Method for constructing size effect equation of type I fracture of compacted clay considering friction effect

Abstract

The present invention discloses a method for constructing a compacted clay type I fracture size effect equation considering the friction effect, and relates to the technical field of core wall crack research of earth-rock dams. The present invention at least includes the following steps: S1: determining the friction coefficient between the support material and the sample; S2: conducting a compacted clay type I fracture size effect test; S3: calibrating the singular terms, non-singular terms, and third high-order terms of the stress field at the tip of the type I fracture sample. The present invention considers the influence of support friction in the traditional three-point bending test, introduces DIC to calibrate the coefficients of the Williams series expansion after considering the support friction, and constructs a compacted clay type I fracture toughness size effect equation considering the friction effect in combination with the MMTS criterion, so as to realize the prediction of compacted clay type I fracture toughness and type I fracture process zone at the engineering scale.

Description

Technical Field

[0001] The invention relates to the technical field of core wall crack research of earth-rock dams, and in particular to a method for constructing a compacted clay type I fracture size effect equation considering friction effect. Background Art

[0002] High-pile earth-rock dams are widely used in my country, but the problem of cracks in their core walls has not been well solved. Type I fracture failure is the most basic and common failure mode in fracture science. Since the friction of the support will lead to an increase in the peak load, it will seriously affect the fracture toughness test results. In addition, due to the existence of the crack tip fracture process zone (FPZ), the fracture test results under indoor single-size specimen conditions are difficult to accurately characterize the crack resistance and crack propagation mechanism of the core wall of the earth-rock dam at the engineering scale, and therefore cannot reasonably guide the construction and safety assessment of large-scale engineering structures.

[0003] Therefore, a new solution to the above problems needs to be proposed. Summary of the invention

[0004] The purpose of the present invention is to provide a method for constructing a compacted clay type I fracture size effect equation taking into account the friction effect, so as to solve the technical problems raised in the background technology.

[0005] To achieve the above object, the present invention provides the following technical solution: a method for constructing a compacted clay type I fracture size effect equation considering the friction effect, comprising at least the following steps:

[0006] S1: Determine the friction coefficient between the support material and the sample;

[0007] S2: Conducting size effect tests on compacted clay type I fractures;

[0008] S3: Extract the crack tip scattered spot coordinates and deformation information at a certain loading moment of the mode I fracture test based on digital image technology;

[0009] S4: The deformation information obtained in S3 is imported into the matrix calculation, that is, the coefficients of the Williams series expansion are determined by matrix calculation, that is, the singular terms, non-singular terms, and the third high-order terms of the stress field at the tip of the type I fracture specimen are calibrated to realize the connection between the support friction, the specimen size, and the dimensionless crack length;

[0010] S5: Based on the MMTS criterion, a size effect equation for the mode I fracture toughness of compacted clay considering the friction effect is constructed to achieve the prediction of the mode I fracture toughness and mode I fracture process zone of compacted clay at an engineering scale.

[0011] Furthermore, the S1 at least comprises the following steps:

[0012] Build the triangle's bottom support;

[0013] Place the sample on the bottom support, gradually lift one end of the bottom support, and stop lifting when the sample starts to slide down;

[0014] Measure the inclination angle α at this time, and the friction coefficient between the support material and the sample is as follows (1):

[0015] f = tanα(1).

[0016] Furthermore, the S2 at least includes the following steps:

[0017] First, the compacted clay is mixed with water and soil according to the optimal moisture content;

[0018] The mixed soil is placed into custom-made moulds of different sizes and compacted in three layers, with each layer being fully shaved to prevent delamination;

[0019] After compaction, demould the specimen;

[0020] The cracks are then preformed on a cutting machine;

[0021] Then the specimen was coated with putty powder and then sprayed with speckle to complete the preparation of soil sample;

[0022] The soil fracture instrument was used to carry out experimental application on soil samples;

[0023] The fracture size effect test was carried out using observation equipment.

[0024] S3 performs full-field strain analysis on the photos taken by DIC through VIC-2D software to extract the coordinates of the scattered plate points in the analysis area and the deformation information at a certain loading moment. S3 at least includes the following steps:

[0025] The photos taken by the DIC high-speed camera were imported into the VIC-2D software, and the analysis area was divided around the crack tip;

[0026] After full-field strain analysis of the analysis area, K points are selected at the crack tip. The K value is determined according to the required number of high-order terms, as shown in the following formula (2):

[0027] K>N+M+2(2)

[0028] In formula (2), N and M are the coefficients of the higher-order terms A and B, respectively. When calculating the third higher-order term, K should be greater than 8. To ensure the accuracy of the calculation, the distance from the crack tip is selected as l, a / 4≤l≤a / 2, and a is the crack length.

