damage evolution under the single passes spinning (a) and multi-pass spinning (b).

fraction of the outer layer was suppressed throughout the multi-pass spinning process since the stress triaxiality was basically less than the cut-off value between the moment b and d, as shown in Fig. 26(b1) and (c1). As per the shear damage in the inner layer, the damage value did not increase sufficiently since the equivalent plastic strain for the inner layer in multi-pass spinning under the total thinning rate of 38% was actually smaller than that in the single-pass spinning process under the thinning rate of 33% (see Fig. 23), thus no cracks appeared during the multi-pass spinning process (see Fig. 20(c)).

However, the shear damage in the inner layer was less than the value in the outer layer, which increased monotonically throughout the spinning process. In the second-pass spinning process, the material strength has been improved due to work hardening after the first pass spinning process. Therefore, the height of the uplift area became smaller relative to the first spinning pass, which suppressed the accumulation of void volume fraction in the uplift deformation stage. However, the shear/void volume fraction in the inner layer increased continuously throughout the spinning process. A similar evolution law could be found in multi-pass spinning, in which the total thinning rate reached 30% (see Fig. 27(b)). As shown in Fig. 27(a) and (b), the rapid accumulation of void volume fraction in the second pass was the major factor that induced the initial cracks in the multi-pass spinning process. Accordingly, the changes in the internal state variables during one spinning period are shown in Fig. 28(a). The stress triaxiality of the outer element was less than the cut-off value between the moment b and d, as shown in Fig. 28(a1), which implied that the void volume fraction and shear damage were suppressed. However, in terms of void volume fraction evolution in Eqs. (3)–(5), the void nucleation and growth in inner layer could not be suppressed since the stress triaxiality of the inner element was higher than the cut-off value between the moment b and d (see Fig. 28(a2)).

5.2. Failure mechanism of multi-pass tube spinning From the experiments the crack occurred during multi-pass spinning at 24.45% total thinning rate (First pass: 12.35% + Second pass: 13.80%), which was less than the forming limit during single-pass spinning (29.85%). To explain this interesting phenomenon, the state variables were analyzed at the failure position as shown in Fig. 27(a). In the first-pass spinning process, the void volume fraction in the outer layer increased in the uplift deformation stage and then decreased to near zero in the thinning deformation stage at the end of the spinning process, while the void volume fraction in the inner layer increased continuously, which was ascribed to the local compression effect of the roller. 15

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Fig. 26. Evolution of matrix equivalent plastic strain, void volume fraction, shear damage, stress triaxiality, and Cut-off value for stress triaxiality in the selected period under thinning deformation stage for the element on the inner and outer surface.

Fig. 27(c) shows the damage evolution when the second-pass spinning process was conducted at a larger thinning rate of 39.5%. Compared with the damage evolution under the total thinning rate of 30%, there were obvious differences in the second-pass spinning process, in which the void volume fraction reduced gradually with the progress of the spinning process. It is possibly caused by the enhanced influence of roller compression on the inner layer since a larger thinning rate was adopted in the second pass. This also indicates that the microvoid would be annihilated in the thinning deformation stage. Meanwhile, the equiv-

alent plastic strain in the inner layer was larger than that under the total thinning rate of 30%. Therefore, the shear damage in the inner layer would be sufficiently developed at a greater thinning rate, thus the value of shear damage could exceed the critical shear damage value (Dc = 0.06). Accordingly, the changes in the internal state variables during one spinning period are shown in Fig. 28(b). Obviously, the void volume fraction for the outer element would be suppressed in the second pass since the stress triaxiality was basically less than the cut-off value during the moment b–d, as shown in Fig. 28(b1). Similarly, the 16

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International Journal of Mechanical Sciences 159 (2019) 1–19

Fig. 27. Damage evolution under the multi-pass spinning: (a) Total thinning rate 25% (12% + 14.8%); (b) Total thinning rate 30% (21.5% + 10.8%); (c) Total thinning rate 39.5% (21.5% + 22.93%).

void volume fraction of inner element was reduced in the period b–d, but the damage value was considerably larger than the outer elements. Besides, the matrix plastic strain in the inner elements increased as the thinning rate increased. Therefore, the shear damage of the inner elements increased continuously in the spinning process, thus it would exceed the critical damage value (Dc = 0.06) and induce cracking in the second pass (see Figs. 11 and 20(d)).

