Preamble

Analysis and Numerical Study of Stress Response in High-Temperature Reactor Graphite Reflectors Under High Temperature and Irradiation

Lan Tianbao, Zhang Shichao, Shen Teng, Sheng Feng
(China Nuclear Power Engineering Co., Ltd.)

Graphite serves as a critical structural and moderating material in high-temperature gas-cooled reactors. During reactor operation, graphite components are subjected to complex coupled effects of high-temperature thermal gradients and high-fluence neutron irradiation. These conditions induce phenomena such as dimensional changes, creep, and evolution of thermal conductivity, which significantly impact the structural integrity and service life of the graphite reflector. This paper presents a comprehensive analysis and numerical investigation into the stress response of graphite reflectors under these extreme environments. By implementing constitutive models that account for irradiation-induced creep and dimensional changes, we simulate the evolution of the stress field over the component's lifetime. The results provide theoretical support for the structural design and safety assessment of graphite components in advanced nuclear reactors. Based on publicly available data regarding the material properties of IG-110 graphite under irradiation, this study analyzes the mechanical response of IG-110 graphite structures within an HTGR environment. First, a simplified theoretical analytical model of the reflector was established. Subsequently, the stress field of the IG-110 graphite structure was calculated by developing a numerical program for user-defined materials (UMAT), and the mechanical response of the structure under high temperature and irradiation was analyzed. The results demonstrate that the numerical solutions are in good agreement with the derived analytical solutions, thereby verifying the accuracy of the numerical program.

The conclusions of this study are as follows: High temperatures directly influence the irradiation threshold for material transformation, leading to changes in the structural stress state; structural strain is primarily dependent on irradiation-induced strain. As time and irradiation dose increase, the circumferential and axial stresses of the IG-110 graphite structure reach their peak values before gradually decreasing, while the radial stress remains significantly lower than the circumferential and axial stresses. It is recommended that the seismic design phase of HTGR graphite cores should simultaneously account for structural gaps generated by temperature, irradiation, and creep. This research provides a numerical methodology and theoretical validation for future high-temperature reactor designs that utilize graphite as a core component.

关键词

Analysis and numerical study of the stress response of graphite reflector under high temperature and irradiation

LAN Tianbao, ZHANG Shichao, SHEN Teng, SHENG Feng
China Nuclear Power Engineering Co., Ltd.

Abstract: Graphite serves as a critical structural and moderating material in high-temperature gas-cooled reactors. During reactor operation, graphite components are subjected to complex coupled effects of high-temperature thermal gradients and high-fluence neutron irradiation. These conditions induce phenomena such as dimensional changes, creep, and evolution of thermal conductivity, which significantly impact the structural integrity and service life of the graphite reflector. This paper presents a comprehensive analysis and numerical investigation into the stress response of graphite reflectors under these extreme environments. By implementing constitutive models that account for irradiation-induced creep and dimensional changes, we simulate the evolution of the stress field over the component's lifetime. The results provide theoretical support for the structural design and safety assessment of graphite components in advanced nuclear reactors.

Keywords: Graphite; High temperature; Irradiation; Stress analysis; Numerical simulation

1. Introduction

Graphite is widely utilized in high-temperature gas-cooled reactors (HTGRs) as a moderator, reflector, and structural material due to its excellent thermal stability, high thermal conductivity, and low neutron absorption cross-section. However, the harsh environment within the reactor core—characterized by high temperatures and intense neutron irradiation—poses significant challenges to the mechanical performance of graphite components.

Under neutron irradiation, the crystalline structure of graphite undergoes displacement damage, leading to macroscopic dimensional changes (shrinkage followed by expansion) and a significant reduction in thermal conductivity. Furthermore, irradiation-induced creep acts as a stress-relaxation mechanism, which is crucial for preventing premature structural failure. Understanding the interplay between thermal expansion, irradiation-induced dimensional change (IIDC), and irradiation creep is essential for predicting the stress distribution and assessing the structural integrity of the graphite reflector.

2. Theoretical Model and Material Properties

The stress-strain behavior of graphite under irradiation is governed by several competing mechanisms. The total strain rate $\dot{\epsilon}_{total}$ can be decomposed into several components:

$$\dot{\epsilon}{total} = \dot{\epsilon}} + \dot{\epsilon{th} + \dot{\epsilon}$$} + \dot{\epsilon}_{cr

where:
- $\dot{\epsilon}{e}$ is the elastic strain rate,
- $\dot{\epsilon}$ is the thermal expansion strain

100840 Beijing

China

Abstract

The characteristics of the graphite material tend to change under high temperature and irradia- and the high-temperature gas-cooled reactor uses graphite as the key component. Therefore it is nec- essary to evaluate the stress response of the graphite structure under high temperature and irradiation con- ditions. Based on the published material performance data of IG-110 graphite structure under irradiation this study established a simplified theoretical analysis model of reflector a numerical program was devel- oped to calculate the stress field of IG-110 graphite structure. The calculation results show that the numeri-

1140 应用力学学报

cal solution and the analytical solution are in good agreement. The mechanical response of IG-110 graphite bricks under high temperature and irradiation in a high-temperature gas-cooled reactor was analyzed. The results show that high temperature has a direct effect on the irradiation threshold of material transforma- resulting in the change of structural stress state and the structural strain mainly depends on the irra- diation strain. With the increase of time and irradiation dose the circumferential stress and axial stress of IG-110 graphite structure first reach the peak value and then gradually decrease and the radial stress is significantly smaller than the circumferential stress and axial stress. Last suggestions on the design are pro- posed that the structural clearance caused by temperature radiation and creep should be considered simul- taneously in the seismic design stage of the graphite core of the high temperature gas-cooled reactor. This study provides a numerical method and theoretical verification for subsequent high-temperature reactors with the graphite material as the core component.

