Method for predicting thermally induced cracks in metal additive manufacturing process

Abstract

The invention relates to a method for predicting thermally induced cracks in a metal additive manufacturing process, and belongs to the technical field of metal additive manufacturing process optimization. High-precision heat flow coupling temperature field data is used for carrying out space-time interpolation in a finite element grid, and in finite element thermosetting coupling simulation, special mechanical properties of a pasty area are incorporated into constitutive calculation, so that simulation is more suitable for an actual physical process, and calculation precision and result credibility are greatly improved.

Description

TECHNICAL FIELD

[0001] The present application relates to a metal additive manufacturing process thermal crack prediction method, belonging to the technical field of metal additive manufacturing process optimization. BACKGROUND

[0002] Metal additive manufacturing is a low-cost, short-cycle, design and manufacturing integrated advanced manufacturing technology, which has shown broad application potential in aerospace, nuclear power and medical fields. However, due to the complex molten pool flow caused by high energy input and the complex solidification process of metal, the forming quality is faced with the problems of shape and property control, among which the thermal crack caused by high temperature gradient and cooling rate is one of the important problems affecting the forming quality. The existence of thermal cracks makes the metal have local stress concentration when bearing, which greatly weakens or even destroys the mechanical properties of the printed components, so it is necessary to fully evaluate the cracking possibility before printing. Therefore, it is necessary to develop a prediction method for the thermal cracking possibility of metal additive manufacturing.

[0003] The thermal cracks of metal additive manufacturing are mainly divided into three types, namely solid-state cracks, liquefaction cracks and solidification cracks. Solid-state cracks are due to the formation of brittle phase at high temperature by printing thermal cycle of the previous deposited layer, which leads to the decrease of toughness and the formation of brittle cracks. Liquefaction cracks are formed in the heat affected zone, which is located at the outer edge of the molten pool. Due to the difference in melting point of various metals in the alloy, there are partially melted metals in the heat affected zone, forming liquid channels, and then cracks are formed under the stretching of thermal stress. Solidification cracks, also the most common thermal cracks, occur during solidification, which are mainly caused by the inability of grain boundaries to withstand the shrinkage caused by cooling. Among the above three kinds of cracks, solidification cracks are the most common and have the most serious impact on the forming quality.

[0004] Currently, the main methods for assessing the likelihood of hot cracking in metal additive manufacturing are theoretical models and numerical simulations. Regarding theoretical models, existing research includes liquid backfilling models and bridging cracking models. The liquid backfilling model posits that solidification cracking occurs because the liquid film between grains is insufficient to fill the intergranular voids. The bridging cracking model, on the other hand, considers solidification cracking due to stress exceeding the critical stress for intergranular bridging, resulting in brittle fracture. Theoretical models, starting from the microstructure of the mushy region, consider the physical mechanism of solidification cracking and provide a risk coefficient for cracking in the printed result. However, because theoretical models cannot obtain the dynamic stress field of the mushy region, they cannot obtain the specific geometric morphology of cracks caused by stress competition. In terms of numerical simulations, existing research uses phenomenological models or phase-field models to obtain temperature fields, combined with elastoplastic fracture phase-field finite element methods to simulate thermally induced cracks in the additive manufacturing process. However, for the unstable and high-temperature-gradient problems of additive manufacturing, using high-precision heat-fluid coupling temperature field data will inevitably yield calculations of hot cracks that more closely approximate reality. Furthermore, existing research is limited to calculations based on traditional physical property parameters and thermal softening functions. However, due to the complex solidification behavior of alloys, the solid near the mushy region possesses unique mechanical properties. Therefore, to obtain more accurate crack calculation results, it is necessary to incorporate the mechanical properties during the solidification process into the constitutive calculation. In summary, there is an urgent need for an elasto-plastic fracture phase field calculation model that can consider both the actual mechanical properties of the alloy mushy region and calculate the actual crack morphology.

