Inversion method of I-type crack cohesion constitutive model parameters

Abstract

The embodiment of the invention provides an I-type crack cohesion constitutive model parameter inversion method, which belongs to the technical field of calculation, and specifically comprises the following steps: step 1, carrying out an I-type fracture test on a sample rock to obtain a relation curve between a load and crack mouth opening displacement; 2, determining the elastic modulus and the initial fracture toughness of the sample rock according to the relation curve; 3, selecting a plurality of data points from the relation curve; 4, substituting the elastic modulus and the crack initiation fracture toughness into a target formula to solve each data point to obtain a corresponding cohesion value and crack surface opening displacement; 5, fitting the cohesion value corresponding to each data point and the crack surface opening displacement, and determining the cohesion constitutive relationship of the sample rock; and step 6, substituting the constitutive relation into a forward calculation formula, comparing whether a calculation result is consistent with the relation curve, and outputting the constitutive relation when the calculation result is consistent with the relation curve. Through the scheme disclosed by the invention, the determination efficiency, accuracy and adaptability are improved.

Description

Technical Field

[0001] The disclosed embodiments relate to the field of computing technology, and more particularly to an inversion method for parameters of a mode I crack cohesive constitutive model. Background Art

[0002] Currently, stress intensity factor theory is often used to describe the growth of Mode I cracks in rock. However, subsequent research has shown that the stress intensity factor theory alone is insufficient to describe tensile crack growth in rock. This is because rock is a typical quasi-brittle material. When a crack in a rock body is subjected to tensile stress, a concentrated region of microcracks is generated at the crack tip. This concentrated region of microcracks resists crack propagation in the rock and blunts the originally sharp crack tip, causing the Mode I crack growth characteristics of the rock to deviate from the predictions of the stress intensity factor theory. Currently, the rock mechanics community believes that a method combining stress intensity factor theory with a cohesive constitutive model can not only accurately describe Mode I crack growth in rock but also obtain the size-independent Mode I fracture toughness of rock. Therefore, the key issue in using this method to describe Mode I crack growth in rock is how to obtain the constitutive relations of the cohesive model for the rock specimen. Traditionally, the cohesive constitutive model of rock mode I cracks is often determined by direct tensile testing. However, deep hard rock has high brittleness. When using direct tensile testing to determine the cohesive constitutive model of deep hard rock mode I cracks, it is difficult to obtain the full stress-strain curve of the rock under direct tensile testing, which brings substantial difficulties to the determination of the cohesive constitutive model.

[0003] It can be seen that there is an urgent need for an inversion method for determining the parameters of the cohesive constitutive model of mode I crack with high measurement efficiency, accuracy and adaptability. Summary of the Invention

[0004] In view of this, an embodiment of the present disclosure provides an inversion method for parameters of a mode I crack cohesive constitutive model, which at least partially solves the problems of poor measurement efficiency, accuracy and adaptability in the prior art.

[0005] The present disclosure provides a method for inverting parameters of a mode I crack cohesive constitutive model, including:

[0006] Step 1: Perform a mode I fracture test on the sample rock to obtain a curve showing the relationship between load and crack mouth opening displacement;

[0007] Step 2: Determine the elastic modulus and fracture toughness of the sample rock according to the relationship curve;

[0008] Step 3, selecting multiple data points from the relationship curve;

[0009] Step 4: Substitute the elastic modulus and the fracture toughness into the target formula to solve each data point to obtain the corresponding cohesion value and crack opening displacement;

[0010] Step 5: Fit the cohesion value and crack opening displacement corresponding to each data point to determine the cohesion constitutive relationship of the sample rock;

[0011] Step 6: Substitute the constitutive relationship into the forward calculation formula, compare the calculation results with the relationship curve to see if they are consistent, and output the constitutive relationship if they are consistent.

