Method for selecting forging loading mode based on crystal phase field method from fine-grain strengthening perspective

Abstract

This invention specifically presents a method for selecting the optimal forging loading mode from the perspective of grain refinement strengthening based on the crystal phase field method, solving the problem of difficulty in characterizing the atomic-scale grain refinement evolution process in traditional experiments. The method includes the following steps: Step 1: Establishing a dual-mode crystal phase field model; Step 2: Determining the numerical method: Spatially discretizing the constructed Cahn-Hillard kinetic equation using a semi-implicit Fourier spectroscopy method to obtain the atomic density field evolution form at the next time step; Step 3: Setting initial condition parameters according to the various forging loading modes to be compared; Step 4: Simulation calculation and result export; Step 5: Comparing and analyzing the crystal evolution process under different forging loading modes obtained in Step 4. This invention focuses on the growth state of nanocrystalline polycrystalline materials under different stress loading conditions and analyzes the influence of different loading conditions on grain refinement.

Description

Technical Field

[0001] This invention relates to the field of crystal simulation technology, specifically a method for selecting the optimal forging loading mode from the perspective of grain refinement strengthening based on the crystal phase field method. Background Technology

[0002] In engineering applications, grain shape and size are factors affecting material properties. In engineering production, grain refinement strengthening is a common method for strengthening materials. Experiments show that fine-grained metals at room temperature have higher strength, hardness, plasticity, and toughness than coarse-grained metals. This is because the plastic deformation of fine grains under external force is dispersed within more grains, resulting in more uniform plastic deformation and less stress concentration. Furthermore, since grain boundaries hinder dislocation movement, as grain size decreases, the number of grain boundaries increases, further enhancing the resistance to dislocation movement, thus giving the material higher plasticity, toughness, and strength.

[0003] In actual production processes, the grain size of workpieces is often refined by changing different loading methods during forging, such as multi-directional forging. Different forging methods have different effects on the grain size of the workpiece; however, based on current experimental observation methods, it is difficult and costly to observe this phenomenon at the atomic scale.

[0004] Currently, computer simulations offer unparalleled advantages over traditional experiments in addressing atomic-scale material microstructure problems. Methods such as finite element method (FEM), Monte Carlo simulation, molecular dynamics, and traditional phase-field methods can all be used to investigate the relationship between crystal structure and properties. Compared to other simulation methods, the crystal phase-field method can track the evolution of the local time-averaged atomic density field to address atomic-scale evolution processes, naturally describing dislocation and grain boundary structures, and achieving long diffusion timescales that molecular dynamics simulations cannot reach. Therefore, using the crystal phase-field method to study the influence of microstructure on the mechanism of polycrystalline plastic deformation is crucial.

[0005] Therefore, it is necessary to invent a method for optimizing forging loading from the perspective of grain refinement strengthening based on the crystal phase field method. The crystal phase field method is used to simulate the influence of different loading methods on the grain refinement effect. Based on the grain refinement degree of the simulation results, the optimal refinement method is selected and finally applied to actual production. Summary of the Invention

[0006] To address the challenge of characterizing the atomic-scale grain refinement evolution process in traditional experiments, this invention provides a method for selecting the optimal forging loading mode from a grain refinement strengthening perspective based on the crystal phase field method. The purpose of this invention is to investigate the grain refinement evolution process of nanocrystalline polycrystalline samples under different strain loading conditions and to elucidate the influence of different loading methods on grain refinement.

[0007] This invention is achieved using the following technical solution:

[0008] A method for selecting the optimal forging loading mode from the perspective of grain refinement strengthening based on the crystal phase field method, the method comprising the following steps:

[0009] Step 1: Establish a two-mode crystal phase-field model, which includes the Helmholtz free energy equation and the Cahn-Hillard kinetic equation; wherein, the Helmholtz free energy equation is a dimensionless free energy functional equation that incorporates a periodic atomic density function.

[0010] Step 2: Determine the numerical method: Use the semi-implicit Fourier spectral method to spatially discretize the constructed Cahn-Hillard dynamic equations to obtain the evolution form of the atomic density field at the next time step;

[0011] Step 3: Set initial condition parameters according to the various forging loading methods to be compared; the forging loading method is a two-dimensional system loading method, and adopts the assumption of equal area conditions;

[0012] Step 4: Simulation Calculation and Result Export: Calculate the crystal evolution process under various forging loading methods and export the results, thereby obtaining the crystal evolution process under different forging loading methods;

[0013] Step 5: Compare and analyze the crystal evolution process under different forging loading methods obtained in Step 4, and select the optimal forging loading method from the perspective of fine grain strengthening.