[0029] Furthermore, the step 4 at least includes the following steps:

[0030] The step 4 at least comprises the following steps:

[0031] The coefficients of the Williams series expansion are determined by the overdetermined finite element method. This method can be determined by using the displacement field and coordinate field of the crack tip node. Combined with S3, the displacement field and coordinate field of K nodes at the crack tip are extracted using VIC-2D software and substituted into the Williams series expansion to obtain a set of overdetermined equations. The optimal solution is obtained using the least squares method.

[0032] The progressive displacement field at the crack tip is shown in equations (3) and (4):

[0033]

[0034] Where u and v represent displacements in the x and y directions, respectively. The shear modulus μ = E / 2(1+v). The Kolosov constant is κ = (3-v) / (1+v) in plane stress problems and κ = 3-4v for plane strain problems.

[0035] When n=0, as shown in the following equations (5) and (6):

[0036]

[0037] Substituting the selected node coordinates into equation (7) yields a system of equations;

[0038]

[0039] It can be simplified into formula (8):

[0040] [U] 2k×1 =[C] 2k×(N+M+2) [X] (N+M+2)×1 (8)

[0041] Where: [U] is the displacement vector of the node, [C] is the node position coordinate matrix of the angular function f(r,θ) and g(r,θ);

[0042] Using the least squares method, we can get formula (9):

[0043] [C] T [U]=[C] T [C][X][X]=([C] T [C]) -1 [C] T [U](9)

[0044] After solving equation (7), we can obtain A n With B n Wait for Williams higher-order terms.

[0045] Furthermore, the S5 at least includes the following steps:

[0046] The first-order coefficients A1 and B1 of the Williams series expansion are related to the stress intensity factor K I and K II， The second-order coefficient A2 corresponds to the T stress, and the second-order term corresponding to B2 is zero, as shown in the following formula (10):

[0047]

[0048] According to the MMTS criterion, the crack propagates along the direction θ0 perpendicular to the maximum tangential stress, and the critical distance r from the crack tip is c Tangential stress σ θθ Reaching the critical value σ θθc When , the crack starts from the tip along the θ0 direction. Using the Williams series expansion, the tangential stress component σ at the crack tip θθ Formula (11):

[0049]

[0050] In the formula, K I and K II are type I and type II stress intensity factors, respectively;

[0051] T is independent of the crack tip distance r c The non-singular stress term is usually called T stress;

[0052] A3 and B3 are the coefficients of the third-order term in the Williams series expansion. 3、 The expressions of B3 and T are shown in formula (12):

[0053]

[0054] Where A3 * 、B3 * 、T * It is a dimensionless parameter that has nothing to do with the specimen size and load size, and only depends on the crack length a / R and the support distance S / R. Substituting the dimensionless parameter and ignoring the higher-order term O(r), the critical distance r from the crack tip is c The tangential stress component at can be expressed as formula (13):

[0055]

[0056] According to the MTS standard, the fracture occurs at the angle θ0 where the tangential stress is the largest. The angle θ0 can be determined by differentiating θ by equation (11) and then solving equation (13), as shown in equation (14):

[0057]

[0058] According to the second assumption of the MTS criterion, when the maximum tangential stress σ along θ0 is θθ and the critical distance r from the crack tip c Reaching the critical value σ θθc When the fracture occurs, the critical stress σ θθc can be assumed to be the tensile strength f t , as shown in formula (15):

[0059]

[0060] Where K If and K IIf is the stress intensity factor K I and K II The critical values ​​of , which characterize the fracture resistance of materials under mixed-mode loading;

[0061] The critical distance r in the equation c , must be determined by equations (11) and (13) to obtain the fracture initiation angle θ0 and predict the onset of fracture. Equation (16) takes into account the influence of high-order terms used to determine the elastic equivalent FPZ length. This equation can be written as:

[0062]

[0063] Where: γ=6(A3 * / K I * );

[0064] MMTS standard assumptions c The value (or FPZ size) depends on the size of the specimen, so the r c The size dependence of isotope is used to analyze the effect of size on the mixed mode fracture behavior of compacted clay, as shown in formula (17):

[0065]

[0066] where the coefficients A and B are determined by linear regression using the effective Mode I fracture toughness of the material obtained from specimens of different sizes, by first converting the Mode I fracture toughness (K If ) is replaced in equation (16) to determine the r corresponding to each specimen size. c value, the calculated r c Substitute the values ​​into equation (18) and calculate based on the linear regression of Y = MX + Q;

[0067]

[0068] Substituting the regression coefficients A and B into equation (17), we can obtain r cThe relationship between R and r can be obtained. c The relationship curve of R is c Substituting the value into formula (15), the predicted K can be calculated eff The value can also be obtained to realize the prediction of mode I fracture toughness and mode I fracture process zone of compacted clay at engineering scale.