[0012] According to a specific implementation of the embodiment of the present disclosure, step 2 specifically includes:

[0013] Analyze the slope C of the initial loading section and the cracking load P of the relationship curve ini , and the elastic modulus E and the fracture toughness at cracking are obtained

[0014] According to a specific implementation of the embodiment of the present disclosure, the expression of the target formula is

[0015]

[0016] Where w is the crack opening displacement, which is a function of the crack plane coordinate x, where x = 0 corresponds to the position of the crack mouth; E is the elastic modulus; σ is the cohesive force, which is a function of the crack opening displacement w; F is the dimensionless stress intensity factor related to the specimen geometry; P is the load; D is the height of the specimen; B is the thickness of the specimen; a is the crack length; a e is the end of the cohesive force action area; G is the Green's function or weight function related to the specimen crack geometry, CMOD is the crack mouth opening displacement, CMOD = w(0); is the fracture toughness at crack initiation.

[0017] According to a specific implementation of the embodiment of the present disclosure, the expression of the crack surface opening displacement is:

[0018]

[0019] Where a0=a, a n+1 =a e ;

[0020] The expression of the cohesion value is:

[0021]

[0022] Where σ0=σ t and w0=0,σ t is the tensile strength of rock.

[0023] According to a specific implementation of an embodiment of the present disclosure, a method for solving the target formula includes a least squares method, a gradient descent method, or a Gauss-Newton method.

[0024] The inversion scheme of the parameters of the mode I crack cohesion constitutive model in the embodiment of the present disclosure includes: step 1, performing a mode I fracture test on the sample rock to obtain a relationship curve between the load and the crack mouth opening displacement; step 2, determining the elastic modulus and the crack initiation fracture toughness of the sample rock according to the relationship curve; step 3, selecting multiple data points from the relationship curve; step 4, substituting the elastic modulus and the crack initiation fracture toughness into the target formula to solve each data point to obtain the corresponding cohesion value and crack surface opening displacement; step 5, fitting the cohesion value and crack surface opening displacement corresponding to each data point to determine the cohesion constitutive relationship of the sample rock; step 6, substituting the constitutive relationship into the forward calculation formula, comparing the calculation result with the relationship curve to see whether it is consistent, and outputting the constitutive relationship when they are consistent.

[0025] The beneficial effects of the embodiments of the present disclosure are as follows: through the scheme of the present disclosure, a load-crack mouth opening displacement (P-CMOD) curve is obtained through a three-point bending test, and combined with a piecewise linear assumption and an inversion algorithm, the technical bottleneck of the traditional direct tensile test that is difficult to obtain a rock cohesion constitutive model is overcome; a piecewise linear function is used to describe the relationship between the crack surface opening displacement (w) and the cohesion (σ), and only a small number of key data points need to be selected to invert the complete cohesion constitutive model, which significantly reduces the experimental complexity and computational cost while ensuring the accuracy of parameter determination; it is not only applicable to deep hard rock, but can also be extended to other types of rock materials by adjusting the test parameters, and has wide engineering applicability. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] In order to more clearly illustrate the technical solutions of the embodiments of the present disclosure, the following briefly introduces the drawings required for use in the embodiments. Obviously, the drawings described below are only some embodiments of the present disclosure. For ordinary technicians in this field, other drawings can be obtained based on these drawings without any creative work.

[0027] Figure 1 A schematic flow chart of an inversion method for parameters of a mode I crack cohesive constitutive model provided in an embodiment of the present disclosure;

[0028] Figure 2 A schematic diagram of a piecewise linear w(x) of crack mouth opening displacement provided by an embodiment of the present disclosure;

[0029] Figure 3 A schematic diagram of a piecewise linear σ(w) of a cohesion value provided in an embodiment of the present disclosure;

[0030] Figure 4The embodiment of the present disclosure provides an initial loading section slope C and a cracking load P ini Schematic diagram of the determination method;

[0031] Figure 5 A flow chart of an inversion method for parameters of a deep hard rock mode I crack cohesive constitutive model provided in an embodiment of the present disclosure;

[0032] Figure 6 A load-to-crack-mouth opening displacement curve P-CMOD and a schematic diagram of data point selection provided in an embodiment of the present disclosure;

[0033] Figure 7 A schematic diagram of a three-point bending specimen provided in an embodiment of the present disclosure, wherein (a) is a schematic diagram of the specimen preparation process, and (b) is a photograph of the prepared specimen;

