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Experimental and numerical investigation on the shearing process of stainless steel thin-walled tubes in the spent fuel reprocessing
Thin-Walled Structures 145 (2019) 106407
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Full length article
Experimental and numerical investigation on the shearing process of stainless steel thin-walled tubes in the spent fuel reprocessing
T
Jianpeng Dong, Shilong Wang∗, Jie Zhou, Chi Ma, Sibao Wang, Bo Yang State Key Laboratory of Mechanical Transmissions, University of Chongqing, Chongqing, 400044, PR China
ARTICLE INFO
ABSTRACT
Keywords: Shear Spent fuel reprocessing Constitutive equation Fracture mechanism Finite element method
The SUS304 stainless steel thin-walled tube (SUS304 tube) is an important part of the simulated spent fuel assembly (SSFA). During the shearing process, owing to the high ductility and malleability of the SUS304 tube, the local stress concentration easily occurs on the shear tool, resulting in such damages as tool tipping and breaking. However, the investigation on the shearing of the SUS304 tube has been scarce so far. In this contribution, the modified Gurson-Tvergaard-Needleman model (GTN-J model) was adopted to investigate the fracture mechanism of the SUS304 tube under the complex stress state. It is proved that the GTN-J model performs perfectly in the simulation of the entire shearing process by comparing the results of shear experiments with simulations. Moreover, The results reveal that the local stress concentration of the shear tool is mainly caused by irregular plastic deformation of the SUS304 tube.
1. Introduction As a kind of safe and clean energy, nuclear power has been widely applied to deal with the deepening energy crisis. Despite many benefits of nuclear energy, it is an enormous challenge for human to dispose of an increasing number of radioactive nuclear waste (spent fuel assembly). The spent fuel assembly (SFA) is a highly radioactive rod, which cannot maintain nuclear fission and must be removed from the nuclear reactor. However, substantial usable nuclear elements (such as uranium and plutonium) still remain in the SFA. Consequently, the closed cycle (i.e. separation and recovery of nuclear elements from the SFA after shearing) is extensively used in the treatment of the SFA because it can improve the utilization rate of nuclear elements and reduce the amount of the radioactive waste. Moreover, when the closed cycle is applied, the time for reducing the radioactivity to the harmless level has been shortened from several hundred thousand years to around one thousand years. During the closed cycle, the SFA needs to be cut into short segments to facilitate the subsequent chemical extraction. Additionally, due to its high-level radioactivity, the SFA is generally substituted by the simulated spent fuel assembly (SSFA) in shear experiments. The SSFA is composed of SUS304 stainless steel core tubes (SUS304 tubes) and Al2O3 ceramics, as shown in Fig. 1. In order to meet the high-reliability requirements for shearing equipment under intense radiation, the mechanical structure shown in Fig. 2 is often adopted in shear experiments.
∗
During the shearing process, brittle Al2O3 ceramics, are crushed into small particles. These particles cause slight abrasive wear of the shear tool, which only slightly affects the tool life. However, the material characteristics of SUS304 tubes (the high ductility and malleability) easily induce local stress concentration on the shear tool and consequently result in such damages as tool tipping and breaking. Therefore, the investigation on the shearing process of the SUS304 tube has an important significance for the optimization of the process parameters and the improvement of the tool life. In the last few years, some studies have been carried out on the forming and fracture of metal tubes such as rotary draw bending [1,2], deep drawing [3], electromagnetic tube compression [4], tube spinning [5,6], and tube hydroforming [7]. However, there are few studies involving the mechanism of ductile fracture during the shearing process of thin-walled metal tubes. Moreover, there are still no systematic damage theories to predict accurately the shearing process of tubes. Liu et al. suggested employing the Gurson-Tvergaard-Needleman (GTN) model to predict the cutting force in shearing of the blow-out preventer drill pipe by comparing with several other theoretical models [8]. It should be mentioned that the GTN model is a classical model of microscopic damage mechanics. Currently, microscopic damage mechanics (e.g. GTN model) has been widely applied to tube processing because it can reveal the causes of the crack formation, the crack propagation, and the macroscopic fracture from the evolution progress of micro-defects in materials [2,7,9–11]. Therefore, the microscopic
Corresponding author. E-mail address: [email protected] (S. Wang).
https://doi.org/10.1016/j.tws.2019.106407 Received 22 April 2019; Received in revised form 9 August 2019; Accepted 14 September 2019 0263-8231/ © 2019 Elsevier Ltd. All rights reserved.
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The aim of this paper is to investigate the mechanism of ductile fracture during the shearing process of the SUS304 tube, which promotes the further study of this field. Furthermore, the contact area between the shear tool and the tube during the shearing process is so small that it is impossible to install sensors. Thus the strain and stress of the tube at different stages cannot be obtained experimentally. In this paper, the GTN-J model is applied to simulate the shearing process of SUS304 tube to overcome the mentioned difficulty. The organization of this paper is as follows: The GTN-J constitutive model and numerical implementation method for the finite element method (FEM) are expounded in Section 2. The shear experiments of the SUS304 tube is presented in Section 3. The finite element model is introduced in Section 4, and the crack propagation during the shearing process is analyzed by comparing the simulations and experimental results. Meanwhile, the fracture morphology and the stress condition of the shear tool are described. Finally, the main conclusions are summarized in Section 5.
Fig. 1. Simulated spent fuel assembly.
2. Constitutive model and numerical algorithm 2.1. GTN-J model To predict the ductile fracture process more accurately, Tvergaard and Needleman [13] proposed a modified yield surface equation (i.e. GTN model) (Eq. (1)) based on the Gurson damage model. The interaction between adjacent voids was considered in the GTN model.
=
eq
2
+ 2q1 f cosh
y
3 q2 m 2 y
1
(q1 f ) 2
(1)
where eq is the von Mises equivalent stress and eq = (3/2) S: S , S is the deviatoric stress tensor, y is the flow stress of matrix material, m is the hydrostatic stress and m = Trace ( )/3 , q1 and q2 are two modified parameters, and f (f ) denotes a piecewise function of the void volume fraction.
Fig. 2. Mechanical shear mode of the SFA.
f
damage mechanics was adopted in this study. So far, several classical models have been proposed in the framework of micromechanics. Gurson introduced the void volume fraction into the plastic yield criterion on the basis of previous studies and proposed the classical Gurson damage model (Gurson model) [12]. On this basis, Tvergaard and Needleman revised Gurson model and developed the classical GTN model [13]. Li et al. applied the GTN model to simulate the incremental sheet forming and the results were as well as expected [14]. However, there is still a limitation in the GTN model. Achouri et al. found that the GTN model was unable to capture shear damage initiation and performed terrible under shear-dominated loading [15]. Accordingly, Nashon and Hutchinson introduced new parameter items into the GTN model and formed N-H model [16]. The model could improve the defect that shear damage cannot accumulate under low-stress triaxiality (e.g. shear-dominated loading). Xue used a similar method and Xue model was eventually obtained [17]. But unfortunately, Nielsen and Tvergaard [18] recognized that the above modifications (N-H model and Xue model) cause excessive and spurious damage in the cases of moderate to high triaxiality. Thus, another modified GTN model (GTN-Z model) was developed by Zhou et al. to solve the problems [19]. Although the GTN-Z model improved in predicting the ductile damage under low triaxiality, Wu et al. proved that GTN-Z model presented much earlier failure prediction [6]. To overcome the drawback, Jiang et al. introduced two damage variables (the void volume fraction and the shear damage parameter) into the GTN model and formed a new extended GTN model (GTN-J model) [20]. It was found that the simulated load-displacement curves of 2024-T3 aluminum alloy specimens using the GTN-J model under compression (i.e. under negative stress triaxiality) were consistent with the experimental results.
f =
fc +
f fu
fc
fF
fc
(f
fc )
fu
fc
fc < f < fF f
fF
(2)
where f denotes the void volume fraction of the material, fc denotes the void volume fraction when the void begins to accelerate coalesce, fF denotes the void volume fraction in the final fracture of the material, and fu denotes the value at complete failure and fu = 1/ q1. Based on the Lemaitre's strain equivalence principle, Jiang et al. characterized the cumulative damage by introducing a new damage variable [20]. The modified yield equation (Eq. (3)) is expressed as:
=
2 eq 2 y (1
Dshear )2
+ 2q1 f cosh
3 2
q2 y (1
m
Dshear )
1
(q1 f ) 2 (3)
where Dshear (Dshear ) denotes a piecewise function of the shear damage variable and it is expressed as:
Dshear Dshear =
Dc + 1
1 Dc DF Dc
Dshear (Dshear
Dc
Dc ) Dc < Dshear < DF Dshear
DF
(4)
where Dshear denotes the cumulative shear damage of the material, Dc denotes the critical shear damage, and DF denotes the shear damage for the complete failure of the material. Two damage variables are used in the GTN-J model to characterize the fracture mechanism of the material under complex stress states. (1) The void volume fraction is used to characterize the fracture failure caused by the void volume growth and aggregation; (2) The shear damage is used to characterize the fracture failure dominated by the 2
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internal shearing effects. It should be noted that the void volume fraction and the shear damage are implicitly coupled through the stress and strain fields. Different damage parameters represent different failure modes.
Table 1 GTN-J model parameters.
