CN117150813A - Phase field simulation method for predicting morphology evolution of dual-phase titanium alloy structure - Google Patents
Phase field simulation method for predicting morphology evolution of dual-phase titanium alloy structure Download PDFInfo
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Abstract
The invention relates to the technical field of metallurgical casting, in particular to a phase field simulation method for predicting the morphology evolution of a dual-phase titanium alloy structure. The invention can introduce the dissolution of alpha phase in the solid solution process and the generation of alpha phase in the cooling process into numerical simulation, and can truly reproduce the two-phase titanium alloyThe change of the structure morphology of the solid phase transition can simulate the evolution process of the microstructure more accurately, and reliable information is provided for improving and optimizing the mechanical properties of the dual-phase titanium alloy. The phase field method is used as an effective numerical simulation method, and can quantitatively study the influence of key factors such as interfacial energy of a phase interface, anisotropy, elastic strain energy and the like on the growth of the microscopic morphology.
Description
Technical Field
The invention relates to the technical field of metallurgical casting, in particular to a phase field simulation method for predicting the morphology evolution of a dual-phase titanium alloy structure.
Background
Titanium alloy is used as a traditional engineering alloy, has the advantages of low density, high strength, high temperature resistance, corrosion resistance and the like, and is widely applied to the fields of aerospace, navigation, medical treatment and the like. In order to optimize the mechanical properties of the titanium alloy, the later heat treatment system is very important besides regulating and controlling the element composition in the titanium alloy. For alpha+beta type dual-phase titanium alloy, the heat treatment generally comprises two stages of solution treatment and aging treatment, is a typical heat treatment scheme of the titanium alloy, and the solution temperature, the solution time and the cooling rate after the solution are key parameters of the heat treatment process, which can obviously influence the characteristics of phase proportion, tissue morphology and the like in the titanium alloy, further influence the mechanical properties of the alloy such as yield strength, tensile strength and the like, so that the selection of a proper heat treatment process in the processing process of the titanium alloy has important significance for regulating and controlling the microstructure and mechanical properties of the titanium alloy.
However, the conventional method of searching for a proper heat treatment process by experiment is time-consuming and labor-consuming, and is used for designing and optimizing the performance of auxiliary alloy, predicting the tissue morphology evolution of the titanium alloy at different solid solution temperatures and different cooling speeds through computer simulation, and clarifying the influence rule of the solid solution temperatures and the cooling speeds after solid solution on the tissue characteristics of alpha phase, which has important guiding significance for further improving the service performance of the titanium alloy.
Researchers have simulated the structure morphology evolution process of the dual-phase titanium alloy at different heating rates through a phase field simulation method (a phase field simulation method for predicting the structure morphology evolution and alloy element distribution of the dual-phase titanium alloy at different heating rates, application (patent) number: CN 202011257462.3) reproduces the alpha-beta transformation process in the dual-phase titanium alloy, and clarifies the structure morphology evolution law at different heating rates. However, the simulated titanium alloy transformation process is too simplified, and firstly, the existence of 12 alpha variants is not considered, and the elastic strain energy of the 12 alpha variants and the beta matrix is not considered, so that the accuracy of a simulation result is affected; secondly, the continuous cooling process after solid solution is not simulated, and the continuous cooling process after solid solution can also have great influence on the structure morphology of the titanium alloy; third, the simulation result is only two-dimensional simulation result at present, and has a certain difference from the actual three-dimensional situation.
In order to solve the problems, the invention provides a phase field simulation method for predicting the tissue morphology evolution of the dual-phase titanium alloy at different solid solution temperatures and different cooling speeds, comprehensively considers the elastic strain energy of 12 alpha variants and beta matrixes, expands to a three-dimensional condition, and can more accurately simulate the tissue morphology evolution process of the titanium alloy in the solid solution process and the subsequent continuous cooling process.
Disclosure of Invention
In order to improve and optimize the mechanical properties of the dual-phase titanium alloy, the invention provides a phase field simulation method for predicting the tissue morphology evolution of the dual-phase titanium alloy at different solid solution temperatures and different cooling speeds, and the method can reproduceThe solid state transformation process provides an effective prediction method for regulating and controlling the microstructure characteristics such as alpha/beta phase composition, size, component distribution and the like.
