A crystal plasticity-cellular automaton coupled multiscale framework for simulating dynamic recrystallization behavior of nickel-based superalloys

Abstract

The application provides a crystal plasticity-cellular automaton coupling multiscale framework for simulating dynamic recrystallization behavior of nickel-based superalloy, which is used for accurately predicting microstructure-mechanical coupling evolution of dynamic recrystallization of the nickel-based superalloy, and overcomes the fundamental incompatibility between the state transition of CA mutation and the continuous field variable of CPFE by introducing a continuous transformation fraction, and steps include: based on experimental data and statistical analysis, establishing a representative volume element (RVE), at each increment step k, solving a balanced mechanical field according to a boundary condition in a CPFE module, mapping key variables to a CA module, updating recrystallization cell state variables and feeding back to the CPFE module after nucleation and grain boundary migration judgment in the CA module, and iterating until the deformation is finished. The framework can accurately predict grain structure evolution, stress-strain response, dislocation density and texture evolution, and is suitable for nickel-based superalloy, titanium alloy and the like, and provides support for microstructure design and performance control of metal hot working.

Description

Technical Field

[0001] This invention relates to the field of metal plastic forming, and in particular to a crystal plasticity-cellular automata coupled multiscale framework for simulating discontinuous dynamic recrystallization. Background Technology

[0002] Grain refinement is one of the core objectives of microstructure control in metallic materials. Based on the Hall-Page relationship, refined grains can effectively improve key mechanical properties such as strength. Dynamic recrystallization (DRX) is a core process for achieving a match between the target grain structure and excellent mechanical properties during hot working. In polycrystalline metals, DRX is mainly achieved through three mechanisms: continuous dynamic recrystallization (CDRX), geometrical dynamic recrystallization (GDRX), and discontinuous dynamic recrystallization (DDRX). Among them, the nucleation sites of DDRX are concentrated at grain boundaries (GBs), and grain evolution is completed through large-angle grain boundary migration. This mechanism is the dominant dynamic recrystallization mode in low stacking fault energy polycrystalline alloys (such as steel and nickel-based superalloys).

[0003] Achieving precise control of DRX requires elucidating the intrinsic relationship between mechanical behavior and microstructure evolution, as well as reliably calculating and predicting DRX dynamics. Existing DRX simulation methods, such as Monte Carlo methods, phase-field methods, and cellular automata (CA), are mostly limited to isotropic deformation scenarios and cannot accurately characterize the effects of non-uniform deformation at the grain scale and crystallographic anisotropy on DRX dynamics and microstructure evolution.

[0004] Crystal plasticity is an efficient tool for characterizing grain-level mechanical behavior. Coupled with a microstructure evolution model, it can simulate the DRX process at the scale of polycrystalline aggregates and fully incorporate key factors such as non-uniform grain deformation and crystallographic orientation. This coupling method can establish the relationship between microstructure evolution dominated by crystallographic mechanisms and macroscopic stress-strain field, providing core support for analyzing the intrinsic relationship between mechanical behavior and DRX.

[0005] However, existing coupling schemes between Crystal Plasticity Finite Element (CPFE) and Calibration (CA) still have core defects: First, the discretization characteristics of CA make it inherently incompatible with the continuous field properties of continuum mechanics when characterizing interface states; Second, existing cross-scale state variable bidirectional mapping coupling schemes often fail to achieve synchronous updates of the mechanical field in CPFE and the microstructure field in CA, resulting in the inability to achieve precise coupling at the physical level between the influence of deformation history on recrystallization and the feedback effect of recrystallization on material rheological stress. This physical distortion of the bidirectional mapping scheme severely restricts the model's ability to predict the dynamic recrystallization behavior of nickel-based superalloys. Summary of the Invention

[0006] The purpose of this invention is to address the shortcomings of the aforementioned background technology by developing a novel multi-scale framework. This framework combines the continuous-field CA model with the CPFE model through an enhanced bidirectional mapping scheme, while introducing a continuous-field description (transformation fraction χ). It embeds the physically continuous microstructure evolution mechanism into the CPFE integration points, ensuring numerical stability and seamlessly coupling microstructure evolution with macroscopic mechanical response. This enables accurate prediction of grain structure evolution, stress-strain response, dislocation density, and crystallographic texture development.

[0007] To achieve the above objectives, the present invention provides a multi-scale framework for continuous field-based crystal plasticity-cellular automata coupling in dynamic recrystallization, comprising the following steps:

[0008] S1. Based on EBSD experimental data and statistical analysis, the initial grain structure and orientation distribution were obtained, and a representative volume element (RVE) was established in Abaqus.

