Method for avoiding fatigue failure caused by oxide inclusion in steel

Abstract

The invention relates to the technical field of steel part fatigue fracture failure technology and metallurgy, in particular to a method for avoiding fatigue failure caused by oxide inclusion in steel. The method for avoiding the fatigue failure caused by the oxide inclusion in the steel comprises the following steps: S1, obtaining the radius row of the maximum oxide inclusion in the detected steel; s2, calculating the safety thickness rsulfide of a sulfide shell layer correspondingly attached to the maximum oxide inclusion radius row according to the maximum oxide inclusion radius row in the step S1; and S3, the corresponding cooling rate RC of the casting blank in the solidification process is calculated according to the rsulfide obtained in the S2. Fatigue failure caused by oxide inclusion in steel is avoided by regulating and controlling the cooling rate parameter of the continuous casting process, and corresponding experimental evidence and theoretical basis are provided for material selection, design and reasonable inclusion size control for prolonging the service life of engineering materials.

Description

Technical Field

[0001] The invention relates to the technical field of steel fatigue fracture failure technology and metallurgy, and in particular to a method for avoiding fatigue failure caused by oxide inclusions in steel. Background Art

[0002] With the rapid development of high-end equipment in my country, more stringent requirements have been put forward for steel performance, such as fatigue resistance and long life. However, during the smelting stage of steel, it is inevitable that large-sized oxide inclusions will still be produced, and the presence of these brittle oxide inclusions is an important cause of cracks in the service process of steel, and ultimately leads to premature fatigue failure. Therefore, in recent years, the control of large-sized oxides in steel has also received more and more attention.

[0003] The prior art has developed a variety of methods to reduce the damage of oxide inclusions to fatigue fracture of steel; although it can be achieved by reducing the size of oxide inclusions, reducing the probability of oxide inclusions, and converting oxide inclusions into other substances that will not cause fatigue fracture of steel. For example: Chinese patent CN118064679A discloses a method for rare earth modified MnS inclusions in 20CrMo gear steel. In the middle of RH vacuum treatment, lanthanum-iron alloy is added, which can make La element react with S element in steel to generate a series of rare earth sulfides, so that the number of long strip MnS inclusions is greatly reduced or even disappears, and is converted into fine round MnS inclusions attached to the rare earth sulfides, and more dispersed in the molten steel. The generated rare earth sulfides are closer to the properties of the steel matrix, reducing the stress caused by deformation during hot working, improving the directional problem of material mechanical properties, and also improving the fatigue performance of steel and extending the service life. However, the cost of rare earth elements is high, and the economic applicability is poor, which is not conducive to large-scale application in industrial large-scale production.

[0004] Chinese patent CN112485118A discloses a method for estimating the critical inclusion size of fatigue failure of high-strength steel. It provides the mechanism of fatigue fracture of high-strength steel induced by inclusions from the perspective of energy. High-strength steels with different strength and toughness matching are obtained by using different heat treatment systems for tensile, fracture toughness and elastic performance tests, and the critical inclusion size of fatigue failure of high-strength steel is estimated by energy criterion. However, this method cannot provide a way to control the inclusion size to be smaller than the critical size and it is difficult to predict whether the control mechanism is suitable for industrial production.

[0005] Although Chinese patent CN118127402A discloses a method for controlling inclusions in martensitic stainless steel with high fatigue performance and corrosion resistance, this method controls the main inclusions Al2O3, CaO-Al2O3, CaS, and TiN in the stainless steel to dispersed Mg-Al-RE-O type inclusions. The control method is very complicated, the operation is difficult, and a large amount of high-cost alloy elements are added.

[0006] Obviously, the above methods are not very practical for reducing the damage of oxide inclusions to fatigue fracture of steel. The process is complicated, the cost is high and the efficiency is low.

[0007] Another effective method is to allow sulfides to precipitate and grow with oxides as the core during the cooling and solidification of the continuous casting billet, and finally obtain composite inclusions in the form of oxides + sulfides. This method reduces the stress concentration between the composite inclusions and the matrix by fully wrapping a layer of soft sulfides around the brittle oxides, effectively preventing the initiation and expansion of cracks on the oxides, thereby improving the fatigue performance of the steel.

[0008] However, there has been no quantitative calculation and effective guidance for the safe thickness range and control strategy of sulfide shell precipitation, so it is difficult to completely overcome the technical difficulties of fatigue failure caused by oxides in production. Considering that steel billets usually need to undergo a quenching heat treatment process before they are finally processed into parts to optimize their mechanical properties. However, due to the different thermal expansion coefficients of composite inclusions and the steel matrix, after quenching heat treatment, certain residual stresses will be generated at the interface between the composite inclusions and the matrix, and concentrated at the interface, which is an important cause of premature fatigue failure of steel parts.

[0009] Therefore, it is necessary to accurately calculate the safe thickness of the outer sulfide shell so that the oxide can be fully wrapped by the sulfide to reduce the risk of fatigue fracture of steel parts caused by the oxide during service. At the same time, the cooling rate of the continuous casting billet during the solidification process should be reasonably controlled to ensure that a sufficiently thick sulfide shell is precipitated on the oxide inclusions, thereby greatly reducing the stress concentration at the interface between the inclusions and the matrix, avoiding the initiation of fatigue cracks, and extending the service life.

[0010] However, there is currently no simple and effective method to calculate the safe thickness of the sulfide shell and its corresponding control strategy to know the production process design to avoid fatigue failure caused by oxide inclusions in steel.

[0011] Therefore, the present invention proposes an accurate calculation method for the safe thickness of the sulfide shell and the cooling rate of the corresponding solidification process, thereby completely avoiding fatigue failure caused by oxide inclusions in steel. Summary of the invention

[0012] In order to solve many technical problems in the prior art, such as premature fatigue failure of steel caused by hard oxide inclusions in service, modification of oxide inclusions and control of safe thickness of sulfide shell precipitation, and regulation of cooling rate during the solidification stage of continuous casting of ingots, the embodiment of the present invention provides a method for avoiding fatigue failure caused by oxide inclusions in steel. The technical solution is as follows:

[0013] A method for preventing fatigue failure caused by oxide inclusions in steel, the method for preventing fatigue failure caused by oxide inclusions in steel comprising the following steps:

[0014] S1. Get the radius r of the largest oxide inclusion in the detected steel oxide ;

[0015] S2, the radius r of the largest oxide inclusion passing through S1 oxide Calculate the safe thickness r of the sulfide shell formed on it sulfide ;

[0016] S3, r obtained according to S2 sulfide Calculate the cooling rate R corresponding to the solidification process of the casting C .

