A direct method for calculating the stress intensity factor at the crack tip

Abstract

The present application relates to the field of computational mechanics, and specifically discloses a direct method for calculating the stress intensity factor at a crack tip, which comprises the steps of: 1. obtaining the stress at the crack tip: obtaining the stress at the crack tip within a specific range at the crack tip; within the specific range, the stress at the crack tip can be calculated using a method containing only the stress intensity factor K I , K II The terms of and T stress are approximately expressed, and the stress at the crack tip shows a linear relationship in the logarithmic coordinate system; Step 1, calculate the stress intensity factor at the crack tip: According to the stress at the crack tip, the stress intensity factor K at the crack tip is calculated by fitting using the least squares method. I and K II The method provided in the present application only needs to obtain the stress at the crack tip, and the stress intensity factor at the crack tip can be obtained by fitting calculation. No integral method is used, and no intermediate variables are introduced. The calculation method is simple and efficient.

Description

Technical Field

[0001] The present application relates to the field of computational mechanics, and more particularly to a direct method for calculating crack tip stress intensity factor. Background Art

[0002] In the field of engineering structures and materials science, crack initiation and propagation is a complex and important research topic. The formation of cracks is usually affected by the combined effects of multiple factors. First, at the microscopic scale of the material, defects such as grain structure and pores will lead to stress concentration, which in turn promotes crack initiation and propagation. Secondly, fatigue damage will also lead to crack formation to a certain extent. Under cyclic loading and unloading, even at stresses far below the yield limit of the material, microscopic defects will gradually evolve into macroscopic cracks with the accumulation of time. In addition, the geometric asymmetry and connection method of engineering structures during the design and manufacturing process will also lead to the appearance of cracks. For example, local temperature and stress changes in the structure during welding will lead to a decrease in the local strength of the structure, causing cracks to initiate and propagate in these areas. Finally, some non-standard operating conditions, such as overload and vibration changes, will subject the structure to stresses beyond the design limit, causing crack initiation and propagation. Therefore, in order to ensure the long-term stable operation of engineering structures, it is very important to correctly evaluate and control the state of cracks.

[0003] The stress intensity factor at the crack tip plays a core role in the study of crack propagation and fracture behavior. It can not only predict the fracture of materials under specific stress conditions, but also is widely used in engineering practice to determine the state of cracks. The key role of the calculation of the stress intensity factor at the crack tip in engineering applications is mainly reflected in the following aspects: First, it has a profound impact on the crack propagation behavior of engineering materials for the selection of suitable engineering materials. By accurately calculating the stress intensity factor, engineers can evaluate the sensitivity of different materials to crack propagation. Secondly, in-depth insight into the microscopic damage and crack propagation mechanisms of materials is crucial for the development of new materials that are more durable and high-performance. The stress intensity factor helps to reveal the behavioral characteristics of materials at the microscopic level, thereby promoting the progress and innovation of materials science. Finally, in many key application fields, such as aviation, bridges, and nuclear power facilities, it is particularly important to ensure the long-term safety and reliability of structures. Through the study of the stress intensity factor at the crack tip, engineers can promptly and accurately determine when damaged components need to be repaired or replaced, thereby greatly improving the safe operation level of key structures. In summary, the development of an efficient method for calculating the crack tip stress intensity factor is not only of great significance for improving the accuracy of engineering design and material selection, but also has irreplaceable value for ensuring the long-term safety and reliable operation of engineering structures.

[0004] The traditional method uses the interaction integral method to calculate the stress intensity factor at the crack tip. The main idea is to use the J integral to superimpose the known auxiliary stress field in the existing stress field to form a new stress field, and then obtain the stress intensity factor through numerical calculation. The traditional interaction integral method based on the J integral is used to calculate the stress intensity factor. The integral calculation is performed at the crack tip, and the stress at the crack tip has stress concentration. Therefore, the calculation accuracy of the traditional method will be affected by the integral area. At the same time, the calculation of the traditional method also requires the use of auxiliary stress states, and the calculation is more complicated. Summary of the invention

[0005] In order to develop a simple and efficient method for calculating the crack tip stress intensity factor, the present application provides a direct method for calculating the crack tip stress intensity factor.