2.1.1. Accumulation of the void volume fraction f The evolution criterion of the void volume fraction in the GTN-J model (Eq. (5)) is consistent with that of the GTN model. The accumulation of the void volume fraction is derived from two parts, namely, the growth of existing voids and the nucleation of new voids.
f = fgrowth + fnucleation
fgrowth = (1
f)
p m
fN
1 2
exp
SN 2
(
N
SN
)
2
0
m
where fN denotes the void nucleation coefficient, N denotes the mean equivalent plastic strain for the void nucleation, and SN denotes the standard deviation of the void nucleation. The cumulative plastic strain of matrix material mp is expressed as: p m
=
: (1
p
f )(1
Dshear )
(9)
y
where denotes the macroscopic plastic strain rate tensor. A new stress state function was proposed in the GTN-J model to ensure that the shear damage accumulates normally under various stress states. p
2 )(1
(1
2)
w = (1 1
c 1) + c 1
(1 + ) c1 T k
2
c1 T k
T< k
T
k 0
T>0
(10)
p eq
0.01 0.035 0.1 4 0.5 0.2
0.001 Mn(%)
(12)
where S(%) and Mn(%) denote the mass fractions of sulfur (S) and manganese (Mn) in the material, respectively. According to Table 2, the initial void volume fraction of SUS304 stainless steel is calculated and f0 0.002 . Zhang et al. used the GTN model to simulate the SUS304 tube hydro-bulging test [23]. The results showed that the simulations agreed well with the experimental results when N = 0.54 , SN = 0.09 , fN = 0.032 , fc = 0.11, and fF = 0.156 . Therefore, the above values are adopted in the GTN-J model. Jiang et al. used the GTN-J model to simulate the torsion of 2024-T3 aluminium alloy thin-walled tube [20]. It was found that the simulated torsion curve was consistent with the experimental curve when
2.1.2. Accumulation of the shear damage Dshear The rule of shear damage accumulation follows the form of the N-H model.
S:
Initial shear damage Critical shear damage Shear damage at failure Shear constant of materials Stress state function coefficient Stress state function coefficient
f0 = 0.054 S(%)
3 ), where c1 and k are constant, is the Lode parameter and = 27J3/(2 eq J3 denotes the third invariant of deviator stress tensor and J3 = det(S) , T denotes the stress triaxiality and T = m / eq .
Dshear = k w Dshear w
0.54 0.09 0.032
1) Finite element fitting method Firstly, the initial values of the parameters are set. Then the values are corrected and optimized through the least square method. Finally, the final values are determined when a good fitting degree between the simulated results and the experimental results is obtained [21]. 2) Microscopic analysis method Some parameters are directly obtained by the microscopic observation [22]. As many parameters in the GTN-J model cannot be obtained by the microscopic observation, the first method is adopted in this paper. To reduce the optimization time, the initial values of parameters are obtained referring to the previous studies. The basic property parameters of SUS304 stainless steel were obtained by the tensile experiment [23]. The parameters were as follows: Young's modulus E = 195.85GPa , Poisson's ratio v = 0.29, initial yield strength 0 = 390.3MPa , and the material strain hardening constant K = 1235 and = 0.38. Tvergaard suggested that the classical values of q1 and q2 were 1.5 and 1 for metal materials, respectively [24]. Thus the constants q1 and q2 could be determined. According to Franklin et al., the initial void volume fraction of the material was related to the content of manganese sulphide (MnS) inclusions [25] and its function was expressed as:
(8)
>0
0.156
0.03
Mean equivalent plastic strain for void nucleation Standard deviation of the void initiation Void nucleation coefficient
SN fN D0 Dc DF kw c1 k
0 m
Void volume fraction at failure
Critical void volume fraction
N
(7)
p m
GTN-J model constant GTN-J model constant Initial void volume fraction
195.85 GPa 0.29 390.3 MPa 1235 0.38 1.5 1 0.002
fc
where A denotes the void nucleation coefficient, mp denotes the cumulative plastic strain of the material, and w denotes the stress state function (Eq. (10)). The void nucleation occurs only under the tension stress, and thus the void nucleation coefficient is expressed as:
A=
Young's modulus Poisson's ratio Initial yield strength Material strain hardening constant
fF
(6)
w )A
E v
f0
where p denotes the plastic strain rate tensor and I denotes the second order identity tensor. Different from the GTN model, the accumulation of the void volume fraction induced by shear effects should be excluded from the rate of new voids nucleation. The evolution form in the GTN-J model is expressed as:
fnucleation = (1
Value
0
(5)
I
Parameters
K n q1 q2
The rate of void growth is related to plastic strain: p:
Notations
(11)
where k w is the shear constant of the material.
Table 2 Chemical composition of SUS304 stainless steel (wt%).
2.2. Determination of the parameters in the GTN-J model The values of 19 parameters in the GTN-J model should be identified in advance, as shown in Table 1. There are two methods to determine damage parameters. 3
C
Mn
P
S
Si
Cr
Ni
Fe
≤0.08
≤2.00
≤0.045
≤0.03
≤1.00
18–20
8–10.5
other
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Fig. 3. The algorithm flow chart at a single incremental step.
D0 = 0.01, Dc = 0.025, DF = 0.1, k w = 4 , c1 = 0.5, and k = 0.2 . Since the parameters related to shear damage cannot be obtained directly, the above values are set to the initial values in the GTN-J model. Then six parameters are calibrated and optimized. The final values are
determined if the minimum difference between the experimental results and the simulated results is achieved. The parameters characterizing the GTN-J model of SUS304 stainless steel are summarized in Table 1. 4
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2.3. Numerical algorithm
is still in the elastic stage and the current stress and strain are taken as the inputs at the time t + t . Otherwise, the shearing process will enter the stage of plastic deformation (step 2). Step 2: Call the stress-strain updating algorithm of the GTN-J model to calculate the stress, the strain, and other variables. It is necessary for each iteration to judge whether the values of the nonlinear equations F1 and F2 in the algorithm are less than or equal to the set value toler = 10 6 and the number of iterations is less than the set value N . If the two mentioned conditions are satisfied, the current stress and strain are updated and the updated data is used as the inputs at the next step
The GTN-J model is implemented in the VUMAT user subroutine of ABAQUS, and the fully implicit backward Euler algorithm is used in the procedure [15]. The numerical algorithm of the model is divided into three steps, as shown in Fig. 3. Step 1: Set the initial values of the parameters, and evaluate the elastic trial stress of the element at the current time t . Then, calculate the GTN-J yield equation (Eq. (1)) to determine whether it is less than or equal to zero. If the above condition is satisfied, the shearing process
Fig. 4. Boundary conditions for three tests: (a) tension; (b) pure shear; (c) compression.
Fig. 5. Single element test for GTN-J model：(a) tension; (b) compression; (c) pure shear; (d) pure shear(D0 = 0.01, k w = 4 in GTN-J model). 5
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zero, and then dDshear = 0 is obtained by Eq. (11). That is, the shear damage parameter will not work in the algorithm. Hence, the GTN-J model degenerates into the GTN model. Since ABAQUS software has its own GTN model algorithm, the stress and the void volume fraction calculated by the GTN-J model in this paper are compared with those obtained by the GTN model in ABAQUS to verify the accuracy of the GTN-J model. A single 8-node element with one integration point (C3D8R element in ABAQUS) is used to simulate the three boundary conditions as shown in Fig. 4, including the tension, the pure shear, and the compression state. The input parameters of the material for the single element test are shown in Table 1. The stress and void volume fraction obtained by the GTN-J model under three conditions are completely consistent with those obtained by GTN model as shown in Fig. 5. Thus the accuracy of the GTN-J model is verified. The void volume fraction does not vary with the equivalent plastic strain under shear loading (as shown in Fig. 5c) because the GTN model does not take the effect of shear loading into account. The GTN-J model (as shown in Fig. 5d) solves the problem by introducing the shear damage. The shear damage accumulates with the equivalent plastic strain and the stress gradually decreases when it exceeds the critical value Dc . The stress drops to zero and the element fails completely when the shear parameter DF is reached.
Fig. 6. SUS304 tube.
3. Experiment 3.1. Experimental material Fig. 7. Flow stress curve of SUS304 thin plate specimen.
The structure of the SUS304 tube is shown in Fig. 6 and the chemical composition of SUS304 stainless steel is listed in Table 2. Assuming that the strain hardening effect of the SUS304 stainless steel (SUS304) is isotropic [26], the stress-strain flow curve is obtained by the standard specimen tensile test. Zhang et al. [23] gave the flow stress curve of SUS304 thin plate specimen under tension (plate thickness 0.9 mm), as shown in Fig. 7. And on this basis, the following relationn , wherein K = 1235 and n = 0.38. ship was obtained: y = K
(step 3). Otherwise, the iteration is repeated until the condition is satisfied. It should be noted that the program is terminated and the constitutive algorithm needs to be checked if the above conditions cannot be satisfied simultaneously. Step 3: Judge whether the void volume fraction or the shear damage exceeds the critical value. If one of them exceeds the critical value, the state of the element is judged as fracture failure and the element is deleted. Otherwise, the element will be continued to calculate at the time t + t .
3.2. Experimental device 3.2.1. Shear device As shown in Fig. 8, the shear device consists of four parts: the shear tool, the holder, the support block, and the fixed tool. All the parts are made of 45 steel and the structural diagram of the shear device is shown
2.4. Single element test If the initial values of Dshear and k w in the GTN-J model are set to
Fig. 8. Shear device. 6
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Fig. 9. Structural diagram of the shear device. Table 3 Structural parameters of the device. Type
a
b
α
d
D
Dimension/mm
2
16
20°
14
16
Fig. 11. SUS304 tube after shearing.
