A method for calculating the strength of precipitation-hardened high-entropy alloys

Abstract

This invention relates to a method for calculating the strength of precipitation-hardening high-entropy alloys. It effectively combines lattice distortion theory, grain boundary strengthening theory, dislocation strengthening theory, and precipitation-hardening theory considering the size and spatial distribution of precipitates to establish a relevant strength analysis theoretical model. This model considers the influence of lattice distortion, grain boundaries, dislocations, precipitates, and their size and spatial distribution on the properties of precipitation-hardening high-entropy alloys, enabling quantitative calculation and analysis of the alloy's strength. The strength obtained using the proposed method shows good agreement with experimental results. The strengthening mechanism analyzed in this invention is of great significance for studying and analyzing the mechanical properties of precipitation-hardening high-entropy alloys under different microscopic conditions. By controlling the elemental content of the alloy and changing the radius and volume fraction of the precipitates using the proposed analytical method, the influence of precipitates on the strength of precipitation-hardening high-entropy alloys can be investigated, thus providing theoretical guidance for the design of high-performance precipitation-hardening high-entropy alloys.

Description

Technical Field

[0001] This invention relates to the field of strength calculation of precipitation-hardened high-entropy alloys, specifically to lattice distortion theory, grain boundary strengthening theory, dislocation strengthening theory, and precipitation strengthening theory. A theoretical model is established to consider the size and spatial distribution of precipitates, thereby enabling the analysis of the strength of precipitation-hardened high-entropy alloys. Background Technology

[0002] High-entropy alloys, possessing high strength, excellent structural stability, and corrosion resistance, have attracted significant attention over the past decade. Unlike traditional binary and ternary alloys, high-entropy alloys are composed of five or more elements in almost equiatomic proportions. To further enhance the strength of high-entropy alloys, the formation of precipitates has become a research hotspot in recent years. Recent studies have shown that the addition of Ti and Al elements to FeCoNi-based and FeCoCrNi-based high-entropy alloys leads to the formation of spherical precipitates in the matrix, thereby increasing the yield strength of the high-entropy alloy. Previous experiments have shown that precipitate-hardened FeCoNi-based and FeCoCrNi-based high-entropy alloys exhibit superior properties compared to traditional FeCoNi-based and FeCoCrNi-based high-entropy alloys, and therefore, they were selected as the research materials for this invention.

[0003] To date, no theoretical model has been established that considers the size and spatial distribution of precipitates in high-entropy alloys. Furthermore, there is no unified theoretical formula to quantify the contribution of various strengthening mechanisms to the strength of precipitate-hardened high-entropy alloys. The precipitate strengthening formula proposed in this invention is based on the normal distribution of precipitate size and the actual spatial distribution of precipitates in the matrix, providing an effective method for accurately analyzing precipitate strengthening in high-entropy alloys during precipitation-hardening. Based on experimental and theoretical analysis results, this invention analyzes the strengthening mechanism of precipitates, establishes a relevant theoretical model for precipitate strengthening, and is of great significance for studying the influence of precipitate size and spatial distribution on the strengthening of precipitates in high-entropy alloys, and for analyzing the contribution of various strengthening mechanisms to precipitate-hardened high-entropy alloys. Summary of the Invention

[0004] The purpose of this invention is to propose a quantitative calculation and analysis method for the strength of precipitation-hardening high-entropy alloys based on experimental data combined with theories of lattice distortion, grain boundary strengthening, dislocation strengthening, and precipitation-phase strengthening considering size and spatial distribution. This invention, while considering the precipitation-phase strengthening model of precipitation-hardening high-entropy alloys and achieving quantitative analysis of their strength, allows for the control of precipitate size and volume fraction to obtain the precipitate strength under specific conditions, providing guidance for the design of higher-performance precipitation-hardening high-entropy alloys.

[0005] The technical solution of this invention is:

[0006] The material parameters of the high-entropy alloy used were determined, including relevant parameters of elemental physical properties. The materials used in this invention are FeCoCrNiTiAl and FeCoNiTiAl high-entropy alloys, and their material parameters are shown in Table 1.

