Effects of Matrix Creep Properties on Effective Irradiation Swelling of U-10Mo/Zr Dispersion Nuclear Fuels

A meso-mechanical model is established for the homogenized irradiation swelling of U-10Mo/Zr dispersion fuels, with an equivalent sphere chosen as the representative volume element (RVE). In the simulation, a mechanistic model for the fission gas swelling of U-10Mo particles and the creep model for Zr matrix with different values of the creep amplification factor are included. Based on the developed method, the results of effective irradiation swelling of U-10Mo/Zr dispersion fuels are obtained by finite element simulation. Additionally, the effects of matrix creep properties on the effective irradiation swelling are investigated. The numerical results indicate that 1) the Zr matrix has the function of restraining and compensating the irradiation swelling of fuel particles by the mechanical interactions between the fuel particles and the matrix; 2) with the increase in the creep amplification factor of the matrix, the effective irradiation swelling increases while the stresses in the fuel particles and the matrix decrease. The enhanced creep rate of the matrix is apt to result in less restraint of the effective irradiation swelling, but reduces the risk of radial crack initiation and propagation in the matrix; 3) based on the results of finite element simulation, a mathematic model for the effective irradiation swelling of U-10Mo/Zr dispersion fuels is fitted to correlate the effective irradiation swelling with different values of the matrix creep amplification factor under the considered irradiation conditions. The creep property of the matrix should be optimized because of its evident effects on the effective irradiation swelling and stresses of the U-10Mo/Zr dispersion fuels.