[0034] Figure 8 A load-crack mouth opening displacement curve obtained from a three-point bending test provided in an embodiment of the present disclosure;

[0035] Figure 9 The obtained loading section slope C and cracking load P are provided in the embodiment of the present disclosure. ini Schematic diagram of the results;

[0036] Figure 10 A schematic diagram of a granite cohesive constitutive model result obtained by inversion provided in an embodiment of the present disclosure;

[0037] Figure 11 A comparison diagram of a load-crack mouth opening displacement curve calculated based on the cohesive force constitutive model results and the test results provided in an embodiment of the present disclosure. DETAILED DESCRIPTION

[0038] The embodiments of the present disclosure are described in detail below with reference to the accompanying drawings.

[0039] The following describes the embodiments of the present disclosure through specific examples, and those skilled in the art can easily understand other advantages and effects of the present disclosure from the contents disclosed in this specification. Obviously, the described embodiments are only a part of the embodiments of the present disclosure, rather than all of the embodiments. The present disclosure can also be implemented or applied through other different specific embodiments, and the details in this specification can also be modified or changed in various ways based on different viewpoints and applications without departing from the spirit of the present disclosure. It should be noted that, in the absence of conflict, the following embodiments and features in the embodiments can be combined with each other. Based on the embodiments in the present disclosure, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present disclosure.

[0040] It should be noted that various aspects of the embodiments within the scope of the appended claims are described below. It should be apparent that the aspects described herein can be embodied in a wide variety of forms, and any specific structure and / or function described herein is merely illustrative. Based on this disclosure, it should be understood by those skilled in the art that an aspect described herein can be implemented independently of any other aspect, and two or more of these aspects can be combined in various ways. For example, any number of aspects described herein can be used to implement an apparatus and / or practice a method. In addition, other structures and / or functionalities other than one or more of the aspects described herein can be used to implement this apparatus and / or practice this method.

[0041] It should also be noted that the illustrations provided in the following embodiments are only schematic illustrations of the basic concept of the present disclosure. The illustrations only show components related to the present disclosure and are not drawn according to the number, shape and size of components in actual implementation. In actual implementation, the type, quantity and proportion of each component can be changed at will, and the component layout type may also be more complicated.

[0042] Additionally, in the following description, specific details are provided to provide a thorough understanding of the examples. However, one skilled in the art will appreciate that the aspects described can be practiced without these specific details.

[0043] The embodiments of the present disclosure provide an inversion method for parameters of a mode I crack cohesive constitutive model, which can be applied to the process of determining rock parameters in rock mechanics and engineering scenarios.

[0044] See also Figure 1 , is a flow chart of an inversion method for parameters of a mode I crack cohesive constitutive model provided by an embodiment of the present disclosure. Figure 1 As shown, the method mainly includes the following steps:

[0045] Step 1: Perform a mode I fracture test on the sample rock to obtain a curve showing the relationship between load and crack mouth opening displacement;

[0046] In specific implementation, for a stable crack expansion, when the crack is in the critical expansion state, The rock crack propagation condition can be expressed as:

[0047]

[0048] Where, is the crack tip stress intensity factor, is the fracture toughness at crack initiation, is the crack growth resistance, is the cohesive stress intensity factor generated by the cohesive force in the fracture process zone, a is the crack length, and a eIt is the end of the area where the cohesive force acts.

[0049] When the crack propagates steadily, the crack propagation resistance can be expressed as the stress intensity factor generated by the equivalent load p at the crack tip of the specimen:

[0050]

[0051] Where, is the equivalent stress intensity factor generated by the load p at the crack tip of the specimen, which can be calculated by the following formula:

[0052]

[0053] Where P is the load, B is the thickness of the specimen, D is the height of the specimen, and F is the dimensionless stress intensity factor of the specimen. The F value can be calculated using finite element software. In the field of rock mechanics, ABAQUS or ANSYS is often used to calculate F. The specific steps are to establish a finite element model with the same shape and arbitrary size as the test, and then apply the load P on the specimen. According to the output of the finite element software, Finally, the geometric shape factor F of the specimen can be expressed as:

[0054]

[0055] Only by changing the value of a and other configuration parameters in proportion can the F value under different a values ​​be obtained.