3.3. Experimental scheme The shear device was installed on the MTS, as shown in Fig. 8. The shear tool was held on the clamping tool of the MTS and the fixed tool was held on the lower part of MTS. The initial position of the shear tool was adjusted to 3~5 mm away from the SUS304 tube. The shear speed was set to 50 mm/s. During the shearing process, the displacement and force of the shear tool were acquired. The experiment was conducted three times in total and the fracture of the SUS304 tube after shearing is shown in Fig. 11. 4. Results and discussions 4.1. Finite element model
Fig. 10. Material testing machine.
To reduce the computing time, the model is simplified according to the characteristics of the model without affecting the simulated results.
in Fig. 9. The clearance between the shear tool and the holder is 0.2 mm and the other structural parameters are shown in Table 3.
1) As the stiffness of the shear device is obviously greater than that of the SUS304 tube, all parts of the shear device are set as discrete rigid bodies in the model. Meanwhile, the structural size of the shear device is reduced and only the structure near the shear zone is retained, as shown in Fig. 12a and Fig. 12b. 2) The discrete rigid body is meshed by R3D4 elements. The element size of the rigid body does not affect the simulated results, but it should be noted that the element size needs to be larger than that of
3.2.2. Material testing machine The shear experiments were completed by the material testing machine Landmark 370.1 (MTS), as shown in Fig. 10. The measurement error is less than 0.5%. The loading range is 0–100 KN and the loading speed is 0–50 mm/s for the MTS. During the shear experiments, the load-displacement curve of the shear tool was acquired. 7
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Fig. 12. Finite element model of the shear device: (a) the whole finite element model and selection of mesh types for each part; (b) constitutive models for different regions of SUS304 tube; (c) meshing method for the SUS304 tube; (d) generating mesh on the section of the tube; (e) the entire mesh swepting along the axial direction of the tube.
Table 4 Mesh number and computing time of the different element size. Element size 4
Number/ × 10 Computing time/h
500 μm × 500 μm
250 μm × 250 μm
200 μm × 200 μm
100 μm × 100 μm
1.7 50 h
11 252 h
21 500 h(estimated)
116 4000 h(estimated)
the SUS304 tube to prevent node penetration. Thus, the mesh size is set to 1 mm × 1 mm, as shown in Fig. 12a. 3) The SUS304 tube is a hollow-cylindrical structure. When the tube is meshed, the section of the tube is meshed firstly, and then the entire mesh is swept along the axial direction of the SUS304 tube, as shown in Fig. 12d and Fig. 12e. Therefore, it is impossible to refine the mesh only for the shear zone. Moreover, the finer the mesh, the longer the computing time. Table 4 shows the number of mesh and the time cost for different element sizes. Finally, the C3D8R element is chosen for the SUS304 tube and the mesh size is set to 500 μm × 500 μm, as shown in Fig. 12a. 4) To improve the computing efficiency, the GTN-J model is only applied to the shear zone of the SUS304 tube. Additionally, the left region of the SUS304 tube is set to the rigid element because there is basically no deformation due to the existence of the support block. Similarly, the Mises plasticity is applied to the right region to reduce the computing time, as shown in Fig. 12c. 4.2. Results The simulated load-displacement curve is compared with three experimental curves in Fig. 13. The right side of the figure shows the sections of the SUS304 tube at different stages during the shearing
Fig. 13. Load-displacement curve during the shearing process.
8
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Fig. 14. Comparisons of simulated and experimental fracture morphology.
process. From the overall trend, the simulated curve is perfectly consistent with the experimental curves. Meanwhile, the simulated fracture morphology is identical with the actual experimental section, as shown in Fig. 14. According to the position of the shear tool, the shearing process can be divided into five stages, as shown in Fig. 15.
4.3. Discussion 4.3.1. Comparisons between the simulated curve and experimental curves The simulated curve is a little smaller than the experimental curves in the second and third stages, as illustrated in Fig. 13. The main reason is that the strain hardening of the SUS304 tube in the forming process is not considered. Fig. 17 shows the forming process of the SUS304 tube. The strength of the stainless steel sheet is further enhanced by the roll forming during the process, but only the strain hardening of the sheet is taken into account in the simulation. The flow stress curve of the SUS304 tube was obtained by the uniaxial tensile test of sheet specimens. Therefore, the simulated results are lower than the experimental results in the second and third stages. In the fifth stage of the shearing process, the displacement of the shear tool for the complete fracture in the simulation is about 1.5 mm smaller than the experimental results. The reasons are as follows: 1) The strainhardening of the SUS304 tube during the forming process is ignored in the simulation, so the occurrence of complete fracture is earlier than the experimental results. 2) The wall thickness of the shear zone may be thicker than the simulated standard value due to the machining error. Thus, the cracks propagation is required to the larger deformation.
①Start shearing: The shear tool had not touched the top surface of the SUS304 tube at that time. ②Initial crack generation: The initial cracks of the SUS304 tube were generated when the shear tool reached the position a and the first peak value of the shear force F1 appeared. ③Phase I tearing process: With the movement of the shear tool, the cracks propagated along the path a b1 (a b2) . Two peak values of the shear force appeared at the beginning (F2 ) and end (F3 ). ④Phase II tearing process：The shear tool kept moving downward and then the cracks propagated along the path b1 c /b2 c . The shear force started to drop. ⑤Fracture: The left and right cracks continued propagating with the movement of the shear tool and eventually converged at the position c . Therefore, the tube broke and the last peak value of shear force (F4 ) was reached.
4.3.2. Crack evolution Fig. 18 shows the shearing process of the SUS304 tube. The analysis of the crack evolution is shown in Fig. 19.
Fig. 16 shows the stress distribution of the SUS304 tube at different stages of the shearing process.
Fig. 15. Crack evolution of the shearing process. 9
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Fig. 16. Stress distribution of the SUS304 tube at different stages: (a) Start shearing; (b) Initial crack generation; (c) Phase I tearing process; (d) Phase II tearing process; (e) Fracture.
Fig. 17. Process flow of the SUS304 tube.
10
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Fig. 18. Crack evolution process of the SUS304 tube: ①Start shearing; ②Initial crack generation; ③Phase I tearing process; ④Phase II tearing process; ⑤Fracture.
1) Before shearing: The shear tool has not touched the top surface of the SUS304 tube at that time. 2) Crack path c1 c2 : The contact area between the shear tool and the SUS304 tube is flattened as the shear tool moves. In this case, the initial crack is generated from the edge of the flattened area once the damage exceeds the allowable limit. The fracture surface is approximate to a trapezoid, as shown in Fig. 19c. 3) Crack path c2 c3 : The SUS304 tube is extruded and deformed laterally along the red arrow with the shear tool moving, as shown in Fig. 19d. Meanwhile, the cracks extend successively along the axial direction besides the hoop direction and thickness direction, and then a burr (a purple protrusion in Fig. 19a) is formed at the third stage of the shearing process. 4) Crack path c3 c4 : The cracks on the left and right sides continue propagating along the hoop direction and thickness direction. In particular, when the shear tool arrives at the middle of the SUS304 tube, the position of the crack tips starts to exceed the top surface of the shear tool (the phenomenon can be observed at the third and fourth stages in Fig. 18). Finally, the cracks on both sides converge at the point c4 . 5) Crack path c4 c5: The cracks no longer propagate due to the softening of the material at that time. The right part (see the fifth stage in Fig. 18) deforms downward until the shear tool is contact with the residual part without fracture. Then the residual part is extruded
and torn. Finally, the cracks converge at the point c5. The fracture surface is a small trapezoid, as shown in Fig. 19e. 4.3.3. Damage accumulation To further state the damage evolution during the entire shearing process, five elements were chosen along the hoop direction of the SUS304 tube, representing five stages in the shearing process, as shown in Fig. 20a. The failure strain of each element is basically the same value and all of them are around 0.35 (see Fig. 20b). The void volume fraction of all the elements decreases initially, as can be seen in Fig. 20c. It indicates that whatever stage in the shearing process, the element is compressed at first. Compared with the void volume fraction, the shear damage of all the elements exceeds the critical value, as shown in Fig. 20d. That is, the failure of the element is induced by shear damage rather than microvoids. This indicates that material failure is mainly dominated by shear loading. It is important to note that such variables as the equivalent strain, the void volume fraction, and the shear damage no longer increase during the moment a-b in the fifth stage (see E_5 in Fig. 20b–d). The reason is that the cracks no longer propagate due to the softening of the material during the moment a-b. The above three variables continue increasing until the shear tool is contact with the residual part of the SUS304 tube (see the fifth stage in Fig. 18). 11
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Fig. 19. Crack evolution analysis of the SUS304 tube: (a) crack evolution path; (b) stress state at different stages; (c) initial crack generation; (d) crack generation analysis at the intersection of the second and third stage; (e) final fracture.
Fig. 20. Damage evolution at different stages: (a) the location of the selected element; (b) the equivalent strain; (c) the void volume fraction; (d) shear damage.
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shear damage is considered in the GTN-J model. Fig. 21 shows the simulated load-displacement curves with and without the void volume fraction. It can be found that the curve without the void volume fraction is basically the same as that with the void volume fraction. The failure strain of each element without the void volume fraction is consistent with that with the void volume fraction, as shown in Fig. 22a. The effect of void volume fraction on the evolution of shear damage during the shearing process is shown in Fig. 22b. Obviously, the void volume fraction has little effect on the evolution of shear damage under shear-dominated conditions. In conclusion, the effect of the void volume fraction to the failure process could be ignored under shear-dominated conditions. 4.3.5. Fracture morphology The section of the SUS304 tube after shearing has not been defined by scholars yet. Thus, through referring to the fracture morphology of the blanking, the section of the tube is divided into the following four parts, namely, initial tearing zone, burr, burnish zone, and fracture tearing zone, as shown in Fig. 23.