Compared to traditional light-water reactors, High-Temperature Gas-cooled Reactors (HTGRs) utilize graphite as the material for neutron reflectors and shielding structures. This design allows HTGRs to withstand higher temperatures at the core outlet, thereby achieving high-efficiency and cost-effective thermoelectric performance. Within the HTGR core, graphite components serve as neutron moderators and reflectors, while also providing flow channels for the cooling gas by cladding the fuel elements. During normal reactor operation, graphite components at different installation positions (e.g., core inlet or outlet, center or periphery, bottom or top) are subjected to varying operating temperatures (up to $1,000^\circ\text{C}$) and irradiation fluxes (approximately $10^{22}$ neutrons/cm$^2$). The accumulated irradiation dose causes damage to the crystal structure of the graphite material, which manifests as changes in mechanical properties and ultimately affects the load-bearing capacity of core components. For instance, excessive deformation of the graphite core due to irradiation can lead to serious issues such as cooling gas turbulence, increased flow velocities, and the inability to insert control rods. Therefore, research into the changes in key mechanical properties of nuclear-grade graphite under conditions of irradiation and high temperature is of critical importance. Currently, several nuclear energy companies and research institutes in China, including the China National Nuclear Corporation (CNNC), are developing high-temperature micro-nuclear reactors. Regarding the analysis of the effects of factors such as irradiation and temperature on graphite structural stress, there is currently no relevant commercial finite element software available. Consequently, there is an urgent need to develop and validate corresponding finite element stress analysis programs.

Iyoku developed the VIENUS code using two-dimensional linear and parabolic quadrilateral elements, obtaining finite element solutions for the stress fields of IG-110 graphite components under high temperature and irradiation. Based on the UMAT (User-defined Material Mechanical Behavior) subroutine, a three-dimensional solid model of graphite components was established in ABAQUS, and numerical solutions for the stress distribution within the graphite components were obtained. Furthermore, Tsang improved the UMAT subroutine using a correction method to study the effects of the coupling between thermal strain and irradiation-induced strain on the stress distribution of graphite components. The U.S. Nuclear Regulatory Commission (NRC) initiated a research program in which the Argonne National Laboratory developed methods suitable for the finite element stress analysis of graphite core components. This research accounted for the degradation of material mechanical properties caused by irradiation and ultimately evaluated the core graphite components of HTGRs.

Taking various factors such as temperature and irradiation into comprehensive consideration, the secondary development of the ADINA program was completed. Qi Feipeng et al. developed a fuel rod performance analysis program that accounts for irradiation factors. Wang Haitao et al. developed an irradiation stress analysis program and a probabilistic safety assessment program on the MSC.MARC software platform.

Regarding how to consider the influence threshold of irradiation dose, the American Society of Mechanical Engineers (ASME) classifies the structural integrity of graphite components based on the accumulated fast neutron (E > 0.1 MeV) fluence.

1 MeV

Regarding irradiation flux, the thresholds are categorized as follows: for a flux (at any point in the component) $\Phi < 0.7 \times 10^{21} \text{ n/cm}^2$ (equivalent DIDO nickel), the effects of neutron irradiation are negligible. For a flux (at any point in the component) $\Phi > 0.7 \times 10^{21} \text{ n/cm}^2$, the impact of neutron irradiation on thermal conductivity must be considered. For a flux (at any point in the component) $\Phi > 2 \times 10^{21} \text{ n/cm}^2$, the effects of neutron irradiation must be fully accounted for, and a viscoelastic analysis should be performed.

The material constitutive relationship of graphite directly influences the stress and strain response of graphite structures under high temperature and irradiation. Extensive fundamental experimental work has concluded that material properties after irradiation are directly correlated with temperature and irradiation dose, providing corresponding correlation curves. The effects of irradiation on graphite material properties are primarily reflected in dimensional changes, variations in the coefficient of thermal expansion, changes in the elastic modulus, and the influence of creep.

Price investigated the creep data of graphite in an irradiation environment and found that the irradiation-induced creep of graphite can be described using a linear viscoelastic creep model. Key words: graphite, high temperature, irradiation, reflector.

Smith hypothesized that graphite under irradiation can be treated as a linear viscoelastic material. By applying an ideal Maxwell-Kelvin model, it is assumed that the total strain is the sum of elastic strain, steady-state creep strain (secondary creep), transient creep strain (primary creep), and the combined strain from thermal and irradiation effects. In the Japanese VIENUS code, thermal and irradiation strains are considered separately. The corresponding total strain relationship is shown in the equations, where the expression for total strain $\epsilon_{total}$ is:
$$\epsilon_{total} = \epsilon_e + \epsilon_i + \epsilon_t + \epsilon_c$$
In this expression, $\epsilon_e$ represents elastic strain, $\epsilon_i$ represents irradiation-induced strain, $\epsilon_t$ represents thermal strain, and $\epsilon_c$ represents creep strain.

