Effects of fuel particle size and fission-fragment-enhanced irradiation creep on the in-pile behavior in CERCER composite pellets

Accepted Manuscript Effects of fuel particle size and fission-fragment-enhanced irradiation creep on the inpile behavior in CERCER composite pellets Yunmei Zhao, Shurong Ding, Xunchao Zhang, Canglong Wang, Lei Yang PII:

S0022-3115(16)30460-3

DOI:

10.1016/j.jnucmat.2016.10.035

Reference:

NUMA 49972

To appear in:

Journal of Nuclear Materials

Received Date: 21 July 2016 Revised Date:

26 September 2016

Accepted Date: 19 October 2016

Please cite this article as: Y. Zhao, S. Ding, X. Zhang, C. Wang, L. Yang, Effects of fuel particle size and fission-fragment-enhanced irradiation creep on the in-pile behavior in CERCER composite pellets, Journal of Nuclear Materials (2016), doi: 10.1016/j.jnucmat.2016.10.035. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Effects of fuel particle size and fission-fragment-enhanced irradiation creep on the in-pile behavior in CERCER composite pellets Yunmei Zhao1, Shurong Ding1,*, Xunchao Zhang2, Canglong Wang2, Lei Yang2

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1 Institute of Mechanics and Computational Engineering, Department of Aeronautics and Astronautics, Fudan University, Shanghai 200433, China 2 Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China

Abstract

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The micro-scale finite element models for CERCER pellets with different-sized fuel particles are developed. With consideration of a grain-scale mechanistic irradiation swelling model in the fuel particles and the irradiation creep in the matrix, numerical simulations are performed to explore the effects of the particle size and the fission-fragment-enhanced irradiation creep on the thermo-mechanical behavior of CERCER pellets. The enhanced irradiation creep effect is applied in the 10um-thick fission fragment damage matrix layer surrounding the fuel particles. The obtained results indicate that (1) lower maximum temperature occurs in the cases with smaller-sized particles, and the effects of particle size on the mechanical behavior in pellets are intricate; (2) the first principal stress and radial axial stress remain compressive in the fission fragment damage layer at higher burnup, thus the mechanism of radial cracking found in the experiment can be better explained.

1 Introduction

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Keywords: particle size effect; fission-fragment-enhanced irradiation creep; numerical simulation; cracking mechanism

Minor Actinides (MA, including Np, Am and Cm, etc.) are the major contributor to the high level radio-toxicity wastes in the spent fuels[1], which cover 0.1% share of nuclear wastes

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produced in nuclear power plants. Therefore, transmutation of MA is considered as the primary option to realize a significant reduction of the hazardous radio-toxicity waste. An accelerator Driven System[2] (ADS, subcritical reactors) equipped with a neutron source has been put

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forward as an effective means dedicated to MA transmutation. As an advanced nuclear fission energy system, ADS provides a possible way to reduce the radio-toxicity of nuclear wastes and generate nuclear energy as well. The fuel element loaded with MA and fissile nuclides is the core component in the ADS

reactors. The element is structurally similar to the fuel rods used in the current plants, where a number of cylindrical pellets are wrapped by a cladding with a pellet-cladding gap designed to prevent the mechanical interaction. The pellet used in ADS fuels is a composite having the ceramic fuel particles with MA dispersively incorporated into a matrix. Two kinds of pellet candidates[3-5] are preferred for the targets in the Project EUROTRANS: one is CERCER with (Pu, MA)O2-x particles embedded in ceramic MgO, another is CERMET with the same inclusions embedded in the metallic Mo92 matrix. During the early Experimental Feasibility of Targets for TRAnsmutation (EFFTTRA) framework[1, 6, 7], UO2 is employed as the particle choice in 1 *: To whom correspondence should be addressed. Email: [email protected] ,[email protected]

ACCEPTED MANUSCRIPT CERCER targets and MgO’s good irradiation stability makes it suitable for the matrix material. Nuclear fuels in a CERCER pellet experience complex behavior under an extreme irradiation environment. Firstly, nuclear fission in fuel particles results in elevated heterogeneous temperatures, and the generated fission products including fission solids and gases lead to considerable irradiation swelling. After a critical fission density, recrystallization[8, 9] happens in particle grains. Other than the scenario in homogeneous UO2 pellets, the dispersed particles are firmly restrained by the MgO matrix. As a result, the accumulated gaseous product cannot be

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released, which causes larger irradiation swelling to induce a strong mechanical interaction between the fuel particles and the matrix together with large deformation. Besides, with increasing burnup, porosity resulting from the generated gas bubbles plays an essential effect on the thermal conductivity and Young’s modulus of particles[10]. Meanwhile, irradiation creep occurs in the MgO material under the continuous attack of fission fragments and fast neutrons. A thin ceramic

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matrix layer[11-13] surrounding the UO2 particles has enhanced irradiation creep induced by both the fission fragments and the fast neutrons. In this study, we call the thin layer as a fission-enhanced layer. An experiment[6] found that radial cracking in the fission-enhanced layer

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were prevented, and the enhanced creep was considered as the possible reason.

In order to optimize the composite CERCER pellets, some irradiation experiments[14] were implemented for the cases with different-sized fuel particles. Micro-dispersed (particle diameter<10um) composite fuel pellets were not recommended because of the observation of large cracking profile in Post Irradiation Examinations (PIE)[6,15]. Meanwhile, macro-dispersed inclusions with a diameter order of 100-300um demonstrated generally good mechanical behavior, and they were preferable for the composite pellet.

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As mentioned above, existence of the fission-enhanced matrix layer is crucial to the cracking failure mechanism in CERCER pellets, and the particle diameter is also an essential influencing factor on their thermo-mechanical behavior. So, a detailed assessment of the effects of fission particle size and the fission-fragment-enhanced irradiation creep needs to be performed. Thus, we can realize an optimization design of CERCER fuels, and can explain the cracking mechanism in

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the matrix. A numerical simulation research should be implemented to evaluate the above two effects because the irradiation examination with high cost is hard to be performed. In the numerical simulation, the complex irradiation effects should be involved in. The effects of the fuel

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particle diameters on the mechanical behavior in plate-type dispersion nuclear fuel was investigated through a numerical simulation method[16], while the relative study for CERCER pellets is not found.

