Friction and Contact Mechanics of Nuclear Graphite in High-Temperature Gas-Cooled Reactors: A Review

Driven by the goals of “carbon peaking and carbon neutrality”, nuclear energy, with its high energy density and near-zero carbon emissions during operation, has become an essential base-load power source in the future energy mix [1]. From experimental prototypes in the 1950s to Generation III advanced light water reactors, nuclear reactor technology has evolved to Generation IV, which is characterized by inherent safety, economic competitiveness, and proliferation resistance (Table 1). The Generation IV International Forum (GIF), established in 2001, selected six candidate reactor types, among which the Very High Temperature Reactor (VHTR) and its transitional type, the High Temperature Gas Cooled Reactor (HTGR), are considered among the most promising because they use helium cooling, TRISO (tristructural isotropic) coated particle fuel, and a graphite core, endowing them with the inherent safety feature of no core meltdown even under loss of all coolant and unattended conditions [2].

China holds a leading international position in HTGR technology. The 10 MW High Temperature Gas Cooled Test Reactor (HTR-10) (Figure 1, [3]), independently designed by the Institute of Nuclear and New Energy Technology of Tsinghua University, reached criticality in 2000 and full power operation in 2003, and successfully conducted three experiments demonstrating inherent safety, including station blackout and main helium blower shutdown. Taking HTR-10 as a prototype, the world’s first modular HTGR commercial demonstration plant, HTR-PM, began construction in Rongcheng, Shandong Province, in 2012, was connected to the grid in December 2021, and formally entered commercial operation after completing a 168-h continuous operation assessment in December 2023, signifying that China has fully mastered the complete technology of modular HTGRs [1].

Graphite plays an irreplaceable role in HTGRs because of its excellent neutron moderating power (moderating ratio ~200), high temperature mechanical strength (strength increases with temperature up to 2500  °C), high thermal conductivity (~125 W/(m·K)), very low thermal expansion coefficient (<4.5 × 10⁻⁶ /K), and good dimensional stability under irradiation [4]. In HTR-PM (Figure 2), approximately 1200 tons of structural graphite and about 80 tons of matrix graphite for fuel elements are used, constituting virtually all components inside the core except the fuel particles and control materials. The contact interface mechanical behavior of these graphite components—including the friction coefficient between graphite surfaces and the contact stiffness of spherical fuel elements—directly determines the packing structure of the pebble bed, its flow characteristics, the coolant flow field distribution, and the design parameters of the fuel handling system, which are crucial to reactor safety and performance [5].

However, owing to the experimental challenges posed by the harsh conditions of high temperature, high purity helium environment, and irradiation, research on the friction coefficient and contact stiffness of nuclear graphite specifically for HTGRs has long been extremely scarce. Most of the available data were obtained from ordinary industrial graphite under room temperature air conditions and cannot directly support the precise design requirements of the core. This paper, by integrating two recent systematic experimental studies targeting this field, starts from the material characteristics of nuclear graphite, comprehensively reviews the current state of research on graphite tribology and contact mechanics, and systematically presents the latest experimental findings and mechanistic analyses of the friction coefficient of nuclear graphite and the stiffness coefficient of fuel spheres in a high temperature helium environment.

The technological origin of HTGRs can be traced back to the 1950s’ Magnox gas cooled reactors, which used natural uranium as fuel and carbon dioxide as coolant, and subsequently evolved through Advanced Gas cooled Reactors (AGR) to advanced reactors cooled by helium and moderated by graphite. Internationally, experimental and prototype reactors such as the Dragon reactor in the UK, the AVR and THTR-300 in Germany, the HTTR in Japan, and the GT-MHR cooperatively developed by the USA and Russia have validated the feasibility of HTGRs from an engineering perspective [6]. HTR-PM adopts two pebble bed modular reactors coupled to a single steam turbine generator set, with the core loaded with approximately 420,000 spherical fuel elements of 60 mm diameter. The designed core outlet temperature is 750  °C, and the thermal power is 250 MWth [7].

Graphite performs three main functions in HTGRs [8,9]:

The friction behavior at graphite interfaces directly affects the above functions: friction between fuel spheres and structural graphite can generate graphite dust, which, once accumulated to a certain extent, may impact normal reactor operation [10]; the friction force between fuel spheres in the pebble bed is a critical parameter for calculating the packing structure and flow characteristics, directly determining the boundary conditions for neutronics and thermal hydraulic calculations [11]; simultaneously, the friction coefficient is a core input for optimizing the design of the fuel handling system [12].

Nuclear graphite is a type of artificial graphite produced through specific raw material selection and optimized preparation processes, with performance requirements far exceeding those of ordinary industrial graphite: besides meeting nuclear purity (extremely low content of neutron poisons such as boron and cadmium), it must possess high density (>1.7 g/cm³), good isotropy, and excellent irradiation resistance [13].

