Method for analyzing carbon concentration distribution in carburized steel material

Abstract

Provided is a method for analyzing a carbon concentration distribution in a carburized steel material with high analysis accuracy. The method includes analyzing, using a diffusion equation, the carbon concentration distribution in a steel material in which carbides are formed by carburizing. The diffusion equation is represented by formula (I): J =-D (gradC) and formula (II): ∂C / ∂t = -divJ (In formula (I), J is the diffusion flux of carbon; D is the diffusion coefficient of carbon in the steel material; C is the carbon concentration in the steel material; and grad is the gradient of position. In formula (II), C is the carbon concentration in the steel material; t is time; div is divergence; and J is the diffusion flux of carbon). In formula (I), D is calculated from formula (III): D = Dγ (∂Cγ / ∂C) or formula (IV) (In formula (III), Dγ is the diffusion coefficient of carbon in austenite; C is the carbon concentration in the steel material; Cγ is the carbon concentration in austenite. In formula (IV), mγ is the mobility of carbon in austenite; R is a gas constant; T is temperature; C is the carbon concentration in the steel material; Cγ is the carbon concentration in austenite; and γγ is the activity coefficient [-] of carbon in austenite).

Description

Analysis method for carbon concentration distribution in carburized steel

[0001] The present invention relates to a method for analyzing the carbon concentration distribution in carburized steel material.

[0002] In the case of steel products (for example, drivetrain products such as gears), carburizing is performed on products that have been machined into a desired shape to ensure necessary properties such as wear resistance and fatigue strength of the product surface. In recent years, reduced-pressure carburizing, typified by reduced-pressure carburizing, has become more common from the perspective of reducing environmental impact.

[0003] In reduced-pressure carburizing, the processed product is heated at a certain temperature for a certain time in a heating furnace in an atmosphere where the gas is evacuated and reduced pressure and a carburizing gas containing carbon is injected, and then the processed product is cooled by immersing it in a cooling medium such as oil. This reduced-pressure carburizing process has a carburizing period, during which the injection of carburizing gas continues, and a diffusion period, during which the injection of carburizing gas is stopped.

[0004] In vacuum carburizing, if too much carbon penetrates the surface layer of a product and the carbon does not diffuse sufficiently deeper than the surface layer, coarse carbides are formed, which can reduce the fatigue strength and other properties of the product. To avoid such problems, the following techniques have been disclosed.

[0005] In Patent Document 1 (JP 2008-208403 A), a carburizing pattern is set, and the carbon concentration of any cell in the surface layer of the target part is calculated using a diffusion equation.Then, the set carburizing conditions are evaluated and pass / fail is determined by comparing the calculated carbon concentration with the desired carbon concentration.

[0006] Patent Document 2 (Japanese Patent No. 6583600) describes a reduced-pressure carburizing method that can suppress carburizing variations in carburized parts. The carburizing process includes an early carburizing process from the start of the carburizing process to crossover time te, and a later carburizing process from crossover time te to carburizing time ta. In the early carburizing process, the actual carburizing gas flow rate is set to be equal to or greater than the theoretical carburizing gas flow rate at the reference time ta / 5 from the start of the carburizing process, and equal to or less than the theoretical carburizing gas flow rate at 20 seconds from the start of the carburizing process. In the later carburizing process, the actual carburizing gas flow rate is set to be within the range of 1.00 to 1.20 times the theoretical carburizing gas flow rate.

[0007] JP 2008-208403 A Japanese Patent No. 6583600 A

[0008] Effect of excessive edge carburization on fatigue strength in vacuum carburizing, Iron and Steel, Iron and Steel Institute of Japan, 2010, Vol. 96, No. 6, pp. 48-53

[0009] In Patent Document 1, the carbon concentration distribution is analyzed using the diffusion coefficient included in the diffusion equation as a constant. However, strictly speaking, the diffusion coefficient is a value that changes depending on the carbon concentration in γ, and therefore the analysis accuracy of the carbon concentration distribution is low.

[0010] Patent Document 2 sets the conditions for reduced pressure carburization treatment in order to suppress variations in carburization of carburized parts, but does not analyze the distribution of carbon concentration in steel material that undergoes reduced pressure carburization treatment.

[0011] An object of the present invention is to improve the accuracy of analysis of carbon concentration distribution.

[0012] In order to solve the above problems, the present invention provides the following aspects: [Aspect 1] A method for analyzing carbon concentration distribution in a carburized steel material, comprising analyzing the carbon concentration distribution in a steel material in which carbides are formed by carburization using a diffusion equation, wherein the diffusion equation is expressed by the following equations (I) and (II): (In formula (I), J is the diffusion flux of carbon, D is the diffusion coefficient of carbon in the steel material, C is the carbon concentration in the steel material, and grad is the gradient of the position. In formula (II), C is the carbon concentration in the steel material, t is time, div is divergence, and J is the diffusion flux of carbon), and D in formula (I) is represented by the following formula (III) or (IV): (In formula (III), D γ is the diffusion coefficient of carbon in austenite, C is the carbon concentration in the steel, and C γ is the carbon concentration in austenite, and in formula (IV), m γ is the mobility of carbon in austenite, R is the gas constant, T is the temperature, C is the carbon concentration in the steel, and C γ is the carbon concentration in austenite, and γ γ is the activity coefficient of carbon in austenite [-]). [Aspect 2] In the carbide precipitation region in the steel material formed by the carburizing treatment, the diffusion coefficient D of formula (I), the diffusion flux J of formula (II), the diffusion coefficient D of formula (III) γ , and the mobility m of formula (IV) γ and multiplying at least one of the carbon concentration distributions analyzed from the diffusion equation by a correction coefficient, the correction coefficient being set to satisfy a condition that an error between the carbon concentration distribution analyzed from the diffusion equation and the carbon concentration distribution measured in advance is equal to or less than a threshold. [Aspect 3] When calculating D in formula (I) from formula (III), the right side of any of formulas (I), (II), and (III) is multiplied by the following formula (V): λ = V γ ｎ ...(V) (in the formula, V γ is the volume fraction of austenite [-], and n is an integer.), and when D in formula (I) is calculated from formula (IV), multiplying the right side of any of formulas (I), (II), and (IV) by the tortuosity λ represented by formula (V). The method for analyzing a carbon concentration distribution according to aspect 1 or 2, comprising:

