Simulation Method of Regular Lattice Spring Model for Deformation and Fracture of Anisotropic Materials

Abstract

This invention relates to the field of numerical simulation technology, specifically to a simulation method for anisotropic material deformation and fracture using a regular lattice spring model. The method involves discretizing a continuum into a set composed of regular lattice particles and bonds, and using this set to construct a simulation model. The parameters of the lattice particles and bonds in the simulation model are calculated and configured. The simulation model with configured parameters is iteratively solved to obtain the deformation and fracture evolution process. Termination conditions are determined for the deformation and fracture evolution process to obtain displacement, stress, and crack distribution. In this invention, each bond in the regular lattice spring model consists of a normal spring, a tangential spring, and a normal-tangential coupled spring. The normal-tangential coupled spring is innovatively used by the inventor to release the model's Poisson's ratio constraint and improve simulation accuracy. The spring stiffness calculation formula, derived from the dual control conditions of strain energy equivalence and shear stress reciprocity, allows the model to conveniently simulate anisotropic material deformation and fracture problems using engineering elastic constants.

Description

Technical Field

[0001] This invention relates to the field of numerical simulation technology, and in particular to a simulation method for a regular lattice spring model of anisotropic material deformation and fracture. Background Technology

[0002] Traditional continuum mechanics methods have achieved great success in describing the deformation of solid materials, fluid flow, heat transfer in media, and multiphysics coupling problems, but they face difficulties in describing the fracture of solid materials. As an effective alternative to traditional continuum mechanics methods, discrete particle models have been widely used to describe the mechanical behavior of solids, fluids, and particulate matter.

[0003] Lattice models (LMs) are a subset of discrete particle models that treat a continuum as a set of interacting particles with a fixed set of neighbors. They can simulate the elastic deformation and fracture failure behavior of materials within a consistent framework without requiring conversion during the analysis at the onset of fracture. Considering the different lattice topologies, existing lattice models can be divided into regular lattice models and irregular lattice models. Regular lattice models are more likely to satisfy elastic homogeneity, but the regularity of the lattice leads to a certain preference for crack propagation directions. Conversely, irregular lattice models offer lower pre-determinacy of crack directions, but generally cannot satisfy elastic homogeneity. Recent research shows that the crack path preference problem of regular lattice models can be mitigated or avoided by developing reasonable failure criteria, and the elastic homogeneity of irregular lattice models can be strengthened or ensured by using Voronoi lattice tessellation. However, it is important to emphasize that the calculation processes for lattice parameters differ significantly between the two models, and regular lattice models, due to their simpler regularity, are easier to form a unified calculation framework for lattice parameters.

[0004] Considering the differences in the interaction modes between lattice particles, existing lattice models can be mainly divided into lattice spring models, lattice beam models, and lattice potential models. The lattice spring model describes the interaction between lattice particles by setting springs between them. These springs can be normal springs, tangential springs, rotational springs, or combinations of two or three of these. The simplest model uses only normal springs to transmit the interaction force between particles, while the most widespread model includes all three types of springs. The lattice beam model is considered an extension of the lattice spring model, mainly describing the interaction between particles by setting Euler-Bernoulli beams or Tymoshenko beams between them. Lattice models based on Euler-Bernoulli beam theory are more widely used, while lattice models based on Tymoshenko beam theory are used when the beam elements are short and deep to provide a more accurate response. The lattice potential model describes the interaction between lattice particles by introducing a potential function. To effectively simulate the elastic deformation and crack propagation of materials, a lattice spring model is typically required to include normal, tangential, and / or rotational springs, or alternatively, a lattice beam model or lattice potential model may be used. Lattice models that only consider the contributions of normal and tangential stiffness have easily determined stiffness parameters, but they face difficulties in describing complex deformations and fractures. Lattice models that simultaneously consider the contributions of normal, tangential, and rotational stiffness have a strong ability to describe complex deformations and fractures, but the contribution of rotational stiffness is controversial and difficult to determine. Lattice models that consider potential interactions also have a strong ability to describe complex deformations and fractures, but the interactions between lattice particles often fail to satisfy classical physical laws (such as Newton's third law of motion). Summary of the Invention

[0005] The purpose of this invention is to provide a simulation method for the deformation and fracture of anisotropic materials using a regular lattice spring model, aiming to solve the problem that existing lattice models are difficult to accurately and conveniently describe the deformation and fracture evolution process of anisotropic continuums.

