Method for calculating stress intensity factor of kinked opening crack tip

Abstract

The application discloses a method for calculating stress intensity factor of a bending open crack tip, and is based on the distributed dislocation technology, combines the Gerasoulis numerical method to give the numerical solution of the distributed dislocation function on the crack surface, and according to the superposition principle, gives the theoretical solution of the complex potential function of the bending open crack problem under the action of the far-field pressure and water pressure, and gives the numerical solution method of the bending segment tip stress intensity factor K under the condition that the water pressure is linearly distributed. Finally, the bending segment tip stress dimensionless intensity factor K is calculated, and the numerical solution of the existing method is compared to verify the effectiveness of the method.

Description

Technical Field

[0001] This invention belongs to the field of geotechnical mechanics, specifically relating to a method for calculating the stress intensity factor at the tip of a bent-type open crack. Background Technology

[0002] In the field of civil engineering, accurate calculation of the stress intensity factor at the crack tip is a prerequisite for judging crack initiation and propagation, and therefore plays a crucial role in the safety assessment of cracked structures. Thus, it has always been a complex and important research topic in fracture mechanics. Crack initiation and propagation phenomena are prevalent in many civil engineering problems, such as cracking and leakage in municipal bridge and pipeline engineering. These are essentially the result of crack initiation and propagation under water pressure or other loads. Numerous experimental and theoretical studies have shown that under far-field compressive-shear loads, crack tips often develop wing cracks that deflect along the direction of the compressive load. In addition, secondary cracks may also occur. Therefore, bending and bifurcating cracks are very common natural phenomena in deep formations. Fissure water in the formation and fracturing fluid injected into perforations can promote crack initiation, which is classified into open, partially closed, and completely closed states depending on the interaction between crack surfaces. Under the action of fracturing fluid, most cracks initiate in an open state. Similarly, in most hydraulic fracturing tests, there is no interaction between pre-existing cracks, so the initiation of cracks can also be considered as a problem of crack initiation in an open state. Therefore, this application focuses on the initiation of bending cracks in an open state under the combined action of far-field compressive load and water pressure.

[0003] In terms of indoor experiments and numerical simulations, many scholars have studied the crack initiation phenomena of single cracks and multiple cracks (parallel and collinear, parallel and non-collinear, and at arbitrary positions) under water pressure. In terms of theoretical research, many scholars have studied the crack initiation of single cracks, multiple cracks, and perforation cracks under water pressure.

[0004] Due to the existence of complex conditions such as bends and bifurcated cracks in strata, it is necessary to study their initiation under water pressure. However, current research on the initiation of bend cracks under water pressure is limited. Liu et al. and Zhao et al. gave an expression for the stress intensity factor when the main crack has antisymmetric bends at both ends based on the equivalent crack length theory. However, when the bend crack dip angle is arbitrary and the length of the bend is close to the length of the main crack, this theory cannot well reflect the interaction between the bend and the main crack. Similarly, this method cannot give the complex potential function of the stress in the entire field, nor can it give an expression for the stress intensity factor when the water pressure is non-uniformly distributed. Summary of the Invention

[0005] To address the aforementioned problems, embodiments of the present invention propose a method for calculating the stress intensity factor at the tip of a bent-type open crack.

[0006] The method for calculating the stress intensity factor at the tip of a bent-type open crack according to the present invention includes the following steps:

[0007] S1. Establish a mechanical model for a folded opening crack: Based on the theory of distributed dislocations, the folded segment of the crack is equivalent to a continuous dislocation distributed in a certain function form. Then, based on the superposition principle, the problem of an opening crack under the combined action of far-field compressive load and water pressure is decomposed into multiple problems.

[0008] S2. Establish the complex potential function for the bending-type opening crack problem: solve the multiple decomposed problems;

[0009] S3. Solving for the stress intensity factor at the tip of the bending crack.

[0010] The described folded tensile crack is a tensile crack subjected to the combined action of far-field compressive load and water pressure. The main crack segment of this tensile crack has a length of 2c, and the folded segment has a length of [missing information]. The main crack segment makes an angle β with the horizontal direction, and the crack bend segment makes an angle θ with the main crack. The crack is subjected to far-field vertical and lateral compressive loads. , The effect is that, simultaneously, there is a uniformly distributed water pressure p on the main crack surface inside the crack. w The crack is in an open state, and the non-uniform water pressure λp is present in the bend section of the crack. w The bending segment is in an open state, and λ is a continuous function of z.

[0011] The opening crack is decomposed into problem a + problem b, where the crack in problem a is subjected to far-field vertical and lateral compressive loads. , The effect is that the crack surface is unconstrained, and since the crack surfaces in the pre-cracked specimens during the experiment all have a certain spacing, the problem a is allowed to proceed as described in question a. As long as it is guaranteed , (in the formula) , The Type I stress intensity factors for subproblems a and b, respectively, ensure that the crack is in the open state in problem a; in problem b, the crack is not subjected to far-field loads, and the main crack is subjected to uniformly distributed water pressure p. w The bending section is subjected to a non-uniformly distributed water pressure λp. w effect.