[0069] Compared with the prior art, the present invention has the following beneficial effects:

[0070] The present invention considers the influence of support friction in the traditional three-point bending test, introduces DIC to calibrate the coefficients of the Williams series expansion after considering the support friction, and constructs the size effect equation of type I fracture toughness of compacted clay considering the friction effect in combination with the MMTS criterion, so as to realize the prediction of type I fracture toughness and type I fracture process zone of compacted clay at engineering scale. BRIEF DESCRIPTION OF THE DRAWINGS

[0071] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings required for describing the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other accompanying drawings can be obtained based on these accompanying drawings without paying creative work.

[0072] Figure 1 Schematic diagram of the friction coefficient testing method of the present invention;

[0073] Figure 2 Schematic diagram of the process of sample preparation method of the present invention;

[0074] Figure 3 is a schematic diagram of the loading and data acquisition system of the present invention;

[0075] Figure 4 This is a schematic diagram of the division and analysis areas of the first embodiment of the present invention;

[0076] Figure 5 It is a schematic diagram of selecting the tip nodes of horizontal and vertical strain cracks according to the first embodiment of the present invention;

[0077] Figure 6 A line diagram of selected horizontal and vertical strain crack tip nodes according to the first embodiment of the present invention;

[0078] Figure 7 This is a schematic diagram of the relationship curve between rc and R when the friction effect is considered and when the friction effect is ignored in the present invention;

[0079] Figure 8 K is the K with friction effect considered and friction effect ignored in the present invention. If Schematic diagram of the relationship curve with R;

[0080] Fig. 9 This is a schematic diagram of an SCB sample according to Example 2 of the present invention;

[0081] Fig.10 This is a schematic diagram of the division and analysis area according to the second embodiment of the present invention;

[0082] Fig.11 This is a schematic diagram of selecting the tip nodes of horizontal and vertical strain cracks according to the second embodiment of the present invention;

[0083] Fig.12 It is a local structural schematic diagram of the curve of the dimensionless intensity factor changing with β under different friction coefficients of the present invention. DETAILED DESCRIPTION

[0084] The technical solutions in the embodiments of the present invention will be described clearly and completely below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all the embodiments.

[0085] Embodiment 1:

[0086] The method for constructing the size effect equation of type I fracture of compacted clay considering the friction effect includes at least the following steps:

[0087] S1: Determine the friction coefficient between the support material and the sample;

[0088] S2: Conducting size effect tests on compacted clay type I fractures;

[0089] S3: Extract the crack tip scattered spot coordinates and deformation information at a certain loading moment of the mode I fracture test based on digital image technology;

[0090] S4: The deformation information obtained in S3 is imported into the matrix calculation, that is, the coefficients of the Williams series expansion are determined by matrix calculation, that is, the singular terms, non-singular terms, and the third high-order terms of the stress field at the tip of the type I fracture specimen are calibrated to realize the connection between the support friction, the specimen size, and the dimensionless crack length;

[0091] S5: Based on the MMTS criterion, a size effect equation for the mode I fracture toughness of compacted clay considering the friction effect is constructed to achieve the prediction of the mode I fracture toughness and mode I fracture process zone of compacted clay at an engineering scale.

[0092] See also Figure 1 , S1 at least comprises the following steps:

[0093] Build the triangle's bottom support;

[0094] Place the sample on the bottom support, gradually lift one end of the bottom support, and stop lifting when the sample starts to slide down;

[0095] Measure the inclination angle α at this time, and the friction coefficient between the support material and the sample is as follows (1):

[0096] f = tanα(1).

[0097] See also Figure 2 , S2 at least includes the following steps:

[0098] First, the compacted clay is mixed with water and soil according to the optimal moisture content;

[0099] The mixed soil is placed into custom-made moulds of different sizes and compacted in three layers, with each layer being fully shaved to prevent delamination;

[0100] After compaction, demould the specimen;

[0101] The cracks are then preformed on a cutting machine;

[0102] Then the specimen was coated with putty powder and then sprayed with speckle to complete the preparation of soil sample;

[0103] The soil fracture instrument was used to carry out experimental application on soil samples;

[0104] The fracture size effect test was carried out using observation equipment.

[0105] The soil fracture instrument consists of a loading rod, a movable support with adjustable spacing, a support rod that can be adjusted up and down (to meet the needs of specimens of different sizes), and a data collector. The entire test instrument includes a displacement and load information acquisition system. The displacement range is 40mm with an accuracy of 0.01mm; the load range is 5000N with an accuracy of 0.01N.

[0106] The observation equipment includes an industrial camera (1 unit), a fill light (1 unit) and image acquisition software. The post-processing software for the speckle image is VIC-2D, which can accurately calculate the displacement and strain of the sample surface. Figure 3 shown.

[0107] S3 uses VIC-2D software to perform full-field strain analysis on the photos taken by DIC, extracts the coordinates of the scattered plate points in the analysis area and the deformation information at a certain loading moment, and S3 includes at least the following steps:

[0108] Import the images taken by the DIC high-speed camera into the VIC-2D software and divide the analysis area around the crack tip. Figure 4 ;

[0109] After full-field strain analysis of the analysis area, K points are selected at the crack tip. The K value is determined according to the required number of high-order terms, as shown in the following formula (2):

[0110] K>N+M+2(2)

[0111] In formula (2), N and M are the coefficients of the higher-order terms A and B, respectively. When calculating the third higher-order term, K should be greater than 8. To ensure the accuracy of the calculation, the distance from the crack tip is selected as l, a / 4≤l≤a / 2, a is the crack length, refer to Figure 5 .