[0006] To achieve the above objectives, this invention provides a simulation method for a regular lattice spring model of anisotropic material deformation and fracture, comprising the following steps:

[0007] The continuum is discretized into a set of regular lattice particles and bonds, and the set is used to construct a simulation model;

[0008] Calculate and configure the parameters of the lattice particles and bonds in the simulation model;

[0009] The simulation model with configured parameters is solved iteratively to obtain the deformation and fracture development and evolution process;

[0010] The termination conditions of the deformation and fracture development process are determined to obtain the displacement, stress and crack distribution.

[0011] In the regular lattice spring model, the lattice particles have the same size, shape (regular hexagon or disk), and mass, and interact with each other by massless bonds, such as... Figure 2 As shown;

[0012] In the regular lattice spring model, each bond consists of three springs: a normal spring, a tangential spring, and a normal-tangential coupling spring. Each spring has its own independent stiffness, such as... Figure 3 As shown;

[0013] In the regular lattice spring model, the bonds have strength, and they break when the force exceeds a set strength.

[0014] Specifically, simulation models are constructed by discretizing the continuum into a collection of many regular lattice particles and bonds, such as... Figure 2 and Figure 6 As shown. Based on symmetry, Figure 2 The normal vectors of the keys whose normals are parallel to sections I-I, II-II, and III-III, and the spring stiffness are shown in Table 1.

[0015] Table 1. ARLSM Key Normal Vectors and Key Spring Stiffnesses

[0016]

[0017] In the step of calculating and configuring the parameters of the lattice particles and bonds of the simulation model, the parameters include physical parameters and boundary conditions;

[0018] The physical parameters include the density and damping of lattice particles, and the spring stiffness and strength of bonds;

[0019] The boundary conditions include the physical forces (including gravity) and velocities of lattice particles, and the internal forces and deformations of bonds.

[0020] The spring stiffness of the key is precisely calculated by the relationship between spring stiffness and engineering elastic constant, as shown in equations (1), (2) and (3);

[0021] The relationship between the spring stiffness and the engineering elastic constant is derived from the dual control conditions of strain energy equivalence and shear stress reciprocity, as shown in equations (12) and (14).

[0022] The shear stress reciprocity condition is used as a boundary condition to determine the number of spring stiffness parameters that are greater than the number of control equations, as shown in equation (13).

[0023] Specifically, the spring stiffness of the key in the ARLSM is precisely calculated using engineering elastic constants, and the calculation formula is as follows:

[0024]

[0025] In the formula, C 11 C 22 C 66 C 12 C 16 and C 26 Let t be the engineering elastic constant in the constitutive model of the anisotropic continuum, and t be the model thickness.

[0026] When the constitutive properties of an anisotropic continuum degenerate into those of an isotropic continuum, equation (1) degenerates into plane stress.

[0027] plane strain, In the formula, E and ν are the engineering elastic modulus and Poisson's ratio in the constitutive model of an isotropic continuum, respectively.

[0028] In ARLSM, the relationship between spring stiffness and engineering elastic constants is derived from the dual control conditions of strain energy equivalence and shear stress reciprocity. The shear stress reciprocity was innovatively added by the inventor as a boundary condition to determine the number of spring stiffness parameters that are greater than the number of control equations.

[0029] Assuming lattice particle i and its surrounding J i Individual lattice particles are connected (e.g.) Figure 2 As shown), the average strain energy density around lattice particle i is calculated as follows:

[0030]

[0031] The elastic constant can be obtained from the formula Based on strain energy density calculations

[0032]

[0033] Equivalent control conditions for strain energy in ARLSM

[0034]

[0035] It is worth noting that Equation (12) is a variant of the strain energy equivalence condition, which is the most essential equivalence condition that has been adopted by many scholars.

[0036] In ARLSM, the mutual equality of shear stresses is a boundary condition, i.e.

[0037] σ 12 =C 1211 ε 11 +C1222 ε 22 +(C 1212 +C 1221 )ε 12 =C 2111 ε 11 +C 2122 ε 22 +(C 2112 +C 2121 )ε 12 =σ 21 (13)

[0038] Considering that equation (13) holds true for any strain, the reciprocal control condition of shear stress in ARLSM can be obtained as follows:

[0039]

[0040] Substituting equation (9) into equations (12) and (14) respectively, we obtain nine linearly independent equations.