[0013] According to the present invention, even in steel in which a multi-phase region of austenite and carbide is formed during carburization (e.g., reduced pressure carburization, high concentration carburization, etc.), the carbon concentration distribution can be analyzed with high accuracy based on a diffusion equation using the carbon concentration gradient in the steel as the driving force. Furthermore, by correcting the parameters used in the calculation of the diffusion coefficient in the steel and the diffusion simulation (in other words, the diffusion flow rate, diffusion coefficient, and / or mobility included in the diffusion equation) with a correction coefficient, it becomes easier to ensure accuracy when analyzing the carbon concentration in steel that has been subjected to carburization.

[0014] 1 is a flowchart illustrating a method for correcting a diffusion simulation for analyzing carbon concentration in a steel material. FIG. 1 is a perspective view of a steel material sample. FIG. 2 is a cross-sectional view of a steel material sample. FIG. 3 is a diagram illustrating the relationship between analysis results and actual measured values ​​in Examples 1-2 and Comparative Examples 1-2 for carbon concentration depending on depth from the steel material surface. FIG. 4 is a diagram illustrating the relationship between analysis results and actual measured values ​​in Examples 3-4 and Comparative Examples 3-4 for carbon concentration depending on depth from the steel material surface. FIG. 5 is a diagram illustrating the relationship between analysis results and actual measured values ​​in Example 5 for carbon concentration depending on depth from the steel material surface. FIG. 6 is a diagram illustrating the relationship between analysis results and actual measured values ​​in Example 7 for carbon concentration depending on depth from the steel material surface. FIG. 7 is a diagram illustrating the relationship between analysis results and actual measured values ​​in Example 8 for carbon concentration depending on depth from the steel material surface. FIG. 9 is a diagram illustrating the relationship between analysis results and actual measured values ​​in Example 9 for carbon concentration depending on depth from the steel material surface. FIG. 10 is a diagram illustrating the relationship between analysis results and actual measured values ​​in Example 10 for carbon concentration depending on depth from the steel material surface. 1 is a diagram showing the relationship between the analysis results and the actual measured values ​​according to Example 11 for the carbon concentration depending on the depth from the steel material surface. FIG. 1 is a diagram showing the relationship between the analysis results and the actual measured values ​​according to Example 12 for the carbon concentration depending on the depth from the steel material surface. FIG. 1 is a diagram showing the relationship between the analysis results and the actual measured values ​​according to Example 13 for the carbon concentration depending on the depth from the steel material surface. FIG. 2 is a diagram showing the relationship between the analysis results and the actual measured values ​​according to Example 14 for the carbon concentration depending on the depth from the steel material surface. FIG. 3 is a diagram showing the relationship between the analysis results and the actual measured values ​​according to Example 15 for the carbon concentration depending on the depth from the steel material surface. FIG. 4 is a diagram showing the relationship between the analysis results and the actual measured values ​​according to Example 16 for the carbon concentration depending on the depth from the steel material surface. FIG. 5 is a diagram showing the relationship between the analysis results and the actual measured values ​​according to Example 17 for the carbon concentration depending on the depth from the steel material surface. FIG. 6 is a diagram showing the relationship between the analysis results and the actual measured values ​​according to Example 18 for the carbon concentration depending on the depth from the steel material surface. FIG. 7 is a diagram showing the relationship between the analysis results and the actual measured values ​​according to Example 19 for the carbon concentration depending on the depth from the steel material surface. FIG. 8 is a diagram showing the relationship between the analysis results and the actual measured values ​​according to Example 20 for the carbon concentration depending on the depth from the steel material surface.10 is a diagram showing the relationship between the analysis results and the actual measured values ​​according to Example 21 for the carbon concentration depending on the depth from the steel material surface; FIG. 11 is a diagram showing the relationship between the analysis results and the actual measured values ​​according to Example 22 for the carbon concentration depending on the depth from the steel material surface; FIG. 12 is a diagram showing the relationship between the analysis results and the actual measured values ​​according to Example 23 for the carbon concentration depending on the depth from the steel material surface; and FIG. 13 is a diagram showing the relationship between the analysis results and the actual measured values ​​according to Example 24 for the carbon concentration depending on the depth from the steel material surface.

[0015] DETAILED DESCRIPTION OF THE INVENTION In the present specification, the symbol [-] means that there is no unit.

[0016] First Embodiment The distribution of carbon concentration inside a steel material after carburizing can be analyzed based on a diffusion simulation (a diffusion equation based on the carbon concentration gradient in the steel material) and calculation of the diffusion coefficient, which will be described later.

[0017] (Diffusion Simulation Based on Diffusion Equation) In carburizing, carbon diffuses through the steel material, and Fick's law holds. In this case, the carbon diffusion flux J is expressed by the following diffusion equation (1), and the change in carbon concentration over time (∂C / ∂t) is expressed by the following diffusion equation (2).

