Underground high-pressure water conveyance tunnel lining crack analysis method

Abstract

An underground high-pressure water delivery tunnel lining crack analysis method aims at the characteristics of a large-range crack-free area and a local crack sensitive area of an underground high-pressure water delivery tunnel lining, the crack-free area (only needing boundary discretization) and a near-field dynamics focusing crack area are discretized through a proportional boundary finite element method, and the high cost of pure near-field dynamics global discretization is avoided; the problem of low efficiency of a traditional method is overcome, and analysis efficiency is remarkably improved; coupling is realized based on a common point force balance condition, a transition area or a complex shape function does not need to be arranged in a'crack-free area-crack area 'of a lining, manual setting errors are avoided, modeling complexity is reduced, and applicability of the method in actual tunnel engineering is enhanced. The method can accurately predict the preset crack and spontaneous crack propagation behavior of the underground high-pressure water tunnel lining, is especially suitable for a stress concentration type crack scene around a hole, and provides reliable support for lining design, anti-seepage arrangement and operation and maintenance.

Description

Technical Field

[0001] This invention relates to the field of analysis technology for cracks in the lining of underground high-pressure water conveyance tunnels, and particularly to a method for analyzing cracks in the lining of underground high-pressure water conveyance tunnels. Background Technology

[0002] In the field of underground water conservancy engineering, underground high-pressure water conveyance tunnels are the core water conveyance carriers of long-distance water transfer and pumped storage systems. The integrity of their lining structure directly determines the safety of the project and the efficiency of water conveyance. The lining is subjected to high-pressure internal water pressure, temperature fluctuation stress, and localized maintenance loads over long periods, making it prone to crack initiation in stress concentration areas (such as construction joints, around openings, and on the inner surface). If these cracks continue to propagate, they will not only cause high-pressure water leakage and increase water conveyance energy consumption, but may also lead to softening of the surrounding rock and instability of the lining, seriously threatening the safe operation of the project. Therefore, accurately calculating the crack initiation threshold and propagation path of underground high-pressure water conveyance tunnel linings is a critical requirement for ensuring the safety of project design and operation.

[0003] Existing technologies for analyzing cracks in the lining of underground high-pressure water conveyance tunnels have significant limitations: While extended finite element method (EPM) and meshless methods can handle some crack problems, simulating spontaneous cracking under high-pressure internal water pressure and crack propagation at construction joints requires frequent mesh refinement or complex interpolation, making modeling cumbersome and prone to errors due to mesh distortion. Pure near-field dynamics methods can naturally capture the discontinuity of cracks, but due to their nonlocal nature, they require discretizing a massive number of material points, resulting in extremely high computational costs, and are prone to surface effect errors when handling lining boundary conditions. Simplified analytical methods, while computationally inexpensive, are based on idealized assumptions such as "homogeneous elastic body" and "regular cracks," failing to reflect the stress field around non-homogeneous interfaces and complex pores in actual engineering, and thus lacking sufficient accuracy in simulating spontaneous cracks and multiple crack propagation.

[0004] Existing coupling methods are also difficult to adapt to the needs of lining analysis: near-field dynamics-finite element coupling requires the setting of a transition region, but the lining cross-sectional space is limited, and the transition region easily occupies the effective analysis space, resulting in complex modeling and low efficiency; near-field dynamics-boundary element coupling relies on the fundamental solution of the material and calculates singular integrals, which limits its applicability when dealing with heterogeneous interfaces and cracks around pores. Scaled boundary finite element method ( SBFEM While this method can efficiently solve the mechanical response of large-scale crack-free regions by only requiring discrete boundaries, its coupling method with near-field dynamics has not yet been reported, failing to provide an integrated solution of "large-scale efficient calculation + local high-precision simulation" for lining crack analysis. Therefore, there is an urgent need to propose a low-cost, simple-to-model, and high-precision method for analyzing cracks in the lining of underground high-pressure water conveyance tunnels to overcome the limitations of existing technologies. Summary of the Invention

[0005] The main purpose of the present application is to provide a high-pressure underground water tunnel lining crack analysis method, aiming to solve the technical problems of high calculation cost, complex modeling and insufficient precision in the existing high-pressure underground water tunnel lining crack analysis.