A number of numerical attempts have been made to model the complex behavior in nuclear

fuels. The recrystallization theory for UO2 and UMo fuels was put forwarded by J.Rest[9,17]. Yi. Cui [18,19] modified the fission gas diffusion equations with the intergranular resolution effect introduced, and the corresponding semi-analytical expressions for the fission gas swelling was obtained. Three-dimensional simulations on the coupled thermo-mechanical in-pile behavior are still limited. The thermo-mechanical behavior was explored at the initial stage[8,20] without taking into account of irradiation swelling of the fuel particles. S.Ding[21] simulated the in-pile thermo-mechanical behavior in CERCER pellet to a deep burnup with a consideration of irradiation creep in the matrix, while the swelling empirical formulas for the homogeneous UO2

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ACCEPTED MANUSCRIPT was adopted. Ding et al.[22-25] further developed self-defined subroutines UMAT and UMATHT in ABAQUS to define the time-temperature dependent constitutive relations in nuclear materials. The multi-scale thermo-mechanical behavior in CERCER fuels was analyzed in Y. Zhao’s study[24] using the fission gas swelling expressions of Yi. Cui, in which the calculated temperature and swelling of the pellet were demonstrated to be consistent with the experimental data[6], while the cracking mechanism in the matrix was discussed without considering the enhanced irradiation creep in the matrix. Some important influencing factors such as fission rate

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and particle volume fraction were assessed in CERCER fuels, in which the porosity-induced degradation of thermal conductivity and Young’s modulus was considered [25].

In the present study, based on our previous works, the three-dimensional constitutive relations for the fuel particles and matrix are briefly presented. With the subroutines UMAT and UMATHT, a coupled temperature-displacement analysis is carried out in ABAQUS, considering

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the fission-enhanced irradiation creep in the matrix. Several three-dimensional finite element models are developed for CERCER pellets with different-sized fuel particles. The corresponding numerical computations are conducted focusing on two crucial influencing factors, i.e. (1) the

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particle size and (2) the fission-enhanced irradiation creep in the matrix. Moreover, for the sake of guaranteeing a good representation of the experimental work, the micro-scale finite element model is built with ellipsoidal particles. The effects of the particle size on the thermo-mechanical behavior are obtained. The mechanism is better interpreted for the cracking reported in the experimental observations[6].

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2 The Material Models

Taking the CERCER pellet as the research object, recrystallization-involved gas swelling in UO2 and the irradiated creep in MgO are mainly considered. The calculation also allows for the porosity-induced degradation of thermal conductivity and Young’s modulus in particles. The other thermo-mechanical properties of the fuel particles and matrix are used as the ones presented in Ref.[21].

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2.1 Thermo-mechanical properties with fission bubble induced porosity

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Porosity refers to a percentage of the differential particle volume taken up by the pores. The initial particle porosity caused by fabrication is set as p0 = 5% . As the burnup increases, the porosity variation is mainly due to the fission gas products. The relation between porosity and gas swelling can be derived and expressed as[25] p gas =

∆ V gas V

∆ V gas V0 sw _ gas = = = V0 + ∆ V 1 + ∆ V 1 + sw _ eng V0 ∆ V gas

p = p 0 + p gas (1 − p 0 ) where

pgas is

(1)

(2)

the gas-induced porosity, sw _ gas is the volumetric gas swelling strain and

sw_eng is the total volumetric swelling strain with respect to the initial volume. (1)Thermal conductivity The thermal conductivity model for fuel particles at high burnup is given as [11] 3

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K = K0 ×FD×FP×FR×FM

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(3) where K in W/m.K is the thermal conductivity of UO2; K0 is the thermal conductivity for un-irradiated fuels; FD characterizes the effect of dissolved fission products; FR quantifies the precipitated solid fission products; FR describes the effect of irradiation damage; FM is the modified factor for the effect of porosity stemming from fission products with the following expression as 1− p (4) FM = 1 + (s − 1) p where p signifies the porosity in Eq. (2), s is the pore shape factor with the value set as 1.5 for spherical bubbles; other specific expressions for the parameters in Eq.(3) can be found in Ref.[21]. (2) Elastic modulus and Poisson’s ratio The model used to characterize the temperature and porosity related Young’s modulus of UO2 is given as [11]

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E =2.26×105[1−1.131×10−4(T −273.15)][1−2.62p] (5) where E in MPa is the elastic modulus, T in K is temperature, p denotes the porosity, as given in 2.1 Irradiation swelling in UO2

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Eqs.(1-2). Poisson’s ratio of UO2 particles is a constant which is set as 0.316.

The total irradiation swelling of the particle point is composed of two contributions from fission solid products and fission gas products. For the solid swelling, its correlation with fission density is expressed as[26]

∆Vsolid = 2.5 ×10−29 × Fd V

(6)

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where F d is the fission density in fission/m 3 , V refers to the original volume. Another swelling contribution stems from the fission gas products, which evolves complexly with the burnup. The theoretical model of fission gas diffusion with the consideration of resolution and recrystallization effects can be found in Appendix of Ref.[25]. The semi-analytical solution

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for the fission gas model[18] is adopted for the preparation of gas-swelling calculation. Before recrystallization ( F ≤ Fdx , Fdx = 4 × 10 24 ( f& ) 2/15 ; Fdx in fissions/m3 denotes the critical fission density; f& is the fission rate in fissions/m3s), the fission gas swelling is expressed as the sum of the intragranular bubble swelling and the intergranular bubble swelling, that is ∆Vgas ∆Vi nt ra ∆Vint er (7) = + V V V

∆Vi nt ra 4π 3 = rb cb V 3

(8)

∆Vint er 2π Rb3Cb = V rgr 0

(9)

where Eq.(8) denotes the intragranular bubble swelling; rb is the radius of the intragranular bubble;

cb is the average concentration of intragranular bubble; Eq. (9) denotes the intergranular bubble swelling; C b is the grain-boundary bubble concentration; Rb denotes the intergranular bubble radius and

rgr 0 is the original grain radius. The calculation of the variables in the above expressions can

be found in Ref.[25], while the radius Rb will be further worked out through a nonlinear equation.