[0009] S2, at increment step k, the equilibrium mechanical field is solved in the CPFE module according to the boundary conditions;

[0010] S3, at increment step k, after the CPFE module has finished solving, the necessary state variables ( , Mapped to CA unit;

[0011] S4, at increment step k, after the CPFE module finishes mapping state variables to the CA module, the number of nucleation units is calculated. And identify all potential nucleation points;

[0012] S5, at increment step k, after the CA module confirms all potential nucleation points, it iterates through all potential nucleation points and compares a random number with the nucleation energy ratio, satisfying the condition ( The element in the nucleation point is identified as the nucleation point, and the initial transformation fraction of the nucleation point is... ;

[0013] S6, at increment step k, after the nucleation point is confirmed, each unit in the cell space is judged one by one. If it is adjacent to a fully recrystallized grain boundary ( If the unit is a neighboring grain boundary, then the migration direction and migration amount of the neighboring grain boundary are calculated. The number of units that were absorbed was converted to a fraction of 10. ;

[0014] S7, After the CA module finishes its calculation, update the state variables of the recrystallization unit. , , , , );

[0015] S8 maps the updated cell state variables back to the CPFE integration points;

[0016] S9, return to the finite element model, repeat S2~S8, and perform the calculation for the next incremental step until the deformation ends.

[0017] Furthermore, the main parameter to be solved in S2 is the stress tensor. Crystal rotation matrix Shear deformation rate of slip system Slip system decomposition shear resistance Dislocation density of slip systems .

[0018] slip rate It can be represented as:

[0019] (1)

[0020] in, This represents the density of movable dislocations. It's the frequency of attempts. It is the Helmholtz free energy barrier. It is the magnitude of the Burgers vector. It is Boltzmann's constant. It is the activation volume. It is a decomposition of shear stress. It is the critical decomposition shear stress (CRSS).

[0021] Dislocation density evolution rate Represented as:

[0022] (2)

[0023] in, This is the proliferation coefficient. The thermally activated annihilation coefficient, It is the dislocation density.

[0024] Critical shear stress (CRSS) includes three different enhancement mechanisms, which can be represented as:

[0025] (3)

[0026] in, The first term represents intrinsic lattice resistance, the second term represents Taylor hardening, and the third term corresponds to Hall-Page strengthening. It is the Taylor factor. Represents shear modulus, It is the Hall-Page coefficient. This represents the maximum distance a dislocation can slip in a single crystal.

[0027] Furthermore, in S4, at grain boundaries within representative volumetric units (RVEs), the nucleation rate depends on thermomechanical conditions ( and ):

[0028] (4)

[0029] in, It is the nucleation rate coefficient, and m is the rate sensitivity index. It is the activation energy for crystal nucleus formation. In the time increment ( The number of newly formed crystal nuclei within the RVE is calculated as follows:

[0030] (5)

[0031] in, It is the grain boundary area of ​​each cell. It represents the number of cells at the grain boundary.

[0032] The driving energy and critical nucleation energy of DDRX nucleation are defined as follows:

[0033] (6)

[0034] in, This represents the critical dislocation density.

[0035] The nucleation probability of the unit ( ) is defined as:

[0036] (7)

[0037] in, For the storage energy of the cell, The maximum stored energy of a potential nucleation point, if a certain unit and When this occurs, the cell will be considered a potential nucleation point, thereby determining all potential nucleation points.

[0038] Furthermore, in S5, all potential nucleation points are traversed, and a random number is generated simultaneously. If the conditions are met Then the potential nucleation point undergoes nucleation (assigned an initial state). (as shown in equation (16)) until the condition is met. Quantity requirements.

[0039] Furthermore, the velocity (v) of GBM in S6 is given by the product of M and the driving force (P):

[0040] (8)

[0041] The driving force P consists of two components:

[0042] (9)

[0043] The first term reflects the difference in storage energy related to dislocation density at the grain boundary, while the second term is the resistance pressure caused by the grain boundary curvature. It is a dislocation line energy, It is the dislocation density difference. It is grain boundary energy. It is local curvature.

[0044] To accurately calculate the impact of grain boundary curvature on interface migration in continuous-field cellular automata models, this invention proposes a robust and mesh-independent local interface curvature estimation method: a local surface fitting method based on the transition state of adjacent cells, utilizing field values ​​from a 3×3 integration point mesh to accurately calculate the curvature at the center point.

[0045] Consider a local region containing nine integration points, numbered P1 to P9 from left to right and top to bottom, where point P5 is the central integration point traversing GBM. Each integration point P... i They all have global coordinates and a GBM status value .here, Represents the recrystallized grains that have migrated, while This represents the grains that are being consumed.

[0046] First, establish a local coordinate system. Its origin is located at P5. The relationship between local and global coordinates is as follows:

[0047] (10)

[0048] In this local coordinate system, the distribution of GBM state values ​​λ can be approximated by a quadratic surface:

[0049] (11)

[0050] Among them, a, h, c, g, e, and f are obtained by considering the nine points. The coefficients are determined by least-squares fitting. The center point is P5 (where...). curvature at ) The following can be derived from the second derivative of the fitted surface:

[0051] (12)

[0052] in, It is a regularization parameter used to avoid numerical instability when the gradient is very small. Furthermore, if This region is considered to have near-zero curvature. ).

[0053] Transition probability The definition is similar to the nucleation criterion:

[0054] (13)

[0055] Where, N n This represents the number of neighboring cells that have undergone transformation, where N is the number of neighboring cells. A unit cell at a grain boundary is randomly selected when that unit cell satisfies the driving force... At that time, generate a random number. If the conditions are met Then the cell grows driven by DRX grain boundary migration.