[0006] To achieve the above-mentioned purpose, the present application provides a high-pressure underground water tunnel lining crack analysis method, which comprises the following steps: S1, first, according to the crack distribution, the elastic body is divided into two calculation regions: for the structure complete crack-free calculation region, the proportional boundary finite element is adopted SBFEM Modeling is carried out, and the proportional boundary finite element only needs to discretize the boundary of the crack-free calculation region to complete the representation of the crack-free calculation region; for the calculation region with existing cracks or potential crack paths, hereinafter the calculation region with existing cracks or potential crack paths is collectively referred to as the crack calculation region, near-field dynamics PD Modeling is carried out, and the near-field dynamics naturally describes the initiation and expansion behavior of the crack through the non-local interaction between material points; S2, by introducing the proportional boundary coordinate system, the momentum equation in the Cartesian coordinate system is converted into the first-order ordinary differential control equation in the proportional boundary coordinate system, and then the stiffness matrix of the crack-free calculation region is solved: first, the analytical method is adopted in the radial direction, and the numerical discretization is adopted in the hoop direction, and the coordinates in the Cartesian coordinate system are converted into the coordinates in the proportional boundary coordinate system. Subsequently, based on the virtual work principle, the momentum equation in the Cartesian coordinate system is derived, which is converted into the first-order ordinary differential control equation form in the proportional boundary coordinate system. Finally, by solving the eigenvalue of the first-order ordinary differential control equation, the stiffness matrix of the crack-free calculation region is obtained; S3, in the crack calculation region, the elastic body is discretized into a series of material points, each material point is connected with other material points within the neighborhood range through a bond, and the counterforce between the material points is constructed based on the bond-based near-field dynamics BBPD ) theory of near-field dynamics. By matrix processing the counterforce between the material points, the micro-stiffness matrix between any two material points is derived. According to the finite element assembly rule, the micro-stiffness matrices of all material points are integrated into the overall stiffness matrix of the crack calculation region; S4, at the common nodes between the crack-free calculation region and the crack calculation region, the direct coupling of the two calculation regions is realized based on the force balance condition: by dividing the stiffness matrices of the two regions according to the internal nodes and the common nodes respectively, then superimposing and combining the stiffness matrices corresponding to the common nodes, finally forming a unified overall stiffness matrix; S5, using an implicit solution algorithm combined with load increment steps for nonlinear calculation: in each load step, first solve the elastic body displacement, then traverse all the keys in the crack calculation area, and judge the fracture of the key according to the current displacement. When the fracture of the key is detected, update the overall stiffness matrix of the elastic body and recalculate iteratively in the current load step until the elastic body reaches an equilibrium state and no new key fracture occurs, and then continue to apply the next load increment. This process is repeated until the calculation of all specified load steps is completed.

[0007] Advantages of the present application: (1) According to the characteristics of the underground high-pressure water tunnel lining "large area without cracks + local crack sensitive area", the proportion boundary finite element method is used to discretize the crack-free area (only the boundary needs to be discretized) and focus on the crack area by near-field dynamics, avoiding the high cost of pure near-field dynamics global discretization, while overcoming the inefficiency of traditional methods, significantly improving the analysis efficiency; (2) Based on the public point force balance condition, coupling is realized without setting a transition area or complex shape function in the lining "crack-free area-crack area", avoiding artificial setting errors, reducing modeling complexity, and enhancing the applicability of the method in actual tunnel engineering; (3) It can accurately predict the pre-existing cracks and spontaneous crack propagation behavior of the underground high-pressure water tunnel lining, especially suitable for hole peripheral stress concentration crack scenarios, providing reliable support for lining design, anti-seepage arrangement and operation and maintenance. BRIEF DESCRIPTION OF DRAWINGS

[0008] Figure 1 The present application adopts SBFEM-PD the flowchart of the underground high-pressure water tunnel lining crack analysis method using implicit coupling; Figure 2 The schematic diagram shown in Figure 1 ; Figure 3 The model diagram of the example; Figure 4 The crack form schematic diagram of the square plate with holes. DETAILED DESCRIPTION

[0009] It should be understood that the specific embodiments described herein are only used to explain the present application and do not limit the present application. The implementation, functional characteristics and advantages of the present application will be further described with reference to the embodiments and the accompanying drawings.

