A new method for calculating residual stress in plate by layer cutting method

Abstract

The present application provides a new method for calculating the internal residual stress of plate by layer cutting method, and mainly relates to the internal residual stress analysis technical field of metal material. According to the basic principle of the deflection calculation of the bending beam in material mechanics, the present application deduces the residual stress calculation formula of the residual stress sample with rectangular section at different depths through the layer cutting method; since the additional deformation is caused to the sample by each layer cutting, the stress correction is needed. These additional stresses include two parts, one part is the stress σ a formed by the bending deformation of the remaining sample, and the other part is the stress σ b released by the remaining sample due to the stress release of the milled material; for the calculation value of the stress σ a formed by the bending deformation, the curvature radius of the middle layer of the sample is calculated by assuming that the measurement point and the support points at both ends of the sample are on the same circle, and then the additional stress σ b introduced by the bending deformation of each milling layer of the sample is calculated according to the curvature radius.

Description

Technical Field

[0001] This invention relates to the field of residual stress analysis technology in metallic materials, specifically a new method for calculating the residual stress inside a sheet metal using the layer-by-layer method. Background Technology

[0002] When a component is under load, if the stress in certain local areas exceeds the yield limit, these areas will undergo plastic deformation, while the rest of the component remains within the elastic deformation range. If the load is then removed, the plastically deformed parts cannot return to their original dimensions, which will inevitably hinder the recovery of the elastically deformed parts, thus causing internally interacting stresses. These stresses are called residual stresses.

[0003] During the production of aluminum alloy sheets, processes such as rolling, heat treatment, and pre-stretching are involved, resulting in residual stress within the sheet. This stress varies with the sheet thickness but remains generally balanced, allowing the sheet to maintain a certain dimensional shape. For sheet samples with residual stress, if a thin layer is removed from one of the extruded surfaces of the sample, the residual stress in the remaining sheet thickness will redistribute, causing the sample to undergo a certain degree of bending deformation to form a new equilibrium.

[0004] The principle of the layer-by-layer method for testing residual stress in plate samples is as follows: Using one rolled surface of the sheet metal sample as a reference plane, the other rolled surface opposite the reference plane is defined as the deflection surface. Thin layers of equal thickness are removed layer by layer from the deflection surface. The deformation deflection at the reference plane and the sample thickness are measured at the mid-length of the sample. Based on the material's Young's modulus, Poisson's ratio, and the sample's deformation deflection and thickness, the residual stress at different thicknesses of the sample and sheet metal is calculated using formulas established from the deflection model. The deflection method (also known as the peeling method) is used in the testing standards of Boeing and Airbus. Boeing's BMS7-323 standard uses the deflection method to test the processing deformation deflection of longitudinal samples of aerospace sheet metal, judging product acceptance based on the deflection value, without calculating the residual stress of the sample or sheet metal.

[0005] Airbus' AITM1-0040 standard uses the layer-by-layer method to test the machining deformation deflection of the specimen, and then calculates the residual stress distribution inside the specimen and the sheet metal. Airbus' formula for calculating the residual stress of the specimen is as follows: In the formula u(i) L Let be the longitudinal internal stress of the i-th layer of the sample, and E be the elastic modulus of the material. For support distance (e.g.) Figure 2 In l ), Let be the deflection of the sample before milling the i-th layer. The deflection of the sample after milling the i-th layer (e.g.) Figure 2 Ω n ), The thickness of the sample before milling the i-th layer. The thickness of the sample after milling the i-th layer. The stress correction amount consists of two parts: one part is the stress correction introduced by the moment release of the milled layer, and the other part is the stress correction introduced by the stress release of the milled layer. The stress correction in the formula is too complex and its clear physical meaning is difficult to discern. Summary of the Invention

[0006] To address the shortcomings of existing technologies, this invention provides a novel method for calculating residual stress within sheet metal using the layer-by-layer method. This method utilizes the deflection curve formula for rectangular beams in mechanics of materials, substituting boundary conditions to calculate the deflection curve equation for a rectangular specimen. Furthermore, it derives the relationship between deflection and bending moment at the midpoint of the specimen's length.

