A residual stress finite element verification method considering focusing ion beam ring cutting taper

Abstract

This invention discloses a finite element method for verifying residual stress considering the taper of a focused ion beam circumferential cut, belonging to the field of residual stress detection and verification technology. The method first measures the actual taper angle, top and bottom diameters, and height of the circumferentially cut structure; then, it uses the FIB-DIC method to obtain the measured normal strain distribution; next, it establishes a frustum model with the actual taper and an ideal cylindrical model without the taper in the finite element software, applying a reference load to simulate stress release; finally, it extracts the simulated normal strain of the two models, and aligns the simulated strain with the measured strain using a least-squares linear transformation, calculating the root mean square error and coefficient of determination for quantitative comparison. This invention reduces the root mean square error of the model with the taper by more than 40% compared to the model without the taper, and increases the coefficient of determination from 0.64 to 0.87, improving the agreement between simulation and measurement, and providing an effective verification method for the reliability of FIB-DIC residual stress measurement technology.

Description

Technical Field

[0001] This invention relates to the field of residual stress detection and verification technology, and in particular to a finite element method for residual stress verification that takes into account the circumferential taper of a focused ion beam. Background Technology

[0002] Residual stress is a key intrinsic factor determining the mechanical properties, fatigue life, and resistance to stress corrosion of metallic materials, and its distribution directly affects the service safety and durability of components. With the development of micro-nano manufacturing and precision equipment, accurate characterization of micro-area residual stress has become an indispensable technical link in new material research and development, process optimization, and life assessment of key components. However, traditional residual stress measurement methods (such as X-ray diffraction and neutron diffraction) face limitations at the micro-nano scale, such as insufficient spatial resolution or excessive destructiveness. There is an urgent need to develop micro-area stress detection technologies that combine high spatial resolution and full-field measurement capabilities.

[0003] Focused ion beam-digital image correlation (FIB-DIC) is a micro-nano scale residual stress measurement method that has emerged in recent years. Its procedure involves: first, depositing or etching a speckle pattern on the surface to be measured using FIB; then, performing a circumferential cutting on the target area to release local stress and induce surface displacement; acquiring high-resolution images before and after the circumferential cutting using a scanning electron microscope; calculating the displacement field using a digital image correlation algorithm; and finally, obtaining the residual stress / strain distribution based on the constitutive relations of elasticity. This technology boasts significant advantages such as sub-micron spatial resolution and full-field non-contact measurement, and has shown promising application prospects in the micro-area stress analysis of metallic materials such as high-temperature alloys, titanium alloys, and ultra-high-strength steel.

[0004] However, the actual morphology of FIB circumferential cutting is not an ideal cylinder. Due to the Gaussian distribution of the ion beam, and the intensification of redeposition effect and sidewall ion reflection with increasing cutting depth, the processed micro-cylinder cross-section exhibits a positive taper (i.e., frustum-shaped) geometric feature, narrower at the top and wider at the bottom. This non-ideal morphology has a dual impact: on the one hand, the geometric configuration itself induces additional strain contributions during stress release, which overlap with the actual residual stress release signal; on the other hand, when using finite element simulation to verify or invert FIB-DIC measurement results, if an ideal cylinder model is used, the geometric mismatch will lead to a systematic deviation between the simulated strain field and the measured field, making it impossible to accurately reproduce the actual stress release process, thus severely weakening the reliability and persuasiveness of the finite element verification results. Therefore, it is urgent to establish a finite element model construction and comparative verification strategy that can accurately reflect the circumferential taper characteristics, so as to objectively evaluate the validity of FIB-DIC measurement data and provide technical support for the accurate characterization of micro-region residual stress. Summary of the Invention

[0005] The purpose of this invention is to propose a finite element method for verifying residual stress considering the taper of the focused ion beam circumferential cutting. By accurately incorporating the taper generated by actual processing into the finite element geometric model and establishing a comparative verification mechanism between the model with and without taper, the simulated normal strain and the measured normal strain of FIB-DIC can be accurately compared, thereby achieving reliable verification of the FIB-DIC measurement results.