[0010] As Figure 1 , Figure 2 shown, Figure 1 is an underground high-pressure water tunnel lining crack analysis method of the present application. The detailed method specifically includes the following steps: S1. First, the elastic body is divided into two computational regions based on the crack distribution: For the structurally intact, crack-free computational region, a scaled boundary finite element method is used. SBFEM For modeling, scaled boundary finite element method only needs to discretize the boundary of the crack-free computational region to complete the representation of the crack-free computational region; for computational regions with existing cracks or potential crack paths, which will be collectively referred to as cracked computational regions below, near-field dynamics is used. PD By modeling, near-field dynamics naturally describes the initiation and propagation behavior of cracks through the nonlocal interactions between material points; S2. By introducing a scaled boundary coordinate system, the momentum equation in the Cartesian coordinate system is transformed into a first-order ordinary differential governing equation in the scaled boundary coordinate system, thereby solving for the stiffness matrix of the crack-free computational region: First, an analytical method is used in the radial direction, while numerical discretization is used in the circumferential direction to convert the coordinates in the Cartesian coordinate system to those in the scaled boundary coordinate system. Then, based on the principle of virtual work, the momentum equation in the Cartesian coordinate system is derived, transforming it into a first-order ordinary differential governing equation in the scaled boundary coordinate system. Finally, by solving the eigenvalues ​​of the first-order ordinary differential governing equation, the stiffness matrix of the crack-free computational region is obtained. S3. In the crack calculation region, the elastic body is discretized into a series of material points. Each material point is connected to other material points in its neighborhood through bonds, and the bond-basis peridynamics is based on peridynamics. BBPD Theoretically constructing the forces between material points. By matrixing the forces between material points, the micro-stiffness matrix between any two material points is derived. According to the finite element assembly rules, the micro-stiffness matrices of all material points are integrated into the overall stiffness matrix of the crack calculation region; S4. At the common node at the boundary between the crack-free calculation region and the crack calculation region, the two calculation regions are directly coupled based on the force equilibrium condition: by dividing the stiffness matrices of the two regions into blocks according to the internal nodes and the common node respectively, and then superimposing and combining them on the stiffness matrix corresponding to the common node, a unified overall stiffness matrix is ​​finally formed. S5. An implicit solution algorithm combined with load increment steps is used for nonlinear calculation: Within each load step, the elastic body displacement is first solved, and then all keys in the crack calculation region are traversed. The fracture status of the keys is determined based on the current displacement. When a key fracture is detected, the overall stiffness matrix of the elastic body is updated, and iterative recalculation is performed within the current load step until the elastic body reaches an equilibrium state and no new key fracture occurs. Then, the next load increment is applied. This process is repeated until the calculation of all specified load steps is completed.

[0011] In the step S1, the two-dimensional elastic body is divided into a crack-free calculation region (divided by a scaled boundary finite element) and a crack calculation region (divided by near-field dynamics), and expressions of momentum equations, geometric equations and constitutive equations of the elastic body are given: wherein, is a gradient operator, is stress, is an external load, composed of a volume force load and a surface force load is strain, is displacement, D is an elastic matrix.