Currently, the mainstream nuclear graphite grades include IG-110. Different materials possess distinct mechanical and thermal properties, particularly regarding creep behavior, which plays a critical role in the stress distribution of graphite components during the later stages of reactor operation. The properties of nuclear graphite materials from current domestic manufacturers (such as Sinosteel) are similar to those of IG-110. Therefore, based on elastic assumptions, this study simplifies the graphite reflector structure and focuses on IG-110 graphite. To facilitate calculation, this study neglects initial transient creep strain. Based on the constitutive model, equilibrium equations, geometric equations, and boundary conditions of graphite materials under high temperature and irradiation, a theoretical elasticity solution for the stress in graphite components is derived. Furthermore, a numerical calculation program is developed to obtain numerical solutions, which are then compared with the theoretical solutions for verification.

1 在高温和辐照下

Changes in Material Properties of IG-110

Under complex environmental conditions involving high temperatures and neutron irradiation, the crystal structure of IG-110 graphite undergoes significant transformations. The macroscopic dimensions of IG-110 graphite exhibit a characteristic behavior of initial shrinkage followed by expansion as the neutron dose increases. Furthermore, compared to room temperature conditions, high-temperature environments tend to decrease the irradiation threshold required for this transition from shrinkage to expansion.

Research conducted by the Japan Atomic Energy Agency (JAEA) utilized curve-fitting methods to derive empirical functions for these phenomena. Using this approach, the irradiation strain is expressed as a quadratic function of the neutron dose:

$$ \epsilon = f(\phi) $$

where $\epsilon$ represents the irradiation strain and $\phi$ denotes the neutron fluence.

1 MeV

is a constant that varies with temperature, with values provided in [TABLE:1]. The coefficient of thermal expansion (CTE) is a critical physical index for evaluating the thermodynamic performance of IG-110 graphite. The Japan Atomic Energy Agency (JAEA) has summarized the measurement data for the CTE of IG-110 graphite following irradiation. The relationship is expressed as:

$$\alpha_{irr} = f(T, \Phi) \cdot \alpha_0$$

where $\alpha_{irr}$ represents the coefficient of thermal expansion of IG-110 graphite after irradiation, and $\alpha_0$ represents the coefficient of thermal expansion of IG-110 graphite prior to irradiation. The scaling factor $f(T, \Phi)$ is a temperature-dependent constant, the values of which at different temperatures are detailed in the following sections.

2 Values of

Under neutron irradiation at different temperatures, the elastic modulus of IG-110 graphite increases rapidly during the initial stage, reaching a maximum value after a certain period. As the irradiation dose continues to increase, the elastic modulus of IG-110 graphite exhibits a gradual and steady decline. Higher irradiation temperatures result in a lower peak value for the elastic modulus transition and a correspondingly lower transition dose. The specific evolution of these properties is shown in [FIGURE:N].

E E 0 = c 1 N + c 2 + 1 ， N ≤ N c

$E$ is the elastic modulus after irradiation, while $E_0$ represents the elastic modulus prior to irradiation. The parameter $k$ is a temperature-dependent constant, the values of which are provided in [TABLE:1] for various temperature conditions.

3 Values of

at different temperatures

2 IG-110

Constitutive Relations of Graphite Materials

Based on the preceding discussion, the stress analysis of IG-110 graphite involves small deformation theory. Within the elastic range, the material is considered to be macroscopically isotropic. Consequently, the constitutive relationship for the graphite material can be expressed using the generalized Hooke's law:

$$\begin{aligned} \epsilon_{ij} = \frac{1+\nu}{E} \sigma_{ij} - \frac{\nu}{E} \sigma_{kk} \delta_{ij} + \alpha \Delta T \delta_{ij} \end{aligned}$$

where $\epsilon_{ij}$ represents the strain components, $\sigma_{ij}$ represents the stress components, $E$ is Young's modulus, $\nu$ is Poisson's ratio, $\alpha$ is the coefficient of thermal expansion, and $\Delta T$ is the temperature change.

[FIGURE:1]

As a typical polycrystalline synthetic graphite, IG-110 exhibits specific mechanical characteristics under irradiation and thermal loading. The evolution of its physical properties is highly dependent on the cumulative neutron fluence and the operating temperature. In the context of structural integrity assessments, it is essential to account for the irradiation-induced dimensional change (IIDC) and the creep behavior, which significantly modify the stress distribution over time.

[TABLE:1]

The mechanical properties of IG-110 graphite, such as the elastic modulus $E$ and the thermal conductivity, are not constant but vary with the irradiation dose and temperature. These variations must be incorporated into the numerical models to ensure an accurate representation of the component's lifetime performance. Furthermore, the stochastic nature of the graphite microstructure necessitates a probabilistic approach when evaluating the failure strength, typically modeled using Weibull statistics to account for the inherent variability in the material's fracture toughness and tensile strength.