[0006] To achieve the above objectives, this invention proposes a finite element method for verifying residual stress considering the circumferential taper of a focused ion beam, comprising the following steps: Step S1: Perform focused ion beam circumferential cutting on the material to be tested to obtain a micro-ring core structure, and measure and record the actual geometric parameters of the structure after circumferential cutting; the actual geometric parameters include at least the taper angle, the diameter of the upper end face, the diameter of the lower end face, and the height of the structure; Step S2: The displacement field caused by circumferential cutting is measured using the focused ion beam-digital image correlation method (FIB-DIC), and the measured normal strain distribution is calculated based on the displacement field. ; Step S3: Based on the measured actual geometric parameters, establish two finite element geometric models, including one with the same taper angle. The frustum model and the ideal cylinder model without taper; Step S4: Assign intrinsic mechanical properties to the material under test to the two finite element geometric models; Step S5: Apply equivalent loads and set boundary conditions on the two finite element geometric models, perform static analysis, and solve for the simulated normal strain distribution. ; Step S6: Extract the simulated normal strain distribution of the two finite element geometric models respectively. A linear transformation is then performed, and the simulated normal strain distribution is fitted using the least squares method to convert it into a predicted strain distribution. ; Step S7: Calculate the predicted strain distribution for the two finite element geometric models respectively. Compared with the measured normal strain distribution The root mean square error and coefficient of determination between them; if the root mean square error of the model with tapered frustum is less than that of the ideal cylinder model without tapering, and the coefficient of determination is greater than that of the ideal cylinder model without tapering, then it is determined that the simulation and the actual measurement are more consistent after considering the tapering, the FIB-DIC measurement results are valid, and the measured residual stress is the true residual stress of the material.

[0007] Preferably, in step S1, the actual geometric parameters are obtained by characterizing the circumferentially cut structure using a scanning electron microscope or an atomic force microscope.

[0008] The preferred formula for calculating the taper angle is as follows: ; in, The taper angle has a range of values. , For the diameter of the upper end face, The diameter of the lower end face. This refers to the structural height.

[0009] Preferably, in step S2, the measured normal strain distribution The details for obtaining it are as follows: Step S21: Before the focused ion beam circumferential cutting process, prepare a speckle pattern on the surface of the area to be tested; Step S22: After circumferential cutting, the speckle area is imaged using a high-resolution scanning electron microscope; Step S23: Calculate the displacement field using a digital image correlation algorithm, and obtain the measured normal strain distribution based on the displacement field. Alternatively, the measured residual stress distribution can be obtained by further calculation based on the constitutive relation. .

[0010] Preferably, in step S3, the mesh size of the finite element geometric model is 0.02 μm to 0.5 μm, and the total number of elements in the model is 10,000 to 100,000; the diameter of the upper end face of the frustum model is equal to The diameter of the lower end face is equal to Height equals H Taper angle equals The diameter of the ideal cylindrical model is equal to... .

[0011] Preferably, in step S4, the intrinsic mechanical properties include elastic modulus, Poisson's ratio, density, and constitutive relation parameters. The intrinsic mechanical properties are the intrinsic properties of the material under conditions without damage from focused ion beam processing.

[0012] Preferably, in step S5, the equivalent load is a compressive stress of -1 MPa or a tensile stress of 1 MPa applied to the outer surface of the model; the boundary condition is a fixed constraint set on the bottom surface of the model to simulate the connection state between the micropillar and the matrix.

[0013] Preferably, in step S6, the formula for the linear transformation is as follows: ; The measured normal strain distribution was fitted using the least squares method. With simulated normal strain distribution Obtain the coefficient And k, then convert the simulated normal strain distribution into the predicted strain distribution, as shown in the following formula: ; in, , k The coefficients are to be fitted.

