A method for deriving a criterion of directional solidification hot crack sensitivity under unified permeability

Abstract

The present application provides a unified permeability directional solidification hot crack sensitivity criterion derivation method, and the method is specifically as follows: a directional solidification multi-region unified permeability calculation model is established, and the mass conservation equation and the Darcy equation are coupled, that is, the present application realizes the unified calculation of the permeability under different dendrite morphologies, and the calculation result is closer to the flow field simulation result in the numerical value; compared with the prediction result of the traditional RDG criterion, the pressure drop order of magnitude closer to the actual cracking of the liquid film is obtained; under different grain boundary energies and grain sizes, the curve morphology is close to the test result, and the double-peak effect in the hot crack sensitivity evaluation test is reproduced by the numerical simulation for the first time.

Description

Technical Field

[0001] This invention belongs to the technical field of basic research on hot cracking, and in particular relates to a method for deriving a criterion for sensitivity to directional solidification hot cracking under uniform permeability. Background Technology

[0002] Hot cracking is a common problem in the casting, welding, and additive manufacturing processes of key aerospace structural materials such as aluminum alloys, nickel-based alloys, and titanium-aluminum intermetallic compounds. It arises from the difficulty in feeding within the narrow, elongated liquid phase channels between dendrites, leading to an increased pressure drop between the dendrites. To avoid this problem, a hot cracking susceptibility assessment using hot cracking criterion is necessary before fabrication. This optimizes alloy composition, reduces the tendency for hot cracking during material preparation, and improves the quality of structural manufacturing. The widely accepted hot cracking susceptibility criterion is the RDG criterion.

[0003] The RDG criterion assesses the tendency for hot crack initiation by calculating the interdendritic pressure drop. However, its application currently faces several challenges: First, the RDG criterion employs a two-dimensional uniform array of columnar dendrites in its derivation, using the Carman-Kozeny approximation for permeability calculation in columnar growth. However, in actual experiments and simulations, cellular growth and grain boundaries contribute more significantly to the pressure drop, and a permeability model based on this morphology currently does not exist. Second, the permeability equations for cellular and columnar morphologies differ, requiring additional algorithms or manual differentiation between cellular and columnar regions during permeability and pressure drop calculations, which is difficult to implement. Therefore, proposing a unified permeability model for cellular and columnar regions and using this model to correct the traditional RDG criterion is a crucial and necessary research step, significantly improving the accuracy of hot crack assessment. Summary of the Invention

[0004] The purpose of this invention is to solve the technical problem that the traditional RDG criterion cannot represent the thermal crack sensitivity of cellular intervals and has limited overall accuracy. It proposes a method for deriving a criterion for directional solidification thermal crack sensitivity under a unified permeability.

[0005] This invention is achieved through the following technical solution: This invention proposes a method for deriving a criterion for the sensitivity of directional solidification hot cracking under a unified permeability, the method comprising the following steps:

[0006] Step 1: Initialization of the multi-region permeability model: Based on the dendrite growth range, establish and initialize the multi-region permeability model.

[0007]

[0008]

[0009]

[0010] Where u is the liquid flow velocity, F is the friction force, dp / dx is the pressure gradient, u' is the apparent flow rate, μ is the liquid viscosity, and K is the apparent permeability;

[0011] Step 2: Establishment of a Unified Permeability Model for Directional Solidification: Based on the dendrite geometry, a mesh is generated along the temperature gradient direction. The upper and lower limits of the mesh are the geometric midlines of adjacent dendrites, respectively. The mesh width is 0.8 micrometers. The solid-liquid interface is approximated using the parallel temperature gradient method to obtain a unified flow unit. Based on the Hagen-Poiseuille theory, the multi-region permeability initialization model mentioned in Step 1, and the flow conservation equation, the permeability calculation formula within the unified flow unit is derived, in the following form:

[0012]

[0013] Where dY is the cell grid width, f l To determine the liquid phase volume fraction within a unit, the geometry of the unified flow unit is replaced based on the characteristic dendrite spacing and solidification path under different temperature gradients. This expands the permeability calculation within the unified flow unit, and the expanded result is as follows:

[0014]

[0015] Where K(T) is the apparent permeability at a certain temperature during directional solidification, λ(T) is the average primary dendrite arm spacing at that temperature, and f l (T) represents the liquid volume fraction at this temperature. This model is a uniform apparent permeability model for directional solidification processes.