In order to achieve the above purpose, the technical scheme adopted by the invention is as follows:
a phase field simulation method for predicting morphology evolution of a dual-phase titanium alloy structure, the phase field simulation method comprising the steps of:
s1, determining the numbers of conservation sequence parameters and non-conservation sequence parameters used by a titanium alloy solid-state phase-change phase field model according to the alloy components of the titanium alloy and the characteristics of the titanium alloy solid-state phase change;
s2, obtaining a Gibbs free energy function expression of alpha and beta phases of the dual-phase titanium alloy system according to a Redlich-Kister-Muggianu (RKM) sub-regular solution model;
s3, connecting the Gibbs free energy function expressions of alpha and beta phases of the titanium alloy system through an interpolation function to construct a bulk chemical free energy density function of the system;
s4, fitting balance components at different temperatures through polynomials to serve as input of subsequent phase field simulation; introducing interface energy into a total chemical free energy item through a gradient item, and constructing a total chemical free energy function expression of the system;
s5, according to the crystal orientation relation ((0001) followed by beta-alpha phase transition of the titanium alloy α //{110} β Anddetermining the stress-free strain tensor of the first alpha variant, and obtaining the stress-free strain tensor of the second alpha variant, the third alpha 0 variant, the fourth alpha 1 variant, the fifth alpha variant, the sixth alpha variant, the seventh alpha variant, the eighth alpha variant, the ninth alpha variant, the tenth alpha variant, the eleventh alpha variant and the twelfth alpha variant according to certain spatial rotation symmetry operation;
s6, determining an elastic strain energy expression when the titanium alloy is subjected to beta-alpha phase transition according to the stress-free strain tensor values of the 12 alpha variants obtained in the S5; combining the elastic strain energy expression and the total chemical free energy function expression to obtain a function expression of the total energy of the system;
s7, establishing an evolution dynamics partial differential equation of the conservation-order parameters and the non-conservation-order parameters, converting the partial differential equation into Fortran language, and solving by using a computer;
s8, randomly nucleating at a specific temperature and isothermally maintaining for a period of time to obtain an initial tissue morphology before solution treatment, keeping the initial tissue morphology before solution treatment to be the same in configuration, and calculating to obtain a microstructure morphology evolution process of an alpha phase at different solution temperatures;
s9, taking the microstructure morphology after solution treatment as an initial microstructure configuration of cooling treatment, and calculating to obtain an alpha-phase microstructure morphology evolution process under different cooling speeds;
and S10, carrying out visual treatment on the microstructure morphology evolution results of the alpha phase in the S8 and the S9 to obtain the influence rules of different solid solution temperatures and different cooling speeds on the microstructure morphology evolution process of the alpha phase.
In some embodiments, the phase field simulation method is a phase field simulation method of the tissue morphology evolution of the dual-phase titanium alloy under the conditions of different solid solution temperatures and different cooling speeds.
In some embodiments, in the step S1, the conservation-order parameter is the concentration of each other alloy element except for the titanium element in the nominal component of the titanium alloy, and the number of the conservation-order parameter is N-1, where N is the total number of all elements in the titanium alloy; the number of the non-conservation-order parameters is 12, and the value of the non-conservation-order parameters is between 0 and 1.
In some embodiments, in the step S2, under the assumption of Redlich-Kister-muggeian (RKM) sub-regular solution model, the gibbs free energy function expression of α phase/β phase is:
wherein the method comprises the steps ofIs->Is the mole fraction of component i in the alpha/beta phase, f i α,β Free energy corresponding to a single element in each phase, < >>And-> The contributions of the mixing entropy and the mixing enthalpy to the free energy of bulk chemistry are respectively corresponding; />The interaction parameters for components i and j in the respective phases can be obtained from existing experimental data.
In some embodiments, in the step S3, the bulk chemical free energy density function of the β and α phases of the titanium alloy is:
wherein the method comprises the steps ofFor interpolation function +.>Free energy curves for connecting the α and β phases; />And->Respectively represent the equilibrium molar chemical free energy of alpha phase and beta phase, which are related to the system temperature T and the alloy element X i Closely associating, and performing fitting approximation through Landau polynomials; η (eta) p And eta q (p, q=1, 2, …, 12) represents the structural field parameters of 12 alpha phase variants; omega pq η p η q Characterizing energy barriers, ω, between two-phase equilibrium states as an energy dual-potential well function pq Is the energy barrier height.
In some embodiments, in the step S4, the expression of the gradient term is:
κ i and kappa (kappa) η The front and back terms respectively represent the contribution of alloy components and structural non-uniformity to interface energy as gradient energy coefficient tensors;representing the composition gradient; />Representing the gradient of the structural field variable.
In some embodiments, the structural gradient term coefficient κ η The expression of (2) is:
r is a matrix representing the inclination degree of interface energy in space, and kappa 1 、κ 2 And kappa (kappa) 3 Representing the structural gradient energy coefficients on the broad, edge and side surfaces of the three planes of the alpha variant, respectively. When the interface thickness is constant, the interface energy is proportional to the square root of the gradient energy coefficient.
In some embodiments, the total chemical free energy function of the system is expressed as:
in some embodiments, the expression of the stress-free strain tensor of the first α variant in step S5 is:
in some embodiments, the rotationally symmetric operation matrix corresponding to the stress-free strain tensor of the second α variant in step S5 is:
in some embodiments, the expression for the stress-free strain tensor of the second alpha variant is:
wherein R is T Is the transposed matrix of R. And obtaining the stress-free strain tensors of all alpha variants through spatial rotation symmetry operation.