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ACCEPTED MANUSCRIPT After recrystallization ( F ≥ Fdx ), the recrystallization is supposed to take place in the original grain. Thus, the total fission gas swelling can be written as   ∆Vgas  ∆V   ∆V   ∆V  (10) =(1−Vr )  i nt ra  + int er   +Vr  int er   V  V  V  r   V  rgrx r gr gr  

rgr

where V r represents the volume fraction of the recrystallized area; grain radius which do not include the recrystallized part;

represents the current

rgrx represents the

radius of the fine

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grains. It can be seen that the unrecrystallized region still experiences intragranular and intragranular swelling. While in the recrystallized part the gas swelling is supposed to be only contributed from the intergranular one due to the gas depletion phenomenon in the recrystallized area[26] and almost all the generated gas atoms move to the grain boundaries without time consuming. The used calculation formula for variables in the above expressions can be found in Ref.[25]. And the intergranular bubble radius should be acquired through the nonlinear iteration as well.

2.2 Irradiation Creep Model in MgO

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Under the continuous attack of the fast neutrons and high energy fragments, the MgO matrix experiences irradiation creep. The used creep strain rate model is expressed as

ε&cr = c0ψσ strain rate(/s); c0 is

(11)

where ε& cr is the irradiation creep the creep coefficient that its value is − 27 −2 − 1 2.1 × 10 (MPa n cm ) in this paper, which is obtained through numerical simulations as

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presented in our previous research[24]. σ denotes the equivalent Mises Stress in the unit of MPa and ψ is the fast neutron flux (n.cm-2s−1) . The above creep model demonstrates a linear relationship between irradiation creep and the equivalent stress for ceramic materials [27]. In this study, the effective fast neutron flux (including the effect of fission fragments) in the fission-enhanced matrix layer is supposed one thousand times[28] of that in the other matrix layer.

3 Three-dimensional incremental constitutive relations[25]

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To realize a precise simulation of the irradiation-induced thermo-mechanical coupling behavior in ADS composite pellets, the specific three-dimensional mechanical constitutive relationships with irradiation-related properties involved are built in the first place. The entire simulation time can be divided into plenty of time increments. The incremental mechanical constitutive relations and the corresponding stress update algorithms for particle and matrix materials are primarily presented in this section. Then the mechanical constitutive relations can be described in subroutines UMAT based on the algorithms, and eventually the subroutines are used in the finite element simulation. Considering the large deformation description, a rotating coordinate system is used to describe the three-dimensional constitutive relations and stress update algorithms. So, the incremental constitutive relation during a time interval [t,t + ∆t ] is given firstly. For an integration point, the constitutive relation in an incremental form for an integration point can be acquired as[25]

∆σ ij = 2G (T + ∆T , t + ∆t ) ∆ε ije + λ (T + ∆T , t + ∆t ) ∆ε kke δ ij + 2∆Gε ij ( ) + ∆λε kk( )δ ij et

where

e t

(12)

∆σij are the Cauchy stress increments and ∆εije are the elastic logarithmic strain

increments; λ and G are the Lame coefficients corresponds to time t + ∆t , which are both temperature and time related; ∆G and ∆λ are the increments of the Lame coefficients. Afterwards, based on the incremental constitutive relations, the stress update algorithms for 5

ACCEPTED MANUSCRIPT the particle and matrix materials can be developed. Firstly, for UO2 particles, elastic strains together with thermal ones, irradiation swelling ones and plastic ones are considered taking place during a time increment. As a result, the logarithmic elastic strain increments at a material point can be determined as

∆εije = ∆εijtotal − ∆εijth − ∆ε ijsw − ∆εijp where

(13)

∆εijtotal denotes the total strain increments.

∆ε ijsw =

ln (1 + SW (t + ∆t ) ) 3 volumetric

δ ij − ε ijsw( t )

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∆ε ijsw in Eq.(13) represents the irradiation swelling strain increments with the expression as (14)

swelling engineering strain with SW ( t + ∆ t ) denotes the ∆V ∆Vsolid ∆Vgas , while the detailed expressions can be found in section 2.1 and SW(t +∆t) = = + V V V Appendix of Ref.[25], which are obtained according to the grain-scale theoretical analysis;

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where

ε ijsw ( t ) depicts the logarithmic form of irradiation swelling strains at time t , which have already

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been known. It is known in Eq.(9) that the intergranular swelling is directly related to the radius Rb of intergranular bubble. While Rb satisfies the nonlinear equation, given as ∆  2γ  4π R3  b g (Rb ) =   − hsbv Nb  − Nb kT = 0   Rb   3

(15)

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where R b signifies the intergranular bubble radius in the recrystallized or unrecrystallized region. Newton-Raphson(N-R) iteration method is adopted to work out the convergent solution of the equation. Thus, SW (t + ∆ t ) can be calculated once the convergence is achieved and eventually the irradiation swelling strain increments Meanwhile,

∆ε ijsw can be obtained.

∆εijth in Eq.(13) denotes the thermal strain increments, which is a spherical

tensor[23] related to the temperature variation ∆T . And ∆ε ijp [23] denotes the contribution of

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plastic strain increments, which should be taken into account when the corresponding Mises stress satisfies the yield condition; otherwise it is set as zero. The specific stress update algorithms can be found in our previous work[23]. Secondly, for MgO matrix, the total strain increments in the integration points are considered to include the elastic ones, the thermal expansion ones and the creep ones. As a result, the increments of elastic strain

where ∆ε ij

total

∆εije can be written as ∆εije = ∆εijtotal −∆εijth −∆εijcr

(16)

denotes the total strain increments, while the superscripts th and cr

describe the thermal strain increments and the creep strain increments [25], separately. The detailed stress update algorithm can be found in Ref.[25].

