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In the early 1980s, new types of carbon materials developed rapidly and played a crucial role in many industrial processes; they are regarded as the most valuable materials for research in the field of materials science in the 21st century. There are four common types of carbon materials: (1) Carbon products ； ⑵Carbon paste types ； ⑶Graphite products ； ⑷Special graphite products. These materials possess the unique properties of ceramics, metals, and polymers alike. In terms of their macroscopic structure, they can take on fibrous, granular, membranous, or bulk forms; most of them are dense, while a few are porous, such as expanded graphite blocks. This variety of low-dimensional, two-dimensional, and three-dimensional structures enables them to exhibit additional special properties. With the progress and development of technology, the modern industrial sector imposes more stringent requirements on materials, such as high temperature resistance, corrosion resistance, light weight, oxidation resistance, good mechanical properties, and high stability. Porous carbon materials refer to carbon materials with various microporous structures, ranging from nanoscale pores that are extremely small to micrometer-scale pores that are large enough to support the growth and activity of microorganisms. This unique microscopic structure endows them with special properties superior to those of other materials, meeting the requirements of modern technology. Moreover, the preparation method for porous carbon materials is extremely simple: coal, coconut shells, asphalt, and similar substances can be used as precursors, and through physical and chemical activation methods, they can be transformed into carbon materials rich in pores, thus granting them wide applicability. In industrial production, porous carbon materials are commonly used as carriers for activators and adsorbents, and are widely applied in fields such as adsorption separation, electromagnetic shielding, and sealing. Expanded graphite, also known as flexible graphite, is a soft graphite material with a worm-like microstructure. It is produced by subjecting natural graphite flakes to oxidation and then high-temperature expansion. It is also referred to as graphite worms (the SEM images showing its appearance and surface pore structure are presented in Figures 1-1 and 1-2). It was first developed by the United Carbon Corporation (UCC) in the United States in 1963, came into production in 1968, and was launched on the market in 1970; thus it has only a few decades of history to date. Although research on some of its properties is still in the exploratory stage, and its full range of excellent application capabilities has not yet been realized, it has already attracted great interest among researchers since its development, with countries around the world competing to invest in the research and development of expanded graphite. China began researching, producing, and applying expanded graphite materials in 1978, but it still lags far behind foreign countries both in terms of technology and theory; its research mainly focuses on improving the structure and properties of expanded graphite. With the acceleration of China’s industrialization process, it will undoubtedly provide a better environment for systematic research on expanded graphite and its large-scale production. Expanded graphite is a high-purity form of graphite. Natural graphite belongs to the hexagonal crystal system; its crystals take on a hexagonal plate-like or flaky shape, and their aggregates appear as scales. It possesses a typical layered structure, with carbon atoms forming covalent bonds within the same layer (referred to as the α direction), whose bond energy is 586 KJ/mol. Between the layers (in the direction perpendicular to α, known as the c direction), the atoms are held together by weak van der Waals forces, with a bond energy of 16.7 KJ/mol. This three-dimensional orderly arrangement of layers results in structural differences between the in-layer direction and the inter-layer direction, which in turn gives graphite crystals anisotropic properties, such as differences in electrical, thermal, and mechanical properties. Therefore, this polycrystalline graphite structure of expanded graphite also exhibits such anisotropic characteristics, which must be taken into account during analysis. The loose and porous structure of expanded graphite gives it a high compressive resilience, making it an increasingly popular new type of sealing material. It is widely used for dynamic and static sealing in various mechanical devices. In particular, developed countries have gradually banned asbestos, a traditional sealing material that is carcinogenic, since the 1990s, which has accelerated the use of expanded graphite as a sealing material. Within five years, our country introduced three consecutive production lines from the United States and Japan to carry out research and production on this new type of sealing material. However, due to the high sulfur content, high volatility, low strength of the produced expanded graphite, as well as outdated production techniques, the products lack competitiveness in the international market. To overcome its weaknesses of low strength and susceptibility to oxidation in air, the international community is researching new preparation methods as well as developing expanded graphite composite materials. Expanded graphite is combined with binders such as HB03 (or its salts), H3PO4 (or its salts), organic silicones, and resins; in some cases, it is also immersed in acids such as nitric acid (or BaMoO3) or tetrafluoro fluids to improve its high-temperature oxidation resistance and lubricity. Applications and development of the sealing performance of expanded graphite: In the late 1960s, the American company Union Carbide used expanded graphite to create sealing materials, giving rise to the second generation of such materials. In 1971, in order to address the problem of leakage in nuclear energy valves, the Rockwell Valve Engineering Design Center in the United States conducted a comparative study on various types of sealing structures produced by nine domestic companies at 64 predetermined test sites. The results show that expanded graphite material has the best sealing effect. Therefore, some developed countries—such as the United States, Japan, and France—have conducted further in-depth research on this sealing material, accelerating the development of expanded graphite materials in the field of sealing. As a result, this highly efficient and energy-saving new type of porous carbon material has become a widely regarded sealing material worldwide. China began to study expanded graphite materials and their composite products in 1978, and the development pace has been rapid in recent years. Huang Zhuoming and others studied the application of flexible graphite in dynamic seals for pumps, demonstrating that flexible graphite packing seals are superior to packing seals such as oil-impregnated gauze, asbestos packing, and rubber packing. Moreover, flexible graphite exhibits good corrosion resistance and chemical