INTRODUCTION
Dispersion nuclear fuels, with fissionable fuel particles dispersed in the non-fissionable matrix , are widely used in research and test reactors, advanced nuclear systems, and disposal of nuclear wastes (Duyn, 2003;Carmack et al., 2006;Lombardi et al., 2008;Gong et al., 2013). Tristructural isotropic particle-based dispersion fuels have been used in high-temperature gas-cooled reactors and have a promising application prospect in advanced pressurized water reactors (Xiang et al., 2014;Zhang T. et al., 2021;Lou et al., 2022). With the non-fissionable matrix encompassing fuel particles, dispersion nuclear fuels possess a more stable swelling behavior and allow higher burnup than traditional homogeneous fuels (Neeft et al., 2003a;Savchenko et al., 2006). Moreover, certain properties of dispersion fuels such as high strength, high thermal conductivity, good corrosion resistance, and good performances under transient conditions can be designed by a proper choice of fuel and matrix materials (Holden, 1967;Savchenko et al., 2007;Savchenko et al., 2010).
In the 1980s, the US Department of Energy Office launched the Reduced Enrichment for Research and Test Reactors (RERTR) program to reduce the threat of nuclear proliferation worldwide, through the conversation of research and test reactors from highly enriched uranium to low-enriched uranium fuels Liu et al., 2011). With the development of the RERTR program, U-Mo/Al dispersion fuels have attracted numerous attention as the low-enriched uranium fuels for research and test reactors (Van den Berghe and Lemoine, 2014), due to their advantages of high uranium density, low neutron capture cross sections, good irradiation stability, and high thermal conductivity Jian et al., 2019a). Early experimental results demonstrated that U-Mo/Al dispersion fuels have good irradiation performance under low power and low burnup (Kim et al., 2002;Meyer et al., 2002). However, excessive local swelling occurred in U-Mo/Al dispersion fuels under high power and high burnup irradiation conditions (Leenaers et al., 2004;Van den Berghe and Lemoine, 2014), due to the formation of an amorphous interaction layer (IL) between U-Mo fuel particles and the Al alloy matrix (Van den Berghe et al., 2008;Oh et al., 2016). Under the irradiation conditions, the IL grows and shows poor fission gas solubility and high diffusivity of fission gas atoms, leading to large bubbles, which induces excessive irradiation swelling (Lee et al., 1997;Huber et al., 2018;Saoudi et al., 2022). To prevent the growth of the IL, some methods were implemented, such as modifying the Al matrix by adding Si (Leenaers et al., 2011;Keiser et al., 2014), coating U-Mo fuel particles with Si or ZrN, and developing monolithic fuels with a Zr layer inserted between U-Mo fuel foil and Al alloy clad (Leenaers et al., 2013;Keiser et al., 2015;Leenaers et al., 2015). An alternative method to eliminate the IL of U-Mo/Al is replacing the Al alloy matrix with some other inert alloys. Zirconium alloys are promising alternative matrixes for U-Mo-based fuels because they have a low reactivity with U and Mo, high melting point, high strength, good corrosion resistance, and low neutron capture cross-section for thermal neutrons (Cox, 2005;Nakamura et al., 2007;Gonzalez et al., 2015). Moreover, zirconium alloys have long been used as fuel matrix and cladding in nuclear reactors (Moorthy, 1969;Savchenko et al., 2007;Savchenko et al., 2010) with extensive knowledge of their material properties and processing technologies, so they would be a good choice to minimize the qualification testing required to introduce a new matrix material for U-Mo-based dispersion fuels (Hagrman et al., 1995;Hayes and Kassner, 2006;Hales et al., 2016;Pasqualini et al., 2016).
Under the irradiation conditions, U-Mo fuel particles undergo irradiation swelling due to the accumulation of solid and gaseous fission products, namely, fission solid swelling and fission gas swelling (Kim and Hofman, 2011). As irradiation swelling is considered the important factor dominating the behavior and service life of nuclear fuels, it is necessary to investigate and model the swelling behavior of U-Mo/Zr dispersion fuels to evaluate its serviceability. Varied knowledge of irradiation swelling behavior of U-Mo alloy has been accumulated through experimental investigation (Kim and Hofman, 2011;Van den Berghe et al., 2012). The fission solid swelling of U-Mo alloy is proportional to the fission density (Kim and Hofman, 2011). The fission gas swelling is complex and related to the fission density, the grain size, the temperature, the external hydrostatic pressure, and the progression of irradiation-induced recrystallization. Kim and Hofman built an empirical model for fission gas swelling by fitting the experiment data (Kim and Hofman, 2011). The empirical model is simple, but it could not reflect the critical influencing parameters mentioned above. Rest built a mechanistic model reflecting the influence of those parameters except the external hydrostatic pressure (Rest, 2005). Cui et al. modified the mechanistic model with consideration of external hydrostatic pressure and re-solution of intergranular gas atoms (Cui et al., 2015). Jian et al. (Jian et al., submitted) further developed the fission gas swelling mechanistic model for U-10Mo fuels, which has been validated by comparing the model results, including the fission gas swelling, the evolution rules of the bubble density, and the bubble size with the experimental results. The newly developed fission gas swelling model could reflect the dependence of external hydrostatic pressure more reasonably. To better understand and study the irradiation swelling behavior of U-Mo/Zr dispersion fuels, it is necessary to adopt the improved mechanistic model of fission gas swelling for fuel particles. For dispersion fuels with numerous fuel particles, it is difficult and expensive to establish a finite element model for the whole fuel element with all particles involved (Schappel et al., 2018). An effective way is to homogenize the thermal-mechanical performances of the dispersion fuels based on homogenization theory, including its effective irradiation swelling behavior (Rest and Hofman, 1997;Zhao et al., 2014;Liu et al., 2018). Zhang J. et al. established an equivalent spherical model to investigate the effective irradiation swelling of PuO 2 /Zr dispersion fuels (Zhang J. et al., 2021). The simulation results indicate that the creep behavior of the Zr matrix has an evident influence on the effective irradiation swelling of PuO 2 /Zr dispersion fuels. As the creep properties of zirconium alloys are related to alloy compositions, microstructure, process conditions, and service conditions (Krishnan and Asundi, 1980;Hayes and Kassner, 2006;Kim et al., 2021), it is essential and necessary to investigate the effects of Zr matrix creep properties on the effective irradiation swelling of U-Mo/Zr dispersion fuels and provide a reference for the optimization design and advanced manufacture. Meanwhile, the relative research studies are currently limited.
In this study, the three-dimensional finite element simulation for the developed equivalent spherical model is performed for the further homogenization of the effective irradiation swelling of U-10Mo/Zr dispersion fuels. The newly improved mechanistic model of fission gas swelling (Jian et al., submitted) is adopted, which could reflect the significant influences of external hydrostatic pressure. The effects of Zr matrix creep properties on the effective irradiation swelling of U-10Mo/Zr dispersion fuels are investigated with different creep amplification factors imposed on a typical creep model. The irradiation swelling model and creep model are given in Material Properties. The three-dimensional finite element model and the validity of the fission gas swelling model are presented in Results and Discussion. The effective irradiation swelling results are obtained by finite element simulation, and the influence mechanisms are analyzed in Finite Element Modelling. Based on the simulated results, a mathematical model of effective irradiation swelling is described with different values of creep amplification factors.
The acronyms and symbols used in the paper are summarized in Appendix A for better readability.

