Nuclear Engineering and Technology

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Original Article

Study of the mechanical properties and effects of particles for oxide dispersion strengthened Zircaloy-4 via a 3D representative volume element model

Dong-Hyun Kim

^a

, Jong-Dae Hong

^a

, Hyochan Kim

^a^,^*

, Jaeyong Kim

^a

, Hak-Sung Kim

^b^,^c

aAdvanced 3D Printing Technology Development Division, Korea Atomic Energy Research Institute, 989-111, Daedeok-Daero, Yuseong-gu, Daejeon, 34057, Republic of Korea

bDepartment of Mechanical Engineering, Hanyang University, 222, Wangsimni-ro, Seongdong-gu, Seoul, 04763, Republic of Korea

cInstitute of Nano Science and Technology, Hanyang University, 222, Wangsimni-ro, Seongdong-gu, Seoul, 04763, Republic of Korea

a r t i c l e i n f o

Article history:

Received 28 June 2021 Received in revised form 20 October 2021 Accepted 31 October 2021 Available online 2 November 2021

Keywords:

Accident tolerant fuel

Oxide dispersion strengthened Zircaloy-4 cladding

3D representative volume element model Mechanical properties

Cladding deformation

a b s t r a c t

As an accident tolerant fuel (ATF) concept, oxide dispersion strengthened Zircaloy-4 (ODS Zry-4) cladding has been developed to enhance the mechanical properties of cladding using laser processing technology.

In this study, a simulation technique was established to investigate the mechanical properties and effects of Y2O3particles for the ODS Zry-4. A 3D representative volume element (RVE) model was developed considering the parameters of the size, shape, distribution and volume fraction (VF) of the Y2O3particles.

From the 3D RVE model, the Young's modulus, coefficient of thermal expansion (CTE) and creep strain rate of the ODS Zry-4 were effectively calculated. It was observed that the VF of Y2O3particles had a significant effect on the aforementioned mechanical properties. In addition, the predicted properties of ODS Zry-4 were applied to a simulation model to investigate cladding deformation under a transient condition. The ODS Zry-4 cladding showed better performance, such as a delay in large deformation compared to Zry-4 cladding, which was also found experimentally. Accordingly, it is expected that the simulation approach developed here can be efficiently employed to predict more properties and to provide useful information with which to improve ODS Zry-4.

1. Introduction

After the Fukushima nuclear accident in 2011, a development of ATF cladding became a hot topic for enhanced accident tolerance of current light water reactor systems while maintaining or improving their performance during normal operation. Worldwide, various concepts of ATF cladding have been proposed [1e4]. At the Korea Atomic Energy Research Institute, an oxide dispersion strengthened Zircaloy-4 (ODS Zry-4) cladding is under development using laser processing technology for the purpose of increasing the strength of Zry-4 cladding [5e7]. When dispersing Y₂O₃particles on the surface of Zry-4 cladding, an ODS layer is formed, which can improve ballooning and rupture resistance under severe accident conditions. From several experiments, it was conﬁrmed that Y₂O₃ particles contribute to a considerable enhancement in the strength

as well as the burst resistance of the cladding while maintaining a proper volume fraction (VF) and distribution of the Y2O3particles.

However, detailed investigations of these effects of Y2O3particles on the mechanical properties was not be conducted because an experimental approach for such an investigation is costly and time- consuming despite the fact that a comprehensive study is required to improve the properties of ODS Zry-4. In addition, in research related to nuclear fuel codes, the high-precision properties of ODS Zry-4 are required for accurate predictions of in- and out-of-reactor performance of ODS Zry-4 [8e10]. This explains why an efﬁcient means by which to understand the mechanical properties and effect of Y2O3particles is needed.

ODS Zry-4 is composed of the Zry-4 matrix and Y2O3particles.

This structure is a typical feature of reinforced composite, where the particles of one material are dispersed in the matrix of a second material [11]. To achieve desirable properties, quality levels and performance capabilities of the composites, the parameters of the particles, such as the size, shape, distribution, and the VF are mainly considered as important factors during the fabrication process

*Corresponding author.

E-mail address:[email protected](H. Kim).

Contents lists available atScienceDirect

Nuclear Engineering and Technology

j o u r n a l h o m e p a g e :w w w . e l s e v i e r . c o m / l o c a t e / n e t

https://doi.org/10.1016/j.net.2021.10.044

Nuclear Engineering and Technology 54 (2022) 1549e1559

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[12,13]. For an efficient understanding of the effects of these parameters, computer-simulation-based techniques have been developed [14,15]. A representative volume element (RVE) model, one of the simulation modelling techniques based on the finite element method, has the advantage of being able to characterize large and complex microstructures of composites by assembling many such small elements periodically. By developing a three dimensional (3D) RVE model that reflects the characteristics of the particle parameters, various studies have been successfully conducted to investigate the mechanical properties of composites.