[0007] Subsequently, it was determined that this bending moment was provided by the residual stress within the milled thin layer. The bending moment provided by the residual stress within the thin layer can be expressed by the internal stress within the thin layer, the thickness of the thin layer, and the lever arm between the thin layer and the center of rotation. From this, an equilibrium equation was established to calculate the residual stress value within the thin layer. Since each layer-by-layer milling causes additional deformation to the specimen, stress correction is necessary. These additional stresses consist of two parts: one part is the stress generated by the bending deformation of the remaining specimen. Another part is due to the stress release of the milled material, which leads to the equivalent stress release of the remaining specimen. .

[0008] To achieve the above objectives, the present invention employs the following technical solution: A new method for calculating residual stress inside sheet metal using the layer-by-layer method: 1-Sample Type The plate material was sampled and processed into rod-shaped specimens with rectangular cross-sections. The length of the rectangular cross-section specimens is assumed to be... L , width is b Thickness is t .

[0009] 2-Deformation and Deflection Calculation 2.1 - From the initial state of the sample to the removal of the first layer of shavings The rectangular cross-section specimen is placed on a measuring platform with supports at both ends, with a support distance of [missing information]. l A schematic diagram of the layer thickness dimensions for layer milling using the layer milling method is shown below. Figure 1 A schematic diagram of the deformation of the sample after layer-by-layer cutting is shown below. Figure 2 , Figure 1In the figure, t(0) is the initial thickness of the sample, t(1) is the remaining thickness of the sample after the first layer is milled, and t(2)-t(10) are deduced in turn. Figure 2 The figures show the shape of the specimen before and after deformation when milling the nth layer. The dashed line represents the shape before deformation, and the solid line represents the shape after deformation. ω(i) is the deflection of the middle part of the specimen after milling the i-th layer. The shape changes of the specimen before and after machining the layer-by-layer surface are as follows: Figure 3 As shown. Figure 3 The solid line in the figure represents the sample position before the layer-by-layer machining, and the dashed line represents the sample position after the thin layer of the layer-by-layer machining is removed.

[0010] According to the deflection curve equation of a rectangular beam in mechanics of materials: (1) In equation (1), ω The displacement of the specimen at the center of the two support points is also called deflection. M Let E be the bending moment, E be the elastic modulus, I be the moment of inertia of the rectangular beam about the central axis z, and C and D be the integration constants.

[0011] according to Figure 1 The coordinate system in the figure has two boundary conditions, namely ; Substituting these two boundary conditions into equation (1), we get: , Then the equation of the deflection curve is: (2) make Then equation (2) can be transformed into: (3) Since the plate sample is initially flat, and the bending deformation occurs only after a thin layer is milled off, the balancing torque M before bending deformation is provided by the milled thin layer, and this balancing torque is provided by the residual stress within the thin layer. Let the residual stress within the thin layer be σ, the thickness of the sample before milling be t(0), and the thickness of the sample after milling the first thin layer be t(1). Then the torque generated by the residual stress of the removed thin layer is: (4) Since the torque M in equations (3) and (4) is equal, therefore equation (4) and Substituting into equation (3), we get: (5) 2.2 - Case where the i-th layer of the sample is removed during machining Let the thickness of the specimen before milling be t(i-1), and the deflection at the center deflection measurement point of the specimen be... The thickness of the sample after milling is t(i), and the deflection at the center deflection measurement point of the sample is... .

[0012] Even after the i-th layer of the sample is removed, it still obeys the deflection curve equation of a rectangular beam, i.e., equation (3) still holds. However, the deformation deflection at this time is... The moment of inertia I is as follows: (6) Then equation (3) is transformed into (7) The torque generated by the residual stress of the thin layer that has been removed is: (8) Since the torque M in equations (3) and (4) is equal, substituting equations (8) and (6) into equation (7) yields: (9) In equation (9) The residual stress within the i-th layer before the i-th layer is removed. Since the specimen containing the i-th layer is obtained by milling the initial specimen layer by layer from the 1st layer to the (i-1)th layer, the specimen containing the i-th layer is actually a deformed specimen after layer removal. Since each layer removed will generate an additional stress on the i-th layer material, the residual stress calculated by equation (9) should be reduced by the additional stress on the i-th layer from the earlier removed layers to obtain the initial residual stress of the specimen. These superimposed additional stresses include two parts: one part is the stress formed by the bending deformation of the remaining specimen. Another part is due to the stress release of the milled material, which leads to the equivalent stress release of the remaining specimen. .