[0017] Optionally, the radius r of the largest oxide inclusion in the detected steel is obtained in S1 oxide The method is to calculate the radius r of the largest oxide inclusion in the tested steel according to the national standard rating results. oxide , or directly obtain the radius r of the largest oxide inclusion from the inclusion data obtained by various microscopic observation and analysis methods oxide , or calculate the radius r of the largest oxide inclusion in steel by the national standard extreme value analysis method oxide .

[0018] Optionally, in S1, the radius r of the largest oxide inclusion in the tested steel is calculated according to the national standard rating result. oxide The national standard GB / T 10561-2023 evaluates the grade of Ds-type large-size oxide inclusions from 0.5 to 5, with a total of 10 grades at intervals of 0.5. The grade N is related to the oxide radius r oxide The relationship is shown in the following formula:

[0019] r oxide =0.5[10 0.302N+0.972 ] (1)

[0020] Among them, the calculation result in the square brackets in the formula is rounded off to get the maximum oxide inclusion radius in the tested steel;

[0021] The radius r of the largest oxide inclusion is directly obtained from the inclusion data obtained by various microscopic observation and analysis methods. oxide To directly obtain the maximum oxide inclusion size r using the ASPEX inclusion automatic scanner oxide ;

[0022] Calculate the radius r of the largest oxide inclusion in steel by using the national standard extreme value analysis method oxide The maximum size of oxide inclusions in steel is calculated by the extreme value analysis method of GB / T40281-2021. oxide .

[0023] Optionally, in S2, when the outer layer of oxide envelops the sulfide to obtain a composite inclusion, the radius r of the composite inclusion is complex Satisfy the following formula:

[0024] r complex =r oxide +r sulfide (2).

[0025] Optionally, in S2, when the thermal expansion coefficient of the composite inclusion is greater than that of the steel matrix, residual stress will be generated at the interface between the composite inclusion and the matrix during the quenching heat treatment process, which is calculated as follows:

[0026] (3)

[0027] Among them, σ R It is the residual stress generated at the interface between the composite inclusion and the steel matrix after quenching; C is related to the Young's modulus and Poisson's ratio of the composite inclusion and the matrix, and is always greater than 0; α M is the thermal expansion coefficient of the substrate; △T is the temperature difference of quenching heat treatment; α complex is the thermal expansion coefficient of the composite inclusion.

[0028] Optionally, the thermal expansion coefficient α of the composite inclusion in S2 complex It can be calculated by the following formula:

[0029] (4)

[0030] Among them, α oxide is the thermal expansion coefficient of the oxide, α sulfide is the thermal expansion coefficient of the sulfide.

[0031] Optionally, if the thermal expansion coefficient α of the composite inclusion in S2 is complex If the stress is not greater than that of the matrix, no residual stress will be generated at the interface between the matrix and the steel matrix, that is, α M ≤α complex When R =0, so the safe thickness of the sulfide shell should satisfy the following formula:

[0032] (5).

[0033] Optionally, by combining equation (5) with equation (1), we can obtain:

[0034] (6).

[0035] Optionally, the thickness of the sulfide shell in S3 is determined by the cooling rate during the solidification process of the ingot, and the relationship between the thickness of the sulfide precipitation shell and the cooling rate is calculated using the Scheil solidification model;

[0036] The diffusion of element i in the steel liquid is the limiting link for the formation of sulfides. Therefore, the diffusion of element i controls the growth rate of sulfides. It can be obtained that:

[0037] (7)

[0038] Among them, J i is the diffusion flux of limiting element i; D i is the diffusion coefficient of limiting element i in the liquid phase, ρ Fe is the density of Fe liquid; M i is the molar mass of limiting element i; w[i] L is the mass fraction of limiting element i at the solidification front of molten steel; w[i] eq is the mass fraction of limiting element i at equilibrium; r sulfide is the radius of the sulfide;

[0039] In the model, the inclusions are assumed to be spherical after precipitation. According to the mass balance, the relationship between the diffusion flux of element i and the shell thickness of the sulfide is:

[0040] (8)

[0041] Combining formula (1), we can get:

[0042] (9)

[0043] By integrating equation (9), we can obtain that the precipitation growth of sulfide is as follows: sulfide expression: (10)

[0044] In formula (10), M sulfide is the molar mass of sulfide; ρ Fe is the density of Fe liquid, 7070kg / m 3 ;M i is the molar mass of limiting element i; ρ sulfideis the density of sulfide; D i is the diffusion coefficient of limiting element i;

[0045] In formula (10), w[i] L is the mass fraction of limiting element i at the solidification front of molten steel, which can be calculated using the Scheil microsegregation model as follows:

[0046] (11)

[0047] Where w[i]0 is the content of solute i in the molten steel before solidification begins; k0 is the equilibrium distribution coefficient of element i; f s is the solid phase ratio of molten steel, which is calculated as follows:

[0048] (12)

[0049] Among them, T0 is the melting point of pure iron, which is 1809K; T start is the temperature at which sulfide begins to precipitate; T S is the solidus temperature in steel;

[0050] In formula (10), w[i] eq It can be calculated using the following formulas (13) to (16):

[0051] [M] + [S] = MS, lgK=A / TB (13)

[0052] (14)

[0053] (15)

[0054] (16)

[0055] Wherein, formula (13) is the reaction equilibrium equation for the reaction of elements M and S to form sulfide MS, and in formula (14) α M , α S , α MS are the activities of [M], [S] and MS respectively; f in equations (15) and (16) M and f S are the activity coefficients of [M] and [S] respectively; and are the interaction coefficients of other elements [j] in the molten steel on [M] and [S] respectively;

[0056] In formula (10), is the solidification time, calculated according to the following formula:

[0057] (17)

[0058] Among them, T start is the temperature at which sulfide begins to precipitate; T S is the solidus temperature in steel; R C is the cooling rate during solidification;

[0059] To achieve a safe thickness r sulfide , Substituting equation (17) into equation (10), the cooling rate R C The following formula should be satisfied:

[0060] (18)

[0061] Among them, M sulfide and M i are the molar masses of sulfide and limiting element i, respectively; ρ Fe and ρ sulfide are the densities of Fe liquid and sulfide respectively; D i is the diffusion coefficient of limiting element i in the liquid phase; w[i] L is the mass fraction of limiting element i at the solidification front of molten steel; w[i] eq is the mass fraction of limiting element i at equilibrium; T start is the temperature at which sulfide begins to precipitate; T S is the solidus temperature in steel.