Some of these are outlined below.

Wang et al. analyzed effects of the particle shape and alignment on the elastic properties of a composite using a 3D RVE model [16].

From elastic simulations, it was conﬁrmed that the particle shapes have a strong effect on the elastic properties; a cylinder shape was the most effective reinforcement, followed by ellipsoid, disc and sphere shapes in that order. With regard to alignment, the stiffness increased when the particles were well aligned along the loading axis, except with a disc shape of the particles, which showed opposite results. Ma et al. investigated effects of the particle size and VF on the elastic modulus, yield strength and elongation of a composite using the 3D RVE model [17]. It was found that a decrease of the particle size improved the yield strength and elongation, while an increase of the particle volume reduced the elongation considerably. An increase in the particle VF contributes to an increase of the Young's modulus, in good agreement with results of a well-known analytical solution referred to as the Halpin-Tsai equations. Barbera et al. studied the creep-fatigue behavior of a composite by applying a linear matching method to a 2D RVE model [18]. The time-dependent creep deformation of the composite was calculated, and the interactions between fatigue and creep damage under a constant load and at a cyclic temperature were efﬁciently investigated.

Here, the mechanical properties and effects of the particles on the aforementioned properties were investigated by establishing a 3D RVE model for ODS Zry-4. In addition, the predicted properties were applied to a simulation model of cladding deformation under a transient condition, followed by a comparison with an experimental case.

2. Development of the 3D RVE model for ODS Zry-4 2.1. Modelling area and parameters

Before developing the 3D RVE model, the modelling area and parameters for the ODS Zry-4 were deﬁned based on the microstructure of fabricated ODS Zry-4 cladding. Fig. 1shows a cross- sectional SEM image of the ODS Zry-4 cladding [19]. There is an apparent distribution of the Y₂O₃particles on the Zry-4 matrix on

the surface of the cladding, referred to as the ODS layer. The ODS layer was considered as the modelling area in this study because the remaining area of the cladding except the ODS layer has no Y2O3

particles. In the ODS layer, the distribution of the particles has generally heterogeneous characteristics. Specific pattern distribution of the particles as well as decrease in Y2O3particle volume fraction with thickness were not observed. The heterogeneous distribution caused variations in volume fraction of Y₂O₃particles inside the ODS layer. Then, the volume fractions were measured in several regions with a measurement dimension of 55m^{m inside} the ODS layer. The results were in range of 5e15%, which was considered as modelling parameter of the volume fraction. The shape of the Y2O3particles was confirmed as spherical. The average diameter of the Y₂O₃particles was 200 nm. Based on these results, the all modelling parameters for the Y2O3 particles could be established. In terms of the interfacial property between the Zry-4 matrix and the Y2O3particles, the perfect bonding assumption was employed based on high-magnification and high-temperature ob- servations of the ODS Zry-4 cladding [20]. In these results, visible interfacial defects such as cracks and pores were not observed from room temperature to 1000C.Table 1summarizes the modelling parameters of ODS Zry-4 for the development of the 3D RVE model.

2.2. Construction of the 3D RVE model

To construct the 3D RVE model containing numerous particles with a heterogeneous distribution efﬁciently, a python script modelling technique, accompanied with the commercial ﬁnite element simulation software ABAQUS [21], was employed, as shown inFig. 2. As the input data, the edge length of a cube for the Zry-4 matrix, the diameter of the sphere for the Y2O3particles, and the target VF of the Y₂O₃particles were given. As each dimension of the cube and sphere, 1.6mm and 200 nm were employed, respectively. Based on the given dimensions, a cube instance was initially produced, followed by the production of a sphere instance located inside the cube instance. Then, a random coordinate that allows a new sphere instance to be located inside the cube instance was generated for the heterogeneous distribution of the Y2O3particles.

The new sphere instance was produced and translated into the

Fig. 1.Cross sectional image of ODS Zry-4 cladding [19].

Table 1

Modelling parameters of ODS Zry-4 for 3D RVE model.

Parameters Values

Particle distribution Heterogeneous

Particle shape Spherical

Particle diameter 200 nm

Particle VF 5e15%

Interfacial property between particle and matrix Perfect bonding

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Fig. 2.Algorithm for construction of 3D RVE model.