In some embodiments, the expression of elastic strain energy is:
wherein the method comprises the steps ofIs a unit inverted lattice vector,>for structural order parameter eta p Fourier transform of (r)/(r)>Is->Is a complex conjugate function of>Is the interaction potential of two bodies, and the unit is J/mol;
the saidThe expression under the boundary conditions of strain control is:
wherein C is ijkl Characterizing the elastic constant tensor of the system;the phase change stress corresponding to the variation p is expressed in GPa.
In some embodiments, theThe expression of (2) is:
wherein the method comprises the steps ofPhase change strain which is the p-th variant; />Is the tensor of the green's function in GPa -1 The method comprises the following steps:
in some embodiments, the total chemical free energy function of the system is expressed as:
F＝F chem +E el
in some embodiments, in the step S7, the evolution dynamics partial differential equation (generally referred to as Cahn-hillard equation) of the conservation-order parameter is:
wherein M is ij Is chemical mobility.
The evolution dynamics partial differential equation of the non-conservation order parameter (commonly referred to as the Allen-Cahn equation) is:
wherein M is η Characterizing interface mobility as an interface dynamic parameter;to solve for the number of non-zero structural field variables at the domain.
In some embodiments, chemical mobility M ij The expression of (2) is:
in the middle ofThe chemical mobility, which is a single phase, can be obtained from the following expression:
wherein δ is a kronecker function, the value of which is 1 if the two subscripts are equal, otherwise 0;then the atomic mobility of chemical element k in the p-phase.
In some embodiments, in the step S8, an explicit nucleation algorithm is used and coupled to the phase field model, where the expression of the nucleation probability is:
P(r,t)＝P 0 (r,t)exp(-ΔG * (r,t)/k B T)
wherein P (r, t) is nucleation rate, P 0 (r, t) is a proportionality constant, ΔG * (r, t) is the nucleation work, k B The Boltzmann constant is given, and T is the temperature of the system.
In some embodiments, at each computational grid point, a random number P is given with a value between 0 and 1 r If the nucleation rate at the current temperature has a value greater than the random number at the calculated grid point, i.e., P (r, t)>P r Then it is considered that a nucleation event occurred at the computational grid point, and a random perturbation of between 0 and 0.1 is added to each of the 12 alpha variants at the computational grid point to simulate the nucleation process.
Compared with the prior art, the invention has the following beneficial effects:
1. the phase field simulation method for predicting the microstructure evolution of the dual-phase titanium alloy provided by the invention can predict the microstructure evolution of the dual-phase titanium alloy at different solid solution temperatures and different cooling speeds, comprehensively considers the elastic strain energy of 12 alpha variants and beta matrixes, and can accurately simulate the microstructure evolution process of the titanium alloy in the solid solution process and the subsequent continuous cooling process under the three-dimensional condition.
2. The phase field simulation method for predicting the dual-phase titanium alloy tissue morphology evolution has universality, and the microstructure evolution process of the dual-phase titanium alloy solid solution and continuous cooling process under different alloy systems can be simulated by only modifying the contents of the corresponding Gibbs free energy expression, chemical mobility and the like for different types of titanium alloy systems.
Drawings
FIG. 1 is a flowchart showing the creation of a numerical model program in the present invention.
FIG. 2 is a graph showing free energy of Ti-13.750Al-0.451Mo (at.%) alloys at different temperature intervals in an embodiment of the present invention; in the figure, the abscissa represents temperature (unit: K) and the ordinate represents Gibbs free energy value (unit: J/mol).
FIG. 3 is a polynomial fit of the balance components of Al and Mo elements in the α and β phases of a Ti-13.750Al-0.451Mo (at.%) alloy at different temperatures, FIG. 3 (a) is a polynomial fit of the balance components of Al element in the β phase, FIG. 3 (b) is a polynomial fit of the balance components of Mo element in the β phase, FIG. 3 (c) is a polynomial fit of the balance components of Al element in the α phase, and FIG. 3 (d) is a polynomial fit of the balance components of Mo element in the α phase; in the figure, the abscissa represents temperature divided by 1000 (unit: K), and the ordinate represents atomic percent (unit: at%).
FIG. 4 is a three-dimensional structure morphology graph of Ti-13.750Al-0.451Mo (at.%) alloy after nucleation by addition of a random perturbation at 1208K and then holding at constant temperature for 3.125 s.
