CN115470672A - Spatial residual stress reconstruction method - Google Patents

Abstract

The invention discloses a spatial residual stress reconstruction method, which comprises the following steps of firstly, obtaining residual stress data of a component to be measured by a traditional residual stress measurement technology; then, fitting calculation is performed on the measured experimental data by means of a least square method and a radial basis function. And finally, reconstructing the spatial multidimensional residual stress field by using the intrinsic strain theory under a finite element frame. In the reconstruction process, the intrinsic strain is introduced into the finite element frame in a thermal strain mode, and the requirements of relevant mechanical conditions are also ensured while calculation is carried out by means of finite element software. The method provided by the invention can reflect more complete multi-dimensional space residual stress field information by using limited experimental data. The method has the advantages that the residual stress with a complex distribution form under the multidimensional condition is effectively and accurately predicted by adopting the radial basis function, the defects of the traditional residual stress measurement technology are overcome, and the method has important significance in further research on the residual stress problem in the key fields of advanced manufacturing and the like.

Description

A Spatial Residual Stress Reconstruction Method

technical field

The invention belongs to the field of residual deformation field measurement, and more specifically relates to a spatial residual stress reconstruction method.

Background technique

Residual stress originates from advanced material processing and manufacturing processes such as plastic deformation, thermal mismatch, and phase transformation. The existence of residual stress will reduce machining accuracy, increase internal stress, reduce bearing capacity and fatigue strength, and easily cause cracks. Due to the significant impact of residual stress on the long-term performance of structures, the measurement, simulation and reconstruction of residual stress have always been a research hotspot in the engineering community.

The techniques for measuring residual stress mainly use experimental methods, which can be divided into destructive and non-destructive methods. Traditional experimental methods are direct and reliable, but the amount of data that can be obtained is limited and requires a certain amount of economic cost and human input. The measurement results of the general experimental method are a limited number of measuring point data. These values are discrete, and the data are usually fitted to obtain an approximate complete stress distribution. However, the stress field distribution obtained by this method has certain limitations. For example, the stress is difficult to meet the self-balancing conditions, and it is inconvenient to input it into the finite element program for analysis. Due to the limitation of the amount of data, most of the existing residual stress measurement methods cannot fully reflect the overall spatial residual stress distribution of the component.

Contents of the invention

In view of the above defects or improvement needs of the prior art, the present invention provides a spatial residual stress reconstruction method, thereby solving the technical problem that the existing residual stress measurement is difficult to fully reflect the overall spatial residual stress distribution of the component.

In order to achieve the above object, according to the first aspect of the present invention, a spatial residual stress reconstruction method is provided, including:

S1. Obtain the stress data t=[t(x ₁ ),t(x ₂ ),…,t( _xi ),…,t(x _M )] ^T of the component to be tested, where x _i is the component to be tested , t( _xi ) is the stress at _xi , and M is the number of measurement points;

S2. Establish the finite element model of the component to be tested and carry out grid division, input Young’s modulus and Poisson’s ratio, set the temperature change, and sequentially use ξ ₁ (x), ξ ₂ (x),…,ξ _k ( x) input the finite element model for the coefficient of thermal expansion to obtain the stress value at each measuring point

表征所述待测构件的未知本征应变分布，N为基函数总数，c_k为待求系数，ξ_k(x)为基函数，N≤M；Among them, with

Characterize the unknown intrinsic strain distribution of the component to be measured, N is the total number of basis functions, c _k is the coefficient to be found, ξ _k (x) is the basis function, N≤M;

S3, solve c _k according to the formula C=[S ^T S] ^-1 St to obtain the intrinsic strain distribution of the component to be measured, and import it into the finite element model as the thermal expansion coefficient to obtain the stress of the component to be measured Field distribution; where C=[c ₁ ,c ₂ ,...,c _k ,...,c _N ] ^T .

Preferably, the basis function adopts a radial basis function.

Preferably, the radial basis function is any one of Gaussian radial basis function, inverse quadratic function or inverse multi-quadratic function.