[0005] Existing technologies struggle to simultaneously simulate the mechanical properties of the mushy region and the geometry of thermally induced cracks: theoretical models cannot simulate stress competition and crack geometry, while existing numerical methods suffer from low accuracy in the temperature field and unreasonable constitutive properties in the mushy region. Summary of the Invention

[0006] Based on this, the present invention proposes a large deformation elastoplastic fracture phase field thermal cracking calculation model that uses a high-fidelity temperature field coupled with thermal flow and can take into account the mechanical properties of the mushy region.

[0007] The technical solution of this invention is: A method for predicting thermally induced cracks in a metal additive manufacturing process includes the following steps: Step S1: Establish and solve the thermal-fluid coupling model; Step S2: Establish a coupling model of solid motion and fracture phase field and improve it by introducing the solidification curve of the alloy; Step S3: Solve the model improved in step S2, using the solution results from step S1 during the solution process; Step S4: Output the result of solving the improved model in step S3, which is the final thermally induced crack morphology.

[0008] In step S1, the method for establishing and solving the heat-flow coupling model is as follows: For the molten pool flow process in metal additive manufacturing, a heat-flow coupling model considering physical mechanisms such as heat transfer, convection radiation, evaporation energy loss, and Marangoni effect is first established.

[0009] The first step is to establish the governing equations as shown below.

[0010]

[0011] The boundary conditions are:

[0012] The heat transfer boundary conditions are:

[0013] The second step is to use the finite volume method to solve the above heat-fluid coupling problem and obtain temperature field data, which will be used in the subsequent heat-structure coupling solution.

[0014] In step S2, the method for establishing the coupling model of solid motion and fracture phase field is as follows: Step 1: For the thermo-structure coupling fracture problem, the following governing equations are established:

[0015] The first formula considers the solid motion under fracture damage, the second formula is the driving effect of stress on cracks, and the third formula is the thermal strain caused by temperature changes. The three are coupled with each other, that is, temperature changes bring thermal stress, and thermal stress drives crack growth, while cracks degrade the stiffness of the material.

[0016] Step 2: Define the crack driving force in the phase-field model: The crack is driven by the normal elastic work and all plastic work obtained from strain spectrum decomposition. To ensure that the crack will not self-heal, the historical maximum driving force is introduced. and using damage threshold To control the timing of cracking. The formula is as follows:

[0017]

[0018] In step S2, the method for improving the solidification curve of the alloy introduced into the established model is as follows: Step 1: For the alloy used in the simulation, first obtain the mechanical properties during solidification based on its solidification curve. Specifically, calculate the solidification curve of the alloy (i.e., the relationship between the solid volume fraction and temperature during solidification) based on the weight percentage (wt%) of each component in the alloy. These data will provide the necessary mechanical property basis for subsequent solidification crack prediction, ensuring the accuracy and reliability of the simulation results.

[0019] Step 2: The bridging zone cracking theory posits that during the final stage of grain solidification, the grains resist the thermal stress caused by cooling contraction through bridging generated during solidification. The critical stress for intergranular bridging is proportional to the solid fraction; therefore, a fracture threshold is used in the phase-field model. To control the critical stress, the corresponding formula is as follows:

[0020] The bridging cracking theory posits that grain bridging begins when the solid fraction reaches 94%, and solidification is complete when it reaches 100%. This represents the critical stress at which grains begin to bridge. This represents the critical stress at which the grains are completely solidified. The calculation uses a known temperature; by referring to the solidification curve and finding the volume fraction at that temperature, and then substituting this value into the formula above, the critical stress of the material at that temperature can be obtained. and fracture threshold .

[0021] In step S3, the method for establishing the solution format for the improved model from step S2 is as follows: Step 1: Based on the weighted residual method, the weak forms of the solid motion equations and phase field control equations are:

[0022] in, and The weight functions are for the displacement field and the phase field, respectively. This represents the gradient of the phase field weighting function.