[0056] At the same time, in formula (1), It can be expressed as:

[0057]

[0058] Where σ is the cohesive force, w is the crack opening displacement, and G is the weight function or Green's function. The G value can be calculated using finite element software.

[0059] In formula (5), according to the linear superposition principle, the crack opening displacement can be calculated by the following formula:

[0060]

[0061] Where E is the elastic modulus. Let x = 0 in equation (6), and the crack mouth opening displacement CMOD of the specimen can be expressed as:

[0062]

[0063] If the crack does not extend, a=a ini (a ini is the initial crack length of the specimen before the crack propagates), CMOD can be expressed as:

[0064]

[0065] Step 2: Determine the elastic modulus and fracture toughness of the sample rock according to the relationship curve;

[0066] In specific implementation, according to formula (9), the elastic modulus E of the sample can be expressed as:

[0067]

[0068] Where C is the slope of the initial loading section of the load-crack mouth opening displacement curve, which can be directly determined by the load-crack mouth opening displacement curve. The fracture toughness of the same sample is The crack initiation load P can also be obtained from the end of the initial loading section of the load-crack mouth opening displacement curve. ini To determine, when the crack load P ini When it is known, let a=a ini and P=P ini , the fracture toughness of the sample It can be expressed as:

[0069]

[0070] The specific method for determining C is as follows:

[0071] Determine the initial loading section of the load-crack mouth opening displacement, use a linear function to fit the initial loading section of the load-crack mouth opening displacement, and the slope of the fitted linear function is the C value.

[0072] The fracture toughness P of the sample ini The determination method is as follows:

[0073] As the crack mouth opening displacement CMOD increases, the specimen load first increases linearly with the load. Since the fitting function of the initial segment of the load-crack mouth opening displacement curve has been obtained in the previous step, the load value at the moment when the fitting function and the load-crack mouth opening curve separate is the crack initiation load P. ini .

[0074] Specific C and P ini The determination method can refer to Figure 4 It should be noted that if the curve does not have an initial linear segment, it is considered that P ini =0.

[0075] Combining equations (1) to (5), the load P of the specimen can be expressed as:

[0076]

[0077] Step 3, selecting multiple data points from the relationship curve;

[0078] Step 4: Substitute the elastic modulus and the fracture toughness into the target formula to solve each data point to obtain the corresponding cohesion value and crack opening displacement;

[0079] In specific implementation, since the load-crack mouth opening displacement curve (P-CMOD) of the specimen is known, the following inversion method can be used to obtain the cohesive constitutive model of the rock (σ-w relationship).

[0080] First, it is assumed that the displacement w on the crack surface can be described by a piecewise linear function, see Figure 2 :

[0081]

[0082] Where a0=a, a n+1 =a e ,w(a i ) needs to satisfy the following formula:

[0083]

[0084] At the same time, it is believed that the constitutive relationship of the cohesive force model of the specimen can also be described by a piecewise linear function, see Figure 3 :

[0085]

[0086] Where σ0=σ t And w0=0. σ t The tensile strength of rock can be determined by the Brazilian splitting test or the direct tensile test.

[0087] Step 5: Fit the cohesion value and crack opening displacement corresponding to each data point to determine the cohesion constitutive relationship of the sample rock;

[0088] In specific implementation, according to formulas (8), (12), (13), (14) and (15), it can be seen that when the sample geometry, fracture parameters, and the constitutive relation σ-w of the cohesive force model are known, the P-CMOD curve of the sample can be derived by changing the crack length a. Similarly, when the P-CMOD curve of the sample is known, the constitutive relation σ-w of the cohesive force model of the sample can be obtained by inversion of the above formula. The specific calculation flow chart is as follows: Figure 5 The detailed calculation steps are as follows:

[0089] (I) Determine the elastic modulus E and the fracture toughness of the sample and tensile strength σ t ;