[0018] The parameters included in the formulas (1) and (2) are defined as follows: J: carbon diffusion flux [m mol % / s] (vector notation); D: carbon diffusion coefficient in steel [m ２ / s]. - C: Carbon concentration in the steel. The carbon concentration C may be a molar concentration [mol%] (or [-]) or a mass concentration [mass%] (or [-]). Here, the concentration is a molar concentration [mol%], but the unit of the diffusion flux J changes depending on the unit of concentration. - t: Time [s]. - grad: Position gradient. For example, in the case of a three-dimensional object with the x-, y-, and z-axes as rectangular coordinate axes, grad C = (∂C / ∂x, ∂C / ∂y, ∂C / ∂z) (the same applies below). - div: Divergence. For example, in the case of a three-dimensional object with the x-, y-, and z-axes as rectangular coordinate axes, div J = ∂Jｘ / ∂x + ∂J ｙ / ∂y+∂J ｚ / ∂z, and J ｘ、 J ｙ、 J ｚ are the x, y, and z components of the carbon diffusion flux J (vector notation), respectively (the same applies below).

[0019] (Diffusion Simulation for Carbide Formation) In high-concentration carburizing, which forms carbides by controlling the carbon potential in the atmospheric gas high, and in the above-mentioned reduced-pressure carburizing, carbides form during carburizing. In this case, to calculate the carbon diffusion flux based on Equation (1), a carbon diffusion coefficient corresponding to the multi-phase region of austenite and carbides is required. This diffusion coefficient varies significantly depending on the carbon concentration, which varies depending on the carburization state, the concentration of dissolved carbon in austenite, which varies depending on the steel components other than carbon, and the amount of carbide formation. In this case, to improve the analytical accuracy, it is necessary to derive diffusion coefficients at different carbon concentrations and temperatures for each steel type and set them in the simulation. For example, a method of deriving a diffusion coefficient solely through experiments, such as the decarburization method, is not practical because it requires an excessive number of experiments. Therefore, the present inventors have discovered a method for calculating the carbon diffusion coefficient in a steel containing a multi-phase region using the diffusion coefficient and mobility of carbon in austenite, as described below, and then calculating the carbon diffusion flux based on Equation (1).

[0020] (Calculation of Diffusion Coefficient of Multi-Phase Region Using Diffusion Coefficient in Austenite) Even in the case of a steel material containing a multi-phase region including austenite and carbide, the diffusion flux of carbon can be calculated based on Equation (1) by calculating the diffusion coefficient D in the steel material using Equation (3).

[0021] The parameters included in equation (3) are defined as follows: D: Diffusion coefficient of carbon in steel [m ２ / s]. - Dγ: Diffusion coefficient of carbon in austenite [m ２ / s]. - C: Carbon concentration in steel. - Cγ: Carbon concentration in austenite. - ∂: Partial differential symbol.

[0022] The carbon concentration may be a molar concentration [mol %] (or [-]) or a mass concentration [mass %] (or [-]).

[0023] (Calculation of Diffusion Coefficient of Multi-Phase Region Using Mobility in Austenite) Even for an object that includes a multi-phase region containing austenite and carbides, the diffusion coefficient in the steel material can be calculated using Equation (4), thereby allowing the carbon diffusion flux to be calculated based on Equation (1).

[0024] The parameters included in equation (4) are defined as follows: D: Diffusion coefficient of carbon in steel [m ２ / s]. - m γ : Mobility of carbon in austenite [m ２ mol / J s]. - R: Gas constant [8.314 J / (K mol)]. - T: Temperature [K]. - C: Carbon concentration in steel. - Cγ: Carbon concentration in austenite. - γ γ : Activity coefficient of carbon in austenite [-]. - ∂ is the partial differential symbol.

[0025] The carbon concentration may be a molar concentration [mol %] (or [-]) or a mass concentration [mass %] (or [-]).

[0026] As described above, the diffusion coefficient D in equation (1) can be determined from equation (3) or (4).

[0027] (Diffusion Simulation Using Thermodynamic Calculation) The details will be described later, but the parameters included in the formulas (3) and (4) can be obtained by the following method. The carbon concentration Cγ in austenite and the carbon activity coefficient γ γ may be obtained by thermodynamic calculation. The diffusion coefficient Dγ in formula (3) may be obtained from a numerical value obtained in advance by an experiment using a steel material to be carburized or from data reported as experimental data. The mobility m of carbon in austenite included in formula (4) γmay be obtained by thermodynamic calculation using a thermodynamic database and a diffusion database. The carbon concentration C included in the formulas (3) and (4) is a value obtained over time by a diffusion simulation.

[0028] (Setting the Labyrinth Factor) The calculation of the diffusion coefficient in steel using the above-mentioned formula (3) or (4) assumes a continuous austenite single phase and ignores the influence of the volume fraction of carbides. Therefore, when targeting a steel containing a multi-phase region containing austenite and carbides, in order to make the analysis more precise, the right side of formula (3) or (4) in the multi-phase region may be multiplied by the labyrinth factor λ (where λ≠1) shown in the following formula (5) to calculate the diffusion coefficient D of carbon in the steel. Since "may be calculated (omitted) to make the analysis more precise," multiplication by the labyrinth factor λ is not essential. In the single-phase region, λ=1. Although mathematically self-evident, the right side of formula (1) may be multiplied by the labyrinth factor λ without multiplying the right side of formula (3) or (4) by the labyrinth factor λ. Alternatively, instead of multiplying the right side of equation (3) or (4) by the labyrinth degree λ, the right side of equation (2) may be multiplied by the labyrinth degree λ. γ ｎ ...(5)

[0029] The parameters included in equation (5) are defined as follows: λ: labyrinth degree [-]; V γ : Austenite volume fraction [-], which may be calculated by thermodynamic calculation. - n: An integer. Generally, n is 1 or 2, but it may be an integer of 3 or more if analytical accuracy is ensured. The upper limit of the integer that can ensure analytical accuracy differs depending on the type of steel, etc., so it cannot be uniquely defined.