[0016] Step 3: Derivation of the sensitivity criterion for directional solidification hot cracking under uniform permeability: 1. Based on the morphology of a two-dimensional dendrite uniform array, a local area is selected as a typical analysis unit. The unit must contain two adjacent dendrites and interdendritic fluid. 2. A mass conservation equation is established based on the mass flow direction within the analysis unit, including fluid feeding along the dendrite direction and shrinkage deformation perpendicular to the dendrite direction. 3. The mass conservation equation and the Darcy diffusion equation are solved simultaneously. The permeability of the Darcy equation is determined using a uniform apparent permeability model for the directional solidification process. This yields the sensitivity criterion for directional solidification hot cracking under uniform permeability.

[0017]

[0018] Where ΔP max The interdendritic pressure drop is used to measure hot crack susceptibility, β is the shrinkage coefficient, μ is the apparent viscosity, G is the temperature gradient, and T is the temperature gradient. S and T L These correspond to the solid-liquid phase boundary temperatures, respectively, where E(T) is the strain rate integral, and vT f is the directional solidification traction rate. s (T) represents the solidification path, numerically the same as 1-f. l (T) are equal.

[0019] Furthermore, the permeability model derived in step two uses λ(T) and f l (T) can be used as a variable to achieve unified calculation of permeability at cellular growth, columnar growth and grain boundary sites.

[0020] Furthermore, the combined process in step three adopts the directional solidification unified permeability model proposed in step two.

[0021] The beneficial effects of this invention are:

[0022] This invention derives a formula for calculating permeability within cellular units under two-dimensional conditions by combining Hagen-Poiseuille theory and the flow conservation equation. This solves the problem of the lack of a quantitative model for calculating permeability in the cellular growth process and grain boundary region, and provides a possibility for calculating permeability values ​​under multiple growth modes.

[0023] This invention divides the calculation units according to the temperature gradient, sets the upper limit of the unit as the geometric center line of the upper dendrite, the lower limit of the unit as the geometric center line of the lower dendrite, and the unit width as 0.8 micrometers. Based on this unit, a formula for calculating the permeability within a unified flow unit is derived, laying the foundation for the proposal of a unified permeability model.

[0024] This invention utilizes the average primary dendrite spacing λ(T) and the liquid phase volume fraction f at various temperatures. l (T) represents the averaged grid geometry of each region. The permeability calculation formula within a unified flow unit is extended to achieve simultaneous calculation of permeability values ​​in cellular growth regions, columnar growth regions, and grain boundary regions under directional solidification conditions, thus solving the problem of manual location and partitioning calculation in traditional methods.

[0025] This invention modifies the traditional RDG criterion by employing a unified permeability model for directional solidification, combined with the mass conservation equation and the Darcy diffusion equation, thereby achieving high-precision calculation of pressure drop in mixed growth regions and solving the problem of the lack of corresponding mathematical and physical calculation models in cellular growth and grain boundary regions.

[0026] After testing, compared with the traditional RDG criterion, the directional solidification hot cracking sensitivity criterion under unified permeability exhibits higher accuracy and theoretical reliability, mainly in the following aspects: First, the permeability variation trend predicted by this invention is the same as the result of the permeability equation used in the traditional RDG criterion, but numerically it is closer to the flow field simulation result; Second, the pressure drop predicted by this invention is on the order of less than 1 MPa, while the pressure drop obtained using the traditional RDG criterion is on the order of hundreds of MPa, even exceeding the material fracture strength. The pressure drop predicted by this invention is closer to the actual cracking order of the liquid film; Third, the pressure drop distribution trend and λ curve predicted by this invention are closer to the theoretical analysis results, and the curve morphology is close to the experimental results under different grain boundary energies and grain sizes; Finally, by extending the calculation scale to the actual directional solidification grain scale, this invention reproduces the bimodal effect in the hot cracking sensitivity assessment test for the first time using numerical simulation. Attached Figure Description

