Method for measuring crack length in high-temperature fatigue crack propagation process

Abstract

The invention relates to the technical field of crack detection, and discloses a method for measuring the crack length in a high-temperature fatigue crack propagation process, which comprises the following steps of: grinding and polishing the surface of a sample by using abrasive paper, selecting 45 steel of the same material as a high-temperature crack propagation test object, the method comprises the following steps: taking a CT sample as a raw material, preprocessing the surface of the CT sample, applying a fatigue load to the CT sample to carry out a high-temperature fatigue crack propagation test, observing and recording surface cracks of the sample, carrying out the fatigue crack propagation test, and measuring the fracture crack length based on a visual method of image recognition and an image measuring instrument. According to the method, the crack length in the current high-temperature fatigue crack growth process can be directly obtained through image shooting of the crack area by the image shooting equipment and calculation by comparing the Photoshop software with the crack area algorithm of the scale, and the crack length on the surface of the sample obtained through the visual method based on image recognition is good in precision and high in accuracy. And ideal effects can be achieved at different high temperatures.

Description

Technical Field

[0001] The invention relates to the technical field of crack detection, in particular to a method for measuring crack length during high-temperature fatigue crack extension. Background Art

[0002] Fatigue crack growth performance refers to the ability of a material to expand and develop cracks under alternating loads or cyclic stresses. Fatigue is the damage and destruction caused by changes in the microstructure inside the material during repeated loading. Fatigue crack growth is an important stage in this process, involving the initiation, expansion and eventual failure of the crack, and is crucial to the structural integrity assessment of engineering components.

[0003] The traditional method of testing fatigue crack growth performance is to perform cyclic tensile fatigue loading on the specimen. In a high temperature environment, the crack length of the specimen can be measured in real time by visual inspection, compliance method or potential method. Visual inspection is a detection method that observes and judges cracks with the naked eye with the help of measuring instruments or image processing software.

[0004] The measurement accuracy of the potential method is easily disturbed under high temperature conditions, the measurement results have large errors, and will be affected by the material structure; the compliance method can test the crack length of different experimental materials under different environments, but it requires a special high-temperature opening displacement gauge. These devices are often expensive and technically complex. Therefore, commonly used crack measurement methods such as the potential method and the compliance method are no longer applicable. In view of this, we propose a method for measuring crack length during high-temperature fatigue crack propagation. Summary of the invention

[0005] In view of the shortcomings of the prior art, the present invention provides a method for measuring crack length during high-temperature fatigue crack propagation, which solves the problem that traditional crack propagation measurement technology in the prior art cannot be directly applied to high-temperature environments and requires specially designed and calibrated high-temperature testing equipment, such as high-temperature testing machines and high-temperature microscopes, to ensure the accuracy and reliability of the measurement.

[0006] To achieve the above objectives, the present invention is implemented by the following technical solutions: A method for measuring crack length during high temperature fatigue crack growth, comprising the following steps: S1: Polish the sample surface with sandpaper 45 steel was selected as the test object for high-temperature crack growth. Different types of sandpaper were used to polish the surface of the CT specimens according to the mesh count from low to high. A high-temperature resistant white paint layer was sprayed on the surface of the specimens. At the same time, a laser marking machine was used to engrave high-definition rulers on both sides of the crack growth path. S2: Apply fatigue load to CT specimens for high temperature fatigue crack growth test and observe and record surface cracks of specimens The CT specimen is mounted on the fatigue testing machine through a fixture, with the painted side with the scale facing the observation window. The crack front shape curve is obtained by adjusting the test frequency and load method. The displacement between the front and the root of the specimen notch is measured under an optical microscope or laser scanner, and the actual crack length is determined. S3: Visual method based on image recognition The fatigue crack image of the sample surface observed by the mobile phone was input into Photoshop through the high-definition ruler engraved on the sample surface, and the crack extension length was measured by comparison based on the laser-printed ruler. S4: Fracture crack length measurement based on image measuring instrument The image measuring instrument uses the nine-point method to measure the distance from the two dividing lines to the root of the notch, and compares the results obtained by the image measuring instrument with the crack length data obtained by visual detection.