FIG. 5 is a graph of microstructure morphology of Ti-13.750Al-0.451Mo (at.%) alloys at different solution temperatures in an embodiment of the invention; fig. 5 (a) is a structure morphology graph after 3.125s solid solution at 1233K, fig. 5 (b) is a structure morphology graph after 3.125s solid solution at 1253K, fig. 5 (c) is a structure morphology graph after 3.125s solid solution at 1273K, and fig. 5 (d) is a structure morphology graph after 3.125s solid solution at 1290K.
FIG. 6 is a graph showing the volume fraction of the alpha phase as a function of the solid solution temperature and the solid solution time, with the solid solution time (unit: s) on the abscissa and the volume fraction of the alpha phase on the ordinate.
FIG. 7 is a graph of the microstructure morphology of a Ti-13.750Al-0.451Mo (at.%) alloy at various cooling rates in an embodiment of the invention. Wherein FIG. 7 (a) is a tissue topography after cooling for 3.125s at a cooling rate of 5K/s, FIG. 7 (b) is a tissue topography after cooling for 3.125s at a cooling rate of 10K/s, FIG. 7 (c) is a tissue topography after cooling for 3.125s at a cooling rate of 15K/s, and FIG. 7 (d) is a tissue topography after cooling for 3.125s at a cooling rate of 20K/s.
Fig. 8 is a graph showing the volume fraction of the α -phase as a function of cooling rate and cooling time, with the abscissa representing cooling time (unit: s) and the ordinate representing the volume fraction of the α -phase.
FIG. 9 is a flow chart of a phase field simulation method for predicting the tissue morphology evolution of a dual-phase titanium alloy at different solution temperatures and different cooling rates.
Detailed Description
The following description of the technical solutions in the embodiments of the present invention will be clear and complete, and it is obvious that the described embodiments are only some embodiments of the present invention, but not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
The phase field simulation method for predicting the tissue morphology evolution of the dual-phase titanium alloy at different solid solution temperatures and different cooling speeds comprises the following steps:
s1, determining the numbers of conservation sequence parameters and non-conservation sequence parameters used by a titanium alloy solid-state phase-change phase field model according to the alloy components of the titanium alloy and the characteristics of the titanium alloy solid-state phase change;
s2, obtaining a Gibbs free energy function expression of alpha and beta phases of the dual-phase titanium alloy system according to a Redlich-Kister-Muggianu (RKM) sub-regular solution model;
s3, connecting the Gibbs free energy function expressions of alpha and beta phases of the titanium alloy system through an interpolation function to construct a bulk chemical free energy density function of the system;
s4, fitting balance components at different temperatures through polynomials to serve as input of subsequent phase field simulation; introducing interface energy into a total chemical free energy item through a gradient item, and constructing a total chemical free energy function expression of the system;
s5, according to the crystal orientation relation ((0001) followed by beta-alpha phase transition of the titanium alloy α //{110} β Anddetermining the stress-free strain tensor of the first alpha variant, and obtaining the stress-free strain tensor of the second alpha variant, the third alpha 0 variant, the fourth alpha 1 variant, the fifth alpha variant, the sixth alpha variant, the seventh alpha variant, the eighth alpha variant, the ninth alpha variant, the tenth alpha variant, the eleventh alpha variant and the twelfth alpha variant according to certain spatial rotation symmetry operation;
s6, determining an elastic strain energy expression when the titanium alloy is subjected to beta-alpha phase transition according to the stress-free strain tensor values of the 12 alpha variants obtained in the S5; combining the elastic strain energy expression and the total chemical free energy function expression to obtain a function expression of the total energy of the system;
s7, establishing an evolution dynamics partial differential equation of the conservation-order parameters and the non-conservation-order parameters, converting the partial differential equation into Fortran language, and solving by using a computer;
s8, randomly nucleating at a specific temperature and isothermally maintaining for a period of time to obtain an initial tissue morphology before solution treatment, keeping the initial tissue morphology before solution treatment identical in configuration, changing different solution treatment temperatures, and calculating to obtain microstructure morphology evolution processes of alpha phase when the solution time is identical at different solution temperatures;
s9, taking the microstructure morphology after solution treatment as an initial microstructure configuration of cooling treatment, changing different cooling speeds, and calculating to obtain an alpha-phase microstructure morphology evolution process when cooling times are the same at the different cooling speeds;
and S10, carrying out visual treatment on the microstructure morphology evolution results of the alpha phase in the S8 and the S9 to obtain the influence rules of different solid solution temperatures and different cooling speeds on the microstructure morphology evolution process of the alpha phase.
The present invention will be described in detail below with reference to specific embodiments for better explaining the present invention.