[0112] Step 4 includes at least the following steps:

[0113] The coefficients of the Williams series expansion are determined by the overdetermined finite element method. This method can be determined by using the displacement field and coordinate field of the crack tip node. Combined with S3, the displacement field and coordinate field of K nodes at the crack tip are extracted using VIC-2D software and substituted into the Williams series expansion to obtain a set of overdetermined equations. The optimal solution is obtained using the least squares method.

[0114] The progressive displacement field at the crack tip is shown in equations (3) and (4):

[0115]

[0116] Where u and v represent displacements in the x and y directions, respectively. The shear modulus μ = E / 2(1+v). The Kolosov constant is κ = (3-v) / (1+v) in plane stress problems and κ = 3-4v for plane strain problems.

[0117] When n=0, as shown in the following equations (5) and (6):

[0118]

[0119] Substituting the selected node coordinates into equation (7) yields a system of equations;

[0120]

[0121] It can be simplified into formula (8):

[0122] [U] 2k×1 =[C] 2k×(N+M+2) [X] (N+M+2)×1 (8)

[0123] Where: [U] is the displacement vector of the node, [C] is the node position coordinate matrix of the angular function f(r,θ) and g(r,θ);

[0124] Using the least squares method, we can get formula (9):

[0125] [C] T [U]=[C] T [C][X][X]=([C] T[C]) -1 [C] T [U](9)

[0126] After solving equation (7), we can obtain A n With B n Wait for Williams higher-order terms.

[0127] S5 at least includes the following steps:

[0128] The first-order coefficients A1 and B1 of the Williams series expansion are related to the stress intensity factor K I and K II， The second-order coefficient A2 corresponds to the T stress, and the second-order term corresponding to B2 is zero, as shown in the following formula (10):

[0129]

[0130] According to the MMTS criterion, the crack propagates along the direction θ0 perpendicular to the maximum tangential stress, and the critical distance r from the crack tip is c Tangential stress σ θθ Reaching the critical value σ θθc When , the crack starts from the tip along the θ0 direction. Using the Williams series expansion, the tangential stress component σ at the crack tip θθ Formula (11):

[0131]

[0132] In the formula, K I and K II are type I and type II stress intensity factors, respectively;

[0133] T is independent of the crack tip distance r c The non-singular stress term is usually called T stress;

[0134] A3 and B3 are the coefficients of the third-order term in the Williams series expansion. 3、 The expressions of B3 and T are shown in formula (12):

[0135]

[0136] Where A3 * 、B3 * 、T * It is a dimensionless parameter that has nothing to do with the specimen size and load size, and only depends on the crack length a / R and the support distance S / R. Substituting the dimensionless parameter and ignoring the higher-order term O(r), the critical distance r from the crack tip is c The tangential stress component at can be expressed as formula (13):

[0137]

[0138] According to the MTS standard, the fracture occurs at the angle θ0 where the tangential stress is the largest. The angle θ0 can be determined by differentiating θ by equation (11) and then solving equation (13), as shown in equation (14):

[0139]

[0140] According to the second assumption of the MTS criterion, when the maximum tangential stress σ along θ0 is θθ and the critical distance r from the crack tip c Reaching the critical value σ θθc When the fracture occurs, the critical stress σ θθc can be assumed to be the tensile strength f t , as shown in formula (15):

[0141]

[0142] Where K If and K IIf is the stress intensity factor K I and K II The critical values ​​of , which characterize the fracture resistance of materials under mixed-mode loading;

[0143] The critical distance r in the equation c , must be determined by equations (11) and (13) to obtain the fracture initiation angle θ0 and predict the onset of fracture. Equation (16) takes into account the influence of high-order terms used to determine the elastic equivalent FPZ length. This equation can be written as:

[0144]

[0145] Where: γ=6(A3 * / K I * );

[0146] MMTS standard assumptions c The value (or FPZ size) depends on the size of the specimen, so the r c The size dependence of isotope is used to analyze the effect of size on the mixed mode fracture behavior of compacted clay, as shown in formula (17):

[0147]

[0148] where the coefficients A and B are determined by linear regression using the effective Mode I fracture toughness of the material obtained from specimens of different sizes, by first converting the Mode I fracture toughness (K If) is replaced in equation (16) to determine the r corresponding to each specimen size. c value, the calculated r c Substitute the values ​​into equation (18) and calculate based on the linear regression of Y = MX + Q;

[0149]

[0150] Substituting the regression coefficients A and B into equation (17), we can obtain r c The relationship between R and r can be obtained. c The relationship curve of R is c Substituting the value into formula (15), the predicted K can be calculated eff The value can also be obtained to realize the prediction of mode I fracture toughness and mode I fracture process zone of compacted clay at engineering scale.

