CN109991387B - Phase field analysis method for simulating gamma-TiAl alloy under non-isothermal condition - Google Patents

Phase field analysis method for simulating gamma-TiAl alloy under non-isothermal condition Download PDF

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CN109991387B
CN109991387B CN201910166621.XA CN201910166621A CN109991387B CN 109991387 B CN109991387 B CN 109991387B CN 201910166621 A CN201910166621 A CN 201910166621A CN 109991387 B CN109991387 B CN 109991387B
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王刚
王富宝
吉丽
曾德长
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South China University of Technology SCUT
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Abstract

The invention belongs to the technical field of phase field methods, and discloses a phase field analysis method for simulating a gamma-TiAl alloy under a non-isothermal condition. Determining initialization conditions through thermodynamic and kinetic parameters of an alloy system; according to the crystal structure of intermetallic compounds in the alloy, constructing a chemical free energy model by using a sub-lattice model, and simultaneously calculating a corresponding atom occupancy ratio by using a gradient descent method; coupling the relation between each phase of chemical free energy and the structure field variable eta by using the corrected potential well function, and simultaneously adding elastic energy to construct a system free energy function; and substituting the constructed function into a phase field equation, calculating result values of the structural field variable and the component field variable, performing visualization processing to obtain a corresponding tissue evolution diagram, and analyzing the result. The invention can simulate the evolution process of crystal grains at different temperatures and the evolution law of structures at different cooling speeds. Provides reference for researching the phase transformation process of the intermetallic compound.