[0014] Preferably, in step S7, the root mean square error (RMSE) is calculated as follows: ; in, The number of data points For the first i FIB-DIC measured normal strain values ​​at each measurement point For the first i The finite element simulation predicts the normal strain value at each measurement point after least squares linear transformation. The determination coefficient R 2 The calculation method is as follows: ; in, This represents the average value of the measured normal strain.

[0015] Preferably, the fit criterion is: the RMSE of the model containing the frustum of cones is reduced by at least 20% compared to the ideal cylindrical model, and the RSE of the model containing the frustum of cones is... 2 It improves upon the ideal cylindrical model by at least 0.10.

[0016] Therefore, this invention proposes a finite element method for verifying residual stress considering the circumferential taper of a focused ion beam, which has the following advantages: (1) The present invention accurately introduces the actual taper generated by FIB circumferential cutting into the finite element geometric model, avoiding simulation deviation caused by model geometry distortion, so that the finite element simulation results and FIB-DIC measured results can directly correspond and verify each other, significantly improving the accuracy and credibility of verification.

[0017] (2) This invention establishes a cone model with a tapered frustum and an ideal cylinder model without a tapered frustum simultaneously, and uses linear transformation and least squares method to numerically align the simulated strain with the measured strain. It also utilizes the root mean square error (RMSE) and the coefficient of determination (R²) to further refine the model. 2 The fit between the two models was quantitatively evaluated, and the necessity of introducing the taper was quantitatively demonstrated.

[0018] (3) This invention does not depend on the type of material or the specific stress state. As long as the actual geometric shape after processing can be accurately measured, it can be applied to the residual stress assessment of various metal materials (including high temperature alloys, titanium alloys, ultra-high strength steel, etc.).

[0019] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0020] Figure 1 This is a flowchart of a finite element method for verifying residual stress considering the circumferential taper of a focused ion beam, according to the present invention. Figure 2 In this embodiment of the invention, the FIB circumferential cutting process results in a tapered angle. Actual SEM images of the micropillar structure; Figure 3 The same taper angle is established in the embodiments of the present invention. A schematic diagram of the ANSYS finite element geometric model and mesh generation; Figure 4 The normal strain obtained by finite element simulation in the embodiment of the present invention. Full field distribution cloud map; Figure 5 The FIB-DIC measured normal strain in this embodiment of the invention The full-field distribution cloud map; where (a) is the full-field distribution cloud map when ξ=0, (b) is the full-field distribution cloud map when ξ=0.1, (c) is the full-field distribution cloud map when ξ=0.2, (d) is the full-field distribution cloud map when ξ=0.4, (e) is the full-field distribution cloud map when ξ=0.5, and (f) is the full-field distribution cloud map when ξ=0.8; Figure 6 The FIB-DIC measured normal strain in this embodiment of the invention Curve showing the variation of normalization depth ξ; Figure 7 The normal strain of the ANSYS simulation models with and without a 4° taper is shown in the embodiments of this invention. Evolutionary curve comparison chart. Detailed Implementation

[0021] To make the technical solutions, advantages, and objectives of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below. The described embodiments are only some, not all, of the embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the described embodiments of the present invention without creative effort are within the protection scope of the present invention.

[0022] Unless otherwise defined, the technical or scientific terms used in this invention shall have the ordinary meaning as understood by one of ordinary skill in the art to which this invention pertains.

[0023] Example 1 This embodiment uses 1500 MPa grade ultra-high strength steel as the test material, and verifies the FIB-DIC measurement results of its micro-region residual stress using the method of this invention. Figure 1 As shown, the specific steps are as follows: Step S1: Perform focused ion beam circumferential cutting on the material to be tested to obtain a micro-ring core structure, and measure and record the actual geometric parameters of the structure after circumferential cutting; the actual geometric parameters include at least the taper angle, the diameter of the upper end face, the diameter of the lower end face, and the height of the structure; 1500 MPa grade ultra-high strength steel was selected as the material to be tested. Using a focused ion beam system, the target micro-region on the surface of the material was cut in a ring under the conditions of 30 kV accelerating voltage and 50 pA beam current to obtain a micro-pillar structure with a diameter of about 5 μm and a height of about 8 μm.