[0014] Furthermore, the establishment of the two-mode crystal phase-field model in step one includes the following:

[0015] The two-mode crystal phase-field model is constructed based on the traditional phase-field method. The periodic atomic density function is introduced into the Helmholtz free energy equation of the two-mode crystal phase-field model. The order parameter adopts the periodic local time-averaged atomic density. The free energy equation is as shown in equation (1):

[0016]

[0017] In the formula, r is the position vector of the atom, φ represents the atomic density, and r1 is a parameter used to control the relative magnitude of the amplitudes of the two reciprocal lattice vectors, which is related to the relative stability of different crystal structures; Let be the Laplace operator; q0 and q1 are the magnitudes of the wave vectors of the principal mode and the secondary mode, respectively, q0 = |K0|, q1 = |K1|, K0 and K1 represent the reciprocal lattice vectors of the nearest neighbor and the second nearest neighbor, respectively; α, λ, g, and r1 are all constants;

[0018] To facilitate the writing of the model program, substitute equations (2) and (3) into equation (1) to achieve scaling of the parameters in the dual-mode crystal phase field model and obtain the dimensionless free energy functional equation as shown in equation (4).

[0019]

[0020]

[0021]

[0022] Simplifying equation (4), we obtain the dimensionless free energy functional equation as shown in equation (5):

[0023]

[0024] In the formula, F is the dimensionless free energy function; ψ represents the dimensionless atomic density; ε characterizes the temperature of the system and is a negative value. A smaller absolute value of ε corresponds to a higher temperature. Here, R1 is the Laplace operator; R1 is a parameter related to the relative stability of different crystal structures. When R1 approaches 0, the model tends to be a single-mode model; Q1 is a constant, which is the ratio of the magnitude of the second nearest neighbor reciprocal vector to the magnitude of the nearest neighbor reciprocal vector. Its value is generally determined by the choice of crystal structure.

[0025] Since the atomic density field ψ in the crystal phase-field model is a conserved order parameter, the system evolution obeys the Cahn-Hillard dynamic equation of the conserved field. After dimensionless processing, the dynamic equation is as shown in equation (6):

[0026]

[0027] In the formula, F is the dimensionless free energy function, which is calculated by formula (5); t is the time variable.

[0028] Furthermore, the determination of the numerical method in step two is achieved through the following steps:

[0029] The dimensionless dynamic equation (6) is solved using a semi-implicit Fourier spectrum algorithm, and the space is discretized by Fourier transform to obtain the evolution form of the atomic density field at the next moment as shown in equation (7):

[0030]

[0031] In the formula, ψ k Represents the Fourier transform; ψ k,t+1 Let represent the atomic density in Fourier space at time t+1; k represents the Fourier space vector, and satisfies k 2 =|k| 2 .

[0032] Furthermore, in step three, the following assumptions are made when setting the initial condition parameters: Under strain loading, the grains will deform, and the deformation simulation of the two-dimensional system adopts the assumption of equal area conditions, as detailed below:

[0033] The assumption of equal area is given by equation (8):

[0034] S=Δx×Δy=Δx′×Δy′ (8)

[0035] In the formula, Δx and Δy are the initial spatial step size, and Δx' and Δy' are the deformed spatial step size;

[0036] When stretching along the y-axis, it is assumed that the spatial step size in the y-direction has an increment d at each step, and in The strain loading rate is dimensionless, and

[0037] After stretching along the y-axis for n steps, the spatial step size in the x-direction is calculated using equation (9), and the spatial step size in the y-direction is calculated using equation (10):

[0038]

[0039]

[0040] When stretching along the x-axis, it is assumed that the spatial step size in the x-direction has an increment d at each step, and in The strain loading rate is dimensionless, and

[0041] When stretching along the x-axis for n steps, the spatial step size in the x-direction is calculated using equation (11), and the spatial step size in the y-direction is calculated using equation (12):

[0042]

[0043]

[0044] This invention employs a dual-mode PFC (Phase-field-crystal) model that introduces two critical wavelengths that dampen the dynamics of the Helmholtz free energy, thereby enabling the simulation of more crystal types and allowing for better study of complex lattice structures. The influence of different loading conditions on grain refinement is analyzed for the growth state of nanocrystalline polycrystalline materials under varying stress loading conditions.