The preparation process includes raw material selection, mixing and kneading, forming, carbonization, graphitization, and purification (Figure 3). The filler usually employs graphitizable carbonaceous precursors such as petroleum coke, pitch coke, or Gilsonite coke; coal tar pitch is mostly used as the binder, which is converted into solid carbon bridges connecting the filler particles at high temperatures. Impregnation–carbonization cycles are utilized to increase the density of the green body. The forming process has a decisive influence on anisotropy: extrusion tends to align the long axes of grains along a plane perpendicular to the extrusion axis, producing one with grain (WG) direction and two against grain (AG) directions; molding produces two WG and one AG directions; whereas isostatic pressing applies pressure uniformly in all directions, yielding products with extremely low anisotropy and is therefore widely adopted for modern nuclear graphite. Graphitization treatment is performed at 2500–3000  °C, transforming carbon atoms from a short range ordered amorphous state to a long range ordered hexagonal layered crystalline state, eliminating defects and improving thermal conductivity. Purification usually employs high temperature halogen gases to remove impurities such as boron [14].

The microstructure of nuclear graphite exhibits typical multiscale characteristics [15]. At the macroscopic level, it consists of millimeter-sized filler particles, a binder phase, and pores (total porosity ~20%) (Figure 4, [16]); the type and particle size distribution of the filler determine the texture of the graphite. At the mesoscopic level, polarized light microscopy can reveal isochromatic domains—regions where crystallite orientations are roughly aligned—ranging in size from micrometers to tens of micrometers; meanwhile, calcination cracks and Mrozowski cracks are visible (Figure 5, [16]). These microcracks are generated by the anisotropic shrinkage of microcrystallites during cooling and play an important regulatory role in the mechanical and thermal behavior of graphite [17]. At the atomic scale, carbon atoms form hexagonal network planes through sp² hybridization. Within the plane, strong covalent bonds (bond length 0.2461 nm) prevail, while the interlayer spacing is 0.3354 nm, with the layers held together by weak van der Waals forces. This strong-in-plane/weak-out-of-plane bonding gives graphite crystals significant anisotropy, with strength and elastic modulus parallel to the basal plane being much higher than those perpendicular to it [18]. Furthermore, high resolution transmission electron microscopy has revealed finer features such as turbostratic structures, quinoline insolubles, and wrinkled structures. These microstructures can evolve under irradiation and affect the macroscopic properties of graphite [19,20].

Both HTR-10 and HTR-PM adopt molded spherical fuel elements, which are fabricated by a quasi isostatic pressing method in silicone rubber molds [21]. The element consists of an inner fuel zone of 50 mm diameter and an outer fuel free shell approximately 5 mm thick, with the matrix material the same in both zones and with no physical interface. The matrix is typically composed of 64% natural graphite powder, 16% artificial graphite powder, and 20% phenolic resin [22]. TRISO coated fuel particles (Figure 6) are homogeneously dispersed in the fuel zone at a volume fraction of only 5–10%; therefore, the overall mechanical behavior of the fuel sphere can be approximately treated as that of a homogeneous solid graphite sphere [5].

In the core, spherical fuel elements bear their own weight and the mechanical loads transmitted by the overlying pebble bed. Nuclear graphite is a typical quasi brittle material with no obvious linear segment on the stress–strain curve and residual stress upon unloading; however, during subsequent loading cycles after experiencing a certain maximum load, the material exhibits stable elastic hysteretic behavior, and the gradient of the reloading curve can be used to define the stiffness coefficient under those conditions [23]. This stiffness coefficient is a key parameter for quantitatively evaluating the deformation of fuel spheres, calculating the contact force chain distribution in the pebble bed, and performing whole core structural integrity simulations. Nevertheless, to date, experimental measurements of the contact stiffness of fuel spheres under actual high temperature helium environments have been lacking, creating an urgent need to establish reliable measurement and analysis methods.

Factors affecting the friction coefficient of graphite can be divided into two categories: intrinsic and extrinsic. Intrinsic factors, determined by the type of raw materials and the preparation process, are fixed before the graphite components are put into service; extrinsic factors are the dominant variables that govern the dynamic changes in friction behavior during reactor operation.