D.-H. Kim, J.-D. Hong, H. Kim et al. Nuclear Engineering and Technology 54 (2022) 1549e1559

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generated coordinate. Next, the distance between the existing sphere instances was calculated to check for collisions between the sphere instances. In the event of a collision, random coordinates of a new sphere instance are regenerated. When no collision has occurred, the total VF of the sphere instances in the cube instance is calculated. Until the calculated VF reaches the given target VF, the process described above is repeated. Then, all sphere and cube instances are merged as a single instance, which is then meshed by a tetra element. Finally, periodic boundary conditions (PBCs) developed by Xia et al. [22] are applied to the meshed instance. The PBCs are set of boundary conditions that can simulate a large or bulk system simply with a 3D RVE model. The PBCs are expressed as follows:

u^jþ_i ðx;y;zÞ u^j_i ðx;y;zÞ ¼ c^j_i ði; j¼1;2;3Þ (1)

where u^jþ_i and u^j_i represent the displacements for one pair of nodes at the opposite boundary surface;iis the normal direction in this casex,y, andz, to three pairs of the boundary surface;jdenotes the three directions of the nodal displacement; andc^j_irepresents the difference in the nodal displacement.

From the process outlined above, various 3D RVE models could be effectively obtained. Finally, three 3D RVE models for ODS Zry-4 were established with different VFs of the particles, in this case 5, 10, and 15% based on the modelling parameters, as shown inFig. 3.

2.3. Mechanical simulations

From the mechanical simulations in this section, the Young's modulus, coefﬁcient of thermal expansion (CTE) and creep strain rate for the ODS Zry-4 with respect to the VF of the Y₂O₃particles were predicted, after which the relationship between the

properties of the Zry-4 and ODS Zry-4 were investigated. Three types of mechanical simulations were conducted, as follows. First, uniaxial force in the x direction (Fx) was applied to the opposite surfaces of the 3D RVE model to determine the Young's modulus.

While the 3D RVE model was under uniaxial loading, extra degrees of freedom, indicated here asyandz, were free from constraints;

the work done (W) by the force can be expressed using the following equation:

W¼1

2Fxε⁰_x; (2)

whereε⁰xis the average strain in the x direction.

The strain energy stored in the 3D RVE model can be expressed in terms of the average stress (s⁰x) and strain via the following equation:

Strain energy¼1 2 ð

V

s

⁰xε⁰_xdV¼1

2 V

s

⁰xε⁰_x (3)

Here,Vis the volume of the 3D RVE model.

From the relationship between the work and strain energy, the Young's modulus (E) can be obtained with the equation below.

E¼

s

⁰x

ε⁰_x ¼ Fx

Vε⁰_x (4)

Next, the CTE can be obtained by applying the temperature difference to the 3D RVE model, as follows:

a

¼ε⁰_x

D

^T ⁽⁵⁾

Fig. 3.3D RVE models with different VFs of 5, 10, and 15%.

Table 2

Material properties of 3D RVE model for mechanical simulation.

Parameters Zry-4 [Reference] Y2O3[Reference]

Elastic modulus (GPa) 91.32 [23] 157.0 [24]

Poisson's ratio 0.37 [23] 0.29 [24]

CTE (m^/K) ^{6.0 [23]} ^{8.1 [25]}

Norton creep constants - A (MPa^-ns¹) - Q (J/mol) - n ()

A¼8736.81

Q¼321,000þ24.67$(Temp - 923.15K) n¼5.89 [26,27]

A¼Q¼n¼0

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Fig. 4.(a) Displacement results of 3D RVE models with VF of 5, 10, and 15% and (b) calculated Young's modulus from 3D RVE and analytical models.

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Finally, the creep strain can be calculated by applying both the constant uniaxial force and temperature to the 3D RVE model during the desired simulation time. As the temperature condition, 1073.15 K (800C) was used in this study because a high temperature creep property has a signiﬁcant effect on the cladding deformation in the event of a severe accident.

Table 2 shows the material properties of the Zry-4 and Y2O3

particles utilized for the 3D RVE model to conduct the mechanical simulations described above [23e25]. The reference temperature of the material properties is 293.15 K. For efﬁciency of calculation, isotropic structure assumption was employed although the Zry-4, which has a hexagonal close-packed crystal structure in an alpha phase at the room temperature, has anisotropic characteristics in the real case. Speciﬁcally, a Norton creep law, which has been adopted to describe the high-temperature behavior of Zry-4 in various studies [26,27], was used here to express the creep behavior of the Zry-4 in the alpha phase, as follows:

ε_¼A$exp

Q RT

s

ⁿ; (6)

wheresis the effective stress (MPa),Tis the temperature (K),Ris the ideal gas constant of 8.314 (Jmol¹K¹),Ais the strength co- efﬁcient (MPa^-ns¹),Qis the activation energy (Jmol¹), andnis the stress exponent.