4 Finite Element Modeling 4.1 FE Models A pellet consisting of an MgO matrix and the embedded UO2 particles is regarded as the research object of this study. Similar to our previous studies[21], the geometric models are given on the base of the assumption that the particles are distributed periodically along both axial and 6

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circumferential directions, as depicted in Fig.1. First of all, in order to explore the effect of particle size, the spherical-shaped particles with the radius being 200um(Model 1), 112um(Model 2) and 50um(Model 3) are considered. And the influence of the particle size is assessed with a constant particle volume fraction of 6.23% and a pellet radius of 4.15mm in each model. With consideration of the periodicity and symmetry of the pellet configuration, the finite element models are correspondingly set up, as illustrated in Fig.2. Secondly, Model 4 in Fig.3 is established with the consideration of a 10um-thick fission-enhanced layer (colored in yellow), to investigate its effect on the thermo-mechanical behavior in the pellet. Besides, the particles in Model 4 are ellipse-shaped with the length of its longer axis being 400um and the length ratio of the longer axis to the shorter axis being 5:4. And the longer axis is in the radial direction of the pellet, similar to the scenario in the experiment [6]. Attacked by the fission fragment and fast neutrons, the effective fast neutron flux is used to present the enhanced irradiation creep effect, in this work a value of 2.5×1016 n/cm2s is set which is one thousand times of the one in the other matrix zone.

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4.2 Load and Boundary Conditions

4.3 Mesh generation

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The bonding between the fuel particles and the matrix are perceived as perfect in the FE models demonstrated in Fig.2 and Fig.3. The used boundary conditions to determine the temperature and stress fields are same for the four finite element models. The applied pressure on the exterior surface is 10MPa, along with a constant temperature of 873K set on the surface. The other surfaces are treated with the symmetric boundary conditions. The adopted irradiation conditions for the models are referenced from the experiment operation in Ref.[6]. The fission rate of the particles is 2.5 ×1020 fission/m3s and the corresponding heat generation can be calculated as a value of 8W/mm3. The fast neutron flux is set as 2.5×1013 n/cm2s in Model 1 and Model 2 and Model 3, and the one in the fission-enhanced layer of Model 4 has been heightened as mentioned above.

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The grid partitioning to the models are performed with the element named C3D10MT in ABAQUS, and the mesh details are summarized in Table 1. It can be depicted in Fig.2 and Fig.3, that the mesh is refined in the fission-enhanced region. A mesh convergence in all the simulations is obtained. Simulations in the study are implemented with UMAT and UMATHT to define the special thermo-mechanical constitutive relations, as discussed in Ref.[24]. The strengthened irradiation creep effect in the fission-enhanced matrix layer of Model 4 is involved with the enlarged effective fast neutron flux in Eq. (11).

Fig.1 Schematic figure of a composite pellet with periodically-distributed particles

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(a)

(b)

(c)

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Fig.2 FE models for different particle radii (a) 200um in Model 1; (b) 112um in Model 2; (c)

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50um in Model 3

Fig.3 FE model for Model 4 with consideration of fission-enhanced matrix layer (highlighted in yellow)

Table 1 Gird information in different models

Model

Model 1

Model 2

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FE

(r=200um)

(r=112um)

Model 3

Model 4

(r=50um)

(ellipse-shaped inclusion)

151337

172334

379124

242951

Node

217143

252451

559832

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Element

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5 Results and discussions The coupled temperature-displacement analysis is successfully performed with the developed subroutines. The correctness and effectiveness of the subroutines, including UMAT and UMATHT, were validated and clarified in our previous work [24]. All the simulations are carried out for deep burnup, which contains two stages. The initial burnup stage is corresponding to a short time stage within 0.01day, during which the temperature variation induced effect is only considered. Another is called a higher burnup stage with the simulation time across [0.01d~230d], in which the main irradiation-induced effects are involved, including the particle swelling and irradiation creep in the matrix. In this section, analysis based on the obtained simulation results will be demonstrated specifically. Numerical results related to the variations of particle diameters are given in Section 5.1. Investigation regarding the fission-enhanced matrix layer is reported in Section 5.2. And the used paths and nodes to output the calculation results are displayed in Fig. 4.

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(a)

(b)

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Fig.4 Outputted paths and nodes (a) in Model 2 and (b) in Model 4

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5.1 Effect of particle size on the thermo-mechanical behavior in pellets 5.1.1 Temperature field

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Fig.5(a) gives the contour plot of temperature distribution in Model 2 on 115th day. It is depicted that the central particle in the pellet experiences the highest temperature, and the temperature decreases gradually from the center to its rim due to heat generation in the fuel particles. The temperature distribution along a radial path (Path 1 in Fig.4(a)) on 115th day for the considered particle diameter cases are outputted and given in Fig.5(b) to assess the particle size on the temperature evolution. It can be found that the case with a smaller particle size indicates a lower pellet temperature field, with the peak temperature value reaching 1126.92K (Model 1), 1097.55K (Model 2) and 1020.12K(Model 3) on 115th day. Besides, the maximum temperature evolution with burnup in each model is also compared and exhibited in Fig.6(a). It is clearly noticeable that the temperature increases with burnup for each case, whereas the temperature rise is observed smaller for the case with a smaller particle size. From the initial stage to 230th day, the central temperature increases by 125.7K (Model 1), 103.97K (Model 2) and 59.04K (Model 3), respectively. The above comparison of the obtained results indicates that the fuel pellet with smaller particles performs much lower and smoother temperature distribution. One plausible explanation can be that with a certain fuel volume fraction, the heat transfer effectiveness between the particles and the matrix is higher in the model with smaller-size particles. For a better understanding, a path (Path 2 in Fig.4(a)) on the particle surface is selected to explore the surface heat flux of particles in each model. As depicted in Fig.6 (b), the particle surface heat flux increases dramatically with the particle size. Given that the heat generation in each model is identical, the case with larger number of smaller-size particles means to have more particle-matrix interface areas. And consequently the total heat energy transfer from the fuel particle to its adjacent matrix can be significantly improved, and the corresponding heat flux and temperature gradient is lower. Moreover, the distance between two neighboring particles is shortened in the small-size-particle model and results in a smoother temperature distribution between the particles, which is consistent with the results in Fig. 5(b). It can be seen that the temperature field in pellets is strongly affected by the particle size. Higher temperature variation and temperature gradient would also lead to large thermal stresses, which could damage the pellet integrity at the early burnup stage. Hence, the particle size ought to be designed properly to guarantee the in-pile safety.