[0056] Furthermore, in S7, to ensure consistency between mechanics and thermodynamics, the following assumptions are defined: 1) Nucleation-induced phase transition: new nuclei are strain-free and exhibit random crystallographic orientation; 2) Grain boundary migration-induced phase transition: regions consumed by recrystallized grains inherit the crystal orientation and elastic deformation state of the intruding grains; 3) State reset: the plastic slip and accumulated dislocations of fully recrystallized materials are restored to their initial state, and their lattice is restored.

[0057] Furthermore, to overcome the fundamental incompatibility between the binary state transition rule (0→1 abrupt change) of the CA module and the continuous field variables of the CPFE, a continuous field description is introduced at the sub-mesh scale of the finite element undergoing recrystallization. The core is to define a continuous transition fraction (χ), which evolves continuously from 0 to 1 within the element, representing the asymptotic process from the deformed state to the fully recrystallized state.

[0058] For an element undergoing nucleation, once a stable nucleus appears, a virtual circular region is constructed, the diameter of which is equal to the current equivalent side length of the finite element. The transformation fraction is defined as the ratio of the nucleus diameter to the element's equivalent edge length.

[0059] (14)

[0060] in, The radius of the crystal nucleus is denoted as .

[0061] For a cell that has undergone GBM, a rectangular region is defined along the migration direction, with a length of [missing information]. The transformation score is defined as the transfer distance and The ratio:

[0062] (15)

[0063] in, This represents the migration distance.

[0064] By iterating through the core cells sequentially, the cells are divided into the following three categories:

[0065] (1) The unit is in the initial state :

[0066] Critical nucleation radius It is given by the following formula:

[0067] (16)

[0068] in, It is the dislocation density difference between the crystal nucleus and the surrounding matrix. It's grain boundary energy. The new crystal nucleus's... The initial value is:

[0069] (17)

[0070] in, It is the area per unit.

[0071] The state variable is then updated as follows:

[0072] (18)

[0073] Where NCL represents nucleation, and the superscript indicates nucleation. This indicates a material point that is undergoing transformation. This represents the weighted average orientation function. and These represent the orientations of the distorted and recrystallized portions, respectively. This refers to the tensile portion in elastic deformation. It is the velocity gradient of elastic deformation recovery. The increment step size for the nucleation process. It is a reference deformation gradient transformation used to maintain configuration compatibility, by Decomposition yields: Any internal state variable of a material point The volume average is calculated as a volume-weighted average of the crystalline and distorted portions;

[0074] (2) If the unit satisfies If the nucleation transition state is reached, the evolution process can be represented as follows:

[0075] (19)

[0076] in, This represents the growth rate of the crystal nucleus.

[0077] Update conversion score (Simultaneous equations (14) and (19)) and state variables:

[0078] (20)

[0079] (3) If the unit satisfies Then the state variable will be updated to the fully recrystallized state:

[0080] (twenty one)

[0081] Among them, superscript This indicates a fully recrystallized state. After DRX is completed, the internal state variables are reset, representing the formation of new defect-free grains. Therefore, the internal variables... Accumulated plastic slip ( ) and dislocation density ( Defined as:

[0082] (twenty two)

[0083] in, This represents the dislocation density at a point in a fully recrystallized material.

[0084] By iterating through the cells of GBM sequentially, the cells are divided into the following two categories:

[0085] (1) If the unit satisfies If the unit is in a grain boundary migration transition state, the evolution process can be represented as:

[0086] (twenty three)

[0087] in, The migration rate represents the inner boundary.

[0088] Update conversion score (Simultaneous equations (15) and (23)) and state variables:

[0089] (twenty four)

[0090] Where Nei refers to adjacent units;

[0091] (2) If the unit Then update the state variable to the fully recrystallized state:

[0092] (25)

[0093] The above-described solution of the present invention has the following beneficial effects:

[0094] A coupled framework of crystal plasticity finite element-cellular automata (CPFE-CA) with good physical consistency and excellent numerical stability was established. The model describes the nucleation and grain boundary migration processes as the evolution of a continuous transformation fraction χ, realizing a gradual characterization of the recrystallization process and avoiding the state abrupt change problem inherent in traditional discrete cellular automata models. This improvement significantly enhances the numerical stability and physical realism of cross-scale simulations.

[0095] This model achieves high-precision prediction of macroscopic mechanical behavior and accurately predicts key multi-scale microstructure evolution characteristics. It not only accurately simulates the evolution of dynamic recrystallization volume fraction and average grain size, but also accurately predicts dislocation density distribution, grain morphology evolution, and texture evolution. In particular, the simulation results for the initial formation of the necklace-like recrystallization structure and subsequent grain growth process are in high agreement with experimental observations.

