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Quantification of the constitutive relationship of high-energy heavy-ion irradiated SS316L using the small punch test
Journal of Nuclear Materials 531 (2020) 152014
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Journal of Nuclear Materials journal homepage: www.elsevier.com/locate/jnucmat
Quantification of the constitutive relationship of high-energy heavyion irradiated SS316L using the small punch test Xianlong Zhang a, Chonghong Zhang a, b, *, Zhaonan Ding a, b, Yuguang Chen a, b, Liqing Zhang a a b
Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, 730000, China School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing, 100049, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 23 September 2019 Received in revised form 17 January 2020 Accepted 18 January 2020 Available online 22 January 2020
Heavy-ion irradiation has been widely used to simulate neutrons-irradiation induced damage effects, due to its higher damage rate and lower activation of the post-irradiation manipulation. The high-energy ion beam is capable of producing a thickness of dozens of microns damaged region in steel, making it possible to evaluate the macroscopic mechanical properties of the irradiated specimens. In the present paper, a method to quantify the constitutive relationships of high-energy heavy-ion irradiated steels is proposed. Ni22þ ions with a kinetic energy of 357.86 MeV provided by a cyclotron were used to produce a quasi-homogeneous atomic displacement damaged layer (about 25 mm in thickness) in specimens of 316 L stainless steel. The temperature of the specimens were kept at about 50 C during ion irradiation. Two damage levels of 0.16 and 0.33 displacement per atom (dpa) were approached. Small punch test of the unirradiated and irradiated f3 mm disk samples was carried out to obtain the load-deflection curves. A series of finite element simulation of SPT of the laminated irradiated samples, in combination with sequential programming algorithm, was performed to characterize the constitutive relationships of the irradiation damaged layer of the samples. Finite element simulations with obtained constitutive relationships show agreement with the experimental results. Nanoindentation tests were carried out to verify the identified constitutive relationships. The nanoindentation results show an irradiation induced hardening in good agreement with that from the obtained constitutive relationships. © 2020 Elsevier B.V. All rights reserved.
Keywords: Heavy-ion irradiation Steel Small punch test Finite element method Constitutive relationship
1. Introduction To ensure the safety and evaluate the lifetime of nuclear reactors, it is fundamentally essential to accurately survey the mechanical properties and model the deformation and failure behavior of the structures [1]. Irradiation hardening and embrittlement is one of the major concern for the safe service of in-pile structure components. Energetic heavy-ion beams have long been used to simulate the radiation effects (void swelling, irradiation hardening) by fast neutrons in materials [2,3], due to their merits of higher atomic displacement rate, similar primary cascade damage structure and much lower radioactivity of the irradiated specimens. Standard mechanical test methods are always destructive because
* Corresponding author. Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, 730000, China. E-mail address: [email protected] (C. Zhang). https://doi.org/10.1016/j.jnucmat.2020.152014 0022-3115/© 2020 Elsevier B.V. All rights reserved.
large specimens need to be extracted and machined from actual serving components. Therefore, it is convenient to use miniature specimens to characterize the mechanical properties. What’s more, due to the limited irradiation volume in test reactors, spallation neutron sources and high-energy ion accelerators, it is compulsory to develop miniaturized specimen techniques to evaluate the mechanical properties of the irradiated material. Small punch test (SPT) is one of the most widely used among all the small specimen test technique. Owing to the small sample size (generally 8 mm or 3 mm in diameter) of SPT, slices of surveillance component or remnants of Charpy impact test (undeformed zone) can be used [4]. A disc sample is firmly gripped between two dies and deformed vertically into a circular hole by a spherical punch, the force to punch the rod and the maximum deflection of the sample are recorded during the test [5e7]. The sample is exposed to a complicated biaxial stress state, making the test more suitable for representing the actual stress state observed in the reactor pressure vessel, rather than uniaxial stress state in conventional uniaxial tensile tests.
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Six different stages are identified in a typical SPT load-deflection curve (LDC) as depicted in Fig. 1, the regimes are: (I) elastic bending of the disk, (II) plastic bending, (III) membrane stretching, (IV) plastic instability, (V) crack initiation and (VI) fracture. The sample strength and ductility can not be obtained straightforwardly due to the complicated stress state. The LDC data is usually interpreted by semi-empirical formulations [4,5,8]. It is worthwhile to mention that precisely distinguish of the boundaries between different stages is subjective and different formulations for evaluation of the strength have been proposed in Ref. [9]. Much effort has been paid to identify material properties as yield stress and ultimate stress from the SPT experiment [10]. Moreover, the constitutive relationship of materials is necessary in modeling a complex structure of a component in order to obtain the actual stress state in the process of operation. Finite element method (FEM) simulations are dedicated to evaluating the role of different parameters on SPT results, and further extract and interpret the mechanical properties based on experimental data [11e14]. However, FEM simulation is a reverse process to obtain the constitutive relationship of the tested material. It means that a constitutive relation of the test material is necessary to in FE modeling. Characterization of material properties from a given LDC curve involves comprehensive numerical computation [15]. Artificial neutron network and non-linear optimization techniques are used to identify the mechanical parameters from the LDCs [16e19]. The penetration depth of the energetic ions in metals varies from several to tens of microns depending on the energy and species of ions. Conventional mechanical characterization methods of ion-irradiated materials include nanoindentation [21,22], microhardness [21,23], and micro compression of nano pillars [24]. Irradiation hardening data can be obtained by transverse comparison, but embrittlement information can not be obtained by the above method [23,24]. In our previous research, mechanical properties of high-energy heavy-ion irradiated samples with 60 mm in thickness were tested by SPT [25]. It was found that such thin samples were already in the membrane stretching state from the beginning of the test so that limited information about the yield load from the LDCs can be obtained [26,27]. In the present study, we propose a comprehensive method to extract constitutive relationships of the irradiation damaged layer from a laminated specimen composed by a heavy-ions irradiated
damaged layer and an unaffected substrate. Constitutive relationships of the irradiation damaged layer from the irradiated specimens are determined by combining FEM simulations and sequential programming algorithm. Finally, the results are verified by FEM simulation and nanoindentation test. The methodology combining experiment and numerical simulation is schematically presented in Fig. 2.
Fig. 1. Typical six stages of the LDC of the SPT and corresponding material parameters of different stages [20].
Fig. 2. Procedure of identification of constitutive relations of the heavy-ion irradiated materials.
2. Experimental details and numerical simulation The material used in this research is the commercial low carbon content austenitic stainless steel 316 L provided by Goodfellow Company. The nominal compositions can be found in literature [13].
2.1. Uniaxial tensile test of the unirradiated specimen A uniaxial tensile test was conducted to obtain the constitutive relationship of the material, which is necessary for FE modeling of the SPT. Sheet specimens with a width of 12.5 mm, a thickness of 0.5 mm and the length of the parallel section of 65 mm within gauge length of 150 mm was used for uniaxial tensile test, as presented in Fig. 3a. The uniaxial tensile was a standard tensile test under Chines standard GB/T 228.1e2010 and was performed at room temperature. The electro-mechanical testing machine was controlled at a constant crosshead velocity to make sure a strain rate of 0.0015 s1, which was comparable with the strain rate of the first stage of the small punch test. The elongation of the interest section was measured by a clip gauge attached to the sample. The nominal and true stress-strain curves are presented in Fig. 3b. The true stress-strain curve is fitted by modified Hollomon’s law as depicted in Equation (1),
s ¼ k1 εn þ k2 ε
(1)
where s and ε are true stress and true strain respectively k1, k2 and n are fitting parameters, k1 ¼ 345:9, k2 ¼ 1502, and n ¼ 0:06086.