stability in alkaline solutions, dilute sulfuric acid, concentrated sulfuric acid, and organic solvents ； Jiang Fang studied the distribution of fluid pressure in packing seals using expanded graphite fillers, and determined the influence of the number, structure, size, and stiffness of the packing rings on the curve of fluid leakage pressure in such seals. He also proposed that using packing rings of the same specifications but different densities assembled within the same stuffing box can result in a more reasonable distribution of fluid leakage pressure ； Gu Boqin and others conducted research on the high-temperature performance of stainless steel-flexible graphite wound gaskets. The experimental results showed that the compression amount increased as the test temperature and the compressive stress on the gasket increased; the resilience of the gasket decreased with rising temperature. Its compression-resilience curve was nonlinear and non-conservative. Moreover, since such gaskets are composed of both metallic and non-metallic materials, creep occurs even at room temperature, with the amount of creep increasing as the temperature rises ； Zhou Tingnian and others used expanded graphite impregnated with sucrose solution, which was then treated at high temperatures to achieve carbon infiltration in the interlayer structure of the graphite; this improved the compressive properties of the expanded graphite material. They also proposed that using this carbon-infiltrated material as a matrix, combined with stainless steel wire, could result in sandwich composite materials with a tensile strength increase of around 40% ； After several years of research, Henan Metallurgical Equipment Joint Venture Company has made new breakthroughs in the technology of expanded graphite flat seals. This has enhanced the reliability of sealing, facilitated the replacement of high-temperature sealing materials, and achieved the goal of increasing production while reducing costs. As a new type of functional carbon material, expanded graphite (EG for short) is a porous, worm-like substance obtained by intercalation, washing, drying, and high-temperature expansion of natural graphite flakes. In addition to possessing the excellent properties of natural graphite itself, such as heat and cold resistance, corrosion resistance, and self-lubrication, EG also has characteristics that natural graphite lacks, including softness, compressive resilience, adsorption capacity, compatibility with the ecological environment, biocompatibility, and radiation resistance. As early as the early 1860s, Brodie discovered expanded graphite by treating natural graphite with chemical reagents such as sulfuric acid and nitric acid followed by heating; however, its application did not begin until a century later. Since then, many **have successively carried out research and development on expanded graphite, achieving significant scientific breakthroughs. Since its preparation method was first developed in 1963 by the United Carbon Corporation (UCC) in the United States, expanded graphite has attracted widespread attention from scholars around the world due to its excellent properties. To enhance and optimize these properties, its preparation methods have been gradually improved; the manufacturing processes have become simpler, and the pollution generated during production has decreased. The development of this material is moving in the direction of being pollution-free, sulfur-free, and possessing high strength. The following is a brief introduction to several common preparation methods: (1) Chemical oxidation method – The chemical oxidation method is a traditional approach used in industrial production to synthesize expandable graphite. When the heating temperature reaches around 100°C, an oxidizing acid solution is prepared by mixing an oxidizer with concentrated sulfuric acid in a certain ratio. Natural flake graphite is immersed in this solution, allowing the sulfate ions present in it to penetrate between the layers of graphite and combine with the carbon atoms that have been oxidized. After reacting for a certain period of time, the material is washed and dried to yield expandable graphite; further expansion at high temperatures then results in the formation of expanded graphite. However, when applying this method, it should be noted that process parameters such as the soaking time, the ratio of oxidant to intercalant, the amount of oxidant used, and the temperature applied during expansion all affect the expansion effect. For example, if too little oxidizing agent is used, the interlayer reactions of graphite will not proceed sufficiently, and the degree of oxidation required for complete expansion will not be achieved; on the other hand, if too much oxidizing agent is used, graphite will be over-oxidized, again failing to produce the desired expansion effect. The reaction equation for this method is: (2) Electrochemical method. Tsinghua University and Nangong Guanghua Chemical Plant use the electrochemical method to produce expandable graphite. This not only reduces production costs and environmental pollution, but also improves the product quality. In this method, the selected auxiliary anode material is combined with graphite to form an anode chamber, which is then immersed in an electrolyte containing an intercalant. A current of appropriate magnitude is applied, and after oxidation over a certain period of time, the material is taken out. After washing and drying, expandable graphite is obtained, which is further expanded at high temperatures to yield expanded graphite material. The expanded graphite produced by this method generally has a low sulfur content; no additional oxidants are required, and the progress of the reaction can be controlled simply by adjusting electrochemical parameters such as current and voltage. The simplicity and controllability of this method make large-scale synthesis of expanded graphite possible. However, it also has its drawbacks. For example, the reaction takes a long time and is costly. (3) The soaking solution replacement method: As shown in reaction equation (1-1), expanded graphite produced by the conventional chemical oxidation process contains a high sulfur content, and the sulfur content affects the service life of the material; therefore, the sulfur content in the material should be reduced as much as possible. Dun Huijuan and others immersed the expandable graphite material prepared through traditional processes in a solution made by mixing oxalic acid and nitric acid. The following reaction occurred: It can be seen that as a result of this reaction, some oxalate ions and oxalic acid molecules intercalated between the graphite layers, displacing some sulfate ions and sulfuric acid molecules. Consequently, **the sulfur content in the expandable graphite material was reduced**. (4) Addition of non-strong acid intercalants: Non-strong acid intercalants can enter the interlayers of graphite and partially replace sulfuric acid, thereby reducing the sulfur content in expandable graphite. Wang Shenmin and colleagues used a small amount of concentrated sulfuric acid and potassium permanganate as mixed