[0062] Optionally, in order to obtain the safe thickness of the sulfide shell in S3, the safe thickness r sulfide , Substituting equation (6) into equation (18), the cooling rate R C The following formula should be satisfied:

[0063] (19)

[0064] Among them, M sulfide and M i are the molar masses of sulfide and limiting element i, respectively; ρ Fe and ρ sulfide are the densities of Fe liquid and sulfide respectively; D i is the diffusion coefficient of limiting element i in the liquid phase; w[i] L is the mass fraction of limiting element i at the solidification front of molten steel; w[i] eq is the mass fraction of limiting element i at equilibrium; T start is the temperature at which sulfide begins to precipitate; T S is the solidus temperature in steel.

[0065] Compared with the prior art, the above technical solution has at least the following beneficial effects:

[0066] The above scheme, the present invention proposes a method for avoiding fatigue failure caused by oxide inclusions in steel, which can solve the problem that there has been no quantitative calculation and effective guidance on the safe thickness range and control strategy for the precipitation of sulfide shells in steel inclusions in the prior art, resulting in many technical problems such as oxide inclusions in steel easily causing fatigue failure during service.

[0067] The present invention first obtains the radius r of the largest oxide inclusion in the steel oxide Then, based on the principle of residual stress generation during quenching, the safe thickness of the sulfide shell wrapped around its outer layer is obtained. Finally, the corresponding control strategy is given in combination with the relationship between the precipitation and growth of sulfides and the cooling rate of the ingot solidification process. This solves the problem that the safe thickness of sulfides is difficult to accurately calculate and control during the production process. Subsequently, automatic inclusion scanning and rotary bending fatigue tests are carried out to verify the effectiveness of the present invention.

[0068] The invention obtains the radius of the largest oxide inclusion in the detected steel by calculating the rating result according to the national standard or directly obtains the radius r of the largest oxide inclusion in the steel by statistically analyzing the inclusion data obtained by various microscopic observation and analysis methods. oxide The obtained values ​​are accurate and reliable, which is further proved after actual technical verification.

[0069] The solidification cooling rate R of the present invention C Safe thickness of sulfide shell r sulfide There is a relationship defined by the following formula:

[0070] , which has also been verified by actual technology to be consistent with the actual situation.

[0071] In the present invention, if the thermal expansion coefficient α of the composite inclusion is complex If the stress is not greater than that of the matrix, no residual stress will be generated at the interface between the matrix and the steel matrix, that is, α M ≤α complex When R =0, so the safe thickness of the sulfide shell should satisfy the following formula:

[0072] , which has also been verified by actual technology to be consistent with the actual situation.

[0073] In summary, compared with other traditional methods, the method of the present invention avoids fatigue failure caused by oxide inclusions in steel by adjusting the cooling rate parameters of the continuous casting process, providing corresponding experimental evidence and theoretical basis for material selection, design and reasonable control of inclusion size to improve the service life of engineering materials; and the method can effectively improve the production efficiency and product quality of steel preparation, reduce the unqualified rate, is suitable for production line regulation of mass production of steel, and is conducive to large-scale industrial production and promotion. BRIEF DESCRIPTION OF THE DRAWINGS

[0074] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the following briefly introduces the drawings required for use in the description of the embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.

[0075] Figure 1 This is a typical composite inclusion morphology of oxide + sulfide shell in a GCr15 bearing steel metallographic sample in a method for avoiding fatigue failure caused by oxide inclusions in steel in Example 1 of the present invention;

[0076] Figure 2 It is a large-sized composite inclusion and energy spectrum result that causes failure fracture in the fatigue fracture source area of ​​GCr15 bearing steel in a method for avoiding fatigue failure caused by oxide inclusions in steel in Example 1 of the present invention;

[0077] Figure 3 This is a statistical result diagram of the thickness of all sulfide shells of GCr15 bearing steel in a method for avoiding fatigue failure caused by oxide inclusions in steel in Example 1 of the present invention;

[0078] Figure 4 is a curve showing the relationship between the solidification cooling rate and the thickness of the sulfide shell of GCr15 bearing steel in a method for avoiding fatigue failure caused by oxide inclusions in steel in Example 1 of the present invention;

[0079] Figure 5 It is a stress intensity factor-fatigue life (K-N) diagram of fatigue failure caused by composite inclusions in GCr15 bearing steel in a method for avoiding fatigue failure caused by oxide inclusions in steel in Example 1 of the present invention;

[0080] Figure 6 This is a relationship diagram between the oxide radius of GCr15 bearing steel and the thickness of the sulfide containment layer in a method for avoiding fatigue failure caused by oxide inclusions in steel in Example 1 of the present invention. DETAILED DESCRIPTION

[0081] The technical solution of the present invention is described below in conjunction with the accompanying drawings.

[0082] In the embodiments of the present invention, words such as "exemplarily" and "for example" are used to indicate examples, illustrations or explanations. Any embodiment or design described as "example" in the present invention should not be interpreted as being more preferred or more advantageous than other embodiments or designs. Specifically, the use of the word "example" is intended to present the concept in a specific way. In addition, in the embodiments of the present invention, the meaning expressed by "and / or" can be both, or it can be either of the two.

[0083] In the embodiments of the present invention, "image" and "picture" can sometimes be used interchangeably. It should be noted that when the difference between them is not emphasized, the meanings they want to express are the same. "of", "corresponding" and "corresponding" can sometimes be used interchangeably. It should be noted that when the difference between them is not emphasized, the meanings they want to express are the same.

[0084] In the embodiments of the present invention, sometimes a subscript such as W1 may be written as a non-subscript such as W1. When the difference is not emphasized, the meanings to be expressed are consistent.

[0085] In order to make the technical problems, technical solutions and advantages to be solved by the present invention more clear, a detailed description will be given below with reference to the accompanying drawings and specific embodiments.

[0086] A method for preventing fatigue failure caused by oxide inclusions in steel, the method for preventing fatigue failure caused by oxide inclusions in steel comprising the following steps:

[0087] S1. Get the radius r of the largest oxide inclusion in the detected steel oxide ;

[0088] S2, the radius r of the largest oxide inclusion passing through S1 oxide Calculate the safe thickness r of the sulfide shell formed on it sulfide ;

[0089] S3, r obtained according to S2 sulfide Calculate the cooling rate R corresponding to the solidification process of the casting C .