[0030] (Calculation method for diffusion simulation) Mesh data is created in which the surface layer of the steel material to be carburized is divided into a plurality of cells. The size of each cell can be set appropriately, for example, to 1 to 500 μm. The cell size can be uniform, or can be gradually increased from the surface of the steel material toward the interior (in other words, gradually decreased from the interior of the steel material toward the surface). The types of mesh and elements are not particularly limited, and can be selected appropriately, for example, triangles and quadrangles in two dimensions, and tetrahedrons and hexahedrons in three dimensions. Either primary or secondary elements can be used.

[0031] When targeting a portion or shape (e.g., a flat portion) where carbon diffuses in one dimension during carburizing, one-dimensional modeling may be used. When the steel material has a round bar or cylindrical shape, it can be treated as one-dimensional by using a cylindrical coordinate system for the mesh data. When targeting a steel material where carbon diffuses in two or three dimensions during carburizing (e.g., a steel material with an edge-shaped portion), two- or three-dimensional modeling is required. When targeting an axisymmetric steel material such as a notched round bar test piece, it can be treated as two-dimensional by using a cylindrical coordinate system for the mesh data. The analysis time (step time) of the diffusion simulation is not particularly limited, but can be, for example, 0.001 to 1.0 seconds.

[0032] It is assumed that the steel surface is in equilibrium with graphite during the carburizing period of reduced-pressure carburizing. Therefore, based on the chemical composition excluding carbon of the steel to be carburized, the equilibrium phase and equilibrium composition at the carburizing temperature in equilibrium with graphite are determined by well-known thermodynamic calculations. The equilibrium phase and equilibrium composition determined by thermodynamic calculations can be used to determine the carbon concentration in the steel, the solute C concentration in austenite, the volume fraction of austenite, and the activity coefficient of carbon in austenite.

[0033] On the other hand, periods when carbon does not penetrate from the steel surface, such as the diffusion period of reduced-pressure carburizing, are considered to be closed systems. Cementite (θ) may precipitate on the surface or inside the steel due to carburizing. In this case, the carbon in the steel is distributed between cementite and austenite. Therefore, the equilibrium phase and equilibrium composition in the steel at the temperature and chemical composition at that position can be determined by the thermodynamic calculation described above, and the solute C concentration in austenite, the volume fraction of austenite, and the activity coefficient of carbon in austenite can be determined.

[0034] Diffusion coefficient D of carbon in austenite of steel γ The mobility m of carbon in austenite in steel may be determined by a previous experiment using the steel to be carburized, or may be reported experimental data. γ can be obtained, for example, by thermodynamic calculation using a thermodynamic database and a diffusion database. γ is required when calculating the diffusion coefficient D in steel from equation (3). For example, the diffusion coefficient D of carbon in austenite γ As the above, the following formula (6) shown in Non-Patent Document 1 may be used.

[0035] The definitions of the parameters included in equation (6) are as follows: γ : Diffusion coefficient in austenite [μm ２ / s]. - C γ : Carbon concentration in austenite. - Carbon concentration C γ may be a molar concentration [mol%] (or [-]) or a mass concentration [mass%] (or [-]). - T: Temperature [K].

[0036] Numerical analysis methods for carrying out a simulation based on a diffusion equation include, but are not limited to, the finite difference method, the finite element method, the finite volume method, and the like.

[0037] Second Embodiment In this embodiment, the formulas (1) to (4) described in the first embodiment are corrected to perform a diffusion simulation, thereby making it easier to ensure accuracy when analyzing the carbon concentration in a steel material that has been subjected to a carburizing treatment. A method for correcting the diffusion simulation will be described below with reference to the flowchart in FIG.

[0038] In step S101, the distribution of carbon concentration inside the steel material when carburizing treatment (e.g., reduced pressure carburizing treatment) is performed is analyzed based on the diffusion simulation. The distribution of carbon concentration can be analyzed by applying the carburizing treatment conditions (temperature and time) to the diffusion simulation.

[0039] In step S102, the steel is carburized, and then the carbon concentration distribution within the steel is measured. The steel used in step S102 is the same as the steel used in step S101, and the carburizing conditions used in step S102 are the same as the carburizing conditions used in step S101. Acetylene, for example, can be used as the carburizing gas used in the carburizing process. Any known measurement method can be used as the carbon concentration measurement method; for example, an electron probe microanalyzer (EPMA) can be used to measure the carbon concentration distribution in the steel.

[0040] The order in which the processes of steps S101 and S102 are performed is not particularly limited.

[0041] In step S103, a correction coefficient k[-] is determined for correcting the diffusion simulation used in the processing of step S101 so that the error Ec between the carbon concentration distribution analyzed in the processing of step S101 and the carbon concentration distribution measured in the processing of step S102 is equal to or less than a threshold value Eth.

[0042] As the error Ec, for example, the mean absolute error (MAE) or the mean square error (MSE) can be used. The threshold value Eth can be set appropriately taking into consideration the analytical accuracy of the carbon concentration when the correction coefficient k is not set. Here, the smaller the threshold value Eth, the more accurate the analytical accuracy of the carbon concentration can be improved. For example, when an edge-shaped steel material is to be analyzed, the analytical accuracy can be improved by setting the threshold value Eth to 18% or less. Furthermore, the threshold value Eth is preferably 15% or less, and more preferably 13% or less.

[0043] As described above, the diffusion equation (corresponding to equations (1) and (2)) used in the diffusion simulation and the calculation formula for the diffusion coefficient D in the steel material (corresponding to equations (3) and (4)) include the diffusion flow rate J of carbon, the diffusion coefficient D, Dγ, and the mobility m γ The diffusion simulation can be corrected by multiplying at least one of these parameters by a correction coefficient k. The correction coefficient k is a value greater than 0 and less than 1. It goes without saying that the correction coefficient k is set in the carbide precipitation region in the steel material formed by the carburizing treatment.

