Calculation method for fracture toughness and fatigue crack propagation rate of single-sided notched tensile specimens with deflection cracks

Abstract

This invention relates to a method for calculating the fracture toughness and fatigue crack propagation rate of a single-sided notched tensile specimen considering a deflection crack, comprising: S1, establishing a two-dimensional plane stress model of the single-sided notched tensile specimen containing a deflection crack; S2, meshing the two-dimensional plane stress model; S3, calculating the stress intensity factor under different initial crack lengths, crack deflection angles, and crack propagation lengths. K I , K II Softness C and normalized stress intensity factor f 11 , f 22 Normalized Softness u S4. Expressions are established through curve fitting to relate the normalized fracture parameters to the ratio of projected crack length to specimen width and the crack deflection angle. This invention provides formulas for calculating the compliance and stress intensity factors of single-sided notched tensile specimens with deflection cracks under different initial crack lengths, deflection crack angles, and crack propagation lengths. This enables accurate calculation of fracture toughness and fatigue crack propagation rate, providing technical support for high-precision design based on damage tolerance.

Description

Technical Field

[0001] This invention belongs to the field of fracture mechanics technology, specifically relating to a method for calculating the fracture toughness and fatigue crack propagation rate of a single-sided notched tensile specimen considering deflection cracks. Background Technology

[0002] Accurate calculation of material fracture toughness and fatigue crack propagation rate is a prerequisite for damage tolerance design. Single-sided notched tensile specimens, as a type of low-constraint specimen, are widely used for testing the mechanical properties of materials, especially in pipes, thin-walled structures, and other low-constraint stress conditions. However, real-world structural materials, such as composite materials, magnesium alloys, titanium alloys, and single-crystal alloys, are anisotropic materials, and they will also develop crack deflection under pure type I loading; currently, there is no universally applicable solution.

[0003] Currently, many scholars use the finite element method (FEM) to calculate fracture parameters such as compliance and stress intensity factor under different geometric dimensions to address this type of deflection crack problem. However, such FEM calculations under a single or limited number of geometric dimensions cannot be directly used by other scholars or engineers, creating unnecessary burdens. On the other hand, using the calculation results of straight cracks, i.e., non-deflection cracks, for equivalent treatment would lead to dangerous or overly conservative test results.

[0004] To address this issue, our team filed a patent application in 2023 entitled "Calculation Method for Fatigue Crack Propagation Length and Rate of Compact Tensile Specimens Considering Crack Deflection," which was subsequently granted (Publication No.: CN116738780B). This patent application employs the finite element method combined with data regression analysis to establish compliance and stress intensity factor formulas for compact tensile specimens with deflection cracks, enabling predictions for different geometric dimensions. These formulas have been successfully applied to calculate crack length and fatigue crack propagation rate. The research results demonstrate that considering crack deflection is of great significance for developing a compliance-based crack propagation length prediction method and achieving accurate calculation of fatigue crack propagation rate.

[0005] However, due to the significant differences in structure and loading between these two types of specimens, the formulas developed based on compact tensile specimens cannot be directly applied to single-sided notched tensile specimens. Therefore, it is urgent to establish calculation formulas applicable to single-sided notched tensile specimens using a similar method. Summary of the Invention

[0006] The purpose of this invention is to address the deficiencies and shortcomings of existing technologies by providing a method for calculating the fracture toughness and fatigue crack propagation rate of a single-sided notched tensile specimen considering deflection cracks, establishing formulas relating compliance and stress intensity factors to structural dimensions, and achieving accurate calculation of fracture toughness and fatigue crack propagation rate.