Fig. 21. Comparisons of load-displacement curve with and without the void volume fraction.
1) Initial tearing zone: The initial tearing zone is a rough surface formed by the cracks. Theoretically, the cracks first take place at the point c1. However, actually, the top surface of the SUS304 tube is flattened under compression at the beginning. The edge of the flattened area (rectangular area) is torn when the stress exceeds the allowable stress of the material. The fracture surface is approximate
4.3.4. Effect of void volume fraction to the failure process To illustrate the effect of void volume fraction to the failure process, the void volume fraction remains constant throughout the shearing process (i.e. f = f0 during the shearing process). That's to say, Only
Fig. 22. Comparisons of damage evolution at different stages: (a) the equivalent strain; (b) shear damage. Note: The straight line represents the simulated curve with shear damage and void volume fraction. The symbol represents the simulated curve with shear damage (without void volume fraction).
Fig. 23. Fracture morphology of the SUS304 tube. 13
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to a trapezoid shape. 2) Burr: The crack path will cause a protrusion in the phase I tearing process (Crack path c2 c3 c4 in Section 4.3.2). 3) Burnish zone: The cracks propagate along the path c3 c4 , as shown in Fig. 19a. The crack path is smooth. 4) Fracture tearing zone: The right part of the SUS304 tube deforms downward, as shown in Fig. 18. Finally, the residual part is torn. Thus the fracture surface is similar to the initial tearing zone, but the size is only one-fifth of it.
4.4. Comparisons of sheet blanking Due to the lack of research on the tube shearing process, the design and optimization of the device for shearing of the tube in engineering are carried out by referring to sheet blanking process. However, there are obvious differences between sheet blanking and tube shearing process. It is unreasonable to develop the device for tube shearing by referring to the sheet blanking process. Therefore, the differences between sheet blanking and tube shearing process are compared from five aspects in this paper to promote the theoretical research of tube shearing process.
4.3.6. Stress condition of the shear tool The stress condition of the shear tool changes constantly with the deformation of the SUS304 tube during the shearing process, as shown in Figs. 24 and 25. It can be seen from Fig. 25a that the middle of the shear tool is subjected to the maximum stress during the initial crack generation. Subsequently, the position of the maximum stress gradually moves to both sides in the third and fourth stages of the shearing process, as shown in Fig. 25b and c. Finally, it is noted that the position of maximum stress returns to the middle of the shear tool in the fifth stage.
4.4.1. Crack evolution Fig. 26 shows the crack path during the sheet blanking. Obviously, the incipient cracks almost occur on the side of the punch and the die simultaneously. During the blanking process, the cracks propagate downward at the position a and upward at the position b until the two cracks intersect with each other (see the blue arrow in Fig. 26). However, the cracks only initiate at the top of the SUS304 tube in the shearing process because the deformation of SUS304 tube is dispersed through the hollow structure and cannot be transmitted to the bottom of the tube. The crack evolution of the tube shearing is divided into five stages, thus the fracture mechanism is more complicated. 4.4.2. Load-displacement curve The overall trend of the load-displacement curve of sheet blanking and tube shearing are compared in Fig. 27. For the tube shearing, the overall fluctuation of the curve is higher. Meanwhile, there are four peaks in the whole process. However, the curve of sheet blanking is smooth and only one peak value of the force appears throughout the blanking process [27].
Fig. 24. The stress condition of the shear tool during the shearing process.
Fig. 25. The stress condition changes with the movement of the shear tool: the maximum stress position at (a) initial crack generation; (b) the beginning of the phase I tearing process; (c) the end of the phase I tearing process; (d) phase II tearing process; (e) fracture. 14
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Fig. 28. Fracture section of sheet blanking.
Fig. 26. Crack evolution of sheet blanking.
Fig. 29. The stress location of the punch during the blanking process.
the punch surface is stressed uniformly during the entire blanking process [28], as shown in Fig. 29. However, the stress condition of the shear tool constantly varies during the shearing process of the SUS304 tube, as shown in Figs. 24 and 25. 4.4.5. Model simplification The sheet blanking can be simplified into a two-dimensional axisymmetric model. However, the whole three-dimensional model must be adopted to ensure the accuracy of the simulation in the tube shearing. 5. Conclusions Due to the limitations of various modified GTN models in simulating complex stress states, the GTN-J model is applied to simulate the shearing process of SUS304 tube. The stress-strain algorithm of the model is embedded in ABAQUS/Explicit to realize its numerical solution. Based on the simulations and experimental results, the conclusions are drawn as follows:
Fig. 27. Comparison of the load-displacement curve.
4.4.3. Fracture morphology Fig. 28 shows the blanking section of the sheet [27]. From top to bottom, it can be divided into four parts, including the rollover, the burnish zone, the fracture zone, and the burr. The fracture morphology of the SUS304 tube retains the burnish zone, the fracture zone, and the burr except for the rollover, as shown in Fig. 23. During the shearing process, the contact area between the shear tool and the SUS304 tube is flattened and then the cracks occur at the edge of the flattened area. Accordingly, the phenomenon of rollover is not obvious. Therefore, the initial tearing zone is used to substitute for rollover.
(1) The simulated load-displacement curve and fracture morphology of the tube is basically consistent with the experimental results, which verifies the effectiveness of the GTN-J model. (2) The fracture mechanism of the shearing process of the SUS304 tube is obtained for the first time. The shearing process of SUS304 tube is divided into five stages in detail, including the start shearing, the initial crack generation, the phase I tearing process, the phase II tearing process, and the fracture. Meanwhile, the crack evolution and the damage accumulation during the shearing process are analyzed. (3) The fracture morphology of the SUS304 tube is firstly proposed, including the initial tearing zone, the burr, the burnish, and the fracture tearing zone.
4.4.4. Stress condition of the shear tool It can be found that the punch surface always contacts with the upper surface of the sheet during the blanking process. Consequently, 15
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(4) The stress condition of the shear tool varies constantly during the shearing process, which provides a theoretical basis for the design of the shear tool. (5) The differences between the tube shearing and sheet blanking are depicted from five aspects, including the load curve, the fracture morphology, the crack evolution, the stress condition of the shear tool, and the model simplification. (6) Shortcomings and prospects: The effect of strain hardening in the forming of the SUS304 tube is not considered, and thus the constitutive model can be further improved to achieve the minimum difference between the simulated results and the experimental results.
[9] J. Jackiewicz, Use of a modified Gurson model approach for the simulation of ductile fracture by growth and coalescence of microvoids under low, medium and high stress triaxiality loadings, Eng. Fract. Mech. 78 (3) (2011) 487–502. [10] V. Lemiale, J. Chambert, P. Picart, Description of numerical techniques with the aim of predicting the sheet metal blanking process by FEM simulation, J. Mater. Process. Technol. 209 (5) (2009) 2723–2734. [11] F.F. Li, M.W. Fu, J.P. Lin, X.N. Wang, Experimental and theoretical study on the hot forming limit of 22MnB5 steel, Int. J. Adv. Manuf. Technol. 71 (1) (2014) 297–306. [12] A.L. Gurson, Continuum theory of ductile rupture by void nucleation and growth: Part I—yield criteria and flow rules for porous ductile media, ASME, J. Eng. Mater. Technol. 99 (1) (1977) 2–15. [13] V. Tvergaard, A. Needleman, Analysis of the cup-cone fracture in a round tensile bar, Acta Metall. Mater. 32 (1) (1984) 157–169. [14] J.C. Li, S.P. Li, Z.Y. Xie, W. Wang, Numerical simulation of incremental sheet forming based on GTN damage model, Int. J. Adv. Manuf. Technol. 81 (9–12) (2015) 2053–2065. [15] M. Achouri, G. Germain, P. Dal Santo, D. Saidane, Numerical integration of an advanced Gurson model for shear loading: application to the blanking process, Comput. Mater. Sci. 72 (2013) 62–67. [16] K. Nahshon, J.W. Hutchinson, Modification of the Gurson model for shear failure, Eur. J. Mech. A Solid. 27 (1) (2008) 1–17. [17] L. Xue, Constitutive modeling of void shearing effect in ductile fracture of porous materials, Eng. Fract. Mech. 75 (11) (2008) 3343–3366. [18] K.L. Nielsen, V. Tvergaard, Ductile shear failure or plug failure of spot welds modelled by modified Gurson model, Eng. Fract. Mech. 77 (7) (2010) 1031–1047. [19] J. Zhou, X.S. Gao, J.C. Sobotka, B.A. Webler, B.V. Cockeram, On the extension of the Gurson-type porous plasticity models for prediction of ductile fracture under shear-dominated conditions, Int. J. Solids Struct. 51 (18) (2014) 3273–3291. [20] W. Jiang, Y.Z. Li, J. Su, Modified GTN model for a broad range of stress states and application to ductile fracture, Eur. J. Mech. A Solid. 57 (2016) 132–148. [21] S. Afshan, Micromechanical Modelling of Damage Healing in Free Cutting Steel, Imperial College London, 2013. [22] G.B. Broggiato, F. Campana, L. Cortese, Identification of material damage model parameters: an inverse approach using digital image processing, Meccanica 42 (1) (2007) 9–17. [23] W.W. Zhang, S. Cong, Failure analysis of SUS304 sheet during hydro-bulging based on GTN ductile damage model, Int. J. Adv. Manuf. Technol. 86 (1–4) (2016) 427–435. [24] V. Tvergaard, On localization in ductile materials containing spherical voids, Int. J. Fract. 18 (4) (1982) 237–252. [25] A.G. Franklin, Comparison between a quantitative microscope and chemical methods for assessment of non-metallic inclusions, J. Iron Steel I. 207 (1969) 181. [26] S. Wang, Z. Chen, C. Dong, Tearing failure of ultra-thin sheet-metal involving size effect in blanking process: analysis based on modified GTN model, Int. J. Mech. Sci. 133 (2017) 288–302. [27] Y. Abe, Y. Okamoto, K. Mori, H. Jaafar, Deformation behaviour and reduction in flying speed of scrap in trimming of ultra-high strength steel sheets, J. Mater. Process. Technol. 250 (2017) 372–378. [28] P.J. Zhao, Z.H. Chen, C.F. Dong, Experimental and numerical analysis of micromechanical damage for DP600 steel in fine-blanking process, J. Mater. Process. Technol. 236 (2016) 16–25.