[0012] Problem a is decomposed into the following three problems: Problem a1 is a crack-free solid subjected to a far-field load; Problem a2 is a solid with a main crack and continuous dislocations distributed along the bending segment, and the main crack surface is unconstrained; Problem a3 is a solid with only the main crack segment, and the main crack surface is subject to unknown constraints. and effect;

[0013] Problem b is broken down into the following three problems: Problem b1 is about a crack-free solid; Problems b2 and b3 are the same as problems a2 and a3, respectively;

[0014] To solve problem b2, we need to first solve the interaction problem between a single dislocation and the crack, i.e., problem d—the crack half-length c and the crack surface is unconstrained, the dislocation coordinate is z0 and the line connecting the right end point (c,0) of the crack makes an angle θ with the positive x-axis;

[0015] Problem d includes problem d1 and problem d2. Problem d1 involves only dislocation interactions in a solid, while problem d2 involves unknown constraints on the crack surface. and effect.

[0016] The complex potential function representation of problem b2 is as follows:

[0017]

[0018] In the formula,

[0019]

[0020] In the formula, c is the half-length of the main crack segment, z0 is the dislocation coordinate, and z0 is in The range of values ​​is given, where z is the coordinate of any point in the solid, and θ is the angle between the line connecting the dislocation and the right end of the crack (c,0) and the positive x-axis. Let e ​​be the length of the bend, e be the natural logarithm, and i be the imaginary unit. For the complex potential function related to stress in subproblem b2, Let be the complex potential function related to strain in subproblem b2.

[0021] The complex potential function of problem b3 is:

[0022]

[0023] In the formula: This is the general term formula for the coefficients of a fractional binomial. ;

[0024] When n is odd, N = (n+1) / 2; when n is even, N = n / 2; k represents the k-th term of the polynomial. c is the half-length of the main crack segment, and z is the complex coordinate of the point in space.

[0025] The main crack surface and the bending segment satisfy two sets of boundary conditions according to the force equilibrium. The boundary conditions of the main crack segment in problem b can be expressed as follows:

[0026]

[0027] In the formula,

[0028]

[0029] In the formula, , , These represent the normal stress components in the y-axis direction in Cartesian coordinates for subproblems b1~b3, respectively. , , These represent the shear stress components in Cartesian coordinates for subproblems b1~b3, respectively. w Where is the water pressure, and i is the imaginary unit. Let n be the Legendre polynomial of the nth term on the interval [-1, 1]. , , Let be the real and imaginary parts of the complex coefficients of the nth Legendre polynomial in subproblem b, respectively. Let be the unknown coefficients to be solved in problem b.

[0030] The process for solving the stress intensity factor at the tip of the bending crack includes the following steps:

[0031] S301. Input the relevant mechanical and geometric parameters according to the problem to be solved;

[0032] S302. Determine the unknown parameters based on the boundary conditions of the main crack segment;

[0033] S303. Determine the coordinates of the interpolation nodes and coordination points of the bent segment;

[0034] S304. Write a program function to discrete the singular integral equation;

[0035] S305. Solving for unknown coefficients from the boundary conditions of the bent segment: Substitute the node coordinates into the discrete program function and solve the linear equation system;

[0036] S306. Solve for the stress intensity factor at the crack tip of the bent section.

[0037] The formula for the stress intensity factor at the crack tip of the bent section is:

[0038]

[0039] In the formula, , These are the stress intensity factors for crack type I and type II, respectively. , Let be the algebraic sums of the Type I and Type II stress intensity factors for subproblems a and b, respectively. , Let be the values ​​of the coefficients of the Gaussian integral of the dislocation distribution function for subproblems a and b at 1, respectively.

[0040] The K Ⅰ When K > 0, Ⅰ This is the Type I stress intensity factor for the problem; otherwise, this method cannot be used for calculation.

[0041] The beneficial effects of this invention are that it employs DDT technology to study the initiation problem of bending cracks opening due to water pressure under far-field pressure, and provides the complex potential function of the stress across the entire field based on the superposition principle. The effectiveness of the method is verified by calculating the stress intensity factor K at the tip of the bending segment under uniaxial tension and comparing it with existing numerical solutions. Furthermore, by considering the water pressure as a linear distribution in the boundary conditions of the bending segment crack, a numerical method is provided to calculate the stress intensity factor K at the tip of the bending segment under the condition of linear water pressure distribution. Attached Figure Description

[0042] Figure 1 This is a schematic diagram of the decomposition of the problem of opening cracks (with bending segments) under compressive load and water pressure, and a geometric model of the crack size.

[0043] Figure 2 This is a schematic diagram showing the decomposition of problems a and b of the present invention.

[0044] Figure 3 This is a schematic diagram of the dislocation-crack interaction problem of the present invention.

[0045] Figure 4 This is a flowchart illustrating the calculation of the stress intensity factor K at the tip of a bending crack under far-field pressure and water pressure according to the present invention.

[0046] Figure 5 This is a schematic diagram of the calculation model for the verification test of this invention. Detailed Implementation

[0047] Embodiments of the present invention are described in detail below, examples of which are illustrated in the accompanying drawings. The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain the present invention, and should not be construed as limiting the present invention.