[0007] Table 1. Physical parameters of each element.

[0008]

[0009] Based on the theory of lattice distortion and combined with Vegard's law, the lattice frictional stress σ in high-entropy alloys Lattice It can be represented as

[0010]

[0011] n is the number of different types of elements in the high-entropy alloy, and c i It is the concentration of element i. This represents the individual contribution of the i-th element in the high-entropy alloy to the overall lattice frictional stress.

[0012]

[0013] Where A is a dimensionless parameter related to the processing technology and the material, and the shear modulus of the high-entropy alloy is... Additionally, the mismatch parameter δmp i It can be represented as:

[0014]

[0015] For FCC high-entropy alloys, ξ = 1. The value of β depends on the dislocation type; β = 3 when screw dislocations dominate, and β = 16 when edge dislocations dominate. Assuming the quinary high-entropy alloy ijklm consists of a quaternary matrix jklm and a mixed element i, then the size mismatch δr of the mixed element i in the high-entropy alloy ijklm is... i,ijklm and modulus mismatch δG i,ijklm The following formula represents,

[0016]

[0017]

[0018] and These represent the average size mismatch and average modulus mismatch of the high-entropy alloy ijklm, respectively. and These represent the average dimensional mismatch and average modulus mismatch of the matrix jklm, respectively.

[0019]

[0020]

[0021] δr ij and δG ij These represent the size mismatch and modulus mismatch between atoms i and j, respectively.

[0022] δr ij =2(r i -r j ) / (r i +r j (8)

[0023] δG ij =2(G i -G j ) / (G i +G j (9)

[0024] The above derivation ultimately leads to the lattice distortion enhancement.

[0025] Based on the Hall-Petch formula, grain boundary strengthening σ grain It can be represented as:

[0026]

[0027] k HP For Hall-Petch parameters, d g This refers to the grain size.

[0028] Grain boundary strengthening can ultimately be achieved using the above formula.

[0029] Based on the classical dislocation strengthening theory, dislocation strengthening σ in high-entropy alloys dislocation It can be represented as:

[0030]

[0031] Where M is the Taylor factor, α is an empirical constant, b is the Burgers vector, and ρ is the dislocation in the grain.

[0032]

[0033] Assume the initial dislocation density ρ0 = 2 × 10 8 m -2 dt is the time step. Dislocation density at each time step.

[0034]

[0035] ε p It is plastic strain, ψ is the scaling factor, and k 20It is the dynamic recovery constant, ε′ p,equ ε0′ is the equivalent plastic strain rate, ε0′ is the reference strain rate, and m1 is the dynamic recovery constant.

[0036]

[0037] σ Mises It is the Mises equivalent stress, σ flow It is flow stress, and the rate sensitivity index m0 = 20.

[0038]

[0039] Where σ d It is a deviatoric stress, which can be expressed as:

[0040] σ d =σ d ′dt (16)

[0041] σ d ′ is the deviatoric stress rate, which can be calculated by the following formula:

[0042]

[0043] Where σ Cauchy ′ is the Cauchy stress rate, which can be expressed as:

[0044]

[0045] σ sph ′ is the spherical stress rate, which can be expressed as:

[0046]

[0047] Where σ x ′、σ y ′、σ z The Cauchy stress rate σ′ can be obtained by calculating the following formula:

[0048]

[0049] Where C is the stiffness matrix, ε Cauchy ′ is the Cauchy strain rate, ε C ′ can be represented as:

[0050] ε Cauchy ′=(S 11 S 12 S 12 0 0 0) T ε e ′ / S 11 (twenty one)

[0051] S is the compliance matrix, and S = C -1 ,ε e ′ is the elastic strain rate, ε e ′ is obtained separately from strain rate ε′:

[0052] ε′=ε e ′+ε p ' (twenty two)

[0053] ε p ′ is the plastic strain rate.

[0054] The above derivation can ultimately yield the dislocation reinforcement term.