Acknowledgments The research was supported by the National Nature Science Foundation of China (No. 51605057) and the Fundamental Research Funds for the Central Universities of China (No. 2018CDQYJX0013). References [1] M. Elyasi, M. Paluch, M. Hosseinzadeh, Predicting the bending limit of AA8112 tubes using necking criterion in manufacturing of bent tubes, Int. J. Adv. Manuf. Technol. 88 (9–12) (2017) 3307–3318. [2] A. Zardoshtian, H. Sabet, M. Elyasi, Improvement of the rotary draw bending process in rectangular tubes by using internal fluid pressure, Int. J. Adv. Manuf. Technol. 95 (1–4) (2018) 697–705. [3] M. Dilmec, M. Arap, Effect of geometrical and process parameters on coefficient of friction in deep drawing process at the flange and the radius regions, Int. J. Adv. Manuf. Technol. 86 (1) (2016) 747–759. [4] H. Savadkoohian, A.F. Arezoodar, B. Arezoo, Analytical and experimental study of wrinkling in electromagnetic tube compression, Int. J. Adv. Manuf. Technol. 93 (1–4) (2017) 901–914. [5] X.X. Wang, M. Zhan, J. Guo, B. Zhao, Evaluating the applicability of GTN damage model in forward tube spinning of aluminum alloy, Metals 6 (6) (2016) 136. [6] H. Wu, W.C. Xu, D.B. Shan, B.C. Jin, An extended GTN model for low stress triaxiality and application in spinning forming, J. Mater. Process. Technol. 263 (2019) 112–128. [7] J. Yuenyong, M. Suthon, S. Kingklang, P. Thanakijkasem, S. Mahabunphachai, V. Uthaisangsuk, Formability prediction for tube hydroforming of stainless steel 304 using damage mechanics model, J. Manuf. Sci, E-T. ASME 140 (1) (2017) 011006. [8] Y. Liu, B. Tang, L. Hua, H. Mao, Investigation of a novel modified die design for fine-blanking process to reduce the die-roll size, J. Mater. Process. Technol. 260 (2018) 30–37.
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Full length article
Experimental and numerical investigation on the shearing process of stainless steel thin-walled tubes in the spent fuel reprocessing
T
Jianpeng Dong, Shilong Wang∗, Jie Zhou, Chi Ma, Sibao Wang, Bo Yang State Key Laboratory of Mechanical Transmissions, University of Chongqing, Chongqing, 400044, PR China
ARTICLE INFO
ABSTRACT
Keywords: Shear Spent fuel reprocessing Constitutive equation Fracture mechanism Finite element method
The SUS304 stainless steel thin-walled tube (SUS304 tube) is an important part of the simulated spent fuel assembly (SSFA). During the shearing process, owing to the high ductility and malleability of the SUS304 tube, the local stress concentration easily occurs on the shear tool, resulting in such damages as tool tipping and breaking. However, the investigation on the shearing of the SUS304 tube has been scarce so far. In this contribution, the modified Gurson-Tvergaard-Needleman model (GTN-J model) was adopted to investigate the fracture mechanism of the SUS304 tube under the complex stress state. It is proved that the GTN-J model performs perfectly in the simulation of the entire shearing process by comparing the results of shear experiments with simulations. Moreover, The results reveal that the local stress concentration of the shear tool is mainly caused by irregular plastic deformation of the SUS304 tube.
1. Introduction As a kind of safe and clean energy, nuclear power has been widely applied to deal with the deepening energy crisis. Despite many benefits of nuclear energy, it is an enormous challenge for human to dispose of an increasing number of radioactive nuclear waste (spent fuel assembly). The spent fuel assembly (SFA) is a highly radioactive rod, which cannot maintain nuclear fission and must be removed from the nuclear reactor. However, substantial usable nuclear elements (such as uranium and plutonium) still remain in the SFA. Consequently, the closed cycle (i.e. separation and recovery of nuclear elements from the SFA after shearing) is extensively used in the treatment of the SFA because it can improve the utilization rate of nuclear elements and reduce the amount of the radioactive waste. Moreover, when the closed cycle is applied, the time for reducing the radioactivity to the harmless level has been shortened from several hundred thousand years to around one thousand years. During the closed cycle, the SFA needs to be cut into short segments to facilitate the subsequent chemical extraction. Additionally, due to its high-level radioactivity, the SFA is generally substituted by the simulated spent fuel assembly (SSFA) in shear experiments. The SSFA is composed of SUS304 stainless steel core tubes (SUS304 tubes) and Al2O3 ceramics, as shown in Fig. 1. In order to meet the high-reliability requirements for shearing equipment under intense radiation, the mechanical structure shown in Fig. 2 is often adopted in shear experiments.
∗
During the shearing process, brittle Al2O3 ceramics, are crushed into small particles. These particles cause slight abrasive wear of the shear tool, which only slightly affects the tool life. However, the material characteristics of SUS304 tubes (the high ductility and malleability) easily induce local stress concentration on the shear tool and consequently result in such damages as tool tipping and breaking. Therefore, the investigation on the shearing process of the SUS304 tube has an important significance for the optimization of the process parameters and the improvement of the tool life. In the last few years, some studies have been carried out on the forming and fracture of metal tubes such as rotary draw bending [1,2], deep drawing [3], electromagnetic tube compression [4], tube spinning [5,6], and tube hydroforming [7]. However, there are few studies involving the mechanism of ductile fracture during the shearing process of thin-walled metal tubes. Moreover, there are still no systematic damage theories to predict accurately the shearing process of tubes. Liu et al. suggested employing the Gurson-Tvergaard-Needleman (GTN) model to predict the cutting force in shearing of the blow-out preventer drill pipe by comparing with several other theoretical models [8]. It should be mentioned that the GTN model is a classical model of microscopic damage mechanics. Currently, microscopic damage mechanics (e.g. GTN model) has been widely applied to tube processing because it can reveal the causes of the crack formation, the crack propagation, and the macroscopic fracture from the evolution progress of micro-defects in materials [2,7,9–11]. Therefore, the microscopic
Corresponding author. E-mail address: [email protected] (S. Wang).
https://doi.org/10.1016/j.tws.2019.106407 Received 22 April 2019; Received in revised form 9 August 2019; Accepted 14 September 2019 0263-8231/ © 2019 Elsevier Ltd. All rights reserved.
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The aim of this paper is to investigate the mechanism of ductile fracture during the shearing process of the SUS304 tube, which promotes the further study of this field. Furthermore, the contact area between the shear tool and the tube during the shearing process is so small that it is impossible to install sensors. Thus the strain and stress of the tube at different stages cannot be obtained experimentally. In this paper, the GTN-J model is applied to simulate the shearing process of SUS304 tube to overcome the mentioned difficulty. The organization of this paper is as follows: The GTN-J constitutive model and numerical implementation method for the finite element method (FEM) are expounded in Section 2. The shear experiments of the SUS304 tube is presented in Section 3. The finite element model is introduced in Section 4, and the crack propagation during the shearing process is analyzed by comparing the simulations and experimental results. Meanwhile, the fracture morphology and the stress condition of the shear tool are described. Finally, the main conclusions are summarized in Section 5.
Fig. 1. Simulated spent fuel assembly.
2. Constitutive model and numerical algorithm 2.1. GTN-J model To predict the ductile fracture process more accurately, Tvergaard and Needleman [13] proposed a modified yield surface equation (i.e. GTN model) (Eq. (1)) based on the Gurson damage model. The interaction between adjacent voids was considered in the GTN model.
=
eq
2
+ 2q1 f cosh
y
3 q2 m 2 y
1
(q1 f ) 2
(1)
where eq is the von Mises equivalent stress and eq = (3/2) S: S , S is the deviatoric stress tensor, y is the flow stress of matrix material, m is the hydrostatic stress and m = Trace ( )/3 , q1 and q2 are two modified parameters, and f (f ) denotes a piecewise function of the void volume fraction.