[0023] Step 2: Discretize using finite element mesh to obtain the displacement field nodal forces and phase field residuals. The calculation method is as follows:

[0024] Step 3: Perform time integration of the displacement field using the frog-leap time scheme and the phase field using the forward Euler scheme, as shown in the following formula:

[0025] In step S3, the improved model from step S2 is solved, and the solution results from step S1 are used in the solution process as follows: Step 1: Input the temperature field data obtained by thermal-fluid coupling solution into the finite element nodes by time-space interpolation, and then obtain the temperature field data at the Gaussian point by finite element shape function interpolation.

[0026] Step 2: Calculate the experimental elastic strain increment using the following formula.

[0027]

[0028] Calculate the test elastic stress using the following formula.

[0029]

[0030] Then, a plastic constitutive solution was performed, and the true elastic strain and true stress were obtained after iterative convergence using a radial backtracking algorithm. The solidification fraction was then indexed by temperature on the solidification curve. Then, calculate the current fracture threshold using the following formula. .

[0031]

[0032] Then, determine the current crack driving force using the following formula.

[0033]

[0034]

[0035] Step 3: Calculate the nodal forces of the displacement field and the nodal residuals of the phase field according to the above solution format.

[0036] Step 4: Perform time integration on the displacement field and phase field according to the above solution format, and update the phase field value at the Gaussian point using the finite element shape function.

[0037] At this point, the calculation for this time step is complete. The physical time step is then pushed forward one step, and steps one through four are repeated until the physical time step reaches its termination time.

[0038] In step S4, the method for outputting the solution result of the improved model in step S3 is as follows: Based on the solution results of step S3, the final thermally induced crack morphology is obtained, and state variable information such as stress and strain is selectively output.

[0039] Beneficial effects The present invention provides a method for predicting thermally induced cracks in metal additive manufacturing processes. It uses high-precision thermal-fluid coupled temperature field data for spatiotemporal interpolation in a finite element mesh. In the finite element thermo-solid coupled simulation, the special mechanical properties of the mushy region are incorporated into the constitutive calculation, making the simulation more consistent with the actual physical process and greatly improving the calculation accuracy and the reliability of the results. Attached Figure Description

[0040] Figure 1 A schematic diagram of a thermal flux coupling model for the metal additive manufacturing process; Figure 2 This is a schematic diagram of the time step in an explicit finite element method solution. Figure 3 Solidification curves of AA6061 and AlSi10Mg; Figure 4 The results of thermal-fluid coupling calculations for the metal additive manufacturing process of AA6061; Figure 5 The AA6061 thermally induced crack was predicted using this method; Figure 6 The image shows a thermally induced crack in AA6061 material captured during the experiment. Figure 7 The image shows the AlSi10Mg thermally induced cracks predicted using this method. Detailed Implementation

[0041] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0042] A method for predicting thermally induced cracks in a metal additive manufacturing process includes the following steps: Step S1: Establish and solve the thermal-fluid coupling model; Step S2: Establish a coupling model of solid motion and fracture phase field and improve it by introducing the solidification curve of the alloy; Step S3: Solve the model improved in step S2, using the solution results from step S1 during the solution process; Step S4: Output the result of solving the improved model in step S3, which is the final thermally induced crack morphology.

[0043] In step S1, the method for establishing and solving the heat-flow coupling model is as follows: For the molten pool flow process in metal additive manufacturing, a heat-flow coupling model considering physical mechanisms such as heat transfer, convection and radiation, evaporation energy loss, and the Marangoni effect is first established. A schematic diagram is shown below. Figure 1 As shown.

[0044] The first step is to establish the governing equations as shown below.

[0045]

[0046] The boundary conditions are:

[0047] The heat transfer boundary conditions are:

[0048] The second step is to use the finite volume method to solve the above heat-fluid coupling problem and obtain temperature field data, which will be used in the subsequent heat-structure coupling solution.

[0049] In step S2, the method for establishing the coupling model of solid motion and fracture phase field is as follows: Step 1: For the thermo-structure coupling fracture problem, the following governing equations are established:

[0050] The first formula considers the solid motion under fracture damage, the second formula is the driving effect of stress on cracks, and the third formula is the thermal strain caused by temperature changes. The three are coupled with each other, that is, temperature changes bring thermal stress, and thermal stress drives crack growth, while cracks degrade the stiffness of the material.