[0044] After arbitrarily setting the correction coefficient k, the carbon concentration is analyzed based on the diffusion simulation to determine the error Ec. If this error Ec is equal to or less than the threshold value Eth, the correction coefficient k used to obtain this error Ec is determined as the correction coefficient k for correcting the diffusion simulation. On the other hand, if the error Ec is greater than the threshold value Eth, the correction coefficient k is reset, and the carbon concentration is analyzed based on the diffusion simulation to determine the error Ec. Then, the correction coefficient k continues to be searched for until the error Ec becomes equal to or less than the threshold value Eth.

[0045] In step S104, the diffusion simulation is corrected using the correction coefficient k determined in the process of step S103. Specifically, the parameters incorporated in the diffusion simulation (diffusion flux J, diffusion coefficient D, D γ , and mobility m γ) by the determined correction coefficient k to correct the diffusion simulation, and the carbon concentration in the carburized steel is analyzed using this corrected diffusion simulation. Here, even if the carburizing conditions, the shape (location) of the steel, and the steel type change, the diffusion simulation can be corrected using the previously determined correction coefficient k. However, analysis accuracy is improved by performing the processes of steps S101 to S103 for each carburizing condition, shape (location) of the steel, and steel type, and determining a correction coefficient k corresponding to each.

[0046] According to this embodiment, by correcting the diffusion simulation, it becomes easier to ensure accuracy when analyzing the carbon concentration in a steel material after carburizing. In particular, as will be understood from the examples described later, it is possible to accurately analyze the distribution of carbon concentration not only in a single-phase region where carbides do not precipitate, but also in a multi-phase region containing austenite and carbides, for example.

[0047] (Correction of Diffusion Simulation) When correcting the diffusion simulation as described above, the diffusion flux J, diffusion coefficients D, D included in the diffusion equation and the calculation formula for the diffusion coefficient in the steel material are used. γ and mobility m γ At least one of the following equations (1a), (2a), (3a), and (4a) is multiplied by a correction coefficient k. The following equations (1a), (2a), (3a), and (4a) respectively represent the parameters (diffusion flux J, diffusion coefficient D, D) of the above equations (1), (2), (3), and (4). γ , and mobility m γ ) multiplied by a correction coefficient k.

[0048] When analyzing the carbon concentration distribution using equations (1), (2), and (3), at least one of the diffusion coefficients D, Dγ, and the diffusion flux J is multiplied by a correction coefficient k as shown in equations (1a), (2a), and (3a). Since "at least one" is used, the carbon concentration distribution may be analyzed using equations (1a), (2), and (3), or the carbon concentration distribution may be analyzed using equations (1), (2a), and (3), or the carbon concentration distribution may be analyzed using equations (1), (2a), and (3a), or the carbon concentration distribution may be analyzed using equations (1a), (2a), and (3a), or the carbon concentration distribution may be analyzed using equations (1a), (2a), and (3a), or the carbon concentration distribution may be analyzed using equations (1a), (2a), and (3a). The correction coefficient k included in each equation may have a different value.

[0049] When analyzing the distribution of carbon concentration using equations (1), (2), and (4), the diffusion coefficient D and the mobility m are used as shown in equations (1a), (2a), and (4a). γ At least one of the diffusion flux J and the diffusion flux J is multiplied by the correction coefficient k. Since "at least one" is used, the carbon concentration distribution may be analyzed using formulas (1a), (2), and (4), or the carbon concentration distribution may be analyzed using formulas (1), (2a), and (4), or the carbon concentration distribution may be analyzed using formulas (1), (2), and (4a), or the carbon concentration distribution may be analyzed using formulas (1a), (2a), and (4), or the carbon concentration distribution may be analyzed using formulas (1a), (2), and (4a), or the carbon concentration distribution may be analyzed using formulas (1a), (2), and (4a). The correction coefficient k included in each formula may be a different value. Needless to say, the description of the "labyrinthiness" in the first embodiment can be applied to the second embodiment.

[0050] Examples 1 to 4 and Comparative Examples 1 to 4 Examples 1 to 4 are examples corresponding to the first embodiment, and Comparative Examples 1 to 4 are comparative examples for Examples 1 to 4.

[0051] (Steel Samples and Carburizing Conditions) Steel samples having the compositions shown in the table below (the balance being Fe and unavoidable impurities) were prepared as steel samples to be subjected to reduced pressure carburizing treatment.

[0052] Figure 2 is a perspective view of a steel sample. As shown in Figure 2, the shape of the steel sample was a rectangular prism with a trapezoidal cross section (steel having an edge-shaped portion). The long side L, short side S, and height H of the trapezoidal cross section were 16 mm, 10 mm, and 10 mm, respectively. The rectangular prism length K was 50 mm. The trapezoidal cross section had two angles of 90° and one angle of α (60°).

[0053] The reduced-pressure carburizing treatment involved the following steps, carried out in this order: (i) a heating step in which a steel sample was heated to the carburizing temperature in a reduced-pressure heating furnace; (ii) a soaking step in which the steel sample was soaked at the carburizing temperature; (iii) a carburizing step in which carburizing gas (acetylene gas) was introduced into the furnace and the steel sample was carburized at the carburizing temperature; (iv) a diffusion step in which the absorbed carbon was diffused into the steel sample; (v) a temperature-reducing step in which the temperature of the steel sample was reduced to a holding temperature; and (vi) a holding step in which the temperature of the steel sample was held at the holding temperature. Note that no carburizing gas was introduced in any steps other than the carburizing step. The reduced-pressure carburizing treatment conditions A shown in the table below were set as the conditions for the reduced-pressure carburizing treatment.