[0090] (II) Obtain (P from P-CMOD curve i,CMOD i ) data points, such as Figure 6 As shown;

[0091] (III) Assume that the crack tip starts from a ini Move to a0, substitute a=a0, P=P1 and CMOD=CMOD1 into equations (6), (7), (8), (12) to solve for w1 and σ1;

[0092] (IV) Assume that the crack tip moves from a0 to a'0, set a = a'0, P = P n+1 and CMOD = CMOD n+1 Substitute into equations (6), (7), (8), and (12) to solve for w n+1 and σ n+1 , where the new a' i (i=1,2,…,n) is recalculated by formula (13);

[0093] (V) Repeat step (IV) until σ n+2 ≤0.

[0094] In the above calculation steps, the end of the fracture process zone is always a constant value, that is, a n+1 ≡a e ≡a ini .

[0095] Step 6: Substitute the constitutive relationship into the forward calculation formula, compare the calculation results with the relationship curve to see if they are consistent, and output the constitutive relationship if they are consistent.

[0096] In specific implementation, the load-loading point displacement curve can be obtained by forward calculation using Equations (8) and (12) based on the obtained constitutive relationship, i.e., the parameters of the cohesive force constitutive model. The calculated results are compared with the relationship curve to see whether they are consistent. If they are consistent, the constitutive relationship is output.

[0097] The inversion method for the parameters of the cohesive constitutive model of a mode I crack provided in this embodiment obtains a load-crack mouth opening displacement (P-CMOD) curve through a three-point bending test. By combining the piecewise linear assumption with the inversion algorithm, the method overcomes the technical bottleneck of obtaining the rock cohesive constitutive model through traditional direct tensile testing. The relationship between the crack mouth opening displacement (w) and the cohesive force (σ) is described by a piecewise linear function. Only a small number of key data points need to be selected to invert the complete cohesive constitutive model, which significantly reduces the experimental complexity and computational cost while ensuring the accuracy of parameter determination. The method is not only applicable to deep hard rock, but can also be extended to other types of rock materials by adjusting the test parameters, and has wide engineering applicability.

[0098] The method disclosed herein will be further described below with reference to a specific embodiment. According to the method recommended by ASTM E399-09, Figure 7 The three-point bending specimen shown in the figure has a height of D = 45 mm, a length of L = 189 mm, and a thickness of B = 25 mm. The prefabricated crack is cut by a diamond wire saw with a diameter of 0.3 mm. The prefabricated crack length is a ini =13.5mm, and the crack width is approximately 0.4mm. Using this configuration allows for a stable crack growth process in subsequent fracture experiments, allowing for a complete P-CMOD curve of the specimen during loading. For this specimen, the geometric parameters F and G can be expressed as:

[0099]

[0100] The specimen processing and machining accuracy must strictly adhere to the methods recommended by the International Institute of Rock Mechanics. The flatness of the upper and lower surfaces, as well as the front and back surfaces, along the thickness direction of the specimens was controlled to within 0.01 mm. The stone used in the test was G654 granite, which is widely distributed in Zhangzhou, Fujian. Under natural lighting conditions, the rock sample appears gray-blue, with no visible macrocracks. Single-polarized and crossed-polarized microscopic images show that the main components of the rock sample are plagioclase (68%), quartz (18%), biotite (7%), hornblende (5%), pyroxene (1%), and other trace minerals (1%). Uniaxial compression and Brazilian splitting tests on the rock showed a uniaxial compressive strength of 174.9 MPa and a tensile strength of 9.8 MPa, indicating that the rock is a typical hard rock and meets the test requirements. The span of the three-point bending test was S = 4.0D. A rolling three-point bending fixture was used to prevent potential friction from affecting the test results. A clamp-on extensometer (maximum range 2.5 mm) was placed at the bottom of the specimen to measure the crack mouth (crack surface at the bottom of the specimen) during loading. The specimen load, P, was measured by a load cell mounted on the press frame. To avoid inertial effects and achieve stable crack growth, an axial displacement rate of 0.05 mm / min was used. Testing was terminated when the specimen load fell below 10 N.