[0045] This invention reveals the grain boundary evolution behavior of polycrystalline crystals under strain loading during growth. The invention utilizes a two-mode crystal phase-field model, employs a semi-implicit Fourier spectroscopy method to solve the kinetic equations, and couples uniaxial strain loading into the model. Driven by kinetics, calculations are performed on different directions at the same loading temperature to obtain the grain boundary evolution behavior of polycrystalline crystals under a series of variable conditions. Attached Figure Description

[0046] Figure 1 This is a schematic diagram of unidirectional (Y) strain application in an embodiment of the present invention;

[0047] Figure 2 This is a schematic diagram of multi-directional (YX) strain application in an embodiment of the present invention;

[0048] Figure 3 This is a schematic diagram of multi-directional (YXY) strain application in an embodiment of the present invention;

[0049] Figure 4 This is a diagram showing the evolution of a polycrystalline crystal under uniaxial (Y) strain application in an embodiment of the present invention.

[0050] Figure 5 This is a diagram showing the evolution of a polycrystalline crystal under multi-directional (YX) strain application in an embodiment of the present invention.

[0051] Figure 6 This is a diagram showing the evolution of a polycrystalline crystal under multi-directional (YXY) strain application in an embodiment of the present invention. Detailed Implementation

[0052] A method for selecting the optimal forging loading mode from the perspective of grain refinement strengthening based on the crystal phase field method, the method comprising the following steps:

[0053] Step 1: Establish a two-mode crystal phase-field model, which includes the Helmholtz free energy equation and the Cahn-Hillard kinetic equation;

[0054] The Helmholtz free energy equation is a dimensionless free energy functional equation that incorporates a periodic atomic density function.

[0055] This step includes the following:

[0056] The two-mode crystal phase-field model is constructed based on the traditional phase-field method. The periodic atomic density function is introduced into the Helmholtz free energy equation of the two-mode crystal phase-field model. The order parameter adopts the periodic local time-averaged atomic density. The free energy equation is as shown in equation (1):

[0057]

[0058] In the formula, r is the position vector of the atom, φ represents the atomic density, and r1 is a parameter used to control the relative magnitude of the amplitudes of the two reciprocal lattice vectors; Let be the Laplace operator; q0 and q1 are the magnitudes of the wave vectors of the principal mode and the secondary mode, respectively, q0 = |K0|, q1 = |K1|, K0 and K1 represent the reciprocal lattice vectors of the nearest neighbor and the second nearest neighbor, respectively; α, λ, g, and r1 are all constants;

[0059] Substituting equations (2) and (3) into equation (1) allows for scaling of the parameters in the two-mode crystal phase-field model, resulting in the dimensionless free energy functional equation as shown in equation (4).

[0060]

[0061]

[0062]

[0063] Simplifying equation (4), we obtain the dimensionless free energy functional equation as shown in equation (5):

[0064]

[0065] In the formula, F is the dimensionless free energy function; ψ represents the dimensionless atomic density; ε characterizes the temperature of the system and is a negative value. A smaller absolute value of ε corresponds to a higher temperature. Here, is the Laplace operator; R1 is a parameter related to the relative stability of different crystal structures. When R1 approaches 0, the model tends to be a single mode; Q1 is a constant, which is the ratio of the magnitude of the second nearest neighbor reciprocal vector to the magnitude of the nearest neighbor reciprocal vector.

[0066] Since the atomic density field ψ in the crystal phase-field model is a conserved order parameter, the system evolution obeys the Cahn-Hillard dynamic equation of the conserved field. After dimensionless processing, the dynamic equation is as shown in equation (6):

[0067]

[0068] In the formula, F is the dimensionless free energy function, which is calculated by formula (5); t is the time variable.

[0069] Step 2: Determine the numerical method: Use the semi-implicit Fourier spectral method to spatially discretize the constructed Cahn-Hillard dynamic equations to obtain the evolution form of the atomic density field at the next time step;

[0070] This step is implemented using the following steps:

[0071] The dimensionless dynamic equation (6) is solved using a semi-implicit Fourier spectrum algorithm, and the space is discretized by Fourier transform to obtain the evolution form of the atomic density field at the next moment as shown in equation (7):

[0072]

[0073] In the formula, ψ k Represents the Fourier transform; ψ k,t+1 This represents the atomic density in Fourier space at time t+1;

[0074] k represents a vector in the Fourier space, and satisfies k 2 =|k| 2 .