[0054] (Measurement of Carbon Concentration) Steel samples were subjected to reduced pressure carburization under the above-mentioned carburization conditions, and the carbon concentration (actual measured value) of each of the steel samples after the treatment was measured.

[0055] Figure 3 is a cross-sectional view of a steel sample. As shown in Figure 3, the cross-section at the center of the rectangular column length of the steel sample (i.e., the center of length K in Figure 2) was used as the inspection position, and the carbon concentration was measured in the measurement direction MD indicated by the arrow. The measurement direction MD was in two directions. Specifically, the measurement direction MD was the line that bisects the right angle portion of the cross-section (hereinafter also referred to as the "90° corner"), and the measurement direction MD was the line that bisects the corner of the cross-section with angle α (hereinafter also referred to as the "60° corner"). Needless to say, both measurement directions MD pass through the edge vertex (the vertex of the trapezoidal cross-section).

[0056] (Diffusion Simulation) A modeling method in the diffusion simulation will be described below. In this example, analysis software (commercially available, manufactured by MSC Software, MSC.Marc) based on a known finite element method was used, which incorporates a program for calculating the carbon diffusion flux J using the diffusion equation of the above-mentioned formula (1) (a diffusion equation in which the gradient of the carbon concentration in the steel material is used as the driving force).

[0057] (I) Mesh Data Two-dimensional cross sections of the area including the area within a radius of 6 mm from the edge vertex of the 90° corner and the 60° corner were modeled, and mesh data was created so that the mesh on the steel surface would be the finest. Specifically, the mesh size was set within the range of 1 to 500 μm. The mesh type was a quadrilateral mesh.

[0058] (II) Boundary conditions During the carburizing period (carburizing process) of reduced pressure carburizing, the carbon concentration at equilibrium with graphite obtained by thermodynamic calculation was given at a position corresponding to the surface of the steel. Furthermore, after the diffusion period (diffusion process), it was assumed that no carbon penetrated from the surface of the steel, and the system was treated as a closed system.

[0059] (III) Set Temperature The temperature conditions for the reduced pressure carburizing treatment were set in the calculation model as shown in the above reduced pressure carburizing conditions. However, for the sake of simplicity of calculation, the temperature in the temperature-lowering step was fixed at 880°C and set in the analysis model.

[0060] (IV) Diffusion coefficient D in steel The diffusion coefficient D in steel was calculated based on formula (3) or (4) and used in the diffusion simulation. The carbon concentration C in austenite included in formula (3) or (4) γ , and the activity coefficient in austenite γ γ The diffusion coefficient D in austenite was calculated by thermodynamic calculation using the components at each position in the steel material and the set temperature described in (III) above. γ was calculated based on equation (6) and used in the diffusion simulation. The mobility of carbon in austenite, m γ The values ​​used were calculated by thermodynamic calculations using a thermodynamic database and a diffusion database.

[0061] (V) Labyrinth factor The labyrinth factor was calculated based on formula (5) and used in the diffusion simulation. That is, the labyrinth factor calculated based on formula (5) was multiplied by the right side of formula (3) or (4) to calculate the diffusion coefficient D in the steel material. The volume fraction V of austenite included in formula (5) γ The value calculated by thermodynamic calculation was used as . n was set to 2.

[0062] (VI) Thermodynamic Calculations Thermodynamic calculations were performed using commercially available thermodynamic calculation software Thermo-Calc (manufactured by Thermo-Calc Software Inc.) The calculations used commercially available thermodynamic databases TCFE10 and MOBFE1 (both manufactured by Thermo-Calc Software Inc.).

[0063] (VII) Correction Coefficient k In this example, in order to correspond to the first embodiment, analysis using multiplication by the correction coefficient k was not performed.

[0064] The conditions for Examples 1 to 4 and Comparative Examples 1 to 4 are summarized in Table 3 below. In Comparative Examples 1 and 3, the diffusion coefficient D in the steel material is calculated by the equation (6) for the diffusion coefficient in austenite (i.e., the diffusion coefficient D γ In Comparative Examples 2 and 4, the diffusion coefficient D in the steel material was set to a constant (43 μm ２ / s), and the simulation was performed. This constant is based on literature values. 4 and 5 show the simulation results. Note that the state of the surface layer under these carburizing conditions is such that the coarse carbides precipitated in the carburizing process are dissolved in the diffusion process, and no coarse carbides remain. As is clear from Examples 1 to 4, it was found that the accuracy of analyzing the carbon concentration distribution can be improved by calculating the diffusion coefficient in the steel using equation (3) or (4) and performing a diffusion simulation based on a diffusion equation using the carbon concentration gradient in the steel as the driving force.

[0065] Examples 5 to 8 Examples 5 to 8 correspond to the first and second embodiments.

[0066] (Steel Sample and Carburizing Conditions) The steel composition and sample shape used in the reduced-pressure carburizing treatment were the same as in Examples 1 to 4. The reduced-pressure carburizing treatment included the following steps, in this order: (i) a heating step in which the steel sample was heated to the carburizing temperature in a reduced-pressure heating furnace; (ii) a soaking step in which the steel sample was soaked at the carburizing temperature; (iii) a carburizing step in which carburizing gas (acetylene gas) was introduced into the furnace and the steel sample was carburized at the carburizing temperature; (iv) a diffusion step in which the absorbed carbon was diffused into the steel sample; (v) a temperature-reducing step in which the temperature of the steel sample was reduced to a holding temperature; and (vi) a holding step in which the temperature of the steel sample was held at the holding temperature. Note that no carburizing gas was introduced in any steps other than the carburizing step. The reduced-pressure carburizing treatment conditions B or C shown in the table below were set as the conditions for the reduced-pressure carburizing treatment. (Measurement of carbon concentration) The steel samples were subjected to reduced pressure carburization under the above-mentioned carburization conditions, and the carbon concentration (actual measured value) of each steel sample after the reduced pressure carburization treatment was measured. The carbon concentration measurement method was the same as in Examples 1 to 4.