[0007] The technical problem solved by this invention is achieved through the following technical solution: A method for calculating the fracture toughness and fatigue crack propagation rate of a single-sided notched tensile specimen considering deflection cracks. The method establishes calculation formulas for compliance and stress intensity factor under different initial crack lengths, crack deflection angles, and crack propagation lengths, thereby achieving accurate calculation of the fracture performance and fatigue crack propagation rate of a single-sided notched tensile specimen considering crack deflection. The steps of the method are as follows: S1. A two-dimensional plane stress model of a single-sided notched tensile (SENT) specimen with a deflection crack is established using the finite element analysis software Abaqus. The elastic modulus of the material is input. E Compared to Poisson n A uniformly distributed load of equal magnitude is applied to the top and bottom surfaces of the SENT specimen. One corner of the SENT specimen is constrained to prevent rigid body translation while allowing in-plane rotation, in order to simulate pin loading conditions rather than clamping conditions. S2. The mesh is generated using an 8-node reduced integral plane stress element (CPS8R). The mesh at the crack tip is refined and singular elements are introduced. A focused ring mesh is used with a minimum radial dimension of 0.005 mm. S3. The finite element analysis software Abaqus calculates the stress intensity factor under different initial crack lengths, crack deflection angles, and crack propagation lengths using the interaction integral method. K I , K II Softness C The stress intensity factor is obtained by the derivative relationship between CMOD and load; after normalization, it is obtained. f 11 , f 22 and normalized flexibility u ; S4. For different crack deflection angles β (0°~90°), the normalized fracture parameters and the ratio of projected crack length to specimen width were established by curve fitting. a p / W and crack deflection angle β Empirical expressions between them: ; in: f The normalized stress intensity factor includes the normalized stress intensity factor. f 11 , f 22 ; The projected crack length; a 0 represents the initial crack length; Da This refers to the crack propagation length;β The angle of crack deflection; W The width of the sample; x i These are the fitting parameters; ; in: u To normalize the softness, These are the fitting parameters.

[0008] Furthermore, the specific structural dimensions of the single-sided notched tensile (SENT) specimen containing the deflection crack are as follows: Specimen width W Fixed at 10 mm; high H / W =5; The crack is in the middle of the sample, and the initial crack length is... a 0 corresponds to a 0 / W =0, 0.2, 0.3, 0.4, 0.5; Crack deflection angle β They are 0°, 15°, 30°, 45°, 60°, 75°, and 90° respectively; Crack propagation length Da Covering the remaining ligament area; Uniformly distributed stress s nom =0.1 MPa; elastic modulus E =200000 MPa, Poisson's ratio n =0.3.

[0009] Moreover, the aforementioned flexibility C Normalized stress intensity factor f 11 , f 22 and normalized flexibility u The calculation steps are as follows: Select two symmetrical nodes at the crack and ensure the initial spacing between them. D Within the recommended range, let the coordinates of these two nodes during the loading process be (x1, y1) and (x2, y2), respectively. Then, the mathematical expression for the crack opening displacement (CMOD) between them is: ; CMOD compliance is obtained from crack opening displacement (CMOD). C The expression: ; in: WB is the sample width, and B is the sample thickness. s nom The nominal stress of a uniformly distributed load; Type I and II normalized stress intensity factors f 11 , f 22 The calculation formula is: ; ; in: K I and K II These are Type I and Type II stress intensity factors, calculated using the interaction integration method, and can be directly output from Abaqus's .dat file. Normalized Softness u The calculation formula is: ; in: E The elastic modulus of the material. B The thickness is the sample thickness.

[0010] The advantages and beneficial effects of this invention are as follows: 1. This invention provides formulas for calculating compliance and stress intensity factors of single-sided notched tensile specimens with deflection cracks under different initial crack lengths, deflection crack angles, and crack propagation lengths. The formulas are highly practical and universal.