According to the literatures [28e30], temperature range of the alpha phase was considered below about 815C for the high temperature creep of the Zry-4. Based on the above, it was considered that 800C for the creep simulation was appropriate to investigate high temperature creep strain of the Zry-4 alpha phase.

In terms of the creep properties for the Y2O3particles, it was assumed that there was no creep deformation of the Y2O3below the 1073.15 K (800C) based on the following facts because many literatures on creep properties of Y₂O₃have covered high temperature ranges of over 1323.15 K (1050 C) [31,32]. It has been reported that the creep deformation of heat resistant ceramic materials such as the Y₂O₃becomes noticeable at 40e50% of the melting temperature [33]. Melting temperature of the Y2O3 is 2683.15 K (2410C) [34,35], and 40% of the melting temperature of Y₂O₃ is 1219.15 K (964C), which is higher than the considered temperature in this study. Therefore, the assumption that creep deformation of Y2O3does not occur is feasible.

3. Prediction results of the mechanical properties of ODS Zry- 4 from the 3D RVE model

Fig. 4(a) shows the displacement results of the 3D RVE models with particle VF of 5, 10, and 15% under uniaxial loading at the reference temperature of 293.15 K. It was found that the displacement was well formed along the loading direction. From these results, the Young's modulus of the 3D RVE models was calculated, which is shown as the red line inFig. 4(b). The VF of 0% represents the results of the Zry-4 that does not contain Y2O3particles. It was observed that the Young's modulus of the 3D RVE model is strongly dependent on the particle VF, where the Young's modulus increased as the particle VF increased. When comparing the particle VF outcomes of 0% and 15%, the difference was up to approximately 8%. It was expected that the increase in the Young's modulus with the VF stemmed from the fact that the Young's modulus of the dispersed Y2O3particles is higher than that of the Zry-4 matrix, as shown inTable 2. It can be considered that the more Y2O3particles are dispersed into the Zry-4 matrix, a higher Young's modulus of ODS Zry-4 can be obtained. In addition, the calculated result of the 3D RVE model was compared with that of a HalpineTsai (H-T) analytical model, which is represented as the black line inFig. 4(b). The H-T model is expressed by the following equation [36]:

E_{ODS Zry4}¼E_Zry4

1þ2$s$q$VFY2O3

1q$VFY2O3

(7)

q¼

EY2O3

EZry4

! 1

EY2O3

EZry4

! þ2s

(8)

Here,E_Zry₄is the Young's modulus of Zry-4,E_Y₂_O₃is the Young's modulus of the Y₂O₃ particles,s is the aspect ratio of the Y₂O₃ particles, andVF_Y₂_O₃is the VF of the Y2O3particles.

The two results show a good agreement, with a few differences.

It was expected that the differences stemmed from the fact that the analytical model is based on the assumption of a perfectly oriented reinforcement in the composite, parallel to the applied loading, while the 3D RVE model can reﬂect the heterogeneous distribution of the particles. This is one of the advantages of the developed 3D RVE model.

Fig. 5shows the calculated results of the CTE from the 3D RVE model compared to those of the Turner analytical model at the reference temperature of 293.15 K [37]. The analytical model is expressed as follows:

a

ODS Zry4¼

a

Zry4$VFZry4$KZry4þ

a

Zry4$VFY2O3$KY2O3

VF_Zry4$KZry4þVF_Y₂_O₃$KY2O3

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In this equation,aZry4andaY2O3are the CTEs of the Zry-4 and Y₂O₃particles,K_Zry₄andK_Y₂_O₃are the bulk moduli of the Zry-4 and Y₂O₃particles, andVF_Zry₄andVF_Y₂_O₃are the VF of the Zry-4 and Y₂O₃particles, respectively.

The bulk modulus in each case was estimated from the relationship between the Young's modulus and the Poisson's ratio in Table 2, as follows.

K¼ E

3ð12vÞ (10)

From the comparison between the results of the 3D RVE and analytical models, it was observed that the CTE was also affected by Fig. 5.Calculated CTE from 3D RVE and analytical models.

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the VF of the Y₂O₃particles, while they showed similar behavior.