[0041]

[0042] In the formula, the 6 equations marked with (*1) are obtained based on the commonly used strain energy equivalence condition, and the 3 equations marked with (*2) are obtained based on the supplementary shear stress equivalence condition.

[0043] Substituting the contents of Table 1 into equation (15), we can solve the system of equations to obtain the formulas for calculating the spring stiffness of the key in ARLSM, as shown in equations (1), (2) and (3).

[0044] The core steps of the iterative solution include the Law of Motion and the Force-Displacement Law, such as... Figure 4 As shown;

[0045] The evolution of deformation and fracture includes quasi-static problem simulation and dynamic problem simulation.

[0046] Specifically, during the ARLSM cyclic solution process, lattice particles move according to the Law of Motion and interact through bond forces within the framework of the Force-Displacement Law. In the Law of Motion phase, the velocities and displacements of each lattice particle are updated using the current time step and the forces and torques calculated in the previous cycle. In the Force-Displacement Law phase, the bond forces are updated based on the current state of the lattice particles, and the bond state (intact, broken, or sheared) is updated according to the failure criterion. As the simulation progresses, the model state advances over time through a series of computational cycles.

[0047] The updates of the lattice particle velocities and displacements are achieved by equations (16) and (17), respectively. For the quasi-static problem, ARLSM uses a locally adaptive damping scheme to obtain the static solution. The locally adaptive damping acts on each lattice particle and is proportional to the unbalanced force, as shown in equation (16).

[0048]

[0049]

[0050] In the formula, and Let be the displacements of lattice particle i at times t and t+Δt, respectively. and Let be the velocities of lattice particle i at times t-Δt / 2 and t+Δt / 2, respectively. m is the resultant force (including applied external force) acting on lattice particle i. i Let be the mass of lattice particle i, and Δt be the time step. Furthermore, α is the dimensionless local damping coefficient, independent of mechanical properties and boundary conditions; it is typically set to 0.7 in static analysis and 0 or a small value in dynamic analysis.

[0051] The bond force update is achieved by equation (18). The normal component of the force vector of bond ij. and tangential components Calculated as

[0052]

[0053] In the formula, Let be the normal spring stiffness of key ij. Let be the tangential spring stiffness of key ij. Let the normal and tangential coupling spring stiffness of key ij be, such as Figure 3 As shown; Let be the normal component of the deformed vector of key ij. Let be the tangential component of the deformation vector of key ij, as shown in equation (5).

[0054] It is worth noting that ARLSM considers the real-time interactions between lattice particles and bonds by solving the complete dynamic equations, thereby simulating the deformation and fracture evolution of a continuum. The dynamic equations can be expressed as follows:

[0055]

[0056] In the formula, u is the displacement vector, M is the diagonal mass matrix, C is the damping matrix, K is the stiffness matrix, and F(t) is the resultant external force vector.

[0057] As an explicit time-stepping scheme, the second-order accurate Verlet algorithm is used to solve the above equations of motion. This solution is conditionally stable; the time step to maintain computational stability must not exceed a critical value related to the minimum characteristic period of the entire system. ARLSM employs a general scheme to estimate the critical time step. ARLSM is a collection of lattice particles and springs; assuming the degrees of freedom in the ARLSM are uncoupled, the critical time step can be estimated as follows:

[0058]

[0059] In the formula, m i K is the mass of lattice particle i. ij is the global stiffness matrix of bonds ij surrounding lattice particle i. It is important to note that, to ensure absolute stability of the calculation, the final critical time step needs to be multiplied by a safety factor (0.8 in ARLSM).

[0060] The deformation and fracture development process uses a modified nonlinear strength criterion as the basis for fracture calculation and judgment.