[0090] In particular, the radius r of the largest oxide inclusion in the detected steel is obtained in S1 oxide The method is to calculate the radius r of the largest oxide inclusion in the tested steel according to the national standard rating results. oxide , or directly obtain the radius r of the largest oxide inclusion from the inclusion data obtained by various microscopic observation and analysis methods oxide , or calculate the radius r of the largest oxide inclusion in steel by the national standard extreme value analysis method oxide .

[0091] In particular, the radius r of the largest oxide inclusion in the tested steel is calculated in S1 according to the national standard rating results. oxide The national standard GB / T 10561-2023 evaluates the grade of Ds-type large-size oxide inclusions from 0.5 to 5, with a total of 10 grades at intervals of 0.5. The grade N is related to the oxide radius r oxide The relationship is shown in the following formula:

[0092] r oxide =0.5[10 0.302N+0.972 ] (1)

[0093] Among them, the calculation result in the square brackets in the formula is rounded off to get the maximum oxide inclusion radius in the tested steel;

[0094] The radius r of the largest oxide inclusion is directly obtained from the inclusion data obtained by various microscopic observation and analysis methods. oxide To directly obtain the maximum oxide inclusion size r using the ASPEX inclusion automatic scanner oxide ;

[0095] Calculate the radius r of the largest oxide inclusion in steel by using the national standard extreme value analysis method oxide The maximum size of oxide inclusions in steel is calculated by the extreme value analysis method of GB / T40281-2021. oxide .

[0096] In particular, when the outer layer of oxide in S2 encloses the sulfide to obtain a composite inclusion, the radius r of the composite inclusion is complex Satisfy the following formula:

[0097] r complex =r oxide +r sulfide (2).

[0098] In particular, when the thermal expansion coefficient of the composite inclusions in S2 is greater than that of the steel matrix, residual stress will be generated at the interface between the composite inclusions and the matrix during the quenching heat treatment process, which is calculated as follows:

[0099] (3)

[0100] Among them, σ R It is the residual stress generated at the interface between the composite inclusion and the steel matrix after quenching; C is related to the Young's modulus and Poisson's ratio of the composite inclusion and the matrix, and is always greater than 0; α M is the thermal expansion coefficient of the substrate; △T is the temperature difference of quenching heat treatment; α complex is the thermal expansion coefficient of the composite inclusion.

[0101] In particular, the thermal expansion coefficient α of the composite inclusions in S2 complex It can be calculated by the following formula:

[0102] (4)

[0103] Among them, α oxide is the thermal expansion coefficient of the oxide, α sulfide is the thermal expansion coefficient of the sulfide.

[0104] In particular, if the thermal expansion coefficient α of the composite inclusion in S2 complex If the stress is not greater than that of the matrix, no residual stress will be generated at the interface between the matrix and the steel matrix, that is, α M ≤α complex When R =0, so the safe thickness of the sulfide shell should satisfy the following formula:

[0105] (5).

[0106] In particular, combining equation (5) with equation (1) yields:

[0107] (6).

[0108] In particular, the thickness of the sulfide shell in S3 is determined by the cooling rate during the solidification process of the ingot, and the Scheil solidification model is used to calculate the relationship between the thickness of the sulfide precipitation shell and the cooling rate;

[0109] The diffusion of element i in the steel liquid is the limiting link for the formation of sulfides. Therefore, the diffusion of element i controls the growth rate of sulfides. It can be obtained that:

[0110] (7)

[0111] Among them, J i is the diffusion flux of limiting element i; D i is the diffusion coefficient of limiting element i in the liquid phase, ρ Fe is the density of Fe liquid; M i is the molar mass of limiting element i; w[i] L is the mass fraction of limiting element i at the solidification front of molten steel; w[i] eq is the mass fraction of limiting element i at equilibrium; r sulfide is the radius of the sulfide;

[0112] In the model, the inclusions are assumed to be spherical after precipitation. According to the mass balance, the relationship between the diffusion flux of element i and the shell thickness of the sulfide is:

[0113] (8)

[0114] Combining formula (1), we can get:

[0115] (9)

[0116] By integrating equation (9), we can obtain that the precipitation growth of sulfide is as follows: sulfide expression: (10)

[0117] In formula (10), M sulfide is the molar mass of sulfide; ρ Fe is the density of Fe liquid, 7070kg / m 3 ;M i is the molar mass of limiting element i; ρ sulfide is the density of sulfide; D i is the diffusion coefficient of limiting element i;

[0118] In formula (10), w[i] L is the mass fraction of limiting element i at the solidification front of molten steel, which can be calculated using the Scheil microsegregation model as follows:

[0119] (11)

[0120] Where w[i]0 is the content of solute i in the molten steel before solidification begins; k0 is the equilibrium distribution coefficient of element i; f s is the solid phase ratio of molten steel, which is calculated as follows:

[0121] (12)

[0122] Among them, T0 is the melting point of pure iron, which is 1809K; T start is the temperature at which sulfide begins to precipitate; T S is the solidus temperature in steel;

[0123] In formula (10), w[i] eq It can be calculated using the following formulas (13) to (16):

[0124] [M] + [S] = MS, lgK=A / TB (13)

[0125] (14)

[0126] (15)

[0127] (16)

[0128] Wherein, formula (13) is the reaction equilibrium equation for the reaction of elements M and S to form sulfide MS, and in formula (14) α M , α S , α MS are the activities of [M], [S] and MS respectively; f in equations (15) and (16) M and f S are the activity coefficients of [M] and [S] respectively; and are the interaction coefficients of other elements [j] in the molten steel on [M] and [S] respectively;

[0129] In formula (10), is the solidification time, calculated according to the following formula:

[0130] (17)

[0131] Among them, T start is the temperature at which sulfide begins to precipitate; T S is the solidus temperature in steel; R C is the cooling rate during solidification;

[0132] To achieve a safe thickness r sulfide , Substituting equation (17) into equation (10), the cooling rate R C The following formula should be satisfied:

[0133] (18)

[0134] Among them, M sulfide and M i are the molar masses of sulfide and limiting element i, respectively; ρ Fe and ρ sulfide are the densities of Fe liquid and sulfide respectively; D i is the diffusion coefficient of limiting element i in the liquid phase; w[i] L is the mass fraction of limiting element i at the solidification front of molten steel; w[i] eq is the mass fraction of limiting element i at equilibrium; T start is the temperature at which sulfide begins to precipitate; T S is the solidus temperature in steel.