[0055] Precipitation enhancement is controlled by two mechanisms, namely the cleavage enhancement mechanism σ. Shear And Orowan bypassing the reinforcement mechanism σ Orowan Cut through the enhancement mechanism σ Shear Coherent reinforcement σ CS Modulus mismatch reinforcement σ MS and ordered reinforcement σ OS Composition. Their expressions are:

[0056]

[0057] σ CS =M·α ε (G·ε L ) 3 / 2 (rf / (0.5Gb)) 1 / 2 (twenty four)

[0058] σ MS =0.0055M(ΔG) 3 / 2 (2f / G) 1 / 2 (r / b) 3m / 2-1 (25)

[0059] σ OS =0.81Mγ APB / (2b)·(3πf / 8) 1 / 2 (26)

[0060] σ Shear =σ CS +σ MS +σ OS (27)

[0061] In the FCC structure, α ε =2.6; m=0.85; ε L It is a constrained lattice parameter mismatch, expressed as ε. L= 2 / 3·Δa / a, where Δa is the constant lattice difference between the precipitate and the FCC matrix, Δa / a = 0.0026; ΔG is the shear modulus difference between the precipitate and the high-entropy alloy, ΔG = |G HEA -G Precipitate |;γ APB It is the antiphase domain boundary energy of the precipitated phase.

[0062] As attached Figure 1 Given σ Shear and σ Orowan As shown by the curve, there exists a mechanism for transforming the critical size r of the precipitate phase. Critical When r ≤ r Critical When r > r, precipitation phase reinforcement is a tangential reinforcement mechanism; Critical At that time, the precipitation phase is enhanced to allow Orowan to bypass the enhancement mechanism. From the appendix Figure 2 As can be seen from (a) and (b), the dislocation occurs when r ≤ r Critical When the precipitate is cut, the cut-through reinforcement mechanism is triggered; when the dislocation is r > r Critical When the precipitate is cut through, the Orowan bypass reinforcement mechanism is triggered, which explains the spatial distribution of the precipitate. (See attached image.) Figure 2 As shown in (c), segment AB is the region that cuts through the reinforcement mechanism, and segment OA is the region that Orowan bypasses the reinforcement mechanism. During calculation, each region needs to be differentiated into n equal parts. For example, in segment AB, h... i+1 =h i +dh(i=1,2,3…n), where dh=AB / n; in segment OA, h j+1 =h j +dh(j=1,2,3…n), where dh=OA / n. Therefore, formula (23-27) can be rewritten as:

[0063]

[0064] for and for

[0065] From the appendix Figure 2 (c) It can be seen that the triggering probability of different precipitation phase enhancement mechanisms can be calculated by dividing the radius of the precipitation phase. The triggering probability when cutting through the enhancement mechanism region is:

[0066]

[0067] Within the Orowan bypass mechanism region, the probability of triggering is:

[0068] p2 = 1 - p1 (30)

[0069] Combining formulas (28-30), consider the spatial distribution of precipitates for enhancement. It can be represented as:

[0070]

[0071] The size distribution is due to the different average sizes of the precipitates distributed within the matrix. In FeCoCrNiTiAl high-entropy alloys, the size distribution of the precipitates follows a normal distribution, with the probability formula as follows:

[0072]

[0073] Where σ is the standard deviation.

[0074] Similar to calculating the spatial distribution of precipitated phase enhancement, r Critical This influences the triggered precipitation enhancement mechanism. When the average radius r of the precipitation phase... ave >r Critical At that time, the spatial distribution consists of two regions. It includes the region that cuts through the reinforcement mechanism (r≤r). Critical ) and Orowan bypass mechanism region (r>r) Critical By using calculus, the radius region of the precipitate size distribution is divided into n equal parts, and the formula for precipitate strengthening based on size distribution can be expressed as:

[0075]

[0076] in It is all regions that have been cut through the enhancement mechanism (r≤r) Critical The sum of the components; It is all regions that have been cut through the enhancement mechanism (r > r) Critical The sum of the components.