oxidants, along with a small amount of ferric chloride as an intercalant, to determine relatively ideal conditions for the production of low-sulfur expandable graphite. This production method is simple, features fast reaction rates, and causes minimal environmental pollution; moreover, its key properties such as mechanical strength, thermal weight loss rate, and sulfur content are superior to those of similar products available domestically and internationally. Song Keming et al. prepared sulfur-free expandable graphite using potassium permanganate and similar substances as oxidants, and a mixture of concentrated sulfuric acid and propionic acid as an intercalant. F. Kang synthesized sulfur-free expandable graphite in formic acid solution; this expandable graphite contains only elements such as carbon, hydrogen, and oxygen between its layers, and becomes almost pure carbon after expansion. This method provides a viable way to produce high-purity, non-corrosive flexible graphite. Properties of expanded graphite: Expanded graphite is obtained by chemically treating natural graphite flakes and then modifying them through instantaneous expansion at high temperatures. It not only possesses the excellent chemical properties of natural graphite but also features a range of unique mechanical properties, making it an ideal sealing material with a wide range of applications and good sealing performance. The main properties are described as follows: (1) Density: Graphite flakes are processed into expanded graphite, which has a loose and porous microscopic structure, through high-temperature expansion. Its bulk density is generally 0.002–0.005 g/cm3, while the density of the finished product is 0.8–1.8 g/cm3; as a result, it has a low weight and good plasticity ； (2) Purity: Carbon is the most abundant element in expanded graphite products, with a fixed carbon content of around 98%; in some specially treated expanded graphite materials, the carbon content can exceed 99%, which meets the high-purity requirements for seals in many industrial sectors. (3) Stability: Theoretically, expanded graphite products can be used at temperatures ranging from -200°C to 3000°C, and as sealing fillers they can be used safely at temperatures from -200°C to 800°C. It ensures that it does not become brittle or age at low temperatures, and does not soften, deform, or decompose at high temperatures. Moreover, expanded graphite has a low coefficient of thermal expansion, which enhances its stability and enables it to provide good sealing even under conditions of high temperatures or sudden temperature changes. (4) Radiation resistance: It does not undergo significant changes under long-term exposure to neutron rays, gamma rays, alpha rays, beta rays, etc. (5) Impermeability: Its loose and porous structure results in a large surface area, which allows for the formation of extremely thin gas and liquid films that prevent the penetration of substances; it exhibits good impermeability to many gases and liquids. (6) Self-lubricating property: Expanded graphite retains the planar layered structure of natural graphite flakes; under external force, the planar layers can slide relative to each other, thereby enabling self-lubrication and effectively preventing wear on shafts or valve stems. (7) Corrosion resistance: Expansive graphite possesses stable chemical inertness; aside from strong oxidizing agents such as aqua regia, nitric acid, sulfuric acid, and halogens, as well as certain other specific media, it can withstand most acid, alkali, salt solutions, seawater, steam, and organic solvents. (8) Compressibility and resilience: Microscopically, expanded graphite products contain numerous compressible, closed tiny pores. Under external force, these pores can be compressed; meanwhile, due to the tension generated by the air within these pores, the material exhibits excellent resilience. (9) Flexibility: Expanded graphite is lightweight and flexible, easy to process; it can be cut using ordinary cutting tools, and can be rolled or bent in various ways, allowing for many different finished shapes. Research and Progress on Expansive Graphite and Its Composite Sealing Materials. Expansive graphite is obtained by treating natural graphite through certain physicochemical methods; its microcrystalline structure is the same as that of ordinary graphite, thus it possesses all the properties of graphite. At the same time, it has a large specific surface area and high surface activity. It can be pressed into shape without the need for any bonding agents or high-temperature sintering. Expansive graphite particles are placed in a pressurized device to be compressed into long, thin sheets or strips. These sheets or strips of expansive graphite are then wound together and shaped under pressure in a mold, thereby producing various types of mechanical seal components such as flexible graphite cartons, flexible graphite packing rings, expansive graphite stainless steel wound gaskets, and expansive graphite wave-tooth composite gasket sheets. Additionally, expansive graphite can also be used to manufacture sheets or graphite-based equipment. However, due to the drawback of its lower mechanical strength, expanded graphite faces certain limitations in certain application areas, which in turn has accelerated the development of its composite materials. As countries around the world impose increasingly strict restrictions on asbestos-containing sealing materials, expanded graphite and its composite sealing materials, as high-performance non-asbestos materials, hold great potential for application and development, attracting growing attention. The use of expanded graphite and its composites in sealing applications is widespread, thanks to the high and low temperature resistance, corrosion resistance, impermeability, and excellent compressive resilience inherent in expanded graphite. Sealing gaskets made from pure flexible graphite, flexible graphite rings, flexible graphite woven packing, as well as various flexible graphite composites, all based on this material, are widely used in elastic and non-elastic sealing systems across various applications. For example, high-temperature sealing of pipes, valves, and equipment flanges in various furnaces within the metallurgical industry ； Corrosion-resistant and radiation-resistant seals used in industries such as machinery, electronics, chemistry, aerospace, nuclear energy, petroleum, and chemical processing ； Long-lasting sealing for railway diesel engines, automobiles, and various diesel engine gasket sets, as well as intake and exhaust pipe gaskets. The common expanded graphite products available on the market are shown in the figure below. The main forms of expanded graphite and its composite materials include the following: (1) Gasketing – Expanded graphite can be used as static gaskets, either in the form of sheets that have been cut from larger plates or in a molded format; its production is relatively simple. As a non-asbestos sealing gasket that is currently widely used, both pure flexible graphite gaskets and their composite counterparts provide excellent sealing performance. They can be extensively applied