[0135] In particular, in order to obtain the safe thickness of the sulfide shell in S3, the safe thickness r sulfide , Substituting equation (6) into equation (18), the cooling rate R C The following formula should be satisfied:

[0136] (19)

[0137] Among them, M sulfide and M i are the molar masses of sulfide and limiting element i, respectively; ρ Fe and ρ sulfide are the densities of Fe liquid and sulfide respectively; D i is the diffusion coefficient of limiting element i in the liquid phase; w[i] L is the mass fraction of limiting element i at the solidification front of molten steel; w[i] eq is the mass fraction of limiting element i at equilibrium; T start is the temperature at which sulfide begins to precipitate; T S is the solidus temperature in steel.

[0138] In order to verify a method for avoiding fatigue failure caused by oxide inclusions in steel, the solidification cooling rate R C Safe thickness of sulfide shell r sulfide There is a relationship defined by the following formula (1):

[0139] (1)

[0140] This verification scheme takes GCr15 bearing steel as an example and specifically includes the following steps:

[0141] (1) Experimentally determine the types of core oxides and outer sulfides in steel, and calculate the thickness of the sulfide shell:

[0142] For the selected GCr15 bearing steel, the total preparation area is 810mm 2 The metallographic samples were subjected to automatic statistical analysis of inclusions under a scanning electron microscope. At the same time, the rotating bending fatigue specimens were prepared and fatigue tests were carried out to observe and count the composite inclusions in the fatigue source area. The geometry of the fatigue specimens and the test steps were processed and carried out in accordance with the national standard GB / T 4337-2015. The test was completed on a QBWP-6000J rotating bending fatigue testing machine with a stress ratio of R=-1.

[0143] Figure 1 The SEM images of metallographic inclusions were automatically analyzed. Combined with the results of energy spectrum, the core oxide components of the composite inclusions in the steel were mainly Al-Mg-O, and the outer sulfides were mainly MnS. Figure 2 This is a large-sized composite inclusion observed in the fatigue fracture source area that caused the failure fracture. Similarly, the oxide is Al-Mg-O and the outer sulfide is MnS. In addition, the shell thickness of all oxide outer sulfides is counted and plotted on Figure 3 It was found that the shell thickness of sulfides was about 2 μm.

[0144] (2) From the known solidification cooling rate R CCalculate the sulfide shell thickness r sulfide :

[0145] In this embodiment, the composition of the GCr15 bearing steel used is shown in Table 1. The size of the ingot is 300×340 mm, and it is divided into a chill layer, a columnar crystal zone, a mixed crystal zone, and a central equiaxed crystal zone from the surface to the core. The cooling rate is from fast to slow, and there is an obvious cooling rate gradient. Since the fatigue specimens are cut from the columnar crystal zone, the mixed crystal zone, and the central equiaxed zone of the ingot, it is necessary to focus on the precipitation radius of sulfides in these three regions, and the corresponding cooling rate can be calculated using the average secondary dendrite spacing of each region, as shown in Table 2.

[0146] Table 1 GCr15 steel composition (mass fraction)

[0147]

[0148] Table 2 Secondary dendrite spacing and calculated cooling rate of ingot

[0149]

[0150] Since the diffusion of element S in molten steel is the limiting link for the formation of MnS, the diffusion of element S controls the growth rate of sulfides, so it can be obtained:

[0151] (2)

[0152] Among them, J S is the diffusion flux of the limiting element S; D S is the diffusion coefficient of the limiting element S in the liquid phase, ρ Fe is the density of Fe liquid; M S is the molar mass of the limiting element S; w[S] L is the mass fraction of the limiting element S at the solidification front of the molten steel; w[S] eq is the mass fraction of the limiting element S at equilibrium; r MnS is the radius of the sulfide.

[0153] In the model, the inclusions are assumed to be spherical after precipitation. According to the mass balance, the relationship between the diffusion flux of element S and the shell thickness of MnS is:

[0154] (3)

[0155] Combining formula (1), we can get:

[0156] (4)

[0157] By integrating equation (3), we can obtain the shell thickness r of MnS: MnSexpression:

[0158] (5)

[0159] In formula (4), M MnS is the molar mass of MnS, 0.087 kg / mol; ρ Fe is the density of Fe liquid, 7070kg / m 3 ;M S is the molar mass of S, 0.032 kg / mol; ρ MnS is the density of MnS, 3990 kg / m 3 ;D S is the diffusion coefficient of S, ;

[0160] In formula (4), w[S] L is the mass fraction of the limiting element S at the solidification front of the molten steel, which can be calculated using the Scheil microsegregation model as follows:

[0161] (6)

[0162] Wherein, w[S]0 is the content of solute S in the molten steel before solidification begins; k0 is the equilibrium distribution coefficient of element S, 0.035; f s is the solid phase ratio of the molten steel, which is calculated as follows:

[0163] (7)

[0164] Among them, T0 is the melting point of pure iron, which is 1809K; T start is the temperature at which sulfide begins to precipitate; T S is the solidus temperature in steel. Calculated by the thermodynamic software Thermal-Calc, in this case T start and T S They are 1625K and 1605K respectively.

[0165] In formula (4), w[S] eq It can be calculated using the following formulas (7) to (10):

[0166] [Mn]+[S]=MnS,lgK=8627 / T-4.745 (8)

[0167] (9)

[0168] (10)

[0169] (11)

[0170] Wherein, equation (7) is the chemical reaction equilibrium equation of the formation of MnS from element Mn and S and the corresponding standard Gibbs free energy change, α Mn , α S and α MnS are the activities of [Mn], [S] and MnS respectively; f Mn and f s are the activity coefficients of [Mn] and [S], respectively; and are the interaction coefficients of other elements [j] in the molten steel to [Mn] and [S], respectively, and their values ​​are shown in Table 3 below:

[0171] Table 3 Interaction coefficients of elements in molten steel

[0172]

[0173] In formula (4), is the local coagulation time, calculated according to the following formula (11):

[0174] (12)

[0175] Among them, T start is the temperature at which MnS begins to precipitate, T S is the solidus temperature in steel, which are 1625K and 1605K respectively, R C is the cooling rate during solidification.