Fig. 6(a) shows the calculated creep strain results from the 3D RVE model with respect to the particle VF under the 800C condition for 50 s. It was found that the amount of creep strain for ODS Zry-4 with the VF of 5, 10 and 15% was signiﬁcantly reduced when

compared with that of the Zry-4. This creep-resistant behavior of ODS Zry-4 can be explained byFig. 6(b). It shows the Von Mises stress distribution of the Zry-4 matrix and Y2O3particles for the three ODS Zry-4 models with VF of 5, 10 and 15% with identical simulation times. It was observed that the stress of the Zry-4 matrix Fig. 6.(a) Calculated creep strain from 3D RVE models and (b) stress distribution of Zry-4 matrix and Y2O3particles of 3D RVE model.

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decreased as the particle VF increased. When constantly loaded, the Y2O3particle, which has a higher Young's modulus that that of the Zry-4 matrix, can carry a certain amount of mechanical load. This can relieve the amount of stress that the Zry-4 matrix carries relatively. Then, a greater VF of the Y2O3particles can allow a more efﬁcient stress transfer from the Zry-4 matrix and the Y2O3parti- cles, which results in an increase of the overall creep resistance of ODS Zry-4 because the creep behavior is highly dependent on the stress of the Zry-4 matrix according to equation(6). Further, it can be expected that the high creep resistance of the Y₂O₃particles can impede the creep deformation of the Zry-4 matrix due to the perfect bonding assumption. These results represent an advantage of the developed 3D RVE model in that analytical models cannot predict these outcomes.

Fig. 7shows the summarized results of the normalized Young's modulus, CTE and creep strain rate of ODS Zry-4 according to those of Zry-4 with respect to the VF. It was observed that the VF of particles have the strongest effect on the creep strain rate, followed by the Young's modulus and CTE. Speciﬁcally, the creep strain rate properties with the VF are noticeable because a low creep strain rate can be advantageous for suppressing large deformation of the cladding under a severe accident condition. Accordingly, in the following section, the predicted material properties of ODS Zry-4 were applied to a simulation to investigate how these properties affect the overall behavior of cladding under a transient condition.

4. Application of the predicted mechanical properties for an investigation of ODS Zry-4 cladding deformation

4.1. ‘EIGEN’experiment and simulation model

As an application of the predicted mechanical properties of ODS Zry-4, a simulation model for the ‘EIGEN’ experiment was employed [38]. As shown in Fig. 8 (a), the ‘EIGEN’ experiment serves to investigate cladding deformation, such as ballooning and rupture, under a transient condition. A cladding specimen is internally heated using a tungsten heater while constantly pressurized by helium gas under a gas atmosphere. During the experiment, the in-situ deformation and temperature of the cladding are measured by a 3D digital image correlation and a thermocouple, respectively. As shown inFig. 8(b), the‘EIGEN’simulation model was composed of a tungsten heater, alumina pellet, and the cladding parts. Two cladding models were considered: Zry-4 and ODS

Zry-4 cladding models. The ODS Zry-4 cladding model was composed of two layers of the Zry-4 and an ODS layer. For an efﬁcient modelling, thickness of the ODS layer was assumed to be 100mm although there was a few deviations in the real case. In addition, the interface between the two layers assumed to be a perfect bond. The predicted mechanical properties from the 3D RVE model were applied to the ODS layer. For the transient condition, where the material properties change with respect to the temperature, temperature-dependent properties for ODS Zry-4 are needed. These properties were obtained by multiplying the normalized factors inFig. 7 by the temperature-dependent mechanical properties of Zry-4 [39] under the assumption that the temperature effect would be similar in both Zry-4 and ODS Zry-4 because Zry-4 accounts for most of the ODS Zry-4. The creep properties for the ODS Zry-4 were described as multiplying Norton creep constant value A of the Zry-4 by the normalized factor of blue line inFig. 7because deriving the Norton creep constant for the ODS Zry-4 requires many simulation results at various temperature and stress conditions for dataﬁtting. Details of the‘EIGEN’experiment and simulation model can be found in the literature [38].

For the experiments, the Zry-4 and ODS Zry-4 claddings were employed as the test specimens. During the experimental process, the cladding pressurized at 6 MPa was rapidly heated from 300C until it burst. For the simulation, the body heatﬂux calculated from experimental results of the power history of the tungsten heater is utilized to heat the cladding model pressurized at 6 MPa. Four cladding models of Zry-4 and ODS Zry-4 with VF of 5, 10 and 15%

were employed in the simulation.