[0101] Figure 8 The load-crack mouth opening displacement curve of the three-point bending specimen obtained by measurement. Figure 9 For this sample according to Figure 4 The obtained loading section slope C and crack initiation load P ini As a result, the initial loading section slope obtained here is C = 114.55 kN / mm, and the cracking load P ini =0.746kN. According to formula (10) and formula (11), the elastic modulus E and the fracture toughness of the sample can be calculated. 54.31GPa and 0.960MPa·m 0.5 .

[0102] From the load-crack mouth opening displacement, select the crack initiation point (P ini ,CMOD ini ) The other 8 data points are (1.39735kN, 0.02332017mm), (1.35037kN, 0.02802017mm), (1.29065kN, 0.03152017mm), (1.20454kN, 0.03582017mm), (1.1365kN, 0.03902017mm), (1.01876kN, 0.04422017mm), (0.94475kN, 0.04732017mm), and (0.88576kN, 0.04972017mm).

[0103] Substitute the data points into equations (6), (7), (8), and (12) step by step to solve (σ i ,w i )(i=1,…,8), the results of the cohesive constitutive model σ-w are as follows Figure 10 As shown. As w increases, σ shows a linear degradation characteristic, so the result obtained by linear function fitting is:

[0104] σ=-0.3247w+9.4983 (18)

[0105] In the above formula, the unit of w is μm and the unit of σ is MPa.

[0106] According to the obtained cohesive constitutive model parameters, the load-loading point displacement curve is calculated using equations (8) and (12) as shown in the figure below: Figure 11 As shown, it is completely consistent with the test curve.

[0107] It should be understood that various parts of the present disclosure can be implemented in hardware, software, firmware, or a combination thereof.

[0108] The above description is merely a specific embodiment of the present disclosure, but the scope of protection of the present disclosure is not limited thereto. Any changes or substitutions that can be easily conceived by a person skilled in the art within the technical scope disclosed in this disclosure should be included in the scope of protection of the present disclosure. Therefore, the scope of protection of the present disclosure should be based on the scope of protection of the claims.

Claims

1. A method for inverting parameters of a mode I crack cohesive constitutive model, characterized in that: include: Step 1: Perform a mode I fracture test on the sample rock to obtain a curve showing the relationship between load and crack mouth opening displacement; Step 2: Determine the elastic modulus and fracture toughness of the sample rock according to the relationship curve; Step 3, selecting multiple data points from the relationship curve; Step 4: Substitute the elastic modulus and the fracture toughness into the target formula to solve each data point to obtain the corresponding cohesion value and crack opening displacement; Step 5: Fit the cohesion value and crack opening displacement corresponding to each data point to determine the cohesion constitutive relationship of the sample rock; Step 6: Substitute the constitutive relationship into the forward calculation formula, compare the calculation results with the relationship curve to see if they are consistent, and output the constitutive relationship if they are consistent.

2. The method according to claim 1, characterized in that The step 2 specifically includes: Analyze the slope C of the initial loading section and the cracking load P of the relationship curve ini , and the elastic modulus E and the fracture toughness at cracking are obtained 3. The method according to claim 2, characterized in that The expression of the target formula is Where w is the crack opening displacement, which is a function of the crack plane coordinate x, where x = 0 corresponds to the position of the crack mouth; E is the elastic modulus; σ is the cohesive force, which is a function of the crack opening displacement w; F is the dimensionless stress intensity factor related to the specimen geometry; P is the load; D is the height of the specimen; B is the thickness of the specimen; a is the crack length; a e is the end of the cohesive force action area; G is the Green's function or weight function related to the specimen crack geometry, CMOD is the crack mouth opening displacement, CMOD = w(0); is the fracture toughness at crack initiation.

4. The method according to claim 3, characterized in that The expression of the crack opening displacement is: Among them, a0=a, a n+1 = a e ; The expression of the cohesion value is: Where σ0=σ t and w0=0,σ t is the tensile strength of rock.

5. The method according to claim 4, characterized in that Methods for solving the target formula include least squares method, gradient descent method or Gauss-Newton method.

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