[0077] Finally, considering the precipitate size distribution and spatial distribution, the precipitate strengthening formula is:

[0078]

[0079] The above derivation allows us to obtain precipitate strengthening that takes into account both size and spatial distribution.

[0080] By coupling the four strengthening mechanisms, the strength of precipitation-hardened high-entropy alloys can be obtained.

[0081] σ Strength =σ Lattice +σ Grain +σ Dislocation +σ Precipitate (35)

[0082] Finally, by processing and analyzing the calculation results, we can obtain the contribution of each strengthening mechanism to the strength under different processing techniques.

[0083] Beneficial effects

[0084] This invention proposes a method for calculating the strength of precipitation-hardened high-entropy alloys. It considers lattice distortion, grain boundary strengthening, dislocation strengthening, and precipitation strengthening based on size and spatial distribution to analyze the strength of precipitation-hardened high-entropy alloys. This method is based on a solid theoretical foundation, a clear modeling process, and a clear physical meaning.

[0085] This invention takes FeCoCrNiTiAl and FeCoNiTiAl high-entropy alloys as examples. Using experimental data of these two high-entropy alloys, the qualitative and quantitative relationship of alloy strength is calculated through the strength theory model in the calculation method. The results are in good agreement with the experiments, thus obtaining the contribution of each strengthening mechanism and providing theoretical guidance for analyzing the precipitate strengthening of high-entropy alloys. Attached Figure Description

[0086] Figure 1 These are the corresponding precipitation enhancement mechanisms under different precipitation radii.

[0087] Figure 2 This is a schematic diagram of dislocations bypassing the precipitate phase. (a) Spherical precipitate phase; (b) Dislocation shearing or bypassing the precipitate phase; (c) Delineating the radius of the precipitate phase.

[0088] Figure 3 This is a comparison of experimental results, classical models, and current models. (a) FeCoCrNiTiAl high-entropy alloy; (b) FeCoNiTiAl high-entropy alloy.

[0089] Figure 4 The results are: (a) the effect of different strengthening mechanisms in FeCoCrNiTiAl high-entropy alloys; (b) the percentage of different strengthening mechanisms in the yield strength of FeCoCrNiTiAl high-entropy alloys; (c) the effect of different strengthening mechanisms in FeCoNiTiAl high-entropy alloys; and (d) the percentage of different strengthening mechanisms in the yield strength of FeCoNiTiAl high-entropy alloys. Detailed Implementation

[0090] The following is in conjunction with the appendix Figure 2 The corresponding precipitation enhancement mechanisms and associated precipitation radii are given. Figure 3 The given schematic diagram of dislocation bypassing precipitates, a theoretical model and specific examples of a precipitate-hardened high-entropy alloy considering four effects: lattice distortion, grain boundaries, dislocations, and precipitates, are presented. The technical solution is further elaborated. This invention is not limited to the following examples; all designs utilizing the design concept of this invention are within the scope of protection of this invention.

[0091] Lattice distortion is an inherent characteristic of high-entropy alloys caused by the mismatch in size and modulus of the constituent elements, which strengthens the mechanical properties of the material. Grain boundaries hinder dislocations and thus strengthen the mechanical properties of the material. When dislocations move, they tend to intersect each other, forming cleavage steps and causing dislocation entanglement, thereby hindering dislocation movement and making it difficult to continue plastic deformation, thus increasing the strength of the material. Precipitated phases are new phases generated by the desolvation and precipitation of the matrix phase, which strengthen the mechanical properties of the material by hindering dislocation movement.

[0092] Specific steps: Collect phase transformation experimental data of FeCoCrNiTiAl and FeCoNiTiAl high-entropy alloys, and obtain microstructure data under different processing techniques, as shown in Tables 1-4.

[0093] Table 1. Physical parameters of each element.

[0094]

[0095] Table 2. Processing technology and atomic percentage of FeCoCrNiTiAl high-entropy alloy.

[0096]

[0097] Table 3. Processing technology and atomic percentage of FeCoNiTiAl high-entropy alloy.