Fig. 2. Mechanical shear mode of the SFA.
f
damage mechanics was adopted in this study. So far, several classical models have been proposed in the framework of micromechanics. Gurson introduced the void volume fraction into the plastic yield criterion on the basis of previous studies and proposed the classical Gurson damage model (Gurson model) [12]. On this basis, Tvergaard and Needleman revised Gurson model and developed the classical GTN model [13]. Li et al. applied the GTN model to simulate the incremental sheet forming and the results were as well as expected [14]. However, there is still a limitation in the GTN model. Achouri et al. found that the GTN model was unable to capture shear damage initiation and performed terrible under shear-dominated loading [15]. Accordingly, Nashon and Hutchinson introduced new parameter items into the GTN model and formed N-H model [16]. The model could improve the defect that shear damage cannot accumulate under low-stress triaxiality (e.g. shear-dominated loading). Xue used a similar method and Xue model was eventually obtained [17]. But unfortunately, Nielsen and Tvergaard [18] recognized that the above modifications (N-H model and Xue model) cause excessive and spurious damage in the cases of moderate to high triaxiality. Thus, another modified GTN model (GTN-Z model) was developed by Zhou et al. to solve the problems [19]. Although the GTN-Z model improved in predicting the ductile damage under low triaxiality, Wu et al. proved that GTN-Z model presented much earlier failure prediction [6]. To overcome the drawback, Jiang et al. introduced two damage variables (the void volume fraction and the shear damage parameter) into the GTN model and formed a new extended GTN model (GTN-J model) [20]. It was found that the simulated load-displacement curves of 2024-T3 aluminum alloy specimens using the GTN-J model under compression (i.e. under negative stress triaxiality) were consistent with the experimental results.
f =
fc +
f fu
fc
fF
fc
(f
fc )
fu
fc
fc < f < fF f
fF
(2)
where f denotes the void volume fraction of the material, fc denotes the void volume fraction when the void begins to accelerate coalesce, fF denotes the void volume fraction in the final fracture of the material, and fu denotes the value at complete failure and fu = 1/ q1. Based on the Lemaitre's strain equivalence principle, Jiang et al. characterized the cumulative damage by introducing a new damage variable [20]. The modified yield equation (Eq. (3)) is expressed as:
=
2 eq 2 y (1
Dshear )2
+ 2q1 f cosh
3 2
q2 y (1
m
Dshear )
1
(q1 f ) 2 (3)
where Dshear (Dshear ) denotes a piecewise function of the shear damage variable and it is expressed as:
Dshear Dshear =
Dc + 1
1 Dc DF Dc
Dshear (Dshear
Dc
Dc ) Dc < Dshear < DF Dshear
DF
(4)
where Dshear denotes the cumulative shear damage of the material, Dc denotes the critical shear damage, and DF denotes the shear damage for the complete failure of the material. Two damage variables are used in the GTN-J model to characterize the fracture mechanism of the material under complex stress states. (1) The void volume fraction is used to characterize the fracture failure caused by the void volume growth and aggregation; (2) The shear damage is used to characterize the fracture failure dominated by the 2
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internal shearing effects. It should be noted that the void volume fraction and the shear damage are implicitly coupled through the stress and strain fields. Different damage parameters represent different failure modes.
Table 1 GTN-J model parameters.
2.1.1. Accumulation of the void volume fraction f The evolution criterion of the void volume fraction in the GTN-J model (Eq. (5)) is consistent with that of the GTN model. The accumulation of the void volume fraction is derived from two parts, namely, the growth of existing voids and the nucleation of new voids.
f = fgrowth + fnucleation
fgrowth = (1
f)
p m
fN
1 2
exp
SN 2
(
N
SN
)
2
0
m
where fN denotes the void nucleation coefficient, N denotes the mean equivalent plastic strain for the void nucleation, and SN denotes the standard deviation of the void nucleation. The cumulative plastic strain of matrix material mp is expressed as: p m
=
: (1
p
f )(1
Dshear )
(9)
y
where denotes the macroscopic plastic strain rate tensor. A new stress state function was proposed in the GTN-J model to ensure that the shear damage accumulates normally under various stress states. p
2 )(1
(1
2)
w = (1 1
c 1) + c 1
(1 + ) c1 T k
2
c1 T k
T< k
T
k 0
T>0
(10)
p eq
0.01 0.035 0.1 4 0.5 0.2
0.001 Mn(%)
(12)
where S(%) and Mn(%) denote the mass fractions of sulfur (S) and manganese (Mn) in the material, respectively. According to Table 2, the initial void volume fraction of SUS304 stainless steel is calculated and f0 0.002 . Zhang et al. used the GTN model to simulate the SUS304 tube hydro-bulging test [23]. The results showed that the simulations agreed well with the experimental results when N = 0.54 , SN = 0.09 , fN = 0.032 , fc = 0.11, and fF = 0.156 . Therefore, the above values are adopted in the GTN-J model. Jiang et al. used the GTN-J model to simulate the torsion of 2024-T3 aluminium alloy thin-walled tube [20]. It was found that the simulated torsion curve was consistent with the experimental curve when
2.1.2. Accumulation of the shear damage Dshear The rule of shear damage accumulation follows the form of the N-H model.
S:
Initial shear damage Critical shear damage Shear damage at failure Shear constant of materials Stress state function coefficient Stress state function coefficient
f0 = 0.054 S(%)
3 ), where c1 and k are constant, is the Lode parameter and = 27J3/(2 eq J3 denotes the third invariant of deviator stress tensor and J3 = det(S) , T denotes the stress triaxiality and T = m / eq .
Dshear = k w Dshear w
0.54 0.09 0.032
1) Finite element fitting method Firstly, the initial values of the parameters are set. Then the values are corrected and optimized through the least square method. Finally, the final values are determined when a good fitting degree between the simulated results and the experimental results is obtained [21]. 2) Microscopic analysis method Some parameters are directly obtained by the microscopic observation [22]. As many parameters in the GTN-J model cannot be obtained by the microscopic observation, the first method is adopted in this paper. To reduce the optimization time, the initial values of parameters are obtained referring to the previous studies. The basic property parameters of SUS304 stainless steel were obtained by the tensile experiment [23]. The parameters were as follows: Young's modulus E = 195.85GPa , Poisson's ratio v = 0.29, initial yield strength 0 = 390.3MPa , and the material strain hardening constant K = 1235 and = 0.38. Tvergaard suggested that the classical values of q1 and q2 were 1.5 and 1 for metal materials, respectively [24]. Thus the constants q1 and q2 could be determined. According to Franklin et al., the initial void volume fraction of the material was related to the content of manganese sulphide (MnS) inclusions [25] and its function was expressed as:
(8)
>0
0.156
0.03
Mean equivalent plastic strain for void nucleation Standard deviation of the void initiation Void nucleation coefficient
SN fN D0 Dc DF kw c1 k
0 m
Void volume fraction at failure
Critical void volume fraction
N
(7)
p m
GTN-J model constant GTN-J model constant Initial void volume fraction
195.85 GPa 0.29 390.3 MPa 1235 0.38 1.5 1 0.002
fc
where A denotes the void nucleation coefficient, mp denotes the cumulative plastic strain of the material, and w denotes the stress state function (Eq. (10)). The void nucleation occurs only under the tension stress, and thus the void nucleation coefficient is expressed as:
A=
Young's modulus Poisson's ratio Initial yield strength Material strain hardening constant
fF
(6)
w )A
E v
f0
where p denotes the plastic strain rate tensor and I denotes the second order identity tensor. Different from the GTN model, the accumulation of the void volume fraction induced by shear effects should be excluded from the rate of new voids nucleation. The evolution form in the GTN-J model is expressed as:
fnucleation = (1
Value
0
(5)
I
Parameters
K n q1 q2
The rate of void growth is related to plastic strain: p:
Notations
(11)
where k w is the shear constant of the material.
Table 2 Chemical composition of SUS304 stainless steel (wt%).
2.2. Determination of the parameters in the GTN-J model The values of 19 parameters in the GTN-J model should be identified in advance, as shown in Table 1. There are two methods to determine damage parameters. 3
C
Mn
P
S
Si
Cr
Ni
Fe
≤0.08
≤2.00
≤0.045
≤0.03
≤1.00
18–20
8–10.5
other
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Fig. 3. The algorithm flow chart at a single incremental step.
D0 = 0.01, Dc = 0.025, DF = 0.1, k w = 4 , c1 = 0.5, and k = 0.2 . Since the parameters related to shear damage cannot be obtained directly, the above values are set to the initial values in the GTN-J model. Then six parameters are calibrated and optimized. The final values are
determined if the minimum difference between the experimental results and the simulated results is achieved. The parameters characterizing the GTN-J model of SUS304 stainless steel are summarized in Table 1. 4
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2.3. Numerical algorithm
is still in the elastic stage and the current stress and strain are taken as the inputs at the time t + t . Otherwise, the shearing process will enter the stage of plastic deformation (step 2). Step 2: Call the stress-strain updating algorithm of the GTN-J model to calculate the stress, the strain, and other variables. It is necessary for each iteration to judge whether the values of the nonlinear equations F1 and F2 in the algorithm are less than or equal to the set value toler = 10 6 and the number of iterations is less than the set value N . If the two mentioned conditions are satisfied, the current stress and strain are updated and the updated data is used as the inputs at the next step
The GTN-J model is implemented in the VUMAT user subroutine of ABAQUS, and the fully implicit backward Euler algorithm is used in the procedure [15]. The numerical algorithm of the model is divided into three steps, as shown in Fig. 3. Step 1: Set the initial values of the parameters, and evaluate the elastic trial stress of the element at the current time t . Then, calculate the GTN-J yield equation (Eq. (1)) to determine whether it is less than or equal to zero. If the above condition is satisfied, the shearing process
Fig. 4. Boundary conditions for three tests: (a) tension; (b) pure shear; (c) compression.