as static seals at flange connections in pipelines, valves, pressure vessels, pumps, and the like. (2) Expansive graphite wound gasket: An expansive graphite wound gasket is a composite sealing gasket formed by alternately winding flexible graphite strips with V-shaped or W-shaped metal strips. It exhibits excellent resistance to high and low temperatures as well as corrosion, and has good sealing performance. It is currently one of the most effective sealing materials for use under high-temperature and high-pressure conditions. Although numerous experimental studies have been conducted on it, there are very few studies involving numerical simulation. Therefore, this paper focuses on investigating the performance of this composite gasket under various operating conditions. (3) Flexible graphite packing: Flexible graphite packing is primarily made by winding flexible graphite around a core and then compressing it under pressure. There are currently many different types of such packing available on the market, including flexible graphite rings, flexible graphite packers, graphite gaskets, and reinforced graphite packers, which are widely used in static and dynamic sealing systems across various industries. Flexible graphite woven packing is a composite material made by using fibers as a framework and surrounding them with a layer of wires woven from flexible graphite. It retains most of the original properties of expanded graphite; it is unaffected by acids and bases, possesses high tensile strength, good compressive properties, and excellent thermal conductivity. It is also easy to install and remove, making it an ideal sealing material for industrial applications such as high-temperature and high-pressure steam engines as well as equipment used in oil refining. The preparation and molding of expanded graphite composite materials. The unique and excellent properties of expanded graphite make it very popular among researchers; however, certain inherent weaknesses that cannot be ignored also limit its application in a wider range of fields. For example, the porous and loose structure of expanded graphite gives it certain water and oil absorption properties; if pure expanded graphite is used as a sealing material for car engine gaskets, severe leaks will occur due to water and oil absorption ； Expanded graphite is obtained by subjecting natural graphite to high-temperature treatment; its internal bonds are held together only by weak van der Waals forces, resulting in relatively low strength ； It has poor wear resistance and is not resistant to erosion; therefore, it cannot be used directly as a sealing material for certain valves and fittings. Therefore, the search for flexible graphite composites has become the main focus for researchers around the world who are studying expanded graphite, and significant progress has already been made. Metal-expanded graphite composite materials: Many traditional metal materials possess good plasticity, toughness, interconnectivity, and resistance to corrosion. They also have high stiffness and excellent resistance to erosion, which makes them effective in compensating for some of the weaknesses of pure expanded graphite materials. As a result, the use of metal-expanded graphite composite materials is very popular in the sealing industry. Currently, there are two main types: (1) Metal-embedded expanded graphite composite gaskets, also known as reinforced graphite sheets or high-strength graphite gaskets. Commonly used inner reinforcing materials include 304, 316 stainless steel, or tinplate. The reinforcement methods include using strip reinforcements, flat plate reinforcements, or mesh plate reinforcements. 304-reinforced graphite is widely used in the sealing systems of equipment such as pressure vessels, pumps, pipes, valves, and generators; it offers good resistance to high and low temperatures, high strength, and excellent compressive resilience. (2) Coating the surface of matrix materials with a protective layer using transition metals has always been the simplest and most practical method used in many industries to enhance the properties of these materials. This approach has naturally also attracted the interest of researchers working on expanded graphite; by coating the surface of flexible graphite sheets with metals such as chromium, molybdenum, and tungsten, they have created various graphite products with excellent corrosion resistance and high strength. The method is quite simple: the surface of the graphite sheet is degassed under vacuum conditions, after which it is immersed in solutions of corresponding salts such as sodium chromate, sodium molybdate, and sodium tungstate. After a certain period of time, drying it yields the flexible graphite composite material. Due to its simplicity and ease of use, this method is widely applied in daily production. However, how to improve the controllability of its performance in order to save materials and time remains a research focus that scientists are striving to address. The sealing mechanism and deformation equation of gaskets: Mechanical equipment can be classified into static seals and dynamic seals based on the relative movement of their sealing surfaces. Static seals are the most commonly used type of seal in daily production, such as in pipe flanges, threaded connections, and the sealing of pressure vessels and their lids. Based on their working principles, static seals can be further divided into sealings using bolted flange gaskets, self-tightening seals, threaded connection gaskets, and sealants. Among these, sealings using bolted flange gaskets are the most widely used type of static seal today. Research on this type of sealing structure focuses on two main aspects: one is the performance testing of the sealing materials, with an emphasis on studying the methods for preparing these materials as well as the scientific validity and accuracy of the experimental instruments and techniques used ； On the other hand, extensive analysis has been conducted on the various causes of system seal failure, and the factors affecting the leakage rate of gaskets made from different materials in different forms have been studied separately. To prevent fluid leakage, the basic approach is to increase the resistance to fluid flow at the seal. When the resistance encountered by the fluid as it passes through the seal is greater than the pressure difference between the sides of the seal, the fluid is sealed in place. The resistance to the flow of the medium through the seal is achieved by means of the specific pressure applied to the compression surface; the greater the specific sealing pressure acting on this surface, the greater the resistance to the flow of the medium through the seal, which in turn contributes to a better seal. Strength failure: In the case of bolted flange connection systems, if a strength failure occurs, a strength analysis should be conducted separately for each of the components that make up such systems, namely the flanges, bolts, and gaskets. (1) Strength failure of flanges: The material selection and structural design of flanges have a direct impact on their strength. When selecting