[0067] (Diffusion Simulation) A modeling method in the diffusion simulation will be described below. In this example, analysis software (commercially available, manufactured by MSC Software, MSC.Marc) based on a known finite element method was used, which incorporates a program for calculating the diffusion flux of carbon using the diffusion equation of the above-mentioned formula (1) (a diffusion equation in which the gradient of the carbon concentration in the steel material is used as the driving force).

[0068] (I) Mesh data: The same as in Examples 1 to 4.

[0069] (II) Boundary conditions were the same as in Examples 1 to 4.

[0070] (III) Set temperature: The same as in Examples 1 to 4.

[0071] (IV) Diffusion coefficient D in steel The diffusion coefficient D in steel was calculated based on formula (3) and used in the diffusion simulation. The carbon concentration C in austenite included in formula (3) γThe diffusion coefficient D in austenite was calculated by thermodynamic calculation using the components at each position in the steel material and the set temperature described in (III) above. γ was calculated based on equation (6) and used in the diffusion simulation.

[0072] (V) Labyrinth factor The labyrinth factor was calculated based on equation (5) and used in the diffusion simulation. That is, the labyrinth factor calculated based on equation (5) was multiplied by the right side of equation (3). The volume fraction V of austenite included in equation (5) γ The value calculated by thermodynamic calculation was used for n. n was set to 2.

[0073] (VI) Thermodynamic Calculation The same as in Examples 1 to 4.

[0074] (VII) Calculation of Error The mean absolute error (MAE) between the analytical value and the measured value of the carbon concentration was calculated in the range of 0.25 mm from the edge apex of the 60° corner or 90° corner along the measurement direction MD.

[0075] (VIII) Setting of Correction Coefficient k In Example 5, the correction coefficient k was not set. That is, Example 5 corresponds to the first embodiment, and the carbon concentration distribution was analyzed using equations (1), (2), and (3). In Examples 6 to 8, the correction coefficient k was set in the region where carbide precipitated during the carburization period (carburization process) after the diffusion period (diffusion process) when carbide solid-solved. The correction coefficient was determined so as to improve the analytical accuracy under the conditions shown in Example 6. Specifically, the correction coefficient (0.77) was determined so that the mean absolute error (MAE) was 17% or less. Note that the correction coefficient k was multiplied only to the right-hand side of equation (3). That is, the carbon concentration distribution was analyzed using equations (1), (2), and (3a).

[0076] 6 to 9 show the simulation results. Note that under these carburizing conditions, the surface layer remains, although the coarse carbides precipitated during the carburizing process have partially dissolved after the diffusion process. The area with significantly high carbon concentration near the surface corresponds to the area where coarse carbides remain. As is clear from the analysis results of the carbon concentration distribution in Examples 5 to 8, the simulations in this example demonstrated excellent analytical accuracy for the carbon concentration distribution. Furthermore, as can be seen from the analysis results of Example 5, even without a correction factor, the carbon concentration distribution can be analyzed with high accuracy in the area where no coarse carbides remain (e.g., in Example 5, the area from approximately 0.3 mm from the edge apex to the interior of the steel). Furthermore, the simulations in Examples 6 to 8, corrected using the correction factor k, were able to analyze the carbon concentration distribution with high accuracy, even in the area with high carbon concentration near the surface, corresponding to the area where coarse carbides remain.

[0077] Examples 9 to 24 Examples 9 to 24 correspond to cases where the labyrinthiness λ in the first and second embodiments is set differently from Examples 1 to 4 and Examples 5 to 9.

[0078] (Steel Sample and Carburizing Conditions) The steel composition and sample shape subjected to reduced pressure carburizing treatment were the same as in Examples 1 and 2. The reduced pressure carburizing conditions were either reduced pressure carburizing condition A, reduced pressure carburizing condition B, or reduced pressure carburizing condition C described in Examples 1 to 4 and Examples 5 to 9.

[0079] (Measurement of Carbon Concentration) The same as in Examples 1 to 4 was carried out.

[0080] (Diffusion Simulation) A modeling method in the diffusion simulation will be described below. In this example, analysis software (commercially available, manufactured by MSC Software, MSC.Marc) based on a known finite element method was used, which incorporates a program for calculating the diffusion flux of carbon using the diffusion equation of the above-mentioned formula (1) (a diffusion equation in which the gradient of the carbon concentration in the steel material is used as the driving force).

[0081] (I) Mesh data: The same as in Examples 1 to 4.

[0082] (II) Boundary conditions were the same as in Examples 1 to 4.

[0083] (III) Set temperature: The same as in Examples 1 to 4.

[0084] (IV) Diffusion coefficient D in steel The diffusion coefficient D in steel was calculated based on formula (3) and used in the diffusion simulation. The carbon concentration C in austenite included in formula (3) γ The diffusion coefficient D in austenite was calculated by thermodynamic calculation using the components at each position in the steel material and the set temperature described in (III) above. γ was calculated based on equation (6) and used in the diffusion simulation.

[0085] (V) Labyrinth Factor In Examples 9 to 12, the labyrinth factor λ was not set. In Examples 13 to 16, the labyrinth factor was calculated based on Equation (5) and used in the diffusion simulation. That is, the labyrinth factor calculated based on Equation (5) was multiplied to the right side of Equation (3). A value calculated by thermodynamic calculation was used for the austenite volume fraction Vγ included in Equation (5). n was set to 1. In Examples 17 to 24, the labyrinth factor was calculated based on Equation (5) and used in the diffusion simulation. That is, the labyrinth factor calculated based on Equation (5) was multiplied to the right side of Equation (3). A value calculated by thermodynamic calculation was used for the austenite volume fraction Vγ included in Equation (5). n was set to 5 or 8.