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ACCEPTED MANUSCRIPT r=200µm(Model 1) r=112µm(Model 2) r=50µm (Model 3)

Temperature(K)

1120

1050

980

910 1

(a)

2 3 Distance(mm)

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0

(b)

Fig.5 (a) Contour plot of temperature distribution on 115th day in Model 2; (b) temperature variation on

1120 1080 1040

Heat Flux (W/mm2)

Temperature(K)

1160

0.64 0.56

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r=200µm(Model 1) r=112µm(Model 2) r=50µm (Model 3)

1200

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115th day along Path 1 in cases with different particle diameters

r=200µm(Model 1) r=112µm(Model 2) r=50µm (Model 3)

0.48 0.40 0.32 0.24 0.16

1000 0

50

100 150 burnup(days)

250

0.08

0.0

0.2 0.4 Distance(mm)

0.6

(b)

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(a)

200

Fig.6 (a) evolution of maximum temperature with increasing burnup and (b) the heat flux distribution on the select particle surface on 115th day in the considered particle diameter cases

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5.1.2 Irradiation Swelling and Gas-induced Porosity

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The irradiation swelling model involving the gas resolution and recrystallization effects has been developed and presented in the Appendix of Ref.[25]. The contour plot in Fig.7 (a) displays the engineering swelling strains on 115th day in the particles of Model 2(r=112 um ), which indicates that the central particle swelling reaches the maximum. Meanwhile, evolution of the volumetric irradiation swelling on the central particle (Node P in Fig.4(a)), varying with particle diameters at different burnup levels is given in Fig.8(a). First of all, it is worth mentioning that the swelling evolution follows the same pattern that it speeds up after the critical fission density (97.2d) due to the recrystallization. Secondly, the volumetric swelling rises as the particle size increases. Since the fission solid induced swelling is only burnup-dependent, the particle size merely affects the gas swelling contribution in these models. On one hand, before recrystallization, it is observed that the particle swelling difference between the models with different particle sizes is little. For example, the total swelling on 95th day is revealed to be 10.51% (Model 1), 10.17% (Model 2) and 9.89% (Model 3) in Fig.8(a), respectively. On the other hand, after recrystallization, the increasing recrystallization-induced gas swelling is quite dominant, which yields gradually large swelling differences in different particle size models. The gas contribution in the recrystallized area of the corresponding grains is shown in Fig.8(b). It can be noticed that on 230th day, the intergranular gas swelling from the recrystallized area reaches 30.18%(Model 1), 29.04%(Model 2), 26.25%(Model 3); while the total swelling magnitude achieves 42.61% (Model 1), 41.45% (Model 2) and 38.68% (Model 3). On 230th day, 10

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the fission solid swelling becomes 12.42% and the gas swelling from the unrecrystallized area in three models is decreasing to zero because the recrystallization process has been completed and the original grain is totally occupied by the recrystallized area with fine grains. It can be seen that the particle size has a relatively large influence on the gas swelling after recrystallization. As mentioned in our previous work[24], the fission gas swelling is closely associated with the bubble size evolution. So the bubble radius evolutions of Node P in different particle diameter models are given in Fig.9. It can be depicted that in the unrecrystallized region, the intergranular bubble radius R b has a small magnitude value and it increases first and then decreases until the recrystallization process ends (see Fig.9(a)). Simultaneously, the bubble radius R b x in recrystallized region keeps growing after the critical fission density. The intergranular bubble radius in the recrystallized and unrecrystallized regions goes up with increasing the particle size and the magnitude reaches 0.0232um in Model 1, 0.0229um in Model 2 and in 0.0221um in Model 3 on 230th day. Besides, higher temperature happens in the cases with bigger particle diameters, which facilitates the formation of larger gas bubbles as explained in Ref.[24]. As is known, that pores are formed in the fuel particles due to the generated gas bubbles. Fig.7 (b) demonstrates the contour plot of particle porosity distribution on 115th day in Model 2. It reveals that the center particle holds the largest porosity where the maximum gas swelling also exhibits. The effect of the particle size on the porosity has been assessed through the outputted porosity of Node P at different burnup levels, as shown in Fig. 10. It is noted that the porosity evolution has a similar characteristic compared to the irradiation swelling in Fig.8(a). Moreover, the fission-gas-induced porosity increases more quickly after recrystallization and at a certain irradiation time, the porosity is heightened for the larger particle size case. On 230th day, the gas-induced porosity reaches up to 21.16% in Model 1, 20.59% in Model 2 and 19.02% in Model 3. Simultaneously, the formed porosity will also result in a degradation of thermal conductivity, ultimately increasing the fuel temperature. The effect of porosity on the thermal-elastic property of particles had been investigated in Ref.[25].

(a)

(b)

Fig.7 Contour plot on 115th day in Mode 2(r=112 µm ) (a) swelling strain distribution; (b) gas-induced

r=200µm(Model 1) r=112µm(Model 2) r=50µm (Model 3) fission solid swelling

0.4

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porosity distribution

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ACCEPTED MANUSCRIPT Fig.8 (a) Evolution of irradiation swelling of Node P with increasing irradiation time; (b) evolution of gas swelling in the recrystallized region of Node P in the considered particle diameter cases

Rbx(mm)

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Fig.9 Radius of the integranular bubble (a) in the unrecrystallized region and (b) in the recrystallized

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Fig.10 Porosity evolution of Node P with irradiation time in different particle diameter cases

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5.1.3 Irradiated creep in the matrix Path 3 in Fig.4(a) along the radial direction with a length of 1.5cm, across the particle-matrix interfaces and the matrix between the particles, is chosen to output the equivalent creep strains. So the creep evolution can be captured for the three models with different particle diameters. The equivalent creep strains on the selected path at the initial stage and on 115th day are provided in Fig.11, respectively. The plot with the solid icons is corresponding to the creep results at the particle-matrix interfaces, while the plot with the hollow icons presents the creep distribution in the matrix between the fuel particles. One can see that nearby the interfaces between the particles and the matrix much larger magnitude of creep strains occur and the magnitudes increase with the particle diameter. At the starting point of the path the largest equivalent creep strains appear. On 0.04th day the maximum value reaches approximately 3.3162e-5 in Model 1, 3.06021e-5 in Model 2 and 2.60639e-5 in Model 3. And on 115th day, the magnitude goes up to 0.09414, 0.09269 and 0.08887. Whereas, it can be found from Fig.11(b) that the equivalent creep strains in the matrix between the two adjacent particles become smaller compared to the ones at the interfaces. As we know that the equivalent creep strain in this work is majorly affected by the Mises 12