[0098]

[0099] Table 4. Constants A, dislocation density, grain size, precipitate radius, and precipitate volume fraction in FeCoCrNiTiAl and FeCoNiTiAl high-entropy alloys.

[0100]

[0101] The material parameters involved in the method of this invention are shown in Table 5.

[0102] Table 5 Parameters of Various Materials

[0103]

[0104] By fitting the relevant experimental data of FeCoCrNiTiAl and FeCoNiTiAl high-entropy alloys, the following results were obtained: Figure 3 (a,b) Comparison of experimental results, classical model, and current model. (See appendix) Figure 3 As can be seen from (a,b), after considering the size and spatial distribution of the precipitated phase, the current model has higher computational accuracy than the classical model.

[0105] From the appendix Figure 4As can be seen, precipitation phase strengthening plays a dominant role in improving yield strength, accounting for more than 50% of the strength. The maximum contribution of precipitation phase strengthening reaches 89% (see appendix). Figure 4 (b) Although the precipitate radii and volume fractions in TP1 and TP2 differ, their processing methods are almost identical (Table 2), so their precipitation-strengthened and yield strengths are virtually indistinguishable. However, TP3 exhibits a higher yield strength compared to TP2, primarily due to dislocation strengthening, which accounts for 27% of the yield strength of TP3, whereas TP2 shows no dislocation strengthening.

[0106] From the appendix Figure 4 As can be seen from c, precipitate strengthening also plays a crucial role in yield strength, contributing more than 43% to it. Its highest contribution rate reaches 81% (see appendix). Figure 4 d). Similar to the case in FeCoCrNiTiAl high-entropy alloys, T3 has a higher dislocation density compared to T1 and T2, which makes the dislocation strengthening effect more significant, with the contribution of dislocation strengthening reaching as high as 29% (see appendix). Figure 4 d).

[0107] According to theoretical modeling results, there is no obvious competition between grain boundary strengthening and precipitate strengthening (see appendix). Figure 4 a and appendix Figure 4 c). Therefore, the optimal choice is to adjust the processing technology to reduce the grain size to increase grain boundary strengthening and increase the dislocation density to enhance dislocation strengthening, thereby improving the yield strength of the precipitate-hardened high-entropy alloy.

[0108] Therefore, this invention provides a high-precision strength analysis of FeCoCrNiTiAl and FeCoNiTiAl high-entropy alloys, and can effectively analyze the contribution of each strengthening mechanism, providing a reliable theoretical model for analyzing the strength of precipitation-hardened high-entropy alloys.