X. Zhang et al. / Journal of Nuclear Materials 531 (2020) 152014
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Fig. 3. (a) Uniaxial tensile test of plate sample, and (b) the nominal (blue) and true (black) stress-strain curve and fitted (red) curve. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
2.2. High-energy Ni22þ beam irradiation experiment Two specimens of the 316 L steel with dimensions of 5 10 0.2 mm3 were subsequently mechanical ground using silicon carbide papers down to 2400 grit, followed by polishing with colloidal silica suspension with a grain size of 1mm: The specimens were used for the heavy-ion irradiation at a terminal of the Sector-focused Cyclotron (SFC) at HIRFL (Heavy-ion Research Facility in Lanzhou) at Institute of Modern Physics, Chinese Academy of Sciences. The samples were mounted on a specimen stage cooled with liquid nitrogen at the terminal, and were irradiated with 357.86 MeV Ni22þ at about 50 C to two doses of 0.16 and 0.33 displacement per atom (dpa), respectively. The flux of the ion beam was about 8:8 1010 Ni-ions/cm2/s. The samples were mounted on a specimen stage cooled with liquid nitrogen, a thermocouple was mounted on the specimen stage to monitor the temperature of the specimens.FEM simulation was carried out to calculate the temperature of the ion-beam irradiated surface of the specimen, it turn out that the ion-beam irradiated surface is about several degrees higher than the thermocouple. By using aluminum foils with ten different thickness in a rotating energy degrader, a quasi-uniform damaged layer was produced in the samples [25]. The damaged layer was approximately 25 mm in thickness. The
Fig. 4. Damage profiles of the high-dose Ni-ions irradiated steel specimen according to estimation with SRIM code. The right column shows the thickness of the aluminium foils used in the energy degrader.
profiles of the atomic displacement damage of the high-dose samples are shown in Fig. 4, according to estimation with SRIM code [28]. More details of the facility and the irradiation conditions are described in our previous papers [23,25].
2.3. Small punch test The schematic of the SPT device set-up is shown in Fig. 5. A disk sample is clamped between the upper and the lower die. The upper die diameter is 1.05 mm (a small tolerance of 0.05 mm), the lower die diameter is 1.5 mm. The apparatus is designed to use 3 mm diameter disc samples with a thickness of 0.2 mm. Disk small punch specimens of 3 mm diameter were employed instead of traditional 8 mm ones due to the limited spot area of ionirradiation and the radioactivity of the ion-irradiated specimens. The rod and the punch are driven by an MTS-E43 electro servo testing machine with a constant displacement rate of 0.02 mm/min. The force and the displacement of the punch are recorded by the machine and an LVDT attached to the punch rod respectively. Instead of a single piece punch recommended by CEN Workshop Agreement [29], a ceramic sphere ball with a radius of 0.5 mm and a rod made of low-carbon steel with a length of 40 mm and a radius of 0.5 mm are used as the punch. Besides the wear problem, the permanent plastic deformation of the punch made of steel was observed in our previous research and FEM simulation.
Fig. 5. Schematic of the small punch test configuration.
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Since the displacement measured by LVDT attached to the punch rod includes the elastic deformation of the rod and the ceramic ball during SPT, the displacement must be corrected for the system compliance by Equation (2) [13,30], where dsp is the displacement of the tip of the punch, dmeasure is the displacement measured by the LVDT, F is the load and C is the compliance of the punch system.
dsp ¼ dmeasure F C
(2)
Fig. 6 shows the experimental LDC of the unirradiated sample from SPT, a standard LDC with 6 stages is obtained as 8 mmdiameter sample in the literature [20]. The compliance corrected curve agrees well with the FEM simulation result up to a displacement of 0.5 mm, exhibiting the combination of the elastic bending and the plastic deformation (stages I and II, as depicted in Fig. 1). The first two stages containing all the elastic and plastic information (characterized by the elastic modulus E, Poisson ration n, strength coefficient K and strain hardening exponent n) of the test material up to 0.2 mm in the LDC as shown in Fig. 6. While the rest of the curve is affected by friction coefficient m, initial void volume fraction f0 , void nucleation fN , void growth fc , coalescence and crack initiation fF [13,16], which is beyond the scope of this paper. Under the hypothesis of irradiation hardening, FE modeling examples show that the upward position of the irradiation damaged layer during SPT demonstrates more distinguishable LDC from the unirradiated specimen than that with the irradiation layer downward (not shown here). The heavy-ion irradiated specimens were stamped into 3 mm diameter SPT disk samples. All the irradiated samples were tested with the irradiation layer upward at room temperature. 2.4. Nanoindentation test
material as a reference with the same control mode and penetration depth. 2.5. Finite element modeling of SPT Finite element modeling of the SPT was developed using ABAQUS 6.14e2 standard code. Because of the specimen geometry and SPT setup, axisymmetric simplification was used as described in Ref. [13]. The upper die, lower die, and the punch were assumed to be rigid bodies, the friction coefficient was set to m ¼ 0:2 between the specimen and the dies/punch[13,20]. The unirradiated specimen was assumed as isotropic and meshed with 100 20 uniform CAX8R reduced integration elements, the number of elements was selected by mesh converge analysis to be sufficient to get the same LDCs as a finer mesh. While the irradiated specimen was assumed as laminar composites with an undetermined layer thickness of 25mm and an unirradiated layer thickness of 175 mm. The schematic of FE modeling of the irradiated specimen is shown in Fig. 7. The maximum effective strain in SPT is higher than the ultimate strain in the uniaxial tensile test due to different constraint conditions and stress state. Extrapolation of the input stress-strain data results in a better agreement of the experiment with FEM results [8,30]. The maximum strain of the specimen was set to 0.8 in all the FE modeling in this paper. The elastic modulus of the unirradiated material is set to 205 GPa according to the uniaxial tensile test. The nanoindentation data indicates that the elastic modulus of the alloy is not affected by ion irradiation, as depicted in Fig. 8. For the irradiated portion of the irradiated sample, assumed parameters for constitutive relationship are list in Table 1. All parameters combined into 13 10 10 ¼ 1300 stress-strain curves, while the material properties of the unirradiated portion remain constant. 2.6. Identification procedure
Nanoindentation test was carried out using a Berkovich tip indenter under depth control mode using a G200 nanoindenter. The continuous stiffness measurement (CSM) method was used to obtain the depth profile of hardness and elastic modules up to a maximum displacement of 2000 nm. Arrays consisting of 9 indentations were conducted in both the irradiated and unirradiated specimens. The nanoindenter tip was calibrated by a fused silica
As described in section 2.5, 1300 constructed constitutive relationships were assigned to the irradiated portion of the irradiated specimen in turn. The FEM simulations of the irradiated specimen SPT were implemented by Python script running in ABAQUS. After a series of FEM simulation, 1300 LDCs were obtained by the Python code through ABAQUS and each LDC corresponds to one of the
Fig. 6. Experimental LDC before and after compliance correction, in comparison with LDC obtained from FEM simulation.
Fig. 7. Axisymmetric FEM model for an irradiated specimen, the top layer is the irradiated portion and the rest of the sample is the unirradiated portion.
X. Zhang et al. / Journal of Nuclear Materials 531 (2020) 152014
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SPT experimental results of the irradiated samples and the FEM simulation database. Stages (I) and (II) of the LDC (d < 0.2 mm) were used here for the identification of the constitutive relationship of the irradiated material, due to fact that the first two stages contain the elastic-plastic information of the tested material as described in section 2.3. The identification procedure is carried out by minimizing the quadratic sum of the least square error measure (e) between the experimental results and the curves in the FEM calculated database by Equation (3) [31], e is a target function which measures the least squares errors between the experimental results and the curves in the FEM calculated database.
e¼
50 Fsim ðui ; pÞ Fexp ðui Þ 2 1 X 50 i¼1 Fexp ðui Þ
(3)
3. Results Fig. 8. Nanoindentation result of elastic modulus as a function of indentation depth for both the unirradiated and irradiated specimens.