[0006] According to the work of Berto and Lazzarind, "Berto F., Lazzarin P., On higher order terms in the crack tip stress field. International Journal of Fracture. 2010, 161(2), 221-226", the general form of the crack tip stress function can be expressed as:

[0007]

[0008] In the formula, a1, a2, a3, a4, a5, a6, and a7 represent seven parameters related to the crack tip, r is the distance from a certain point to the crack tip in the polar coordinate system, and θ is the angle of rotation from the positive x-axis to the r segment in the polar coordinate system. represents the high-order term of the stress function. ω represents the rotation angle of the crack tip, and θ=ω+π. The local coordinates of the crack tip are defined as Figure 1 shown.

[0009] The first three parameters a1, a2 and a3 in formula (1) are respectively related to the type I stress intensity factor K I , Type II stress intensity factor K II And T stress, which can be expressed as:

[0010] and a3 = T / 4

[0011] Then, the seven parameters in formula (1) are calibrated. After calibration, it is found that the axial stress σ at the crack tip is rr , hoop stress σθθ and shear stress σ rθ It exhibits a linear change characteristic within a certain range from the crack tip. This range is determined as the range where η is less than 0.1%, where η represents the ratio of the distance from the interpolation point to the crack tip to the crack length. And within this range, only K is considered. I , K II The influence of T stress can also better obtain the crack tip stress. Figure 2-4 The comparison diagrams of the crack tip stress in the directions of θ=0°, 90° and -90° are shown respectively. Figure 2-4 It can be seen that in the range of η less than 0.1%, the axial stress σ at the crack tip rr , hoop stress σ θθ and shear stress σ rθ It can be approximated by considering only K I , K II It is expressed as the crack tip stress affected by the T stress, and in the logarithmic coordinate system, the stress shows a good linear relationship within this range.

[0012] In addition, according to the research of Mirlohi and Aliha, "Mirlohi SN, Aliha MRM. Crack growth path prediction for the angled cracked plate using higher order terms of Williams series expansion [J]. Engineering Solid Mechanics, 2013, 1 (3): 77-84. (Mirlohi SN, Aliha MRM, Prediction of crack growth path in cracked plate using higher order terms of Williams series expansion, Engineering Solid Mechanics, 2013, Vol. 1, No. 3, Pages: 77-84)", when η is very close to 0, without considering other higher order terms, the crack initiation angle can be calculated very accurately using only the first two terms of the Williams series. The first two terms of the Williams series correspond to the stress intensity factor K at the crack tip. I , K II and T stress. The crack tip stress obtained from the first two terms of the Williams series can be expressed as:

[0013]

[0014] In the formula, σ x +σ y It represents the sum of horizontal and vertical stresses at the crack tip.

[0015] At the same time, since the sum of horizontal and vertical stresses is equal to the sum of radial and circumferential stresses, that is:

[0016] σ x +σ y =σ rr +σ θθ (3)

[0017] In summary, when η is less than 0.1%, the crack tip stress can be calculated by considering only K I , K II and T stress, and in the logarithmic coordinate system, the stress shows a good linear relationship within this range. Based on this characteristic of crack tip stress, the present invention proposes a direct method for calculating the stress intensity factor at the crack tip. Compared with the traditional method of solving the stress intensity factor K I and K II This method does not require numerical integration and does not introduce any intermediate variables. This method only needs to obtain the stress at the crack tip and obtain the stress intensity factor at the crack tip through fitting calculation. The calculation method is simple, so it is called a direct method for solving the stress intensity factor.

[0018] A direct method for calculating the crack tip stress intensity factor provided in this application adopts the following technical solution:

[0019] A direct method for calculating the stress intensity factor at a crack tip comprises the following steps:

[0020] Obtaining the crack tip stress: Obtain the stress at the crack tip within a specific range at the crack tip; within the specific range, the crack tip stress can be calculated using only the stress intensity factor K I , K II The terms of and T stress are approximately expressed, and the crack tip stress shows a linear relationship in the logarithmic coordinate system;

[0021] Calculate the stress intensity factor at the crack tip: According to the stress at the crack tip, the stress intensity factor K at the crack tip is calculated by fitting using the least squares method. I and K II .

[0022] Further, in the step of obtaining the crack tip stress, the specific range is a range where η is less than 0.1%, and η represents the ratio of the distance from the interpolation point to the crack tip to the crack length.

[0023] Further, in the step of obtaining the crack tip stress, the obtained crack tip stress includes the horizontal stress σ at the crack tip x and vertical stress σ y .