materials, it is essential to take into full account factors such as the corrosiveness of the internal medium, the appropriate operating temperature of the material, and the maximum stress it can withstand; these are the basic requirements for material selection. The structural design and strength verification of the flange are carried out in accordance with GB150; flanges designed in this way can generally ensure sufficient strength, thereby reducing the probability of strength failure. (2) Strength failure of bolts: The strength failure of bolts is largely caused by the following factors: insufficient material strength, corrosion by the surrounding medium, high-temperature creep relaxation, internal defects in the bolts, and excessive pre-tension. These factors cause the bolts to bend or twist, preventing them from performing their tightening function and thus leading to the failure of the connected structure. Given the high probability of bolt strength failure and its severe consequences, it must attract great attention from designers. (3) Strength failure of gaskets: Gaskets are the most important sealing elements in an entire sealed connection, and their performance directly affects the tightness of the connection structure. The deformability and rebound characteristics of gaskets are important factors determining their sealing performance. Gaskets with high deformability generally require only a low bolt preload to fill the uneven gaps on the compression surface, thereby facilitating the establishment of an initial seal ； Gaskets with good rebound properties are better able to adapt to fluctuations caused by the pressure and temperature of the medium under operating conditions. Since the gasket is in direct contact with the medium, it must also possess properties such as corrosion resistance, impact resistance, and fatigue resistance, in order to maintain excellent sealing performance and a long service life even when subjected to actual operating conditions. Seal failure in bolted flange connections is primarily manifested as leakage; once leakage occurs, it can lead to accidents. In mild cases, it results in waste of energy and equipment as well as disruptions to normal production, causing economic losses. In severe cases, it may trigger explosions that result in casualties and cause irreversible environmental damage. Therefore, it is even more important to minimize leaks as much as possible and to raise the sealing requirements for sealing systems such as containers and pipes. An analysis of the causes of leaks in actual production reveals that there are generally two ways in which fluid can leak at seals. One is “permeation leakage”. This type of leakage occurs mainly because the gasket material itself contains numerous capillaries; fluid can permeate through these capillaries, resulting in leakage. Generally, the higher the internal pressure and temperature of the medium, and the lower its viscosity, the greater the likelihood of permeation leakage. In addition to the effects of these fluid properties and the operating conditions, the type of gasket structure as well as the properties of the materials are also key factors. This issue occurs most often with non-metallic gaskets, as these materials have many tiny gaps within them, a loose structure, and poor sealing properties. When a very small pressure is applied to it, the medium can easily leak through a large number of capillaries. Generally, this can be achieved by adding certain fillers to these permeable gasket materials. For example, improvement can be achieved by soaking in solutions such as silicic acid or phosphoric acid; alternatively, combining it with certain impermeable materials to form composite materials can also reduce “permeation leakage”. The second is \"interface leakage\", which occurs along the gap between the gasket and the flange’s compressed surface; the amount of leakage is primarily determined by the size of this interface gap. There are mainly two factors that cause such leakage: (1) the unevenness of the sealing surface during processing, or insufficient and uneven compression, which results in gaps on the sealing surface. (2) There is a pressure difference or concentration difference on both sides of the sealing surface. In practical operation, pressure differences and concentration differences on both sides of the sealing surface are inevitable. To reduce such interfacial leakage, it is necessary to improve the precision of processing, while also applying an appropriate compressive force to the gasket, making full use of the compressive elastic deformation generated by the gasket to fill in the uneven gaps in the seal. The deformation coordination equations for bolted flange connection systems are currently based on elastic strength analysis. The main design and calculation methods for flange connection systems are the Waters theory and the Timosenko theory; the latter is more suitable for integral flange connection systems than the former. This is because, in addition to the conditions that need to be considered in the Waters theory, the Timosenko theory also takes into account factors such as medium pressure, internal and external temperatures, the elastic modulus of materials, and the linear expansion coefficient – all of which can change with temperature and thus affect the leakage rate of the connection system. This paper analyzes the bolted flange connection system as a whole using Timosenko theory. The basic assumptions are as follows: (1) The flange shell is simplified to an infinitely long elastic foundation beam, while the flange ring is regarded as a circular plate that does not twist but only bends by a certain angle under the action of external torque; (2) The flange and bolts are considered to be linearly elastic, without plastic yielding or creep, and the effect of bolt holes on flange deformation is not taken into account ； (3) Ignoring the radial displacement of the flange at the junction with the cylinder and the membrane stress generated by internal pressure, it is assumed that the rotation angles of the shell and the flange, as well as the resulting radial displacements, are equal. The selection of bolted flange connection structures depends on the production processes and equipment installation requirements. In chemical equipment or pipeline connections, detachable connection structures are commonly used, such as threaded connections, socket connections, and flange connections. Among these, flange connections are the most widely used detachable type, which makes it important to study the stress and deformation characteristics of flange connection structures. The materials and structural types of flanges, bolts, and gaskets should be selected appropriately based on the pressure within the medium and the type of medium. The figure below shows the general configuration of a flange connection structure. Metal toothed gaskets are typically made by using precision lathes to create concentric grooves on both sides of a metal stainless-steel flat gasket. Depending on the medium involved, materials such as flexible graphite, PTFE, asbestos-free sheets, or other soft metals can be used and