[0012] In the step S2, the specific steps are as follows: SBFEM By introducing a scaled boundary coordinate system in the crack-free calculation region, the relationship between the coordinates in the Cartesian coordinate system and the coordinates in the scaled boundary coordinate system is: wherein, and represent the coordinates of a point to be solved in the Cartesian coordinate system, represents a radial coordinate in the scaled boundary coordinate system, represents a circumferential coordinate in the scaled boundary coordinate system; is a shape function; and are node coordinates in the Cartesian coordinate system. The transformation equation of the differential operator in the Cartesian coordinate system and the scaled boundary coordinate system is represented as: In matrix form, it is represented as: wherein, J represents a Jacobian matrix; The two-dimensional elastic differential operator is represented in scaled coordinates: wherein , are coefficient matrices, and are specifically represented as: wherein, and​ denotes the partial derivative of x and y, i.e. and ; The displacement and stress are expressed in the Cartesian coordinate system as: where, denotes the function of the displacement in the Cartesian coordinate system with respect to the scaled boundary coordinate system , denotes the first-order partial derivative of the displacement with respect to the coordinate , denotes the function of the displacement in the Cartesian coordinate system with respect to the scaled boundary coordinate system and ; denotes the function of the displacement in the Cartesian coordinate system with respect to the scaled boundary coordinate system and ; denotes the function of the displacement in the Cartesian coordinate system with respect to the scaled boundary coordinate system and ; , denotes the strain matrix, which is expressed as: The virtual work form of the momentum equation is: where, denotes the variation, denotes the volume, denotes the function of the displacement in the Cartesian coordinate system with respect to the scaled boundary coordinate system , denotes the function of the external load in the Cartesian coordinate system with respect to the scaled boundary coordinate system , and the first-order ordinary differential control equation in the scaled boundary coordinate system is obtained by simplifying: where, denotes the second-order partial derivative of the displacement in the Cartesian coordinate system with respect to the scaled boundary coordinate system , and the superscript denotes the transpose symbol of the matrix, , , are all coefficient matrices, and their specific expressions are: The matrix expression of the first-order ordinary differential control equation is as follows: wherein, denotes the first-order partial derivative of the state vector with respect to the state vector , denotes the state vector, and the Hamilton matrix is specifically expressed as: The first-order ordinary differential control equation is solved by using an eigenvalue problem, and is specifically expressed as: wherein and are the eigenvalue and eigenvector of the Hamilton matrix, denotes the upper half of the Hamilton matrix; , , , , denotes four parts into which the Hamilton matrix is uniformly divided.

[0013] Finally, the stiffness matrix of the crack-free calculation region is: .

[0014] In the step S3, the following specific steps are adopted: using the key-based near-field dynamic theory, assuming that the crack calculation region is composed of a plurality of material points, each material point a has a volume of V a . Each material point a interacts with other material points δ h in the adjacent region with a radius of a' , and the set of all material points in the crack calculation region is called the influence domain H a . The momentum equation of each material point a is as follows: wherein, is the displacement of the material point ; is the displacement of the material point a ; is the interaction force between the material point a and the material point a' .​ For a material point a the volumetric force density, For a material point the volumetric force density, In the key-based near-field dynamic theory, where, and are the relative position and relative displacement of material points a and For a micro-elastic brittle material, is expressed as: where, is a history-dependent scalar-valued function; represents the micro-modulus function; is a kernel function characterizing the long-range force strength between two material points; is the elongation of the bond, and the expressions of each term are as follows: where, represents the elastic modulus; represents the Poisson's ratio; s c represents the critical elongation.

[0015] If the elongation of the bond is greater than the critical elongation of the bond s c , the bond between the two material points will break. The critical elongation of the bond s c is determined by the breaking energy G 0 , which can be written as: The damage in PD is defined as follows: The momentum equation is matrixed as follows: where, and represent two material points; represents the volume of material point ; and represent material point and The displacement; Material point The volumetric force density; In order to achieve PD and SBFEM The coupling is addressed by proposing a microstiffness matrix in PD. k ,expression At the same time, it is represented in the local coordinate system as The forces between the global coordinate system and the local coordinate system and displacement The relationship is achieved through coordinate transformation matrix T express: in, l and m It is the coordinate transformation factor between the local coordinate system and the global coordinate system; Therefore, material points in the global coordinate system Relative to material point The microstiffness matrix is: Material Point The stiffness matrix is ​​formed by combining the stiffness matrix with all material points in the neighborhood. And further combine the stiffness matrices of all material points. : In step S4, the specific steps are as follows: First, in SBFEM and PD In the coupled model constructed in the model, the forces at the common points have the following relationship: in, Indicates the distance between two material points; The result obtained in S2 SBFEM stiffness matrix according to SBFEM The unique points and common points are divided into blocks as follows: in, The rows and columns are all corresponding SBFEM Unique features; The corresponding row SBFEM Unique points, corresponding common points; the rows correspond to common points, and the columns correspond to SBFEM unique points; the rows and columns of K all correspond to common points. denote SBFEM displacements of unique points; denote displacements of common points; denote SBFEM forces of unique points; forces of common points; partition the stiffness matrix of S3 into PD wherein the rows of K correspond to common points, and the columns correspond to the rows and columns of K all correspond to common points. the rows correspond to common points, and the columns correspond to PD unique points; the rows correspond to PD unique points, and the columns correspond to common points; the rows and columns of K all correspond to PD unique points. denote displacements of common points; denote PD displacements of unique points; denote volume force densities of common points; denote PD volume force densities of unique points.