[0007] Preferably, the chemical mass fraction of the sample used as the test sample in S1 is: 0.45% C, 0.25% Cr, 0.65% Mn, 0.25% Ni, 0.0035% P, 0.035% S, 0.27% Si; its main material mechanical properties are: elastic modulus 200-220 GPa, yield strength 340-370 MPa, tensile strength 550-650 MPa, and the crack propagation direction is perpendicular to the rolling direction. The sample is formed by wire cutting, and the sample thickness is consistent with the plate thickness of 10 mm.

[0008] Preferably, during the sandpaper grinding and polishing process in S1, a fixed fixture is used to fix the CT sample, and after the fixation is completed, the surface of the CT sample is polished by a program-controlled grinding component.

[0009] Preferably, the image optimization algorithm is used in S1 to optimize the clarity and accuracy of the laser engraved scale, and the optimization modeling process involves power, frequency, and scanning speed in laser marking, and the mark engraving quality includes line clarity and dimensional accuracy.

[0010] Preferably, in the S2 test, the fatigue crack extension test load is determined to be 10 kN according to the crack length and stress intensity factor range at the end of the prefabricated crack, and the frequency is kept unchanged at 10 Hz. When the visual crack length is 10 mm, the load is increased to 20 kN and the sample surface image is recorded by taking photos. When the visual crack length is 20 mm, the test is terminated, the sample surface image is recorded by taking photos, and then the sample is broken.

[0011] Preferably, the CT specimen is mounted on a fatigue testing machine by a fixture, with the painted side with the scale facing the observation window, the test temperature is set to 350-450 degrees, and a 5 mm long crack is prefabricated with parameters of f=10 Hz and Kmax=20 MPa·m0.5 before the test.

[0012] Preferably, in S2, a stress-strain model of the CT specimen at different frequencies and loads is established based on finite element simulation software, and the optimal test frequency and load parameter combination is reversely solved according to the target crack front shape curve through an algorithm, and these parameters are automatically input into the control system of the fatigue testing machine for automatic control.

[0013] Preferably, an image measurement system is established in S3, cracks and scales in the image are automatically identified by an algorithm, and the scale of the image is automatically calibrated according to the image imported into Photoshop, and the position and boundary of the crack are identified by using a target detection algorithm in the process.

[0014] Preferably, the data verification comparison in S4 preferentially eliminates measurement results with lower confidence, whose confidence is determined based on the discrete state between all measurement data, and after eliminating the measurement values ​​with lower confidence, the average value of multiple image measurement values ​​is selected as the final crack length measurement result.

[0015] Preferably, during the measurement process, the image measuring instrument can be optionally replaced with a scanning electron microscope or an optical microscope for detection, and the mobile phone photography can be optionally replaced with a high-definition camera or a CCD optical camera.

[0016] The present invention provides a method for measuring crack length during high temperature fatigue crack propagation. It has the following beneficial effects: 1. The present invention can directly obtain the crack length in the current high-temperature fatigue crack extension process by capturing images of the crack area with an image recording device and calculating the crack area algorithm with a comparison scale using Photoshop software. The crack length on the sample surface obtained by a visual method based on image recognition has good accuracy and can achieve ideal results at different high temperatures.

[0017] 2. The present invention can obtain the actual crack extension length by using the microscopic image of the sample fracture, and compare it with the visual crack length. Through this method, the actual situation of crack extension can be more accurately evaluated, thereby providing more reliable data support for the fatigue life prediction of the material. In addition, the measurement method also has the advantages of simple operation and low cost, and is easy to promote and use in laboratories and industrial sites.

[0018] 3. The present invention finds the best combination of marking parameters by establishing a relationship model between laser marking parameters (such as power, frequency, scanning speed, etc.) and scale engraving quality (such as line clarity, dimensional accuracy). By optimizing the laser marking parameters, the present invention can ensure the accuracy and clarity of scale engraving, thereby improving the accuracy of crack length measurement and providing a more accurate quantitative basis for the analysis of high-temperature fatigue crack propagation.