4.2. Experimental and simulation results of the‘EIGEN’test

Fig. 9(a) shows the‘EIGEN’experimental results of the Zry-4 and ODS Zry-4 claddings. The cladding temperature and hoop strain were measured from the axial center of the cladding specimen. First, it was found that the ODS Zry-4 cladding had higher resistance to an increase of the hoop strain, which resulted in a delay of the time at which a large amount of deformation occurred compared to that of the Zry-4 cladding. It was also observed that the difference in the deformation behavior caused the difference in the subsequent temperature behavior. The delayed large deformation can be an advantage in terms of nuclear safety because such a delay can provide extra time to conduct emergency activities under an accident condition. In addition, excess interaction between the surrounding claddings due to instances of considerable deformation can be reduced.

Fig. 9(b) shows the simulation result of the Zry-4 and ODS Zry-4 cladding models with VF of 5, 10 and 15%. These results were obtained from the axial center of the cladding model as well. It was observed that the ODS Zry-4 cladding models had greater resistance to large deformation compared to that of the Zry-4 cladding, as found in the experimental results. As the VF of ODS Zry-4 was increased, the delay of the time to large deformation increased. It was considered that the low creep strain rate of the ODS layer may have had a considerable inﬂuence on the overall deformation of the ODS Zry-4 cladding under the high temperature condition. The CTE of the ODS layer, which is slightly higher than that of the Zry-4 layer, could affect the stress of the Zry-4 layer in the cladding.

However, the thickness of the Zry-4 layer is about 4.8 times greater than that of the ODS layer in the cladding. In addition, it has been reported that the CTE does not affect cladding deformation as much as creep and irradiation deformation during incore residence time [40]. Then, it can be expected that the CTE value of the ODS layer does not signiﬁcantly affect the overall performance of the cladding.

Based on the simulation results, it can be considered that 15% is Fig. 7.Normalized mechanical properties of ODS Zry-4 by those of Zry-4 with respect

to VF.

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Fig. 8.(a) Schematics of experiment and (b) simulation models for 'EIGEN'.

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the most promising fraction in terms of reactor safety because the cladding deformation can be maximally suppressed, which signif- icantly contribute to maintaining coolability of the reactor under accident condition.

For comparison of the experimental and simulation results, it can be assumed that the volume fraction of experimental specimen is between 5 and 15% because there was a variation in the volume fraction of Y2O3 particle inside the ODS layer as mentioned in section 2.1. It was conﬁrmed that the experimental result was located within the simulation results of the volume fraction with 5e15% for the ODS Zry-4. Therefore, it can be considered that the 3D RVE model can reliably predict the mechanical properties of ODS Zry-4 with respect to the volume fraction, which are difﬁcult to obtain by the conventional experimental approach.

5. Conclusion

In this work, a 3D RVE model was developed to investigate the mechanical properties and effect of Y2O3particles on ODS Zry-4.

Based on the modelling parameters of the size, shape, distribution and VF for the Y2O3 particles, the 3D RVE model was con- structed with a Zry-4 matrix and Y2O3 particles. From a Python

script modelling technique with PBCs, various 3D RVE models with different particle VFs were established. As applying force and temperature conditions to the 3D RVE models, the mechanical simulations were conducted to predict the mechanical properties of ODS Zry-4. The Young's modulus, CTE and creep strain rate of ODS Zry-4 with respect to the particle VF were efficiently calculated. Considering the heterogeneous distribution of Y2O3particles as used here, it was confirmed that the 3D RVE model overall offers more advantages when predicting those mechanical properties than analytical models because analytical models use a homoge- neous distribution assumption in general. From the results of the 3D RVE model, ODS Zry-4 had a higher Young's modulus and CTE outcomes with a lower creep strain rate when compared to those of Zry-4. It was observed that the VF had a significant effect on the mechanical properties. The predicted material properties were applied to an‘EIGEN’simulation for an investigation of cladding deformation under a transient condition. In the simulation results, the ODS Zry-4 cladding showed higher ballooning resistance than the Zry-4 cladding, which was observed in the corresponding experimental results. Therefore, it can be concluded that the 3D RVE model can predict the material properties of ODS Zry-4 reliably. The 3D RVE model can therefore provide more useful information to improve the ODS Zry-4 cladding and can be utilized in the design and evaluation of other ODS alloys.

Declaration of competing interest

The authors declare that they have no known competing ﬁnancial interests or personal relationships that could have appeared to inﬂuence the work reported in this paper.

Acknowledgement

This work has been carried out supported by the Korea Foun- dation of Nuclear Safety R&D Program (2101050-0121-CG100).

References

[1] B. Cheng, Y.-J. Kim, P. Chou, Improving accident tolerance of nuclear fuel with coated Mo-alloy cladding, Nucl. Eng.Technol. 48 (1) (2016) 16e25.