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stress. At the early burnup stage, the thermal stress contribution is dominated and at higher burnup， the remarkable irradiation swelling occurring in the fuel particles will directly strengthen the mechanical interaction between the fuel particles and the matrix. It can be obtained from Fig.11(b) that the maximum equivalent creep strain on Path 3 is weakened for the three cases, and it can be predicted that the Mises stresses in the matrix tend to be closer with increasing burnup. The corresponding mechanical behavior evolution in the matrix will be discussed in Section 5.1.5.

Fig.11 Distribution of the equivalent creep strains along Path 3 (a) on 0.04th day and (b) on 115th day

5.1.4 Deformation behavior

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Excessive radial deformation in pellets could narrow the pellet-cladding gap or even makes the pellets contact with the cladding to result in strong mechanical interaction between them, which will bring a safety hazard during operation. Hence, the pellet deformation behavior should be properly investigated. Contour plots of the displacement magnitude and the x-direction displacement component on 115th day in Model 2 are given in Fig.12. It is worth to mention that within a Cartesian coordinate framework, the x-direction displacement U1 along the radial path (Path 1in Fig.4(a)) in Fig.12(a) is in accord with its radial-direction one in Fig.12(b). Therefore, the x-directional displacement component along Path 1 is exported to investigate the radial deformation of pellets. In the first place, the radial deformation evolution with burnup in Model 2 is given in Fig. 13(a). It can be obtained that (1) the deformation between the matrix-particle interfaces displays a smaller value than those at the adjacent particle positions, which attributes to the large compressive strains appearing in the matrix, as explained already in Ref.[24]; (b) the radial deformation increases as the irradiation time increases and on 230th day the radial displacement at the pellet’s outer surface reaches up to 0.07539mm (1.82%). Additionally, the effect of particle size on the pellet deformation is studied and the comparison of the radial displacement is denoted in Fig.13(b). One can observe that the deformation distribution for different particle sizes is similar to the one in Fig.13(a). The radial displacement at the outer surface of the pellet attains up to 0.07916(1.91%) in Model 1, 0.07539(1.82%) in Model 2 and 0.05495(1.32%) in Model 3 on 230th day. Thus, the pellet deformation increases with the fuel particle size, which stems from larger thermal-induced expansion and irradiation swelling in fuel particles.

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(a)

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Fig. 12 Contour plots of (a) the displacement magnitude and (b) x-directional displacement in

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Fig. 13 (a) Evolution of pellet radial deformation with burnup in Model 2; (b) distribution of pellet

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radial deformation in different particle radius cases on 230th day

5.1.5 Stress in the Matrix

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Under the in-pile service condition, evolution of stresses in the matrix can be intricate. Firstly, at the initial stage, high temperature gradient and different thermal expansions in the fuel particles and the matrix are the leading cause of stresses in the matrix, such that the stresses at this stage are called thermal stresses. Afterwards, the stresses will evolve resulting from the combined effects of irradiation swelling in the particles and irradiation creep in the matrix. Based on the radial cracks observed in irradiation experiments, the following stress analysis is given through a path along the radial direction (Path 2 in Fig.4(a)). Besides, it is well known that the first principal stress accounts for the failure of ceramic materials. And it has been illustrated in Ref.[21] that the planes with the maximum tensile stresses are perpendicular to the pellet axial direction. As a result, the distribution and evolution of the first principal stress along a radial direction (Path 2) in the matrix are mainly investigated in this section. Fig.14 gives the first principal stress distribution along Path 2 in the matrix at different burnup in the considered three cases with different particle sizes. The compared results indicate that the stress evolution shows a similar characteristic. The initial results(0.001d) reveal the largest first principal stress appears at the bottom of the path and then the stress decreases (before 11.5th day). On 57.5th day the distribution is changed with the largest stress appearing at the two ends of 14

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the path with the stresses enlarged as a whole. It can be explained with the fact that during this irradiation time interval, the irradiation creep relaxing effect in the matrix competes with the particle swelling contribution which could enhance the mechanical interaction between the fuel particles and the matrix. Before 11.5th day, the creep relaxing effect dominates, and then the irradiation swelling in the fuel particles gradually plays an important role. At higher irradiation time (103.5d) it can be observed that the stresses are prone to increasing because the particle swelling results in an enhanced load to the matrix. Although the stresses in different models share the similar distribution rule, it can be observed from Fig.14 that the stress evolution varies with the particle size. In order to gain a specific knowledge of the impact of the particle size, the first principal stresses along Path 3 (with a length of 1.5cm) at the initial stage and on 115th day are provided in Fig.16, separately. One can see that the stresses at the interfaces are distinctively larger than the ones locating between the particles. The initial thermal stresses in Fig.15 (a) show that the maximum value increases with the particle size reaching up to 915.4MPa in Model 1, 868.1MPa in Model 2 and 747.5MPa in Model 3. This phenomenon is due to the non-homogeneous thermal expansion strains in the fuel particles and the matrix. For a better understanding, the thermal strains in the particle and the matrix at the initial stage are displayed in Fig.16, respectively. It can be found that (1) thermal expansion strains in the particles present relatively larger values than the ones in the matrix because of the large thermal expansion coefficient of particles; (2) the thermal expansion strains increase along with the size of particles due to the higher temperature. It is important to notice from Fig. 15(b) that the first principal stresses reveal no obvious difference in different models, while the location of the maximum stress changes.