[0013] It can be calculated as follows: Figure 4 This is a schematic diagram for calculating the radius of curvature of the additional deformation. Since the thickness of the sample before milling (before removing the i-th layer) is t(i-1), the deflection at the center deflection measurement point of the sample is... The radius of curvature of the neutral layer of the sample is ρ. Then, the following relationship exists between the right triangles OAC: (10) Will , , , Substituting into equation (10), we get: (11) Because a rectangular cross-section elastic beam has the following relationship: (12) (13) Substituting equations (13) and (11) into equation (12), we get: (14) It can be calculated as follows: For the first layer of peeling, there is no bending deformation stress correction or peeling stress release stress correction; therefore, the internal stress of the first layer is... That is: (15) stress .

[0014] Second layer: = (16) Third layer: (17) nth level: (18) The residual stress on the cut surface of the i-th layer is: - (19).

[0015] Compared with the prior art, the beneficial effects of the present invention are: In existing technology, the formula for calculating the residual stress of a specimen according to Airbus' AITM1-0040 standard includes the following stress correction term: The stress correction in this formula is too complex to discern a clear physical meaning. Because calculating the stress of each layer requires iterative accumulation, the calculation process is extremely complex, and Airbus even designed a special calculation program for it.

[0016] Based on the fundamental principles of bending beam deflection calculation in mechanics of materials, this invention derives a formula (Equation 19) for calculating the residual stress of a rectangular cross-section specimen at different depths using the layer-by-layer method. This formula comprises three terms: the first is the principal stress calculation formula; the second is a stress correction term introduced by the bending deformation of the specimen after layer-by-layer milling; and the third is a stress correction term for the stress released within the milled layer and distributed to the remaining portion of the specimen. The physical meaning is clear, and only the third term involves cyclic accumulation calculations, resulting in a much smaller total computational load and a simpler calculation method, typically achievable using Excel spreadsheet tools. Attached Figure Description

[0017] Figure 1 This is a schematic diagram of layer thickness milling using the layer milling method; Figure 2 This is a schematic diagram of the deformation of the sample after layer-by-layer chipping; Figure 3 This is a schematic diagram showing the shape changes of the sample layer before and after the surface trimming process; Figure 4 This is a schematic diagram for calculating the radius of curvature of the additional deformation. Detailed Implementation

[0018] The present invention will be further described in conjunction with the accompanying drawings and specific embodiments. It should be understood that these embodiments are for illustrative purposes only and are not intended to limit the scope of the invention. Furthermore, it should be understood that after reading the teachings of this invention, those skilled in the art can make various alterations or modifications to the invention, and these equivalent forms also fall within the scope defined in this application.

[0019] Combined with appendix Figure 1-4 A new method for calculating the internal residual stress of sheet metal using the layer-by-layer method, including a new formula for calculating residual stress using the layer-by-layer method: Calculation process of residual stress calculation formula 1. Sample type For sheet metal, it is more convenient to sample and process it into rectangular cross-section rod-shaped specimens, which are also easier to clamp and process. The dimensions of the rectangular cross-section specimen are assumed to be [length missing]. L , width is b Thickness is t .

[0020] 2. Deformation deflection calculation formula 2.1 The situation from the initial state of the sample to the removal of the first layer of shavings The rectangular cross-section specimen is placed on a measuring platform with supports at both ends, with a support distance of [missing information]. l A schematic diagram of the layer thickness dimensions for layer milling using the layer milling method is shown below. Figure 1 A schematic diagram of the deformation of the sample after layer-by-layer cutting is shown below. Figure 2 , Figure 1 In the figure, t(0) is the initial thickness of the sample, t(1) is the remaining thickness of the sample after the first layer is milled, and t(2)-t(10) are deduced in turn. Figure 2 The figures show the shape of the specimen before and after deformation when milling the nth layer. The dashed line represents the shape before deformation, and the solid line represents the shape after deformation. ω(i) is the deflection of the middle part of the specimen after milling the i-th layer. The shape changes of the specimen before and after machining the layer-by-layer surface are as follows: Figure 3 As shown. Figure 3 The solid line in the figure represents the sample position before the layer-by-layer machining, and the dashed line represents the sample position after the thin layer of the layer-by-layer machining is removed.