[0048] The method for calculating the stress intensity factor at the tip of a bent-type open crack according to the present invention includes the following steps:

[0049] S1. Establish a mechanical model for a bending-type tension crack.

[0050] For the solution domain, satisfying ▽ 2 ▽ 2The plane stress or strain problem with U(x,y)=0 can be transformed into a problem of solving two complex potential functions Φ(z) and Ω(z) on the complex plane that satisfy certain conditions, according to the complex potential theory.

[0051] In the formula, ▽ 2 ▽ 2 It is a biharmonic operator, expressed as: U is the Airy stress function, and x and y are the abscissa and ordinate of the Cartesian coordinate system, respectively. It is a complex potential function related to stress. Let Ω(z) be a complex potential function related to strain, and let Ω(z) be a user-defined complex potential function. , The relationship is ,in , Represents the complex potential function In conjugate complex numbers The function value is taken as its conjugate complex number. Represents the complex potential function In conjugate complex numbers The function value is taken as its conjugate complex number.

[0052] Intradomain stress components (σ) x ,σ y The following relationship exists between the displacement components (u,v) and the complex potential function.

[0053] (1)

[0054] In the formula, G is the shear modulus of the material, and z = x + iy is a complex variable in the complex plane. Let be the conjugate of the complex variable z, and κ be the Krosov constant, characterizing the coupling strength between tensile deformation and transverse constraint. For plane stress problems, κ = (3-v) / (1+v), and for plane strain problems, κ = 3-4v, where ν is the material's Poisson's ratio, and Re represents taking the real part of the complex number. The complex potential function is related to stress. , Represents the complex potential function In conjugate complex numbers The function value is taken as its conjugate complex number. for The function value of the first derivative at z, and its complex conjugate. The complex potential function is related to strain. , Represents the complex potential function In conjugate complex numbers The function value is taken as its conjugate complex number. σ represents the shear stress component of the stress tensor in Cartesian coordinates; x σ y Let i represent the normal stress components in the x and y directions of the stress tensor in Cartesian coordinates, where i is the imaginary unit.

[0055] According to the theory of distributed dislocations, the bend in the crack can be equivalently represented as a continuous series of dislocations distributed in a certain functional form. Then, based on the superposition principle, the problem of an open-type crack under the combined action of far-field compressive load and water pressure can be described as follows: Figure 1 The decomposition is shown.

[0056] The length of the main crack segment in the figure is 2c, and the length of the bend segment is... The angle between the main crack segment and the horizontal direction is β, and the angle between the crack bend segment and the main crack is θ. In this problem, the crack is subjected to far-field vertical and lateral compressive loads. , The effect is that the blue area inside the crack indicates a uniformly distributed water pressure p on the main crack surface. w The crack is in an open state, and the blue portion in the bend indicates a non-uniformly distributed water pressure λp. w The bending segment is in an open state, and λ is a continuous function of z. w λp represents the water pressure uniformly distributed in the main crack segment. w This indicates that the water pressure is not uniformly distributed in the bend section.

[0057] To simplify the solution process, the problem can be decomposed into problem a + problem b according to the superposition principle. In problem a, the crack is subjected to far-field vertical and lateral compressive loads. , The effect is that the crack surface is unconstrained, and since the crack surfaces in the pre-cracked specimens during the experiment all have a certain spacing, the problem a is allowed to proceed as described in question a. As long as it is guaranteed This ensures that the crack in the problem remains open. In problem b, the crack is not subjected to far-field loads, and the main crack is subjected to uniformly distributed water pressure p. w The bending section is subjected to a non-uniformly distributed water pressure λp. w effect.

[0058] Next, we will solve problems a and b separately. To simplify the solution process, we will only provide a detailed solution for problem b. Problem a can be solved by referring to the solution process of problem b, decomposing the problem accordingly. Problems a and b can be solved as follows: Figure 2 The decomposition is shown.

[0059] Figure 2 This is a schematic diagram showing the decomposition of problems a and b. The main crack length in the diagram is 2c, and the bending section is... The angle between the main crack segment and the horizontal direction is β, and the angle between the dislocation distribution line and the positive x-axis is θ. Problem a is decomposed into the following three problems: Problem a1 is a crack-free solid subjected to a far-field load; Problem a2 is a solid with a main crack and continuous dislocations distributed along the bending segment, and the main crack surface is unconstrained; Problem a3 is a solid with only a main crack segment, and the main crack surface is subject to unknown constraints. and Function. Problem b is decomposed into the following 3 problems: Problem b1 is a crack-free solid; Problems b2 and b3 are the same as problems a2 and a3, respectively.

[0060] The water pressure in the main crack segment is approximately uniformly distributed and equal to the injected water pressure, while the water pressure in the wing crack decreases rapidly. To more accurately describe the water pressure distribution on the crack surface, the calculation model assumes that the fluid pressure is uniformly distributed in the main crack segment and that the water pressure magnitude is p. w The water pressure on the crack surface of the bend section exhibits a linear distribution. The water pressure at different locations on the crack surface of the bend section and the water pressure in the main crack section are characterized by a linear function λ(s). Relationship between them.