[0096] This study reveals the coupling mechanism between dynamic recrystallization and non-uniform deformation. Through analysis of the evolution of microscopic field variables, it was found that the recrystallization region becomes a local softening zone, which can significantly release elastic strain energy and plastic strain energy. This local energy release, in turn, has a reaction effect on the dynamic recrystallization process, reducing the driving force for subsequent nucleation and grain boundary migration, thus forming a dynamic coupling feedback system.

[0097] The continuous field CPFE-CA coupling framework established in this invention not only provides a reliable multi-scale tool for simulating the dynamic discontinuous recrystallization behavior of nickel-based superalloys, but can also be extended to other polycrystalline metal materials with dynamic discontinuous recrystallization as the main softening mechanism, providing theoretical basis and simulation support for microstructure design and performance control in hot working processes. Attached Figure Description

[0098] Figure 1 This is a flowchart illustrating the calculation of the bidirectional mapping between CPFE and CA based on continuous transformation fractions in an embodiment of the present invention.

[0099] Figure 2 This is a method for calculating the local curvature of grain boundary migration in an embodiment of the present invention;

[0100] Figure 3 The following is a schematic diagram of the core mechanism of the crystal plasticity-cellular automaton multi-scale coupling model in an embodiment of the present invention: (a) is the multi-scale mapping strategy between crystal plasticity and cellular automaton modules, (b) is the subgrid nucleation process, and (c) is the subgrid grain boundary migration process.

[0101] Figure 4 This is a schematic diagram of a multi-scale scheme for intra-unit state transitions during dynamic recrystallization in an embodiment of the present invention;

[0102] Figure 5This is a graph evaluating the prediction accuracy of the true stress-strain curve in an embodiment of the present invention.

[0103] Figure 6 Correlation analysis of experimental and simulation results in the embodiments of the present invention: where (a, c) are the volume fraction of dynamic recrystallization, and (b, d) are the average grain size;

[0104] Figure 7 In an embodiment of the present invention, increasing true strain and Distribution;

[0105] Figure 8 The simulation results of grain structure and dislocation density in the embodiments of the present invention are shown.

[0106] Figure 9 This is a flowchart of the method steps of the present invention. Detailed Implementation

[0107] The following detailed implementation examples illustrate the technical framework of this invention. Those skilled in the art can clearly grasp the core advantages and practical application value of this invention in simulating dynamic recrystallization behavior through the content disclosed in this specification. Obviously, the described embodiments are only some implementation scenarios of this invention, not all implementation forms. The technical solutions of this invention can also be implemented or expanded through other different specific embodiments, and the various technical details in this specification can also be adjusted or optimized based on different technical perspectives and application scenarios, without departing from the core inventive concept of this disclosure. Based on the embodiments provided by this invention, all other embodiments obtained by those skilled in the art without creative effort are within the protection scope of this disclosure.

[0108] It should be noted that the illustrations provided in the following embodiments are merely schematic representations of the core principles of the present invention, and only components directly related to the technical solutions disclosed herein are labeled. Furthermore, the provision of specific process parameters, model parameters, and other details in the following description is to facilitate a thorough understanding of the implementation examples by those skilled in the art. However, those skilled in the art should understand that even without these specific details, the technical concept of the present invention can be implemented based on the core technical solutions disclosed herein.

[0109] like Figure 1As shown, this invention proposes a crystal plasticity-cellular automata coupled multi-scale framework for simulating the dynamic recrystallization behavior of nickel-based superalloys. Based on the Abaqus software platform, a user-defined material subroutine was developed. S2 to S8 are all implemented through the user-defined material subroutine, where S2 to S8 represent the specific calculation content of a time increment step. After the calculation of all time increment steps is completed, the calculation of the dynamic recrystallization evolution process is finished. Taking its application in the discontinuous dynamic recrystallization of nickel-based superalloy (FGH4113A) as an example, the relevant parameter calibration values ​​are shown in Table 1.

[0110] Table 1 Parameter Calibration Values

[0111] parameter unit Calibration value elastic parameters <![CDATA[C 11 ]]> (MPa) <![CDATA[5.00×10 3 ]]> <![CDATA[C 12 ]]> (MPa) <![CDATA[3.23×10 3 ]]> <![CDATA[C 44 ]]> (MPa) <![CDATA[1.95×10 3 ]]> Intrinsic parameters <![CDATA[ρ α 0]]> <![CDATA[(μm -2 )]]> <![CDATA[5.60×10 -2 ]]> <![CDATA[τ α c0 ]]> (MPa) 62.12 Hardening parameters <![CDATA[k1]]> (1 / μm) <![CDATA[4.10×10 -1 ]]> ξ (--) <![CDATA[8.64×10 -1 ]]> <![CDATA[g α ]]> (--) <![CDATA[1.47×10 7 ]]> <![CDATA[D α ]]> (MPa) <![CDATA[6.57×10 2 ]]> <![CDATA[K HP ]]> (MPa) <![CDATA[6.33×10 2 ]]> Slip parameters <![CDATA[ρ m ]]> <![CDATA[(μm -2 )]]> <![CDATA[1.53×10 -2 ]]> <![CDATA[ν0]]> (1 / s) <![CDATA[1.01×10 11 ]]> ΔF (J) <![CDATA[4.93×10 -20 ]]> ΔV <![CDATA[(μm 3 )]]> 0.99 DRX related parameters <![CDATA[C n ]]> (--) <![CDATA[2.54×10 -1 ]]> <![CDATA[Q act ]]> (J / mol) <![CDATA[2.15×10 3 ]]> <![CDATA[δD 0b ]]> <![CDATA[(μm 3 / s)]]> <![CDATA[7.51×10 -26 ]]> <![CDATA[Q gb ]]> (J / mol) <![CDATA[9.52×10 3 ]]> <![CDATA[Q b ]]> (J / mol) <![CDATA[1.19×10 2 ]]>