[0051] Step 2: Define the crack driving force in the phase-field model: The crack is driven by the normal elastic work and all plastic work obtained from strain spectrum decomposition. To ensure that the crack will not self-heal, the historical maximum driving force is introduced. and using damage threshold To control the timing of cracking. The formula is as follows:

[0052]

[0053] In step S2, the method for improving the solidification curve of the alloy introduced into the established model is as follows: Step 1: For the alloy used in the simulation, first obtain the mechanical properties during solidification based on its solidification curve. Specifically, calculate the solidification curve of the alloy (i.e., the relationship between the solid volume fraction and temperature during solidification) based on the weight percentage (wt%) of each component in the alloy. These data will provide the necessary mechanical property basis for subsequent solidification crack prediction, ensuring the accuracy and reliability of the simulation results.

[0054] Step 2: The bridging zone cracking theory posits that during the final stage of grain solidification, the grains resist the thermal stress caused by cooling contraction through bridging generated during solidification. The critical stress for intergranular bridging is proportional to the solid fraction; therefore, a fracture threshold is used in the phase-field model. To control the critical stress, the corresponding formula is as follows:

[0055] The bridging cracking theory posits that grain bridging begins when the solid fraction reaches 94%, and solidification is complete when it reaches 100%. This represents the critical stress at which grains begin to bridge. This represents the critical stress at which the grains are completely solidified. The calculation uses a known temperature; by referring to the solidification curve and finding the volume fraction at that temperature, and then substituting this value into the formula above, the critical stress of the material at that temperature can be obtained. and fracture threshold .

[0056] In step S3, the method for establishing the solution format for the improved model from step S2 is as follows: Step 1: Based on the weighted residual method, the weak forms of the solid motion equations and phase field control equations are:

[0057] in, and The weight functions are for the displacement field and the phase field, respectively. This represents the gradient of the phase field weighting function.

[0058] Step 2: Discretize using finite element mesh to obtain the displacement field nodal forces and phase field residuals. The calculation method is as follows:

[0059] Step 3: Perform time integration of the displacement field using the frog-leap-time scheme and the phase field using the forward Euler scheme, as shown in the following formulas. The relationship between the time steps is as follows: Figure 2 As shown;

[0060] In step S3, the improved model from step S2 is solved, and the solution results from step S1 are used in the solution process as follows: Step 1: Input the temperature field data obtained by thermal-fluid coupling solution into the finite element nodes by time-space interpolation, and then obtain the temperature field data at the Gaussian point by finite element shape function interpolation.

[0061] Step 2: Calculate the experimental elastic strain increment using the following formula.

[0062]

[0063] Calculate the test elastic stress using the following formula.

[0064]

[0065] Then, a plastic constitutive solution was performed, and the true elastic strain and true stress were obtained after iterative convergence using a radial backtracking algorithm. The solidification fraction was then indexed by temperature on the solidification curve. Then, calculate the current fracture threshold using the following formula. .

[0066]

[0067] Then, determine the current crack driving force using the following formula.

[0068]

[0069]

[0070] Step 3: Calculate the nodal forces of the displacement field and the nodal residuals of the phase field according to the above solution format.

[0071] Step 4: Perform time integration on the displacement field and phase field according to the above solution format, and update the phase field value at the Gaussian point using the finite element shape function.

[0072] At this point, the calculation for this time step is complete. The physical time step is then pushed forward one step, and steps one through four are repeated until the physical time step reaches its termination time.

[0073] In step S4, the method for outputting the solution result of the improved model in step S3 is as follows: Based on the solution results of step S3, the final thermally induced crack morphology is obtained, and state variable information such as stress and strain is selectively output.

[0074] Example This section describes the effectiveness of the invention based on experimental data. To evaluate the performance of the proposed prediction method, laser selective melting was used, and experiments were conducted with two aluminum alloys: AA6061 and AlSi10Mg.