[0061] Specifically, ARLSM determines the principal stresses of the key (compressive stress is specified as negative in the model, and...) by... The discrimination of bond fracture state and fracture mode is achieved by comparing the parameters with those of the modified nonlinear strength criterion. The discriminant is defined as follows:

[0062]

[0063]

[0064] In the formula, σ tβ The uniaxial tensile strength (UTS) of anisotropic rocks when the plane of anisotropy forms an angle β with the direction of the maximum principal stress, and satisfies the following relationship: The uniaxial compressive strength (UCS) of anisotropic rocks when the plane of anisotropy forms an angle β with the direction of maximum principal stress; c β0 and φ β0 The Mohr-Coulomb shear strength parameter is obtained by conducting triaxial strength tests on rock samples under low confining pressure (confining pressure approaching 0); σ crt The critical confining pressure for anisotropic rocks is statistically almost equal to 1.25 times the maximum UCS obtained by applying loads parallel or perpendicular to the anisotropic plane.

[0065] when When the key is pulled off, it breaks; otherwise, when At that time, the key was cut off.

[0066] In ARLSM, the stress of bond ij is taken as the average of the sum of the local stresses of the two lattice particles i and j connected to it, i.e.

[0067]

[0068] The local stress of lattice particle i is calculated as follows:

[0069]

[0070] In the formula, Let N be the effective volume of lattice particle i, and N be the number of bonds connected around lattice particle i. i Let x be the coordinates of lattice particle i. ij Let F be the coordinates of the bonds ij surrounding lattice particle i. ij Let be the force vector of the bonds ij surrounding lattice particle i.

[0071] Specifically, when the iterative solution is a quasi-static problem simulation, the termination condition of the deformation and fracture development process is determined to obtain the displacement, stress, and crack distribution, including:

[0072] The kinetic energy or unbalanced force during the deformation and fracture development process is used as the termination condition. When the termination condition is less than or equal to the target value, the simulation ends, and the displacement, stress and crack distribution are obtained.

[0073] When the iterative solution is used for dynamic problem simulation, termination conditions are determined for the deformation and fracture development process to obtain displacement, stress, and crack distribution, including:

[0074] The simulation physical time during the deformation and fracture development process is used as the termination criterion. When the termination criterion is greater than or equal to the target value, the simulation ends, and the displacement, stress and crack distribution are obtained.

[0075] This invention discloses a simulation method for anisotropic material deformation and fracture using a regular lattice spring model. The method discretizes the continuum into a set composed of regular lattice particles and bonds, and uses this set to construct a simulation model. The parameters of the lattice particles and bonds in the simulation model are calculated and configured. The simulation model with configured parameters is iteratively solved to obtain the deformation and fracture evolution process. Termination conditions are determined for the deformation and fracture evolution process to obtain displacement, stress, and crack distribution. In this invention, each bond in the regular lattice spring model consists of a normal spring, a tangential spring, and a normal-tangential coupled spring. The normal-tangential coupled spring is innovatively used by the inventors to release the model's Poisson's ratio constraint and improve simulation accuracy. The spring stiffness calculation formula, derived from the dual control conditions of strain energy equivalence and shear stress reciprocity, allows the model to conveniently simulate the deformation and fracture of anisotropic materials using engineering elastic constants, solving the problem that existing lattice models cannot accurately and conveniently describe the deformation and fracture evolution process of anisotropic continuums. Attached Figure Description

[0076] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0077] Figure 1 This is a flowchart of the construction and calculation of the regular lattice spring model of the present invention;

[0078] Figure 2 This is a schematic diagram of the regular lattice spring model lattice discretization and bond network construction of the present invention;

[0079] Figure 3 This is a schematic diagram of the regular lattice spring model key spring of the present invention;

[0080] Figure 4 This is a flowchart of the core steps for solving the regular lattice spring model in this invention.

[0081] Figure 5 This is a theoretical diagram of the bond nonlinearity failure criterion for the regular lattice spring model of the present invention;

[0082] Figure 6This is a schematic diagram of the physical model and numerical model (ARLSM model) of anisotropic rock Brazilian splitting in an embodiment of the regular lattice spring model of the present invention (θ is the anisotropic direction angle);

[0083] Figure 7 This is a displacement, stress, and crack distribution diagram (θ=45°) of an anisotropic rock Brazilian splitting numerical model (ARLSMModel) of the regular lattice spring model embodiment of the present invention.

[0084] Figure 8 This is a flowchart of a simulation method for a regular lattice spring model of anisotropic material deformation and fracture provided by the present invention. Detailed Implementation

[0085] Embodiments of the present invention are described in detail below, examples of which are illustrated in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. 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.