[0176] A point was selected every 5K from the range of 50-120K / min, and the MnS shell thickness corresponding to 15 points was calculated. The calculation results are as follows: Figure 4 As shown, the shell thickness of MnS is in the range of 1.5-2.3 μm.

[0177] (3) Comparison of experimental observations and calculation results of sulfide shell thickness:

[0178] Figure 3 The experimentally calculated MnS shell thickness is about 2 μm. The calculated MnS shell thickness is as follows: Figure 4 The solidification cooling rate R of S3 in the method for avoiding fatigue failure caused by oxide inclusions in steel of the present invention is verified. C and the sulfide shell thickness r sulfide satisfies the derived relationship.

[0179] In order to verify the safe thickness of the sulfide shell in S2 calculated according to the following formula, when it is attached to the outer layer of the oxide, no inclusion-induced fatigue failure will occur.

[0180] (13)

[0181] This example takes GCr15 bearing steel as an example to verify the solidification cooling rate R of S3. C and the sulfide shell thickness r sulfide The materials used to satisfy the derived relationship are the same material. The validity of the S2 relationship (13) of the present invention is verified by a high-cycle rotation bending fatigue test, which specifically includes the following steps:

[0182] (1) Extrapolating the fatigue test results, the critical size of fatigue failure caused by composite inclusions is obtained:

[0183] If the fatigue specimen does not break after more than 10^7 cycles under the applied stress, it is considered that the specimen will no longer break and the test is terminated. This also shows that the size of all composite inclusions in the specimen is smaller than the critical size, and the composite inclusions are not sufficient to cause fatigue fracture. The geometry of the fatigue specimen and the test steps are processed and carried out in accordance with the national standard GB / T 4337-2015. The test is completed on a QBWP-6000J rotary bending fatigue testing machine with a stress ratio of R=-1. After the fatigue test, the fatigue failure data points caused by the composite inclusions are plotted as Figure 5 SN curve, the horizontal axis is lgN f , represents the number of cycles; the vertical axis is the stress intensity factor K, which can be calculated as follows:

[0184] (14)

[0185] Where σ0 is the value of the applied cyclic stress, Represents the diameter of the composite inclusion observed in the source area of ​​the fatigue fracture.

[0186] After fitting the fatigue failure data points caused by composite inclusions, the linear relationship is satisfied: K=-0.76·lgN f +9.7. When lgN f ≥7, that is, the number of cycles is greater than 10 7 , it is believed that the sample will no longer fatigue fracture, so the stress intensity factor needs to meet K≤4.38. Since fatigue tests have found that when the external cyclic load is greater than 1197MPa, fatigue failure is caused by surface cracking. Therefore, fatigue failure caused by composite inclusions only occurs when the external cyclic load is less than 1197MPa. Therefore, the maximum value of the external cyclic load σ0 is taken as 1197MPa. And because K≤4.38, the extrapolation method is calculated to obtain ≤17.0μm, that is, in this example, when the radius of the composite inclusion is less than 8.5μm, fatigue failure will not occur.

[0187] (2) Calculation of critical size of composite inclusions:

[0188] If αM ≥α complex After quenching, the interface between the composite inclusions and the steel matrix will produce a certain amount of residual stress. The subsequent loading of cyclic loads will cause stress concentration at the interface, which will increase the possibility of failure of steel parts in the early stage of service. Therefore, if α can be satisfied M ≤α complex , then no residual stress is generated at the interface between the composite inclusions and the steel matrix after quenching heat treatment, which makes it difficult for cracks to initiate around the composite inclusions, reducing the probability of them being the source of fatigue cracks. Therefore, it can be deduced that the thickness of the containment layer of sulfide should satisfy the following formula:

[0189] (15)

[0190] In this example, since the sample comes from Example 1, and it has been verified that the shell thickness of MnS is about 2 μm, it can be obtained that the radius of the oxide should satisfy:

[0191]

[0192] where α oxide and α MnS The thermal expansion coefficients of oxide and MnS are 6×10 -6 ℃ -1 and 1.61×10 -5 ℃ -1 ; α M is the thermal expansion coefficient of the substrate, 1.25×10 -5 ℃ -1 ;

[0193] The radius of the oxide should be less than 6.7 μm. Since the shell thickness of MnS is about 2 μm, the radius of the composite inclusion should be less than 8.7 μm. At this time, it is believed that the hard Al-Mg-O oxide wrapped by MnS will not produce residual stress at the interface with the matrix after quenching, that is, it will not cause the initiation of fatigue cracks and the fracture of the sample. The relationship between the oxide radius and the thickness of the sulfide containment layer is as follows: Figure 6 shown.

[0194] (3) Comparison of experimental and calculated results of critical size of composite inclusions:

[0195] The extrapolation of fatigue test shows that the critical radius of composite inclusions that will not cause fatigue fracture is 8.5μm, and based on the principle of no residual stress, the critical size of composite inclusions calculated is 8.7μm. The results are very consistent, thus proving the effectiveness of this method.

[0196] Example 1

[0197] A method for avoiding fatigue failure caused by oxide inclusions in the GCr15 steel composition in Table 1, characterized in that the method for avoiding fatigue failure caused by oxide inclusions in the steel comprises the following steps:

[0198] S1. According to the rating results of the national standard GB / T 10561-2023, the radius r of the largest oxide inclusion in the GCr15 steel tested in Table 1 is calculated. oxide ;

[0199] S2, the radius r of the largest oxide inclusion passing through S1 oxide Calculate the safe thickness r of the sulfide shell formed on it sulfide ;

[0200] S3, r obtained according to S2 sulfide Calculate the cooling rate R corresponding to the solidification process of the casting C The calculated range should be met. The final results are shown in Table 4 below.

[0201] Table 4

[0202]

[0203] According to the rating results of the national standard GB / T 10561-2023, since the oxygen content in the steel is only 4.4ppm, it is a high-cleanliness bearing steel with the highest rating of N=1.0, so the solidification cooling rate of the continuous casting billet should be controlled to be less than 17K / min.

[0204] Example 2

[0205] A method for avoiding fatigue failure caused by oxide inclusions in Cr-Mo gear steel, characterized in that the method for avoiding fatigue failure caused by oxide inclusions in steel comprises the following steps:

[0206] S1. The radius r of the largest oxide inclusion in the Cr-Mo gear steel tested is obtained according to the ASPEX inclusion automatic scanner as shown in Table 5 below: oxide ;

[0207] Table 5 CrMo gear steel composition (mass fraction mass%)

[0208]

[0209] S2, the radius r of the largest oxide inclusion passing through S1 oxide Calculate the safe thickness r of the sulfide shell formed on it sulfide ;

[0210] S3, r obtained according to S2 sulfideCalculate the cooling rate R corresponding to the solidification process of the casting C It should not be greater than the calculated critical value. The final results are shown in Table 6 below.