[0151] Example analysis for embodiment 1:

[0152] The present invention is further described by taking the study on the type I fracture size effect of compacted clay of NSCB specimens as an example, but the protection scope of the present invention is not limited to these examples.

[0153] The test clay was taken from Chongqing area, with the maximum dry density ρ max =1.68g / cm 3 , optimal moisture content w op =18.3% for water-soil mixing, and three-point bending test using DTV-3 soil fracture instrument and DIC instrument. After preparing the sample according to step 2 above, the friction coefficient between the sample and the support can be measured using the method of step 1 to obtain f≈0.4.

[0154] Using VIC-2D software, 9 nodes were selected at the crack tip for horizontal and vertical strain analysis to obtain the displacement asymptotic field. The obtained data was imported into formula (7), and the calibration results are shown in the following table:

[0155] Table 1 Calibration results of Williams series expansion coefficients after considering friction effect

[0156]

[0157]

[0158] See also Figure 6 According to step 5, the calibration results in Table 1 and the test data are substituted into formula (16) to obtain the length r of the fracture process zone at this time: c , the calculated r c Substituting into equation (18), we can obtain the linear regression curve of Y=MX+Q, as Figure 6As shown, when the friction effect is ignored, A = 464.67mm, B = 91.74mm. When the friction effect is considered, A = 305.23mm, B = 38.31mm. When the friction effect is ignored, r c The relationship between R and R is shown in formula (19). When considering the influence of friction, r c The relationship with R is shown in formula (20).

[0159]

[0160] Figure 7 Give r when ignoring the effect of friction and considering the effect of friction c The relationship curve between friction and R shows that ignoring friction will overestimate r c The true value of the fracture process zone length (r c∞ ), when considering friction, r c∞ =51.58mm; when friction is ignored, r c∞ =90.06mm; Ignore friction and compare with r without considering friction c∞ It will increase by 74.61%.

[0161] Will Figure 7 R c Substituting the predicted value into formula (15) we can obtain the mode I fracture toughness K If The relationship curve with R, such as Figure 8 As shown, Figure 7 The rule is similar. Ignoring friction will overestimate K. If The value of K after ignoring the friction effect If The change range is only 7.03 kPa·m 0.5 , after considering the influence of friction, K If The change range reaches 13.57 kPa·m 0.5 , an increase of 93.03%.

[0162] See also Figure 8 , as R increases, K considering friction and ignoring friction If The error shows a decreasing trend and finally tends to be horizontal. When R=50mm, the error between the two is 165.85%. When R=25000mm, the error is only 32.82%. By observing the curve ignoring friction, it can be found that the overall trend is first rapidly increased, then slowly decreased, and finally tends to be stable, reaching a peak at about R=400mm.

[0163] Embodiment 2:

[0164] This embodiment is used to further disclose an error correction method for the results of the I-II composite fracture test of compacted clay considering the friction effect on the premise of the above embodiment;

[0165] A method for error correction of compacted clay type I-II composite fracture test results considering friction effect comprises the following steps:

[0166] (1) Determine the friction coefficient between the support material and the specimen;

[0167] (2) Conducting composite type I-II fracture tests on compacted clay;

[0168] (3) Extract the coordinates of the crack tip scattered spots and the deformation information at a certain loading moment of the I-II composite fracture specimen based on the digital image correlation method;

[0169] (4) Matrix calculation is used to determine the coefficients of the Williams series expansion, that is, to calibrate the singular and non-singular terms of the stress field at the tip of the I-II composite fracture specimen;

[0170] (5) Correction of the I-II composite fracture toughness.

[0171] In view of step 1, a method for determining the friction coefficient between the support material and the sample is invented, comprising:

[0172] like Figure 1 As shown, place the sample on the bottom support, slowly lift one end of the bottom support, and stop lifting when the sample just slides down. Measure the inclination angle α at this time. The friction coefficient between the support material and the sample is shown in the following formula:

[0173] f = tanα(1).

[0174] For step 2, a composite type I-II fracture test was conducted based on compacted clay, including:

[0175] See also Fig. 9 , taking SCB sample as an example, Figure 2 As shown, the specimen radius is R, the crack length is a, the crack inclination is β, the thickness is B, and the support spacing is S.

[0176] Sample preparation Figure 3 As shown, the compacted clay is first mixed with water and soil according to the optimal moisture content, the mixed soil is placed in a large rectangular customized mold, the semicircular ring knives are evenly placed in the large rectangular mold, and then compacted in three layers. Each layer is fully scraped to prevent stratification. After compaction, the sample is demolded and cracks are then prefabricated on a cutting machine.