Examples take the Ti-13.750Al-0.451Mo (at.%) alloy as an example:
(1) Selection of order parameters and construction of total chemical free energy function of system
The beta- & gtalpha solid state phase transformation process of Ti-Al-Mo alloys involves both changes in composition (Ti, al, mo) and crystal structure (bcc, hcp). Thus, the microstructure of the α/β two phase needs to be characterized by two types of time-and space-dependent field variables. Wherein the non-uniformity of the chemical element is formed by two independent concentration fields (conservation fields) { X i (r)} i＝Al,Mo Description. Meanwhile, in order to study the spatial orientation of the precipitated phase alpha variants, 12 alpha variants precipitated in the same beta grain are regarded as 12 "phases" with the same components and different orientations, and 12 groups of structural fields (non-conservation fields) are introducedTo describe the non-uniformity of the architecture, corresponding to the spatial distribution of each alpha variant, respectively.
According to the phase field theory, the chemical free energy of any heterogeneous system can be written in the following integral form:
wherein V is m Is molar volume; the first term of integral function is the bulk chemical free energy density of the alloy, which is temperatureDegree T, concentration field variable X i (r) and structural field variable η p A function of (r), while the second and third terms are gradient terms corresponding to concentration and structural field variables, respectively, characterizing the contribution of alloy composition and structural non-uniformity to interface energy; kappa (kappa) i And kappa (kappa) η For the gradient term coefficients, corresponding to the concentration field and the structural field, respectively, wherein:
r is a matrix representing the inclination degree of interface energy in space, and kappa 1 、κ 2 And kappa (kappa) 3 Representing the gradient energy coefficients on the broad, side and side surfaces, respectively. When the interface thickness is constant, the interface energy is proportional to the square root of the gradient energy coefficient.
In a ternary two-phase titanium alloy system, the bulk chemical free energy density is the sum of the free energies of the α -phase and the β -phase:
wherein,and->Respectively represents equilibrium molar chemical free energy of alpha phase and beta phase; /> For interpolation function +.>ω pq η p η q Characterizing energy barriers, ω, between two-phase equilibrium states as an energy dual-potential well function pq Is a wellDeep.
(2) Calculation of stress-free strain tensor and elastic strain energy
When calculating the stress-free strain tensor of the alpha variant, under the establishment of a specific coordinate system, i.e. x [101 ]]-y[-111]-z[-121]Based on the difference of lattice constants of alpha and beta phases on the three orientation axes, the phase change strain tensor of the first alpha variant is obtained, namely
The phase change strain tensors of the remaining eleven alpha variants can be obtained through spatially rotationally symmetric operation; taking the phase change strain tensor of the second α variant as an example, the corresponding rotationally symmetric operation matrix is:
the phase change strain of the second alpha variant is expressed as:
wherein R is T A transposed matrix of R; the rest alpha variants are analogically, and the stress-free strain tensors of all 12 variants can be obtained.
The elastic strain energy expression is:
wherein the method comprises the steps ofIs a unit inverted lattice vector,>for structural order parameter eta p Fourier transform of (r)/(r)>Is->Is a complex conjugate function of>For two-body interaction potential, the unit is J/mol, and under the boundary condition of strain control, the expression is:
wherein C is ijkl Characterizing the elastic constant tensor of the system;the phase change stress corresponding to the variant p is expressed in GPa, and the expression is as follows:
wherein the method comprises the steps ofPhase change strain which is the p-th variant; />Is the tensor of the green's function in GPa -1 The method comprises the following steps:
(3) Establishment of phase field control equation
The evolution dynamics partial differential equation (Cahn-hillard equation) of the conservation-order parameter (concentration field) is:
wherein M is ij Is chemical mobility.
The evolution dynamics partial differential equation (Allen-Cahn equation) of the non-conservation order parameter is:
wherein M is η Characterizing interface mobility as an interface dynamic parameter;to solve for the number of non-zero structural field variables at the domain.
(4) Outcome output and visualization
The method mainly solves a control equation of a structural field and a concentration field based on a semi-implicit Fourier spectrum method and an explicit difference method according to a phase field model and calculation parameters thereof. According to the embodiment of the invention, a program describing the microstructure evolution process of the Ti-13.750Al-0.451Mo (at%) alloy at different solid solution temperatures and different cooling speeds after solid solution is written by adopting Fortran language, and then the program is converted into a more visual image form by utilizing visualization software according to the sequence parameter evolution result output by the program, so that the purpose of visualizing the simulation result of the solid-state phase change process is achieved.
As shown in fig. 1, the specific flow of the numerical model program establishment is as follows: firstly, collecting single-phase Gibbs free energy data under a titanium alloy system, then calculating elastic strain energy, and fitting the total chemical free energy of the system; then, a phase field dynamics model is established, various physical parameters, boundary conditions, interface energy, elastic strain energy and other conditions are input into the model, and a phase field control equation, namely a Cahn-Hilliard equation and an Allen-Cahn equation, is solved by means of an explicit difference method and a semi-implicit Fourier spectrum method; and carrying out iterative solution on the output components and the structural field variables.