[2] H.-G. Kim, I.-H. Kim, Y.-I. Jung, D.-J. Park, J.-H. Park, J.-H. Yang, Y.-H. Koo, Progress of surface modiﬁed Zr cladding development for ATF at KAERI, in:

Proceedings of the 2017 Water Reactor Fuel Performance Meeting, Ramada Plaza Jeju, Jeju Island, Korea, 2017, pp. 10e14.

[3] K.A. Terrani, S.J. Zinkle, L.L. Snead, Advanced oxidation-resistant iron-based alloys for LWR fuel cladding, J. Nucl. Mater. 448 (1e3) (2014) 420e435.

[4] K. Yueh, K.A. Terrani, Silicon carbide composite for light water reactor fuel assembly applications, J. Nucl. Mater. 448 (1e3) (2014) 380e388.

[5] Y.-I. Jung, H.-G. Kim, I.-H. Kim, S.-H. Kim, J.-H. Park, D.-J. Park, J.-H. Yang, Y.- H. Koo, Strengthening of Zircaloy-4 using Y2O3 particles by a laser-beam- induced surface treatment process, Mater. Des. 116 (2017) 325e330.

[6] H.-G. Kim, I.-H. Kim, Y.-I. Jung, D.-J. Park, J.-Y. Park, Y.-H. Koo, Microstructure and mechanical strength of surface ODS treated Zircaloy-4 sheet using laser beam scanning, Nucl. Eng.Technol. 46 (4) (2014) 521e528.

[7] Y.-I. Jung, D.-J. Park, J.-H. Park, H.-G. Kim, J.-H. Yang, Y.-H. Koo, Effect of a surface oxide-dispersion-strengthened layer on mechanical strength of zircaloy-4 tubes, Nucl. Eng.Technol. 50 (2) (2018) 218e222.

[8] D.M. Perez, R. Williamson, S. Novascone, G. Pastore, J. Hales, B. Spencer, Assessment of BISON: A Nuclear Fuel Performance Analysis Code, Idaho Na- tional Laboratory, Idaho Falls, 2013.

[9] T. Glantz, T. Taurines, O. De Luze, S. Belon, G. Guillard, F. Jacq, DRACCAR: a multi-physics code for computational analysis of multi-rod ballooning, coolability and fuel relocation during LOCA transients Part one: general modeling description, Nucl. Eng. Des. 339 (2018) 269e285.

[10] H. Kim, S. Lee, J. Kim, J. Yoon, Development of MERCURY for simulation of multidimensional fuel behavior for LOCA condition, Nucl. Eng. Des. 369 (2020) 110853.

[11] N. Chawla, Y.L. Shen, Mechanical behavior of particle reinforced metal matrix composites, Adv. Eng. Mater. 3 (6) (2001) 357e370.

[12] Z. Wang, M. Song, C. Sun, D. Xiao, Y. He, Effect of extrusion and particle volume fraction on the mechanical properties of SiC reinforced AleCu alloy composites, Mater. Sci. Eng., A 527 (24e25) (2010) 6537e6542.

[13] M. Barmouz, P. Asadi, M.B. Givi, M. Taherishargh, Investigation of mechanical Fig. 9. (a) 'EIGEN' experimental results and (b) simulation results for Zry-4 and ODS

Zry-4 claddings.

(11)

properties of Cu/SiC composite fabricated by FSP: effect of SiC particles' size and volume fraction, Mater. Sci. Eng., A 528 (3) (2011) 1740e1749.

[14] N. Chawla, R. Sidhu, V. Ganesh, Three-dimensional visualization and microstructure-based modeling of deformation in particle-reinforced composites, Acta Mater. 54 (6) (2006) 1541e1548.

[15] H.J. B€ohm, A. Rasool, Effects of particle shape on the thermoelastoplastic behavior of particle reinforced composites, Int. J. Solid Struct. 87 (2016) 90e101.

[16] H. Wang, H. Zhou, R. Peng, L. Mishnaevsky Jr., Nanoreinforced polymer composites: 3D FEM modeling with effective interface concept, Compos. Sci.

Technol. 71 (7) (2011) 980e988.

[17] S. Ma, X. Zhuang, X. Wang, 3D micromechanical simulation of the mechanical behavior of an in-situ Al3Ti/A356 composite, Compos. B Eng. 176 (2019) 107115.

[18] D. Barbera, H. Chen, Y. Liu, Creep-fatigue behaviour of aluminum alloy-based metal matrix composite, Int. J. Pres. Ves. Pip. 139 (2016) 159e172.