[0112] The specific implementation steps of this embodiment include:

[0113] S1. Based on EBSD experimental data and statistical analysis, the initial grain structure and orientation distribution were obtained, and a representative volume element (RVE) was established in Abaqus.

[0114] A quasi-three-dimensional representative volume element (RVE) containing 5 grains (size: 9μm×9μm×0.9μm) was constructed and subjected to isothermal (1383K) compression tests under periodic boundary conditions with a constant strain rate of 0.1s⁻¹, and uniaxially compressed to the true strain levels (A) 0.22, (B) 0.36, (C) 0.51, (D) 0.69 and (E) 0.92.

[0115] S2, at increment step k, the equilibrium mechanical field is solved in the CPFE module based on the boundary conditions:

[0116] The main parameter to be solved is the stress tensor. Crystal rotation matrix Shear deformation rate of slip system Slip system decomposition shear resistance Dislocation density of slip systems .

[0117] Among them, slip rate It can be represented as:

[0118] (1)

[0119] in, This represents the density of movable dislocations. To test the frequency, For Helmholtz free energy base, The Burgers vector (the Burgers vector of nickel-based superalloys is...) ), It is Boltzmann's constant. It is the activation volume. It is a decomposition of shear stress. It is the critical decomposition shear stress (CRSS).

[0120] Dislocation density evolution rate Represented as:

[0121] (2)

[0122] in, This is the proliferation coefficient. The thermally activated annihilation coefficient, It is the dislocation density.

[0123] Critical shear stress (CRSS) includes three different enhancement mechanisms, which can be represented as:

[0124] (3)

[0125] in, The first term represents intrinsic lattice resistance, the second term represents Taylor hardening, and the third term corresponds to Hall-Page strengthening. Taylor factor (for nickel-based superalloys) ), This represents the shear modulus (80 GPa). The Hall-Page coefficient, This represents the maximum distance a dislocation can slip in a single crystal.

[0126] S3, at increment step k, after the CPFE module has finished solving, the necessary state variables ( , These variables are mapped to CA units, and they provide physical drivers for the kernel and GBM rules within the CA module.

[0127] S4, at increment step k, after the CPFE module finishes mapping state variables to the CA module, the number of nucleation units is calculated. And identify all potential nucleation points:

[0128] At grain boundaries within a representative volume unit (RVE), the nucleation rate depends on thermomechanical conditions ( and ):

[0129] (4)

[0130] in, It is the nucleation rate coefficient, and m is the rate sensitivity index. It is the activation energy for crystal nucleus formation. In the time increment ( The number of newly formed crystal nuclei within the RVE is calculated as follows:

[0131] (5)

[0132] in, It is the grain boundary area of ​​each cell. It represents the number of cells at the grain boundary.

[0133] The driving energy and critical nucleation energy of DDRX nucleation are defined as follows:

[0134] (6)

[0135] in, This represents the critical dislocation density.

[0136] The nucleation probability of the unit ( ) is defined as:

[0137] (7)

[0138] in, For the storage energy of the cell, The maximum stored energy of a potential nucleation point, if a certain unit and When this occurs, the cell will be considered a potential nucleation point, thereby confirming all potential nucleation points.

[0139] S5, at increment step k, after the CA module confirms all potential nucleation points, it iterates through all potential nucleation points and generates a random number. If the conditions are met Then the potential nucleation point undergoes nucleation (assigned an initial state). (as shown in equation (16)) until the condition is met. Quantity requirements.

[0140] S6, at increment step k, after the nucleation point is confirmed, each unit in the cell space is judged one by one. If it is adjacent to a fully recrystallized grain boundary ( If the unit is a neighboring grain boundary, then the migration direction and migration amount of the neighboring grain boundary are calculated. The number of units that were absorbed was converted to a fraction of 10. :

[0141] The velocity (v) of GBM is given by the product of M and the driving force (P):

[0142] (8)

[0143] The driving force P consists of two components:

[0144] (9)

[0145] The first term reflects the difference in storage energy related to dislocation density at the grain boundary, while the second term is the resistance pressure caused by the grain boundary curvature. It is a dislocation line energy, It is the dislocation density difference. It is grain boundary energy. It is local curvature.