applied to both sides of the gasket. It can also be used directly without a sealing layer, achieving good sealing performance; however, in high-pressure environments, it may cause damage to the surface of the flange. Simulation of the sealing performance and structural optimization of metal toothed composite gaskets. Overview of metal toothed gaskets: In actual production, the fabrication of metal toothed gaskets is extremely simple; a precision lathe is used to create numerous concentric grooves on both sides of a metal stainless steel flat gasket, thereby forming a toothed metal framework. The special tooth shape determines its sealing mechanism: under compression, stress concentration occurs at the tips of the teeth, resulting in high stresses that create a linear seal of the metal. It can achieve a high level of sealing performance, is relatively safe and reliable. However, factors such as material selection, structural design, and product manufacturing have a significant impact on the sealing capabilities of metal toothed gaskets. Moreover, using such toothed gaskets in high-pressure environments can easily cause damage to the surface of the flanges. As a result, in the current industry, these metal-based gaskets are rarely used alone; instead, composite gaskets formed by combining them are more commonly employed. A metal toothed composite gasket is a type of gasket that uses a metal framework as its base, and depending on the working medium, flexible graphite, PTFE, non-asbestos sheets, or other soft metals are applied to both sides of the framework. In recent years, with the continuous advancement of sealing technology, the properties of expanded graphite materials have gained increasing popularity. As a new type of sealing material, it is increasingly replacing asbestos-based sealing materials and traditional metal sealing materials. Therefore, for researchers working on metal-toothed composite gaskets, using the substrate of toothed sealing gaskets and attaching flexible graphite to their upper and lower contact surfaces to create flexible graphite metal-toothed composite gaskets is a research direction that they are particularly eager to pursue. Numerous experimental studies have shown that this composite gasket not only retains the reliability of the wavy-toothed framework of the original toothed gasket’s overall structure, while maintaining multiple linear seals at its sharp teeth, but also possesses the characteristics of expanded graphite—such as resistance to high and low temperatures and excellent compressibility and resilience—thereby achieving a dual-sealing effect. The sealing mechanism of this composite gasket is such that, under a low preload, the soft and flexible graphite layer on the surface of the gasket is pressed into the grooves of the metal frame. The upper and lower surfaces of the metal frame do not come into direct contact with the flange, thereby preventing damage to the flange. As more stress is applied, the metal frame undergoes elastic deformation, which compresses the expanded graphite and traps it in the annular sealed space between the metal frame and the flange surfaces. Thus, a dual sealing effect is achieved by combining multiple layers of metal materials with flexible graphite. Flexible graphite metal-toothed composite gaskets are particularly suitable for high-temperature, high-pressure environments as well as those with alternating operating conditions. They are widely used at the flange connections of equipment such as heat exchangers, pressure vessels, and reaction kettles, serving as an ideal alternative to traditional metal-lined gaskets. The structural types of metal toothed gaskets are divided into two basic categories depending on the type of sealing surface of the flange: one is the toothed gasket with a positioning ring, which is used for flat flanges ； The second is a toothed gasket without a positioning ring, used for uneven flanges. Figure 4-1 shows the external structure of the positioning ring for the flexible graphite metal toothed composite gasket in actual production. The simulation model for the compressive properties of flexible graphite metal toothed composite gaskets, as well as the types of materials used, are based on the chemical industry standard in China titled \"Toothed Composite Gaskets with Coatings for Steel Pipe Flanges\" (standard number: HG/T20623-2009), which came into effect in July 2009. In this paper, a metal toothed gasket with a DN80 size is selected; its metal framework structure and overall design are shown in Figures 4-2 and 4-3 respectively, while the dimensions of the standard gasket are listed in the table. This paper mainly investigates the effects of the number of teeth N of the metal framework and the thickness S of the expanded graphite coating on the mechanical properties of the composite gasket, under the condition that the nominal diameter remains constant. During the pre-treatment process, 304 stainless steel is used as the material for the metal framework; its elastic modulus is 2.0x10＾11 Pa, Poisson’s ratio is 0.313, shear modulus is 7.45x10＾11 Pa, yield strength is 2.05X 10＾8 Pa, and density is 7850 kg/m3. The element type used is “planel83”” ； The material used for the cover layer is expanded graphite with a density of 1000 kg/m3; its elastic modulus is 1.0x10＾10 Pa, Poisson’s ratio is 0.01, shear modulus is 1.0x10＾7 Pa, and yield strength is 6.0x10＾7 Pa. The element type used is “planel82”. Since the gasket structure is a centrally symmetric shape, the circumferential stresses are neglected, and the three-dimensional geometric model is simplified to a two-dimensional model for axial stress analysis. (1) The Plane182 element and the Planel82 element are 2D 4-node solid elements used for 2D models. Each unit has 4 nodes, and each node has two degrees of freedom, namely translation in the x and y directions. This element can be used as a planar element (plane stress, plane strain, or generalized plane strain) or as an axisymmetric element. It possesses capabilities related to plasticity, hyperelasticity, stress strengthening, large deformations, large strains, as well as mixed force-displacement formulations; it can therefore be used to simulate the deformation of quasi-elastic materials that are hardly compressible and of hyperelastic materials that are completely incompressible. (2) The Plane183 element, also known as the Planel83 element, is a high-order 2D 8-node solid element with quadratic displacement functions, suitable for meshing irregular models. The unit is defined by 8 nodes, each with two degrees of freedom in direction. Similar to the PIAnel82 element, it can also be used as a planar element and an axisymmetric element. It possesses the capabilities for plasticity, flow, stress stiffness, large deformations, and high strains, as well as the ability to use force-displacement hybrid formulas; it can be used to simulate the deformation of nearly incompressible elastoplastic materials, and it supports initial stress options. And it provides different print output options. Considering the irregular shape of the metal framework, a relatively simple free meshing method was adopted in this