Fig. 5. Single element test for GTN-J model：(a) tension; (b) compression; (c) pure shear; (d) pure shear(D0 = 0.01, k w = 4 in GTN-J model). 5
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zero, and then dDshear = 0 is obtained by Eq. (11). That is, the shear damage parameter will not work in the algorithm. Hence, the GTN-J model degenerates into the GTN model. Since ABAQUS software has its own GTN model algorithm, the stress and the void volume fraction calculated by the GTN-J model in this paper are compared with those obtained by the GTN model in ABAQUS to verify the accuracy of the GTN-J model. A single 8-node element with one integration point (C3D8R element in ABAQUS) is used to simulate the three boundary conditions as shown in Fig. 4, including the tension, the pure shear, and the compression state. The input parameters of the material for the single element test are shown in Table 1. The stress and void volume fraction obtained by the GTN-J model under three conditions are completely consistent with those obtained by GTN model as shown in Fig. 5. Thus the accuracy of the GTN-J model is verified. The void volume fraction does not vary with the equivalent plastic strain under shear loading (as shown in Fig. 5c) because the GTN model does not take the effect of shear loading into account. The GTN-J model (as shown in Fig. 5d) solves the problem by introducing the shear damage. The shear damage accumulates with the equivalent plastic strain and the stress gradually decreases when it exceeds the critical value Dc . The stress drops to zero and the element fails completely when the shear parameter DF is reached.
Fig. 6. SUS304 tube.
3. Experiment 3.1. Experimental material Fig. 7. Flow stress curve of SUS304 thin plate specimen.
The structure of the SUS304 tube is shown in Fig. 6 and the chemical composition of SUS304 stainless steel is listed in Table 2. Assuming that the strain hardening effect of the SUS304 stainless steel (SUS304) is isotropic [26], the stress-strain flow curve is obtained by the standard specimen tensile test. Zhang et al. [23] gave the flow stress curve of SUS304 thin plate specimen under tension (plate thickness 0.9 mm), as shown in Fig. 7. And on this basis, the following relationn , wherein K = 1235 and n = 0.38. ship was obtained: y = K
(step 3). Otherwise, the iteration is repeated until the condition is satisfied. It should be noted that the program is terminated and the constitutive algorithm needs to be checked if the above conditions cannot be satisfied simultaneously. Step 3: Judge whether the void volume fraction or the shear damage exceeds the critical value. If one of them exceeds the critical value, the state of the element is judged as fracture failure and the element is deleted. Otherwise, the element will be continued to calculate at the time t + t .
3.2. Experimental device 3.2.1. Shear device As shown in Fig. 8, the shear device consists of four parts: the shear tool, the holder, the support block, and the fixed tool. All the parts are made of 45 steel and the structural diagram of the shear device is shown
2.4. Single element test If the initial values of Dshear and k w in the GTN-J model are set to
Fig. 8. Shear device. 6
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Fig. 9. Structural diagram of the shear device. Table 3 Structural parameters of the device. Type
a
b
α
d
D
Dimension/mm
2
16
20°
14
16
Fig. 11. SUS304 tube after shearing.
3.3. Experimental scheme The shear device was installed on the MTS, as shown in Fig. 8. The shear tool was held on the clamping tool of the MTS and the fixed tool was held on the lower part of MTS. The initial position of the shear tool was adjusted to 3~5 mm away from the SUS304 tube. The shear speed was set to 50 mm/s. During the shearing process, the displacement and force of the shear tool were acquired. The experiment was conducted three times in total and the fracture of the SUS304 tube after shearing is shown in Fig. 11. 4. Results and discussions 4.1. Finite element model
Fig. 10. Material testing machine.
To reduce the computing time, the model is simplified according to the characteristics of the model without affecting the simulated results.
in Fig. 9. The clearance between the shear tool and the holder is 0.2 mm and the other structural parameters are shown in Table 3.
1) As the stiffness of the shear device is obviously greater than that of the SUS304 tube, all parts of the shear device are set as discrete rigid bodies in the model. Meanwhile, the structural size of the shear device is reduced and only the structure near the shear zone is retained, as shown in Fig. 12a and Fig. 12b. 2) The discrete rigid body is meshed by R3D4 elements. The element size of the rigid body does not affect the simulated results, but it should be noted that the element size needs to be larger than that of
3.2.2. Material testing machine The shear experiments were completed by the material testing machine Landmark 370.1 (MTS), as shown in Fig. 10. The measurement error is less than 0.5%. The loading range is 0–100 KN and the loading speed is 0–50 mm/s for the MTS. During the shear experiments, the load-displacement curve of the shear tool was acquired. 7
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Fig. 12. Finite element model of the shear device: (a) the whole finite element model and selection of mesh types for each part; (b) constitutive models for different regions of SUS304 tube; (c) meshing method for the SUS304 tube; (d) generating mesh on the section of the tube; (e) the entire mesh swepting along the axial direction of the tube.
Table 4 Mesh number and computing time of the different element size. Element size 4
Number/ × 10 Computing time/h
500 μm × 500 μm
250 μm × 250 μm
200 μm × 200 μm
100 μm × 100 μm
1.7 50 h
11 252 h
21 500 h(estimated)
116 4000 h(estimated)
the SUS304 tube to prevent node penetration. Thus, the mesh size is set to 1 mm × 1 mm, as shown in Fig. 12a. 3) The SUS304 tube is a hollow-cylindrical structure. When the tube is meshed, the section of the tube is meshed firstly, and then the entire mesh is swept along the axial direction of the SUS304 tube, as shown in Fig. 12d and Fig. 12e. Therefore, it is impossible to refine the mesh only for the shear zone. Moreover, the finer the mesh, the longer the computing time. Table 4 shows the number of mesh and the time cost for different element sizes. Finally, the C3D8R element is chosen for the SUS304 tube and the mesh size is set to 500 μm × 500 μm, as shown in Fig. 12a. 4) To improve the computing efficiency, the GTN-J model is only applied to the shear zone of the SUS304 tube. Additionally, the left region of the SUS304 tube is set to the rigid element because there is basically no deformation due to the existence of the support block. Similarly, the Mises plasticity is applied to the right region to reduce the computing time, as shown in Fig. 12c. 4.2. Results The simulated load-displacement curve is compared with three experimental curves in Fig. 13. The right side of the figure shows the sections of the SUS304 tube at different stages during the shearing
Fig. 13. Load-displacement curve during the shearing process.
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Fig. 14. Comparisons of simulated and experimental fracture morphology.
process. From the overall trend, the simulated curve is perfectly consistent with the experimental curves. Meanwhile, the simulated fracture morphology is identical with the actual experimental section, as shown in Fig. 14. According to the position of the shear tool, the shearing process can be divided into five stages, as shown in Fig. 15.
4.3. Discussion 4.3.1. Comparisons between the simulated curve and experimental curves The simulated curve is a little smaller than the experimental curves in the second and third stages, as illustrated in Fig. 13. The main reason is that the strain hardening of the SUS304 tube in the forming process is not considered. Fig. 17 shows the forming process of the SUS304 tube. The strength of the stainless steel sheet is further enhanced by the roll forming during the process, but only the strain hardening of the sheet is taken into account in the simulation. The flow stress curve of the SUS304 tube was obtained by the uniaxial tensile test of sheet specimens. Therefore, the simulated results are lower than the experimental results in the second and third stages. In the fifth stage of the shearing process, the displacement of the shear tool for the complete fracture in the simulation is about 1.5 mm smaller than the experimental results. The reasons are as follows: 1) The strainhardening of the SUS304 tube during the forming process is ignored in the simulation, so the occurrence of complete fracture is earlier than the experimental results. 2) The wall thickness of the shear zone may be thicker than the simulated standard value due to the machining error. Thus, the cracks propagation is required to the larger deformation.
①Start shearing: The shear tool had not touched the top surface of the SUS304 tube at that time. ②Initial crack generation: The initial cracks of the SUS304 tube were generated when the shear tool reached the position a and the first peak value of the shear force F1 appeared. ③Phase I tearing process: With the movement of the shear tool, the cracks propagated along the path a b1 (a b2) . Two peak values of the shear force appeared at the beginning (F2 ) and end (F3 ). ④Phase II tearing process：The shear tool kept moving downward and then the cracks propagated along the path b1 c /b2 c . The shear force started to drop. ⑤Fracture: The left and right cracks continued propagating with the movement of the shear tool and eventually converged at the position c . Therefore, the tube broke and the last peak value of shear force (F4 ) was reached.
4.3.2. Crack evolution Fig. 18 shows the shearing process of the SUS304 tube. The analysis of the crack evolution is shown in Fig. 19.
Fig. 16 shows the stress distribution of the SUS304 tube at different stages of the shearing process.
Fig. 15. Crack evolution of the shearing process. 9
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Fig. 16. Stress distribution of the SUS304 tube at different stages: (a) Start shearing; (b) Initial crack generation; (c) Phase I tearing process; (d) Phase II tearing process; (e) Fracture.
Fig. 17. Process flow of the SUS304 tube.
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Fig. 18. Crack evolution process of the SUS304 tube: ①Start shearing; ②Initial crack generation; ③Phase I tearing process; ④Phase II tearing process; ⑤Fracture.