[0146] To accurately calculate the impact of grain boundary curvature on interface migration in continuous-field cellular automata models, this invention proposes a robust and mesh-independent local interface curvature estimation method: a local surface fitting method based on the transition states of adjacent cells, such as... Figure 2 As shown, the curvature at the center point is accurately calculated using field values ​​from a 3×3 integration point grid:

[0147] Consider a local region containing nine integration points, numbered P1 to P9 from left to right and top to bottom, where point P5 is the central integration point traversing GBM. Each integration point P... i They all have global coordinates and a GBM status value .here, Represents the recrystallized grains that have migrated, while This represents the grains that are being consumed.

[0148] First, establish a local coordinate system. Its origin is located at P5. The relationship between local and global coordinates is as follows:

[0149] (10)

[0150] In this local coordinate system, the distribution of GBM state values ​​λ can be approximated by a quadratic surface:

[0151] (11)

[0152] Among them, a, h, c, g, e, and f are obtained by considering the nine points. The coefficients are determined by least-squares fitting. The center point is P5 (where...). curvature at ) The following can be derived from the second derivative of the fitted surface:

[0153] (12)

[0154] in, It is a regularization parameter used to avoid numerical instability when the gradient is very small. Furthermore, if This region is considered to have near-zero curvature. ).

[0155] Transition probability The definition is similar to the nucleation criterion:

[0156] (13)

[0157] Where, N n This represents the number of neighboring cells that have undergone transformation, where N is the number of neighboring cells. A unit cell at a grain boundary is randomly selected when that unit cell satisfies the driving force... At that time, generate a random number. If the conditions are met Then the cell grows driven by DRX grain boundary migration.

[0158] S7, After the CA module finishes its calculation, update the state variables of the recrystallization unit. , , , , ):

[0159] To ensure consistency between mechanics and thermodynamics, the following assumptions are defined: 1) Nucleation-induced phase transition: new nuclei are strain-free and exhibit random crystallographic orientations; 2) Grain boundary migration-induced phase transition: regions consumed by recrystallized grains inherit the orientation and elasticity of the intruding grains; 3) State reset: fully recrystallized materials have no plastic slip and accumulated dislocations, and their lattice is restored.

[0160] Figure 3 The diagram shows a multi-scale mapping strategy, which includes sub-grid nucleation and grain boundary migration processes.

[0161] To overcome the fundamental incompatibility between the binary state transition rule (0→1 abrupt change) of the CA module and the continuous field variables of the CPFE, a continuous field description is introduced at the sub-mesh scale of the finite element undergoing recrystallization. The core is to define a continuous transition fraction (χ), which evolves continuously from 0 to 1 within the element, representing the asymptotic process from the deformed state to the fully recrystallized state.

[0162] For elements undergoing nucleation, once a stable nucleus appears, a virtual circular region is constructed, such as... Figure 4 As shown, its diameter is equal to the current equivalent side length of the finite element ( The transformation fraction is defined as the ratio of the nucleus diameter to the element's equivalent edge length.

[0163] (14)

[0164] in, The radius of the crystal nucleus is denoted as .

[0165] For a cell that has undergone GBM, a rectangular region is defined along the migration direction, with a length of [missing information]. The transformation score is defined as the transfer distance and The ratio:

[0166] (15)

[0167] in, This represents the migration distance.

[0168] By iterating through the core cells sequentially, the cells are divided into the following three categories:

[0169] (1) The unit is in the initial state :

[0170] Critical nucleation radius It is given by the following formula:

[0171] (16)

[0172] in, It is the dislocation density difference between the crystal nucleus and the surrounding matrix. It's grain boundary energy. The new crystal nucleus's... The initial value is:

[0173] (17)

[0174] in, It is the area per unit.

[0175] The state variable is then updated as follows:

[0176] (18)

[0177] Where NCL represents nucleation, and the superscript indicates nucleation. This indicates a material point that is undergoing transformation. This represents the weighted average orientation function. and These represent the orientations of the crystalline and recrystallized portions, respectively. This refers to the tensile portion in elastic deformation. It is the velocity gradient of elastic deformation recovery. The increment step size for the nucleation process. It is a reference deformation gradient transformation used to maintain configuration compatibility, by Decomposition yields: Any internal state variable of a material point The volume average is calculated as a volume-weighted average of the crystalline and distorted portions;

[0178] (2) If the unit satisfies If the nucleation transition state is reached, the evolution process can be represented as follows:

[0179] (19)

[0180] in, This represents the growth rate of the crystal nucleus.

[0181] Update conversion score (Simultaneous equations (14) and (19)) and state variables:

[0182] (20)

[0183] (3) If the unit satisfies Then the state variable will be updated to the fully recrystallized state:

[0184] (twenty one)

[0185] Among them, superscript This indicates a fully recrystallized state. After DRX is completed, the internal state variables are reset, representing the formation of new defect-free grains. Therefore, the internal variables... Accumulated plastic slip ( ) and dislocation density ( Defined as:

[0186] (twenty two)

[0187] in, This represents the dislocation density at a point in a fully recrystallized material.

[0188] By iterating through the cells of GBM sequentially, the cells are divided into the following two categories:

[0189] (1) If the unit satisfies If the unit is in a grain boundary migration transition state, the evolution process can be represented as:

[0190] (twenty three)

[0191] in, The migration rate represents the inner boundary.