[0016] The overall stiffness matrix of the calculation region is obtained by assembling the stiffness matrices of SBFEM and PD wherein is the sum of and , i.e. = = .

[0017] The specific steps of the step S5 are as follows: after the overall stiffness matrix in S4 is obtained, displacement constraints and load boundaries are applied. A load step counter m = 0, and an initial load increment P = 0. The load step calculation process is as follows: S5.1, set m = m + 1, and update the load increment P ; S5.2, solve the linear equation set ​​Get all nodes (including SBFEM Nodes and PD Displacement of the material point; S5.3 Calculate the elongation of the point bonds in the crack calculation region using formula S3. ; S5.4, if <s c Proceed to step S5.5; otherwise, update the stiffness matrix of the crack calculation region and return to step S5.1. S5.5 If all load steps have been calculated, exit the analysis; otherwise, return to step S5.4.

[0018] This embodiment uses the crack analysis of a porous structure in the lining of an underground high-pressure water conveyance tunnel as an example to verify the "proportional boundary finite element method" proposed in this invention. SBFEM - Perifield Dynamics PD The effectiveness of the "implicit coupling method" in predicting cracks around lining holes.

[0019] To address the issue of crack initiation and propagation in the lining of underground high-pressure water conveyance tunnels under high internal water pressure, a 50m × 50m underground structure was selected as the research object, and a diameter [not specified] was set at its center. D =8m circular hole, width of SBFEM area on both sides l 1 = 10m and l 3=10m, middle PD area l 2=30m. By simulating the crack initiation and propagation process of a square plate under tension, the crack development law around the hole is analyzed, providing numerical support for the crack-resistant design of the lining with holes.

[0020] like Figure 3 As shown, the square plate is made of an elastic material with an elastic modulus of [missing value]. E= 30 GPa, Poisson's ratio v= 1 / 3 (default setting for plane stress problems with near-field dynamics of bonded bases), critical elongation of the crack computational region s c =8×10 -4 (Derived from material fracture energy). The square plate is divided into two crack-free computational regions and one cracked computational region. The material point spacing in the cracked computational region is... x=0.5m, radius of influence zone =4 x. The displacement load for each step is 2 × 10. -8 m.

[0021] Crack patterns in perforated square plates are as follows Figure 4The crack initiation of the square plate with holes can be observed at the stress sensitive points above and below the circular hole, and the propagation direction is basically perpendicular to the load application direction, which is consistent with the actual propagation law of the crack around the hole of the underground high-pressure water tunnel lining hole, indicating that the coupling method proposed by the application can effectively solve the crack analysis problem of the lining hole structure.