Intrinsic properties of graphite, such as grain size, porosity, and degree of graphitization, significantly affect its friction performance. Zhou Qiang et al. pointed out that graphite micropowder with an average particle size of 4–5 μm exhibits the best lubricating effect; larger grains induce an abrasive effect that increases the friction coefficient, whereas very fine particles tend to form an adsorbed film on the friction surface, also leading to an increase in the friction coefficient [24]. Luo et al. used a standard friction tester to compare the friction behavior of graphites with different grain sizes in air and helium, and found that in room temperature air, the friction coefficient of fine grained graphite decreased with increasing load, while coarse grained graphite showed the opposite trend. In a helium environment, the friction coefficient of fine grained graphite increased compared with that in air, whereas that of coarse grained graphite was close to the value in air, indicating a coupling effect between atmosphere and grain size [25]. Further mechanistic analysis indicated that the adsorbed film on the graphite surface plays a dominant lubricating role in air, whereas in helium, friction relies on microdisplacement lubrication of graphite crystals on the friction surface [26]. Zhu et al. found that carbon materials with a higher degree of graphitization are more likely to form a transfer lubricating film on the friction surface, thereby reducing the friction coefficient [27]. Xu prepared graphite using pitch coke as the filler and confirmed that the friction coefficient gradually increases as the filler grain size decreases, whereas the introduction of natural graphite can further reduce the friction coefficient [28]. Skinner et al. used a scanning electron microscope to observe the sliding of a tungsten needle on the basal and prismatic faces of graphite single crystals and found that the friction coefficient on the prismatic face could be 5 to 20 times higher than that on the basal face, visually demonstrating the tremendous influence of graphite crystal orientation on friction [29].

Moreover, the pore structure also has an important impact on friction behavior. Liu et al. prepared high density graphites with different pore structures using isostatic pressing, compression molding, and thermoforming methods, and found that as the fractal dimension increases, both the friction coefficient and wear amount of graphite increase; thermoformed graphite, with the highest degree of graphitization (75.4%) and the lowest total porosity (15.08%), exhibited the best overall tribological performance [30]. For isotropic nuclear graphite used in HTGRs, the above intrinsic factors are already determined during preparation, and the key variables affecting the friction coefficient during core operation are the extrinsic environmental conditions.

The ambient atmosphere is the most significant extrinsic factor affecting the friction coefficient of graphite, and two theoretical hypotheses—the surface energy mechanism and the dangling bond mechanism—are mainly used to explain its action [31].

The surface energy mechanism explains friction behavior based on the differences in surface energy of different crystal faces of graphite. In a vacuum, the surface energy of the prismatic face of graphite is as high as 5 J/m², whereas that of the basal face is only 0.1–0.2 J/m², meaning the prismatic face has higher adsorption activity [32]. When graphite slides against graphite, the high surface energy in a vacuum results in strong physical adsorption between the two friction surfaces, leading to a very high friction coefficient. In active gases (oxygen, water vapor, organic vapors), gas molecules form a physically or chemically adsorbed passivation film on the graphite surface, reducing the surface energy and weakening the adsorption interaction between the friction surfaces, thereby decreasing the friction coefficient [33]. The classic study by Savage et al. found that when the water vapor partial pressure reached approximately 3 mmHg, the friction coefficient of graphite decreased significantly and the wear rate dropped below 10⁻⁴ mm³/s; oxygen required a higher partial pressure to achieve a similar effect, whereas the friction coefficient and wear rate in gases such as nitrogen and carbon monoxide were similar to those in vacuum [34]. In inert gases (helium, argon), gas atoms intercalate between the graphite layers, weakening the van der Waals forces between layers, making the graphite lamellae more easily separated and rearranged under frictional shear forces, and causing the basal planes to tend to align parallel to the sliding direction, thereby reducing the friction coefficient to an even lower level than in active gases [35]. In summary, according to the surface energy mechanism, the order of the friction coefficient of graphite from highest to lowest in different atmospheres is: vacuum > oxygen > air > water vapor > inert gas [32] (Figure 7).

The dangling bond mechanism explains friction from the perspective of chemical reactions between active carbon atoms generated by friction and gas molecules (Figure 8). During friction, a large number of active carbon atoms with dangling bonds are generated on the graphite surface due to lattice fracture. In vacuum or under inert gases, the dangling bonds on the two friction surfaces directly form “carbon–carbon” bonds, which require extremely high shear forces to break, manifesting as a very high friction coefficient. In oxygen, the dangling bonds react with oxygen to form “carbon–oxygen–carbon” bond bridges, or form oxidized groups on the same surface, weakening the direct bonding between the surfaces and significantly reducing the friction coefficient [36]. In water vapor, carbon atoms tend to form “carbon–oxygen–hydrogen” groups on the same surface, also reducing interlayer adhesion [37]. Xiao et al. used highly oriented pyrolytic graphite (HOPG) in comparative experiments and measured an average friction coefficient of 0.25 perpendicular to the basal plane and only 0.1 parallel to the basal plane. When the atmosphere was changed from air to nitrogen containing water vapor, the friction coefficient in both directions decreased, with a larger decrease in the perpendicular direction. Through X ray photoelectron spectroscopy, they confirmed the difference in the amount of chemisorbed oxygen combined with dangling bonds on the two surfaces, providing strong support for the dangling bond mechanism [36].