1) Before shearing: The shear tool has not touched the top surface of the SUS304 tube at that time. 2) Crack path c1 c2 : The contact area between the shear tool and the SUS304 tube is flattened as the shear tool moves. In this case, the initial crack is generated from the edge of the flattened area once the damage exceeds the allowable limit. The fracture surface is approximate to a trapezoid, as shown in Fig. 19c. 3) Crack path c2 c3 : The SUS304 tube is extruded and deformed laterally along the red arrow with the shear tool moving, as shown in Fig. 19d. Meanwhile, the cracks extend successively along the axial direction besides the hoop direction and thickness direction, and then a burr (a purple protrusion in Fig. 19a) is formed at the third stage of the shearing process. 4) Crack path c3 c4 : The cracks on the left and right sides continue propagating along the hoop direction and thickness direction. In particular, when the shear tool arrives at the middle of the SUS304 tube, the position of the crack tips starts to exceed the top surface of the shear tool (the phenomenon can be observed at the third and fourth stages in Fig. 18). Finally, the cracks on both sides converge at the point c4 . 5) Crack path c4 c5: The cracks no longer propagate due to the softening of the material at that time. The right part (see the fifth stage in Fig. 18) deforms downward until the shear tool is contact with the residual part without fracture. Then the residual part is extruded
and torn. Finally, the cracks converge at the point c5. The fracture surface is a small trapezoid, as shown in Fig. 19e. 4.3.3. Damage accumulation To further state the damage evolution during the entire shearing process, five elements were chosen along the hoop direction of the SUS304 tube, representing five stages in the shearing process, as shown in Fig. 20a. The failure strain of each element is basically the same value and all of them are around 0.35 (see Fig. 20b). The void volume fraction of all the elements decreases initially, as can be seen in Fig. 20c. It indicates that whatever stage in the shearing process, the element is compressed at first. Compared with the void volume fraction, the shear damage of all the elements exceeds the critical value, as shown in Fig. 20d. That is, the failure of the element is induced by shear damage rather than microvoids. This indicates that material failure is mainly dominated by shear loading. It is important to note that such variables as the equivalent strain, the void volume fraction, and the shear damage no longer increase during the moment a-b in the fifth stage (see E_5 in Fig. 20b–d). The reason is that the cracks no longer propagate due to the softening of the material during the moment a-b. The above three variables continue increasing until the shear tool is contact with the residual part of the SUS304 tube (see the fifth stage in Fig. 18). 11
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Fig. 19. Crack evolution analysis of the SUS304 tube: (a) crack evolution path; (b) stress state at different stages; (c) initial crack generation; (d) crack generation analysis at the intersection of the second and third stage; (e) final fracture.
Fig. 20. Damage evolution at different stages: (a) the location of the selected element; (b) the equivalent strain; (c) the void volume fraction; (d) shear damage.
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shear damage is considered in the GTN-J model. Fig. 21 shows the simulated load-displacement curves with and without the void volume fraction. It can be found that the curve without the void volume fraction is basically the same as that with the void volume fraction. The failure strain of each element without the void volume fraction is consistent with that with the void volume fraction, as shown in Fig. 22a. The effect of void volume fraction on the evolution of shear damage during the shearing process is shown in Fig. 22b. Obviously, the void volume fraction has little effect on the evolution of shear damage under shear-dominated conditions. In conclusion, the effect of the void volume fraction to the failure process could be ignored under shear-dominated conditions. 4.3.5. Fracture morphology The section of the SUS304 tube after shearing has not been defined by scholars yet. Thus, through referring to the fracture morphology of the blanking, the section of the tube is divided into the following four parts, namely, initial tearing zone, burr, burnish zone, and fracture tearing zone, as shown in Fig. 23.
Fig. 21. Comparisons of load-displacement curve with and without the void volume fraction.
1) Initial tearing zone: The initial tearing zone is a rough surface formed by the cracks. Theoretically, the cracks first take place at the point c1. However, actually, the top surface of the SUS304 tube is flattened under compression at the beginning. The edge of the flattened area (rectangular area) is torn when the stress exceeds the allowable stress of the material. The fracture surface is approximate
4.3.4. Effect of void volume fraction to the failure process To illustrate the effect of void volume fraction to the failure process, the void volume fraction remains constant throughout the shearing process (i.e. f = f0 during the shearing process). That's to say, Only
Fig. 22. Comparisons of damage evolution at different stages: (a) the equivalent strain; (b) shear damage. Note: The straight line represents the simulated curve with shear damage and void volume fraction. The symbol represents the simulated curve with shear damage (without void volume fraction).
Fig. 23. Fracture morphology of the SUS304 tube. 13
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to a trapezoid shape. 2) Burr: The crack path will cause a protrusion in the phase I tearing process (Crack path c2 c3 c4 in Section 4.3.2). 3) Burnish zone: The cracks propagate along the path c3 c4 , as shown in Fig. 19a. The crack path is smooth. 4) Fracture tearing zone: The right part of the SUS304 tube deforms downward, as shown in Fig. 18. Finally, the residual part is torn. Thus the fracture surface is similar to the initial tearing zone, but the size is only one-fifth of it.
4.4. Comparisons of sheet blanking Due to the lack of research on the tube shearing process, the design and optimization of the device for shearing of the tube in engineering are carried out by referring to sheet blanking process. However, there are obvious differences between sheet blanking and tube shearing process. It is unreasonable to develop the device for tube shearing by referring to the sheet blanking process. Therefore, the differences between sheet blanking and tube shearing process are compared from five aspects in this paper to promote the theoretical research of tube shearing process.
4.3.6. Stress condition of the shear tool The stress condition of the shear tool changes constantly with the deformation of the SUS304 tube during the shearing process, as shown in Figs. 24 and 25. It can be seen from Fig. 25a that the middle of the shear tool is subjected to the maximum stress during the initial crack generation. Subsequently, the position of the maximum stress gradually moves to both sides in the third and fourth stages of the shearing process, as shown in Fig. 25b and c. Finally, it is noted that the position of maximum stress returns to the middle of the shear tool in the fifth stage.
4.4.1. Crack evolution Fig. 26 shows the crack path during the sheet blanking. Obviously, the incipient cracks almost occur on the side of the punch and the die simultaneously. During the blanking process, the cracks propagate downward at the position a and upward at the position b until the two cracks intersect with each other (see the blue arrow in Fig. 26). However, the cracks only initiate at the top of the SUS304 tube in the shearing process because the deformation of SUS304 tube is dispersed through the hollow structure and cannot be transmitted to the bottom of the tube. The crack evolution of the tube shearing is divided into five stages, thus the fracture mechanism is more complicated. 4.4.2. Load-displacement curve The overall trend of the load-displacement curve of sheet blanking and tube shearing are compared in Fig. 27. For the tube shearing, the overall fluctuation of the curve is higher. Meanwhile, there are four peaks in the whole process. However, the curve of sheet blanking is smooth and only one peak value of the force appears throughout the blanking process [27].
Fig. 24. The stress condition of the shear tool during the shearing process.
Fig. 25. The stress condition changes with the movement of the shear tool: the maximum stress position at (a) initial crack generation; (b) the beginning of the phase I tearing process; (c) the end of the phase I tearing process; (d) phase II tearing process; (e) fracture. 14
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Fig. 28. Fracture section of sheet blanking.
Fig. 26. Crack evolution of sheet blanking.
Fig. 29. The stress location of the punch during the blanking process.
the punch surface is stressed uniformly during the entire blanking process [28], as shown in Fig. 29. However, the stress condition of the shear tool constantly varies during the shearing process of the SUS304 tube, as shown in Figs. 24 and 25. 4.4.5. Model simplification The sheet blanking can be simplified into a two-dimensional axisymmetric model. However, the whole three-dimensional model must be adopted to ensure the accuracy of the simulation in the tube shearing. 5. Conclusions Due to the limitations of various modified GTN models in simulating complex stress states, the GTN-J model is applied to simulate the shearing process of SUS304 tube. The stress-strain algorithm of the model is embedded in ABAQUS/Explicit to realize its numerical solution. Based on the simulations and experimental results, the conclusions are drawn as follows:
Fig. 27. Comparison of the load-displacement curve.
4.4.3. Fracture morphology Fig. 28 shows the blanking section of the sheet [27]. From top to bottom, it can be divided into four parts, including the rollover, the burnish zone, the fracture zone, and the burr. The fracture morphology of the SUS304 tube retains the burnish zone, the fracture zone, and the burr except for the rollover, as shown in Fig. 23. During the shearing process, the contact area between the shear tool and the SUS304 tube is flattened and then the cracks occur at the edge of the flattened area. Accordingly, the phenomenon of rollover is not obvious. Therefore, the initial tearing zone is used to substitute for rollover.
(1) The simulated load-displacement curve and fracture morphology of the tube is basically consistent with the experimental results, which verifies the effectiveness of the GTN-J model. (2) The fracture mechanism of the shearing process of the SUS304 tube is obtained for the first time. The shearing process of SUS304 tube is divided into five stages in detail, including the start shearing, the initial crack generation, the phase I tearing process, the phase II tearing process, and the fracture. Meanwhile, the crack evolution and the damage accumulation during the shearing process are analyzed. (3) The fracture morphology of the SUS304 tube is firstly proposed, including the initial tearing zone, the burr, the burnish, and the fracture tearing zone.
4.4.4. Stress condition of the shear tool It can be found that the punch surface always contacts with the upper surface of the sheet during the blanking process. Consequently, 15
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(4) The stress condition of the shear tool varies constantly during the shearing process, which provides a theoretical basis for the design of the shear tool. (5) The differences between the tube shearing and sheet blanking are depicted from five aspects, including the load curve, the fracture morphology, the crack evolution, the stress condition of the shear tool, and the model simplification. (6) Shortcomings and prospects: The effect of strain hardening in the forming of the SUS304 tube is not considered, and thus the constitutive model can be further improved to achieve the minimum difference between the simulated results and the experimental results.
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