A specific example is provided below. For the Ti-13.750Al-0.451Mo (at.%) alloy, the main physical properties in this example are as follows:
TABLE 1 physical parameter values and units
The specific implementation manner of this embodiment is as follows:
(1) Based on a thermodynamic database, obtaining Gibbs free energy parameter information of each phase at different temperatures, obtaining Gibbs free energy function expressions of alpha phase and beta phase at different temperature intervals, and drawing a Gibbs free energy graph at different temperatures by using drawing software, wherein the free energy curves of the alpha phase and the beta phase of the Ti-13.750Al-0.451Mo (at%) alloy at different temperature intervals are shown as figure 2, and as can be seen from figure 2, the free energy of the alpha phase is lower than the free energy of the beta phase in the temperature range of 813-1003K, so that the alpha phase is a stable phase at lower temperature; in the range of 1203-1313K, the free energy difference of two phases gradually decreases along with the temperature rise, and the two curves basically coincide when the temperature is near 1290K, which indicates that the free energy of the alpha phase and the beta phase of the alloy is near 1290K and is equal; in the temperature range of 1313-1503K, the free energy difference between the two phases increases gradually with further temperature rise, and the free energy of the α phase is higher than that of the β phase at this time, indicating that the β phase is a stable phase at this time.
(2) Obtaining balance components of Al atoms and Mo atoms of alpha phase and beta phase in Ti-13.750Al-0.451Mo (at%) alloy at a series of different temperatures through a thermodynamic database, fitting by using a polynomial to obtain a balance component curve shown in figure 3, wherein the R square value is very close to 1, the accuracy of curve fitting is high, and the obtained polynomial expression is brought into a phase field model to approximately obtain the balance components at each temperature; further, as can be seen from fig. 3, the Al atom concentrations in both the α and β phases increase with an increase in temperature, while the Mo atom concentration decreases with an increase in temperature.
(3) Information such as alpha/beta interfacial energy, lattice constants of alpha phase and beta phase, elastic constant tensor and the like in Ti-13.750Al-0.451Mo (at%) alloy is collected, the lattice constant value of the beta phase is 0.3196nm, and the lattice constant of the alpha phase is a α ＝0.2943nm，c α In this case, the default values of the elastic constant tensors for the α and β phases are the same, = 0.2943 nm. The relationship between the orientations of Burgers maintained by the 12 alpha variants and the beta phase is shown in Table 2, and the stress-free strain tensors of the remaining 11 alpha variants are obtained by performing certain rotational symmetry operation on the stress-free strain tensors of the first alpha variant, as shown in Table 3. Then substituting the elastic modulus value and the stress-free strain tensor values of the 12 alpha variants into an elastic energy function expression.
TABLE 2 Burgers orientation relationship between 12 alpha variants and beta phase
Table 3 stress-free strain tensor values for 12 alpha variants of Ti-13.750Al-0.451Mo (at.%) alloy
(4) According to the phase field equation, the physical parameters obtained in the above steps are brought into a phase field model for the Ti-13.750Al-0.451Mo (at.%) alloy system, and the phase field control equation of the structural field and the concentration field, namely the Allen-Cahn equation and the Cahn-Hilliard equation, is solved.
(5) Programming the model and equation established by using Fortran language, bringing initial values and periodic boundary conditions into the model and equation, running a program, obtaining a corresponding result and performing visualization processing. Firstly, giving random disturbance to a simulation system at 1208K, wherein the free energy of beta phase is higher than that of alpha phase, solid phase transformation of beta-alpha can occur at the moment, the tissue morphology after 3.125s of heat preservation is shown in figure 4, and the tissue morphology at the moment is taken as the initial tissue morphology of subsequent solid solution treatment.
(6) According to the above-mentioned tissue morphology obtained after isothermal holding at 1208K for 3.125s as the initial tissue morphology before solution treatment, solid solution was carried out at 1233K, 1253K, 1273K, 1290K for 3.125s, respectively, the tissue morphology after solution treatment was as shown in FIGS. 5 (a) - (d), and the volume fraction of the alpha phase was as a function of the solid solution temperature and the solid solution time, as shown in FIG. 6.
As can be seen from fig. 5 (a) - (d), the number of α phases is reduced to a different extent after the solution treatment, compared with the initial three-dimensional tissue morphology corresponding to fig. 3, and the higher the solution temperature, the greater the extent of reduction in the volume fraction of the α phases. As can be seen from fig. 6, the volume fractions of the α -phase after 3.125s solid solution at 1233K, 1253K, 1273K and 1290K were about 0.57, 0.45, 0.44, 0.43, respectively. The reason why the volume fraction of the α phase decreases with an increase in the solid solution temperature is that the difference in free energy of the α phase and the β phase, i.e., the phase change driving force decreases with an increase in temperature.