Both mechanisms are effective in explaining the reduction of friction in active gas atmospheres; the divergence mainly appears in the case of inert gases: the surface energy mechanism predicts a friction coefficient lower than that in vacuum, whereas the dangling bond mechanism predicts a high friction close to that in vacuum. Numerous experiments have shown that the friction coefficient of graphite in inert gas atmospheres is significantly lower than in vacuum and even lower than in active gas atmospheres. Therefore, under inert gas conditions, the surface energy mechanism is more persuasive [38].

However, contrary to the conventional expectation that inert gases promote lubrication, the literature reveals a more nuanced and often contradictory frictional behavior of graphite in argon. Some studies report remarkably high friction coefficients in such environments. For instance, research on turbostratic graphite under argon has documented coefficients ranging from a surprisingly low 0.15 to an ultralow 0.002, a phenomenon attributed to the self organization and superficial mobility of crystallites at the contact interface. However, the same study found that friction was fairly high (0.32–0.42) when tests were performed under nitrogen, highlighting the gas specific nature of the effect [39].

The mechanistic origins of these varied friction coefficients are rooted in the surface chemistry of graphite. The tribological behavior of graphite in inert environments, particularly argon, is governed by a delicate balance between surface chemistry and mechanical conditions. Paulmier and Zaidi established that the high friction coefficient of graphite in vacuum or inert gases stems from the creation of active “dangling bonds” at the contact interface during sliding [33]. They demonstrated that while freshly crushed graphite exhibits a low concentration of free radicals (5%), exposure to atomic hydrogen produces two opposing effects: a low concentration increases chemical activity and friction, whereas a high, near saturation concentration passivates the surface, reducing both friction and wear. Complementing this, Wild and Mack provided practical engineering data for nuclear reactor components, reporting a notably high friction coefficient of 0.35–0.5 for CrC-coated steel sliding in argon, highlighting the challenging nature of such atmospheres for moving mechanical systems [40]. Collectively, these works underscore that friction in argon is not a fixed material property but a complex response to the interplay of surface passivation, sliding dynamics, and contact history, with the inert gas facilitating both ultra low friction under mild conditions and severe adhesion when surface activation outpaces passivation.

The influence of temperature on the friction coefficient of graphite depends on the atmospheric environment and exhibits markedly different patterns.

In a vacuum environment (Figure 9), the friction coefficient of graphite decreases monotonically as temperature increases. The increase in temperature enlarges the interlayer spacing of graphite crystals, making it easier for graphite microcrystallites on the friction surface to undergo relative sliding and orientation rearrangement, forming a lubricating film with basal planes parallel to the sliding direction, thereby reducing the friction coefficient [41,42].

In inert gases (e.g., helium, argon), the friction coefficient of graphite likewise decreases with increasing temperature (Figure 10). Inert gas atoms do not participate in chemisorption, but as the temperature rises, the thermal motion of the atoms intensifies, and combined with the thermal expansion of the graphite interlayer spacing, gas atoms can more readily intercalate in large numbers between the graphite layers, further weakening the interlayer bonding and enhancing the lubrication effect [43]. However, a study by Driesner et al. found that above 1500  °C the friction coefficient of graphite no longer continued to decrease but rather slightly rebounded, which was attributed to the onset of obstruction of relative sliding by fine particles exfoliated from the graphite at high temperatures [44].

In active gases (air, oxygen, water vapor), the effect of temperature is more complex. The friction surface of graphite relies on physically adsorbed gas molecules (mainly water and oxygen molecules) to provide lubrication. As the temperature increases, the physically adsorbed layer gradually desorbs and fails. Lancaster found that when the friction surface temperature reached 150–185  °C, the lubricating adsorbed film began to collapse and the friction coefficient increased [44]. Zaidi et al. experimentally confirmed that at 200  °C, approximately 50% of the water molecules adsorbed on the graphite surface underwent desorption, and these water molecules are the key source of low friction of graphite in an air environment [32]. Therefore, in active gases, the friction coefficient of graphite generally increases with increasing temperature. However, around 400  °C, some experiments have observed another decrease in the friction coefficient because, at this temperature, the rate of the thermal chemical reaction between graphite and oxygen is sufficiently high, and the generated CO and CO₂ gases cover the nascent friction surface and provide temporary lubrication [45,46].