Based on the above analysis, it can be concluded that during multipass spinning, the material failure was induced by the void coalescence under small thickness reduction and the shear damage under large thickness, respectively. Besides, the rational matching of thinning rates in various spinning pass played a significant role in improving the forming limitation of materials during multi-pass spinning. Over small and large pass thinning rates may induce earlier cracking in the spinning 17

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International Journal of Mechanical Sciences 159 (2019) 1–19

Fig. 28. Evolution of matrix equivalent plastic strain, void volume fraction, shear damage, stress triaxiality and cut-off value for stress triaxiality in the selected period under thinning deformation stage for the element on inner and outer surface under the second pass spinning.

process. When the pass reduction was set at around 21%, the void volume fraction was suppressed and the shear damage was not fully developed as well, resulting in the increase of the tube spinnability of materials, which theoretically confirmed the estimation that the optimal pass thinning rate should be about 20% by Bylya et al. [8].

final fracture. During multi-pass tube spinning, the material failure mechanism changed with the variation of the thinning rate in different spinning passes. Both the cracks occurring at 24.45% total thinning rate (First pass: 12.35% + Second pass: 13.80%), which was even less than the forming limit during single-pass spinning (no more than 29.85%), and at 30.35% total thinning rate (First pass: 21.70% + Second pass: 11.05%) were induced by the void coalescence. However, shear damage was the primary mechanism causing material failure at 39.77% total thinning rate (First pass: 21.50% + Second pass: 23.27%). 3. For multi-pass spinning, when the total thinning rate was 38.0% (First pass: 21.40% + Second pass: 21.12%), the void volume fraction of inner layer increased to the maximum value close to the critical void volume fraction fc in the uplift deformation stage and then decreased in the thinning deformation stage. However, the void volume fraction of the outer layer was suppressed throughout the multipass spinning process. As per the shear damage in the inner layer, the damage value did not increase sufficiently since the equivalent plastic strain for the inner layer in multi-pass spinning with the total thinning rate of 38% was smaller than that in single-pass spinning with the thinning rate of 33%, thus higher thickness reduction could be obtained through the multi-pass spinning process. 4. To increase the spinnability in the multi-pass spin forming process, the thinning strategy for each pass should be designed reasonably. The formability limit may be reduced when smaller pass thinning rate was adopted. The thinning rate in each pass should be

6. Conclusions In this paper, the influence of spinning pass and thinning rate on spin forming of tube blank was investigated by spinning experiment and FE simulation. A modified GTN model extended to shear deformation was implemented in ABAQUS/Explicit to analyze the damage evolution and crack mechanism during single-pass and multi-pass tube spinning. The main conclusions are summarized as follows: 1. By re-defining the shear damage evolution law and modifying the void nucleation function under the negative stress triaxiality, an improved GTN model was proposed and applied to FE simulation on the damage evolution during tube spinning. The reliability of the improved GTN model was verified through the comparison of numerical simulation and experimental observation on the backward tube spinning process of the 2024-T351 alloy. 2. During single-pass tube spinning, void volume fraction first increased to the peak value and then decreased, while shear damage presented an overall increasing tendency with increasing thinning rate, wherein the shear damage should be responsible for the 18

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International Journal of Mechanical Sciences 159 (2019) 1–19

determined in the desired range (about 21%) so that the void volume fraction would be suppressed, and the shear damage would not exceed the critical shear damage in the inner layer of as-spun workpieces. Compared with single-pass spinning, the multi-pass thinning strategy could increase the spinnability limit since it may induce smaller equivalent strain in the inner layers under the same total thinning rate.

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