[0177] The soil fracture instrument consists of a loading rod, a movable support with adjustable spacing, a support rod that can be adjusted up and down (to meet the needs of specimens of different sizes), and a data collector. The entire test instrument includes a displacement and load information acquisition system. The displacement range is 40mm with an accuracy of 0.01mm; the load range is 5000N with an accuracy of 0.01N.

[0178] The observation equipment includes an industrial camera (1 unit), a fill light (1 unit) and image acquisition software. The post-processing software for the speckle image is VIC-2D, which can accurately calculate the displacement and strain of the sample surface. Figure 3 shown.

[0179] For step 3, the coordinates of the crack tip scattered spots of the type I fracture specimen and the deformation information at a certain loading moment are extracted based on the digital image correlation method, including:

[0180] The full-field strain analysis is performed on the photos taken by DIC by using VIC-2D software to extract the coordinates of the scattered plate points in the analysis area and the deformation information at a certain loading moment. S3 at least includes the following steps:

[0181] Import the images taken by the DIC high-speed camera into the VIC-2D software and divide the analysis area around the crack tip. Fig.10 ;

[0182] After full-field strain analysis of the analysis area, K points are selected at the crack tip. The K value is determined according to the required number of high-order terms, as shown in the following formula (2):

[0183] K>N+M+2(2)

[0184] In formula (2), N and M are the coefficients of the higher-order terms A and B, respectively. When calculating the third higher-order term, K should be greater than 8. To ensure the accuracy of the calculation, the distance from the crack tip is selected to be l, a / 4≤l≤a / 2. Fig.11 .

[0185] For step 4, matrix calculation is used to determine the coefficients of the Williams series expansion, that is, to calibrate the singular terms, non-singular terms, and third high-order terms of the stress field at the tip of the type I fracture specimen, including:

[0186] The coefficients of the Williams series expansion can be determined using the overdetermined finite element method, which can be determined using the displacement field and coordinate field of the crack tip node. Combined with step 3, the displacement field and coordinate field of K nodes at the crack tip are extracted using VIC-2D software and substituted into the Williams series expansion to obtain a set of overdetermined equations, and the optimal solution is obtained using the least squares method.

[0187] The progressive displacement field at the crack tip is shown in equations (3) and (4):

[0188]

[0189]

[0190] Where u and v represent displacements in the x and y directions, respectively. The shear modulus μ = E / 2(1+v). The Kolosov constant is κ = (3-v) / (1+v) in plane stress problems and κ = 3-4v for plane strain problems. When n = 0:

[0191]

[0192] Substituting the selected node coordinates into equation (7) yields a system of equations.

[0193]

[0194] It can be simplified into formula (8)

[0195] [U] 2k×1 =[C] 2k×(N+M+2) [X] (N+M+2)×1 (8)

[0196] Where: [U] is the displacement vector of the node, [C] is the node position coordinate matrix of the angle function f(r,θ) and g(r,θ). Using the least squares method, we can get:

[0197] [C] T [U]=[C] T [C][X] [X]=([C] T [C]) -1 [C] T [U] (9)

[0198] After solving equation (7), we can obtain A n With B n Equal to the coefficients of the Williams series expansion.

[0199] For step 5, the I-II composite fracture toughness is corrected. Substitute the singular terms and non-singular terms of the stress field at the tip of the I-type fracture specimen obtained in step 4 into the formula:

[0200] The stress intensity factor (SIF) reflects the strength of the stress field around the crack tip, while the non-singular stress term (T stress) also affects the crack initiation and development. I , K II The calculation formula with T is as follows:

[0201]

[0202] T=4A2 (12)

[0203] Where A1, B1 and A2 are the coefficients of Williams series expansion, which can be expressed as dimensionless intensity factors and rewritten as follows:

[0204]

[0205] In the formula, With T * They are dimensionless parameters and can be calibrated by substitution in Eq.

[0206] For symmetrically loaded specimens, the combination mode of type I loading and type II loading can be easily changed by changing the crack inclination angle β. Different crack inclination angles β correspond to different stress states at this time, which are usually defined by dimensionless parameters, as shown in the following formula:

[0207]

[0208] In order to conveniently describe the I-II composite fracture toughness, the composite fracture toughness (K eff ) is expressed as follows:

[0209]

[0210] According to step 4, we obtain A1, B1 and A2 after considering friction. Substituting them into equations (10), (11) and (12) and then combining equations (13), (14) and (15), we can get the dimensionless parameter K I * , K I * I With T * The calibrated dimensionless parameters are substituted into equations (13), (14) and (15) to obtain the corrected critical mode I fracture toughness and mode II fracture toughness. The corrected fracture toughness can be obtained by substituting the values ​​into equation (17).

[0211] Example analysis for Example 2:

[0212] The present invention is further described by taking the study of compacted clay type I-II composite fracture of SCB specimens as an example, but the protection scope of the present invention is not limited to these examples.