[0061] S2. Establish the complex potential function for the bending-type opening crack problem.

[0062] First of all Figure 2 Problem b2 in the above problem is solved. In this problem, the actual crack is equivalently replaced by a continuously distributed dislocation on the original bending segment, and there is no constraint on the main crack segment. Solving this problem requires first solving the interaction problem between a single dislocation and the crack. This problem can be solved according to the superposition principle as follows: Figure 3 The decomposition is shown.

[0063] Figure 3 The diagram below illustrates the decomposition of the dislocation-crack interaction problem, specifically problem d. In problem d, the crack half-length is c and the crack surface is unconstrained. The dislocation coordinate is z0, and the line connecting the dislocation to the right endpoint (c,0) of the crack makes an angle θ with the positive x-axis. Problem d1 represents a solid with only dislocation interaction, while problem d2 represents a crack surface with unknown constraints. and effect.

[0064] Based on the research results of Lo et al., the complex potential functions for problems d1 and d2 can be given respectively. The complex potential function for problem d1, when the local coordinates are established with the main fracture segment, has the following form:

[0065] (2)

[0066] In the formula:

[0067] (3)

[0068] In the formula: Let be the complex potential function related to stress in subproblem d1. Let z = x + iy be the complex potential function related to strain in subproblem d1, z = x + iy be the complex variable on the complex plane, and κ be the Krosov constant, which characterizes the coupling strength between tensile deformation and transverse constraint. For the plane stress problem, κ = (3 - v) / (1 + v), and for the plane strain problem, κ = 3 - 4v, where v is the material Poisson's ratio. The constant related to the Burgers vector of dislocation displacement is expressed as follows: , for The conjugate complex number of , G is the material shear modulus, e is the natural logarithm, θ is the angle between the line connecting the dislocation and the right end of the crack (c,0) and the positive x-axis, z0 is the dislocation coordinate, and z is the spatial coordinate of any point in the solid. , The Burgers vector is the dislocation in a local coordinate system with the bend segment as the local coordinate system. The entire problem is localized with the main crack segment as the local coordinate system.

[0069] The complex potential function for problem d2 can be given by the unconstrained equilibrium conditions of the crack surface and the solution of the Remiann-Hilbert problem, and its form is as follows:

[0070] (4)

[0071] In the formula:

[0072] (5)

[0073] In the formula:

[0074] (6)

[0075] In the formula: c is the half-length of the main crack segment. It is the weight function of the Gauss integral, and its expression is shown in equation (6). , It is a user-defined function, and its expression is shown in equation (5).

[0076] In summary, the complex potential function of problem d can be obtained by linearly adding the complex potential functions of problems d1 and d2 according to the superposition principle.

[0077] Returning to question b2, since dislocations follow a certain density function distribution in the bend segment, let's assume its density function in the bend segment is α. b (z0), z0 in The range of values. The complex potential function of problem b2 can be expressed in the following form by equation (4):

[0078] (7)

[0079] In the formula:

[0080] (8)

[0081] For problem b3, the unknown constraints on the crack surface and It can be represented as a finite-term series expansion in terms of the following Legendre orthogonal polynomials:

[0082] (9)

[0083] In the formula:

[0084] (10)

[0085] In the formula: , Let be the real and imaginary parts of the complex coefficients of the nth Legendre polynomial of subproblem b, respectively.

[0086] According to the research results of Niu et al., for crack surfaces according to Complex potential function of crack problem with function distribution It has the following forms:

[0087] (11)

[0088] In the formula: This is the general term formula for the coefficients of a fractional binomial.

[0089] (12)

[0090] In the formula: when n is odd, N = (n+1) / 2; when n is even, N = n / 2. Based on the research results of Benvensite et al., they gave the complex potential functions for problems corresponding to n=0~4, and n=4 is sufficient to meet the solution accuracy requirements for most crack problems in isotropic media. For problem b3, subsequent derivation revealed that the unknown constraints satisfy the case where n=0.

[0091] Depend on Figure 2 It can be seen that problems a2 and b2, and problems a3 and b3 have similar forms of expression. Therefore, the complex potential function of the entire problem has the following form according to the superposition principle:

[0092] (13)

[0093] In the formula, For the complex potential function related to stress in the combined problem, Let be the complex potential function related to strain in the integral problem. , Let be the complex potential functions related to stress in problems a and b, respectively. , These are the complex potential functions related to strain in problems a and b, respectively.

[0094] S3. Solving for the stress intensity factor at the tip of the bending crack.

[0095] Problem b can be decomposed into b1~b3. In b2, we consider the case where dislocations are continuously distributed in the bend of the crack, and the dislocation density function in the bend is set as α. b Substituting (z0) into equation (2) and replacing α, we obtain the complex potential function of a continuously distributed dislocation, whose expression is: , Substituting this into equation (7) yields the complex potential function of the dislocations distributed on the bent segment. At this point, the unknown variable to be solved is the unknown dislocation continuous distribution function α. b (z0). As can be seen from equation (9), the subproblem b3 contains orthogonal polynomial coefficients to be solved. and Therefore, solving problem b reduces to solving the dislocation density distribution function α. b (z0) and orthogonal polynomial coefficients and .