Claims

1. A method for calculating the strength of precipitate-hardened high-entropy alloys, which establishes a strength analysis model based on the lattice distortion theory, the grain boundary strengthening theory, the dislocation strengthening theory and the precipitate strengthening theory, characterized in that: the triggered probability in the region of the strengthening mechanism is: the lattice distortion, the grain boundary and the dislocation in the material are usually generated in the process of processing or stretching, and the precipitate is generated in the process of processing; the calculation and analysis of the strength of the precipitate-hardened high-entropy alloys are realized by analyzing the contribution of each strengthening mechanism, and the yield strength of the precipitate-hardened high-entropy alloys under specific conditions can be obtained by adjusting the precipitate size and the precipitate volume fraction, thereby providing guidance for the design of the precipitate-hardened high-entropy alloys with more excellent performance. Consider the precipitation phase hardening high entropy alloy lattice distortion, grain boundary, dislocation and precipitation strengthening effect; characterized in that, When dislocations move, they are easy to intersect with each other, form steps, cause dislocation entanglement, and thus cause obstacles to dislocation movement, making it difficult for continuous plastic deformation, thereby increasing the strength of the material. Based on the classical dislocation strengthening theory, the dislocation strengthening in high-entropy alloys can be expressed as: ; where M is the Taylor factor, is an empirical constant, b is the Burgers vector, and is determined by the lattice parameters, for an FCC lattice , for a BCC lattice ; is the dislocation density in the grain , assuming an initial dislocation density , is the time step, is the dislocation density at each time step; is the plastic strain, is the proportionality factor, is the dynamic recovery constant, is the equivalent plastic strain rate, is the reference strain rate, m is the dynamic recovery constant; The precipitates also have a strengthening effect on the mechanical properties of the material by impeding dislocation motion. The main mechanisms of precipitation strengthening are the cutting through mechanism and the Orowan bypassing mechanism; the cutting through mechanism consists of coherent strengthening , modulus mismatch strengthening and order strengthening ; their expressions are: In FCC structure, ; ; is the constrained lattice parameter mismatch, which is expressed as , is the constant lattice difference between precipitates and FCC matrix, ; is the shear modulus difference between precipitates and high-entropy alloys, ; is the anti-phase domain boundary energy of precipitates; There is a mechanism transition of precipitate critical size When , the precipitate strengthening is a shearing strengthening mechanism, and when , the precipitate strengthening is an Orowan bypassing strengthening mechanism; and when the dislocation cuts through the precipitate , the shearing strengthening mechanism is triggered; and when the dislocation cuts through the precipitate , the Orowan bypassing strengthening mechanism is triggered; the formula considering each precipitate strengthening component of spatial distribution can be written as: For and , , (i = 1, 2, 3…n); for , , (j = 1, 2, 3…n); The four strengthening mechanisms in the strengthening model are accurately calculated by using the inherent parameters of the elements in the material and the existing experimental data. In the Orowan bypass mechanism region, the probability of triggering is: Precipitate strengthening taking into account spatial distribution may be expressed as: In the FeCoCrNiTiAl high-entropy alloy, the size distribution of precipitated phases is normally distributed, and the probability formula is: When the average radius of precipitate The spatial distribution includes the area of cutting through the strengthening mechanism ( ) and the area of Orowan bypass mechanism ( ); by calculus, the precipitate size distribution radius area is divided into n equal parts, and the precipitate strengthening formula of size distribution can be expressed as: wherein is the sum of all the cut-through strengthening mechanism region ( ) component accumulations; is the sum of all the cut-through strengthening mechanism region ( ) component accumulations; The precipitation phase strengthening formula considering the size distribution and spatial distribution of the precipitation phase is ; 3. The method according to any one of claims 1-2, which comprises the following specific steps: 2.The method according to claim 1, wherein determining the basic material parameters required in the model and collecting the relevant physical parameters of the relevant materials; calculating the strength contributed by each strengthening mechanism of the lattice distortion theory, the grain boundary strengthening theory, the dislocation strengthening theory and the precipitate strengthening; the four strengthening mechanisms are coupled and processed, and the strength of the precipitate-hardened high-entropy alloys can be obtained, determining the effective material parameters; the materials used in the application are FeCoCrNiTiAl high-entropy alloys and FeCoNiTiAl high-entropy alloys, the shear moduli of the materials Fe, Co, Cr, Ni, Ti and Al are 82, 75, 115, 76, 44 and 26 GPa respectively, and the atomic radii of the materials Fe, Co, Cr, Ni, Ti and Al are 126, 125, 128, 124, 147 and 143 pm respectively. the difference between the atomic radii and the shear moduli of each main element; according to the Vegard law, the lattice friction stress of the high-entropy alloy is formed by the superposition of the action of each element in the alloy: wherein is the lattice friction stress, is the grain boundary strengthening, is the dislocation strengthening, is the precipitate strengthening; analyze the results of the theoretical calculations.

4. The specific steps of claim 3, wherein, 6. According to claim 3, characterized in that the grain boundary hinders the movement of dislocations, thereby causing the improvement of the strength of the material; based on the Hall-Petch formula, the grain boundary strengthening in the high-entropy alloy can be expressed as:

5. The method of claim 3, calculating the contribution of the lattice distortion to the intensity, wherein, ​ n is the number of element species, is the concentration of element i, is the individual contribution value of the i-th element in the precipitate hardened high-entropy alloy to the overall yield strength. ​ H is a Hall-Petch parameter, is the grain size.

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