[0011] 2. The calculation formulas for flexibility and equivalent stress strength factor provided by this invention can accurately calculate fracture toughness and fatigue crack propagation rate, providing technical support for high-precision design based on damage tolerance. Attached Figure Description

[0012] Figure 1 This is a diagram showing the SENT testing system and sample geometry of the present invention. Figure 2 These are fracture SENT specimen images of composite materials with different fiber orientations according to the present invention; Figure 3 This is a load-displacement curve of the SENT specimen of the present invention; Figure 4 The critical fracture toughness diagrams of SENT samples with different fiber orientations according to the present invention are shown. Figure 5 This is a three-dimensional metallographic image of the ZK60 magnesium alloy of the present invention; Figure 6 This is a sampling direction diagram of the SENT sample of the present invention; Figure 7This is a diagram of the crack path observed in ZK60 magnesium alloy SENT samples from different sampling directions according to the present invention. Figure 8 This is a fatigue crack propagation rate curve of ZK60 magnesium alloy in different sampling directions according to the present invention. Figure 9 For the present invention α Evolution of normalized stress intensity factor along crack path under a sampling direction of 60°. Detailed Implementation

[0013] The present invention will be further described in detail below through specific embodiments. The following embodiments are merely descriptive and not limiting, and should not be used to limit the scope of protection of the present invention.

[0014] An innovative method for calculating the fracture toughness and fatigue crack propagation rate of a single-sided notched tensile specimen considering deflection cracks is proposed. The method establishes calculation formulas for compliance and stress intensity factors under different initial crack lengths, crack deflection angles, and crack propagation lengths, thereby achieving accurate calculation of the fracture performance and fatigue crack propagation rate of a single-sided notched tensile specimen considering crack deflection. The steps of the method are as follows: S1. A two-dimensional plane stress model of a single-sided notched tensile (SENT) specimen with a deflection crack is established using the finite element analysis software Abaqus. The elastic modulus of the material is input. E Compared to Poisson n A uniformly distributed load of equal magnitude is applied to the top and bottom surfaces of the SENT specimen. One corner of the SENT specimen is constrained to prevent rigid body translation while allowing in-plane rotation, in order to simulate pin loading conditions rather than clamping conditions. S2. The mesh is generated using an 8-node reduced integral plane stress element (CPS8R). The mesh at the crack tip is refined and singular elements are introduced. A focused ring mesh is used with a minimum radial dimension of 0.005 mm. S3. The finite element analysis software Abaqus calculates the stress intensity factor under different initial crack lengths, crack deflection angles, and crack propagation lengths using the interaction integral method. K I , K II Softness C The stress intensity factor is obtained by the derivative relationship between CMOD and load; after normalization, it is obtained. f 11 , f 22 and normalized flexibility u ; S4. For different crack deflection angles β(0°~90°), the normalized fracture parameters and the ratio of projected crack length to specimen width were established by curve fitting. a p / W and crack deflection angle β Empirical expressions between them: ; in: f The normalized stress intensity factor includes the normalized stress intensity factor. f 11 , f 22 ; The projected crack length; a 0 represents the initial crack length; Da This refers to the crack propagation length; β The angle of crack deflection; W The width of the sample; x i These are the fitting parameters; f 11 , f 22 The fitting coefficients are shown in Table 1.

[0015] Table 1 f 11 and f 22 Fit coefficient

[0016] ; in: u To normalize the softness, These are the fitting parameters; u The fitting coefficients are shown in Table 2.

[0017] Table 2 u Fit coefficient

[0018] The specific structural dimensions of the single-sided notched tensile (SENT) specimen containing the deflection crack are as follows: Specimen width W Fixed at 10 mm; high H / W =5; The crack is in the middle of the sample, and the initial crack length is... a 0 corresponds to a 0 / W =0, 0.2, 0.3, 0.4, 0.5; Crack deflection angleβ They are 0°, 15°, 30°, 45°, 60°, 75°, and 90° respectively; Crack propagation length Da Covering the remaining ligament area; Uniformly distributed stress s nom =0.1 MPa; elastic modulus E =200000 MPa, Poisson's ratio n =0.3.