[0024] After circumferential cutting, high-resolution scanning electron microscopy was used to perform lateral imaging of the processed micropillars to accurately characterize the actual geometric morphology of the post-circumferential structure. The measured scanning electron microscope images are shown below. Figure 2 As shown, due to the Gaussian distribution characteristics of the ion beam and the redeposition effect during FIB processing, the resulting micropillars are not ideal cylinders, but rather have a truncated cone shape that is narrower at the top and wider at the bottom. The actual geometric parameters of the structure after circumferential cutting were precisely measured and recorded as follows: taper angle... = 4.0°, upper end face diameter = 4.8 μm, lower end face diameter = 5.9 μm, micropillar height H = 8.0 μm.

[0025] The measured actual geometric parameters will serve as the basis for subsequent finite element modeling. In this embodiment, the aforementioned geometric parameters are obtained through lateral imaging using a scanning electron microscope combined with image measurement software; in other embodiments of the present invention, an atomic force microscope can also be used to accurately characterize the geometric morphology of the circumferentially cut structure.

[0026] Step S2: The displacement field caused by circumferential cutting is measured using the focused ion beam-digital image correlation method (FIB-DIC), and the measured normal strain distribution is calculated based on the displacement field. ; Before FIB circumcision, platinum nanodot arrays are deposited on the surface of the area to be tested as speckle patterns using FIB. After circumcision, high-resolution scanning electron microscopy is used to perform secondary electron imaging on the speckle area to obtain reference images and deformed images before and after circumcision. The imaging conditions are kept consistent to ensure image contrast quality.

[0027] Using the NCORR digital image correlation plugin on the MATLAB platform, sequential microscopic images at various circumferential depths were imported. A rectangular region centered on the side of the micropillar with uniform speckle distribution was selected as the Region of Interest (ROI). The displacement of the deposited nanodots within this region was characterized and tracked point-by-point. The in-plane displacement field caused by the circumferential cutting was calculated using digital image correlation algorithms. Then, based on small deformation elasticity theory, the measured normal strain distribution was obtained by calculating the partial derivatives of the displacement field with respect to the coordinates. The strain distribution at different circumferential cutting depths is fitted and analyzed to obtain strain curves evolving along the depth direction. In other embodiments of the invention, the measured residual stress distribution can also be further calculated from the strain field based on the constitutive relation of the material. .

[0028] Step S3: Based on the measured actual geometric parameters, establish two finite element geometric models, including one with the same taper angle. The frustum model and the ideal cylinder model without taper; Based on the measured actual geometric parameters, a three-dimensional geometric model was established in the finite element analysis software ANSYS WORKBENCH. Two finite element geometric models were established for comparison and verification: One is a frustum model (including tapered models) that matches the actual taper, such as... Figure 3 As shown, the diameter of the upper surface of the model is equal to = 4.8 μm, lower end face diameter equal to = 5.9 μm, height equal to H = 8.0 μm, taper angle equal to = 4.0°, completely replicating the ring core structure processed by the actual FIB.

[0029] The second is the ideal cylindrical model without taper (the non-tapered model), whose diameter is equal to... = 4.8 μm, the same height as the frustum model.

[0030] In this embodiment, the global mesh size for both models is set to 0.1 μm, with a total of approximately 35,000 elements and 11,000 nodes. To ensure the accuracy of surface strain extraction, local mesh refinement is performed on the surface region of the micropillar side. In other embodiments of the invention, the mesh size can be adjusted within the range of 0.02 μm to 0.5 μm, and the total number of elements can vary from 10,000 to 100,000, which can be reasonably selected according to the model size and computational resources.