[0213] The test clay in this embodiment was taken from Chongqing area with the maximum dry density ρ max =1.68g / cm3, optimal moisture content w op =18.3% for water-soil mixing, and three-point bending test using DTV-3 soil fracture instrument and DIC instrument. After preparing the sample according to step 2 above, the friction coefficient between the sample and the support can be measured using the method of step 1 to obtain f≈0.4.

[0214] Using VIC-2D software, 9 nodes at the crack tip were selected for horizontal and vertical strain analysis to obtain the displacement asymptotic field. The obtained data was imported into formula (7), and the calibration results were as follows: Fig.12 As shown;

[0215] After substituting the dimensionless intensity factor after considering friction into equations (13), (14) and (15), the results of ignoring friction and considering friction are shown in Tables 1 and 2. The results show that in the I-II composite fracture test, after ignoring friction, the composite fracture toughness will be overestimated at any crack angle. When ignoring friction, when β = 41°, it just reaches pure II loading, but when considering friction, when β = 32°, it has reached pure II loading. However, the support friction cannot be ignored, and the traditional stress intensity factor needs to be recalibrated to correct the composite fracture toughness when considering friction.

[0216] Table 1 Ignore friction test results

[0217]

[0218] Table 2 Considering the friction test results

[0219]

[0220]

[0221] It should be noted that the technical problem to be solved by the second embodiment is

[0222] Cracks are one of the key issues that have not been well resolved in high earth core rockfill dam projects. Studies have shown that the stress state inside the core wall of an earth-rock dam is usually complex, mainly affected by tensile shear stress, which leads to the generation of I-II composite cracks. The expansion of such cracks may lead to the formation of leakage channels in the core wall and even endanger the stability of the dam body. At the same time, the existence of support friction will seriously affect the fracture toughness test results.

[0223] In view of this, the present invention provides an error correction method for compacted clay I-II composite fracture test results considering the friction effect.

[0224] The technical effect of this embodiment is as follows: the present invention takes into account the influence of fulcrum friction in the traditional three-point bending test, introduces DIC to calibrate the coefficients of the Williams series expansion after considering the fulcrum friction, and uses the calibration coefficients to correct the I-II composite fracture test results.

[0225] It will be apparent to those skilled in the art that the invention is not limited to the details of the exemplary embodiments described above and that the invention can be implemented in other specific forms without departing from the spirit or essential features of the invention. Therefore, the embodiments should be considered exemplary and non-limiting in all respects, and the scope of the invention is defined by the appended claims rather than the foregoing description, and it is intended that all variations falling within the meaning and scope of the equivalent elements of the claims be included in the invention. Any reference numeral in a claim should not be considered as limiting the claim to which it relates.

Claims

1. A method for constructing a size effect equation for type I fracture of compacted clay considering friction effect, characterized by: At least the following steps are included: S1: Determine the friction coefficient between the support material and the sample; S2: Conducting size effect tests on compacted clay type I fractures; S3: Extract the crack tip scattered spot coordinates and deformation information at a certain loading moment of the mode I fracture test based on digital image technology; S4: The deformation information obtained in S3 is imported into the matrix calculation, that is, the coefficients of the Williams series expansion are determined by matrix calculation, that is, the singular terms, non-singular terms, and the third high-order terms of the stress field at the tip of the type I fracture specimen are calibrated to realize the connection between the support friction, the specimen size, and the dimensionless crack length; S5: Based on the MMTS criterion, a size effect equation for the mode I fracture toughness of compacted clay considering the friction effect is constructed to achieve the prediction of the mode I fracture toughness and mode I fracture process zone of compacted clay at an engineering scale.

2. The method for constructing a compacted clay type I fracture size effect equation considering friction effect according to claim 1, characterized in that: The S1 at least comprises the following steps: Build the triangle's bottom support; Place the sample on the bottom support, gradually lift one end of the bottom support, and stop lifting when the sample starts to slide down; Measure the inclination angle α at this time, and the friction coefficient between the support material and the sample is as follows (1): f = tanα (1).

3. The method for constructing a compacted clay type I fracture size effect equation considering friction effect according to claim 2, characterized in that: The S2 at least comprises the following steps: First, the compacted clay is mixed with water and soil according to the optimal moisture content; The mixed soil is placed into custom-made moulds of different sizes and compacted in three layers, with each layer being fully shaved to prevent delamination; After compaction, demould the specimen; The cracks are then preformed on a cutting machine; Then the specimen was coated with putty powder and then sprayed with speckle to complete the preparation of soil sample; The soil fracture instrument was used to carry out experimental application on soil samples; The fracture size effect test was carried out using observation equipment.