The influence of load on the friction coefficient of graphite exhibits a two stage characteristic. When the load is low and the graphite is within the elastic deformation range, as the load increases, the rate of increase in the real contact area of the friction surface lags behind the rate of increase in load. This means that the increase in the adhesion force component is smaller than that of the shear force component, resulting in a decrease in the friction coefficient [47]. When the load further increases to the point of causing plastic deformation or particle exfoliation on the graphite surface, plastic flow and abrasive wear lead to an increase in the friction coefficient, with furrows and severe wear characteristics forming on the friction surface [48,49].

The influence of sliding velocity is related to a “critical sliding velocity”. In inert gases, the friction coefficient under low speed sliding is relatively low because the gas has sufficient time to enter between the graphite crystal layers and form a lubricating film. When the sliding velocity exceeds the critical value, the friction surface is rapidly stripped, and the lubricating film cannot form or recover in time, causing the friction coefficient to increase sharply [50]. In active gases, high speed sliding leads to a rapid temperature rise on the friction surface, resulting in the desorption and failure of the adsorbed gas film, thereby increasing the friction coefficient [51].

On the whole, most friction tests in the existing literature were conducted at room temperature. Friction data for graphite at high temperatures (>400  °C) and in inert atmospheres (especially helium) are extremely limited and show a large scatter. As shown in Table 2, the experimental results for the friction coefficient of graphite in helium range from roughly 0.02 to 0.40, with almost no data in the high temperature range. This represents a significant gap relative to the actual operating conditions of HTGRs and is precisely the starting point for the latest systematic experimental research reviewed in this paper.

The contact stiffness of spherical fuel elements is essentially a macroscopic characterization of the contact deformation of two rough solid surfaces under a certain normal load. Its theoretical modeling involves a cross scale mechanical analysis from a single asperity to the entire rough interface.

The contact mechanics of asperities originates from the elastic contact theory proposed by Hertz in 1881, which assumes the asperity to be a homogeneous, isotropic elastic sphere and provides an analytical relationship between normal load and deformation, still serving as the foundation of contact mechanics today [21] (Figure 11). When the load increases to the point where the maximum stress in the contact area reaches a threshold of the material hardness, the asperity yields. The AF model describes the behavior of an asperity in a fully plastic state, assuming that the contact load equals the product of the material hardness and the contact area [52].

Real asperities undergo an elastoplastic transition stage between elasticity and full plasticity. Chang et al. attempted to directly splice the Hertz model with the AF model, but the contact pressure was discontinuous at the critical deformation, exposing the deficiency of a simple two segment division [53]. Zhao et al. used an exponential function to smoothly connect the elastic and plastic segments, ensuring continuity of contact load and area at the critical point, but the contact stiffness remained discontinuous [54]. Xiao et al. used a low order elliptic curve to achieve three continuity boundary conditions for contact load, area, and stiffness, with high accuracy, but the mathematical form was complex and inconvenient for further coupling with the overall interface model [55].

The development of finite element technology has provided a new approach for modeling elastoplastic contact of asperities. Based on numerous finite element simulations, Kogut and Etsion [56] subdivided the elastoplastic deformation stage into the first and second substages and fitted exponential relationships between contact load and contact area and deformation in each stage. This model, which combines high accuracy with a concise functional form, has been widely used in macroscopic contact modeling of interfaces. Jackson and Green, based on more extensive finite element calculation results, modified the expression for the critical elastoplastic deformation, offering better applicability in the large deformation stage [57]. In addition, regarding the tangential behavior of asperities, Mindlin derived the exact solution for the tangential force and tangential deformation of an asperity under partial slip conditions based on the local Coulomb friction law, which remains the core theory for tangential contact analysis of rough surfaces [58].

Contact between actual engineering surfaces involves the collective behavior of many asperities, requiring the combination of a single asperity model with a statistical or fractal description of the surface topography.

The GW model proposed by Greenwood and Williamson is the most classic statistical contact model. It assumes that the heights of surface asperities follow a Gaussian distribution, the deformation of each asperity is independent of the others, and the contributions of all asperities are integrated to obtain the macroscopic contact load and contact stiffness [59]. The GW model is simple and practical, but its key parameters (such as asperity density, radius of curvature, etc.) depend on the resolution and sampling length of the measuring instrument, suffering from a “measuring instrument dependence” problem and lacking universality across different machined surfaces.