[0096] Where α b (z0) is divided into Re[α] b (z0)] and Im[α] b (z0)]. The main crack surface and the bending segment satisfy two sets of boundary conditions according to the force equilibrium. The boundary conditions of the main crack segment in problem b can be expressed as follows:

[0097] (14)

[0098] In the formula:

[0099] (15)

[0100] In the formula, , , These represent the normal stress components in the y-axis direction in Cartesian coordinates for subproblems b1~b3, respectively. , , These represent the shear stress components in Cartesian coordinates for subproblems b1~b3, respectively. w Where is the water pressure, and i is the imaginary unit. Let n be the Legendre polynomial of the nth term on the interval [-1, 1]. , , Let be the real and imaginary parts of the complex coefficients of the nth Legendre polynomial of subproblem b, respectively.

[0101] The boundary conditions for the bent segment in problem b can be expressed in the following form:

[0102] (16)

[0103] In the formula:

[0104] (17)

[0105] In the formula:

[0106] (18)

[0107] In the formula: , As given by equation (5), Let be the distribution function of the dislocation extension bend in problem b. Let be the length of the bent segment, and θ be the angle between the bent segment and the positive x-axis. , Let be the function representing the influence of the load distributed on the crack surface on the bent section in problem b3. Its specific expression will be given later. , , These represent the circumferential stress components in the stress tensor in polar coordinates for subproblems b1 to b3, respectively. , , These are the tangential components of the stress tensor in polar coordinates for subproblems b1 to b3, respectively. , , , For a user-defined function, its expression is shown in equation (18) and equation (5). For user-defined functions The first derivative; For size The complex potential function related to stress when normal stress is applied to the crack surface. For size The complex potential function related to stress when shear stress acts on the crack surface. .

[0108] Note that equation (16) uses the bent segment as the main coordinate axis, and for It consists of two parts: one part is the stress distribution generated by the continuous distribution of dislocations in the bending section, and the other part is the disturbance of the stress distribution of the crack in the bending section by the main crack section.

[0109] Based on the above analysis method, a similar expression can be obtained for problem a. The equilibrium condition of the principal crack in problem a has the following form:

[0110] (19)

[0111] In the formula:

[0112] (20)

[0113] In the formula:

[0114] (twenty one)

[0115] In the formula: , , Let be the normal stress components in the y-axis of the Cartesian coordinate system in subproblems a1~a3. , , For the shear stress components in Cartesian coordinates in subproblems a1~a3, , These represent the normal stress and shear stress components in the far-field Cartesian coordinate system, respectively. , These represent the far-field vertical and lateral pressures, respectively.

[0116] The boundary conditions for the bent segment in problem a can be expressed in the following form:

[0117] (twenty two)

[0118] In the formula:

[0119] (twenty three)

[0120] In the formula:

[0121] (twenty four)

[0122] In the formula: function , This is a user-defined function, see equation (18); , , Let represent the circumferential components of the stress tensor in polar coordinates in subproblems a1~a3. , , For the tangential components of the stress tensor in polar coordinates in subproblems a1~a3; , These are the circumferential and tangential stress components in the far field. Let be the dislocation density function for problem a; , Let be the real and imaginary parts of the complex coefficients of the nth Legendre polynomial in problem a, respectively.

[0123] As can be seen from equations (2) and (7), problems a and b give the complex potential function in integral form, which contains singular integral equations with unknowns to be solved. Therefore, such problems generally cannot be solved in closed form and require appropriate numerical methods to solve them. The following uses piecewise quadratic polynomial interpolation to find the coefficients of the dislocation density function at the crack tip.

[0124] Next, we will continue with problem b as an example to detail the numerical solution process. First, we give the following coordinate transformation formulas for points on the main crack and the bending crack:

[0125] (25)

[0126] In the formula, This represents the parametric form of the spatial coordinates of each point on the main crack in the main crack segment as the main coordinate system. Let z0 be the parametric form of the spatial coordinates of each point on the bend segment, and z0 be the parametric form of the spatial coordinates of the dislocation point on the bend segment. , , These are the corresponding parameter coordinates, in Take the value above.

[0127] To find the Legendre polynomial coefficients in equation (15), substitute equation (15) into equation (14), and then multiply both sides of equation (14) by... And in Integrating over the range, based on the orthogonality of Legendre polynomials, we obtain an integral equation of the following form:

[0128] (26)

[0129] From the orthogonality property of Legendre polynomials, we obtain:

[0130] (27)

[0131] When N=0, we can obtain from equation (11) , Influence function , (Separate the part of the function that does not have an imaginary unit i) The part of the function with the imaginary unit is ).

[0132] The format is as follows:

[0133] (28)

[0134] In the formula, This is the stress-dependent complex potential function when a unit normal stress is applied to the crack surface. This is the complex potential function related to stress when unit shear stress is applied to the crack surface.

[0135] Now only the dislocation distribution function α remains to be solved. b (z0). According to Lo's research, α b (z0) has the following form:

[0136] (29)

[0137] In the formula, weight function The coefficient function of the corresponding Gaussian integral.