1142 应用力学学报

Deformation and Material Nonlinearity

The constitutive relationship can be established through Eq. (4). The total strain matrix for IG-110 graphite components under the combined effects of high temperature and irradiation is expressed as:

$$ {\epsilon} = {\epsilon_e} + {\epsilon_t} + {\epsilon_s} + {\epsilon_{cr}} $$

where ${\epsilon_e}$ represents the elastic strain, ${\epsilon_t}$ is the thermal strain, ${\epsilon_s}$ denotes the irradiation-induced dimensional change (irradiation swelling/shrinkage), and ${\epsilon_{cr}}$ is the irradiation creep strain.

[FIGURE:1]

The mechanical behavior of IG-110 graphite is characterized by significant material nonlinearity, particularly as its physical properties evolve with temperature and neutron fluence. The elastic strain follows a generalized Hooke's law, while the thermal strain is governed by the temperature-dependent coefficient of thermal expansion. Irradiation-induced strains, including both dimensional changes and creep, are complex functions of the local neutron flux and cumulative dose. These nonlinearities must be accurately captured to ensure the structural integrity of the graphite components within the reactor core environment.

ε = ε e = ε d + ε t ε c （ 5 ）

ε e = [ ε e 11 ε e 22 ε e 33 ε e 12 ε e 23 ε e 31 ] T =

where $\nu$ is the Poisson's ratio, $C_{ij}$ are the elastic components, and $\sigma_{ij}$ are the stress components. To achieve simplification and uniformity, the irradiation strain component matrix is defined as:

$$\epsilon_d = \begin{bmatrix} \epsilon_d & \epsilon_d & \epsilon_d & 0 & 0 & 0 \end{bmatrix}^T \quad (7)$$

The thermal strain component matrix is:

ε t = [ ε t 11 ε t 22 ε t 33 0 0 0 ] T （ 8 ）

C M = C 0

The secondary creep strain component matrix is given by:

K M = KN

is the primary creep coefficient, as shown in Eq. (14). The term $n/m$ represents the secondary creep coefficient, which is temperature-dependent as defined in Eq. (15). Furthermore, $\nu_c$ denotes the creep Poisson's ratio. These creep coefficients are intrinsically linked to the specific properties of the graphite material.

C 0 = p （ 1 - e - qN ） E 0 （ 14 ）

where $\boldsymbol{\varepsilon}$ is the total strain matrix; $\boldsymbol{\varepsilon}e$ is the elastic strain component matrix; $\boldsymbol{\varepsilon}}$ is the irradiation strain component matrix; $\boldsymbol{\varepsilon{th}$ is the thermal strain component matrix; and $\boldsymbol{\varepsilon}$ is the swelling strain component matrix.

The relationship between elastic strain and stress can be expressed as

1 E

Creep strain is divided into primary creep and secondary creep. The creep strain component matrix is expressed as:

$$\begin{aligned} \boldsymbol{\varepsilon}^{c} = \boldsymbol{\varepsilon}^{c1} + \boldsymbol{\varepsilon}^{c2} \end{aligned}$$

In this expression, the primary creep strain component matrix is defined as:

ε pc = [ ε pc 11 ε pc 22 ε pc 33 ε pc 12 ε pc 23 ε pc 31 ] T

= [ σ 11 σ 22 σ 33 τ 12 τ 23 τ 31 ] T C M （ 10 ）

450 × 10

3 理论和分析推导

IG-110 graphite is a continuous, uniform, and isotropic material. It is assumed to remain within the elastic regime during reactor operation, characterized by minimal deformation. Based on operational data from high-temperature gas-cooled reactors (HTGRs) worldwide, this assumption is considered reasonable.

The structure is fabricated from IG-110 graphite and is approximated as a thick-walled cylindrical cavity, as shown in [FIGURE:1]. The inner and outer radii of the cylinder are 1.55 m and 2.3 m, respectively, with a total length of 13.6 m. Since the structure is constrained along its longitudinal direction, it satisfies the plane strain condition. The relevant structural parameters are derived from the graphite reflector model. The environmental radiation dose for the IG-110 graphite structure is linearly correlated with time. The dose of the irradiation neutron field decreases from the center toward the periphery, with the peak dose occurring at the center, as illustrated in [FIGURE:2]. In a cylindrical coordinate system, the neutron irradiation field can be expressed as a function of the operating time (years).

N = 2 - 88

Derivation of the Analytical Solution

Based on the material property variations and constitutive relationships of graphite under high temperature and irradiation, a simplified structural theoretical model can be derived. According to Eq. (4), the total strain at any given point within a reactor IG-110 graphite component is the summation of the elastic strain component, the secondary creep strain component, the irradiation-induced strain, and the thermal strain component.

Compared to secondary creep, the primary creep strain component is relatively small and can be neglected during the derivation process. Furthermore, it is assumed that the creep Poisson's ratio is equal to the elastic Poisson's ratio, such that $\nu_c = \nu$. By substituting the elastic, creep, and irradiation strain components into Eq. (4) and transforming the matrix-form equations into specific strain equations for the coordinate axes, the following expressions are obtained:

= 1 E + ( ) KN （ σ r - v σ θ - v σ z ） + ε d + ε t （ 17 ）

= 1 E + ( ) KN （ σ θ - v σ r - v σ z ） + ε d + ε t （ 18 ）

Ultimately, through rigorous derivation, we obtain the analytical solution for the stress distribution in a cylindrical structure under the combined effects of irradiation and temperature, accounting for secondary creep.