[0027] Figure 1 This is a schematic diagram of the grid division strategy for achieving unified calculation of permeability in multiple regions in Example 1;

[0028] Figure 2 This is a schematic diagram showing the comparison and effectiveness verification of the unified calculation of penetration rate in multiple regions in Example 1;

[0029] Figure 3 This is a schematic diagram of the dendrite morphology of the hot crack sensitivity extraction region under single crystal and polycrystalline growth conditions in Example 1;

[0030] Figure 4 This is a schematic diagram of the λ curve morphology under single-crystal and polycrystalline growth conditions in Example 1;

[0031] Figure 5 This is a schematic diagram of the hot cracking sensitivity prediction results and experimental verification in Example 1. Detailed Implementation

[0032] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0033] Hot cracking is a common problem in the casting, welding, and additive manufacturing processes of key aerospace structural materials such as aluminum alloys, nickel-based alloys, and titanium-aluminum intermetallic compounds. To avoid this problem, it is necessary to assess the tendency for hot cracking to occur before preparation using hot cracking susceptibility criteria. However, the commonly used RDG criterion only considers columnar growth regions, ignoring the effects of cellular growth regions and grain boundary regions. This invention proposes a method for deriving a unified permeability hot cracking susceptibility criterion for directional solidification. The specific method is as follows: establish a unified permeability calculation model for multiple regions in directional solidification, and couple the mass conservation equation and the Darcy equation. This invention achieves unified calculation of permeability under different dendritic morphologies, and the calculation results are numerically closer to the flow field simulation results; compared with the prediction results of the traditional RDG criterion, it obtains a pressure drop order of magnitude closer to the actual cracking of the liquid film; the curve morphology under different grain boundary energies and grain sizes is close to the experimental results, and for the first time, it reproduces the bimodal effect in the hot cracking susceptibility assessment test using numerical simulation.

[0034] Example 1

[0035] In this embodiment, a method for deriving a criterion for directional solidification hot cracking sensitivity under uniform permeability is achieved through the following steps:

[0036] Step 1: Initialization of the multi-region permeability model; Based on the described dendrite growth range, establish the initialization model for the multi-region permeability model:

[0037]

[0038]

[0039]

[0040] Where u is the liquid flow velocity, F is the friction force, dp / dx is the pressure gradient, u' is the apparent flow rate, μ is the liquid viscosity, and K is the apparent permeability;

[0041] Step 2: Establishment of a Unified Permeability Model for Directional Solidification: Based on the dendrite geometry, a mesh is created along the temperature gradient direction. The upper and lower limits of the mesh are the geometric midlines of adjacent dendrites, respectively. The mesh width is 0.8 micrometers. The solid-liquid interface is approximated using the parallel temperature gradient method to obtain a unified flow unit. Based on the Hagen-Poiseuille theory, the multi-region permeability initialization model mentioned in Step 1, and the flow conservation equation, the permeability calculation formula within the unified flow unit is derived, in the following form:

[0042]

[0043] Where dY is the cell grid width, f lTo determine the liquid phase volume fraction within a unit, the geometry of the unified flow unit is replaced based on the characteristic dendrite spacing and solidification path under different temperature gradients. This expands the permeability calculation within the unified flow unit, and the expanded result is as follows:

[0044]

[0045] Where K(T) is the apparent permeability at a certain temperature during directional solidification, λ(T) is the average primary dendrite arm spacing at that temperature, and f l (T) represents the liquid volume fraction at this temperature. This model is a uniform apparent permeability model for directional solidification processes.

[0046] The permeability model derived in step two uses λ(T) and f l (T) can be used as a variable to achieve unified calculation of permeability at cellular growth, columnar growth and grain boundary sites.