Preferably, the basis function adopts a modified radial basis function;

In two-dimensional space,

In three-dimensional space,

Among them, r is the radius of the radial basis function, x, y, z are the abscissa, ordinate, and vertical coordinates of any point in space, x _c , y _c , z _c are the abscissa, ordinate of the center point of the radial basis function Coordinates, vertical coordinates, α, β are shape coefficients.

Preferably, the modified radial basis function is any one of a modified Gaussian radial basis function, a modified inverse quadratic function or a modified inverse multi-quadratic function.

Preferably, in step S1, the stress data of the component to be tested is obtained based on a priori method.

Preferably, the prior method is any one of drilling method, ring core method, slit method, XRD method, profilometry and neutron diffraction method.

According to the second aspect of the present invention, a spatial residual stress reconstruction system is provided, including: a computer-readable storage medium and a processor;

The computer-readable storage medium is used to store executable instructions;

The processor is configured to read executable instructions stored in the computer-readable storage medium, and execute the method as described in the first aspect.

Generally speaking, compared with the prior art, the above technical solutions conceived by the present invention can achieve the following beneficial effects:

1. The method provided by the present invention, taking into account the defects of traditional residual stress measurement technology, uses the limited residual stress measurement data of components combined with the intrinsic strain theory to perform a complete reconstruction calculation of component residual stress in multiple dimensions of space, and based on the radial The basis function can effectively calculate the residual stress with complex distribution form in the multi-dimensional situation, which is of great significance to the further study of residual stress.

2. The method provided by the present invention uses finite element software to construct the overall spatial residual stress according to the finite residual stress value of the component to be tested, so as to obtain the distribution of the residual stress field of the component to be measured. There is no need to consider complex material parameters, only the thermal expansion coefficient, Young's modulus and Poisson's ratio need to be entered in the finite element software.

Description of drawings

FIG. 1 is a schematic flow chart of a spatial residual stress reconstruction method provided by an embodiment of the present invention;

Fig. 2 is a schematic diagram of Gaussian radial basis functions in one-dimensional and two-dimensional spaces;

(a) among Fig. 3 the structural representation of the complete finite element model of the component to be tested that the embodiment of the present invention provides; (b) among Fig. 3 is the structural representation of 1/2 finite element model; Among Fig. 3 (c) Schematic diagram of the selection method for the experimental measurement points;

(a) and (b) in FIG. 4 are respectively the contour schematic diagrams of the three-dimensional Gaussian radial basis functions before and after correction provided by the embodiment of the present invention;

(a) and (b) in Fig. 5 are the comparison diagrams between the residual stress reconstruction results and the target values of the paths AB and BC in Fig. 3(b) provided by the embodiment of the present invention, respectively;

(a) in FIG. 6 is the cloud diagram of the target residual stress, and (b) in FIG. 6 is the cloud diagram of the residual stress reconstructed by the method provided by the embodiment of the present invention.

detailed description

In order to make the object, technical solution and advantages of the present invention clearer, the present invention will be further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described here are only used to explain the present invention, not to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not constitute a conflict with each other.

An embodiment of the present invention provides a spatial residual stress reconstruction method, as shown in FIG. 1 , including:

S1. Obtain the stress data t=[t(x ₁ ),t(x ₂ ),…,t( _xi ),…,t(x _M )] ^T of the component to be tested, where x _i is the component to be tested , t( _xi ) is the stress at _xi , and M is the number of measurement points.

Preferably, in step S1, the stress data of the component to be tested is obtained based on a drilling method, a ring core method or a slit method.

Specifically, the stress data of the component to be tested is obtained through traditional residual stress measurement techniques, such as: drilling method, ring core method, slit method, XRD method, profile method, neutron diffraction method, etc.

S2. Establish the finite element model of the component to be tested and carry out grid division, input Young’s modulus and Poisson’s ratio, set the temperature change, and sequentially use ξ ₁ (x), ξ ₂ (x),…,ξ _k ( x) input the finite element model for the coefficient of thermal expansion to obtain the stress value at each measuring point

表征所述待测构件的未知本征应变分布，N为基函数的总个数，c_k为待求系数，ξ_k(x)为基函数，N≤M。Among them, with

To characterize the unknown eigenstrain distribution of the component to be tested, N is the total number of basis functions, c _k is the coefficient to be obtained, ξ _k (x) is the basis function, N≤M.