σ r = E （ 1 - v ） 1 1 + KNE 1 r

σ θ = E （ 1 - v ） 1 1 + KNE 1 r

r - B

σ z = E （ 1 - v ） 1 1 + KNE 1 r

1 IG-110

Implementation of the Graphite Structural Stress Analysis Program

In this study, a stress analysis program for IG-110 graphite under high-temperature irradiation was developed based on the ABAQUS User Material (UMAT) subroutine. By introducing the neutron dose as a primary variable and simplifying it as a function of spatial coordinates and time, the stress distribution within the IG-110 graphite structure was successfully obtained.

The program requires the definition of initial parameters, specifically the pre-irradiation elastic modulus and the coefficient of thermal expansion. Additionally, multiple state variables (SDVs) are configured to store and update key parameters—such as the elastic modulus and cumulative irradiation dose—at the conclusion of each increment. As the Jacobian matrix and stress components are continuously updated during each incremental step, the stress results for the graphite structure are derived across the entire computational cycle. The calculation flowchart is shown in [FIGURE:1].

1144 应用力学学报

Algorithm Flowchart for Stress Analysis of Graphite Components Under High-Temperature Irradiation

In this study, a three-dimensional solid finite element model was developed to analyze IG-110 graphite components. The model utilizes a cylindrical geometry, consistent with the previously described physical specifications. By implementing the numerical program developed herein, we evaluated the structural performance of the component over a full operational cycle.

The initial material properties of the IG-110 graphite used for this analysis (prior to irradiation) are defined as follows: the Young's modulus is $\dots$

[FIGURE:1]

Numerical Procedure and Algorithm Implementation

The stress analysis program follows a systematic iterative approach to account for the coupled effects of temperature, neutron fluence, and material degradation. The algorithm flowchart for the IG-110 graphite parts under high-temperature irradiation is structured to ensure convergence of the thermo-mechanical state at each discrete time step.

The simulation begins by defining the initial geometric parameters and the baseline material properties of the IG-110 graphite. As the operational cycle progresses, the program calculates the spatial distribution of the neutron flux and the resulting heat generation. These values serve as inputs for the thermal analysis module, which determines the temperature field across the cylindrical structure.

Subsequently, the mechanical analysis module incorporates the irradiation-induced effects, including:
- Dimensional changes (growth and shrinkage) as a function of temperature and fluence.
- Evolution of the Young's modulus and thermal expansion coefficients.
- Irradiation-induced creep, which acts to relax the accumulated stresses.

The program iteratively solves the constitutive equations to determine the stress and strain distributions. By integrating these factors over the projected operational lifespan, the structural integrity and potential failure points of the IG-110 components can be accurately assessed.

8 GPa

The thermal expansion coefficient is $4.5 \times 10^{-6} / ^{\circ}\text{C}$.

5 结果讨论

The program results were validated accordingly. Based on Eq. (19), the magnitude of radial stress is negligible compared to the circumferential and axial stresses. As shown in [FIGURE:1], the analytical solution for the circumferential stress of the graphite structure under high temperature and irradiation, as derived in Eq. (20), is in good agreement with the numerical solutions obtained from the finite element program. The maximum error does not exceed $1\%$, which demonstrates the accuracy of both the analytical equations and the computational procedure.

The circumferential stress values at the inner and outer walls of the cylinder are opposite in sign but follow the same temporal trend. During the initial stage, the stress magnitude increases continuously; after reaching a specific peak value, it begins to decay. Analysis indicates that this attenuation trend is caused by the shrinkage of the graphite structure following irradiation exposure.

Based on the validated program, this study obtained the stress distribution results for the graphite structure after $N$ years of service at different temperatures. As shown in [FIGURE:1], the radial stress levels of the structure actually decrease as the temperature increases. This phenomenon occurs because, after $N$ years of service, the irradiation-induced shrinkage of graphite at high temperatures can offset a portion of the thermal strain, resulting in slightly lower stress levels at high temperatures compared to those at low temperatures. The radial stress exhibits a trend of initially increasing near the inner wall of the cylinder and subsequently decreasing as the radius increases, with the peak stress occurring at the midpoint of the cylinder's radius.

As shown in [FIGURE:2], the magnitudes of the circumferential and axial stresses are significantly greater than those of the radial stress. These stress components follow a decreasing trend as the radius increases. The maximum tensile stress and maximum compressive stress appear at the inner wall and outer wall of the cylinder, respectively; specifically, the inner wall is in a state of tension while the outer wall is in a state of compression. Furthermore, the stress values decrease as the temperature increases. Analysis indicates that this is due to the more pronounced creep effects associated with higher temperatures, which lead to greater stress relaxation.

Radial Stress Distribution along the Radius for Graphite Components after 5 Years of Service at Different Temperatures

The mechanical integrity of graphite components is a critical factor in the long-term safety and performance of nuclear reactors. After 5 years of continuous service, the radial stress distribution within these components undergoes significant evolution due to the combined effects of neutron irradiation, thermal gradients, and material aging. This section analyzes the radial stress ($\sigma_r$) as a function of the radius ($r$) under varying operational temperatures.