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(c) Fig. 14 Evolution of the first principal stress with burnup in (a) Model 1; (b) Model 2; (c) Model 3

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Fig. 15 Distribution of the first principal stress along Path3 with different particle radii (a) on 0.003d; (b)

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Fig. 16 Distribution of the thermal strains on 0.003d along Path 3 (a) in matrix; (b) in particles

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Since the fission-enhanced matrix layer experiences an enhanced irradiation creep effect, the stresses there will be more relaxed, which was regarded as a possible reason for the special matrix cracking found in the irradiation experiment [6]. In this section, the equivalent creep strains in the fission-enhanced layer and in the other matrix region are investigated and compared. Evolution of the first principal stresses will be emphatically analyzed in order to interpret the cracking mechanism. 5.2.1 Equivalent irradiation creep distribution

Fig.17 presents the contour plot of equivalent creep strains on 0.3th day in the fission-enhanced layer and the other matrix region, separately. Path 4 (in the other matrix) and Path 5(in the fission enhanced layer) in Fig.4(b) are chosen to study the equivalent creep strain

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evolution.

It can be found from Fig.18 that the distribution characteristics of equivalent creep strains along Path 4 and Path 5 demonstrate similarly that the peak magnitude appears at the lowest point

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of the paths. Also, it is noticeable that the largest equivalent creep strain (~0.01021) in the fission enhanced layer along Path 5 is much larger than the one (~9.25972e-7) along Path 4. It can be understandable that the equivalent fast neutron flux in the fission-enhanced layer has a much larger value to result in a strengthened equivalent creep strain rate.

Meanwhile, at higher burnup, it can be observed in Fig.19 that the equivalent creep strains on both paths are increasing progressively with the irradiation time. However, compared to the one in Fig.19(b), the equivalent creep strains in the fission-enhanced layer (Path 5) show a different

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distribution in Fig.19(a). The peak magnitude transfers to the location nearby the bottom in Fig.19(a), which presents two maximums. Because the irradiation creep behavior is governed by the fast neutron flux and Mises stresses in the matrix, the presented equivalent creep strain distribution and evolution in Fig.19(a) are mainly induced by the stress evolution. Hence, in the following section a detailed investigation of the stress distribution and evolution in different

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matrix zones will be given.

(a)

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Fig.17 Contour plot of the equivalent creep strains on 0.3d (a) in the fission-enhanced matrix and (b) in the other matrix

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Fig.18 Equivalent creep distribution at the initial burnup stage (a) in the fission-enhanced matrix(Path 5) and (b) in

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Fig.19 The equivalent creep strain evolution with the irradiation time (a) in the fission-enhanced matrix(Path 5) and (b) in the other matrix(Path 4)

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5.2.2 Evaluation of the stresses at the matrix-particle interfaces Fig.20 gives the contour plot of the first principal stress in the fission-enhanced matrix layer and the other matrix regions at the initial stage. With the purpose of exploring the first principal

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stresses evolution in different matrix zones, the stresses along Path 4 and Path 5 are outputted and compared, as shown in Fig.21. Firstly, on 0.001th day when the irradiation creep is not involved, the thermal stresses are

predominant in the matrix. It is noted that (1) the first principal stress shows similar distribution characters in the two regions with the maximum magnitude appearing at the lowest positions of the paths; (2) the stresses in the fission-enhanced layer present higher magnitudes than the ones in the other matrix region, with the peak values equal to 843.7MPa and 698.6MPa. It could be explained by the fact that the fission-enhanced layer are in the vicinity of the interface between the fuel particle and the matrix, the mechanical interaction there is strengthened in order to realize deformation compatibility. At higher irradiation time, one can observe from Fig.21(b) that the stresses evolve with the irradiation time, having a similar evolution law as discussed in section 5.1.5. Whereas, the stresses in Fig.21(a) exhibit different evolution characteristics. As previously discussed in Ref.[24], the 18

ACCEPTED MANUSCRIPT matrix stresses at higher burnup are majorly affected from the combined effects of matrix irradiation creep and particle swelling. They have opposite influences on the stress evolution. On one hand, the particle swelling accumulation will strengthen the mechanical interaction between the particles and the matrix. On the other hand, the irradiation creep occurring in the matrix will weaken the stresses in the matrix[24]. Since the equivalent fast neutron flux in the fission-enhanced matrix layer is assumed to be thousand-fold over the one in the other matrix zone, the induced creep influence is playing a leading role in this layer during the irradiation. And it

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explains the phenomenon in Fig.21(a) that the stresses decrease rapidly and turn into compressive ones at long-time irradiation.

Comparing the stress evolution in Fig.21(a) and Fig.21(b), one can see that at higher irradiation time, the first principal stresses in Fig.21(b) remain tensile with the maximum magnitude locating at the two ends, while in fission-enhanced layer there are relatively large integrity of the ceramic matrix than the compressive stresses.

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compressive ones. As we know, the appeared large tensile stresses could be more dangerous to the In order to explore and explain the mechanism of the radial cracks observed in the irradiation

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experiments [6], the axial stress component S33 at different burnup stages is provided in Fig.22. It can be found that (1)on 0.001th day, the axial stresses are all negative, namely compressive stresses exist in the fission-enhanced layer as well as the other matrix region; (2)subsequently, in the matrix excluding the fission-enhanced layer, the compressive stresses in the vicinity of the two ends gradually evolve into tensile stresses and progressively increase with irradiation time; however the maximum compressive stresses maintain at the bottom; it can be observed that at higher burnup the axial tensile stress components are close to the first principal stresses at the path

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ends;(3) at the same time, at higher burnup, the axial stresses in the fission-enhanced layer (Fig.22(a)) evolve into compressive ones; and one can see that at higher burnup the axial compressive stresses also have similar magnitudes as the ones of the first principal stresses at the two path ends (see Fig.21(a)).

Comparing the first principal stresses and axial stresses at high irradiation time, one can find

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that the axial symmetrical plane of the pellet (also along the symmetrical plane of the fuel particles) is the principal plane with a peak magnitude of tensile stresses existing in the matrix surrounding the fission-enhanced layer. Then the radial cracking failure can only occur in the matrix keeping a

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distance away from the interfaces between the fuel particles and the matrix. This phenomenon was found in the irradiation experiments[6]. As the crack was observed propagating along the radial direction in the matrix, the first principal stresses and axial stresses along a radial path between two particles are investigated in the next section.