[0211] Table 6

[0212]

[0213] Example 3

[0214] A method for avoiding fatigue failure caused by oxide inclusions in 65Mn spring steel, characterized in that the method for avoiding fatigue failure caused by oxide inclusions in steel comprises the following steps:

[0215] S1. According to the extreme value analysis method of GB / T 40281-2021, the radius r of the maximum oxide inclusion in the 65Mn spring steel with five different oxygen contents in Table 7 was calculated respectively. oxide ;

[0216] Table 7 65Mn spring steel composition (mass percentage)

[0217]

[0218] S2, the radius r of the largest oxide inclusion passing through S1 oxide Calculate the safe thickness r of the sulfide shell formed on it sulfide ;

[0219] S3, r obtained according to S2 sulfide Calculate the cooling rate R corresponding to the solidification process of the casting C It should not be greater than the calculated critical value. The final results are shown in Table 8 below.

[0220] Table 8

[0221]

[0222] The above scheme, the present invention proposes a method for avoiding fatigue failure caused by oxide inclusions in steel, which can solve the problem that there has been no quantitative calculation and effective guidance on the safe thickness range and control strategy for the precipitation of sulfide shells in steel inclusions in the prior art, resulting in many technical problems such as oxide inclusions in steel easily causing fatigue failure during service.

[0223] The present invention first obtains the radius r of the largest oxide inclusion in the steel oxideThen, based on the principle of residual stress generation during quenching, the safe thickness of the sulfide shell wrapped around its outer layer is obtained. Finally, the corresponding control strategy is given in combination with the relationship between the precipitation and growth of sulfides and the cooling rate of the ingot solidification process. This solves the problem that the safe thickness of sulfides is difficult to accurately calculate and control during the production process. Subsequently, automatic inclusion scanning and rotary bending fatigue tests are carried out to verify the effectiveness of the present invention.

[0224] The invention obtains the radius of the largest oxide inclusion in the detected steel by calculating the rating result according to the national standard or directly obtains the radius r of the largest oxide inclusion in the steel by statistically analyzing the inclusion data obtained by various microscopic observation and analysis methods. oxide The obtained values ​​are accurate and reliable, which is further proved after actual technical verification.

[0225] The solidification cooling rate R of the present invention C Safe thickness of sulfide shell r sulfide There is a relationship defined by the following formula:

[0226] , which has also been verified by actual technology to be consistent with the actual situation.

[0227] In the present invention, if the thermal expansion coefficient α of the composite inclusion is complex If the stress is not greater than that of the matrix, no residual stress will be generated at the interface between the matrix and the steel matrix, that is, α M ≤α complex When R =0, so the safe thickness of the sulfide shell should satisfy the following formula:

[0228] , which has also been verified by actual technology to be consistent with the actual situation.

[0229] In summary, compared with other traditional methods, the method of the present invention avoids fatigue failure caused by oxide inclusions in steel by adjusting the cooling rate parameters of the continuous casting process, providing corresponding experimental evidence and theoretical basis for material selection, design and reasonable control of inclusion size to improve the service life of engineering materials; and the method can effectively improve the production efficiency and product quality of steel preparation, reduce the unqualified rate, is suitable for production line regulation of mass production of steel, and is conducive to large-scale industrial production and promotion.

[0230] It should be understood that the term "and / or" in this article is only a description of the association relationship of associated objects, indicating that there can be three relationships. For example, A and / or B can represent: A exists alone, A and B exist at the same time, and B exists alone. A and B can be singular or plural. In addition, the character " / " in this article generally indicates that the associated objects before and after are in an "or" relationship, but it may also indicate an "and / or" relationship. Please refer to the context for specific understanding.

[0231] In the present invention, "at least one" means one or more, and "plurality" means two or more. "At least one of the following" or similar expressions refers to any combination of these items, including any combination of single items or plural items. For example, at least one of a, b, or c can be represented by: a, b, c, ab, ac, bc, or abc, where a, b, and c can be single or multiple.

[0232] It should be understood that in various embodiments of the present invention, the size of the serial numbers of the above-mentioned processes does not mean the order of execution. The execution order of each process should be determined by its function and internal logic, and should not constitute any limitation on the implementation process of the embodiments of the present invention.

[0233] The above is only a specific embodiment of the present invention, but the protection scope of the present invention is not limited thereto. Any person skilled in the art can easily think of changes or substitutions within the technical scope disclosed by the present invention, which should be included in the protection scope of the present invention. Therefore, the protection scope of the present invention should be based on the protection scope of the claims.

Claims

1. A method for avoiding fatigue failure caused by oxide inclusions in steel, characterized in that: The method for avoiding fatigue failure caused by oxide inclusions in steel is as follows: S1. Get the radius r of the largest oxide inclusion in the detected steel oxide ; S2, the radius r of the largest oxide inclusion passing through S1 oxide Calculate the safe thickness r of the sulfide shell formed on it sulfide ; S3, r obtained according to S2 sulfide Calculate the cooling rate R corresponding to the solidification process of the casting C .

2. The method for avoiding fatigue failure caused by oxide inclusions in steel according to claim 1, characterized in that: The radius r of the largest oxide inclusion in the detected steel is obtained in S1 oxide The method is to calculate the radius r of the largest oxide inclusion in the tested steel according to the national standard rating results. oxide , or directly obtain the radius r of the largest oxide inclusion from the inclusion data obtained by various microscopic observation and analysis methods oxide , or calculate the radius r of the largest oxide inclusion in steel by the national standard extreme value analysis method oxide .

3. The method for avoiding fatigue failure caused by oxide inclusions in steel according to claim 2, characterized in that: In S1, the radius r of the largest oxide inclusion in the tested steel is calculated based on the national standard rating results. oxide The national standard GB / T10561-2023 for the grading of Ds-type large-size oxide inclusions is N from 0.5 to 5, with a total of 10 levels at intervals of 0.