[0086] (VI) Thermodynamic calculations were the same as in the first example.

[0087] (VII) Calculation of Errors Calculation was performed under the same conditions as in Examples 5 to 8 only under the conditions in which coarse carbides remained near the apexes of the edge portions of Examples 10 to 12, 14 to 16, 18 to 20, and 22 to 24.

[0088] (VIII) Setting of Correction Coefficient k In Examples 9, 10, 13, 14, 17, 18, 21, and 22, the correction coefficient k was not set. In other words, these examples correspond to the first embodiment, and the carbon concentration distribution was analyzed using equations (1), (2), and (3). In Examples 11, 12, 15, and 16, the correction coefficient k was set in the region where carbide precipitated during the carburization period (carburization process) after the diffusion period (diffusion process) in which carbide dissolves. The correction coefficient was determined so as to improve the analytical accuracy under the conditions shown in Example 11 or 15. Specifically, in Example 11, the correction coefficient (0.83) was determined so that the mean absolute error (MAE) would be 11% or less. In Example 15, the correction coefficient (0.83) was determined so that the mean absolute error (MAE) would be 18% or less. In Examples 19, 20, 23, and 24, the correction coefficient k was set in the region where carbide precipitated during the carburization period (carburization process) after the diffusion period (diffusion process) when carbide solidified. The correction coefficient was determined so as to improve the analytical accuracy under the conditions shown in Example 19 or 23. Specifically, in Example 19, the correction coefficient (0.50) was determined so that the mean absolute error (MAE) was 40% or less. In Example 23, the correction coefficient (0.37) was determined so that the mean absolute error (MAE) was 40% or less. The correction coefficient k was multiplied only by the right-hand side of Equation (3). In other words, the carbon concentration distribution was analyzed using Equations (1), (2), and (3a).

[0089] 10 to 25 show the simulation results. In Examples 10 to 12, 14 to 16, 18 to 20, and 22 to 24, the state near the corner apex is such that the coarse carbides precipitated in the carburizing process are partially dissolved after the diffusion process, but remain. In Examples 9, 13, 17, and 21, the coarse carbides precipitated in the carburizing process are dissolved in the diffusion process, and almost no coarse carbides remain. Therefore, the range where the carbon concentration is significantly high corresponds to the range where coarse carbides remain. As is clear from the analysis results of the carbon concentration distribution in Examples 9 to 24, the simulation of this example was found to have superior analytical accuracy of the carbon concentration distribution compared to the comparative example.

[0090] Furthermore, as can be seen from the analysis results of Examples 9, 10, 13, 14, 17, 18, 21, and 22, it was found that even when a correction coefficient was not set, the carbon concentration distribution could be analyzed with high accuracy in the range where no coarse carbides remained (for example, the range from about 0.3 mm from the edge apex to the interior of the steel material in Example 10). In addition, it was found that the simulations of Examples 11, 12, 15, 16, 19, 20, 23, and 24, which were corrected with the correction coefficient k, could analyze the carbon concentration distribution with even higher accuracy in the region where the carbon concentration near the surface was high, which corresponds to the range where coarse carbides remained, compared to when a correction coefficient was not set.

[0091] As can be seen from the analysis results of Examples 5 to 8 and Examples 9 to 24, it was found that analysis could be performed with higher accuracy than in the comparative examples, regardless of whether or not the labyrinth level was set and the value of the integer n in the labyrinth level. In other words, the integer n in the labyrinth level is not limited to the common values ​​of 1 or 2, and even if it is a larger integer, the required analysis accuracy (i.e., higher analysis accuracy than in the comparative examples) is guaranteed. The upper limit of the integer n cannot be uniquely defined because it differs depending on the degree of carbide formation, which varies depending on the steel composition. The integer n can be set appropriately so as to obtain the required analysis accuracy.

Claims

1. A method for analyzing carbon concentration distribution in a carburized steel material, comprising: analyzing the carbon concentration distribution in a steel material in which carbides are formed by carburization using a diffusion equation, the diffusion equation being expressed by the following formulas (I) and (II): (In formula (I), J is the diffusion flux of carbon, D is the diffusion coefficient of carbon in the steel, C is the carbon concentration in the steel, and grad is the gradient of the position. In formula (II), C is the carbon concentration in the steel, t is time, div is divergence, and J is the diffusion flux of carbon). D in formula (I) is represented by the following formula (III) or (IV): (In formula (III), D γ is the diffusion coefficient of carbon in austenite, C is the carbon concentration in the steel, and C γ is the carbon concentration in austenite, and in formula (IV), m γ is the mobility of carbon in austenite, R is the gas constant, T is the temperature, C is the carbon concentration in the steel, and C γ is the carbon concentration in austenite, and γ γ is the activity coefficient of carbon in austenite [-]).

2. In the carbide precipitation region in the steel material formed by the carburizing treatment, the diffusion coefficient D of formula (I), the diffusion flux J of formula (II), and the diffusion coefficient D of formula (III) are γ and the mobility m of formula (IV) γ a correction coefficient set to satisfy a condition that an error between the carbon concentration distribution analyzed from the diffusion equation and a carbon concentration distribution measured in advance is equal to or smaller than a threshold value.

3. When calculating D in formula (I) from formula (III), the following formula (V): λ = V γ ｎ ...(V) (in the formula, V γ is a volume fraction of austenite [-], and n is an integer.) and when D in formula (I) is calculated from formula (IV), multiplying the right side of any of formulas (I), (II), and (IV) by the labyrinthness λ represented by formula (V). The method for analyzing a carbon concentration distribution according to claim 1 or 2, comprising the steps of: multiplying the right side of formula (I), (II), and (IV) by the labyrinthness λ represented by formula (V)

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