[0024] Furthermore, according to the stress at the crack tip, the stress intensity factor K at the crack tip is calculated using the least squares method. I The methods include:

[0025] The crack tip stress is expressed as I The linear equation E=K I F+G represents, where E is the crack tip stress, F is a variable related to r, r is the distance from a point to the crack tip in the polar coordinate system, and G is a constant;

[0026] Obtain the data set of crack tip stress E and r, and then use the least squares method to fit the data to obtain the stress intensity factor K I .

[0027] Further, we get the linear equation E = K I The F+G method includes:

[0028] The crack tip stress expression obtained according to the first two terms of the Williams series is:

[0029]

[0030] Among them, σ x +σ y represents the sum of the horizontal and vertical stresses at the crack tip, and θ is the angle of rotation from the positive x-axis to the r segment in polar coordinates;

[0031] When θ=90°:

[0032]

[0033] When θ=-90°:

[0034]

[0035] When r takes the same value, we get:

[0036]

[0037] Rewrite the linear equation E = K I F+G, where E represents (σ x +σ y ) θ＝-90° +(σ x +σ y ) θ＝90° The value of F represents G represents twice the T stress and is a constant.

[0038] Furthermore, according to the stress at the crack tip, the stress intensity factor K at the crack tip is calculated using the least squares method. IIThe methods include:

[0039] The crack tip stress is expressed as II The linear equation M=K II N represents, where M is the crack tip stress, N is a variable related to r, and r is the distance from a point to the crack tip in the polar coordinate system;

[0040] Obtain the data set of crack tip stress M and r, and then use the least squares method to fit the data to obtain the stress intensity factor K II .

[0041] Further, we get the linear equation M = K II The methods of N include:

[0042] According to θ=90° and θ=-90°, σ x +σ y When r takes the same value, we get:

[0043]

[0044] Rewrite the linear equation M = K II N, where M represents (σ x +σ y ) θ＝-90° -(σ x +σ y ) θ＝90° The value of N represents

[0045] The present application also provides a computer-readable storage medium having a computer program stored thereon. When the computer program is executed by a processor, a direct method for calculating a crack tip stress intensity factor is implemented.

[0046] The present application also provides a computer program product, comprising a computer program, which implements a direct method for calculating a crack tip stress intensity factor when the computer program is executed by a processor.

[0047] In summary, this application includes the following beneficial technical effects:

[0048] The method provided in this application only needs to obtain the stress at the crack tip, and the stress intensity factor K at the crack tip can be obtained by fitting. I and K II Compared with the traditional method of solving stress intensity factor, this method does not need to calculate numerical integration method, nor does it introduce any intermediate variables. The calculation method is simple and efficient. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] Figure 1is a schematic diagram of the local coordinate definition of the crack tip;

[0050] Figure 2 It is the stress comparison diagram at the crack tip in the θ=0° direction;

[0051] Figure 3 It is the stress comparison diagram at the crack tip in the direction of θ = 90°;

[0052] Figure 4 It is the stress comparison diagram at the crack tip in the θ=-90° direction;

[0053] Figure 5 is a schematic diagram of a finite width plate with edge cracks in Example 2;

[0054] Figure 6 3 is a comparison chart of the results of calculating the stress intensity factor using the method of the present invention and the traditional method based on J-integral in Example 2. DETAILED DESCRIPTION

[0055] The following is combined with Figure 1-6 This application is described in further detail.

[0056] Example 1

[0057] The present application embodiment discloses a direct method for calculating the stress intensity factor at the crack tip, comprising the following steps:

[0058] Step S1, obtaining the crack tip stress:

[0059] The stress at the crack tip is obtained within a specific range of the crack tip. Within a specific range, the stress at the crack tip can be expressed as I , K II The terms of and T stress are approximately expressed, and the crack tip stress shows a linear relationship in the logarithmic coordinate system.

[0060] Specifically, the specific range is a range where η is less than 0.1%, where η represents the ratio of the distance from the interpolation point to the crack tip to the crack length.

[0061] Furthermore, the obtained crack tip stress includes the horizontal stress σ at the crack tip x and vertical stress σ y .