[19] D.-H. Kim, J.-D. Hong, H. Kim, J. Kim, H.-S. Kim, H. Guim, Measurement and investigation of mechanical properties of a partial oxide dispersion strengthened Zircaloy-4 tube via an instrumented indentation test, Materials Today Communications 27 (2021) 102210.

[20] H.-G. Kim, I.-H. Kim, Y.-I. Jung, D.-J. Park, J.-H. Park, J.-H. Yang, Y.-H. Koo, Microstructure and mechanical characteristics of surface oxide dispersion- strengthened Zircaloy-4 cladding tube, Additive Manufacturing 22 (2018) 75e85.

[21] G. Abaqus, Abaqus 6.11, Dassault Systemes Simulia Corporation, Providence, RI, USA, 2011.

[22] Z. Xia, Y. Zhang, F. Ellyin, A uniﬁed periodical boundary conditions for representative volume elements of composites and applications, Int. J. Solid Struct. 40 (8) (2003) 1907e1921.

[23] K. Geelhood, W. Luscher, C. Beyer, FRAPCON-4.0: Integral Assessment, 2, Pa- ciﬁc Northwest National Laboratory, 2015.

[24] R.G. Munro, Elastic Moduli Data for Polycrystalline Oxide Ceramics, 2002.

[25] C. Curtis, Properties of yttrium oxide ceramics, J. Am. Ceram. Soc. 40 (8) (1957) 274e278.

[26] R.L. Williamson, G. Pastore, S.R. Novascone, B.W. Spencer, J.D. Hales, Modelling of LOCA Tests with the BISON Fuel Performance Code, Idaho National Lab.(INL), Idaho Falls, ID (United States), 2016.

[27] J. Kim, J.W. Yoon, H. Kim, S.-U. Lee, Prediction of Ballooning and Burst for Nuclear Fuel Cladding with Anisotropic Creep Modeling during Loss of Coolant Accident (LOCA), Nuclear Engineering and Technology, 2021.

[28] D. Campello, N. Tardif, M. Moula, M.-C. Baietto, M. Coret, J. Desquines, Iden- tiﬁcation of the steady-state creep behavior of Zircaloy-4 claddings under simulated Loss-Of-Coolant Accident conditions based on a coupled experimental/numerical approach, Int. J. Solid Struct. 115 (2017) 190e199.

[29] D. Kaddour, S. Frechinet, A. Gourgues, J.-C. Brachet, L. Portier, A. Pineau, Experimental determination of creep properties of zirconium alloys together with phase transformation, Scripta Mater. 51 (6) (2004) 515e519.

[30] A.R. Massih, An Evaluation of High-Temperature Creep of Zirconium Alloys:

Data versus Models, 2014.

[31] R. Gaboriaud, High-temperature creep of yttrium sesquioxide: Y 2 O 3, phil- osophical magazine. A, physics of condensed matter, Defects and Mechanical Properties 44 (3) (1981) 561e587.

[32] A. Gallardo-Lopez, A. Domínguez-Rodríguez, C. Estournes, R. Marder, R. Chaim, Plastic deformation of dense nanocrystalline yttrium oxide at elevated temperatures, J. Eur. Ceram. Soc. 32 (12) (2012) 3115e3121.

[33] A.P. Mouritz, Introduction to Aerospace Materials, Elsevier2012.

[34] R. Lefever, J. Matsko, Transparent yttrium oxide ceramics, Mater. Res. Bull. 2 (9) (1967) 865e869.

[35] Y. Nigara, Measurement of the optical constants of yttrium oxide, Jpn. J. Appl.

Phys. 7 (4) (1968) 404.

[36] J. Halpin, S. Tsai, Environmental Factors in Composite Materials Design, US Air Force Technical Report AFML TR, 1967, p. 67423.

[37] P.S. Turner, The Problem of Thermal-Expansion Stresses in Reinforced Plastics, 1942.

[38] D.-H. Kim, G.-H. Choi, H. Kim, C. Lee, S.-U. Lee, J.-D. Hong, H.-S. Kim, Mea- surement of Zircaloy-4 cladding tube deformation using a three-dimensional digital image correlation system with internal transient heating and pres- surization, Nucl. Eng. Des. 363 (2020) 110662.

[39] D.L. Hagrman, G.A. Reymann, MATPRO-Version 11: A Handbook of Materials Properties for Use in the Analysis of Light Water Reactor Fuel Rod Behavior, Idaho National Engineering Lab., Idaho Falls (USA), 1979.

[40] Y. Lee, J.I. Lee, H.C. No, Mechanical analysis of surface-coated zircaloy cladding, Nucl. Eng.Technol. 49 (5) (2017) 1031e1043.