[0138] From equations (25) and (29), equation (16) has the following form: Substituting equation (17) into the left side of equation (16), and replacing the coordinates z and z0 with equation (25), the dislocation distribution function Substituting equation (29) yields equation (30).

[0139] (30)

[0140] In the formula: λ(s) is a continuous function related to the location of the crack surface in the bend section, representing the water pressure distribution at different locations of the crack surface in the bend section and the water pressure p in the main crack section. w The relationship between them.

[0141] Based on Gerasoulisd's piecewise quadratic polynomial interpolation method, the interval [-1, 1]^n (where n is an even number) is divided into equal parts, and the interpolation points are... and coordination sites The expression is as follows:

[0142] (31)

[0143] Then, discretizing equation (30) gives the following form:

[0144] (32)

[0145] In the formula: , For user-defined functions, their expressions are shown in equations (43) to (48). Let J be the coordinates of the coordination point, j = 1, ..., n. Let i be the coordinates of the interpolation point, i=1,…,n+1.

[0146] , The specific derivation process is as follows.

[0147] A general singular integral equation, after coordinate transformation, takes the following form:

[0148] (33)

[0149] In the formula: ; Given a function, the corresponding boundary conditions in this problem; function and Holder continuity is satisfied within the domain. Substituting equation (29) into equation (33) yields equation (34), whose expression is:

[0150] (34)

[0151] Divide the interval [-1, 1] into n equal parts, and take n = 2N (N is a positive integer), h k Indicates adjacent interpolation points t k With t k+1 The spacing between points (k=1,…,2N) is constant in equation (31). ,make Substituting into equation (31), we get:

[0152] (35)

[0153] In the formula: t1 = -1; t 2N+1 =1; i=2,…,2N+1; h k =1 / N, h k The interval between adjacent interpolation points is denoted by i, where i represents the i-th interpolation point, and 2N+1 is the total number of interpolation points.

[0154] According to equation (35), the functions A(t) and U(s,t)A(t) in equation (34) have the following forms:

[0155] (36)

[0156] In the formula: =1,..,N; .

[0157] The first term of equation (34) can be rewritten in the following form:

[0158] (37)

[0159] The second term in equation (34) can be rewritten in a form similar to the above equation.

[0160] In equation (35), for the interval The integral can be written in the following form:

[0161] (38)

[0162] Integrating each part of equation (38), the results are as follows:

[0163] (39)

[0164] In summary, the singular integral (34) has the following discrete form:

[0165] (40)

[0166] When i = 1, 2N+1 , It has the following forms:

[0167] (41)

[0168] (42)

[0169] In the formula, , functions respectively At the 1st and 2N+1th interpolation points (corresponding to the 1st interpolation point) =1, the corresponding value at the (2N+1)th interpolation point =N) User-defined functions , function value ( =1,…,N), , See equation (45); , For function At the 1st and 2N+1th interpolation points (corresponding to the 1st interpolation point) =1, l=N corresponding to the (2N+1)th interpolation point) User-defined function , function value ( =1,…,N), , See equation (47).

[0170] When i ≠ 1, 2N+1 , It has the following forms:

[0171] (43)

[0172] (44)

[0173] In the formula, , For kronecker symbols, This means that if 1 and i have the same remainder when divided by 2, then... =1, otherwise 0; This means that if 0 and i have the same remainder when divided by 2, then... =1, otherwise 0; I indicates when i is even. The corresponding function expressions. E, I, and G are user-defined functions.

[0174] In equation (43), the functions E, I, and G have the following forms:

[0175] (45)

[0176] In the formula, M is a term in equation (38). In the interval The integral value, =1,…,N, and its expression is shown in equation (46).

[0177] (46)

[0178] In the formula: , =1,…,N, holds true for all j.

[0179] In equation (44), the user-defined functions B, C, and D have the following forms:

[0180] (47)

[0181] In the formula, H is a user-defined function, and its expression is shown in formula (48).

[0182] (48)

[0183] In the formula: , =1,…,N, for all Both are valid.

[0184] Since there are 2n+2 unknowns in equation (32), there are only 2n sets of actual equations. According to the research of Lo, He, etc., the stress distribution at the intersection of the main crack and the bending segment has a lower singularity than that at the crack tip. Therefore, the following conditions need to be met:

[0185] (49)

[0186] In summary, by combining equations (32) and (49), we can obtain the relevant dislocation distribution function coefficient function A. b(t) has a unique solution at the discrete point. Problem a can be solved sequentially using the method described above.

[0187] Based on the research results of Lo, Niu, and others, and according to the superposition principle, the stress intensity factor at the tip of a bending crack has the following form:

[0188] (50)

[0189] The calculation process for the stress intensity factor K at the tip of a bending crack under far-field pressure and water pressure is as follows: Figure 4 As shown, it includes the following steps:

[0190] S301. Input the relevant mechanical and geometric parameters according to the problem to be solved: , p w , λ, β, θ, c, The meaning of the input parameters and n: , p represents the far-field vertical and lateral pressures, respectively. w λ is the water pressure in the main crack segment, λ is the water pressure distribution coefficient on the crack surface in the bend segment; β is the angle between the main crack and the horizontal direction; θ is the angle between the bend segment of the crack and the direction of the main crack; c is the half-length of the main crack segment. is the length of the bent segment; n is the number of interpolation nodes.