[0047] Step 3: Derivation of the sensitivity criterion for directional solidification hot cracking under uniform permeability: (1) Based on the morphology of the two-dimensional dendrite uniform array, a local area is selected as a typical analysis unit. The unit is required to contain two adjacent dendrites and interdendritic fluid; (2) Establish a mass conservation equation based on the mass destination within the analysis unit, including fluid feeding along the dendrite direction and shrinkage deformation perpendicular to the dendrite direction; (3) Combine the mass conservation equation and the Darcy diffusion equation. The permeability of the Darcy equation adopts the uniform apparent permeability model of the directional solidification process in Step 2, and the sensitivity criterion for directional solidification hot cracking under uniform permeability can be obtained:

[0048]

[0049] Where ΔP max The interdendritic pressure drop is used to measure hot crack susceptibility, β is the shrinkage coefficient, μ is the apparent viscosity, G is the temperature gradient, and T is the temperature gradient. S and T L These correspond to the solid-liquid phase boundary temperatures, respectively, where E(T) is the strain rate integral, and v T f is the directional solidification traction rate. s (T) represents the solidification path, numerically the same as 1-f. l (T) are equal.

[0050] The combined process in step three adopts the directional solidification unified permeability model proposed in step two.

[0051] The parameters in this invention can be obtained through experiments, literature, numerical simulations, and thermodynamic calculations.

[0052] This criterion comprehensively considers the differences in permeability patterns in cellular, columnar, and grain boundary growth regions, enabling high-precision assessment of directional solidification hot cracking sensitivity. It is the first to reproduce the bimodal effect in hot cracking sensitivity assessment tests using numerical simulation. Based on the traditional RDG hot cracking sensitivity criterion, this criterion derives a unified calculation model for permeability in different solidification regions, selects characteristic dendrite spacing and solidification path as inputs, and assesses hot cracking tendency based on solidification interval integration.

[0053] Figure 1 The meshing strategy for the directional solidification uniform permeability model obtained in this embodiment shows that cellular growth and columnar growth can use the same meshing method to describe dendrite morphology with liquid phase channel units in a unified form. The dendrite morphology is well preserved under this meshing strategy, and the geometry of the liquid phase channel units can be characterized by the average primary dendrite arm spacing and solidification path.

[0054] Figure 2 The results are the calculation results of the directional solidification unified permeability model obtained in this embodiment. ULK represents the Carman-Kozeny approximation result with a fixed primary dendrite arm spacing, VLK represents the Carman-Kozeny approximation result with a varying primary dendrite arm spacing, VLW represents the result of the directional solidification unified permeability model, RDG represents the Carman-Kozeny approximation result with a fixed secondary dendrite arm spacing, and CFD represents the flow field calculation result. It can be seen that the calculation results of the directional solidification unified permeability model in this embodiment, along with the CFD calculation results, exhibit high accuracy under both cellular and columnar growth conditions.

[0055] Figure 3 The dendrite morphology used in this embodiment for extracting hot crack sensitivity and λ curves shows that under single-crystal growth conditions, the length of the liquid phase tank gradually increases with increasing solute concentration, while the maximum number of liquid phase tanks gradually decreases. When the solute concentration reaches 3.0 wt.%, the radius of curvature at the root of the liquid phase tank increases. Under polycrystalline growth conditions, the coagulation temperature gradually decreases with increasing solute concentration, while the eutectic temperature increases with increasing solute concentration. Affected by the eutectic reaction and the length of the mushy region, when the solute concentration is higher than 1.0 wt.%, the length of the isolated grain boundary interval gradually decreases with increasing solute concentration.

[0056] Figure 4The results of hot crack sensitivity prediction under single-crystal and polycrystalline conditions in this embodiment show that, in both cases, the hot crack sensitivity exhibits a λ-curve relationship with the change in solute concentration. Under single-crystal conditions, the peak phase of hot crack sensitivity differs from the prediction result of the RDG criterion. The specific reason is that the RDG criterion ignores the weakening effect of the isolated liquid phase trough at the root of the mushy region due to the dynamic change in the spacing of the primary dendrite arms. Under polycrystalline conditions, the peak phase of hot crack sensitivity is exactly the same as the prediction result of the RDG criterion. The reason is that the grain boundary effect dominates under polycrystalline conditions, that is, the pressure drop at the grain boundary accounts for more than 95% of the total pressure drop. As the solute concentration increases, the proportion of the length of the isolated grain boundary interval gradually decreases until the concentration reaches 3.0 wt.%, at which point the grain boundary effect weakens and the hot crack sensitivity decreases.