Fractal contact models exploit the self-affine fractal characteristics of rough surfaces, using a scale-invariant Weierstrass–Mandelbrot (W–M) function (Figure 12) to describe the surface topography, fundamentally overcoming the scale dependence problem of statistical models. Majumdar and Bhushan combined the W–M function with asperity contact mechanics and proposed the classic MB model, establishing fractal scaling relationships among interface contact load, contact area, and contact stiffness [60,61]. Wang and Komvopoulos further corrected the distribution function of the truncated contact area of asperities in the MB model by introducing a domain extension coefficient, improving calculation accuracy [62]. Yan and Komvopoulos extended the MB model to three dimensional isotropic rough surfaces, providing a more precise description of normal deformation [63]. Subsequently, Zhang, Tian, and others conducted a series of studies on the normal and tangential contact stiffness of mechanical joints based on the MB model and elastoplastic asperity models, and proposed the virtual material method for contact interfaces—introducing a virtual material layer at the interface and setting its elastic parameters to equivalently represent the contact stiffness of the interface, which facilitates implementation in finite element analysis [64]. However, none of these models have specifically considered the unique microstructural features of nuclear graphite (such as Mrozowski cracks and pores) and the influence of high temperature environments on contact behavior. Furthermore, modeling under non-uniform interface stress distribution conditions still relies on complex pre-analysis and lacks generality.

There are three main types of experimental methods for measuring interface stiffness [65]:

The interface displacement method directly relies on the definition of stiffness. By measuring the relative normal displacement of the interface under different normal loads, the contact stiffness is obtained by calculating the slope of the load–displacement curve (Figure 13). The principle of this method is simple, but it imposes extremely high requirements on the proximity of the displacement measurement points to the contact interface. Displacements measured far from the interface will include deformations of the testing system and supporting structure, which require careful correction. Yuan et al. proposed an improved method to eliminate systematic errors arising when measurement points are not located directly on the interface [66].

The impact test method correlates stiffness with the natural frequency and frequency response function of the structure, and the stiffness is inversely derived through experimental modal analysis. The limitation of this method lies in the need for an accurate analytical model of the vibration response, which is very difficult to obtain for complex boundary conditions [67].

The acoustic method utilizes the reflection, transmission, and nonlinear effects of elastic waves at the contact interface to characterize the interfacial contact state. Krolikowski proposed evaluating the contact stiffness by measuring the longitudinal wave transmission coefficient [68]; Dwyer-Joyce et al. measured the contact stiffness of rough surfaces through the longitudinal wave reflection coefficient and found significant plastic flow during the first loading, which tended to elastic stability after multiple loading cycles [69]. The acoustic method has advantages in nondestructive testing and online monitoring, but its application in high temperature, high pressure, and complex geometric structures still faces technical challenges such as sensor installation and signal decoupling.

The structural integrity of nuclear graphite in high temperature reactors is critically influenced by damage accumulation and complex stress states, as explored through complementary experimental, analytical, and numerical approaches. Jia et al. systematically investigated the failure mechanism of line contact structures (e.g., graphite brick-tenon joints) in IG11 graphite, identifying a three stage progressive failure process: initial damage accumulation leading to characteristic yielding without visible cracks, followed by crack initiation and lateral propagation when stresses satisfy the Mohr failure criterion, and finally, the formation of a secondary Y-shaped crack that precipitates complete structural failure. This staged mechanism underscores the critical role of contact geometry-induced stress concentration in triggering damage evolution [70]. To model such damage driven failure, Zou et al. developed a unified continuum damage mechanics (CDM) model that integrates both stress-based and fracture-mechanics-based criteria. Their model successfully predicts failure under both regular and singular stress concentrations, as validated against experiments on L-shape and channel section specimens with varying corner radii, demonstrating its suitability for capturing the transition from diffuse damage to discrete crack formation [71]. Complementing these studies, Yang et al. addressed an operational challenge—molten salt infiltration in TMSR—by analyzing its influence on stress distribution in graphite components under irradiation. Their simulations revealed that dose gradients, permeation zone shapes, and areas significantly alter the principal stress fields, with peak stresses often localized at geometric centers and symmetry modifications induced by non-uniform infiltration patterns [72].

To conclude, these studies highlight that the mechanical behavior of nuclear graphite in contact is governed by a complex interplay among initial damage, stress concentrators, and environmental factors (e.g., irradiation and coolant infiltration), necessitating integrated damage models and geometry-aware stress analyses for reliable reactor core design and safety assessment.

For convenience, the following table (Table 3) summarizes the friction coefficient study.

Summarizing the above status, the following significant deficiencies exist in the research on the friction and contact stiffness of nuclear graphite for HTGRs:

To address the above mentioned gaps, two independent systematic experimental studies have recently completed measurements of the friction coefficient of nuclear graphite and the stiffness coefficient of spherical fuel elements for HTGRs, providing for the first time continuous experimental data from room temperature up to 1300  °C in a high purity helium environment.