(7) According to the above-mentioned tissue morphology obtained after solutionizing for 3.125s at 1290K as the initial tissue morphology of the subsequent cooling treatment process, cooling for 3.125s at cooling rates of 5K/s, 10K/s, 15K/s and 20K/s, respectively, the tissue morphology diagrams after cooling are shown in FIGS. 7 (a) - (d), and the volume fraction of the alpha phase as a function of the cooling rate and the cooling time are shown in FIG. 8.
As can be seen from fig. 7 (a) - (d) and fig. 8, as the cooling rate increases and the cooling time increases, the thickness of the α phase correspondingly increases, and the volume fraction of the α phase correspondingly increases, because the faster the cooling rate, the lower the temperature of the system at the same time of cooling, and the free energy difference of the α phase and the β phase, i.e., the phase change driving force increases.
The phase field simulation method for predicting the microstructure evolution of the dual-phase titanium alloy provided by the invention can predict the microstructure evolution of the dual-phase titanium alloy at different solid solution temperatures and different cooling speeds, comprehensively considers the elastic strain energy of 12 alpha variants and beta matrixes, and can accurately simulate the microstructure evolution process of the titanium alloy in the solid solution process and the subsequent continuous cooling process under the three-dimensional condition.
The phase field simulation method for predicting the dual-phase titanium alloy tissue morphology evolution has universality, and the microstructure evolution process of the dual-phase titanium alloy solid solution and continuous cooling process under different alloy systems can be simulated by only modifying the contents of the corresponding Gibbs free energy expression, chemical mobility and the like for different types of titanium alloy systems.
While the foregoing is directed to the preferred embodiments of the present invention, it will be appreciated by those skilled in the art that various modifications and adaptations can be made without departing from the principles of the present invention, and such modifications and adaptations are intended to be comprehended within the scope of the present invention.
Claims (10)
1. The phase field simulation method for predicting the morphology evolution of the dual-phase titanium alloy structure is characterized by comprising the following steps of:
s1, determining the numbers of conservation sequence parameters and non-conservation sequence parameters used by a titanium alloy solid-state phase-change phase field model according to the alloy components of the titanium alloy and the characteristics of the titanium alloy solid-state phase change;
s2, obtaining Gibbs free energy function expression of alpha and beta phases of the dual-phase titanium alloy system according to a Redlich-Kister-Muggianu sub-rule solution model;
s3, connecting the Gibbs free energy function expressions of alpha and beta phases of the titanium alloy system through an interpolation function to construct a bulk chemical free energy density function of the system;
s4, fitting balance components at different temperatures through polynomials to serve as input of subsequent phase field simulation; introducing interface energy into a total chemical free energy item through a gradient item, and constructing a total chemical free energy function expression of the system;
s5, according to the crystal orientation relation followed by the titanium alloy beta-alpha phase transformation, determining the stress-free strain tensor of the first alpha variant, and obtaining the stress-free strain tensor of the second alpha variant, the third alpha variant, the fourth alpha variant, the fifth alpha variant, the sixth alpha variant, the seventh alpha variant, the eighth alpha variant, the ninth alpha variant, the tenth alpha variant, the eleventh alpha variant and the twelfth alpha variant through certain spatial rotation symmetry operation;
s6, determining an elastic strain energy expression when the titanium alloy is subjected to beta-alpha phase transformation according to the stress-free strain tensor values of the 1 st to twelfth alpha variants obtained in the S5; combining the elastic strain energy expression and the total chemical free energy function expression to obtain a function expression of the total energy of the system;
s7, establishing an evolution dynamics partial differential equation of the conservation-order parameters and the non-conservation-order parameters, converting the partial differential equation into Fortran language, and solving by using a computer;
s8, randomly nucleating at a specific temperature and isothermally maintaining for a period of time to obtain an initial tissue morphology before solution treatment, keeping the initial tissue morphology before solution treatment to be the same in configuration, and calculating to obtain a microstructure morphology evolution process of an alpha phase at different solution temperatures;
s9, taking the microstructure morphology after solution treatment as an initial microstructure configuration of cooling treatment, and calculating to obtain an alpha-phase microstructure morphology evolution process under different cooling speeds;
and S10, carrying out visual treatment on the results of the microstructure morphology evolution process of the alpha phase in S8 and S9 to obtain the influence rules of different solid solution temperatures and different cooling speeds on the microstructure morphology evolution process of the alpha phase.