Claims

1. A method of analyzing a lining crack of an underground high-pressure water conveyance tunnel, characterized by, The method comprises the following steps: S1, first, the elastomer is divided into two calculation regions according to the crack distribution: for the intact crack-free calculation region, a scaled boundary finite element (SBFEM) is used for modeling, and the scaled boundary finite element only needs to discretize the boundary of the crack-free calculation region to complete the representation of the crack-free calculation region; for the calculation region with existing cracks or potential crack paths, the calculation region with existing cracks or potential crack paths is collectively referred to as a crack calculation region in the following, and a peridynamics (PD) is used for modeling, and the peridynamics naturally describes the initiation and propagation behavior of the crack through the non-local interaction between material points; S2, the momentum equation in the Cartesian coordinate system is converted into a first-order ordinary differential control equation in the scaled boundary coordinate system by introducing the scaled boundary coordinate system, and then the stiffness matrix of the crack-free calculation region is solved: first, the analytical method is used in the radial direction, and the numerical discretization is used in the hoop direction, and the coordinates in the Cartesian coordinate system are converted into the coordinates in the scaled boundary coordinate system; then, the momentum equation in the Cartesian coordinate system is derived based on the virtual work principle, and is converted into the first-order ordinary differential control equation form in the scaled boundary coordinate system; finally, the stiffness matrix of the crack-free calculation region is obtained by solving the eigenvalue of the first-order ordinary differential control equation; S3, in the crack calculation region, the elastomer is discretized into a series of material points, each material point is connected to other material points within the neighborhood range through a bond, and the counterforce between the material points is constructed based on the bond-based peridynamics theory of peridynamics; by matrix processing the counterforce between the material points, the micro-stiffness matrix between any two material points is derived; according to the finite element assembly rule, the micro-stiffness matrices of all material points are integrated into the overall stiffness matrix of the crack calculation region; S4, at the common nodes between the crack-free calculation region and the crack calculation region, the direct coupling of the two calculation regions is realized based on the force balance condition: by block processing the stiffness matrices of the two regions according to the internal nodes and the common nodes respectively, and then superimposing and combining the stiffness matrices corresponding to the common nodes, a unified overall stiffness matrix is finally formed; S5, an implicit solution algorithm is used in combination with load increment steps for nonlinear calculation: in each load step, first, the displacement of the elastomer is solved, then all the bonds in the crack calculation region are traversed, and the fracture of the bond is judged according to the current displacement; when it is detected that the bond is fractured, the overall stiffness matrix of the elastomer is updated and iterative re-calculation is performed in the current load step until the elastomer reaches an equilibrium state and no new bond fracture occurs, and then the next load increment is applied; this process is repeated until the calculation of all specified load steps is completed.

2. A method of analyzing a lining crack of an underground high-pressure water conveyance tunnel according to claim 1, characterized in that, In the step S1, the two-dimensional elastomer is divided into a crack-free calculation region and a crack calculation region, and expressions of momentum equations, geometric equations and constitutive equations of the elastomer are given: where is the gradient operator, is the stress, is the external load, consisting of volume force loads and surface force loads , is the strain, is the displacement, D is the elasticity matrix.