[0021] According to the deflection curve equation of a rectangular beam in mechanics of materials: (1) In equation (1), ω The displacement of the specimen at the center of the two support points is also called deflection. M Let E be the bending moment, E be the elastic modulus, I be the moment of inertia of the rectangular beam about the central axis z, and C and D be the integration constants.

[0022] according to Figure 1 The coordinate system in the figure has two boundary conditions, namely ; Substituting these two boundary conditions into equation (1), we get: , Then the equation of the deflection curve is: (2) make Then equation (2) can be transformed into: (3) Since the plate sample is initially flat, and the bending deformation occurs only after a thin layer is milled off, the balancing torque M before bending deformation is provided by the milled thin layer, and this balancing torque is provided by the residual stress within the thin layer. Let the residual stress within the thin layer be σ, the thickness of the sample before milling be t(0), and the thickness of the sample after milling the first thin layer be t(1). Then the torque generated by the residual stress of the removed thin layer is: (4) Since the torque M in equations (3) and (4) is equal, therefore equation (4) and Substituting into equation (3), we get: (5) 2.2 Case where the i-th layer of the sample is removed during machining Let the thickness of the specimen before milling be t(i-1), and the deflection at the center deflection measurement point of the specimen be... The thickness of the sample after milling is t(i), and the deflection at the center deflection measurement point of the sample is... .

[0023] Even after the i-th layer of the sample is removed, it still obeys the deflection curve equation of a rectangular beam, i.e., equation (3) still holds. However, the deformation deflection at this time is... The moment of inertia I is as follows: (6) Then equation (3) is transformed into (7) The torque generated by the residual stress of the thin layer that has been removed is: (8) Since the torque M in equations (3) and (4) is equal, substituting equations (8) and (6) into equation (7) yields: (9) In equation (9) The residual stress within the i-th layer before the i-th layer is removed. Since the specimen containing the i-th layer is obtained by milling the initial specimen layer by layer from the 1st layer to the (i-1)th layer, the specimen containing the i-th layer is actually a deformed specimen after layer removal. Since each layer removed will generate an additional stress on the i-th layer material, the residual stress calculated by equation (9) should be reduced by the additional stress on the i-th layer from the earlier removed layers to obtain the initial residual stress of the specimen. These superimposed additional stresses include two parts: one part is the stress formed by the bending deformation of the remaining specimen. Another part is due to the stress release of the milled material, which leads to the equivalent stress release of the remaining specimen. .

[0024] It can be calculated as follows: Figure 4 This is a schematic diagram for calculating the radius of curvature of the additional deformation. Since the thickness of the sample before milling (before removing the i-th layer) is t(i-1), the deflection at the center deflection measurement point of the sample is... The radius of curvature of the neutral layer of the sample is ρ. Then, the following relationship exists between the right triangles OAC: (10) Will , , , Substituting into equation (10), we get: (11) Because a rectangular cross-section elastic beam has the following relationship: (12) (13) Substituting equations (13) and (11) into equation (12), we get: (14) It can be calculated as follows: For the first layer of peeling, there is no bending deformation stress correction or peeling stress release stress correction; therefore, the internal stress of the first layer is... That is: (15) stress .

[0025] Second layer: = (16) Third layer: (17) nth level: (18) The residual stress on the cut surface of the i-th layer is: - (19).

[0026] Table 1. Comparison of residual stress data calculated using the formula of this invention and the Airbus formula for longitudinal specimens of 7050 alloy T7451 plate (165mm thick). Table 1 compares the residual stress data calculated using the formula of this invention and the Airbus formula for longitudinal specimens of 7050 alloy T7451 plate (165mm thick). As shown in Table 1, except for the data from the 14th layer onwards, the relative values ​​of the calculated residual stress are slightly larger, approximately greater than 10%. The absolute values ​​of the residual stress differences for the first 13 layers are all less than 1MPa. This difference between the two algorithms is due to the different correction methods used to calculate the residual stress. Following the correction approach of this paper... Figure 4 The arc containing BC must be a strictly circular arc for the formula derived in this paper to produce a relatively accurate result. Theoretically, the arc containing BC should be a circular arc, but if it is not a strictly circular arc in reality, the calculation result will be inaccurate.