MATERIAL PROPERTIES
In this section, the irradiation swelling model for U-10Mo fuel particles and the creep strain rate model for the Zr matrix correlated with the creep amplification factor are given. The other material property models for U-10Mo and the Zr matrix, such as the elastic constants and plasticity model are described in Refs. (Zhao et al., 2014;Jian et al., 2019b). The mechanical constitutive relations for fuel particles and the matrix are introduced to commercial software ABAQUS through user-defined subroutines, based on the stress update algorithms in our previous studies (Zhao et al., 2015;Kong et al., 2018).

The Irradiation Swelling Model for U-10Mo Fuels
The irradiation swelling of U-10Mo consists of fission solid swelling and fission gas swelling, expressed as follows: The fission solid swelling is proportional to the fission density, expressed as follows (Kim et al., 2015): where V 0 is the initial fuel volume in m 3 , F d depicts the fission density in fission/m 3 , and ΔV solid is the absolute volume change caused by solid fission products in m 3 . The fission gas swelling is measured by the fraction of identified inter-granular bubbles through SEM, while the intragranular bubbles are invisible (Kim and Hofman, 2011). Thus, the fission gas swelling is only regarded as the contribution of intergranular bubbles in the newly developed model (Jian et al., submitted). The fission gas swelling is described as follows: where V bubble is the total volume of inter-granular bubbles in m 3 ; N bubble is the total number of inter-granular bubbles in the recrystallized and un-recrystallized areas with a new description by Jian et al. (Jian et al., submitted), which could reflect the significant influence of external pressure, and the obtained evolution results of the bubble density and the bubble size have been demonstrated in the same magnitude order as the related experimental results; R is the average radius of inter-granular bubbles.
The average radius of inter-granular bubbles is determined using the modified van der Waals gas law (Jian et al., submitted) as follows: R is the surface tension-induced pressure, in Pa; P h is the external hydrostatic pressure in Pa, involved when positive; h s is the fitting parameter to make the van der Waals equation equivalent to the hard-sphere equation of state in the value of 0.6; b v 8.5 × 10 −29 m 3 /atom is the van der Waals gas constant of Xe; N is the average number of fission gas atoms in each intergranular bubble; k 1.38 × 10 −23 J/K is the Boltzmann constant and T is the temperature in K.
The numbers of inter-granular bubbles in the recrystallized and un-recrystallized areas are defined as N bubble1 and N bubble2 , respectively, expressed as follows (Jian et al., submitted): where C b and C bx are the defined grain-boundary bubble concentration of the un-recrystallization region and the fine grain region in Refs. (Rest, 2005;Cui et al., 2015), respectively; η F d is the modified factor for the gas bubble number; F d0 = 4.7 × 10 27 fissions/m 3 .