[0075] Step 3: Set initial condition parameters according to the various forging loading methods to be compared; the initial condition parameters include atomic density, temperature parameters, and time step.

[0076] The forging loading method is a two-dimensional system loading method, and it adopts the assumption of equal area conditions;

[0077] This embodiment takes one equal-quantity forging with a Y-axis, three equal-quantity forgings from the Y-axis to the X-axis and back to the Y-axis, and two equal-quantity forgings from the YX-axis as examples, as shown in the attached figure. Figure 1 To be continued Figure 3 As shown;

[0078] The initial condition parameters for Y-axis single-stage equal-volume directional forging, YX-axis double-stage equal-volume directional forging, and Y-axis-X-axis-Y-axis triple-stage equal-volume directional forging are shown in Table 1.

[0079] Table 1: Initial parameter settings under different directions of dynamic strain loading

[0080]

[0081] This step makes the following assumptions when setting the initial condition parameters: Under strain loading, the grains will deform, and the deformation simulation of the two-dimensional system adopts the assumption of equal area conditions, as detailed below:

[0082] The assumption of equal area is given by equation (8):

[0083] S=Δx×Δy=Δx′×Δy′ (8)

[0084] In the formula, Δx and Δy are the initial spatial step size, and Δx' and Δy' are the deformed spatial step size;

[0085] When stretching along the y-axis, it is assumed that the spatial step size in the y-direction has an increment d at each step, and in The strain loading rate is dimensionless, and

[0086] After stretching along the y-axis for n steps, the spatial step size in the x-direction is calculated using equation (9), and the spatial step size in the y-direction is calculated using equation (10):

[0087]

[0088]

[0089] When stretching along the x-axis, it is assumed that the spatial step size in the x-direction has an increment d at each step, and in The strain loading rate is dimensionless, and

[0090] When stretching along the x-axis for n steps, the spatial step size in the x-direction is calculated using equation (11), and the spatial step size in the y-direction is calculated using equation (12):

[0091]

[0092]

[0093] In this embodiment, the uniaxial strain loading method is dynamic loading, as detailed below:

[0094] The dynamic strain loading method is as follows: Assuming the coordinates of the grid points in the sample before deformation are represented by (x, y), and the coordinates of the grid points after deformation are (x, y), then... ′ ,y ′ The sample is subjected to a strain of 0.5 g in the x-axis direction. After the stretching action, the coordinate transformation relationship before and after deformation is x ′ =x(1+∈ x ) and y ′ =y / (1+∈ x After strain causes deformation of the sample, its internal atomic density function is as shown in equation (13):

[0095]

[0096] When subjected to strain in the y-axis direction After the stretching action, the coordinate transformation relationship before and after deformation is x ′ =x / (1+∈ y ) and y ′ =y(1+∈ y Its internal atomic density function is shown in equation (14):

[0097]

[0098] Step 4: Simulation Calculation and Result Export: Calculate the crystal evolution process under various forging loading methods and export the results, thereby obtaining the crystal evolution process under different forging loading methods;

[0099] Step 5: Compare and analyze the crystal evolution process under different forging loading methods obtained in Step 4, and select the optimal forging loading method from the perspective of fine grain strengthening.

[0100] Appendix Figure 4 To be continued Figure 6 This represents the evolution of polycrystalline crystals under different loading conditions. Specifically, in this embodiment, stress is applied sequentially in the Y, YX, and YXY directions under dynamic uniaxial strain loading conditions, with a temperature r = 0.2. As the time step gradually increases from 3100 to 7100, (i.e., as time progresses), the grain size gradually decreases.

[0101] By analyzing the appendix Figure 4 To be continued Figure 6 It can be seen that unidirectional (Y-axis direction) loading results in 82 grains; bidirectional (YX-axis direction) loading results in 131 grains; and tridirectional (YXY-axis direction) loading results in 140 grains. From the perspective of grain refinement, multidirectional (YXY-axis direction) loading is the optimal forging method to achieve the best grain refinement and strengthening effect.

[0102] The above description represents the preferred embodiments of the present invention. It should be noted that those skilled in the art can make various improvements and modifications without departing from the principles of the present invention, and these improvements and modifications are also considered to be within the scope of protection of the present invention.