The first study used isostatically pressed nuclear graphite BG80, which is widely used in HTGRs, as the research object, and designed and constructed a graphite friction coefficient testing apparatus capable of simulating the core environment. The apparatus adopted an end face contact configuration of a graphite rod against a graphite rod, driven by a servo motor and loaded by a bellows. It can accurately acquire the driving force and rebound force–displacement curves under a temperature range of 25–1300  °C, a load range of 0–30 kg (corresponding to a contact surface pressure of about 31.25–68.75 kPa, close to the actual stress level of core graphite components), and under different atmospheres (high purity helium/nitrogen) [88].

In terms of data processing, a systematic and rigorous methodology was adopted: first, the system resistance was determined based on the non-return condition of the graphite test pieces, and it was found that this system resistance was only related to the characteristics of the apparatus itself and not affected by external factors such as temperature, atmosphere, and load. Then, by analyzing the rebound force data, a fitting relationship between the bellows elastic force and displacement was established, thereby verifying the stability of the elastic force throughout the test process. Finally, by combining the system resistance and the elastic force to correct the driving force data, the average friction force curves under various working conditions were obtained [88]. A total of 4 test groups were completed in a helium atmosphere and 2 groups in a nitrogen atmosphere, covering the complete temperature range from room temperature to 1300  °C, which fully encompasses the temperature range under normal HTGR operation and some accident conditions [89].

The experimentally obtained static and dynamic friction coefficients between graphite were as follows [90]: in a helium atmosphere (Figure 14), the maximum static friction coefficient was 0.20856 (low temperature range) and the minimum was 0.03796 (high-temperature range); the maximum average dynamic friction coefficient was 0.17852, and the minimum was 0.03942. In a nitrogen atmosphere, the maximum static friction coefficient was 0.25550, and the minimum was 0.04225; the maximum dynamic friction coefficient was 0.23190, and the minimum was 0.03293 [6]. The dynamic friction coefficient did not change significantly with sliding displacement, indicating that within the load range of this test (without reaching plastic deformation of graphite), the friction coefficient was essentially independent of load and displacement [91].

The variation of the friction coefficient with temperature exhibited a clear four stage characteristic [91]:

At temperatures above about 300  °C, since nitrogen molecules could also enter the graphite interlayer space, the friction coefficients in the two atmospheres tended to converge [92]. The above findings provide the first set of quantitative friction data under continuous temperature variation for pebble bed flow calculations and fuel handling system design in HTGRs.

The uncertainty in experimental results can be categorized as Type A and Type B uncertainties. Type A uncertainty is evaluated through statistical analysis of the experimental data, typically using the Bessel method. In this study, the best estimate of the friction coefficient μ is represented by the mean value of n independent measurements. The standard uncertainty, expressed as the standard deviation, quantifies the dispersion of the measurement results. Accordingly, the standard uncertainty u(μ_k) of any individual measurement μ_k is given by the experimental standard deviation s(μ_k), calculated using the Bessel formula. For practical measurements, the Type A uncertainty of the best estimate as follows

Type A uncertainty is obtained solely from the statistical evaluation of measurement results. In contrast, Type B uncertainty can originate from various sources, such as calibration certificates, instrument specifications, technical documentation, prior data, or expert experience.

In the present study, the friction coefficient μ is calculated from the measured friction force and normal load. Therefore, the Type B uncertainty is primarily determined by the uncertainties of the force-measuring instruments and the calibration weights used for loading, as listed in Table 4. The standard Type B uncertainty of an input quantity $${u}_{B}\left(x\right)$$ is determined by the absolute value of its error limit a and a divisor factor, which is typically taken as $$\sqrt{3}\,$$for a rectangular distribution $${u}_{B}\left(x\right)=\frac{a}{\sqrt{3}}$$.

The Type B uncertainty of the mean value $${u}_{B}\left(\bar{\mu }\right)$$ is calculated as:

The uncertainties are evaluated for all friction coefficient results presented here. For other measured quantities, such as the bellows elastic force $${F}_{e}$$ and system resistance $${F}_{r}$$,

the uncertainty calculation differs slightly. While the Type A uncertainty remains applicable according to Equation (1), the Type B uncertainty excludes the contribution from the calibration weights, as these quantities are independent of the applied load. Consequently, Type B uncertainty of system resistance is computed by:

The combined standard uncertainty $${u}_{C}\left(\bar{\mu }\right)$$ of the mean $$\bar{\mu }$$ is then obtained from the law of uncertainty propagation $${u}_{C}\left(\bar{\mu }\right)=\sqrt{{u}_{A}^{2}\left(\bar{\mu }\right)+{u}_{B}^{2}\left(\bar{\mu }\right)}$$.