The Creep Model of Zr Matrix
The effects of different creep properties are investigated based on the creep model of Zircaloy-2 alloy (Macdonald and Thompson, 1976). The creep model of the Zr matrix consists of thermal creep and irradiation creep, expressed as follows: where β is the creep amplification factor in the values of 0.01, 0.1, 1, 10, and 100; _ ε ss and _ ε ir are the thermal creep strain rate and the irradiation creep strain rate, respectively, in 1/s. The thermal creep strain rate is described as follows: where R is the gas constant in J/(mol·K) with a value of 8.314 J/ (mol·K); T is temperature in K; σ m is the Mises equivalent stress in Pa and G is the shear modulus in Pa; Q = 27,000 J/mol is the activation energy; A 0 = 3.14 × 10 26 s −1 and n = 5 are the used material constants in this study.
The irradiation creep strain rate is described as follows: where _ ε ir is the irradiation creep strain rate in 1/s; σ m is the Mises equivalent stress in Pa and ϕ is the fast neutron flux in n/(m 2 ·s), with the constants K = 5.129 × 10 -27 , B = 7.252 × 10 2 , and C = 4.967 × 10 -8 in this study. Figure 1 displays the creep curves of the Zr matrix with different values of β under the condition of 373 K, with a Von Mises stress of 50 MPa and a fast neutron flux of 1.9128 × 10 18 n/ (m 2 ·s). It can be found that the creep strain rate decreases with time. For cases of β = 100 and β = 0.01, the effective creep strains after irradiation of 160 days reach 0.5 and 5 × 10 -5 , respectively.

FINITE ELEMENT MODELING
In this section, the FE (finite element) model for a respective volume element (RVE) of U-10Mo/Zr dispersion fuels with the fuel particle, matrix part, and the applied boundary conditions are presented. The irradiation conditions are chosen according to those of RERTR programs. The effective irradiation swelling of U-10Mo/Zr dispersion fuels is defined and presented in a formula form. To validate the fission gas swelling model, the fission gas swelling results calculated by the model used in this study are compared with the experiment results in the references.

Finite Element Model
The equivalent spherical model has been adopted to study the irradiation swelling behavior of polycrystalline nuclear fuels (Wood and Kear, 1983;Rest, 2010) and the effective irradiation swelling of dispersion fuels (Zhang J et al., 2021). In this study, an equivalent spherical model containing the fuel particle and the matrix is selected as the RVE to obtain the effective irradiation swelling of U-10Mo/Zr dispersion fuels. The initial particle volume fraction of the equivalent spherical model is set as 30% in this study. According to the symmetry in geometry and loading, 1/8 part of the equivalent spherical model is ultimately selected as the FE model shown in Figure 2, including the particle part with a radius of 50 μm and a matrix-shell part.
The bonding between the fuel particle and the matrix is assumed to be perfect with continuous displacements and interfacial stresses. Symmetrical boundary conditions are applied to the surfaces of x = 0, y = 0, and z = 0, which are the symmetrical surfaces of the 1/8 part of the RVE. Constant pressure in a magnitude of 2.5 MPa is applied to the outer surface of the RVE, according to the pressure of the reactor coolant for RERTR (Salvato et al., 2018). A uniform steadystate temperature of 373 K is set. The fission rate in the fuel particle is set as 6 × 10 20 fission/(m 3 ·s) and the fast neutron flux is set as 1.9128 × 10 18 n/(m 2 ·s) in this study, according to irradiation conditions of research and test reactors (Perez et al., 2011). The total irradiation time is 160 days, on which the recrystallization process is completed.
To ensure the computation precision and efficiency, three mesh cases are adopted with 1472, 4,147, and 7,868 elements, respectively. The obtained effective irradiation swelling results of the RVE and Mises stress of the matrix for the case of β = 1 are shown in Figure 3. The maximum relative error of the effective irradiation swelling results and Mises stress between Mesh 2 and Mesh 3 is less than 0.11 and 0.14%, respectively. So the effective irradiation swelling and stress results of Mesh 2 have converged, and Mesh 2 is adopted in the following studies.  The effective irradiation swelling of U-10Mo/Zr dispersion fuels is calculated as follows: where V is the calculated post-irradiation volume of the equivalent sphere; V 0 is the initial spherical volume; r is the post-irradiation outer spherical radius, obtained from the FE model displacement results; r 0 is the initial outer spherical radius.