[0192] Update conversion score (Simultaneous equations (15) and (23)) and state variables:

[0193] (twenty four)

[0194] Where Nei refers to adjacent units;

[0195] (2) If the unit Then update the state variable to the fully recrystallized state:

[0196] (25)

[0197] S8 maps the updated element state back to the CPFE integral point. This information is used to update the constitutive state of the material point, inducing stress relaxation and material softening.

[0198] S9, return to the finite element model, repeat S2~S8, and perform the calculation for the next incremental step until the deformation ends.

[0199] Through this embodiment, the following prediction results can be obtained:

[0200] Figure 5 , Figure 6 The predicted stress-strain curves, dynamic recrystallization fraction, and average grain size shown are in good agreement with the experimental curves. The R-value and AARE of the stress prediction results are 0.9923 and 2.26%, respectively, while the R-value and AARE of the XDRX results are 0.9870 and 10.34%, respectively. The predicted results showed R=0.9882 and AARE=4.49%, which confirms that the proposed CPFE-CA framework can accurately predict the coupled evolution of microstructure and mechanical properties of FGH4113A nickel-based superalloy during dynamic recrystallization.

[0201] Figure 7 Demonstrating different intermediate strain levels and The spatial distribution of [something] in regions where dynamic recrystallization occurs, whether through nucleation or grain boundary migration, [is affected]. and Both stresses and plastic strains are significantly reduced, which can explain the softening effect of dynamic recrystallization: during dynamic recrystallization, the distorted lattice structure is reorganized into a crystalline state with reduced defects. In contrast, the non-recrystallized region retains higher levels of stress and plastic strain. The defect-rich distorted structure (including grain boundaries, dislocations, and vacancies) is replaced by the lattice, resulting in a reduction in cumulative plastic deformation. As new nuclei form or existing grains migrate and grow through grain boundaries, the elastic contraction of the lattice is alleviated, thereby releasing elastic strain and correspondingly reducing stress. .

[0202] During thermomechanical deformation, driven by nucleation and grain boundary migration, dynamic recrystallization significantly alters the microstructure. This is demonstrated by the dynamic recrystallization diagrams of experimental alloys under different true strains, such as... Figure 8As shown, the simulation results describe the evolution of grain structure and the spatial variation of grain dislocation density with increasing macroscopic strain. In the early stage of deformation, nuclei mainly form at the initial grain boundaries and gradually consume the deformed structure. As deformation progresses, the adjacency between recrystallized grains increases, leading to significant mutual consumption and the disappearance of the necklace-like microstructure. Finally, driven by continuous grain boundary migration, the deformed structure further decreases, the size of recrystallized grains increases, and grain boundaries migrate simultaneously to adjacent dynamically recrystallized grains with high energy density and the remaining deformed region. Before reaching the critical strain for the initiation of dynamic recrystallization, the dislocation density within the grains gradually increases. Once nucleation occurs at the grain boundaries, or the region is invaded by migrating grain boundaries of dynamically recrystallized grains, the dislocation density in these regions decreases significantly. The significantly lower dislocation density and minimal lattice distortion of the recrystallized grains inhibit the rapid accumulation of dislocations and enable the material to reharden during subsequent deformation.

[0203] The embodiments described above in this application only illustrate some implementation forms. Although the related descriptions are detailed, they do not constitute a limitation on the scope of protection of this application. It should be noted that those skilled in the art can make several modifications and optimizations without departing from the core inventive concept of this application, and such modifications all fall within the scope of protection of this application. Therefore, the scope of protection of this application should be based on the content of the appended claims.

Claims

1. A crystal plasticity-cellular automata coupled multiscale framework for simulating the dynamic recrystallization behavior of nickel-based superalloys, characterized in that, The process includes the following steps: S1, obtaining the initial grain structure and orientation distribution based on EBSD experimental data and statistical analysis, and establishing a representative volume element (RVE) in Abaqus; S2, at increment step k, solving the equilibrium mechanical field in the CPFE module according to the boundary conditions; S3, at increment step k, after the CPFE module has finished solving, transferring the necessary state variables ( , Mapping to CA units; S4, at incremental step k, after the CPFE module finishes mapping state variables to the CA module, the number of nucleation units is calculated ( ) and confirm all potential nucleation points; S5, at increment step k, after the CA module confirms all potential nucleation points, it iterates through all potential nucleation points and compares a random number with the nucleation energy ratio, satisfying the condition ( The element in the nucleation point is identified as the nucleation point, and the initial transformation fraction of the nucleation point is... S6, at increment step k, after the nucleation point is confirmed, each unit in the cell space is judged one by one. If it is adjacent to a fully recrystallized grain boundary ( If the unit is a neighboring grain boundary, then the migration direction and migration amount of the neighboring grain boundary are calculated. The number of units that were absorbed was converted to a fraction of 10. S7, After the CA module finishes its calculations, update the state variables of the recrystallization unit. , , , , S8, map the updated element state variables back to the CPFE integration points; S9, return to the finite element model, repeat S2~S8, and perform the calculation for the next incremental step until the deformation ends.