[0075] Data and parameter settings: The laser power is 355W; The scanning speed is 2320 mm / s; Laser spot radius 50μm; The coefficient of thermal expansion of solids is 22 × 10⁻⁶. -6 1 / ℃ AlSi10Mg composition (wt%): Al-88.95 Si-10 Mg-0.4 Fe-0.3 Mn-0.2 Cu-0.1 Zn-0.05 AA6061 composition (wt%): Al-97.4 Mg-1.0 Si-0.6 Cu-0.3 Cr-0.2 Fe-0.2 Mn-0.1 Ti-0.1 Zn-0.1 Critical stress at the beginning of bridging 5Mpa Critical stress at complete solidification 10 MPa Calculations show that the solidification curves of the two alloys are as follows: Figure 3As shown, a frame of the heat-fluid coupling calculation results for AlSi10Mg is displayed as follows. Figure 4 As shown, crack prediction calculations are performed on one of the yz planes.

[0076] Prediction results: The crack prediction results of AA6061 alloy are as follows: Figure 5 As shown in the figure, the predicted crack morphology at four moments during solidification is presented in chronological order. It can be seen that the crack grows upwards from the bottom of the molten pool as solidification progresses. This is because AA6061 has a relatively large temperature range (52℃) at the end of solidification, resulting in slow strength recovery and insufficient resistance to tensile stress caused by cooling. The crack grows radially along the molten pool, i.e., along the temperature gradient direction, because the temperature decreases fastest along this direction, leading to the fastest accumulation of thermal stress. Solidification cracking of AA6061 was also observed in the experiment, and the observed crack morphology at the yz section is shown below. Figure 6 As shown in the experimental photograph, the cracks are also radial, which verifies that the proposed prediction method can predict the occurrence of cracks and their approximate morphology.

[0077] Crack prediction results for AlSi10Mg alloy are as follows: Figure 7 As shown, no solidification cracks appeared in AlSi10Mg, consistent with the experimental results. This is because AlSi10Mg has a relatively large temperature range of 11℃ at the end of solidification, resulting in rapid strength recovery sufficient to resist tensile stress caused by cooling. This verifies that the proposed prediction method can reasonably predict the possibility of alloy cracking in additive manufacturing.

[0078] In summary, the above are merely preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for predicting thermally induced cracks in a metal additive manufacturing process, characterized in that... Includes the following steps: Step S1: Establish and solve the thermal-fluid coupling model; Step S2: Establish a coupling model of solid motion and fracture phase field and improve it by introducing the solidification curve of the alloy; Step S3: Solve the model improved in step S2, using the solution results from step S1 during the solution process; Step S4: Output the result of solving the improved model in step S3, which is the final thermally induced crack morphology.

2. The method for predicting thermally induced cracks in a metal additive manufacturing process according to claim 1, characterized in that: In step S1, the established heat-fluid coupling model is as follows: in, Represents partial derivatives, Represents gradient, Represents divergence, Represents density, Represents time, Represents fluid velocity. Represents pressure. Represents dynamic viscosity. Represents gravity. Represents the coefficient of thermal expansion. Represents the current temperature. Represents reference temperature. Represents the liquid volume fraction. Represents enthalpy change. Represents thermal conductivity. Represents specific heat capacity. Represents the enthalpy of solid-liquid phase transition. It represents an empirical constant (assuming the fluid's motion in the mushy region based on the flow state in a porous medium). For the Darcy item, For buoyancy, For latent heat term; The force boundary conditions are: in, For recoil pressure, For surface tension, Represents the recoil pressure coefficient. Represents environmental pressure, Represents latent heat of vaporization. Represents molar mass. This is the universal gas constant. Represents the evaporation temperature. Represents the surface tension coefficient. Represents the surface curvature of the molten pool. The normal vector representing the gas-metal interface; The heat transfer boundary conditions are: in, For heat convection, For thermal radiation, As a heat source, Energy loss due to evaporation, Represents the convective heat transfer coefficient. Represents reference room temperature. Represents the Stefan–Boltzmann coefficient. Represents emissivity.