[0062] Step S2, calculating the stress intensity factor at the crack tip, specifically comprising the following steps:

[0063] S2-1. According to the stress at the crack tip, the stress intensity factor K at the crack tip is calculated by fitting using the least squares method. I , the specific method is:

[0064] S2-1-1. The crack tip stress is calculated by using the stress intensity factor K I The linear equation E=K I F+G represents, where E is the crack tip stress, F is a variable related to r, r is the distance from a point to the crack tip in the polar coordinate system, and G is a constant;

[0065] Get the linear equation E = K I The F+G method includes:

[0066] According to formula (2):

[0067] When θ=90°:

[0068]

[0069] When θ=-90°:

[0070]

[0071] When r takes the same value, we get:

[0072]

[0073] Rewrite the linear equation:

[0074] E=K I F+G (7)

[0075] Where E represents (σ x +σ y ) θ＝-90° +(σ x +σ y ) θ＝90° The value of F represents G represents twice the T stress and is a constant.

[0076] Using the crack tip stress at specific angles θ = 90° and θ = -90° is beneficial to reduce the stress intensity factor K I and K II The mutual influence between them.

[0077] S2-1-2. Obtain the data set of crack tip stress E and r, and then use the least squares method to fit the data to obtain the stress intensity factor K I , the specific method is as follows:

[0078] The fitting error for each data point can be expressed as K I F j +GE j , subscript j is the number of the sampling point. In order to obtain the best fitting curve, the fitting error of the data needs to satisfy the following expression:

[0079]

[0080] Among them, m represents the total amount of data, S represents the sum of square errors, and S min Represents the minimum sum of squared errors.

[0081] In order to find the minimum S min ,have:

[0082]

[0083] in, is the partial differential symbol.

[0084] Solving the above formula, we can get:

[0085]

[0086] Stress intensity factor K at the crack tip I It can be obtained by solving formula (10).

[0087] S2-2. According to the stress at the crack tip, the stress intensity factor K at the crack tip is calculated using the least squares method. II , the specific method is:

[0088] S2-2-1. The crack tip stress is calculated by using the stress intensity factor K II The linear equation M=K II N represents, where M is the crack tip stress, N is a variable related to r, and r is the distance from a certain point to the crack tip in the polar coordinate system.

[0089] Get the linear equation M = K II The methods of N include:

[0090] According to formulas (4) and (5), when r takes the same value, we get:

[0091]

[0092] Rewrite the linear equation:

[0093] M=K II N (12)

[0094] Where M represents (σ x +σ y ) θ＝-90° -(σ x +σ y ) θ＝90° The value of N represents

[0095] S2-2-2. Obtain the data set of crack tip stress M and r, and then use the least squares method to fit the data to obtain the stress intensity factor K. II , the specific method is as follows:

[0096] Using the same fitting method as in step S2-1-2, since there is no T stress term in equation (12), the fitting formula can be simplified to:

[0097]

[0098] Among them, M j Indicates that at the interpolation point r j Where The value of N j Indicates that it corresponds to M j At (σ x +σ y ) θ＝-90° -(σ x +σ y ) θ＝90° The value of .

[0099] Stress intensity factor K at the crack tip II It can be obtained by solving formula (13).

[0100] A specific numerical example is given below:

[0101] Take 10 sets of data at the crack tip: r, (σ x +σ y ) θ＝90° and (σ x +σ y ) θ＝-90° , which can be expressed as follows:

[0102] r = [0.00001, 0.000012, 0.000023, 0.000034, 0.000045, 0.000056, 0.000067, 0.000078, 0.000089, 0.000100]; unit: m;

[0103] (σ x +σ y ) θ＝90° =[0.005055, 0.005056, 0.005057, 0.005058, 0.005059, 0.005060, 0.005061, 0.00505762, 0.005063, 0.005064]; unit: Pa;

[0104] (σ x +σ y ) θ＝-90°=[0.01440, 0.01441, 0.01442, 0.01443, 0.01444, 0.01445, 0.01446, 0.01447, 0.01448, 0.01449]; unit: Pa.

[0105] When calculating, first get:

[0106] (σ x +σ y ) θ＝90° +(σ x +σ y ) θ＝-90° = [0.019455, 0.019516, 0.019577, 0.019638, 0.019699, 0.01976, 0.019821, 0.01982762, 0.019543, 0.019554];

[0107] (σ x +σ y ) θ＝-90° -(σ x +σ y ) θ＝90° = [0.009345, 0.009354, 0.009363, 0.009372, 0.009381, 0.00939, 0.009399, 0.00941238, 0.009417, 0.009426];

[0108] r and (σ x +σ y ) θ＝90° +(σ x +σ y ) θ＝-90° Substitute the ten sets of data into the fitting formula (10) and calculate

[0109] r and (σ x +σ y ) θ＝-90° -(σ x +σ y ) θ＝90° Substitute the ten sets of data into the fitting formula (13) and calculate

[0110] It should be noted that for the crack tip stress, the T stress term is much smaller than the K I and K II The order of the term is higher, that is, it contains K I and K II The value of the term will be much larger than the term containing T stress. Therefore, calculate K I and KII The error produced in the calculation will have a great impact on the calculation of T stress, so this method is not recommended for solving T stress.