Verification of the Used Fission Gas Swelling Model
The irradiation swelling model for the fuel particles is vitally important for calculating the effective irradiation swelling of U-10Mo/Zr dispersion fuels. The fission solid swelling model is directly obtained from the reference (Kim and Hofman, 2011). Meanwhile, the fission gas swelling model considering the critical physical mechanisms will be validated with the experiment results. The plate elements, consisting of Al-alloy cladding and the fuel meat with U-Mo particles dispersed in the aluminum matrix, were irradiated in the advanced test reactor for the test campaigns of RERTR-1~5 (Kim et al., 2013). The fuel temperature during the test was in the range of 339-431 K, and the achieved fission density was in the range of 2.0 × 10 27 -5.7 × 10 27 fission/m 3 . The post-irradiation samples were punched out at the plate center to measure the fission gas swelling by scanning electron microscopy. Ignoring the constraint of the IL on U-10Mo particles, the subjected hydrostatic pressure of the fuel particles in the middle zone of fuel plates could be considered as that of the reactor coolant. So, the temperature is ultimately set as 373 K and the hydrostatic pressure is set as 2.5 MPa in this study. The results of the calculated fission gas swelling are compared with the experimental results of RERTR1~5, shown in Figure 4. It can be seen that the predictions agree well with the experimental results. So, the fission gas swelling model used in this study is effective.

RESULTS AND DISCUSSION
Adopting the FE model in Figure 2, numerical simulation of the irradiation-induced mechanical behavior is implemented, considering different values of β of the Zr matrix. Here, the deformations and stresses of the FE model and the effective irradiation swelling for cases with different values of β are obtained, and the influence mechanisms for effective irradiation swelling are analyzed. A mathematical model correlating the effective irradiation swelling with different values of β is fitted at last.

The Deformation and Stress Analysis
Due to the irradiation swelling of U-10Mo fuel particles under the irradiation environments, the intensive mechanical interactions  are aroused between the fuel particles and the matrix. These mechanical interactions could result in creep and plastic deformations of the matrix. The value of β has a direct influence on the creep deformation of the matrix. Figure 5 gives the contour plots of effective creep strain and equivalent plastic strain of the matrix at the fission density of 8.3 × 10 27 fission/m 3 for the cases of β = 0.01 and β = 1. It is noted that the other cases with different values of β have similar effective creep strain distributions. The effective creep strain field of the matrix is spherically symmetric on the whole, and the largest effective creep strain occurs at the interface of the matrix with the fuel particle. The plastic deformation of the matrix only occurs in the case of β = 0.01, with the maximum value of 0.055%, excluding the other cases with β ranged from 0.1 to 100. Figure 6 compares the distributions of effective creep strain of the matrix along the radial direction at the fission density of 8.3 × 10 27 fission/m 3 , for different values of β. One can see that the effective creep strain of the matrix decreases along the radial direction for all cases and grows with the increase in β. For the cases with β in the range of 0.01-10, the effective creep strain of the matrix has a small difference with a relative deviation of less than 1.2%. When β is amplified to 100, the maximum effective creep strain of the matrix is about 21%, with a relative increase of~20% compared to the result of β = 0.01. Figure 7 gives the contour plots of Mises stress and the first principal stress of the matrix at a fission density of 8.3 × 10 27 fission/m 3 for the case of β = 0.01. The largest Mises stress occurs at the interface of the matrix with the fuel particle, while the largest first principal stress occurs at the outer surface of the matrix. Figure 8 compares the evolution results of Mises stress of Element 2 and the first principal stress of Element 3 for the cases    with different values of β. It can be found that the stresses of the matrix increase quickly in the preliminary stage of irradiation and tend to be stable at higher burnup levels. As β increases from 0.01 to 100, the maximum Mises stress decreases from 402 to 40 MPa, with a relative reduction of 90%, and the maximum first principal stress decreases from 366 to 16 MPa, with a relative reduction of 96%. As neutron radiations will result in the loss of ductility of the matrix and the formation of the fission fragment damage zone (Duyn, 2003), there is a possibility for the brittle fracture of the matrix. Thus, the increase in β has the positive consequence that the stresses of the matrix decrease, which could reduce the risk of stress-induced failure. To further predict the possible form of crack initiation and propagation, Figure 9 compares the distributions of tangential and radial stress of the matrix along the radial direction at the fission density of 8.3 × 10 27 fission/m 3 , for different values of β. One can see that the radial stress of the matrix is compressive, while the tangential stress is tensile stress approximately equal to the first principal stress. The tangential stress of the matrix decreases with the increase in β. Thus, radial cracks are susceptible to generating and propagating in the matrix, and the enhanced creep rate could prevent the stressinduced cracks of the matrix, similar to the conclusion from the experimental investigation (Neeft et al., 2003b).