Description

Phase field analysis method for simulating gamma-TiAl alloy under non-isothermal condition
Technical Field
The invention belongs to the technical field of phase field methods, and particularly relates to a phase field analysis method for simulating a gamma-TiAl alloy under a non-isothermal condition.
Background
At present, most of material researches follow the design idea of material composition-microstructure-performance, and the prediction and regulation of the microstructure of the material are important components of the idea. However, due to the complex shape, structure and size of the microstructure of the material, a lot of trial and error experiments are performed to obtain the material with ideal structure and physical properties by continuously adjusting the process parameters such as components, temperature, time and the like. In order to realize the aim of 'double halving' of the material development period and the development cost, material genome engineering is provided, and the material development process is greatly accelerated by means of computational simulation and high-throughput computing technology. The phase field method has been receiving wide attention as an important simulation method for mesoscale in a multi-scale calculation simulation system. Due to the advantages of no need of tracking an interface and the convenience of coupling a thermodynamic and kinetic function library, the phase field method can truly reproduce the complete process of microstructure evolution in material phase transition and is favorable for deeply understanding the internal mechanism of material phase transition. Moreover, the phase field method has mature application in the processes of dendritic crystal growth, martensite phase transformation, ferroelectric phase transformation and the like.
The TiAl alloy has good high-temperature performance and low density, and is a light high-temperature material which is concerned with. The phase field method research of the TiAl alloy also obtains wide attention, such as 3D reproduction of TiAl lamellar structure, research on twin crystal formation mechanism and the like. However, the current phase-field method mainly focuses on simulation under constant temperature conditions, and does not involve different temperatures and continuous cooling processes, mainly due to the lack of an accurate chemical free energy model.
Disclosure of Invention
Aiming at the defects and shortcomings of the prior art, the invention aims to provide a phase field analysis method for simulating gamma-TiAl alloy under non-isothermal conditions. The method constructs a model capable of accurately describing the free energy of the intermetallic compound, can reproduce the formation process of lamellar structures at different temperatures and different cooling rates, provides a theoretical basis for researching the internal mechanism of TiAl alloy phase change, and provides a visual prediction method for the structure change caused by the temperature change.
The purpose of the invention is realized by the following technical scheme:
a phase field analysis method for simulating a gamma-TiAl alloy under a non-isothermal condition comprises the following steps:
(1) collecting corresponding thermodynamic and kinetic parameters of a gamma-TiAl alloy system, and determining initialization conditions;
(2) according to the crystal structure of intermetallic compounds in the TiAl alloy, constructing a chemical free energy model by using a sublattice model, and calculating a corresponding atomic occupancy ratio by using a gradient descent method;
(3) coupling the relation between each phase of chemical free energy and the structure field variable eta by using the corrected potential well function, and simultaneously adding elastic energy to construct a system free energy function;
(4) substituting the function constructed in the step (3) into a phase field equation, and calculating a result value of a structural field variable and a result value of a component field variable;
(5) and (4) carrying out visualization processing on the structural field variables and the component field variables obtained in the step (4) to obtain a corresponding tissue evolution diagram, and analyzing the result.
Further, the sub-lattice model formula in the step (2) is as follows:
Figure BDA0001986540220000021
wherein G ismIs the chemical free energy of the m phase, c is the atomic percentage of the corresponding element, G0Is the chemical free energy of the corresponding element pure substance, and y is the atomic space ratio of the corresponding element in the sublattice; the other parameters are conventional thermodynamic parameters; the relevant constraints for each atom site-occupancy are as follows:
Figure BDA0001986540220000031
further, the gradient descent method algorithm in the step (2) is repeated and iterated until convergence as follows:
Figure BDA0001986540220000032
wherein: is an assignment symbol, and α is a learning rate constant.
Further, the modified potential well function in step (3) is:
Figure BDA0001986540220000033
the elastic energy is as follows:
Figure BDA0001986540220000034
the system free energy function constructed is:
Figure BDA0001986540220000035
further, the phase field equation in step (4) is as follows:
Figure BDA0001986540220000036
Figure BDA0001986540220000037
in the formula
Figure BDA0001986540220000038
And
Figure BDA0001986540220000039
respectively, the rate of change of the structural field variables and components with respect to time, F is the free energy function of the system, including the chemical free energy and the elastic energy, LηIs the interfacial mobility, M is the diffusion mobility; and solving the partial differential equation by Fourier transform and finite difference method to obtain the result values of the structure field variable eta and the component field variable c.
The method of the invention has the following advantages and beneficial effects:
(1) by combining a sub-lattice model with a gradient descent method, a chemical free energy model of an available intermetallic compound can be quickly and accurately constructed and can be reasonably coupled with a phase field theory;
(2) a new potential well function is provided, in the programming calculation process, calculation errors among different phases are avoided, meanwhile, the balance value of a structure field variable eta is fixed near 1, errors caused by Fourier change are eliminated, and a calculation system is more stable;
(3) the influence of temperature change and various strains on the TiAl alloy sheet formation mechanism can be researched, and qualitative prediction is made for rapidly judging the rationality of the TiAl processing technology.
Drawings
FIG. 1 is a graph showing the calculated gamma-phase atomic site occupation ratios at different Al contents in the examples of the present invention
Figure BDA0001986540220000041
Temperature dependence curves (a) and a2Phase-to-atom space ratio
Figure BDA0001986540220000042
Correlation curve with temperature: (b) Figure (a).
FIG. 2 is a graph showing the free energy curves of the alpha 2 and gamma phases at various temperatures ((a)1300K, (b)1200K, (c)1000K and (d)900K) calculated in the examples of the present invention.
FIG. 3 is a graph of the volume fraction of the gamma phase calculated at different temperatures over simulation time in an example of the present invention;
FIG. 4 is a calculated topographic map of the tissue at 50K/s in an example of the present invention.
FIG. 5 is a graph of the cooling rate versus the thickness of the sheet calculated in the example of the present invention.
Detailed Description
The present invention will be described in further detail with reference to examples and drawings, but the present invention is not limited thereto.
Examples
This example provides a phase field analysis method for simulating γ -TiAl alloys under non-isothermal conditions, wherein the Al content is 46 at.%, and the alloys generate α in the temperature range of 1300K to 900K, for the TiAl alloy system2Ordering phase transition of → γ, forming a lamellar structure. The embodiment simulates the evolution process of the phase change, and the specific implementation manner is as follows:
firstly, determining that a research system is TiAl alloy, and the related phase is alpha2And gamma phase, which are intermetallic compounds, the chemical free energy of which is expressed by using a sublattice model (shown in formula (1)), and other related thermodynamic parameters are shown in table 1.
Figure BDA0001986540220000051
TABLE 1
Figure BDA0001986540220000052
Figure BDA0001986540220000061
Figure BDA0001986540220000071
Secondly, the accurate values of the occupancy ratios of the atoms are conveniently, quickly and accurately found, and because parameters are restricted mutually, a gradient descent method is utilized to iterate step by step (as shown in a formula (2));
Figure BDA0001986540220000072
until it converges to an equilibrium value. For rapid convergence, some equilibrium values at certain concentrations and temperatures calculated by referring to the first principle can be used as initial values, and the final equilibrium value is shown in formula (3);
Figure BDA0001986540220000073
substituting the obtained atomic space ratio into the step one to obtain the atomic space ratio
Figure BDA0001986540220000074
The relationship with concentration and temperature is shown in FIG. 1.
And thirdly, substituting the formula calculated in the step two into the sub-lattice model in the step one to obtain the chemical free energy of each phase, and then obtaining the chemical free energy of the system (formula (5)) through a corrected potential well function (formula (4)). The free energy profiles of the alpha 2 and gamma phases are calculated at different temperatures ((a)1300K, (b)1200K, (c)1000K, (d)900K) respectively as shown in FIG. 2.
Figure BDA0001986540220000081
Figure BDA0001986540220000082
Fourthly, calculating the elastic energy by referring to a model shown in a formula 6;
Figure BDA0001986540220000083
according to the crystallographic orientation relationship between two phases
Figure BDA0001986540220000084
And
Figure BDA0001986540220000085
can obtain alpha2Six different variants are produced in the phase transition → gamma, thus introducing 6 different structural field variables for each representation; while the strains generated by the 6 variants were as follows:
Figure BDA0001986540220000086
Figure BDA0001986540220000087
Figure BDA0001986540220000088
integrating the calculation results, constructing a phase field equation, and performing programming calculation to obtain the change rate of the variables of the structural field and the component field with time; therefore, visual analysis can be carried out, and the influence rule and the internal mechanism of the cooling speed on the TiAl lamellar structure can be obtained. Wherein the relevant simulation parameters are as follows in table 2.
TABLE 2
Figure BDA0001986540220000091
The equilibrium concentrations of the phases at different temperatures calculated in this example can be obtained from fig. 2, and compared to the phase diagram in table 3 below.
TABLE 3
Figure BDA0001986540220000092
It can be seen that the calculated equilibrium concentration is basically consistent with that in the phase diagram, the maximum error does not exceed 0.01, and the accuracy of the method provided by the invention is verified. Fig. 3 is a graph showing the increase of the volume fraction of the gamma phase at different temperatures according to the present embodiment with the simulation time. From FIG. 3, the temperature has a large influence on the TiAl alloy sheet layer, and at 1000K, the sheet layer grows faster, and the other temperatures grow slower; the mutual influence relationship of nucleation and growth is reflected, and the stability of the model at different temperatures is also verified. FIG. 4 is a calculated tissue topography at 50K/s for the present example; fig. 5 is a graph showing the correspondence between the cooling rate and the thickness of the sheet layer calculated in this example. As can be seen from fig. 4 and 5, the γ grains of the TiAl alloy have a lamellar structure at different cooling rates, and the lamellar thickness gradually decreases as the cooling rate increases. In a word, the morphology of the structure under different temperatures and cooling conditions can be accurately predicted by the method, and reference is provided for TiAl alloy experiments.
The above embodiments are preferred embodiments of the present invention, but the present invention is not limited to the above embodiments, and any other changes, modifications, substitutions, combinations, and simplifications which do not depart from the spirit and principle of the present invention should be construed as equivalents thereof, and all such changes, modifications, substitutions, combinations, and simplifications are intended to be included in the scope of the present invention.