[0191] S302. Determine the unknown parameters from equations (26) and (27) and based on the boundary conditions of the main crack segment. and .

[0192] S303. Determine the interpolation node coordinates of the bend segment using parameter n and formula (31). and coordinates of the coordination site For the specific form, see formula (31).

[0193] S304. Combined equations (40)-(48), write the singular integral equation (32) discrete program function , .

[0194] S305. As analyzed above, subproblems a and b each have a coefficient function A of the dislocation distribution function. a (t), A b (t) is to be determined. After numerical discretization using equation (32), the unknown quantity to be solved changes from a coefficient function to a coefficient function in t. i The function value A at (i=1,…,n) a (t i A) b (t iFurthermore, the unknown function value A can be solved by constructing a system of linear equations based on the boundary conditions of the bent segment. a (t i A) b (t i ): Combined formulas (30) and (49), in discrete program functions , Substitute node coordinates , Solve the system of linear equations.

[0195] S306. Substitute the coefficients into formula (50) The stress intensity factor K at the crack tip of the bent section is obtained.

[0196] S307. When When the value is greater than 0, this problem falls under the category of open bending cracks. This is for type I stress intensity factor; otherwise, this method cannot be used for calculation.

[0197] S308. Establish a local coordinate system based on the main crack segment to determine the solution range. Divide the region using the mesh function, with the mesh node coordinates as (x...). m y m ).

[0198] S309. Substitute the coefficients into equation (7) The discrete form of the complex potential function of the stress in the whole field is obtained, and based on the coordinates (x... m y m This allows us to determine the stress distribution across the entire field.

[0199] Numerical verification

[0200] Although the far-field load in this application's model is a compressive load, the crack is in a fully open state under water pressure. Therefore, the effectiveness of the proposed stress intensity factor K calculation method can be verified by comparing it with the numerical solution of the stress intensity factor at the crack bending tip under far-field tensile load. The calculation model is as follows: Figure 5 As shown.

[0201] The length of the main crack segment in the bent crack shown in the figure is 2c, and the length of the bent segment is... The angle between the main crack segment and the horizontal direction is 0°, and the angle between the bending segment and the positive x-axis is θ.

[0202] The stress intensity factor K is dimensionless using the following formula:

[0203] (51)

[0204] In the formula: is the far-field vertical tensile load; c is the half-length of the main crack.

[0205] According to Lo and Jin's research results Figure 5 The model calculation parameters and the stress intensity factor K at the tip A of the bent section, using the solution of this invention and the solution in the literature, are shown in Table 1. Since the dimensionless stress intensity factor K is calculated, the far-field load in the calculation process... The value can be set arbitrarily.

[0206] Table 1 Figure 5 Model calculation parameters and stress intensity factor K at the tip of the bent segment A: Solution of this invention and solution in the literature

[0207] Note: Figure 5 The angle between the main crack segment and the horizontal direction is 0°.

[0208] This invention addresses different ratios of main crack length to bending segment length c / The results of this invention were compared with those of Lo, Jin, and others. The calculated results showed good agreement with the results of different bending angles θ. The maximum relative error occurred at c / At θ=75°, the relative error compared with the Lo result is approximately 1.25%. This demonstrates that the method of this invention can accurately calculate the value of the crack tip stress intensity factor K in the crack opening problem under external force.

[0209] This invention, based on distributed dislocation technology and combined with Gerasoulis' numerical method, provides a numerical solution for the distributed dislocation function on the crack surface. Furthermore, based on the superposition principle, it presents a theoretical solution for the complex potential function of a bending crack opening under far-field pressure and water pressure. A numerical solution for the stress intensity factor K at the tip of the bending segment under linearly distributed water pressure is also provided. Finally, through calculation... Figure 5 The dimensionless intensity factor K of the stress at the tip A of the bent segment is obtained and compared with the numerical solutions of Lo and Lin to verify the effectiveness of the method.

[0210] Although the above embodiments have been shown and described, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Any changes, modifications, substitutions and variations made to the above embodiments by those skilled in the art are within the protection scope of the present invention.

Claims

1. A method for calculating the stress intensity factor at the tip of a bent-type tensile crack, characterized in that, Includes the following steps: S1. Establish a mechanical model for a folded opening crack: Based on the theory of distributed dislocations, the folded segment of the crack is equivalent to a continuous dislocation distributed in a certain function form. Then, based on the superposition principle, the problem of an opening crack under the combined action of far-field compressive load and water pressure is decomposed into multiple problems. S2. Establish the complex potential function for the bending-type opening crack problem: solve the multiple decomposed problems; S3. Solving for the stress intensity factor at the tip of the bending crack.