The Radial Displacement Analysis
Under the irradiation conditions, U-10Mo fuel particles undergo irradiation swelling, presented as positive radial displacements. Figure 10 displays the radial displacement contour plot of the FE model at a fission density of 8.3 × 10 27 fission/m 3 for the cases of β = 0.01 and β = 100. It is noted that similar displacement contour plots appear for the other cases with different values of β. The radial displacements of the fuel particle increase along the radial direction. As β increases from 0.01 to 100, the maximum radial displacement, located at the particle surface to be connected with the matrix, increases from 5.5 to 6.4 μm, with a relative increase of~18%. The radial displacements of the matrix part are smaller than the   March 2022 | Volume 10 | Article 851747 maximum of the particle. Figure 11 compares the distributions of radial displacement along Path 1 at the fission density of 8.3 × 10 27 fission/m 3 , for the cases with different values of β and the case without the constraint of the matrix condition. The case without the matrix part means that the original matrix part in Figure 2 is occupied by the particle material. The radial displacements of the FE model along path1 keep increasing for the case without the matrix. Meanwhile, those of the other cases display a decreasing trend along the radial direction across the matrix part, due to the through-thickness shrinking of the matrix. It demonstrates that the Zr matrix has the effects of restraining and compensating the irradiation swelling of U-10Mo fuel particles, and the effects are promoted with the decrease in β. As analyzed in The Deformation and Stress Analysis, the matrix with lower values of β is susceptible to be fractured, which will result in the reduced restraint effect after failure. So, a compromise should be determined in the actual design and fabrication.