Claims (2)

1. A phase field analysis method for simulating gamma-TiAl alloy under non-isothermal conditions is characterized by comprising the following steps:
(1) collecting corresponding thermodynamic and kinetic parameters of a gamma-TiAl alloy system, and determining initialization conditions;
(2) according to the crystal structure of intermetallic compounds in the TiAl alloy, constructing a chemical free energy model by using a sublattice model, and calculating a corresponding atomic occupancy ratio by using a gradient descent method;
(3) coupling the relation between each phase of chemical free energy and the structure field variable eta by using the corrected potential well function, and simultaneously adding elastic energy to construct a system free energy function;
(4) substituting the function constructed in the step (3) into a phase field equation, and calculating a result value of a structural field variable and a result value of a component field variable;
(5) performing visualization processing on the structural field variables and the component field variables obtained in the step (4) to obtain corresponding tissue evolution diagrams, and analyzing results;
the sub-lattice model formula in the step (2) is as follows:
Figure FDA0003357786860000011
wherein G ismIs the chemical free energy of the m phase, c is the atomic percentage of the corresponding element, G0Is the chemical free energy of the corresponding element pure substance, and y is the atomic space ratio of the corresponding element in the sublattice; the relevant constraints for each atom site-occupancy are as follows:
Figure FDA0003357786860000012
the gradient descent method algorithm in the step (2) is repeated and iterated until convergence as follows:
Figure FDA0003357786860000021
wherein: is an assignment symbol, α is a learning rate constant;
the corrected potential well function in step (3) is:
Figure FDA0003357786860000022
the elastic energy is as follows:
Figure FDA0003357786860000023
the system free energy function constructed is:
Figure FDA0003357786860000024
2. the method for simulating the phase field analysis of the gamma-TiAl alloy under the non-isothermal condition according to claim 1, wherein the phase field equation in the step (4) is as follows:
Figure FDA0003357786860000025
Figure FDA0003357786860000026
in the formula
Figure FDA0003357786860000027
And
Figure FDA0003357786860000028
respectively, the rate of change of the structural field variables and components with respect to time, F is the free energy function of the system, including the chemical free energy and the elastic energy, LηIs the interfacial mobility, M is the diffusion mobility; and solving the partial differential equation by Fourier transform and finite difference method to obtain the result values of the structure field variable eta and the component field variable c.
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