paper; the elements are quadrilaterals with an edge length of 0.1 mm. The gasket models of several sizes used in this simulation, after mesh generation, resulted in approximately 20,000 nodes and around 7,600 elements. Figure 4-4 shows the element mesh distribution for a toothed composite gasket with a tooth pitch of 12 teeth and an expansion graphite coating thickness of 0.4 mm. The ANSYS software for coupled degrees of freedom includes functions for coupled degrees of freedom and constraint equations, enabling accurate representation of details such as rigid regions, hinge connections in certain structures, symmetrically arranged relative sliding boundaries, periodic conditions, and other special internal node connections. Typical coupling degrees used in the model section include: structural symmetry, the creation of hinges between two identical nodes, pins, joints, and sliding connections, so that a certain part of the model exhibits the properties of a rigid body. In the simulations of this section, in order to better reflect the deformation coordination of the expanded graphite composite layer, it is not possible to achieve this using only conventional elements to connect the nodes; rather, it is necessary to apply coupling of degrees of freedom on the upper surface of the composite layer. The X direction is taken as the primary degree of freedom, with the displacements in the X and Y directions of the repeated nodes on that upper surface being constrained to be identical, thereby ensuring the rigid-body behavior of the upper surface. After loading and specifying that the analysis type for the defined structure is static analysis, a multi-step loading method is required to obtain the compression-elasticity curve of the overall structure of the flexible graphite-metal toothed composite gasket. First, boundary conditions need to be applied to the model; in this simulation, the axial freedom of the lower surface of the lower composite layer in all models is constrained, that is, UY=0 is imposed ； Then, a uniform load of 60 MPa is applied to the upper surface of the upper composite layer. The settings for the load step include setting the end time of the load step at 10 s, the number of sub-steps at 100, and specifying that the results should be read at each sub-step. After that, the load step is written, denoted as 1 ； Finally, a load of 0 MPa is applied to the upper surface of the upper composite layer. The settings for the load step include a duration of 20 seconds for that step; all other values are set to their default values. This load step is designated as 2. Load step 1 serves to gradually apply load to the model, while load step 2 is used to remove the load from the model. Once the boundary conditions and load steps are set, calculations can begin; select “Solution—Solve—From LS Files” to solve for multiple load steps. To view the calculation results, use ANSYS’ general post-processing tools: select the desired step for reading, and then choose around ten time steps during the loading process as well as ten time steps during the unloading process for a particular node. The displacement and stress values of that node at these corresponding time steps can be retrieved. The next three figures show the performance of the standard flexible graphite-metal toothed composite gasket (P=1.0 mm) ； S=0.5mm, N=12) Stress and displacement maps of the gasket model at certain moments during loading and unloading. (1) Stress cloud diagram (2) Displacement diagram. Since the deformation of this composite gasket is very small during loading and unloading, the displacement diagram for unloading is not presented here. As can be seen from the above three graphs, during the loading process, as the load step increases – that is, as the load value rises – both the stress and displacement values of the gasket structure continue to increase. During unloading, the stress value at this node decreases as the load decreases, which is consistent with the actual loading and unloading patterns. In the post-processing tool POST26, the changes in stress values at this node during loading and unloading are represented by line graphs. The stress values and total displacements corresponding to each selected load step are then entered into Origin software for further processing, thereby obtaining the pressure-closure curves for standard flexible graphite metal toothed composite gaskets at room temperature. Using the aforementioned simulation method, the pressure-closure characteristics of flexible graphite metal wave-toothed composite gaskets with different structural features are determined one after another, and these characteristic values are also entered into Origin software for processing. The results of this processing are shown in Figure 4-10. It can be observed that the compression curves of standard composite gaskets (with 12 teeth) as well as those with 10 teeth are approximately linear. The compression behavior curves of standard gaskets with an expanded graphite coating thickness of 0.5 mm, and those with coating thicknesses of 0.4 mm and 0.6 mm, are also approximately linear. Figure 4-10(a) shows the pressure closure eigenvalues corresponding to composite gaskets with a gasket thickness of S=0.5 mm, a wave tooth depth of h=0.32 mm, and different tooth pitches and numbers of teeth ； Figure 4-10(b) shows the pressure closure characteristic values for composite gaskets with a tooth pitch of P=1.0 mm, a number of teeth of N=12, and a wave tooth depth of h=0.32 mm, at different gasket thicknesses (these values are provided for numerical analysis later). The leakage rate simulation of flexible graphite metal-toothed composite gaskets: For sealing gaskets, the value of their leakage rate directly determines the quality of their sealing performance. At present, research on flexible graphite-metal toothed composite gaskets is mostly limited to experimental studies. Moreover, even when numerical simulations are used to assess the performance of such gaskets, they focus on the structural properties of the metal framework alone; there is almost no numerical simulation analysis regarding the leakage rate of the entire composite gasket. For the composite gaskets of the various structures mentioned above, the bolt pre-tension forces corresponding to internal pressures of 0.98 MPa, 1.568 MPa, 2.45 MPa, 4 MPa, 6 MPa, and 8 MPa are calculated using the following formula. During the preprocessing stage, the material parameters of the gaskets are entered as compression and linear unloading values corresponding to the compression-rebound values of the stress closure curves for each structure shown in Figure 4-10; all other aspects such as element selection, model establishment, boundary conditions, loading, and solution methods remain unchanged. The figure below shows the stress distribution maps for standard flexible graphite metal-toothed composite gaskets under ten different operating conditions. Under any operating conditions, the stress distribution in the gasket follows the same pattern: it is uneven along the diameter, increasing gradually from the inside toward the outside ； As the pressure inside the medium increases, the