Table 1 Constructed parameters for constitutive relation for irradiated portion of sample. Parameter
min
max
steps
k1 k2 n
0.8 345.9 0.8 1502 0.8 0.6086
2.0 345.9 1.7 1502 1.7 0.6086
13 10 10
input constitutive relationships, and the LDCs were read and processed by MATLAB. The ABAQUS input stress-plastic strain and the calculated LDCs curves are shown in Fig. 9a. To build the data base, the calculated LDCs are interpolated at certain displacement locations. The selected 50 location intervals increase gradually in the whole interest zones. The interpolated load information is stored in 50 1300 matrix Fsim ðu; pÞ and the SPT experimental results of the irradiated samples are stored in 50 2 matrix Fexp ðuÞ. M. Abendroth and M. Kuna [16] found that a direct approximation of the SPT experimental results with the FEM simulation data base leads to better results compared to artificial neural network approximation. The parameters can be obtained by an optimization procedure to minimize the difference between the
3.1. Parameter indentation According to the optimization path described in section 2.6, the minimum e of the irradiated samples of different irradiation doses are found through Equation (3) by MATLAB code, the corresponding material parameters for the irradiation portion of the samples are identified. The identified parameters and corresponding yield stresses are listed in Table 2, and the stress-plastic strain curves of the identified irradiated materials of the two irradiation doses are shown in Fig. 10a. In order to verify the results, the obtained constitutive relationship was used to simulate the SPT experiment by FEM simulation. The results show good agreement with the SPT experiment curves, as shown in Fig. 10b.
Table 2 Constitutive relationship parameters identified from identification procedure. unirr: Dsy ¼ sirr: y sy
Irradiated condition
k1
k2
n
sy (MPa)
Dsy (MPa)
unirradiated 0.16dpa 0.33dpa
345.9 657.2 506.4
1502 1351.6 1448.7
0.06086 0.0548 0.0520
262 408 486
_ 146 224
Fig. 9. The 1300 constructed stress-plastic strain curves for irradiated portion of the irradiated specimens (a), and their corresponding LDCs calculated from FEM simulation (b).
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(a)
(b)
Fig. 10. The stress-plastic strain curves of the irradiated (0.16dpa, 0.33dpa) and unirradiated SS316L (a), and their corresponding LDCs calculated from FEM simulation compared with experimental results (b).
3.2. Hardness from nanoindentation test
4. Discussion
The average nano-hardness as a function of indentation depth of the unirradiated and the irradiated specimens are presented in Fig. 11a. Average of 9 indentation curves for the hardness of the alloys before and after irradiation are shown, with the error bars representing the standard deviation. Significant increase in hardness is observed in the irradiated specimens compared with unirradiated material. According to the hardness profiles, the results strongly demonstrate the indentation size effect. The indentation size effect can be characterized by Nix-Gao model described in Equation (4) [32],
Campitelli [12] found that more than one constitutive relationship may construct the same LDC of SPT by FE modeling. The constitutive relationships which construct the same LDC differ in yield stress and in the range of small strain (<0.05), while are close to each other in the range of larger strain. Therefore, the stressplastic strain curve can be treated as two segments: a linear portion in the range of 0e0.5 and a non-linear strain hardening portion. For the possible constitutive relationships construct the same LDC, the second segments agree while the first segments vary. Once the interception of the constitutive relationships (yield stress of the material) is determined, the constitutive relationship of the material will be identified. Therefore, in order to prove the uniqueness of the results obtained by the proposed reverse method based on SPT and optimization procedure, it is necessary to verify the accuracy of the yield stress of the irradiated material. Therefore nanoindentation tests were employed. According to our previous research, there is a linear relation between Vickers hardness and nanohardness of various polycrystalline (fcc, bcc) materials before and after irradiation [21]. The Berkovich hardness (Hb ) obtained from nanoindentation can be converted to Vickers hardness (Hv ) by Equation (5).
H ¼ Hb
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi h* 1þ h
(4)
where H is the hardness profile with respect to depth h obtained in the nanoindentation test, Hb is the hardness of infinite depth and h* is the characteristic length of the experiment setup. Hb and h* can be obtained by fitting the curve H 2 versus h1 . The least-square fitting of H 2 versus h1 is used to extract Hb and h* for both the unirradiated and irradiated specimens are plotted in Fig. 11b. The nanohardness of the unirradiated, 0.16dpa, and 0.33dpa steel are 2.35 GPa, 2.90 GPa and 3.2 GPa respectively.
(a)
(b)
Fig. 11. Average indentation hardness profiles of the steel before and after irradiation (a), H2 versus h1 plots of the alloys before and after irradiation (b).
X. Zhang et al. / Journal of Nuclear Materials 531 (2020) 152014
Hv ¼ 81:0Hb
7
(5)
The Vickers hardness of 316 L stainless steel before and after irradiation obtained from nanoindentation are tabulated in Table 3. A linear relationship was previously found between the increase in yield stress Dsy and the increase in Vickers hardness DHv in Ref. [33], with a coefficient ranging from 2.7 to 3.6. For the purpose of comparing the results between nanoindentation and SPT, the yield stress and corresponding Vickers hardness of unirradiated samples are used as baseline values for determining the ionirradiated induced hardening. In addition, low temperature ( < 100 C) fast neutron irradiated 316SS tested at room temperature from literatures [33,34] are employed to compare with our results due to similar irradiation doses and test conditions. The change in yield stress characterized by SPT is plotted as a function of corresponding the change in Vickers hardness converted from nanoindentation test as depicted in Fig. 12. A least-square linear fit was applied, the results show a good linear relationship between Dsy and DHv . The correlation Dsy ¼ 3:355 DHv is observed by linear regression with R2 ¼ 0.98. The coefficient is close to the theoretical value of 3.33 MPa/kg/mm2 [35]. The correlation factor of 316SS for the data set used in the literature [34] is 3.43 with R2 ¼ 0.98, the data points from our research show good agreement with linear regression line in Fig. 12. It thus proves the validity of the yield stress as well as the constitutive relationship obtained from the proposed method. The equation Hv ¼ 81:0Hb (Hv with the unit of kgf/mm2 and Hb with the unit of GPa) was employed to establish the correlation between the Berkovich hardness and Vickers hardness the in our manuscript based on our previous research [21], while the theoretical predicted value is 94.5 [35]. The difference between the experimental and theoretical value of the correlation coefficient is probably from the influence of pile up or sink in effect during hardness measurement [21]. If the theoretical value (94.5) of the coefficient was chosen, the correlation coefficient K in the formula Dsy ¼ K DHv changed from 3.355 to 3.259 and hence still kept around the theoretical value (data not shown here in detail). The effect on the outcome is not significant.