The Irradiation Swelling Analysis
Irradiation swelling of nuclear fuels proceeds with the increase in fission density (Kim and Hofman, 2011;Cui et al., 2015). Figure 12 depicts the effective irradiation swelling evolutions of dispersion fuels, calculated using Eq. 11 for the cases with different values of β. The curve of effective irradiation swelling can be divided into two parts with the critical fission density F dx . Beyond F dx , the effective irradiation swelling is accelerated. One can also see that the effective irradiation swelling increases with the rise of β. For the cases with β ranged in 0.01-10, the differences of effective irradiation swelling at the fission density of 8.3 × 10 27 fission/m 3 are small, not more than 2.5%. For the case of β = 100, the effective irradiation swelling is 13%, with a relative increase of 18% compared to that of the case with β = 0.01.
The effective irradiation swelling of dispersion fuels is driven by the irradiation swelling of fuel particles and is approximately equal to the product of the initial particle volume fraction and the sum of fission solid swelling and fission gas swelling, as shown in Figure 13. The evolutions of fission solid swelling and fission gas swelling of Element 1 in the part of the fuel particle are shown in Figure 14. The fission solid swelling in Figure 14A linearly increases with the fission density independent of β, as expressed in Eq. 2. The results of fission gas swelling in Figure 14B rise with the increase in β. The fission gas swelling at the fission density of 8.3 × 10 27 fission/m 3 for β = 0.01 is 3.44%, becoming 10.2% for β = 100, and they are both less than the fission solid swelling of 33%. So, the discrepancies of effective irradiation swelling for U-10Mo/Zr dispersion fuels are mainly dependent on the difference of fission gas swelling for various values of β. The obtained fission gas swelling predictions here are less than the measured results for U-10Mo/Al dispersion fuels in Ref. (Kim et al., 2013). It is mainly due to the occurrence of the IL between the U-10Mo fuel particles and the Al matrix, which weakens the restraint effect of the matrix and can be interpreted as the high creep rate of the IL (Jeong et al., 2020). Moreover, according to our previous studies, the fission gas swelling increases with the decrease in hydrostatic pressure in fuel particles (Cui et al., 2015;Jian et al., 2019a;Zhang J. et al., 2021). The particles selected to measure the fission gas swellings (Kim et al., 2013) are near the middle region of U-10Mo/Al dispersion fuel plates, and the hydrostatic pressures of these particles are close to the coolant pressure of~2.5 MPa, which are much lower than the experienced hydrostatic pressure of the particles in U-10Mo/Zr dispersion fuels, as presented in Fission Gas Swelling of Fuel Particles and the Influencing Mechanisms.

Fission Gas Swelling of Fuel Particles and the Influencing Mechanisms
It can be known from The Irradiation Swelling Model for U-10Mo Fuels that the fission gas swelling behavior of nuclear fuels is influenced by the external hydrostatic pressure. Figure 15 displays the evolution results of hydrostatic pressure of Element 1 for the cases with different values of β. It can be found that the hydrostatic pressure of the fuel particle increases quickly in the preliminary stage of irradiation and tend to be stable at higher burnup levels. As β increases from 0.01 to 100, the maximum hydrostatic pressure decreases from 299 to 24 MPa, with a relative reduction of~92%. Creep deformation of the matrix could relax the mechanical interactions between the fuel particles and the matrix (Zhang J. et al., 2021). A larger value of β will result in enlarged matrix creep deformations, as shown in Figure 6, and the stress relaxation effect will be enhanced to result in lower hydrostatic pressure in the fuel particle, as shown in Figure 15.
According to Eq. 4, the average radius of inter-granular bubbles is correlated with hydrostatic pressure and the average number of fission gas atoms in each inter-granular bubble. Figure 16 depicts the average radius predictions of the intergranular bubbles of Element 1 for the cases with different values of β. One can observe that the average radius of inter-granular bubbles increases with the fission density overall, due to the increase in fission gas atoms. As β increases, the average radius of inter-granular bubbles increases because of the lowered hydrostatical pressure. The predicted radius of inter-granular bubbles is in the same magnitude order with experimental results of U-7Mo fuels (Salvato et al., 2018), so that the external hydrostatical pressure effects could be well reflected. According to Eq. 3, the fission gas swelling also depends on the number of inter-granular bubbles or the bubble density N bubble V0 . The cases with different values of β have the same results of the bubble density. One can find from Figure 17 that the number of intergranular bubbles does not show significant changes before recrystallization. After recrystallization, the number of inter-granular bubbles increases sharply, due to the increased grain-boundary area per unit volume (Rest, 2005). As the fission density increases further, the inter-granular bubbles begin to coalesce  and the predicted number of inter-granular bubbles decreases. The predictions of bubble density match the evolution traits of the experimental results (Salvato et al., 2018). The different fission gas swelling results in Figure 14B for the varied cases of β can be known to stem mainly from the discrepancy of inter-granular bubble size influenced by hydrostatic pressure.