[0019] 4. The present invention uses an algorithm to reversely solve the optimal test frequency and load parameter combination according to the target crack front shape curve to realize automatic control of the test process, and automatically adjusts the test parameters according to the crack growth rate to ensure the continuity and stability of the test. BRIEF DESCRIPTION OF THE DRAWINGS

[0020] Figure 1 This is a flow chart of the method for measuring crack length during high temperature fatigue crack growth; Figure 2 This is a schematic diagram of the sample surface spray painting of the present invention; Figure 3 It is a schematic diagram of the sample surface scale of the present invention; Figure 4 Schematic diagram of a high temperature fatigue crack growth test device used in the present invention; Figure 5 This is a schematic diagram of the sample surface when the crack length is 10 mm using the visual method of the present invention; Figure 6 This is a schematic diagram of the sample surface when the crack length is 20 mm according to the visual method of the present invention; Figure 7 It is a schematic diagram of the CT sample fracture of the present invention; Figure 8 A schematic diagram of an image measuring instrument used in the present invention; Fig. 9 This is a schematic diagram of measuring crack extension using the nine-point method of the present invention. DETAILED DESCRIPTION

[0021] The following will be combined with the drawings in the specification of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0022] Example: Please see attached Figure 1 -Attached Fig. 9 The embodiment of the present invention provides a method for measuring crack length during high temperature fatigue crack propagation, comprising the following steps: S1: Polish the sample surface with sandpaper 45 steel was selected as the object of high temperature crack growth test. After the surface of CT specimen was polished with 240 mesh, 500 mesh and 1000 mesh sandpaper, a high temperature resistant white paint layer was sprayed on the surface of the specimen. Figure 2 As shown in the figure, the contrast of the sample surface is increased to make the crack propagation path more obvious, and a high-temperature resistant white paint layer is sprayed on the sample surface. At the same time, a high-definition ruler is engraved on both sides of the crack propagation path using a laser marking machine. Figure 3 , which is convenient for subsequent visual measurement of crack extension, and the crack length is measured from the root of the notch; S2: Apply fatigue load to CT specimens for high temperature fatigue crack growth test and observe and record surface cracks of specimens The CT specimen is mounted on the fatigue testing machine through a fixture, with the painted side and the ruler facing the observation window, and the test temperature is set to 400 degrees. Before the test, a 5 mm long crack is prefabricated with parameters of f=10Hz and Kmax=20 MPa·m0.5. In the fatigue crack growth test, the crack front shape curve is obtained by adjusting the test frequency and load method, and the displacement between the crack front and the root of the specimen notch is measured under an optical microscope or laser scanner, and the actual crack length is determined; S3: Visual method based on image recognition The fatigue crack image of the sample surface observed by the mobile phone was input into Photoshop through the high-definition ruler engraved on the sample surface, and the crack extension length was measured by comparison based on the laser-printed ruler. S4: Fracture crack length measurement based on image measuring instrument The image measuring instrument uses the nine-point method to measure the distance from the two dividing lines to the root of the notch, and compares the results obtained by the image measuring instrument with the crack length data obtained by visual detection.

[0023] The chemical mass fraction of the sample used as the test sample in S1 is: 0.45% C, 0.25% Cr, 0.65% Mn, 0.25% Ni, 0.0035% P, 0.035% S, 0.27% Si; its main material mechanical properties are: elastic modulus 200-220 GPa, yield strength 340-370 MPa, tensile strength 550-650 MPa, and the crack propagation direction is perpendicular to the rolling direction. The sample is formed by wire cutting, and the sample thickness is consistent with the plate thickness of 10 mm.

[0024] In the sandpaper grinding and polishing process in S1, a fixed fixture is used to fix the CT sample, and after the fixation is completed, the surface of the CT sample is polished by a program-controlled grinding component. The surface of the sample is polished with sandpaper. On the one hand, the surface roughness of the sample meets the characterization requirements, and on the other hand, it is convenient for subsequent spraying of high-temperature resistant paint to observe cracks. A layer of high-temperature resistant white paint is sprayed on the side with higher finish, and a scale is laser printed on both sides of the crack propagation path to facilitate observation of the crack propagation length.

[0025] In S1, an image optimization algorithm is used to optimize the clarity and accuracy of the laser-engraved scale. The optimization modeling process involves power, frequency, and scanning speed in laser marking, and the quality of mark engraving includes line clarity and dimensional accuracy. The following algorithms are used in the process: This algorithm is used to optimize the clarity and accuracy of laser marking machines when engraving high-definition rulers. By establishing a relationship model between laser marking parameters (such as power, frequency, scanning speed, etc.) and ruler engraving quality (such as line clarity and dimensional accuracy), the best combination of marking parameters can be found: Assume the line clarity index is , the dimensional accuracy index is , the laser power is , the frequency is , the scanning speed is .