residual stress in the gasket increases ； When the internal pressure of the medium remains constant, the residual stress in the gasket increases as the pre-tensioning force of the bolts applied increases, which is in line with the mechanical behavior of bolted flange connection systems. The correctness of the numerical analysis method for simulation was verified once again. The stress and deformation values obtained from ANSYS analysis for each operating condition are respectively substituted into the formulas to calculate the leakage rate of the flexible graphite metal toothed composite gaskets under those conditions. The same analysis is conducted on gaskets of other structural types: first, the large-scale commercial finite element analysis software ANSYS is used to determine the stress values and deformations under various operating conditions, which are then inserted into the leakage rate equations for porous media models to calculate the corresponding leakage rates. Finally, the gasket leakage rate values under ten operating conditions, calculated using various models, were imported into the Origin data processing software for comparative analysis. The results are shown in Figure 4-12. Figure (a) shows a comparison of the leakage rates under different internal pressures in various media for composite gaskets with a gasket thickness of S=0.5 mm, a wave tooth depth of h=0.32 mm, different tooth pitches, and different numbers of teeth ； Figure (b) shows a comparison of the leakage rates for composite gaskets with a gasket thickness of S=0.5 mm and a wave tooth depth of h=0.32 mm, under different tooth pitches and numbers of teeth, at a medium internal pressure of 2.45 MPa and under various bolt pre-tension forces ； Figure (C) shows a comparison of the leakage rates under different internal media pressures for composite gaskets with a tooth pitch of P==1.0 mm, a number of teeth of N=12, and various thicknesses of expanded graphite coating layers ； The figure shows a comparison of the leakage rates under different bolt pre-tension forces, with a tooth pitch of P=1.0 mm and a number of teeth of N=12, at an internal pressure of 2.45 MPa (for reference only in subsequent numerical analyses). As can be seen from Figure (a), the effect of the internal pressure on the expansion graphite metal toothed composite gaskets with different numbers of teeth is consistent; that is, as the internal pressure increases, the leakage rates of the composite gaskets with different numbers of teeth all increase ； At the same pressure in the same medium, the composite gasket with N=10 teeth has the lowest leakage rate, while the standard composite gasket (N=12) has the highest leakage rate. As can be seen from Figure (b), the effect of bolt pre-tension force on expanded graphite-metal toothed composite gaskets with different numbers of teeth is also consistent; that is, as the bolt pre-tension force increases, the leakage rate of composite gaskets with various numbers of teeth all decreases ； Under the same pre-tension of bolts of the same size, the composite gasket with N=10 teeth has the lowest leakage rate, while the standard composite gasket (N=12) still has the highest leakage rate. As can be seen from Figures (a) and (b) in summary. For DN8 flexible graphite-metal composite gaskets, the standard metal frame structure is not the best; rather, the composite gasket designed with a thickness of S=0.5mm, a wave tooth depth of h=0.32mm, a tooth spacing of p=1.0mm, and 10 teeth in total exhibits the lowest leakage rate, meaning it has the best sealing performance. The effect of the pressure inside the medium on the expansion graphite metal toothed composite gaskets with different expansion graphite coatings is consistent; that is, as the pressure inside the medium increases, the leakage rate of these composite gaskets decreases for all of them ； At the same pressure in the same medium, composite gaskets with an expanded graphite coating of a certain thickness have the lowest leakage rate; the maximum leakage rate remains at the level observed for standard composite gaskets. As can be seen from Figure (d), the effect of bolt pre-tension on the expansion graphite metal toothed composite gaskets with different expansion graphite coatings is consistent; that is, as the bolt pre-tension increases, the leakage rate of the composite gaskets with these different coatings decreases ； Under the same pre-tension of bolts of the same size, the composite gasket with an expanded graphite coating thickness of S=0.4 mm has the lowest leakage rate, while the composite gasket under the same standards (S=0.5 mm) still has the highest leakage rate. As can be seen from Figures (c) and (d), for DN80 flexible graphite metal composite gaskets, the composite gasket designed with a gasket thickness of S=0.4 mm, a wave tooth depth of h=0.32 mm, a tooth spacing of P=1.0 mm, and 12 teeth exhibits the lowest leakage rate, thereby achieving the goal of structural optimization for the gasket. This article summarizes a composite gasket consisting of a gasket material, expanded graphite, and metal teeth, which has received considerable attention in current research. This type of gasket is obtained by using a metal wave-shaped framework as its base, with a layer of expanded graphite of a certain thickness applied over it. It combines the advantages of high stiffness and reliability associated with the metal framework, along with the excellent compressive resilience and resistance to extreme temperatures of expanded graphite, thereby achieving a dual-sealing effect. It is widely used in the sealing structures of various devices. At present, most of the analyses on it are still limited to experimental studies, with few numerical simulation analyses; therefore, this paper uses ANSYS software to conduct numerical simulation analysis on the sealing performance of such gaskets and achieves structural optimization of the gaskets. The conclusions drawn include: (1) During the loading process, as the load steps increase and the load value rises, both the stress and displacement values of the gasket structure continue to increase; whereas during unloading, the stress value at this node decreases as the load decreases, which is consistent with the actual loading and unloading patterns ； (2) When the thickness of the expanded graphite coating is 0.5 mm, the compression curves of the standard composite gasket (12 teeth) and the composite gasket with 10 teeth are nearly linear ； (3) When the number of teeth on the metal frame is 12, the compression performance curves of the standard gasket (S=0.5 mm) and the composite gaskets with cover layer thicknesses of 0.4 mm and 0.6 mm are approximately linear ； (4) When the thickness of the coating layer on the expanded graphite is 0.5 mm, the composite gasket with N=10 teeth has the lowest leakage rate, whereas the standard composite gasket (N=12) has the highest leakage rate ； (5) When the number of teeth on the metal skeleton is 12, the composite gasket with an expansion graphite coating of that thickness has the lowest leakage rate; the maximum leakage rate remains at the level of that of a standard composite gasket.