Fitted Mathematic Model for the Effective Irradiation Swelling
The effective irradiation swelling is defined as the relative volume variation to the initial volume of the RVE, calculated using Eq. 11.    where SW eff is the effective irradiation swelling; β is the creep amplification factor in the range of 0.01-100; F d is fission density in 10 27 fission/m 3 ; The critical fission density of 3.54 × 10 27 fission/m 3 is dependent on fission rate _ f, calculated as 4 × 10 24 ( _ f) 2/15 . The comparison of finite element simulation results and the fitted model results is shown in Figure 18, for the cases with different values of β. The relative error is less than 0.8% at the fission density of 8.3 × 10 27 fission/m 3 . So the fitted model is acceptable and could be used to obtain the effective irradiation swelling for U-10Mo/Zr dispersion fuels with the values of β ranged in 0.01-100, under the irradiation condition mentioned in Finite Element Model.
It should be mentioned that the homogenization pressures of dispersion fuel foil in the plate-type fuel elements are distributed heterogeneously and vary with the burnup levels and also depend on the matrix creep and the particle volume fractions. As a result, it will be possible for the homogenization pressures to differ greatly from the coolant pressure. Further research should be carried out to obtain a comprehensive mathematic model for usage in the numerical simulations of the irradiation-induced thermal-mechanical behaviors in dispersion fuel elements or assemblies.

CONCLUSION
In this study, a meso-mechanical model is established for the homogenized irradiation swelling of U-10Mo/Zr dispersion fuels. An equivalent sphere is chosen as the RVE, with an initial particle volume fraction of 30% and a temperature of 373 K. A newly developed fission gas swelling mechanistic model for U-10Mo is involved and validated with the experimental results in literatures. To investigate the effect of Zr matrix creep properties on effective irradiation swelling of U-10Mo/Zr dispersion fuels, different creep amplification factors are imposed on a typical creep model. Based on the developed method, the results of effective irradiation swelling for U-10Mo/Zr dispersion fuels are obtained and analyzed. The main conclusions are as follows: 1) The Zr matrix has the effects of restraining and compensating for the irradiation swelling of fuel particles. The effective irradiation swelling of U-10Mo/Zr dispersion fuels increases with the increase of the creep amplification factor of the Zr matrix due to the stress relaxation effect of matrix creep deformations. When the creep amplification factor increases from 0.01 to 100, the effective irradiation swelling at the fission density of 8.3 × 10 27 fission/m 3 increases from 11 to 13%, with a relative increase of 18%. 2) Although the effective irradiation swelling increases with the increase of the creep amplification factor, the stresses in the matrix of U-10Mo/Zr dispersion fuels decrease to result in a reduction of the risk of radial crack initiation and propagation. So, the creep properties of the Zr matrix need to be optimized according to design requirements and service conditions. 3) When the creep amplification factor increases from 0.01 to 100, the maximum hydrostatic pressure of the fuel particle decreases from 299 to 24 MPa, with a reduction of 92%. The different fission gas swelling results for the varied cases of the creep amplification factor stems mainly from the discrepancy of intergranular bubble size influenced by hydrostatic pressure.  4) Based on the results of finite element simulation, a mathematic model for the effective irradiation swelling of U-10Mo/Zr dispersion fuels is fitted and verified. The fitted model is acceptable under the adopted irradiation condition and the selected parameter ranges.
In reality, the initial particle volume fraction and the temperature could be different under different irradiation environments or in different positions. The mechanical interaction between particles and between fuel meat and cladding could also lead to the external equilibrium hydrostatic pressure. So, wide future research studies should be performed.

DATA AVAILABILITY STATEMENT
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding authors.