Specifically, a combination of a series of basis functions is used to assume the unknown intrinsic strain distribution of the component to be tested. The eigenstrain represented by a single basis function is imported into the finite element framework, and the residual stress generated by the eigenstrain is calculated under the finite element framework.

The assumed form of the unknown eigenstrain distribution is:

Among them, N is the total number of basis functions, c _k is the unknown coefficient, and ξ _k (x) is the basis function.

Furthermore, after the intrinsic strain is introduced into the finite element frame in the form of thermal strain, the corresponding residual stress field distribution will be generated. Define the thermal expansion coefficient that changes with space through the user subroutine, and import it into the finite element software as the intrinsic strain distribution for calculation. The result of the stress distribution is the residual stress distribution caused by the intrinsic strain, and then output the corresponding experimental measurement points. The residual stress value at the position.

The method provided by the invention does not need to consider complex material parameters when establishing a finite element model, and only needs to input thermal expansion coefficient, Young's modulus and Poisson's ratio.

S3, solve c _k according to the formula C=[S ^T S] ^-1 St to obtain the intrinsic strain distribution of the component to be measured, and import it into the finite element model as the thermal expansion coefficient to obtain the stress of the component to be measured Field distribution; where C=[c ₁ ,c ₂ ,...,c _k ,...,c _N ] ^T .

Specifically, since only Young's modulus, Poisson's ratio and coefficient of thermal expansion are input as material parameters in the model, it can be considered that the finite element calculation process of the present invention is a complete elastic process. Since the elastic problem is a linear problem, the residual stress T(x) in the finite element model is a linear combination of the residual stress s _k (x) produced by the intrinsic strain ξ _k (x) represented by the basis function in formula (1) given by:

Further, the measured experimental data t is calculated and reconstructed by means of the least square method and basis functions to obtain the unknown coefficient c _k of each basis function. That is, in order to obtain a reconstruction solution that is in good agreement with t, the least squares approximation constructor J is used:

In the formula, t( _xi ) is the stress value at the measurement point x _i obtained based on the traditional residual stress measurement technology, and T( _xi ) is the measurement point (also called the experimental measurement point) x _i output by the finite element model The stress value of , w _i is the weight factor.

In order to obtain a solution that agrees well with the experimental values, we need to find a set of c _k that minimizes the objective error function J. This is achieved by taking the first derivative of the function J with respect to the unknown parameter c _k and making it equal to zero, i.e.:

Formula (4) is further expanded to get:

In order to simplify the calculation, the following matrix is defined:

C＝[c ₁ ,c ₂ ,…,c _k ,…,c _N ] ^T (7)

t＝[t(x ₁ ),t(x ₂ ),…,t(x _i ),…,t(x _M )] ^T (8)

Furthermore, the residual stress reconstruction problem is finally simplified to the problem of solving the unknown coefficients of the basis functions. Combining formulas (6), (7), and (8), formula (5) is simplified into a matrix form, and the specific form is:

C＝[S ^T S] ^-1 St (9)

In the formula, C is an N×1 array composed of unknown coefficients c _k , S is the N of the residual stress s _k ( _xi ) caused by the intrinsic strain ξ _k (x) at the measurement point x _i in the finite element model ×M array, M is the number of measurement points, and t is an M×1 array composed of stress values at measurement points x _i obtained based on traditional residual stress measurement techniques.

Further, after obtaining the unknown coefficient c _k , the final complete intrinsic strain distribution can be obtained according to formula (1). The function instead imports the complete eigenstrain distribution from the final solution. The spatially complete residual stress distribution information can be obtained.

That is to say, by importing the finally obtained intrinsic strain distribution results into the finite element framework, spatially complete residual stress distribution information can be obtained.

Among them, the way of introducing the intrinsic strain into the finite element frame is: importing into the finite element frame in the form of thermal strain.

Preferably, the basis function adopts a radial basis function.

Specifically, the radial basis function is used as the basis function space, and one-dimensional, two-dimensional or three-dimensional function forms are used to meet the requirements of the residual stress field in different dimensions in the space. At the same time, each radial basis function imported into the finite element model is based on The density of experimental measuring points is distributed in space.