[0031] Step S4: Assign intrinsic mechanical properties to the material under test to the two finite element geometric models; The intrinsic mechanical properties of 1500 MPa grade ultra-high strength steel were assigned to the two established finite element geometric models. The material properties are as follows: elastic modulus E = 210 GPa, Poisson's ratio ν = 0.3, density ρ = 7850 kg / m³. 3 In this embodiment, the material is assumed to be an isotropic linear elastic material, and a linear elastic constitutive model is adopted; the material properties are all intrinsic properties of the material in the state without FIB processing damage, and the additional damage or phase transformation effects introduced by FIB processing are not considered.

[0032] Step S5: Apply equivalent loads and set boundary conditions on the two finite element geometric models, perform static analysis, and solve for the simulated normal strain distribution. ; To simulate stress release during the FIB circumferential cutting process, an equivalent load is applied in the finite element model. Specifically, a uniform compressive load with an amplitude of -1 MPa is applied to the original surface of the micropillar (i.e. the surface of the material in the initial residual stress state before circumferential cutting, including the side and top surfaces of the micropillar). The negative value represents compressive stress to simulate the shrinkage effect caused by the release of the original residual tensile stress.

[0033] Fixed constraint boundary conditions are set on the bottom surface of the model to constrain the displacement degrees of freedom in all directions, simulating the rigid connection between the micropillar and the matrix. This boundary condition is consistent with the state in which the bottom of the micropillar remains connected to the matrix during actual FIB machining.

[0034] After setting the loads and boundary conditions, submit the solution to the ANSYS solver for static analysis. The solution type is linear static analysis, yielding the displacement, strain, and stress field distributions for both models. The simulated normal strain distribution for the model with a tapered profile is also shown. Full field distribution cloud map as follows Figure 4 As shown, the FIB-DIC measured normal strain distribution Full field distribution cloud map as follows Figure 5 As shown.

[0035] Step S6: Extract the simulated normal strain distribution of the two finite element geometric models respectively. A linear transformation is then performed, and the simulated normal strain distribution is fitted using the least squares method to convert it into a predicted strain distribution. ; The simulated normal strain distributions of the two finite element geometric models were extracted from the solution results. The extracted region strictly corresponds to the region of interest (ROI) in FIB-DIC, which is the surface normal strain component of the central region of the micropillar's side surface. The extraction depth range is the complete depth interval from the upper end face to the lower end face. The extraction path for models with tapers is along the generatrix direction of the taper surface, while the extraction path for models without tapers is along the generatrix direction of the cylinder surface.

[0036] Extract With the obtained measured normal strain Numerical registration is performed. Since the load applied in the finite element simulation is a reference load of -1 MPa, the actual residual stress level corresponding to the measured values ​​is still unknown. There is a proportional relationship between the two; therefore, a linear transformation of the simulated normal strain is required to achieve numerical alignment. The transformation formula is: ; in, , k The coefficients to be fitted are represented by the measured normal strain distribution. Dependent variable, simulated normal strain distribution Using the least squares method as the independent variable, the following results were obtained: and k Specifically, and Substituting the point-by-point data pairs into the above equation, the optimal fitting coefficients are determined by minimizing the sum of squared residuals.

[0037] In this embodiment, the least squares fitting is performed as follows: The fitting results for the model containing a 4° conical frustum are as follows: , ; The fitting result for the ideal cylindrical model without taper is as follows: , ; The fitted result and k Substituting into the formula, the simulated normal strain distribution is converted into the predicted strain distribution: ; Calculate the predicted strain distribution for the two finite element geometric models respectively. .

[0038] Step S7: Calculate the predicted strain distribution for the two finite element geometric models respectively. Compared with the measured normal strain distribution The root mean square error and coefficient of determination between them; if the root mean square error of the model with tapered frustum is less than that of the ideal cylinder model without tapering, and the coefficient of determination is greater than that of the ideal cylinder model without tapering, then it is determined that the simulation and the actual measurement are more consistent after considering the tapering, the FIB-DIC measurement results are valid, and the measured residual stress is the true residual stress of the material.