5. Conclusions Small punch test combined with a series of finite element simulations and sequential programming algorithm is proposed to characterize the constitutive relationship of high-energy heavy-ion irradiated steels. Finite element simulations utilizing the constitutive relationships obtained by the proposed method assigned to the damaged layer show good agreement with the load-displacement curves of the irradiated samples. Nanoindentation tests were carried out to verify the identified constitutive relationships. The nanoindentation results show irradiation induced hardening in agreement with that from the obtained constitutive relationships. Besides the ion irradiation steels, the proposed method could be used to characterize the constitutive relationships of the laminar composite such as thin coating material and functionally graded material free from the sample size restrictions. Furthermore, the
Table 3 The Vickers hardness obtained from nanoindentation test for the unirradiated and irradiated material. Irradiated condition
Hb (GPa)
Hv (kg/mm2)
DHv (kg/mm2)
Unirradiated 0.16dpa 0.33dpa
2.35 2.91 3.19
190.35 235.71 258.39
_ 45.36 68.04
Fig. 12. The irradiation induced increase in yield stresses obtained from SPT are plotted as a function of corresponding Vickers hardness converted from nanoindentation test. The dashed line is linear regression data of our research.
obtained constitutive relationship of the irradiated layer of the ionirradiated specimen is necessary to calculate the fracture energy of the SPT in order to find out the DBTT in further research. Data statement All data and Python script included in this study are available upon request by contacting the corresponding author. Declaration of competing interest The authors declare no conflict of interest. CRediT authorship contribution statement Xianlong Zhang: Formal analysis, Writing - original draft. Chonghong Zhang: Funding acquisition, Conceptualization, Project administration. Zhaonan Ding: Methodology. Yuguang Chen: Software. Liqing Zhang: Supervision. Acknowledgments This study is sponsored by the National Key Research and Development Program of China (Grant No.2017YFB0702202) and the National Science Foundation of China (Grant No.U1532262). The authors gratefully acknowledge the operation team of HIRFL for their help in the ion irradiation experiments, and Dr Kai Nie for inspiring discussion. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.jnucmat.2020.152014. References [1] Yvon, Carre, Structural materials challenges for advanced reactor systems, J. Nucl. Mater. 385 (2) (2009) 217e222. [2] R.S. Nelson, D.J. Mazey, J.A. Hudson, The use of ion accelerators to simulate fast neutron-induced voidage in metals, J. Nucl. Mater. 37 (1) (1970) 1e12. [3] N.H. Packan, K. Farrell, J.O. Stiegler, Correlation of neutron and heavy-ion damage * : I. The influence of dose rate and injected helium on swelling in pure nickel, J. Nucl. Mater. 78 (1) (1978) 143e155.
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[4] M.F. Moreno, Effects of thickness specimen on the evaluation of relationship between tensile properties and small punch testing parameters in metallic materials, Mater. Des. 157 (2018) 512e522. [5] X. Mao, H. Takahashi, Development of a further-miniaturized specimen of 3 mm diameter for tem disk (ø 3 mm) small punch tests, J. Nucl. Mater. 150 (1) (1987) 42e52. [6] J. Mcnaney, G.E. Lucas, G.R. Odette, Application of ball punch tests to evaluating fracture mode transition in ferritic steels, J. Nucl. Mater. 179e181 (1) (1991) 429e433. [7] X. Jia, Y. Dai, Small punch tests on martensitic/ferritic steels F82H, T91 and Optimax-A irradiated in SINQ Target-3, J. Nucl. Mater. 323 (2) (2003) 360e367. [8] M.F. Moreno, G. Bertolino, A. Yawny, The significance of specimen displacement definition on the mechanical properties derived from Small Punch Test, Mater. Des. 95 (2016) 623e631. [9] Y.Q. Peng, et al., A new method based on energy principle to predict uniaxial stress-strain relations of ductile materials by small punch testing, Int. J. Mech. Sci. 138 (2018) 244e249. [10] J.C. Chica, P.M.B. Diez, M.P. Calzada, Development of an improved prediction method for the yield strength of steel alloys in the Small Punch Test, Mater. Des. 148 (2018) 153e166. [11] M. Bruchhausen, et al., Recent developments in small punch testing: tensile properties and DBTT, Theor. Appl. Fract. Mech. 86 (2016) 2e10. [12] E.N. Campitelli, et al., Assessment of irradiation-hardening on Eurofer97’ and Zircaloy 2 with punch tests and finite-element modeling, Mater. Sci. Eng. 400 (1) (2005) 386e392. [13] E.N. Campitelli, et al., Assessment of the constitutive properties from small ball punch test: experiment and modeling, J. Nucl. Mater. 335 (3) (2004) 366e378. [14] T.E. García, et al., Estimation of the mechanical properties of metallic materials by means of the small punch test, J. Alloys Compd. 582 (2) (2014) 708e717. [15] E. Altstadt, et al., Critical evaluation of the small punch test as a screening procedure for mechanical properties, J. Nucl. Mater. 472 (2016) 186e195. [16] M. Abendroth, M. Kuna, Determination of deformation and failure properties of ductile materials by means of the small punch test and neural networks, Comput. Mater. Sci. 28 (3) (2003) 633e644. [17] M. Abendroth, M. Kuna, Identification of ductile damage and fracture parameters from the small punch test using neural networks, Eng. Fract. Mech. 73 (6) (2006) 710e725. [18] T. Linse, et al., Usage of the small-punch-test for the characterisation of reactor vessel steels in the brittleeductile transition region, Eng. Fract. Mech. 75 (11) (2008) 3520e3533. [19] T. Linse, M. Kuna, H.W. Viehrig, Quantification of brittle-ductile failure
[20]
[21]
[22]
[23] [24]
[25] [26]
[27]
[28] [29] [30]
[31] [32] [33]
[34]
[35]
behavior of ferritic reactor pressure vessel steels using the Small-Punch-Test and micromechanical damage models, Mater. Sci. Eng. 614 (4) (2014) 136e147. I.I. Cuesta, J.M. Alegre, R. Lacalle, Determination of the GursoneTvergaard damage model parameters for simulating small punch tests, Fatig. Fract. Eng. Mater. Struct. 33 (11) (2010) 703e713. Y.T. Yang, et al., A correlation between micro- and nano-indentation on materials irradiated by high-energy heavy ions, J. Nucl. Mater. 498 (2018) 129e136. Y.T. Yang, et al., Damage and recovery behavior of 4H-SiC implanted with He ions, Nucl. Instrum. Methods Phys. Res. Sect. B Beam Interact. Mater. Atoms 449 (2019) 54e57. Z.N. Ding, et al., Hardening of ODS ferritic steels under irradiation with highenergy heavy ions, J. Nucl. Mater. 493 (2017) 53e61. A. Prasitthipayong, et al., Micro mechanical testing of candidate structural alloys for Gen-IV nuclear reactors, Nuclear Materials & Energy 16 (2018) 34e45. C.H. Zhang, et al., Irradiation response of ODS ferritic steels to high-energy Ne ions at HIRFL, J. Nucl. Mater. 455 (1e3) (2014) 61e67. S. Haroush, et al., Evaluation of the mechanical properties of SS-316L thin foils by small punch testing and finite element analysis, Mater. Des. 83 (2015) 75e84. E. Priel, et al., Estimation of yield and ultimate stress using the Small Punch Test method applied to non-standard specimens: a computational study validated by experiments, Int. J. Mech. Sci. 135 (2017). S0020740317314765. J.F. Ziegler, J.P. Biersack, The Stopping and Range of Ions in Matter, 1985. K. Matocha, R. Hurst, Small punch testing - the transition from a code of practice to a European testing standard, Key Eng. Mater. 734 (2017) 3e22. V. Karthik, et al., Small specimen test techniques for evaluation of tensile flow properties - evolution and developmental activities at IGCAR, Kalpakkam, Nucl. Eng. Des. 330 (2018) 538e549. S. Rasche, M. Kuna, Improved small punch testing and parameter identification of ductile to brittle materials, Int. J. Pres. Ves. Pip. 125 (2) (2014) 23e34. W.D. Nix, H. Gao, Indentation size effects in crystalline materials: a law for strain gradient plasticity, J. Mech. Phys. Solid. 46 (3) (1998) 411e425. J.T. Busby, M.C. Hash, G.S. Was, The relationship between hardness and yield stress in irradiated austenitic and ferritic steels, J. Nucl. Mater. 336 (2e3) (2005) 267e278. H.R. Higgy, F.H. Hammad, Effect of fast-neutron irradiation on mechanical properties of stainless steels: AISI types 304, 316 and 347, J. Nucl. Mater. 55 (2) (1975) 177e186. D. Tabor, The physical meaning of indentation and scratch tests, Br. J. Appl. Phys. 7 (5) (1956) 159.