[0027] 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 the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A novel method for calculating internal residual stress in sheet metal using the layer-by-layer method, characterized in that: 1-Sample Type The plate material was sampled and processed into rod-shaped specimens with rectangular cross-sections. The length of the rectangular cross-section specimens is assumed to be... L , width is b Thickness is t ; 2-Deformation and Deflection Calculation 2.1 - From the initial state of the specimen to the removal of the first layer of veneer: Place the rectangular cross-section specimen on a measuring platform with supports at both ends, with a support distance of [missing information]. l After the first layer is milled, the sample deforms; t(0) is the initial thickness of the sample; t(1) is the remaining thickness of the sample after the first layer is milled; t(2)-t(10) are deflected in turn; ω(i) is the deformation deflection of the middle part of the sample after the i-th layer is milled. According to the deflection curve equation of a rectangular beam in mechanics of materials: （1） In equation (1), ω The displacement of the specimen at the center of the two support points, also known as deflection. M Let E be the bending moment, E be the elastic modulus, I be the moment of inertia of the rectangular beam about the central axis z, and C and D be the integration constants. Based on the coordinate system for milling layer thickness using the layer-by-layer milling method, there are two boundary conditions, namely... ; Substituting these two boundary conditions into equation (1), we get: , Then the equation of the deflection curve is: （2） make Then equation (2) can be transformed into: （3） Since the plate sample is initially flat, and the bending deformation occurs only after a thin layer is milled off, the balancing torque M before bending deformation is provided by the milled thin layer, and this balancing torque is provided by the residual stress within the thin layer; let the residual stress within the thin layer be σ, the thickness of the sample before milling be t(0), and the thickness of the sample after milling the first thin layer be t(1), then the torque generated by the residual stress of the removed thin layer is: （4） Since the torque M in equations (3) and (4) is equal, therefore equation (4) and Substituting into equation (3), we get: （5）； 2.2 - Case where the i-th layer of the sample is removed during machining Let the thickness of the specimen before milling be t(i-1), and the deflection at the center deflection measurement point of the specimen be... The thickness of the sample after milling is t(i), and the deflection at the center deflection measurement point of the sample is... ; Even after the i-th layer of the sample is removed, it still obeys the deflection curve equation of a rectangular beam, i.e., equation (3) still holds. However, the deformation deflection at this time is... The moment of inertia I is as follows: （6） Then equation (3) is transformed into （7） The torque generated by the residual stress of the thin layer that has been removed is: （8） Since the torque M in equations (3) and (4) is equal, substituting equations (8) and (6) into equation (7) yields: （9） In equation (9) The residual stress within the i-th layer before the i-th layer is removed; since the specimen containing the i-th layer is obtained by milling the initial specimen layer by layer from the 1st layer to the (i-1)th layer, the specimen containing the i-th layer is actually a deformed specimen after layer removal; since each layer removed will generate an additional stress on the i-th layer material, the residual stress calculated by equation (9) should be reduced by the additional stress on the i-th layer from the early removed layers to obtain the initial residual stress of the specimen; these superimposed additional stresses include two parts, one part is the stress formed by the bending deformation of the remaining specimen. Another part is due to the stress release of the milled material, which leads to the equivalent stress release of the remaining specimen. ; Calculate as follows: Since the thickness of the sample before milling before removing the i-th layer is t(i-1), the deflection at the center deflection measurement point of the sample is... If the radius of curvature of the neutral layer of the sample is ρ, then the following relationship exists between the right triangles OAC: （10） Will , , , Substituting into equation (10), we get: （11） Because a rectangular cross-section elastic beam has the following relationship: （12） （13） Substituting equations (13) and (11) into equation (12), we get: （14） Calculate as follows: For the first layer of peeling, there is no bending deformation stress correction or peeling stress release stress correction; therefore, the internal stress of the first layer is... That is: （15） stress ; Second layer: = （16） Third layer: （17） nth level: （18） The residual stress on the cut surface of the i-th layer is: - （19）。

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