(a)

(b)

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ACCEPTED MANUSCRIPT Fig.20 Contour plot of the first principal stress at the initial stage (a) in the fission-enhanced layer; (b) in the other

0.001d 11.5d 57.5d 115d 172.5d

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Fig.21 The distribution of the first principal stress (a) in the fission-enhanced matrix layer(Path 5) and (b) in the

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Fig.22 The axial stress evolution (a) in the fission-enhanced matrix layer(Path 5) and (b) in the other matrix region(Path 4)

5.2.3 The stresses in the radial path between the fuel particles The contour plots of the first principal stress and axial stress in the matrix between two

particles on 115th day are separately presented in Fig.23. It is clearly noticeable that the stresses in the fission-enhanced layer differ a lot from those in the other matrix zone. For a better understanding, Fig.24 gives the first principal stress and axial stress distribution along a path between two adjacent particles (Path 6 in Fig.4(b)) at different burnup stages, respectively. From Fig.24(a), it can be seen that (1) because the thermal effect is the only considered effect at the initial stage, therefore the particle-matrix interfaces (in the fission-enhanced layer) hold the largest stress magnitude with its value up to 548.1MPa, and the minimum value is equal to 145.6MPa 20

ACCEPTED MANUSCRIPT appearing near the middle of the path; (2) at higher irradiation time, the corresponding stresses in the fission-enhanced layer evolve into compressive stresses due to the strong irradiation creep effect; thus, the maximum tensile stresses transfer to the matrix adjacent to the fission-enhanced layer and the value increases with the irradiation time and shortly exceeds the maximum tensile stress at the initial stage. Correspondingly, it can be observed from Fig.24(b) that (1) at the initial stage, the axial stresses are compressive ones;(2) at higher irradiation time, similar to the ones in Fig.24(a), the axial stresses in the fission-enhanced layer keep compressive; while in the other

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matrix region the axial stresses change from compressive stresses to tensile stresses and increase with the irradiation time. In the crossing point between the fission-enhanced layer and the other matrix, the radial cracking is prone to happen, which agrees with the experimental observation[6]. As analyzed in Section 5.1, the particle size has an important influence on the stress distribution and evolution in the matrix. Here we will further explore the effect of the particle size

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on the stresses in the fission-enhanced layer and the other matrix region. For the two considered cases with the radius of the particle being 112m and 50um, Fig.25 and Fig.26 demonstrate the distribution and evolutions of the first principal stress and axial stress between the two adjacent

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particles near the center of the pellet, respectively. As a whole, the stresses present the same distribution features as the ones given in Fig.24. It can be noticed that (1) the largest stress magnitude exists in the interface between the fission-enhanced layer and the other matrix, and for the case with larger-sized particles the first principal stress is ~1130.9MPa(r=12um) in Fig.25(a) and for the other case the value is ~804.0MPa(r=50um) in Fig.25(b) on 115th day; (2) the first principal stresses in the fission-enhanced layer exhibit compressive ones, which is mainly due to the considerable enhanced creep effect in this region. As provided in Fig.27, it can be found that

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the equivalent creep strains in the fission-enhanced layer are notably larger than the ones in the other region. The maximum equivalent creep strain reaches ~0.145 in Fig.27(a) and 0.121 in Fig.27(b). So the comparatively higher creep strains could protect the fission-enhanced layer from cracking. From Fig.26, one can see again that the maximum axial stresses are nearly identical to the maximum magnitudes of the first principal stress, which implies that the radial cracking is

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possible along the pellet cross section. The relatively smaller axial stresses exist in the cases with the particle radius being 50um, which signifies this particle size should be a good choice for

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CERCER fuel pellets.

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Fig.23 Contour plot of (a) the first principal stress; (b) the axial stress between two adjacent particles on 115th day

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Fig.24 Evolution of (a) the first principal stress and (b) axial stress between two adjacent particles(Path 6)

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Fig.25 Evolution of the first principal stress between two adjacent particles with the particle radius being (a) 112um and (b) 50um

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Fig.26 Evolution of the axial stress between two adjacent particles with the particle radius being (a) 112um and (b) 50um

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Fig.27 Distribution of the equivalent creep strains between two adjacent particles with the particle radius being (a)

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6 Conclusions

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In this work, the effects of the fuel particle size are numerically simulated on the thermo-mechanical in-pile behavior in the composite pellets. Besides, the fission-enhanced matrix layer is developed in the finite element geometric model for the pellets with ellipse-shaped particles. The results are to be used for detailed interpretation of the cracking phenomenon in the irradiation experiment. The irradiation effects such as the grain recrystallization and the fission gas induced porosity variation in the fuel particles are taken into account together with the irradiation creep effects in the pellet matrix. The methodology is implemented in the subroutines

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UMAT and UMATHT to define the complex thermo-mechanical constitutive relations in the ABAQUS standard analysis. The obtained conclusions can be summarized as follows. (1) For the cases with a larger particle radius, the maximum temperature in the pellet center goes up more quickly, and the phenomenon of local temperature increase in the fuel particles appears more evident. radius.

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(2) The particle swelling and radial deformation in the pellet increase with the fuel particle (3) At the initial stage, the maximum magnitude of the first principal stress in the particle-matrix

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interface increases with the particle size. While at the higher burnup stage, the corresponding stress reveals no significant difference for different particle size cases due to the combined effect of matrix creep, particle swelling and distance between two adjacent particles.

(4) Large compressive stresses are observed in the fission-enhanced layer due to the significant irradiation creep effect there. It can be predicted from the first principal stress evolution results that radial cracks will emerge in the matrix outside the fission-enhanced layer, which is consistent with the experimental observation. The maximum axial stresses at higher irradiation time are much smaller so that it is more difficult to cause cracking for the pellet with the particle radius being 50um, compared to the case with a 112um particle radius.

ACKNOWLEDGEMENTS The authors thank for the support of the National Key Research and Development Program of

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Reference

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A grain-scale gaseous swelling model allowing for the development of recrystallization and resolution is adopted for particles. >The influence of fission-gas-induced porosity is considered in the constitutive relations for particles. > A simulation method is developed for the multi-scale thermo-mechanical behavior. > The effects of fuel particle size and fission-fragment-enhanced irradiation creep are investigated in CERCER pellets.