[0026] Assuming line clarity The relationship with each parameter can be expressed by the following linear regression model:

[0027] in , , is the coefficient corresponding to each parameter, is the intercept term, and by collecting a large number of actual burning effect data under different parameter combinations (i.e. The actual measured values), using the least square method and other methods to fit and determine these coefficients; Similarly, for dimensional accuracy P, :

[0028] The coefficients are also determined by fitting the data and the intercept term ; Assume that after many tests, the following set of data is obtained (the actual data volume needs to be larger): |Laser power (W)|Frequency (Hz)|Scanning speed (mm / s)|Line clarity (Actual measurement value) | Dimension accuracy (actual measurement value) |

[0029] |50|10000|50|80 (hypothesized clarity score) |95 (hypothesized accuracy score)| |60|12000|40|85|92| |40|8000|60|75|98| Substitute these data into the above linear regression model formula and calculate the coefficients by the least squares method. as well as , for example, after calculation, we get , , , (These are example values ​​only). Then, based on these coefficients, when a new laser marking task is given, the line clarity and dimensional accuracy under different parameter combinations can be predicted, so that the best parameter combination can be selected to achieve a better burning effect.

[0030] In the S2 test, the prepared high temperature crack extension specimen is installed on an electro-hydraulic servo fatigue testing machine (the test device for measuring the crack length during high temperature fatigue crack extension is as follows: Figure 4 As shown), the furnace is closed for heating and heat preservation for a period of time before the test is carried out. According to the crack length at the end of the prefabricated crack and the range of stress intensity factor, the fatigue crack growth test load is determined to be 10kN, and the frequency is kept unchanged at 10Hz. Figure 5 This is the surface image of the sample when the visual crack length is 10 mm. When the visual crack length is 10 mm, the load is increased to 20 kN and the sample surface image is recorded by taking photos; Figure 6 This is the surface image of the sample when the visual crack length is 20 mm. When the visual crack length is 20 mm, the crack length on the crack surface is observed through the observation window and photographed and recorded. The load is adjusted at the target crack length and a photograph of the sample surface is taken. The test is terminated, the sample surface image is photographed and recorded, and then the sample is broken.

[0031] The CT specimen was mounted on a fatigue testing machine through a fixture, with the painted side with a scale facing the observation window. The test temperature was set at 350-450 degrees. Before the test, a 5 mm long crack was prefabricated with parameters of f=10 Hz and Kmax=20 MPa·m0.5.

[0032] In the S2, a stress-strain model of the CT specimen under different frequencies and loads is established based on finite element simulation software, and the optimal test frequency and load parameter combination is reversely solved according to the target crack front shape curve through an algorithm, and these parameters are automatically input into the control system of the fatigue testing machine for automatic control. The process involves the following algorithms: Based on the finite element simulation software, the stress-strain model of the CT specimen under different frequencies and loads is established. Then, the algorithm is used to reversely solve the optimal test frequency and load parameter combination according to the target crack front shape curve to realize the automated control of the test process: Assume that the target crack front shape curve is expressed by the function express( is the coordinate of a certain direction on the sample surface), and the finite element simulation is used to obtain the and load The crack front shape curve under , define the objective function To measure the difference between the simulation curve and the target curve, the mean square error (MSE) is used as the objective function:

[0033] in is the number of sampling points; For the The coordinates of the sampling points; To find the best combination of test frequency and load parameters is to find the objective function Smallest and The value of is implemented by the optimization algorithm. For the optimization algorithm, the update formula is as follows (with frequency For example, the load Similarly):

[0034] in is the updated frequency value; is the current frequency value; is the learning rate; is the objective function The partial derivative of Through continuous iterative updates and until the objective function Reach a minimum or meet certain convergence conditions.