Preferably, the radial basis function is any one of Gaussian radial basis function, inverse quadratic function or inverse multi-quadratic function.

Specifically, as shown in Figure 2, the basic form of ξ _k (x) of the radial basis function is as follows:

In the formula, X _c is the coordinate of the center point of the radial basis function, X represents the coordinate of any point in the area to be measured, and r is the distance from a point X in the area to be measured to the center of the radial basis function.

Preferably, in the process of reconstructing the residual stress field in multi-dimensional space, it is considered to modify the radial basis function to improve the efficiency and accuracy of the reconstruction process, and the specific modification form is as follows:

In two-dimensional space,

In three-dimensional space,

Among them, r is the radius of the radial basis function, x, y, z are the abscissa, ordinate, and vertical coordinates of any point in space, x _c , y _c , z _c are the abscissa, ordinate of the center point of the radial basis function Coordinates, vertical coordinates, α, β are shape coefficients, which are selected according to the ratio of the distribution distance of experimental measuring points in all directions of space.

The radial basis functions before and after correction are shown in (a) and (b) in Figure 4, respectively.

Preferably, the modified radial basis function is any one of a modified Gaussian radial basis function, a modified inverse quadratic function or a modified inverse multi-quadratic function.

The method provided by the present invention is verified below with a specific example. This verification method is carried out under the condition that the intrinsic strain of the component to be tested is known.

The validity of the method is verified by establishing a three-dimensional finite element model and using the residual stress field calculated by the finite element as the reconstruction target. The specific steps are as follows:

Step 1: In this embodiment, since the eigenstrain of the component to be measured is known, the residual stress field is constructed by finite element simulation, and the stress data output according to the position of the measuring point in (c) in Fig. 3 is used as the component to be measured Stress data t=[t(x ₁ ),t(x ₂ ),…,t(x _i ),…,t(x _M )] ^T , the specific steps are as follows:

(1) Establish a three-dimensional model, as shown in (a) in Figure 3. For the sake of simplicity of calculation, the 1/2 model structure is taken here, as shown in (b) in Figure 3, and meshed.

In this example, the size of the original model is 160mm in length, 80mm in width, and 20mm in thickness; after taking 1/2 structure, the size of the model is 160mm in length, 40mm in width, and 20mm in thickness.

(2) Import intrinsic strain into the finite element model to generate residual stress;

The specific form of the imported eigenstrain is:

In the formula, x is the abscissa of a point in space, y is the ordinate of a point in space, w is the width of the original model of 80 mm, and t is the thickness of the original model of 20 mm.

The intrinsic strain is imported into the finite element model in the form of thermal strain, for example, ε ^* = αΔT, where ε ^* is the imported intrinsic strain, α is the coefficient of thermal expansion, ΔT is the temperature change value, and ΔT is set as a constant.

In this embodiment, complex material parameters do not need to be considered, and only the coefficient of thermal expansion, Young's modulus and Poisson's ratio need to be input. Wherein, the Young's modulus of the present embodiment is 210GPa, and the Poisson's ratio is 0.3.

(3) Import the above formula as the intrinsic strain distribution into the finite element software for calculation, and the obtained stress distribution result is the residual stress distribution caused by the intrinsic strain. The stress results are shown in (a) and (b) in Figure 5 , where Target represents t, and then output the stress value t=[t(x ₁ ),t(x ₂ ),…,t( _xi ),…,t(x _M )] ^T at the corresponding measuring point position.

The distribution of target experimental measuring points of this embodiment is shown in (c) in Figure 3, specifically comprising: get 7 groups of experimental measuring points along the z-axis direction, each group of experimental measuring points z-direction is 24mm apart; along the y-axis direction by 0mm Take 4 groups of experimental measurement points at 20mm, the maximum distance between the measurement points is 8mm; take points at a distance of 4mm along the x-axis direction, and at the same time, according to the ratio of the distance between the experimental measurement points in each direction of space, the radial basis function of this embodiment is corrected The parameter α takes 6 and β takes 2.