4. The method for constructing a compacted clay type I fracture size effect equation considering friction effect according to claim 1, characterized in that: S3 performs full-field strain analysis on the photos taken by DIC through VIC-2D software to extract the coordinates of the scattered plate points in the analysis area and the deformation information at a certain loading moment. S3 at least includes the following steps: The photos taken by the DIC high-speed camera were imported into the VIC-2D software, and the analysis area was divided around the crack tip; After full-field strain analysis of the analysis area, K points are selected at the crack tip. The K value is determined according to the required number of high-order terms, as shown in the following formula (2): K>N+M+2 (2) In formula (2), N and M are the coefficients of the higher-order terms A and B, respectively. When calculating the third higher-order term, K should be greater than 8. To ensure the accuracy of the calculation, the distance from the crack tip is selected as l, a / 4≤l≤a / 2, and a is the crack length.

5. The method for constructing a compacted clay type I fracture size effect equation considering friction effect according to claim 4, characterized in that: The step 4 at least comprises the following steps: The coefficients of the Williams series expansion are determined by the overdetermined finite element method, which uses the displacement field and coordinate field of the crack tip node to determine the coefficients. Combined with S3, the displacement field and coordinate field of K nodes at the crack tip are extracted using VIC-2D software and substituted into the Williams series expansion to obtain a set of overdetermined equations. The optimal solution is obtained using the least squares method. The progressive displacement field at the crack tip is shown in equations (3) and (4): Where u and v represent the displacement in the x and y directions respectively, and the shear modulus μ = E / 2(1+v), Kolosov constant is κ=(3-v) / (1+v) in plane stress problems and κ=3-4v for plane strain problems; When n=0, as shown in the following equations (5) and (6): Substituting the selected node coordinates into equation (7) yields a system of equations; It can be simplified into formula (8): [U] 2k×1 =[C] 2k×(N+M+2) [X] (N+M+2)×1 (8) Where: [U] is the displacement vector of the node, [C] is the node position coordinate matrix of the angular function f(r,θ) and g(r,θ); Using the least squares method, we can get formula (9): [C] T [U]=[C] T [C][X][X]=([C] T [C]) -1 [C] T [U](9) After solving equation (7), we can obtain A n With B n Williams higher-order terms.

6. The method for constructing a compacted clay type I fracture size effect equation considering friction effect according to claim 5, characterized in that: The S5 at least comprises the following steps: The first-order coefficients A1 and B1 of the Williams series expansion are related to the stress intensity factor K I and K II， The second-order coefficient A2 corresponds to the T stress, and the second-order term corresponding to B2 is zero, as shown in the following formula (10): According to the MMTS criterion, the crack propagates along the direction θ0 perpendicular to the maximum tangential stress, and the critical distance r from the crack tip is c Tangential stress σ θθ Reaching the critical value σ θθc When the crack starts from the tip along the θ0 direction, the tangential stress component σ at the crack tip is expanded using the Williams series. θθ Formula (11): In the formula, K I and K II are type I and type II stress intensity factors, respectively; T is independent of the crack tip distance r c The non-singular stress term is usually called T stress; A3 and B3 are the coefficients of the third-order term in the Williams series expansion. 3、 The expressions of B3 and T are shown in formula (12): Where A3 * 、B3 * , T * It is a dimensionless parameter that has nothing to do with the specimen size and load size, and only depends on the crack length a / R and the support distance S / R. Substituting the dimensionless parameter and ignoring the higher-order term O(r), the critical distance r from the crack tip is c The tangential stress component at can be expressed as formula (13): According to the MTS standard, the fracture occurs at the angle θ0 where the tangential stress is the largest. The angle θ0 is determined by differentiating θ by equation (11) and then solving equation (13), as shown in equation (14): According to the second assumption of the MTS criterion, when the maximum tangential stress σ along θ0 is θθ and the critical distance r from the crack tip c Reaching the critical value σ θθc When the fracture occurs, the critical stress σ θθc Assumed to be tensile strength f t , as shown in formula (15): Where K If and K IIf is the stress intensity factor K I and K II The critical values ​​of , which characterize the fracture resistance of materials under mixed-mode loading; The critical distance r in the equation c , must be determined by equations (11) and (13) to obtain the fracture initiation angle θ0 and predict the onset of fracture. Equation (16) takes into account the influence of higher-order terms used to determine the elastic equivalent FPZ length. This equation is written as: Where: γ=6(A3 * / K I * ); MMTS standard assumptions c The value depends on the size of the specimen, therefore, r c The size dependence of isotope is used to analyze the effect of size on the mixed mode fracture behavior of compacted clay, as shown in formula (17): where the coefficients A and B are determined by linear regression using the effective Mode I fracture toughness of the material obtained from specimens of different sizes, by first converting the Mode I fracture toughness (K If ) is replaced in equation (16) to determine the r corresponding to each specimen size. c value, the calculated r c Substitute the values ​​into equation (18) and calculate based on the linear regression of Y = MX + Q; Substituting the regression coefficients A and B into equation (17), we can obtain r c The relationship between R and r can be obtained. c The relationship curve between the predicted r c Substitute the value into formula (15) to calculate the predicted K eff The value can also be obtained to realize the prediction of mode I fracture toughness and mode I fracture process zone of compacted clay at engineering scale.

Images