Claims

1. A method for selecting the optimal forging loading mode from the perspective of grain refinement strengthening based on the crystal phase field method, characterized in that: The method includes the following steps: Step 1: Establish a two-mode crystal phase-field model, which includes the Helmholtz free energy equation and the Cahn-Hillard kinetic equation; wherein, the Helmholtz free energy equation is a dimensionless free energy functional equation that incorporates a periodic atomic density function. Step 2: Determine the numerical method: Use the semi-implicit Fourier spectral method to spatially discretize the constructed Cahn-Hillard dynamic equations to obtain the evolution form of the atomic density field at the next time step; Step 3: Set initial condition parameters according to the various forging loading methods to be compared; the forging loading method is a two-dimensional system loading method, and adopts the assumption of equal area conditions; Step 4: Simulation Calculation and Result Export: Calculate the crystal evolution process under various forging loading methods and export the results, thereby obtaining the crystal evolution process under different forging loading methods; Step 5: Compare and analyze the crystal evolution process under different forging loading methods obtained in Step 4, and select the optimal forging loading method from the perspective of fine grain strengthening; The establishment of the two-mode crystal phase-field model in step one includes the following: The two-mode crystal phase-field model is constructed based on the traditional phase-field method. The periodic atomic density function is introduced into the Helmholtz free energy equation of the two-mode crystal phase-field model. The order parameter adopts the periodic local time-averaged atomic density. The free energy equation is as shown in equation (1): （1） In the formula, It is the position vector of the atom. Represents atomic density, It is a parameter used to control the relative magnitudes of the two reciprocal lattice vector amplitudes; For the Laplace operator; and These are the magnitudes of the primary and secondary mode wave vectors, respectively. , , and These represent the reciprocal lattice vectors of the nearest and second nearest neighbors, respectively; They are all constants; Substituting equations (2) and (3) into equation (1) allows for scaling of the parameters in the two-mode crystal phase-field model, resulting in the dimensionless free energy functional equation as shown in equation (4). （2） （3） （4） Simplifying equation (4), we obtain the dimensionless free energy functional equation as shown in equation (5): （5） In the formula, F is the dimensionless free energy function; This represents dimensionless atomic density; Characterizing the temperature of the system, it is a negative value with a small absolute value. Corresponding to higher temperatures; For the Laplace operator; For parameters related to the relative stability of different crystal structures, when As the value approaches 0, the model tends to become a single-mode model; It is a constant, which is the ratio of the magnitude of the second nearest neighbor reciprocal vector to the magnitude of the nearest neighbor reciprocal vector; Due to the atomic density field in the crystal phase field model It is a conserved order parameter, and the evolution of the system obeys the Cahn-Hillard dynamic equation of the conserved field. After dimensionless transformation, the dynamic equation is as shown in equation (6): （6） In the formula, F is the dimensionless free energy function, which is calculated from equation (5); It is a time variable; The determination of the numerical method in step two is achieved through the following steps: The dimensionless dynamic equation (6) is solved using a semi-implicit Fourier spectrum algorithm, and the space is discretized by Fourier transform to obtain the evolution form of the atomic density field at the next moment as shown in equation (7). （7） In the formula, Indicates in The atomic density in Fourier space at time t; Let represent a vector in Fourier space, and satisfy . .

2. The method for selecting the optimal forging loading mode from the perspective of grain refinement strengthening based on the crystal phase field method according to claim 1, characterized in that: In step three, the following assumptions are made when setting the initial condition parameters: Under strain loading, the grains will deform, and the deformation simulation of the two-dimensional system adopts the assumption of equal area conditions, as detailed below: The assumption of equal area is given in equation (8): （8） In the formula, and Let the initial spatial step size be , and The spatial step size after deformation; When stretching along the y-axis, it is assumed that the spatial step size in the y-direction has an increment at each step. ,and ,in The strain loading rate is dimensionless, and ; After stretching along the y-axis for n steps, The spatial step size in the direction is calculated using equation (9). The spatial step size in the direction is calculated using equation (10): （9） （10） When stretched along the x-axis, assuming The spatial step size in the direction has an increment at each step. ,and ,in The strain loading rate is dimensionless, and ; When stretching along the x-axis for n steps, The spatial step size in the direction is calculated using equation (11). The spatial step size in the direction is calculated using equation (12): （11） （12）。

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