Step 2: Using a combination of a series of basis functions to assume the unknown intrinsic strain distribution of the object to be measured in the component to be measured.

式中，N为基函数总数，c_k为未知系数，ξ_k(x,y,z)为基函数。In this embodiment, the hypothetical form of the unknown intrinsic strain distribution in the component to be measured is:

In the formula, N is the total number of basis functions, c _k is the unknown coefficient, and ξ _k (x, y, z) is the basis function.

The basis function in this embodiment adopts the radial basis function, that is, the radial basis function is used as the basis function space. Further, the radial basis function adopts a three-dimensional Gaussian radial basis function, and the specific form is:

r_k为空间中任意一点到第k个径向基函数的中心点的距离，x为空间任意一点横坐标，y为空间任意一点纵坐标，z为空间任意一点竖坐标，x_kc为第k个径向基函数的中心点横坐标，等于第k个实验测点的横坐标，y_kc为径向基函数中心点纵坐标，z_kc为径向基函数中心点竖坐标，α、β为径向基函数修正系数，根据实验测点在空间各方向上分布间距的比值选取。In the formula,

r _k is the distance from any point in space to the center point of the kth radial basis function, x is the abscissa of any point in space, y is the ordinate of any point in space, z is the vertical coordinate of any point in space, x _kc is the kth The abscissa of the center point of a radial basis function is equal to the abscissa of the kth experimental measuring point, y _kc is the ordinate of the center point of the radial basis function, z _kc is the vertical coordinate of the center point of the radial basis function, and α and β are The radial basis function correction coefficient is selected according to the ratio of the distribution distance of the experimental measuring points in each direction of space.

The intrinsic strain represented by a single basis function is imported into the finite element framework, and the residual stress generated by the single intrinsic strain is calculated under the finite element framework. The specific steps are as follows:

(1) Establish the same three-dimensional model as step 1, and perform grid division;

(2) Apply a predefined initial temperature field and temperature change on the model

(3) Input material parameters;

Same as step 1, only need to input thermal expansion coefficient, Young's modulus and Poisson's ratio. Among them, Young's modulus is taken as 210GPa, and Poisson's ratio is taken as 0.3. According to formula (2), the specific functional form of the distribution of intrinsic strain is the three-dimensional Gaussian radial basis function in step 1.

Since only Young's modulus, Poisson's ratio and thermal expansion coefficient are input as material parameters in the model, it can be considered that the finite element calculation process of the present invention is a complete elastic process. Since the elastic problem is a linear problem, the residual stress T(x) in the finite element model is a linear combination of the residual stress s _k (x) produced by the intrinsic strain ξ _k (x) represented by the basis function in formula (1) given by:

It can be seen from formula (1) that this embodiment needs to distribute N basis functions ξ _k (x, y, z) on the component to be tested, and take the position of the target experimental measurement point as the center point position of each radial basis function. Therefore, the total number of basis function distributions in this embodiment is the same as the total number of selected experimental measurement points, that is, N=M=308.

(4) Import the three-dimensional Gaussian radial basis function as the intrinsic strain distribution into the finite element software for calculation, and the obtained stress distribution result is the residual stress distribution caused by the intrinsic strain, and then output the intrinsic strain in the form of a single basis function distribution The predicted value of residual stress under , that is, s _k (x) in the formula, finally forms the matrix S.

Since a total of N basis functions are required, step 2 requires the Gaussian radial basis functions of N different center point positions to be imported into the finite element model as eigenstrains for calculation, and finally output N sets of stress data to form a stress matrix:

Step 3: take the stress data t obtained in step 1 as the target experimental data of this embodiment, and calculate and reconstruct the obtained target experimental data t by means of the least square method to solve the unknown coefficient c _k of the basis function; in this step , using least squares to approximate the constructor J, and finally get [S ^T S]C=St.

The problem of residual stress reconstruction is finally simplified to the problem of solving the unknown coefficients of basis functions. Since S and t are known, the unknown coefficient c _k can be obtained according to the formula C=[S ^T S] ^-1 St.