2. The crystal plasticity-cellular automata coupled multiscale framework for simulating the dynamic recrystallization behavior of nickel-based superalloys as described in claim 1, characterized in that, Multiscale coupling is achieved through a bidirectional mapping scheme executed at each incremental step, (1) CPFE→CA: state variables related to recrystallization behavior (including dislocation density (ρ), crystallographic orientation ( Mapping CPFE integration points to the corresponding CA units, these variables provide the physical drivers for the kernel formation and GBM rules within the CA module (e.g., ...). (2) CA→CPFE: The microstructure evolution predicted by the CA module (mainly the updated χ and the crystallographic orientation of the recrystallization region) or The information is mapped back to the CPFE integral point, which is used to update the constitutive state of the material point, leading to stress relaxation and material softening.

3. A crystal plasticity-cellular automata coupled multiscale framework for simulating the dynamic recrystallization behavior of nickel-based superalloys as described in claim 1, characterized in that, State variable evolution process: For an element undergoing nucleation, once a stable nucleus appears, a virtual circular region is constructed, the diameter of which is equal to the current equivalent side length of the finite element. The transformation fraction is defined as the ratio of the nucleus diameter to the element's equivalent edge length. ,in, Let be the nucleus radius. For a cell undergoing GBM, a rectangular region is defined along the migration direction, with a length of . The transformation score is defined as the transfer distance and The ratio: ,in, The migration distance is calculated. The cells of the nucleus are traversed sequentially, and the cells are divided into the following three categories: (1) The cells are in the initial state. Critical nucleation radius It is given by the following formula: ,in, It is the dislocation density difference between the crystal nucleus and the surrounding matrix. It's grain boundary energy. The new crystal nucleus's... The initial value is: ,in, It is the area of ​​a single unit. Therefore, the state variable is updated as follows: Where NCL represents nucleation, and the superscript indicates nucleation. This indicates a material point that is undergoing transformation. This represents the weighted average orientation function. and These represent the orientations of the distorted and recrystallized portions, respectively. This refers to the tensile portion in elastic deformation. It is the velocity gradient of elastic deformation recovery. The increment step size for the nucleation process. It is a reference deformation gradient transformation used to maintain configuration compatibility, by Decomposition yields: Any internal state variable of a material point The volume average is calculated as the volume-weighted average of the crystalline part and the distorted part; (2) if the unit satisfies If the nucleation transition state is reached, the evolution process can be represented as follows: ,in, Represents the growth rate of the crystal nucleus. Update the transformation fraction. (Simultaneous equations (14) and (19)) and state variables: (3) If the unit satisfies Then the state variable will be updated to the fully recrystallized state: , where superscript This indicates a fully recrystallized state. After DRX is completed, the internal state variables are reset, representing the formation of new defect-free grains. Therefore, the internal variables... Accumulated plastic slip ( ) and dislocation density ( Defined as: ,in, This represents the dislocation density at a point in a fully recrystallized material. The elements that pass through GBM are sequentially traversed, and the elements are divided into the following two categories: (1) If the element satisfies... If the unit is in a grain boundary migration transition state, the evolution process can be represented as: ,in, The migration rate represents the inner boundary. Update the transition score. (Simultaneous equations (15) and (23)) and state variables: , where Nei refers to the adjacent unit; (2) if the unit Then update the state variable to the fully recrystallized state: .

4. A crystal plasticity-cellular automata coupled multiscale framework for simulating the dynamic recrystallization behavior of nickel-based superalloys as described in claim 4, characterized in that, The velocity (v) of GBM is given by the product of M and the driving force (P): The driving force P consists of two components: The first term reflects the difference in storage energy related to dislocation density at the grain boundary, while the second term is the resistance pressure caused by the grain boundary curvature. It is a dislocation line energy, It is the dislocation density difference. It is grain boundary energy. This refers to local curvature. To accurately calculate the grain boundary curvature driving force during grain boundary migration in a continuous-field cellular automata model, a robust and mesh-independent local interface curvature estimation method is proposed: a local surface fitting method based on the transition state of adjacent units, utilizing field values ​​from a 3×3 integration point grid to accurately calculate the curvature at the center point. Consider a local region containing nine integration points, numbered P1~P9 from left to right and top to bottom, where point P5 is the center integration point undergoing GBM. Each integration point P... i They all have global coordinates and a GBM status value .here, Represents the recrystallized grains that have migrated, while This represents the grains that are being consumed. First, establish a local coordinate system. Its origin is located at P5. The relationship between local and global coordinates is as follows: In this local coordinate system, the distribution of GBM state values ​​λ can be approximated by a quadratic surface: Where a, h, c, g, e, and f are obtained by considering the nine points. The coefficients are determined by least-squares fitting. The center point is P5 (where...). curvature at ) The following can be derived from the second derivative of the fitted surface: ,in, It is a regularization parameter used to avoid numerical instability when the gradient is very small. Furthermore, if This region is considered to have near-zero curvature. ).

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