5. The level N is related to the oxide radius r oxide The relationship is shown in the following formula: r oxide =0.5[10 0.302N+0.972 ] （1） Among them, the calculation result in the square brackets in the formula is rounded off to get the maximum oxide inclusion radius in the tested steel; The radius r of the largest oxide inclusion is directly obtained from the inclusion data obtained by various microscopic observation and analysis methods. oxide To directly obtain the maximum oxide inclusion size r using the ASPEX inclusion automatic scanner oxide ; Calculate the radius r of the largest oxide inclusion in steel by using the national standard extreme value analysis method oxide To calculate the size r of the maximum oxide inclusion in steel by the extreme value analysis method of GB / T 40281-2021 oxide .

4. The method for avoiding fatigue failure caused by oxide inclusions in steel according to claim 3, characterized in that: In S2, when the outer layer of oxide wraps the sulfide to obtain a composite inclusion, the radius of the composite inclusion r complex Satisfy the following formula: r complex =r oxide +r sulfide （2）。 5. The method for avoiding fatigue failure caused by oxide inclusions in steel according to claim 4, characterized in that: In S2, when the thermal expansion coefficient of the composite inclusions is greater than that of the steel matrix, residual stress will be generated at the interface between the composite inclusions and the matrix during the quenching heat treatment process, which can be calculated as follows: （3） Among them, σ R It is the residual stress generated at the interface between the composite inclusion and the steel matrix after quenching; C is related to the Young's modulus and Poisson's ratio of the composite inclusion and the matrix, and is always greater than 0; α M is the thermal expansion coefficient of the substrate; △T is the temperature difference of quenching heat treatment; α complex is the thermal expansion coefficient of the composite inclusion.

6. The method for avoiding fatigue failure caused by oxide inclusions in steel according to claim 5, characterized in that: Thermal expansion coefficient α of composite inclusions in S2 complex It can be calculated by the following formula: （4） Among them, α oxide is the thermal expansion coefficient of the oxide, α sulfide is the thermal expansion coefficient of the sulfide.

7. The method for avoiding fatigue failure caused by oxide inclusions in steel according to claim 6, characterized in that: If the thermal expansion coefficient of the composite inclusion in S2 is α complex If the stress is not greater than that of the matrix, no residual stress will be generated at the interface between the matrix and the steel matrix, that is, α M ≤α complex When R =0, so the safe thickness of the sulfide shell should satisfy the following formula: （5）。 8. The method for preventing fatigue failure caused by oxide inclusions in steel according to claim 7, characterized in that: Combining equation (5) with equation (1), we can obtain: （6）。 9. The method for preventing fatigue failure caused by oxide inclusions in steel according to claim 8, characterized in that: The thickness of the sulfide shell in S3 is determined by the cooling rate during the solidification process of the ingot. The Scheil solidification model is used to calculate the relationship between the thickness of the sulfide precipitation shell and the cooling rate. The diffusion of element i in the steel liquid is the limiting link for the formation of sulfides. Therefore, the diffusion of element i controls the growth rate of sulfides. It can be obtained that: （7） Among them, J i is the diffusion flux of limiting element i; D i is the diffusion coefficient of limiting element i in the liquid phase, ρ Fe is the density of Fe liquid; M i is the molar mass of limiting element i; w[i] L is the mass fraction of limiting element i at the solidification front of molten steel; w[i] eq is the mass fraction of limiting element i at equilibrium; r sulfide is the radius of the sulfide; In the model, the inclusions are assumed to be spherical after precipitation. According to the mass balance, the relationship between the diffusion flux of element i and the shell thickness of the sulfide is: （8） Combining formula (1), we can get: （9） By integrating equation (9), we can obtain that the precipitation growth and shell thickness of sulfide is r sulfide expression: （10） In formula (10), M sulfide is the molar mass of sulfide; ρ Fe is the density of Fe liquid, 7070kg / m 3 ;M i is the molar mass of limiting element i; ρ sulfide is the density of sulfide; D i is the diffusion coefficient of limiting element i; In formula (10), w[i] L is the mass fraction of limiting element i at the solidification front of molten steel, which can be calculated using the Scheil microsegregation model as follows: （11） Where w[i]0 is the content of solute i in the molten steel before solidification begins; k0 is the equilibrium distribution coefficient of element i; f s is the solid phase ratio of molten steel, which is calculated as follows: （12） Among them, T0 is the melting point of pure iron, which is 1809K; T start is the temperature at which sulfide begins to precipitate; T S is the solidus temperature in steel; In formula (10), w[i] eq It can be calculated using the following formulas (13) to (16): [M] + [S] = MS, lgK=A / TB (13) （14） （15） （16） Wherein, formula (13) is the reaction equilibrium equation for the reaction of elements M and S to form sulfide MS, and in formula (14) α M , α S , α MS are the activities of [M], [S] and MS respectively; f in equations (15) and (16) M and f S are the activity coefficients of [M] and [S] respectively; and are the interaction coefficients of other elements [j] in the molten steel on [M] and [S] respectively; In formula (10), is the solidification time, calculated according to the following formula: （17） Among them, T start is the temperature at which sulfide begins to precipitate; T S is the solidus temperature in steel; R C is the cooling rate during solidification; To achieve a safe thickness r sulfide , Substituting equation (17) into equation (10), the cooling rate R C The following formula should be satisfied: （18） Among them, M sulfide and M i are the molar masses of sulfide and limiting element i, respectively; ρ Fe and ρ sulfide are the densities of Fe liquid and sulfide respectively; D i is the diffusion coefficient of limiting element i in the liquid phase; w[i] L is the mass fraction of limiting element i at the solidification front of molten steel; w[i] eq is the mass fraction of limiting element i at equilibrium; T start is the temperature at which sulfide begins to precipitate; T S is the solidus temperature in steel.

10. The method for preventing fatigue failure caused by oxide inclusions in steel according to claim 9, characterized in that: In order to obtain the safe thickness of the sulfide shell in S3, the safe thickness r sulfide , Substituting equation (6) into equation (18), the cooling rate R C The following formula should be satisfied: （19） Among them, M sulfide and M i are the molar masses of sulfide and limiting element i, respectively; ρ Fe and ρ sulfide are the densities of Fe liquid and sulfide respectively; D i is the diffusion coefficient of limiting element i in the liquid phase; w[i] L is the mass fraction of limiting element i at the solidification front of molten steel; w[i] eq is the mass fraction of limiting element i at equilibrium; T start is the temperature at which sulfide begins to precipitate; T S is the solidus temperature in steel.

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