[0035] Assuming the target crack front shape curve exist The range from 0 to 100 (unit: mm) is a known function, for example ; The crack front shape curves at different frequencies and loads were obtained through finite element simulation. , such as when Hz, kN, ; Set the initial frequency Hz, initial load kN, learning rate ; Calculate the objective function The value of , and then calculate the partial derivatives of frequency and load respectively, and then update the value of frequency and load, and repeat this process until the value of the objective function reaches the minimum or meets the convergence condition. For example, after several iterations, the optimal frequency is obtained. Hz, optimal load kN, at this time, the simulated crack front shape curve is closest to the target curve, so that the experimental process can be completely completed through the program established by the algorithm, thereby achieving the measurement of crack length during high temperature fatigue crack propagation of the automatically controlled fatigue testing machine.

[0036] In the S3, an image measurement system is established, and the cracks and rulers in the image are automatically identified by an algorithm, and the scale of the image is automatically calibrated according to the image imported into Photoshop, and the crack extension length is obtained by comparing the scale with the crack extension path; the distance between the crack front shape curve and the sample notch is measured under the image measuring instrument to determine the actual crack length, and the target detection algorithm is used to identify the position and boundary of the crack in the process. In the process, the cracks and rulers in the image are automatically identified by an algorithm, and the scale of the image is automatically calibrated according to the image imported into Photoshop, and then the crack extension length is obtained. The algorithm is as follows: Assume that the fatigue crack image of the sample surface observed by the mobile phone is , the laser printed ruler image is ; First, feature extraction is used to extract feature points from the two images. , get the feature point set for , get the feature point set:

[0037] Then, feature matching is used to match the feature points in the two images. Let the successfully matched feature point pair be:

[0038] Calculate the scale of the image based on the matched feature point pairs. Assume that the matched feature point pairs are Two points in and The actual distance corresponding to (It is known that this is a laser-printed ruler mark), The corresponding two points in and The pixel distance is (calculated by image coordinates), then the scale of the image is It can be calculated as:

[0039] Once the scale is obtained, the pixel length of the identified crack in the image can be used to (determined by identifying the location and boundary of the crack through the target detection algorithm) and calculating the actual extension length of the crack

[0040] Assume that the laser printed ruler image There are two feature points and , the actual distances they correspond to mm (known); Image of fatigue crack on the sample surface observed on a mobile phone In the example, feature extraction and matching algorithms are used to find feature points that match the above two feature points. and , their pixel distance Pixels (calculated from image coordinates); The scale of the image is calculated according to the formula mm / pixel; Assume that the pixel length of the crack in the image is identified by the target detection algorithm Pixels; Then the actual extension length of the crack is calculated according to the formula mm.

[0041] The data verification comparison in S4 preferentially eliminates the measurement results with low confidence. The confidence is determined according to the discrete state between all the measurement data. After eliminating the measurement values ​​with low confidence, the average value of multiple image measurement values ​​is selected as the final crack length measurement result. The effectiveness of this method is verified by comparing the crack extension length of the visual method with the crack extension length of the fracture. Figure 7The figure shows the fracture of the CT specimen under the image measuring instrument. It can be clearly found that there is a clear boundary at the fracture after sudden loading and breaking at the end of the test. Figure 8 The image measuring instrument shown uses the nine-point method to measure the distances from the two dividing lines to the root of the notch, which are 10.015 mm and 20.027 mm respectively. The errors of the crack lengths obtained by the visual method are 0.15% and 1.35% respectively. The results show that the high-temperature crack length measured by the visual method has a higher accuracy.

[0042] During the measurement process, the image measuring instrument can be optionally replaced with a scanning electron microscope or an optical microscope for detection, and the mobile phone photography can be optionally replaced with a high-definition camera or a CCD optical camera.