[0057] Figure 5 The prediction results of the directional solidification hot crack sensitivity criterion proposed in this embodiment under a uniform permeability and its comparison with experimental results show that the hot crack sensitivity is higher under high grain boundary energy and smaller under low grain boundary energy. The overall distribution trend is the same as that of the experimental results. In addition, as the grain size increases, the bimodal phenomenon gradually becomes obvious, and the phase is the same as that of the experimental results.

[0058] Based on the proposed method for deriving the criterion for directional solidification hot crack sensitivity under a unified permeability, the permeability values ​​of different regions are calculated simultaneously with high precision, significantly improving the calculation efficiency and accuracy for hot crack sensitivity. At the same time, the generation mechanism of the λ curve and the bimodal phenomenon are explained, providing theoretical guidance for the selection of components in the manufacturing process.

[0059] The above embodiments are merely illustrative examples for clear explanation and are not intended to limit the implementation. Those skilled in the art will recognize that other variations or modifications can be made based on the above description. It is neither necessary nor possible to exhaustively list all possible implementations. However, obvious variations or modifications derived therefrom are still within the scope of protection of this invention.

Claims

1. A method for deriving a criterion for directional solidification hot cracking sensitivity under uniform permeability, characterized in that, The method includes the following steps: Step 1: Initialization of the multi-region permeability model: Based on the dendrite growth range, establish and initialize the multi-region permeability model. Where u is the liquid flow velocity, F is the friction force, dp / dx is the pressure gradient, u' is the apparent flow rate, μ is the liquid viscosity, and K is the apparent permeability; Step 2: Establishment of a Unified Permeability Model for Directional Solidification: Based on the dendrite geometry, a mesh is generated along the temperature gradient direction. The upper and lower limits of the mesh are the geometric midlines of adjacent dendrites, respectively. The mesh width is 0.8 micrometers. The solid-liquid interface is approximated using the parallel temperature gradient method to obtain a unified flow unit. Based on the Hagen-Poiseuille theory, the multi-region permeability initialization model mentioned in Step 1, and the flow conservation equation, the permeability calculation formula within the unified flow unit is derived, in the following form: Where dY is the cell grid width, f l To determine the liquid phase volume fraction within a unit, the geometry of the unified flow unit is replaced based on the characteristic dendrite spacing and solidification path under different temperature gradients. This expands the permeability calculation within the unified flow unit, and the expanded result is as follows: Where K(T) is the apparent permeability at a certain temperature during directional solidification, λ(T) is the average primary dendrite arm spacing at that temperature, and f l (T) represents the liquid volume fraction at this temperature. This model is a uniform apparent permeability model for directional solidification processes. Step 3: Derivation of the sensitivity criterion for directional solidification hot cracking under uniform permeability:

1. Based on the morphology of a two-dimensional dendrite uniform array, a local area is selected as a typical analysis unit. The unit must contain two adjacent dendrites and interdendritic fluid.

2. A mass conservation equation is established based on the mass flow direction within the analysis unit, including fluid feeding along the dendrite direction and shrinkage deformation perpendicular to the dendrite direction.

3. The mass conservation equation and the Darcy diffusion equation are solved simultaneously. The permeability of the Darcy equation is determined using a uniform apparent permeability model for the directional solidification process. This yields the sensitivity criterion for directional solidification hot cracking under uniform permeability. Where ΔP max The interdendritic pressure drop is used to measure hot crack susceptibility, β is the shrinkage coefficient, μ is the apparent viscosity, G is the temperature gradient, and T is the temperature gradient. S and T L These correspond to the solid-liquid phase boundary temperatures, respectively, where E(T) is the strain rate integral, and v T f is the directional solidification traction rate. s (T) represents the solidification path, numerically the same as 1-f. l (T) are equal.

2. The method according to claim 1, characterized in that: The permeability model derived in step two uses λ(T) and f l (T) can be used as a variable to achieve unified calculation of permeability at cellular growth, columnar growth and grain boundary sites.

3. The method according to claim 2, characterized in that: The combined process in step three adopts the directional solidification unified permeability model proposed in step two.

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