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Quantification of the constitutive relationship of high-energy heavyion irradiated SS316L using the small punch test Xianlong Zhang a, Chonghong Zhang a, b, *, Zhaonan Ding a, b, Yuguang Chen a, b, Liqing Zhang a a b
Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, 730000, China School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing, 100049, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 23 September 2019 Received in revised form 17 January 2020 Accepted 18 January 2020 Available online 22 January 2020
Heavy-ion irradiation has been widely used to simulate neutrons-irradiation induced damage effects, due to its higher damage rate and lower activation of the post-irradiation manipulation. The high-energy ion beam is capable of producing a thickness of dozens of microns damaged region in steel, making it possible to evaluate the macroscopic mechanical properties of the irradiated specimens. In the present paper, a method to quantify the constitutive relationships of high-energy heavy-ion irradiated steels is proposed. Ni22þ ions with a kinetic energy of 357.86 MeV provided by a cyclotron were used to produce a quasi-homogeneous atomic displacement damaged layer (about 25 mm in thickness) in specimens of 316 L stainless steel. The temperature of the specimens were kept at about 50 C during ion irradiation. Two damage levels of 0.16 and 0.33 displacement per atom (dpa) were approached. Small punch test of the unirradiated and irradiated f3 mm disk samples was carried out to obtain the load-deflection curves. A series of finite element simulation of SPT of the laminated irradiated samples, in combination with sequential programming algorithm, was performed to characterize the constitutive relationships of the irradiation damaged layer of the samples. Finite element simulations with obtained constitutive relationships show agreement with the experimental results. Nanoindentation tests were carried out to verify the identified constitutive relationships. The nanoindentation results show an irradiation induced hardening in good agreement with that from the obtained constitutive relationships. © 2020 Elsevier B.V. All rights reserved.
Keywords: Heavy-ion irradiation Steel Small punch test Finite element method Constitutive relationship
1. Introduction To ensure the safety and evaluate the lifetime of nuclear reactors, it is fundamentally essential to accurately survey the mechanical properties and model the deformation and failure behavior of the structures [1]. Irradiation hardening and embrittlement is one of the major concern for the safe service of in-pile structure components. Energetic heavy-ion beams have long been used to simulate the radiation effects (void swelling, irradiation hardening) by fast neutrons in materials [2,3], due to their merits of higher atomic displacement rate, similar primary cascade damage structure and much lower radioactivity of the irradiated specimens. Standard mechanical test methods are always destructive because
* Corresponding author. Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, 730000, China. E-mail address: [email protected] (C. Zhang). https://doi.org/10.1016/j.jnucmat.2020.152014 0022-3115/© 2020 Elsevier B.V. All rights reserved.
large specimens need to be extracted and machined from actual serving components. Therefore, it is convenient to use miniature specimens to characterize the mechanical properties. What’s more, due to the limited irradiation volume in test reactors, spallation neutron sources and high-energy ion accelerators, it is compulsory to develop miniaturized specimen techniques to evaluate the mechanical properties of the irradiated material. Small punch test (SPT) is one of the most widely used among all the small specimen test technique. Owing to the small sample size (generally 8 mm or 3 mm in diameter) of SPT, slices of surveillance component or remnants of Charpy impact test (undeformed zone) can be used [4]. A disc sample is firmly gripped between two dies and deformed vertically into a circular hole by a spherical punch, the force to punch the rod and the maximum deflection of the sample are recorded during the test [5e7]. The sample is exposed to a complicated biaxial stress state, making the test more suitable for representing the actual stress state observed in the reactor pressure vessel, rather than uniaxial stress state in conventional uniaxial tensile tests.
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Six different stages are identified in a typical SPT load-deflection curve (LDC) as depicted in Fig. 1, the regimes are: (I) elastic bending of the disk, (II) plastic bending, (III) membrane stretching, (IV) plastic instability, (V) crack initiation and (VI) fracture. The sample strength and ductility can not be obtained straightforwardly due to the complicated stress state. The LDC data is usually interpreted by semi-empirical formulations [4,5,8]. It is worthwhile to mention that precisely distinguish of the boundaries between different stages is subjective and different formulations for evaluation of the strength have been proposed in Ref. [9]. Much effort has been paid to identify material properties as yield stress and ultimate stress from the SPT experiment [10]. Moreover, the constitutive relationship of materials is necessary in modeling a complex structure of a component in order to obtain the actual stress state in the process of operation. Finite element method (FEM) simulations are dedicated to evaluating the role of different parameters on SPT results, and further extract and interpret the mechanical properties based on experimental data [11e14]. However, FEM simulation is a reverse process to obtain the constitutive relationship of the tested material. It means that a constitutive relation of the test material is necessary to in FE modeling. Characterization of material properties from a given LDC curve involves comprehensive numerical computation [15]. Artificial neutron network and non-linear optimization techniques are used to identify the mechanical parameters from the LDCs [16e19]. The penetration depth of the energetic ions in metals varies from several to tens of microns depending on the energy and species of ions. Conventional mechanical characterization methods of ion-irradiated materials include nanoindentation [21,22], microhardness [21,23], and micro compression of nano pillars [24]. Irradiation hardening data can be obtained by transverse comparison, but embrittlement information can not be obtained by the above method [23,24]. In our previous research, mechanical properties of high-energy heavy-ion irradiated samples with 60 mm in thickness were tested by SPT [25]. It was found that such thin samples were already in the membrane stretching state from the beginning of the test so that limited information about the yield load from the LDCs can be obtained [26,27]. In the present study, we propose a comprehensive method to extract constitutive relationships of the irradiation damaged layer from a laminated specimen composed by a heavy-ions irradiated
damaged layer and an unaffected substrate. Constitutive relationships of the irradiation damaged layer from the irradiated specimens are determined by combining FEM simulations and sequential programming algorithm. Finally, the results are verified by FEM simulation and nanoindentation test. The methodology combining experiment and numerical simulation is schematically presented in Fig. 2.
Fig. 1. Typical six stages of the LDC of the SPT and corresponding material parameters of different stages [20].
Fig. 2. Procedure of identification of constitutive relations of the heavy-ion irradiated materials.
2. Experimental details and numerical simulation The material used in this research is the commercial low carbon content austenitic stainless steel 316 L provided by Goodfellow Company. The nominal compositions can be found in literature [13].
2.1. Uniaxial tensile test of the unirradiated specimen A uniaxial tensile test was conducted to obtain the constitutive relationship of the material, which is necessary for FE modeling of the SPT. Sheet specimens with a width of 12.5 mm, a thickness of 0.5 mm and the length of the parallel section of 65 mm within gauge length of 150 mm was used for uniaxial tensile test, as presented in Fig. 3a. The uniaxial tensile was a standard tensile test under Chines standard GB/T 228.1e2010 and was performed at room temperature. The electro-mechanical testing machine was controlled at a constant crosshead velocity to make sure a strain rate of 0.0015 s1, which was comparable with the strain rate of the first stage of the small punch test. The elongation of the interest section was measured by a clip gauge attached to the sample. The nominal and true stress-strain curves are presented in Fig. 3b. The true stress-strain curve is fitted by modified Hollomon’s law as depicted in Equation (1),
s ¼ k1 εn þ k2 ε
(1)
where s and ε are true stress and true strain respectively k1, k2 and n are fitting parameters, k1 ¼ 345:9, k2 ¼ 1502, and n ¼ 0:06086.