[0019] Moreover, the aforementioned flexibility C Normalized stress intensity factor f 11 , f 22 and normalized flexibility u The calculation steps are as follows: Select two symmetrical nodes at the crack and ensure the initial spacing between them. D Within the recommended range, let the coordinates of these two nodes during the loading process be (x1, y1) and (x2, y2), respectively. Then, the mathematical expression for the crack opening displacement (CMOD) between them is: ; CMOD compliance is obtained from crack opening displacement (CMOD). C The expression: ; in: W B is the sample width, and B is the sample thickness. s nom The nominal stress of a uniformly distributed load; Type I and II normalized stress intensity factors f 11 , f 22 The calculation formula is: ; ; in: K I and K II These are Type I and Type II stress intensity factors, calculated using the interaction integration method, and can be directly output from Abaqus's .dat file. Normalized Softness u The calculation formula is: ; in: E The elastic modulus of the material.B The thickness is the sample thickness.

[0020] Taking a T300 carbon fiber reinforced epoxy resin laminate with unidirectional layup as an example, the method proposed in this invention is used to obtain its accurate critical fracture toughness. The basic mechanical properties of the T300 carbon fiber reinforced epoxy resin laminate are measured: elastic modulus... E 1 = 86300 MPa (fiber direction). E 2 = 7370 MPa (vertical direction), Poisson's ratio n 12 = 0.266, shear modulus G 12 = 6517 MPa.

[0021] The critical fracture toughness of T300 carbon fiber reinforced epoxy resin laminate will be accurately predicted below according to the method of the present invention.

[0022] (1) Sample design and processing T300 composite material SENT specimen (width) W =6 mm, thickness B =2 mm), initial crack length a 0 ≈ 0.5 W The clamping end spacing (pin hole center distance) is 30 mm (e.g. Figure 1 (As shown). The initial single-sided straight notch was machined by wire EDM, with a notch root radius of approximately 0.15 mm. The angle between the sample fiber direction and the notch direction (x-axis) is shown. i Designed for 0°, 30°, 60°, and 90°, the actual measured values ​​after machining are... i = 3.27°, 29.42°, 62.56° and 88.39° (e.g.) Figure 2 (As shown).

[0023] (2) Fracture toughness test process Quasi-static tensile loading was performed using an in-situ mechanical testing system (IBTC-5000), with the axial displacement loading rate fixed at 0.005 mm / s. A speckled pattern was formed by spraying white paint onto one side of the specimen surface. Crack propagation was periodically recorded using a CCD camera (Microvision MV-E800C), and crack opening displacement (CMOD) was measured using digital image correlation (DIC) technology. Loading was then performed until the critical load was reached. P c Data was recorded in real time. The crack started at the root of the notch and deflected along the fiber-matrix interface. The deflection angle was... β With fiber orientation angle i Matching (e.g.) Figure 2 As shown, the white line indicates fiber orientation, and the red arrow indicates the direction of crack propagation.

[0024] (3) Experimental data processing During crack initiation, the notch root experiences a type I-II combined stress state, reaching the critical fracture toughness. G c The calculation formula is: ; in: l and r The calculation formula is: ; ; in: E x , E y , v xy and G xy This represents the material properties of the SENT specimen in the xy coordinate system. The material properties are derived from the material properties through transformation relationships.

[0025] Critical stress intensity factor K Ic and K IIc The calculation formula is: ; ; The formulas for calculating the normalized stress intensity factor are all β and a p / W The function. In the early stages of crack initiation, a p / W The value is equal to a 0 / W Assuming an instantaneous deflection occurs, β Values i ; will the corresponding a 0 / W and i Substituting the values ​​into the formula for calculating the normalized stress intensity factor, we can obtain the corresponding values ​​for each of the four SENT tests. f 11 and f 22 Value, and then further calculation G c .