The static and kinetic friction coefficients exhibit very similar temperature dependent behavior, generally showing an initial increase followed by a decrease and then stabilization. Specifically: below 200 °C, the friction coefficient increases slightly with rising temperature, reaching a maximum around 200 °C; above 200 °C, the friction coefficient decreases significantly with further temperature increase, and stabilizes after 400 °C; when the temperature continues to rise above 900 °C, the kinetic friction coefficient shows a slight increase.

The mechanisms for the effects of temperature on the friction might be interpreted as follows:

Comparing the friction results in helium (He) and nitrogen (N₂), the mechanism of the influence of the environmental atmosphere is interpreted as follows:

Based on the high similarity in friction behavior between N₂ and He atmospheres at elevated temperatures (above 300 °C), it is feasible to use nitrogen (N₂) as a substitute for expensive helium (He) as the working gas in exploratory or validation experiments related to high temperature gas cooled reactor cores. This substitution effectively simulates the tribological behavior of graphite components in the reactor core while significantly reducing experimental costs, offering favorable economic benefits for engineering design and innovative validation.

The second study focused on the stiffness coefficient of spherical fuel elements. An experimental system consisting of a graphite sphere in compressive contact with a graphite rod was established, and the load–displacement response of the fuel sphere was measured at different temperatures in a high temperature helium environment [2]. For theoretical modeling, the fuel sphere contact problem was decomposed into two cases: graphite sphere-on-sphere contact and graphite sphere-on-rod contact. For the former, the Hertz contact theory was used to derive the relationship between deformation and external load; for the latter, the method of dimensionality reduction was employed to establish an analytical force–displacement relationship, and the correctness of the theoretical model was verified by finite element numerical simulation [92]. Finally, a relational expression linking the directly measured quantity in the experiment (the overall system displacement) with the stiffness coefficient of the fuel sphere itself was derived, enabling the accurate extraction of material parameters from the experimental data.

The experiments revealed an important characteristic of the temperature dependence of the fuel sphere stiffness coefficient: it exhibited unidirectional irreversibility. When the temperature was increased from room temperature to a high temperature and then decreased back to the initial temperature, the stiffness coefficient did not return to its original value, demonstrating a significant dependence on thermal history [92]. This phenomenon is attributed to two kinds of mechanisms: (1). The intrinsic elastic modulus of graphite tends to increase with temperature, the interplay between modulus evolution, residual stress relaxation, and microstructural characteristics profoundly dictates the resultant macroscopic rigidity and failure behavior. In coarse-grained Gilsocarbon graphite [93], the dominant stress relaxation and extensive extrinsic toughening promote a more compliant, damage-tolerant response with enhanced failure strains at high temperatures. Conversely, in fine-grained graphites [94], the disproportionate increase in modulus relative to strength, coupled with limited residual stress relief and scarce toughening mechanisms, renders the material stiffer yet more brittle, with reduced failure strains and a marked propensity for catastrophic fracture. (2). Irreversible closure and structural adjustment of microcracks (such as Mrozowski cracks) inside the graphite during thermal cycling, which has important implications for understanding the mechanical behavior evolution of fuel spheres under long term operational thermal cycling. These behaviors underscore that the temperature-dependent rigidity of nuclear graphite cannot be inferred solely from modulus measurements, but must be understood within the broader context of residual stress states, microstructural heterogeneity, and the prevailing toughening mechanisms that collectively dictate the material’s structural integrity under high-temperature reactor operating conditions. In addition, the study used various functional forms, including polynomial and exponential functions, to fit the stiffness coefficient–temperature relationship, yielding concise analytical expressions with recommended ranges of applicability for engineering convenience [2].

The recent work carried out by Si et al. [95] fills the experimental gap in the contact stiffness coefficient of spherical fuel elements in a high temperature helium environment and provides crucial input parameters for fuel element design and core deformation simulations (Figure 15).

Along with the current state of research on the friction and stiffness of graphite, the following aspects are suggested for future investigation:

The friction coefficient and contact stiffness of nuclear graphite materials in HTGRs are indispensable fundamental parameters for core mechanical design, pebble bed flow path analysis, and safety assessments. Based on a systematic review of the characteristics of nuclear graphite materials and the current status of tribology and contact mechanics research, this paper integrates the results of the two latest systematic experimental studies, with the main conclusions as follows:

The authors are grateful for the support of this research by the LingChuang Research Project of China National Nuclear Corporation (Recipient : Qian Wang).

Conceptualization, T.L.; Methodology, M.S. and Z.C.; Writing—Original Draft Preparation, T.L.; Supervision, N.G.; Project Administration, S.J.

Not applicable.

Not applicable.

Data will be available from the corresponding author on request.

This research was funded by [National Major Science and Technology Program, and LingChuang Research Project of China National Nuclear Corporation].

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