X. Zhang et al. / Journal of Nuclear Materials 531 (2020) 152014
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Fig. 3. (a) Uniaxial tensile test of plate sample, and (b) the nominal (blue) and true (black) stress-strain curve and fitted (red) curve. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
2.2. High-energy Ni22þ beam irradiation experiment Two specimens of the 316 L steel with dimensions of 5 10 0.2 mm3 were subsequently mechanical ground using silicon carbide papers down to 2400 grit, followed by polishing with colloidal silica suspension with a grain size of 1mm: The specimens were used for the heavy-ion irradiation at a terminal of the Sector-focused Cyclotron (SFC) at HIRFL (Heavy-ion Research Facility in Lanzhou) at Institute of Modern Physics, Chinese Academy of Sciences. The samples were mounted on a specimen stage cooled with liquid nitrogen at the terminal, and were irradiated with 357.86 MeV Ni22þ at about 50 C to two doses of 0.16 and 0.33 displacement per atom (dpa), respectively. The flux of the ion beam was about 8:8 1010 Ni-ions/cm2/s. The samples were mounted on a specimen stage cooled with liquid nitrogen, a thermocouple was mounted on the specimen stage to monitor the temperature of the specimens.FEM simulation was carried out to calculate the temperature of the ion-beam irradiated surface of the specimen, it turn out that the ion-beam irradiated surface is about several degrees higher than the thermocouple. By using aluminum foils with ten different thickness in a rotating energy degrader, a quasi-uniform damaged layer was produced in the samples [25]. The damaged layer was approximately 25 mm in thickness. The
Fig. 4. Damage profiles of the high-dose Ni-ions irradiated steel specimen according to estimation with SRIM code. The right column shows the thickness of the aluminium foils used in the energy degrader.
profiles of the atomic displacement damage of the high-dose samples are shown in Fig. 4, according to estimation with SRIM code [28]. More details of the facility and the irradiation conditions are described in our previous papers [23,25].
2.3. Small punch test The schematic of the SPT device set-up is shown in Fig. 5. A disk sample is clamped between the upper and the lower die. The upper die diameter is 1.05 mm (a small tolerance of 0.05 mm), the lower die diameter is 1.5 mm. The apparatus is designed to use 3 mm diameter disc samples with a thickness of 0.2 mm. Disk small punch specimens of 3 mm diameter were employed instead of traditional 8 mm ones due to the limited spot area of ionirradiation and the radioactivity of the ion-irradiated specimens. The rod and the punch are driven by an MTS-E43 electro servo testing machine with a constant displacement rate of 0.02 mm/min. The force and the displacement of the punch are recorded by the machine and an LVDT attached to the punch rod respectively. Instead of a single piece punch recommended by CEN Workshop Agreement [29], a ceramic sphere ball with a radius of 0.5 mm and a rod made of low-carbon steel with a length of 40 mm and a radius of 0.5 mm are used as the punch. Besides the wear problem, the permanent plastic deformation of the punch made of steel was observed in our previous research and FEM simulation.
Fig. 5. Schematic of the small punch test configuration.
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Since the displacement measured by LVDT attached to the punch rod includes the elastic deformation of the rod and the ceramic ball during SPT, the displacement must be corrected for the system compliance by Equation (2) [13,30], where dsp is the displacement of the tip of the punch, dmeasure is the displacement measured by the LVDT, F is the load and C is the compliance of the punch system.
dsp ¼ dmeasure F C
(2)
Fig. 6 shows the experimental LDC of the unirradiated sample from SPT, a standard LDC with 6 stages is obtained as 8 mmdiameter sample in the literature [20]. The compliance corrected curve agrees well with the FEM simulation result up to a displacement of 0.5 mm, exhibiting the combination of the elastic bending and the plastic deformation (stages I and II, as depicted in Fig. 1). The first two stages containing all the elastic and plastic information (characterized by the elastic modulus E, Poisson ration n, strength coefficient K and strain hardening exponent n) of the test material up to 0.2 mm in the LDC as shown in Fig. 6. While the rest of the curve is affected by friction coefficient m, initial void volume fraction f0 , void nucleation fN , void growth fc , coalescence and crack initiation fF [13,16], which is beyond the scope of this paper. Under the hypothesis of irradiation hardening, FE modeling examples show that the upward position of the irradiation damaged layer during SPT demonstrates more distinguishable LDC from the unirradiated specimen than that with the irradiation layer downward (not shown here). The heavy-ion irradiated specimens were stamped into 3 mm diameter SPT disk samples. All the irradiated samples were tested with the irradiation layer upward at room temperature. 2.4. Nanoindentation test
material as a reference with the same control mode and penetration depth. 2.5. Finite element modeling of SPT Finite element modeling of the SPT was developed using ABAQUS 6.14e2 standard code. Because of the specimen geometry and SPT setup, axisymmetric simplification was used as described in Ref. [13]. The upper die, lower die, and the punch were assumed to be rigid bodies, the friction coefficient was set to m ¼ 0:2 between the specimen and the dies/punch[13,20]. The unirradiated specimen was assumed as isotropic and meshed with 100 20 uniform CAX8R reduced integration elements, the number of elements was selected by mesh converge analysis to be sufficient to get the same LDCs as a finer mesh. While the irradiated specimen was assumed as laminar composites with an undetermined layer thickness of 25mm and an unirradiated layer thickness of 175 mm. The schematic of FE modeling of the irradiated specimen is shown in Fig. 7. The maximum effective strain in SPT is higher than the ultimate strain in the uniaxial tensile test due to different constraint conditions and stress state. Extrapolation of the input stress-strain data results in a better agreement of the experiment with FEM results [8,30]. The maximum strain of the specimen was set to 0.8 in all the FE modeling in this paper. The elastic modulus of the unirradiated material is set to 205 GPa according to the uniaxial tensile test. The nanoindentation data indicates that the elastic modulus of the alloy is not affected by ion irradiation, as depicted in Fig. 8. For the irradiated portion of the irradiated sample, assumed parameters for constitutive relationship are list in Table 1. All parameters combined into 13 10 10 ¼ 1300 stress-strain curves, while the material properties of the unirradiated portion remain constant. 2.6. Identification procedure
Nanoindentation test was carried out using a Berkovich tip indenter under depth control mode using a G200 nanoindenter. The continuous stiffness measurement (CSM) method was used to obtain the depth profile of hardness and elastic modules up to a maximum displacement of 2000 nm. Arrays consisting of 9 indentations were conducted in both the irradiated and unirradiated specimens. The nanoindenter tip was calibrated by a fused silica
As described in section 2.5, 1300 constructed constitutive relationships were assigned to the irradiated portion of the irradiated specimen in turn. The FEM simulations of the irradiated specimen SPT were implemented by Python script running in ABAQUS. After a series of FEM simulation, 1300 LDCs were obtained by the Python code through ABAQUS and each LDC corresponds to one of the
Fig. 6. Experimental LDC before and after compliance correction, in comparison with LDC obtained from FEM simulation.
Fig. 7. Axisymmetric FEM model for an irradiated specimen, the top layer is the irradiated portion and the rest of the sample is the unirradiated portion.
X. Zhang et al. / Journal of Nuclear Materials 531 (2020) 152014
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SPT experimental results of the irradiated samples and the FEM simulation database. Stages (I) and (II) of the LDC (d < 0.2 mm) were used here for the identification of the constitutive relationship of the irradiated material, due to fact that the first two stages contain the elastic-plastic information of the tested material as described in section 2.3. The identification procedure is carried out by minimizing the quadratic sum of the least square error measure (e) between the experimental results and the curves in the FEM calculated database by Equation (3) [31], e is a target function which measures the least squares errors between the experimental results and the curves in the FEM calculated database.
e¼
50 Fsim ðui ; pÞ Fexp ðui Þ 2 1 X 50 i¼1 Fexp ðui Þ
(3)
3. Results Fig. 8. Nanoindentation result of elastic modulus as a function of indentation depth for both the unirradiated and irradiated specimens.