2. The method for calculating the stress intensity factor at the tip of a bent-type open crack according to claim 1, characterized in that, The described folded tensile crack is a tensile crack subjected to the combined action of far-field compressive load and water pressure. The main crack segment of this tensile crack has a length of 2c, and the folded segment has a length of [missing information]. The main crack segment makes an angle β with the horizontal direction, and the crack bend segment makes an angle θ with the main crack. The crack is subjected to far-field vertical and lateral compressive loads. , The effect is that, simultaneously, there is a uniformly distributed water pressure p on the main crack surface inside the crack. w The crack is in an open state, and the non-uniform water pressure λp is present in the bend section of the crack. w The bending segment is in an open state, and λ is a continuous function of z.

3. The method for calculating the stress intensity factor at the tip of a bent-type open crack according to claim 2, characterized in that, The opening crack is decomposed into problem a + problem b, where the crack in problem a is subjected to far-field vertical and lateral compressive loads. , In problem b, the crack is not subjected to far-field loads, and the main crack is subjected to uniformly distributed water pressure p. w The bending section is subjected to a non-uniformly distributed water pressure λp. w effect.

4. The method for calculating the stress intensity factor at the tip of a bent-type open crack according to claim 3, characterized in that, Problem a is decomposed into the following three problems: Problem a1 is a crack-free solid subjected to a far-field load; Problem a2 is a solid with a main crack and continuous dislocations distributed along the bending segment, and the main crack surface is unconstrained; Problem a3 is a solid with only the main crack segment, and the main crack surface is subject to unknown constraints. and effect; Problem b is broken down into the following three problems: Problem b1 is about a crack-free solid; Problems b2 and b3 are the same as problems a2 and a3, respectively; To solve problem b2, we need to first solve the interaction problem between a single dislocation and a crack, i.e., problem d—the crack half-length c and the crack surface is unconstrained, the dislocation coordinate is z0 and the line connecting the right end point (c,0) of the crack makes an angle θ with the positive x-axis; Problem d includes problem d1 and problem d2. Problem d1 involves only dislocation interactions in a solid, while problem d2 involves unknown constraints on the crack surface. and effect.

5. The method for calculating the stress intensity factor at the tip of a bent-type open crack according to claim 4, characterized in that, The complex potential function representation of problem b2 is as follows: In the formula, In the formula, c is the half-length of the main crack segment, z0 is the dislocation coordinate, and z0 is in The range of values ​​is given, where z is the coordinate of any point in the solid, and θ is the angle between the line connecting the dislocation and the right end of the crack (c,0) and the positive x-axis. Let e ​​be the length of the bend, e be the natural logarithm, and i be the imaginary unit. For the complex potential function related to stress in subproblem b2, Let be the complex potential function related to strain in subproblem b2.

6. The method for calculating the stress intensity factor at the tip of a bent-type open crack according to claim 5, characterized in that, The complex potential function of problem b3 is: In the formula: This is the general term formula for the coefficients of a fractional binomial. When n is odd, N = (n+1) / 2; when n is even, N = n / 2; k is the k-th term of the polynomial. c represents the half-length of the main crack segment, and z represents the complex coordinates of a point in space.

7. The method for calculating the stress intensity factor at the tip of a bent-type open crack according to claim 6, characterized in that, The main crack surface and the bending segment satisfy two sets of boundary conditions according to the force equilibrium. The boundary conditions of the main crack segment in problem b can be expressed as follows: In the formula, In the formula, , , These represent the normal stress components in the y-axis direction in Cartesian coordinates for subproblems b1~b3, respectively. , , These are the shear stress components in Cartesian coordinates for subproblems b1~b3, respectively. w Where is the water pressure, and i is the imaginary unit. Let n be the Legendre polynomial of the nth term on the interval [-1, 1]. , , Let be the real and imaginary parts of the complex coefficients of the nth Legendre polynomial of subproblem b, respectively.

8. The method for calculating the stress intensity factor at the tip of a bent-type open crack according to claim 7, characterized in that, The process for solving the stress intensity factor at the tip of the bending crack includes the following steps: S301. Input the relevant mechanical and geometric parameters according to the problem to be solved; S302. Determine the unknown parameters based on the boundary conditions of the main crack segment; S303. Determine the coordinates of the interpolation nodes and coordination points of the bent segment; S304. Write a program function to discrete the singular integral equation; S305. Solving for unknown coefficients from the boundary conditions of the bent segment: Substitute the node coordinates into the discrete program function and solve the linear equation system; S306. Solve for the stress intensity factor at the crack tip of the bent section.

9. The method for calculating the stress intensity factor at the tip of a bent-type open crack according to claim 8, characterized in that, The formula for the stress intensity factor at the crack tip of the bent section is: In the formula, , These are the stress intensity factors for crack type I and type II, respectively. , Let be the algebraic sums of the Type I and Type II stress intensity factors for subproblems a and b, respectively. , Let be the values ​​of the Gaussian integral coefficients of the dislocation distribution functions of subproblems a and b at 1, respectively.

10. The method for calculating the stress intensity factor at the tip of a bent-type open crack according to claim 9, characterized in that, The K Ⅰ When K > 0, Ⅰ This is the Type I stress intensity factor for the problem; otherwise, this method cannot be used for calculation.

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