3. The method for predicting thermally induced cracks in a metal additive manufacturing process according to claim 2, characterized in that: In step S1, the method for solving the established heat-fluid coupling model is as follows: The temperature field data were obtained by using the finite volume method.

4. The method for predicting thermally induced cracks in a metal additive manufacturing process according to claim 2, characterized in that: In step S2, the established coupling model between solid motion and fracture phase field is as follows: in, d The phase field variable represents the fracture state of the material; d=0 indicates the material is intact, and d=1 indicates the material is completely fractured. This represents the stress corresponding to the positive principal strain tensor. This represents the stress corresponding to the negative principal strain tensor. Represents physical strength. Represents solid acceleration. The work done by the maximum historical stress at a point in the material (including elastic work and plastic work). Represents fracture toughness. This represents the derivative of the degenerate function with respect to the phase field variables. Represents the characteristic length. The artificial viscosity coefficient representing the phase field solution. This represents the divergence with respect to the gradient of the phase field. Represents the increase in thermal strain. Represents the amount of temperature change. Represents the coefficient of thermal expansion. Representative unit: Zhang Liang; in, Represents the strain tensor. Represents the elastic strain tensor. The strain represents the strain corresponding to the positive principal strain tensor of elasticity. Represents the damage threshold energy. This represents the historical maximum value. This represents the strain energy used to drive crack growth. Macaulay brackets are used to assign the original value when the number inside the brackets is greater than zero, and to assign zero when the number inside the brackets is less than zero. Represents Lamé constant, Represents shear modulus, The trace representing the tensor. Represents Mises stress, This represents the equivalent plastic strain increment.

5. The method for predicting thermally induced cracks in a metal additive manufacturing process according to claim 4, characterized in that: In step S2, the process of introducing the solidification curve of the alloy for improvement includes: The relationship between temperature and critical stress is obtained based on the relationship between solid volume fraction and temperature during solidification, and the relationship between critical stress and solid volume fraction. The relationship between critical stress and solid volume fraction is as follows: in, Represents the volume fraction of the solid phase. Represents the critical stress. This represents the critical stress at which grains begin to bridge. This represents the critical stress at which the grains completely solidify. Represents Young's modulus.

6. The method for predicting thermally induced cracks in a metal additive manufacturing process according to claim 5, characterized in that: In step S3, the method for solving the improved model is as follows: The temperature field data obtained by thermal-fluid coupling solution is entered into the finite element nodes by time-space interpolation, and then the temperature field data at the Gaussian point is obtained by finite element shape function interpolation. The experimental elastic strain increment is calculated, followed by stress update and plastic constitutive solution. The true elastic strain and true stress are obtained after iterative convergence using the radial return algorithm. in, Represents the total strain increment. Represents the elastic strain increment. Represents the increment of plastic strain; The formula for calculating the experimental elastic stress is: in, Represents the total elastic strain. This represents the strain corresponding to the negative principal strain tensor. This represents the elastic constitutive tensor after the phase field degenerates stiffness. This represents the elastic constitutive tensor that maintains its original stiffness under pressure; in, Representative node At any moment The first nodal force One portion, Representative node At any moment Phase field nodal residuals, This represents the summation of all Gaussian points within the cell. Let the derivative of the shape function take its value at the Gaussian point. Represents the phase field value at the Gaussian point. This represents the value of the degenerate function at a Gaussian point. This represents the stress corresponding to the positive principal strain tensor at the Gaussian point. This represents the stress corresponding to the negative principal strain tensor at the Gaussian point. Represents the volume of a Gaussian point. This represents the value of the derivative of the degenerate function at the Gaussian point. This represents the value of the nodal shape function at Gaussian points. Represents the phase field gradient at the Gaussian point. Representative moment At the time The time interval, Representative moment At the time The time interval, Representative moment At the time The time interval, Representative node At any moment The first speed One portion, Representative node At any moment The first speed One portion, Represents the node volume. Representative node At any moment The first displacement One portion, Representative node At any moment The first displacement Each component.

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