[0111] The embodiment of the present application further discloses a computer-readable storage medium having a computer program stored thereon. When the computer program is executed by a processor, a direct method for calculating a crack tip stress intensity factor is implemented.

[0112] The embodiment of the present application further discloses a computer program product, including a computer program, which implements a direct method for calculating a crack tip stress intensity factor when executed by a processor.

[0113] Example 2

[0114] This embodiment uses a finite width plate problem with edge cracks to illustrate the reliability of the method of the present invention. Figure 5 , the width b of the plate is 1 meter, the length is 3 meters, and a uniform load of σ = 1 MPa is applied to the top and bottom of the plate. The analytical solution of the stress intensity factor of this problem can be expressed as:

[0115]

[0116] Under different crack lengths a, the stress intensity factor at the crack tip is calculated using the method of the present invention and the traditional method based on J integral. The extended finite element method is used to model the model. When the model mesh density is the same, the results calculated by the two methods are as follows: Figure 6 As shown, it can be seen that the method of the present invention can obtain a solution with similar accuracy to the traditional method. However, the traditional method based on J integral requires integral calculation, while the method of the present invention only requires simple data fitting. Therefore, the method provided by the present invention has good applicability.

[0117] The above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit the same. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that the technical solutions described in the aforementioned embodiments may still be modified, or some of the technical features may be replaced by equivalents. However, these modifications or replacements do not deviate the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A direct method for calculating the stress intensity factor at the crack tip, characterized in that: The following steps are involved: Get crack tip stress: Get the stress at the crack tip within a specific range of the crack tip; Within the specified range, the crack tip stress can be expressed using only the stress intensity factor , and T The stress terms are approximately expressed, and the crack tip stress shows a linear relationship in the logarithmic coordinate system; Calculate the stress intensity factor at the crack tip: According to the stress at the crack tip, the stress intensity factor at the crack tip is calculated by fitting using the least squares method. and ; In the step of obtaining the crack tip stress, the obtained crack tip stress includes the horizontal stress at the crack tip. and vertical stress ; According to the stress at the crack tip, the stress intensity factor at the crack tip is calculated using the least squares fitting method. The methods include: The crack tip stress is expressed as The linear equation Indicates that E is the crack tip stress, F is with r Related variables, r is the distance from a certain point to the crack tip in the polar coordinate system, G is a constant; Obtaining the crack tip stress E dataset, and r The data set is then fitted using the least squares method to obtain the stress intensity factor ; According to the stress at the crack tip, the stress intensity factor at the crack tip is calculated using the least squares fitting method. The methods include: The crack tip stress is expressed as The linear equation Indicates that M is the crack tip stress, N is with r Related variables, r is the distance from a certain point to the crack tip in the polar coordinate system; Obtaining the crack tip stress dataset, and r The data set is then fitted using the least squares method to obtain the stress intensity factor .

2. A direct method for calculating crack tip stress intensity factor according to claim 1, characterized in that: In the step of obtaining the crack tip stress, the specific range is η Less than 0.1% range, η It represents the ratio of the distance from the interpolation point to the crack tip to the crack length.

3. A direct method for calculating crack tip stress intensity factor according to claim 1, characterized in that: Get the linear equation The methods include: The crack tip stress expression obtained according to the first two terms of the Williams series is: in, represents the sum of horizontal and vertical stresses at the crack tip, θ Rotate from the positive x-axis to r The angle of the line segment; when hour: when hour: when r When the values ​​are the same, we get: Rewrite the linear equation ,in express The value of express , G Indicates twice T Stress is a constant.

4. A direct method for calculating crack tip stress intensity factor according to claim 1, characterized in that: Get the linear equation The methods include: according to and hour The expression of r When the values ​​are the same, we get: Rewrite the linear equation ,in express The value of express .

5. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the direct method for calculating the crack tip stress intensity factor according to any one of claims 1 to 4 is implemented.

6. A computer program product, comprising a computer program, characterized in that: When the computer program is executed by a processor, the direct method for calculating the crack tip stress intensity factor according to any one of claims 1 to 4 is implemented.

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