3. A method of analyzing a lining crack of an underground high-pressure water conveyance tunnel according to claim 2, characterized in that, The specific steps of the step S2 are as follows: SBFEM By introducing the proportional boundary coordinate system in the crack-free calculation area, the relationship between the coordinates in the Cartesian coordinate system and the coordinates in the proportional boundary coordinate system is: wherein and denotes the coordinates of the point to be found in the Cartesian coordinate system, denotes the radial coordinate of the proportional boundary coordinate system, denotes the circumferential coordinate of the proportional boundary coordinate system; is a shape function; and are the node coordinates in the Cartesian coordinate system; the transformation equation of the differential operator in the Cartesian coordinate system and the proportional boundary coordinate system is represented as: which is expressed in matrix form as: wherein J represents a Jacobian matrix; The two-dimensional elastic differential operator is expressed in scaled coordinates: wherein , are coefficient matrices, and are specifically represented as: wherein and denotes the partial derivative with respect to x and y, i.e. and ; The displacement and stress are expressed in the Cartesian coordinate system as: wherein denotes the function of the displacement in the Cartesian coordinate system with respect to the scaled boundary coordinate system , denotes the first order partial derivative of the displacement with respect to the coordinate , denotes the function of the displacement in the Cartesian coordinate system with respect to the scaled boundary coordinate system and ; denotes the function of the displacement in the Cartesian coordinate system with respect to the scaled boundary coordinate system and ; denotes the function of the displacement in the Cartesian coordinate system with respect to the scaled boundary coordinate system and ; , denotes the strain matrix, which is expressed as: The virtual work form of the momentum equation is: wherein, denotes the variation, denotes the volume, denotes the function of the displacement in the Cartesian coordinate system with respect to the proportional boundary coordinate system, denotes the function of the external load in the Cartesian coordinate system with respect to the proportional boundary coordinate system, simplifying to the first-order ordinary differential control equation of the proportional boundary coordinate system:​ wherein denotes the second order partial derivative of the displacement in the Cartesian coordinate system with respect to the second order partial derivative of the displacement in the proportional boundary coordinate system, the upper index denotes the transposed symbol of the matrix, , , are coefficient matrices, the specific expressions of which are:​ The matrix form of the first-order ordinary differential control equation is wherein denotes the first order partial derivative of the state vector with respect to the state vector denotes the state vector is the Hamiltonian matrix, which is given by The first-order ordinary differential control equation is solved by using an eigenvalue problem, and is specifically expressed as wherein and eigenvalues and eigenvectors of denotes the upper half of , , , denotes four equal parts Finally, the stiffness matrix of the crack-free calculation region is 。 4. A method of analyzing a lining crack of an underground high-pressure water conveyance tunnel according to claim 3, characterized in that, The specific steps in step S3 are as follows: Using the bond-based near-field dynamics theory, the crack calculation region is assumed to consist of multiple material points, each material point... a Volume is V a Each material point a With the radius of the adjacent region δ h Other material points a' There are interactions between the cracks, and the set of all material points within the computational region is called the influence domain H. a Each material point a The momentum equation is as follows: where is the displacement of material point ; is the displacement of material point a ; is the force on material point a between material points a' ; is the body force density on material point a , denotes the volume of material point ; in the key-based peridynamic theory, is described as follows: wherein, and are the relative position and relative displacement of material points a and material points respectively; for a micro-elastic-brittle material is expressed as: wherein is a history-dependent scalar value function; denotes a micro-modulus function; is a kernel function characterizing the strength of long-range forces between two material points; is the elongation of the bond, the expressions for the individual terms being as follows: wherein, represents the modulus of elasticity; represents the Poisson's ratio; s c represents the critical elongation; If the elongation of the bond is greater than the critical elongation of the bond s c then the bond between the two material points will break; the critical elongation of the bond s c is determined by the breaking energy G 0 which can be written as: The damage in the PD is defined as follows: The momentum equation is matrixed to obtain wherein and denotes two material points; denotes a material point volume; and denotes a material point and displacement of denotes a material point volume force density of To achieve PD and SBFEM coupling, a micro-stiffness matrix k is proposed in the PD, the expression is expressed as ; the relationship between the force and displacement between the global coordinate system and the local coordinate system is represented by the coordinate transformation matrix T : wherein, l and m are coordinate transformation factors between the local coordinate system and the global coordinate system; Thus, the material point in the global coordinate system has a micro stiffness matrix relative to the material point ​ material points composing stiffness matrix with all material points in the neighborhood and further combining stiffness matrices of all material points : 。 5. A method of analyzing a lining crack of an underground high-pressure water conveyance tunnel according to claim 4, characterized in that, The step S4 is implemented as follows: first, in SBFEM and PD a coupling model is constructed, and the force at the common point has the following relationship: wherein, represents the distance between two points of material; The stiffness matrix of S2 obtained in SBFEM is partitioned into individual and common points as follows: SBFEM ​ wherein the rows and columns of the matrix correspond to SBFEM unique points; the rows of the matrix correspond to SBFEM unique points, the columns correspond to common points; the rows of the matrix correspond to common points, the columns correspond to SBFEM unique points; the rows and columns of the matrix correspond to common points; denote SBFEM displacements of unique points; denote displacements of common points; denote SBFEM forces of unique points; denote forces of common points; In S3 PD The stiffness matrix is ​​divided into blocks as follows: wherein, the rows and columns of the matrix correspond to common points; the rows of the matrix correspond to common points, the columns correspond to PD unique points; the rows of the matrix correspond to PD unique points, the columns correspond to common points; the rows and columns of the matrix correspond to PD unique points; denotes displacement of a common point; denotes PD displacement of a unique point; denotes volume force density of a common point; denotes PD volume force density of a unique point; The overall stiffness matrix of the calculation region is obtained by assembling the stiffness matrices of SBFEM and PD ​ wherein is and collectively, i.e. = = .

6. A method of analyzing a lining crack of an underground high-pressure water conveyance tunnel according to claim 5, characterized in that, The specific steps of the step S5 are as follows: after the overall stiffness matrix in S4 is assembled, displacement constraints and load boundaries are applied; a load step counter is set m =0, initial load increment P =0; the load step calculation process is as follows: S5.1, Set up m = m +1, and update the load delta P ; S5.2, solving linear equations obtain the displacement of all nodes; S5.3, calculating the elongation of the crack calculation region material point key by the formula of S3 ; S5.4, if <s c enter step S5.5; otherwise, update the stiffness matrix of the crack calculation region and return to step S5.1; S5.5、If all load steps are calculated, exit the analysis; otherwise, return to step S5.4.

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