Table 1 Constructed parameters for constitutive relation for irradiated portion of sample. Parameter
min
max
steps
k1 k2 n
0.8 345.9 0.8 1502 0.8 0.6086
2.0 345.9 1.7 1502 1.7 0.6086
13 10 10
input constitutive relationships, and the LDCs were read and processed by MATLAB. The ABAQUS input stress-plastic strain and the calculated LDCs curves are shown in Fig. 9a. To build the data base, the calculated LDCs are interpolated at certain displacement locations. The selected 50 location intervals increase gradually in the whole interest zones. The interpolated load information is stored in 50 1300 matrix Fsim ðu; pÞ and the SPT experimental results of the irradiated samples are stored in 50 2 matrix Fexp ðuÞ. M. Abendroth and M. Kuna [16] found that a direct approximation of the SPT experimental results with the FEM simulation data base leads to better results compared to artificial neural network approximation. The parameters can be obtained by an optimization procedure to minimize the difference between the
3.1. Parameter indentation According to the optimization path described in section 2.6, the minimum e of the irradiated samples of different irradiation doses are found through Equation (3) by MATLAB code, the corresponding material parameters for the irradiation portion of the samples are identified. The identified parameters and corresponding yield stresses are listed in Table 2, and the stress-plastic strain curves of the identified irradiated materials of the two irradiation doses are shown in Fig. 10a. In order to verify the results, the obtained constitutive relationship was used to simulate the SPT experiment by FEM simulation. The results show good agreement with the SPT experiment curves, as shown in Fig. 10b.
Table 2 Constitutive relationship parameters identified from identification procedure. unirr: Dsy ¼ sirr: y sy
Irradiated condition
k1
k2
n
sy (MPa)
Dsy (MPa)
unirradiated 0.16dpa 0.33dpa
345.9 657.2 506.4
1502 1351.6 1448.7
0.06086 0.0548 0.0520
262 408 486
_ 146 224
Fig. 9. The 1300 constructed stress-plastic strain curves for irradiated portion of the irradiated specimens (a), and their corresponding LDCs calculated from FEM simulation (b).
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(a)
(b)
Fig. 10. The stress-plastic strain curves of the irradiated (0.16dpa, 0.33dpa) and unirradiated SS316L (a), and their corresponding LDCs calculated from FEM simulation compared with experimental results (b).
3.2. Hardness from nanoindentation test
4. Discussion
The average nano-hardness as a function of indentation depth of the unirradiated and the irradiated specimens are presented in Fig. 11a. Average of 9 indentation curves for the hardness of the alloys before and after irradiation are shown, with the error bars representing the standard deviation. Significant increase in hardness is observed in the irradiated specimens compared with unirradiated material. According to the hardness profiles, the results strongly demonstrate the indentation size effect. The indentation size effect can be characterized by Nix-Gao model described in Equation (4) [32],
Campitelli [12] found that more than one constitutive relationship may construct the same LDC of SPT by FE modeling. The constitutive relationships which construct the same LDC differ in yield stress and in the range of small strain (<0.05), while are close to each other in the range of larger strain. Therefore, the stressplastic strain curve can be treated as two segments: a linear portion in the range of 0e0.5 and a non-linear strain hardening portion. For the possible constitutive relationships construct the same LDC, the second segments agree while the first segments vary. Once the interception of the constitutive relationships (yield stress of the material) is determined, the constitutive relationship of the material will be identified. Therefore, in order to prove the uniqueness of the results obtained by the proposed reverse method based on SPT and optimization procedure, it is necessary to verify the accuracy of the yield stress of the irradiated material. Therefore nanoindentation tests were employed. According to our previous research, there is a linear relation between Vickers hardness and nanohardness of various polycrystalline (fcc, bcc) materials before and after irradiation [21]. The Berkovich hardness (Hb ) obtained from nanoindentation can be converted to Vickers hardness (Hv ) by Equation (5).
H ¼ Hb
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi h* 1þ h
(4)
where H is the hardness profile with respect to depth h obtained in the nanoindentation test, Hb is the hardness of infinite depth and h* is the characteristic length of the experiment setup. Hb and h* can be obtained by fitting the curve H 2 versus h1 . The least-square fitting of H 2 versus h1 is used to extract Hb and h* for both the unirradiated and irradiated specimens are plotted in Fig. 11b. The nanohardness of the unirradiated, 0.16dpa, and 0.33dpa steel are 2.35 GPa, 2.90 GPa and 3.2 GPa respectively.
(a)
(b)
Fig. 11. Average indentation hardness profiles of the steel before and after irradiation (a), H2 versus h1 plots of the alloys before and after irradiation (b).
X. Zhang et al. / Journal of Nuclear Materials 531 (2020) 152014
Hv ¼ 81:0Hb
7
(5)
The Vickers hardness of 316 L stainless steel before and after irradiation obtained from nanoindentation are tabulated in Table 3. A linear relationship was previously found between the increase in yield stress Dsy and the increase in Vickers hardness DHv in Ref. [33], with a coefficient ranging from 2.7 to 3.6. For the purpose of comparing the results between nanoindentation and SPT, the yield stress and corresponding Vickers hardness of unirradiated samples are used as baseline values for determining the ionirradiated induced hardening. In addition, low temperature ( < 100 C) fast neutron irradiated 316SS tested at room temperature from literatures [33,34] are employed to compare with our results due to similar irradiation doses and test conditions. The change in yield stress characterized by SPT is plotted as a function of corresponding the change in Vickers hardness converted from nanoindentation test as depicted in Fig. 12. A least-square linear fit was applied, the results show a good linear relationship between Dsy and DHv . The correlation Dsy ¼ 3:355 DHv is observed by linear regression with R2 ¼ 0.98. The coefficient is close to the theoretical value of 3.33 MPa/kg/mm2 [35]. The correlation factor of 316SS for the data set used in the literature [34] is 3.43 with R2 ¼ 0.98, the data points from our research show good agreement with linear regression line in Fig. 12. It thus proves the validity of the yield stress as well as the constitutive relationship obtained from the proposed method. The equation Hv ¼ 81:0Hb (Hv with the unit of kgf/mm2 and Hb with the unit of GPa) was employed to establish the correlation between the Berkovich hardness and Vickers hardness the in our manuscript based on our previous research [21], while the theoretical predicted value is 94.5 [35]. The difference between the experimental and theoretical value of the correlation coefficient is probably from the influence of pile up or sink in effect during hardness measurement [21]. If the theoretical value (94.5) of the coefficient was chosen, the correlation coefficient K in the formula Dsy ¼ K DHv changed from 3.355 to 3.259 and hence still kept around the theoretical value (data not shown here in detail). The effect on the outcome is not significant.