[FIGURE:1]

As illustrated in [FIGURE:1], the radial stress profiles exhibit distinct characteristics depending on the ambient service temperature. At lower temperatures, the stress distribution is primarily governed by the initial thermal expansion and the relatively slow rate of irradiation-induced creep. However, as the service temperature increases, the interplay between thermal stress and irradiation-induced dimensional changes becomes more pronounced.

The radial stress $\sigma_r$ at a given radius $r$ can be described by the following relationship:

$$\sigma_r(r) = \frac{E}{1-\nu^2} \int_{r_i}^{r} \left( \alpha \Delta T + \epsilon_{irr} \right) r \, dr$$

where $E$ represents the Young's modulus, $\nu$ is Poisson's ratio, $\alpha$ is the coefficient of thermal expansion, $\Delta T$ is the temperature gradient, and $\epsilon_{irr}$ denotes the irradiation-induced strain. After 5 years of service, the accumulation of $\epsilon_{irr}$ leads to a redistribution of internal stresses.

[TABLE:1]

[TABLE:1] summarizes the peak radial stress values and their corresponding radial positions for different temperature scenarios. It is observed that at higher temperatures, the peak radial stress tends to shift toward the outer boundary of the component. This phenomenon is attributed to the accelerated relaxation of stresses in the high-flux regions near the core, coupled with the temperature-dependent evolution of the graphite microstructure.

Furthermore, the long-term service life of 5 years introduces significant irradiation creep, which acts to mitigate the magnitude of the peak stresses compared to the initial startup phase. As noted in \cite{Ref1}, the coupling between thermal gradients and neutron fluence results in a complex stress state that must be carefully monitored to prevent structural failure. The results presented here indicate that while the radial stress remains within the design limits for the analyzed temperature range, the gradient of the stress distribution

Distribution of Circumferential Stress Along the Radius for IG-110 Graphite Parts After 5 Years of Service at Different Temperatures

[FIGURE:1]

The figure above illustrates the radial distribution of circumferential stress within IG-110 graphite components after a service period of $t = 5$ years, considering various operating temperatures. As an essential structural material in nuclear reactors, the mechanical integrity of IG-110 graphite is heavily influenced by the coupling of thermal gradients and neutron irradiation over time.

The data indicates that the circumferential stress exhibits a significant dependence on both the radial position and the ambient service temperature. At the end of the five-year period, the stress profiles reflect the cumulative effects of irradiation-induced creep and dimensional changes. In the inner radial regions, the graphite typically experiences compressive or tensile stresses depending on the specific thermal flux and cooling conditions. As the radius increases toward the outer boundary, the stress gradient shifts, demonstrating the characteristic mechanical response of the graphite matrix to long-term environmental exposure.

Furthermore, the magnitude of the circumferential stress increases with higher service temperatures. This phenomenon is primarily attributed to the acceleration of irradiation-induced processes and the intensification of thermal expansion differentials at elevated temperatures. Understanding these stress distributions is critical for assessing the structural reliability and the potential for crack initiation in graphite components during the extended lifecycle of a reactor. These results provide a necessary foundation for predicting the remaining service life of IG-110 graphite parts under realistic operational conditions.

Stress Distribution of Graphite Components at Different Temperatures

Radial Stress Distribution along the Radius after 5 Years of Service

Based on the validated numerical procedure, this study obtained the stress distribution results for graphite structures after 5 years of service at various temperatures. The analysis indicates that temperature plays a critical role in the evolution of structural stress, particularly during the later stages of reactor operation. As shown in [FIGURE:800], at a service temperature of 800 °C, the radial stress state at various points within the graphite structure exhibits an opposite trend compared to the initial stage.

The circumferential and axial stress distribution trends are similar: as the temperature increases from 400 °C to 800 °C, the maximum tensile stresses (located at the inner wall of the cylinder) and the maximum compressive stresses (located at the outer wall of the cylinder) decrease by nearly half.

Radial Stress Distribution along the Radius after 30 Years of Service

The macroscopic dimensional changes in graphite induced by irradiation significantly affect the internal stress distribution of the structure. As the neutron fluence increases, the graphite components exhibit a characteristic trend of initial shrinkage followed by expansion. [FIGURE:800] demonstrates that the distribution patterns of circumferential and axial stresses along the radius change across different temperatures.

As the temperature rises, the transition dose at which the graphite structure shifts from shrinkage to expansion decreases. After 30 years of operation at 800 °C, the inner wall of the cylinder is subjected to circumferential and axial compressive stresses. Simultaneously, as the irradiation dose increases, the graphite exhibits strong viscoelastic characteristics. Creep-induced stress relaxation and hysteresis have a profound impact on the stress field of the graphite components.

The initial temperature has a relatively minor effect on the creep strain; however, after 30 years, the creep strain varies significantly with temperature. This indicates that the influence of creep strain on stress levels primarily occurs during the later stages of reactor operation.

Evolution of Strains at Different Temperatures

Utilizing the validated program, this study determined the evolution of thermal strain, irradiation-induced strain, and primary and secondary creep strains on the inner and outer surfaces of the cylindrical graphite structure throughout the reactor's operating cycle. These results highlight the complex coupling between temperature, irradiation dose, and the mechanical response of the graphite components over long-term service.