[0026] Depend on Figure 3 The load-displacement curve shows that iCritical loads at 3.27° and 29.42° P c Similar, and i = 62.56° and 88.39° P c Significantly increased. Figure 4 It can be seen that the calculated G c The values ​​are as follows: i At 3.27°, it is approximately 302 N·m. i The same applies when the angle is 29.42°. i = 968.48 N·m at 62.56° i = 1925.99 N·m at 88.39° (error bars represent the standard deviation of repeated measurements).

[0027] To further apply relevant methods and obtain the fatigue crack propagation rate, rolled ZK60 magnesium alloy plate was used as the research object. Basic mechanical properties: elastic modulus... E ≈ 45000 MPa (mean in all directions), Poisson's ratio n = 0.35, yield strength 105-185 MPa. The specimen is cut from the TD-ND plane (e.g., Figure 6 ), sampling direction angle α The angle between the notch direction and the TD direction is set to 0°, 30°, 60°, and 90°. Below, the fatigue crack propagation rate of ZK60 magnesium alloy plates will be accurately predicted according to the method of this invention.

[0028] (1) Sample design and processing SENT specimen size is the same as that of fracture toughness test ( B = 2 mm, W = 6 mm), initial notch length ratio a 0 / W = 0.2. The sample surface was sanded and electropolished (AC II solution) to obtain clear crack paths.

[0029] (2) Fatigue crack propagation rate test process When a sinusoidal tensile cyclic load is applied, the maximum load is reached during all loading cycles. P max Set to 560 N, load ratio P min / P max =0.1, the loading frequency is fixed at 1 Hz, and an image of the sample surface is captured using a camera.

[0030] (3) Experimental data processing After testing, the image sequence was processed using ImageJ software to determine the crack length. a p With the number of loops N The relationship between them was then determined. The fatigue crack propagation rate d was then calculated using the 7-point polynomial method. a p / d N .

[0031] Equivalent stress intensity factor range Δ K eq The formula for characterizing the crack driving force is as follows: ; Where Δ K I and Δ K II The calculation formula is: ; ; f 11 and f 22 The calculation of the normalized stress intensity factor requires the input of the instantaneous value at each time step. a p / W and β value.

[0032] Taking a sampling direction of α=60° as an example, the corrosion pits near the crack opening form a natural speckle pattern, allowing for crack opening displacement (CMOD) measurement using digital image correlation (DIC) without the need for surface painting. a p / W At a position of 0.3, the measured crack curvature is 25.1°. Substituting these data into equation (2), the following calculation is obtained: u =0.3724; The measured CMOD value was then substituted into the normalized compliance value. u The calculation formula is obtained. u =0.3835. The resulting error is approximately 2.98%, indicating that the established compliance calculation formula has good accuracy.

[0033] Crack path as Figure 7 As shown: α At 0°, the crack propagates almost vertically. α = Initial deflection angle at 30° and 60° β The 0° and 40.52° are respectively 29.64° and 40.52°, which gradually decrease as the expansion progresses (tending towards the type I mode); αCracks branch at 90°.

[0034] Using the above formulas, the crack propagation length and crack propagation driving force can be calculated, thus obtaining d under different orientations. a p / d N -Δ K eq Curve (e.g.) Figure 8 As shown in the figure). The results show that the same Δ K eq Below, d a p / d N Follow α The increase and decrease indicate that crack deflection consumes more energy, verifying the accuracy of the driving force calculation using this method.

[0035] Figure 9 by α Taking 60° as an example, considering the actual crack curvature... f 11 and f 22 Compared with the results obtained under the straight crack assumption f 11 β=0° and f 22 β=0° A comparison between the actual deflection and the SIFs assumed by the straight crack shows an initial deviation of 27.1%, which increases with... β The deviation was reduced to 2.6% by 7.22°, highlighting the necessity of considering crack deflection.

[0036] Although embodiments and drawings of the present invention have been disclosed for illustrative purposes, those skilled in the art will understand that various substitutions, variations and modifications are possible without departing from the spirit and scope of the present invention and the appended claims. Therefore, the scope of the present invention is not limited to the contents disclosed in the embodiments and drawings.