[0039] Based on the predicted strain distribution obtained in step S6 Compared with the measured normal strain distribution Calculate the root mean square error (RMSE) and coefficient of determination (R²) between the predicted strain distribution and the measured strain distribution for the two finite element geometric models.2 : ; in, The number of data points For the first i FIB-DIC measured normal strain values ​​at each measurement point For the first i The finite element simulation predicts the normal strain value at each measurement point after least squares linear transformation. ; in, This represents the average value of the measured normal strain.

[0040] The quantitative evaluation indices for the two finite element geometric models are shown in Table 1.

[0041] Table 1. Quantitative evaluation indicators for two finite element geometric models

[0042] As shown in Table 1, the RMSE of the tapered model (0.00040) is significantly lower than that of the non-tapered model (0.00067), decreasing by approximately 40% compared to the non-tapered model; the RSE of the tapered model is... 2 The value of (0.87) is significantly greater than that of the model without taper (0.64), increasing from 0.64 to 0.87. These results indicate that after introducing an actual taper into the finite element model, the simulated normal strain and the FIB-DIC measured normal strain show good agreement in both their variation trends and amplitude levels (compare the evolution curves of the simulated and measured normal strains with normalization depth ξ for the model with taper as shown in the figure). Figure 6 As shown, the simulated normal strain evolution curves of models with and without taper are compared. Figure 7 As shown in the figure, there is a systematic deviation between the simulation results and the measured results of the non-tapered model.

[0043] The RMSE of the tapered model is smaller than that of the non-tapered model, and R 2 The residual stress is greater than that of the non-tapered model, therefore, it is determined that the FIB-DIC residual stress measurement results of this embodiment are valid and reliable, and the measured residual stress is the true residual stress of the micro-region of 1500 MPa grade ultra-high strength steel. This embodiment successfully proves that by accurately incorporating the actual taper of the FIB circumferential section into the finite element model, a high degree of agreement between the finite element simulation and the FIB-DIC measured results can be achieved, effectively solving the technical problem of inaccurate correspondence between simulation and measurement caused by ignoring the taper characteristics.

[0044] It is worth noting that all contents not described in detail in this invention are existing technologies and are well known to those skilled in the art.

[0045] Therefore, this invention provides a finite element method for verifying residual stress considering the circumferential taper of a focused ion beam. By accurately incorporating the actual taper parameters into the finite element model and establishing a quantitative comparison and verification mechanism between a frustum model with taper and an ideal cylindrical model without taper, this method effectively solves the technical problem of inaccurate correspondence between simulation and actual measurement caused by neglecting taper in traditional methods. At the same time, this method does not depend on specific materials or stress states and is applicable to the assessment of micro-area residual stress in various metallic materials, providing a scientific and effective verification means for the reliability of FIB-DIC measurement technology.

[0046] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A finite element method for verifying residual stress considering the circumferential taper of a focused ion beam, characterized in that, Includes the following steps: Step S1: Perform focused ion beam circumferential cutting on the material to be tested to obtain a micro-ring core structure, and measure and record the actual geometric parameters of the structure after circumferential cutting; the actual geometric parameters include at least the taper angle, the diameter of the upper end face, the diameter of the lower end face, and the height of the structure; Step S2: The displacement field caused by circumferential cutting is measured using the focused ion beam-digital image correlation method (FIB-DIC), and the measured normal strain distribution is calculated based on the displacement field. ; Step S3: Based on the measured actual geometric parameters, establish two finite element geometric models, including one with the same taper angle. The frustum model and the ideal cylinder model without taper; Step S4: Assign intrinsic mechanical properties to the material under test to the two finite element geometric models; Step S5: Apply equivalent loads and set boundary conditions on the two finite element geometric models, perform static analysis, and solve for the simulated normal strain distribution. ; Step S6: Extract the simulated normal strain distribution of the two finite element geometric models respectively. A linear transformation is then performed, and the simulated normal strain distribution is fitted using the least squares method to convert it into a predicted strain distribution. ; Step S7: Calculate the predicted strain distribution for the two finite element geometric models respectively. Compared with the measured normal strain distribution The root mean square error and coefficient of determination between them; if the root mean square error of the model with tapered frustum is less than that of the ideal cylinder model without tapering, and the coefficient of determination is greater than that of the ideal cylinder model without tapering, then it is determined that the simulation and the actual measurement are more consistent after considering the tapering, the FIB-DIC measurement results are valid, and the measured residual stress is the true residual stress of the material.