After the unknown coefficient c _k is obtained, the final and complete intrinsic strain distribution can be obtained according to formula (1). The method of importing the intrinsic strain distribution into the finite element is the same as in step 2, that is, importing a single radial basis function in step 2 Instead, import the complete eigenstrain distribution from the final solution. The complete spatial residual stress distribution information of the component to be tested can be obtained.

(b) in Fig. 5 shows the residual stress field obtained after finite element calculation after importing the final eigenstrain, compared with the target residual stress field shown in (a) in Fig. 5, it can be seen that the present invention provides The method reconstructed the three-dimensional residual stress field to be tested very well. Further select the two paths AB and BC in Figure 3(b) to compare the residual stress results. The comparison results are shown in Figure 6(a) and (b). It can be seen that for the two selected paths, the residual The stress reconstruction prediction value is in good agreement with the target value, which further verifies the effectiveness of the present invention.

The method provided by the present invention takes into account the defects of the traditional residual stress measurement technology, uses limited experimental data combined with the intrinsic strain theory to carry out complete reconstruction prediction of component residual stress in multiple dimensions of space, and effectively analyzes the multidimensional It is of great significance for the further study of residual stress to accurately predict the residual stress with a complex distribution form.

An embodiment of the present invention provides a spatial residual stress reconstruction system, including: a computer-readable storage medium and a processor;

The computer-readable storage medium is used to store executable instructions;

The processor is configured to read executable instructions stored in the computer-readable storage medium, and execute the method as described in any one of the above embodiments.

It is easy for those skilled in the art to understand that the above descriptions are only preferred embodiments of the present invention, and are not intended to limit the present invention. Any modifications, equivalent replacements and improvements made within the spirit and principles of the present invention, All should be included within the protection scope of the present invention.

Claims (8)

1. A spatial residual stress reconstruction method, characterized in that, comprising: S1. Obtain the stress data t=[t(x ₁ ),t(x ₂ ),…,t( _xi ),…,t(x _M )] ^T of the component to be tested, where x _i is the component to be tested , t( _xi ) is the stress at _xi , and M is the number of measurement points; S2. Establish the finite element model of the component to be tested and carry out grid division, input Young’s modulus and Poisson’s ratio, set the temperature change, and sequentially use ξ ₁ (x), ξ ₂ (x),…,ξ _k ( x) input the finite element model for the coefficient of thermal expansion to obtain the stress value at each measuring point

Among them, with

Characterize the unknown intrinsic strain distribution of the component to be measured, N is the total number of basis functions, c _k is the coefficient to be found, ξ _k (x) is the basis function, N≤M; S3, solve c _k according to the formula C=[S ^T S] ^-1 St to obtain the intrinsic strain distribution of the component to be measured, and import it into the finite element model as the thermal expansion coefficient to obtain the stress of the component to be measured Field distribution; where C=[c ₁ ,c ₂ ,...,c _k ,...,c _N ] ^T . 2. The method according to claim 1, wherein the basis function is a radial basis function. 3. The method according to claim 2, wherein the radial basis function is any one of a Gaussian radial basis function, an inverse quadratic function or an inverse multi-quadratic function. 4. The method according to claim 1, wherein the basis function adopts a modified radial basis function; In two-dimensional space,

In three-dimensional space,

Among them, r is the radius of the radial basis function, x, y, z are the abscissa, ordinate, and vertical coordinates of any point in space, x _c , y _c , z _c are the abscissa, ordinate of the center point of the radial basis function Coordinates, vertical coordinates, α, β are shape coefficients. 5. The method according to claim 4, wherein the modified radial basis function is any one of a modified Gaussian radial basis function, a modified inverse quadratic function or a modified inverse multi-quadratic function. 6. The method according to claim 1, characterized in that, in step S1, the stress data of the component to be tested is obtained based on a priori method. 7. The method according to claim 6, wherein the prior method is any one of the drilling method, the ring core method, the slit method, the XRD method, the profilometry method, and the neutron diffraction method. 8. A spatial residual stress reconstruction system, comprising: a computer-readable storage medium and a processor; The computer-readable storage medium is used to store executable instructions; The processor is configured to read executable instructions stored in the computer-readable storage medium, and execute the method according to any one of claims 1-7.

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