[0043] Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions and variations may be made to the embodiments without departing from the principles and spirit of the present invention, and that the scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A method for measuring crack length during high temperature fatigue crack growth, characterized in that: The following steps are involved: S1: Polish the sample surface with sandpaper 45 steel was selected as the test object for high-temperature crack growth. Different types of sandpaper were used to polish the surface of the CT specimens according to the mesh count from low to high. A high-temperature resistant white paint layer was sprayed on the surface of the specimens. At the same time, a laser marking machine was used to engrave high-definition rulers on both sides of the crack growth path. S2: Apply fatigue load to CT specimens for high temperature fatigue crack growth test and observe and record surface cracks of specimens The CT specimen is mounted on the fatigue testing machine through a fixture, with the painted side with the scale facing the observation window. The crack front shape curve is obtained by adjusting the test frequency and load method. The displacement between the front and the root of the specimen notch is measured under an optical microscope or laser scanner, and the actual crack length is determined. S3: Visual method based on image recognition The fatigue crack image of the sample surface observed by the mobile phone was input into Photoshop through the high-definition ruler engraved on the sample surface, and the crack extension length was measured by comparison based on the laser-printed ruler. S4: Fracture crack length measurement based on image measuring instrument The image measuring instrument uses the nine-point method to measure the distance from the two dividing lines to the root of the notch, and compares the results obtained by the image measuring instrument with the crack length data obtained by visual detection.

2. The method for measuring crack length during high temperature fatigue crack growth according to claim 1, characterized in that: The chemical mass fraction of the sample used as the test sample in S1 is: 0.45% C, 0.25% Cr, 0.65% Mn, 0.25% Ni, 0.0035% P, 0.035% S, 0.27% Si; The main mechanical properties of the material are: elastic modulus 200-220 GPa, yield strength 340-370 MPa, tensile strength 550-650 MPa, and the crack propagation direction is perpendicular to the rolling direction. The sample is cut and formed by wire cutting, and the sample thickness is consistent with the plate thickness of 10 mm.

3. The method for measuring crack length during high temperature fatigue crack growth according to claim 1, characterized in that: In the sandpaper grinding and polishing process in S1, a fixed fixture is used to fix the CT sample, and after the fixation is completed, the surface of the CT sample is polished by a program-controlled grinding component.

4. The method for measuring crack length during high temperature fatigue crack growth according to claim 1, characterized in that: In S1, an image optimization algorithm is used to optimize the clarity and accuracy of the laser engraved scale. The optimization modeling process involves power, frequency, and scanning speed in laser marking, and the mark engraving quality includes line clarity and dimensional accuracy.

5. The method for measuring crack length during high temperature fatigue crack growth according to claim 1, characterized in that: In the S2 test, the fatigue crack growth test load is determined to be 10 kN according to the crack length and stress intensity factor range at the end of the prefabricated crack, and the frequency is kept unchanged at 10 Hz. When the visual crack length is 10 mm, the load is increased to 20 kN and the sample surface image is recorded by taking photos. When the visual crack length is 20 mm, the test is terminated, the sample surface image is recorded by taking photos, and then the sample is broken.

6. The method for measuring crack length during high temperature fatigue crack growth according to claim 1, characterized in that: The CT specimen was mounted on a fatigue testing machine through a fixture, with the painted side with a scale facing the observation window. The test temperature was set at 350-450 degrees. Before the test, a 5 mm long crack was prefabricated with parameters of f=10 Hz and Kmax=20 MPa·m0.

5.

7. The method for measuring crack length during high temperature fatigue crack growth according to claim 1, characterized in that: In the S2, a stress-strain model of the CT specimen under different frequencies and loads is established based on finite element simulation software, and the optimal test frequency and load parameter combination is reversely solved according to the target crack front shape curve through an algorithm, and these parameters are automatically input into the control system of the fatigue testing machine for automatic control.

8. The method for measuring crack length during high temperature fatigue crack growth according to claim 1, characterized in that: In the S3, an image measurement system is established to automatically identify cracks and scales in the image through an algorithm, and automatically calibrate the scale of the image according to the image imported into Photoshop. During the process, a target detection algorithm is used to identify the position and boundary of the crack.

9. The method for measuring crack length during high temperature fatigue crack growth according to claim 1, characterized in that: The data verification and comparison in S4 preferentially eliminates the measurement results with lower confidence, whose confidence is determined according to the discrete state between all the measurement data, and after eliminating the measurement values ​​with lower confidence, the average value of multiple image measurement values ​​is selected as the final crack length measurement result.

10. The method for measuring crack length during high temperature fatigue crack growth according to claim 1, characterized in that: During the measurement process, the image measuring instrument can be optionally replaced with a scanning electron microscope or an optical microscope for detection, and the mobile phone photography can be optionally replaced with a high-definition camera or a CCD optical camera.

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