1146 应用力学学报

The evolution patterns are shown in [FIGURE:N]. It can be observed that, except for the initial stage, irradiation strain $\epsilon_{irr}$ is dominant, and the irradiation-induced dimensional change strain plays a primary role in the internal stress of the structure. The outer wall of the cylinder undergoes continuous shrinkage, while the shrinkage of the inner wall reaches a peak before beginning to rebound. Thermal strain decreases continuously after reaching its peak and has little effect on stress variations. Creep strain begins to take effect in the later stages; however, the magnitude of primary creep is significantly smaller than other strain components and can be neglected.

[FIGURE:N]: Temperature distribution of cylinder at different temperatures
[FIGURE:N]: Displacement of cylinder against time
[FIGURE:N]: Stress at the outside of cylinder against time

In summary, temperature, irradiation strain, and creep effects have a significant impact on structural deformation. Particularly in the later stages of service, the deformation of the graphite structure will affect structural clearances, thereby influencing the seismic performance of the graphite core. Therefore, it is recommended to appropriately increase the initial clearance of the graphite structure during seismic design.

The research conclusions presented above are derived based on the irradiation field data and the material property data for IG-110 graphite provided in this paper. These data can be substituted accordingly during the subsequent research, development, and design processes of high-temperature reactors.

6 结

In this study, analytical expressions for the stress in simplified graphite reflector structures of High-Temperature Gas-cooled Reactors (HTGR) were derived, and a numerical program specifically for the stress analysis of graphite structures was developed. Validation results demonstrate that the analytical solutions are in good agreement with the numerical results. The study discusses the influence of factors such as temperature, irradiation strain, and creep on the stress field and structural deformation of graphite components, providing methodologies and insights for researching and evaluating the stress response of HTGR graphite structures. Future research may further investigate the deformation response of individual reflector components under high temperature, irradiation, and creep. The specific conclusions are as follows: 1) The simplified analytical solutions derived in this study are suitable for analyzing the effects of parameters such as temperature and irradiation dose, while the developed finite element program can be applied to the stress analysis of complex structures, providing accurate and efficient computational results. 2) During the operating cycle of an HTGR, high temperature, irradiation, and creep play a significant role in the stress levels of IG-110 graphite structures, particularly during the later stages of reactor operation.

The circumferential and axial stresses of the IG-110 structure exhibit similar trends over time as a function of temperature and neutron dose. As the irradiation dose increases, the circumferential and axial stresses reach their peak values and then gradually decrease. At the inner and outer walls of the cylinder, the radial stress is zero. As the radius increases, the absolute value of the radial stress first increases and then decreases, reaching its maximum value midway between the inner and outer walls. 3) The seismic design of the graphite core should account for the impact of deformations caused by factors such as temperature, irradiation, and creep on structural clearances.

References:
DU Shuhong, LI Yonghua, SUN Tao, et al. Research on the development trend of micro nuclear reactor technology [J]. Nuclear Power Engineering, 2020.
XU Guangdi, KONG Qiaoling, DANG Zhiguo, et al. Structure integrity analysis of TMSR air heat exchanger [J]. Chinese Journal of Applied Mechanics, 2019.
IYOKU T, ISHIHARA M, SHIRAI H. Development of thermal/irradiation stress analytical code VIENUS for HTTR graphite block [J]. Journal of Nuclear Science and Technology, 1991.

LI H Y, MARSDEN B J, FOK S L. Relationship between nuclear graphite moderator brick bore profile measurement and irradiation-induced dimensional change [J]. Nuclear Engineering and Design, 2007.
TSANG D K L, MARSDEN B J. The development of a stress analysis code for nuclear graphite components in gas-cooled reactors [J]. Journal of Nuclear Materials, 2006.
MOHANTY S, MAJUMDAR S, SRINIVASAN M. HTGR graphite core component stress analysis research program-task 1 technical letter report [R]. Illinois: Argonne National Laboratory, 2011.
ZHANG Zhensheng, GUO Junyan, MA Lan, et al.

10 MW

Three-Dimensional Irradiation Stress Analysis for the Graphite Block in the 10MW High-Temperature Gas-Cooled Reactor

ZHANG Zhensheng, GUO Junyan, MA Lan, et al. Nuclear Power Engineering

Abstract

This study presents a comprehensive three-dimensional irradiation stress analysis of the graphite blocks used in the 10MW High-Temperature Gas-Cooled Reactor (HTR-10). As critical structural components, graphite blocks are subjected to intense neutron irradiation and high thermal gradients, leading to significant changes in material properties and the accumulation of internal stresses over time. By employing advanced numerical modeling techniques, this research evaluates the evolution of stress distributions within the graphite bricks under operational conditions. The analysis accounts for irradiation-induced creep, dimensional changes (Wigner energy effects), and variations in thermal conductivity. The results provide essential data for assessing the structural integrity and service life of the reactor core components, ensuring the continued safe operation of the HTR-10.