Claims

1. A method for calculating the fracture toughness and fatigue crack propagation rate of a single-sided notched tensile specimen considering deflection cracks, characterized in that: The method establishes formulas for calculating compliance and stress intensity factors under different initial crack lengths, crack deflection angles, and crack propagation lengths, enabling accurate calculation of fracture performance and fatigue crack propagation rate of single-sided notched tensile specimens considering crack deflection. The steps of the method are as follows: S1. A two-dimensional plane stress model of a single-sided notched SENT tensile specimen with a deflection crack was established using the finite element analysis software Abaqus. The elastic modulus of the material was input. E Compared to Poisson ν A uniformly distributed load of equal magnitude is applied to the top and bottom surfaces of the SENT specimen. One corner of the SENT specimen is constrained to prevent rigid body translation while allowing in-plane rotation, in order to simulate pin loading conditions rather than clamping conditions. S2. The mesh is generated using an 8-node reduced integral plane stress element CPS8R. The mesh at the crack tip is refined and singular elements are introduced. A focused ring mesh is used with a minimum radial dimension of 0.005 mm. S3. The finite element analysis software Abaqus calculates the stress intensity factor under different initial crack lengths, crack deflection angles, and crack propagation lengths using the interaction integral method. K I , K II Softness C The stress intensity factor is obtained by the derivative relationship between CMOD and load; after normalization, it is obtained. f 11 , f 22 and normalized flexibility u ; S4. For different crack deflection angles β From 0° to 90°, the normalized fracture parameters and the ratio of projected crack length to specimen width were established through curve fitting. a p / W and crack deflection angle β Empirical expressions between them: ； in: f’ To fit the stress intensity factor; The projected crack length; a 0 represents the initial crack length; Δa This refers to the crack propagation length. β The angle of crack deflection; W The width of the sample; ξ i These are the fitting parameters; ； in: u’ To fit the compliance, These are the fitting parameters.

2. The method for calculating the fracture toughness and fatigue crack propagation rate of a single-sided notched tensile specimen considering deflection cracks according to claim 1, characterized in that: The specific structural dimensions of the single-sided notched SENT tensile specimen containing the deflection crack are as follows: Specimen width W Fixed at 10 mm; high H / W =5; The crack is in the middle of the sample, and the initial crack length is... a 0 corresponds to a 0 / W =0, 0.2, 0.3, 0.4, 0.5; Crack deflection angle β They are 0°, 15°, 30°, 45°, 60°, 75°, and 90° respectively; Crack propagation length Δa Covering the remaining ligament area; Uniformly distributed stress σ nom =0.1 MPa; elastic modulus E =200000 MPa, Poisson's ratio ν =0.

3.

3. The method for calculating the fracture toughness and fatigue crack propagation rate of a single-sided notched tensile specimen considering deflection cracks according to claim 1, characterized in that: The flexibility C Normalized stress intensity factor f 11 , f 22 and normalized flexibility u The calculation steps are as follows: Select two symmetrical nodes at the crack and ensure the initial spacing between them. D Within the recommended range, let the coordinates of these two nodes during the loading process be (x1, y1) and (x2, y2), respectively. Then, the mathematical expression for the crack opening displacement (CMOD) between them is: ； CMOD compliance is obtained from crack opening displacement (CMOD). C The expression: ； in: W B is the sample width, and B is the sample thickness. σ nom The nominal stress of a uniformly distributed load; Type I and II normalized stress intensity factors f 11 , f 22 The calculation formula is: ； ； in: K I and K II These are the Type I and Type II stress intensity factors, calculated using the interaction integration method and directly output from the Abaqus .dat file. Normalized Softness u The calculation formula is: ； in: E The elastic modulus of the material. B The thickness is the sample thickness.

Images