2. The phase field simulation method according to claim 1, wherein in the step S1, the conservation-order parameter is the concentration of each of the remaining alloy elements except for the titanium element in the nominal titanium alloy component, and the number of the conservation-order parameter is N-1, where N is the total number of all the elements in the titanium alloy; the number of the non-conservation-order parameters is 12, and the value of the non-conservation-order parameters is between 0 and 1;
and/or the number of the groups of groups,
in the step S2, under the assumption of Redlich-Kister-Muggianu sub-rule solution model, when the temperature is T, the expression of gibbs free energy function of α phase/β phase is:
wherein the method comprises the steps ofIs->Is the mole fraction of component i in the alpha/beta phase, f i α,β Free energy corresponding to a single element in each phase, < >>And->The contributions of the mixing entropy and the mixing enthalpy to the free energy of bulk chemistry are respectively corresponding; />Interaction parameters for components i and j in the respective phases;
and/or the number of the groups of groups,
in the step S3, the bulk chemical free energy density function of the titanium alloy system β and α phases is:
wherein the method comprises the steps ofFor interpolation function +.>Free energy curves for connecting the α and β phases; />And->Respectively represent the equilibrium molar chemical free energy of alpha phase and beta phase, which are related to the system temperature T and the alloy element X i Closely associating, and performing fitting approximation through Landau polynomials; η (eta) p And eta q (p, q=1, 2, …, 12) represents the structural field parameters of 12 alpha phase variants; omega pq η p η q Characterizing energy barriers, ω, between two-phase equilibrium states as an energy dual-potential well function pq Is the energy barrier height;
and/or the number of the groups of groups,
in the step S4, the expression of the gradient term is:
κ η for structural gradient term coefficients, κ i Is the gradient term coefficient of the alloy composition,representing the composition gradient; />Representing the gradient of the structural field variable.
3. The phase field simulation method according to claim 2, wherein a structural gradient term coefficient κ η The expression of (2) is:
r is a matrix representing the inclination degree of interface energy in space, and kappa 1 、κ 2 And kappa (kappa) 3 Representing the structural gradient energy coefficients on the broad face, the side face and the side face of the alpha variant on three planes respectively; when the interface thickness is constant, the interface energy is proportional to the square root of the gradient energy coefficient;
and/or the number of the groups of groups,
the total chemical free energy function expression of the system is:
4. a phase field simulation method according to claim 3, wherein the expression of the stress-free strain tensor of the first α -variant in step S5 is:
and/or the number of the groups of groups,
the rotationally symmetric operation matrix corresponding to the stress-free strain tensor of the second α variant in step S5 is:
and/or the number of the groups of groups,
the expression for the stress-free strain tensor of the second alpha variant is:
wherein R is T A transposed matrix of R; and obtaining the stress-free strain tensors of all alpha variants through spatial rotation symmetry operation.
5. The phase field simulation method according to claim 4, wherein the expression of elastic strain energy is:
wherein the method comprises the steps ofIs a unit inverted lattice vector,>for structural order parameter eta p Fourier transform of (r)/(r)>Is->Is a complex conjugate function of>Is the interaction potential of two bodies, and the unit is J/mol;
the saidThe expression under the boundary conditions of strain control is:
wherein C is ijkl Characterizing the elastic constant tensor of the system;the phase change stress corresponding to the variation p is expressed in GPa.
6. The phase field simulation method according to claim 5, wherein the phase field simulation method comprisesThe expression of (2) is:
wherein the method comprises the steps ofPhase change strain corresponding to the p-th variant; />Is the tensor of the green's function in GPa -1 The method comprises the following steps:
7. the phase field simulation method of claim 6 wherein the total chemical free energy function of the system is expressed as:
F＝F chem +E el 。
8. the phase field simulation method according to claim 1, wherein in the step S7, the evolution dynamics partial differential equation of the conservation-order parameter is:
wherein M is ij Is chemical mobility;
the evolution dynamics partial differential equation of the non-conservation-order parameter is as follows:
wherein M is η Characterizing interface mobility as an interface dynamic parameter;to solve for the number of non-zero structural field variables at the domain.
9. The phase field simulation method according to claim 8, wherein the chemical mobility M ij The expression of (2) is:
in the middle ofThe chemical mobility, which is a single phase, can be obtained from the following expression:
wherein δ is a kronecker function, the value of which is 1 if the two subscripts are equal, otherwise 0;then the atomic mobility of chemical element k in the p-phase.
10. The phase field simulation method according to claim 9, wherein in the step S8, an explicit nucleation algorithm is used and coupled to the phase field model, and the expression of the nucleation probability is:
P(r,t)＝P 0 (r,t)exp(-ΔG * (r,t)/k B T)
wherein P (r, t) is nucleation rate, P 0 (r, t) is a proportionality constant, ΔG * (r, t) is the nucleation work, k B Is Boltzmann constant, T is the temperature of the system;
at each calculation grid point, a random number P with a value between 0 and 1 is given r If the nucleation rate at the current temperature has a value greater than the random number at the calculated grid point, i.e., P (r, t)>P r Then it is considered that a nucleation event occurred at the computational grid point, and a random perturbation of between 0 and 0.1 is added to each of the 12 alpha variants at the computational grid point to simulate the nucleation process.
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