5. Conclusions Small punch test combined with a series of finite element simulations and sequential programming algorithm is proposed to characterize the constitutive relationship of high-energy heavy-ion irradiated steels. Finite element simulations utilizing the constitutive relationships obtained by the proposed method assigned to the damaged layer show good agreement with the load-displacement curves of the irradiated samples. Nanoindentation tests were carried out to verify the identified constitutive relationships. The nanoindentation results show irradiation induced hardening in agreement with that from the obtained constitutive relationships. Besides the ion irradiation steels, the proposed method could be used to characterize the constitutive relationships of the laminar composite such as thin coating material and functionally graded material free from the sample size restrictions. Furthermore, the
Table 3 The Vickers hardness obtained from nanoindentation test for the unirradiated and irradiated material. Irradiated condition
Hb (GPa)
Hv (kg/mm2)
DHv (kg/mm2)
Unirradiated 0.16dpa 0.33dpa
2.35 2.91 3.19
190.35 235.71 258.39
_ 45.36 68.04
Fig. 12. The irradiation induced increase in yield stresses obtained from SPT are plotted as a function of corresponding Vickers hardness converted from nanoindentation test. The dashed line is linear regression data of our research.
obtained constitutive relationship of the irradiated layer of the ionirradiated specimen is necessary to calculate the fracture energy of the SPT in order to find out the DBTT in further research. Data statement All data and Python script included in this study are available upon request by contacting the corresponding author. Declaration of competing interest The authors declare no conflict of interest. CRediT authorship contribution statement Xianlong Zhang: Formal analysis, Writing - original draft. Chonghong Zhang: Funding acquisition, Conceptualization, Project administration. Zhaonan Ding: Methodology. Yuguang Chen: Software. Liqing Zhang: Supervision. Acknowledgments This study is sponsored by the National Key Research and Development Program of China (Grant No.2017YFB0702202) and the National Science Foundation of China (Grant No.U1532262). The authors gratefully acknowledge the operation team of HIRFL for their help in the ion irradiation experiments, and Dr Kai Nie for inspiring discussion. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.jnucmat.2020.152014. References [1] Yvon, Carre, Structural materials challenges for advanced reactor systems, J. Nucl. Mater. 385 (2) (2009) 217e222. [2] R.S. Nelson, D.J. Mazey, J.A. Hudson, The use of ion accelerators to simulate fast neutron-induced voidage in metals, J. Nucl. Mater. 37 (1) (1970) 1e12. [3] N.H. Packan, K. Farrell, J.O. Stiegler, Correlation of neutron and heavy-ion damage * : I. The influence of dose rate and injected helium on swelling in pure nickel, J. Nucl. Mater. 78 (1) (1978) 143e155.
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X. Zhang et al. / Journal of Nuclear Materials 531 (2020) 152014
[4] M.F. Moreno, Effects of thickness specimen on the evaluation of relationship between tensile properties and small punch testing parameters in metallic materials, Mater. Des. 157 (2018) 512e522. [5] X. Mao, H. Takahashi, Development of a further-miniaturized specimen of 3 mm diameter for tem disk (ø 3 mm) small punch tests, J. Nucl. Mater. 150 (1) (1987) 42e52. [6] J. Mcnaney, G.E. Lucas, G.R. Odette, Application of ball punch tests to evaluating fracture mode transition in ferritic steels, J. Nucl. Mater. 179e181 (1) (1991) 429e433. [7] X. Jia, Y. Dai, Small punch tests on martensitic/ferritic steels F82H, T91 and Optimax-A irradiated in SINQ Target-3, J. Nucl. Mater. 323 (2) (2003) 360e367. [8] M.F. Moreno, G. Bertolino, A. Yawny, The significance of specimen displacement definition on the mechanical properties derived from Small Punch Test, Mater. Des. 95 (2016) 623e631. [9] Y.Q. Peng, et al., A new method based on energy principle to predict uniaxial stress-strain relations of ductile materials by small punch testing, Int. J. Mech. Sci. 138 (2018) 244e249. [10] J.C. Chica, P.M.B. Diez, M.P. Calzada, Development of an improved prediction method for the yield strength of steel alloys in the Small Punch Test, Mater. Des. 148 (2018) 153e166. [11] M. Bruchhausen, et al., Recent developments in small punch testing: tensile properties and DBTT, Theor. Appl. Fract. Mech. 86 (2016) 2e10. [12] E.N. Campitelli, et al., Assessment of irradiation-hardening on Eurofer97’ and Zircaloy 2 with punch tests and finite-element modeling, Mater. Sci. Eng. 400 (1) (2005) 386e392. [13] E.N. Campitelli, et al., Assessment of the constitutive properties from small ball punch test: experiment and modeling, J. Nucl. Mater. 335 (3) (2004) 366e378. [14] T.E. García, et al., Estimation of the mechanical properties of metallic materials by means of the small punch test, J. Alloys Compd. 582 (2) (2014) 708e717. [15] E. Altstadt, et al., Critical evaluation of the small punch test as a screening procedure for mechanical properties, J. Nucl. Mater. 472 (2016) 186e195. [16] M. Abendroth, M. Kuna, Determination of deformation and failure properties of ductile materials by means of the small punch test and neural networks, Comput. Mater. Sci. 28 (3) (2003) 633e644. [17] M. Abendroth, M. Kuna, Identification of ductile damage and fracture parameters from the small punch test using neural networks, Eng. Fract. Mech. 73 (6) (2006) 710e725. [18] T. Linse, et al., Usage of the small-punch-test for the characterisation of reactor vessel steels in the brittleeductile transition region, Eng. Fract. Mech. 75 (11) (2008) 3520e3533. [19] T. Linse, M. Kuna, H.W. Viehrig, Quantification of brittle-ductile failure
[20]
[21]
[22]
[23] [24]
[25] [26]
[27]
[28] [29] [30]
[31] [32] [33]
[34]
[35]
behavior of ferritic reactor pressure vessel steels using the Small-Punch-Test and micromechanical damage models, Mater. Sci. Eng. 614 (4) (2014) 136e147. I.I. Cuesta, J.M. Alegre, R. Lacalle, Determination of the GursoneTvergaard damage model parameters for simulating small punch tests, Fatig. Fract. Eng. Mater. Struct. 33 (11) (2010) 703e713. Y.T. Yang, et al., A correlation between micro- and nano-indentation on materials irradiated by high-energy heavy ions, J. Nucl. Mater. 498 (2018) 129e136. Y.T. Yang, et al., Damage and recovery behavior of 4H-SiC implanted with He ions, Nucl. Instrum. Methods Phys. Res. Sect. B Beam Interact. Mater. Atoms 449 (2019) 54e57. Z.N. Ding, et al., Hardening of ODS ferritic steels under irradiation with highenergy heavy ions, J. Nucl. Mater. 493 (2017) 53e61. A. Prasitthipayong, et al., Micro mechanical testing of candidate structural alloys for Gen-IV nuclear reactors, Nuclear Materials & Energy 16 (2018) 34e45. C.H. Zhang, et al., Irradiation response of ODS ferritic steels to high-energy Ne ions at HIRFL, J. Nucl. Mater. 455 (1e3) (2014) 61e67. S. Haroush, et al., Evaluation of the mechanical properties of SS-316L thin foils by small punch testing and finite element analysis, Mater. Des. 83 (2015) 75e84. E. Priel, et al., Estimation of yield and ultimate stress using the Small Punch Test method applied to non-standard specimens: a computational study validated by experiments, Int. J. Mech. Sci. 135 (2017). S0020740317314765. J.F. Ziegler, J.P. Biersack, The Stopping and Range of Ions in Matter, 1985. K. Matocha, R. Hurst, Small punch testing - the transition from a code of practice to a European testing standard, Key Eng. Mater. 734 (2017) 3e22. V. Karthik, et al., Small specimen test techniques for evaluation of tensile flow properties - evolution and developmental activities at IGCAR, Kalpakkam, Nucl. Eng. Des. 330 (2018) 538e549. S. Rasche, M. Kuna, Improved small punch testing and parameter identification of ductile to brittle materials, Int. J. Pres. Ves. Pip. 125 (2) (2014) 23e34. W.D. Nix, H. Gao, Indentation size effects in crystalline materials: a law for strain gradient plasticity, J. Mech. Phys. Solid. 46 (3) (1998) 411e425. J.T. Busby, M.C. Hash, G.S. Was, The relationship between hardness and yield stress in irradiated austenitic and ferritic steels, J. Nucl. Mater. 336 (2e3) (2005) 267e278. H.R. Higgy, F.H. Hammad, Effect of fast-neutron irradiation on mechanical properties of stainless steels: AISI types 304, 316 and 347, J. Nucl. Mater. 55 (2) (1975) 177e186. D. Tabor, The physical meaning of indentation and scratch tests, Br. J. Appl. Phys. 7 (5) (1956) 159.