2. The finite element method for verifying residual stress considering the circumferential taper of a focused ion beam according to claim 1, characterized in that, In step S1, the actual geometric parameters are obtained by characterizing the circumferentially cut structure using a scanning electron microscope or an atomic force microscope.

3. The finite element method for verifying residual stress considering the circumferential taper of a focused ion beam according to claim 2, characterized in that, The formula for calculating the taper angle is as follows: ； in, It is the taper angle. For the diameter of the upper end face, The diameter of the lower end face. This refers to the structural height.

4. The finite element method for verifying residual stress considering the circumferential taper of a focused ion beam according to claim 3, characterized in that, In step S2, the measured normal strain distribution The details for obtaining it are as follows: Step S21: Before the focused ion beam circumferential cutting process, prepare a speckle pattern on the surface of the area to be tested; Step S22: After circumferential cutting, image the speckle area using a high-resolution scanning electron microscope; Step S23: Calculate the displacement field using a digital image correlation algorithm, and obtain the measured normal strain distribution based on the displacement field. Alternatively, the measured residual stress distribution can be obtained by further calculation based on the constitutive relation. .

5. The finite element method for verifying residual stress considering the circumferential taper of a focused ion beam according to claim 4, characterized in that, In step S3, the mesh size of the finite element geometric model is 0.02 μm to 0.5 μm, and the total number of elements in the model is 10,000 to 100,000; the diameter of the upper end face of the frustum model is equal to The diameter of the lower end face is equal to Height equals H Taper angle equals ; The diameter of the ideal cylindrical model is equal to .

6. The finite element method for verifying residual stress considering the circumferential taper of a focused ion beam according to claim 5, characterized in that, In step S4, the intrinsic mechanical properties include elastic modulus, Poisson's ratio, density, and constitutive relation parameters. The intrinsic mechanical properties are the intrinsic properties of the material under conditions without damage from focused ion beam processing.

7. The finite element method for verifying residual stress considering the circumferential taper of a focused ion beam according to claim 6, characterized in that, In step S5, the equivalent load is a compressive stress of -1 MPa or a tensile stress of 1 MPa applied to the outer surface of the model; the boundary condition is a fixed constraint set on the bottom surface of the model to simulate the connection state between the micropillar and the matrix.

8. The finite element method for verifying residual stress considering the circumferential taper of a focused ion beam according to claim 7, characterized in that, In step S6, the formula for the linear transformation is as follows: ； The measured normal strain distribution was fitted using the least squares method. With simulated normal strain distribution Obtain the coefficient And k, then convert the simulated normal strain distribution into the predicted strain distribution, as shown in the following formula: ； in, , k The coefficients are to be fitted.

9. The finite element method for verifying residual stress considering the circumferential taper of a focused ion beam according to claim 8, characterized in that, In step S7, the root mean square error (RMSE) is calculated as follows: ； in, The number of data points For the first i FIB-DIC measured normal strain values ​​at each measurement point For the first i The finite element simulation predicts the normal strain value at each measurement point after least squares linear transformation. The determination coefficient R 2 The calculation method is as follows: ； in, This represents the average value of the measured normal strain.

10. The finite element method for verifying residual stress considering the circumferential taper of a focused ion beam according to claim 9, characterized in that, The fit criteria are as follows: the RMSE of the model containing the frustum of cones is reduced by at least 20% compared to the ideal cylindrical model, and the RSE of the model containing the frustum of cones is also lower. 2 It improves upon the ideal cylindrical model by at least 0.10.

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