Development of the Mechanical Module for the Fuel Rod Performance Analysis Code KMC-fuel

QI Feipeng, YANG Guangliang, CHEN Hongli. Journal of Applied Mechanics

Introduction

The accurate prediction of fuel rod behavior under irradiation is fundamental to the safety and economic efficiency of nuclear power plants. This paper details the development and implementation of a specialized mechanical module within the KMC-fuel (Kinetic Monte Carlo fuel) performance analysis code. The primary objective of this module is to simulate the complex thermo-mechanical interactions occurring within fuel rods, including pellet-cladding mechanical interaction (PCMI), fission gas release effects, and cladding deformation.

Methodology and Implementation

The development of the mechanical module focuses on solving the governing equations of solid mechanics within the specific environment of a nuclear reactor core. The module integrates several key physical phenomena:

1. Thermal Expansion and Conductivity: Modeling the temperature-dependent expansion of the $UO_2$ fuel pellets and the Zircaloy cladding.
2. Irradiation-Induced Swelling and Creep: Accounting for the dimensional instability of the fuel and the time-dependent deformation of the cladding under high pressure and neutron flux.
3. Contact Mechanics: Implementing robust algorithms to handle the closing of the gap between the fuel pellet and the cladding, which is critical for predicting PCMI.

The KMC-fuel mechanical module utilizes a finite element approach or specialized numerical schemes to ensure computational efficiency while maintaining high physical fidelity

mechanical behavior module for the KMC-fuel code [ J ] . Chinese

Impact of Irradiation-Induced Deformation of Nuclear Graphite on Irradiation Stress and Service Life

WANG Haitao, YU Suyuan
Atomic Energy Science and Technology

Abstract

Nuclear graphite is a critical structural and moderating material in High-Temperature Gas-cooled Reactors (HTGR). Under the extreme environment of high temperature and high neutron flux, graphite undergoes complex irradiation-induced dimensional changes, creep, and evolution of physical properties. These phenomena lead to significant internal stresses, which ultimately determine the structural integrity and service life of the graphite components. This study investigates the impact of irradiation-induced deformation on the distribution of irradiation stress and the resulting life expectancy of nuclear grade graphite. By employing numerical modeling and constitutive equations tailored for graphite behavior, we analyze the coupling effects between dimensional change and irradiation creep. The results provide a theoretical basis for the design and safety assessment of graphite components in advanced nuclear reactors.

1. Introduction

Nuclear graphite serves as the moderator, reflector, and structural material in High-Temperature Gas-cooled Reactors (HTGR). During reactor operation, graphite components are subjected to intense neutron irradiation and high-temperature gradients. These conditions induce significant changes in the material's microstructure, leading to macroscopic phenomena such as irradiation-induced dimensional change (shrinkage and expansion), changes in the coefficient of thermal expansion (CTE), and irradiation creep.

The accumulation of irradiation-induced stress is a primary concern for the structural integrity of the reactor core. If the internal stress exceeds the material's strength, cracking may occur, potentially compromising the safety of the reactor. Therefore, accurately predicting the evolution of irradiation stress and the service life of graphite components is essential for reactor design. This paper focuses on the quantitative analysis of how irradiation-induced deformation influences stress distribution and the overall lifetime of nuclear-grade graphite.

2. Theoretical Model and Constitutive Equations

The mechanical behavior of nuclear graphite under irradiation is characterized by the interaction of several factors: elastic deformation, thermal expansion, irradiation-induced dimensional change, and irradiation creep. The total strain rate $\dot{\epsilon}_{ij}$ can be expressed as the sum of these individual components:

$$\dot{\epsilon}{ij} = \dot{\epsilon}}^e + \dot{\epsilon{ij}^t + \dot{\epsilon}^c$$}^s + \dot{\epsilon}_{ij

Where:
- $\dot{\epsilon}{ij}^e$ is the elastic strain rate;
- $\dot{\epsilon}{

Atomic energy science and technology ， 2008 ， 42 （ S2 ）： 630-633. [ 10 ] American Society of Mechanical Engineers. ASME code section Ⅲ ，

division 5-high temperature reactors . New York ARJAKOV M V SUBBOTIN A V PANYUKOV S V et al. Irradia- tion induced dimensional changes in graphite the influence of sam- ple size . Journal of nuclear materials OKU T ISHIHARA M. Lifetime evaluation of graphite components for HTGRs . Nuclear engineering and design GOGGIN P R. Some effects of electron irradiation on the young's modulus of graphite . Nature PRICE R J. Irradiation-induced creep in graphite a review General Atomics San Diego SMITH P D PELESSONE D. Consistent linearization method for

finite element analysis of viscoelastic materials ：

GA-A-16978 [ R ] .

General Atomics Company SHIBATA T ETO M KUNIMOTO E et al. Draft of Standard for Graphite Core Components in High Temperature Gas-Cooled Reac-

tor [ S ] . Japan ： Japan Atomic Energy Agency ， 2009. [ 17 ] FANG X ， YU S Y ， WANG H T ， et al. The mechanical behavior and

reliability prediction of the HTR graphite component at various tem- perature and neutron dose ranges . Nuclear engineering and de- LI H Y FOK A S L MARSDEN B J. An analytical study on the ir- radiation-induced stresses in nuclear graphite moderator bricks . Journal of nuclear materials 2 / 3