[0086] Please see Figures 1 to 8 This invention provides a simulation method for the deformation and fracture of anisotropic materials using a regular lattice spring model (ARLSM), comprising the following steps:

[0087] S1 discretizes the continuum into a set of regular lattice particles and bonds, and uses the set to build a simulation model;

[0088] Specifically, the lattice particles have the same size, shape (regular hexagon or disk), and mass, and are connected by massless bonds to generate interactions, such as... Figure 2 As shown; the key consists of three springs, namely a normal spring, a tangential spring, and a normal-tangential coupling spring, each with its own independent stiffness, such as... Figure 3 As shown; the key has strength and breaks when subjected to a force exceeding a set strength.

[0089] Simulation model construction is a necessary foundation for the application of simulation methods. This step constructs the simulation model by discretizing the continuum into a set composed of many regular lattice particles and bonds, such as... Figure 2 and Figure 6 As shown. Based on symmetry, Figure 2 The normal vectors of the keys whose normals are parallel to sections I-I, II-II, and III-III, and the spring stiffness are shown in Table 1.

[0090] Table 1. ARLSM Key Normal Vectors and Key Spring Stiffnesses

[0091]

[0092] S2 calculates and configures the parameters of the lattice particles and bonds in the simulation model;

[0093] Specifically, the parameters include physical parameters and boundary conditions; the physical parameters include the density and damping of lattice particles, and the spring stiffness and strength of bonds; the boundary conditions include the body forces (including gravity) and velocities of lattice particles, and the internal forces and deformations of bonds.

[0094] A key task in lattice model calculations is to simulate the corresponding equivalent continuum using correct lattice parameters. In this step, lattice particle parameters are indispensable for completing the simulation calculations, and the calculation and configuration of bond parameters are the key to accurately characterizing the deformation and fracture behavior of anisotropic materials.

[0095] In ARLSM, the deformation effect of the equivalent continuum is achieved by the deformation of the springs, and the spring stiffness of the bonds is the most important component of the ARLSM lattice parameters. The spring stiffness of the bonds in ARLSM is precisely calculated using engineering elastic constants, and the calculation formula is as follows:

[0096]

[0097] In the formula, C 11 C 22 C 66 C 12 C 16 and C 26 Let t be the engineering elastic constant in the constitutive model of the anisotropic continuum, and t be the model thickness.

[0098] When the constitutive properties of an anisotropic continuum degenerate into those of an isotropic continuum, equation (1) degenerates into plane stress.

[0099] plane strain, In the formula, E and ν are the engineering elastic modulus and Poisson's ratio in the constitutive model of an isotropic continuum, respectively.

[0100] In ARLSM, the relationship between spring stiffness and engineering elastic constants is derived from the dual control conditions of strain energy equivalence and shear stress reciprocity. The shear stress reciprocity was innovatively added by the inventor as a boundary value condition to determine the number of spring stiffness parameters that exceed the number of control equations. The derivation process is as follows.

[0101] Let ij be the bond connecting lattice particles i and j. Then, its deformation vector is calculated from the relative displacement vectors of lattice particles i and j.

[0102] δ ij =u j -u i (4)

[0103] In the formula, u i and u j Let n be the displacement vectors of lattice particles i and j, respectively. Let n be the normal vector pointing from lattice particle i to j. ij =(c ij ,s ij ) T The tangential vector is s ij =(-s ij ,c ij ) T Then the normal component of the deformed vector of key ij and tangential components Calculated as

[0104]

[0105] The key ij consists of three springs: a normal spring, a tangential spring, and a normal-tangential coupling spring, and each spring has its own independent stiffness. and like Figure 3 As shown.

[0106] Assuming lattice particle i and its surrounding J i Individual lattice particles are connected (e.g.) Figure 2 As shown), the average strain energy density around lattice particle i is calculated as follows:

[0107]

[0108] In the formula, Let be the effective volume of lattice particle i (the volume of a regular hexagonal prism), r be the radius of the inscribed circle of lattice particle i, and t be the model thickness. It is important to note that when calculating the strain energy density, only half of the strain energy of each bond surrounding lattice particle i is taken to avoid overlap.

[0109] Considering a small rotation problem (neglecting the rotation tensor), the relative displacement between two lattice particles i and j can be expressed as:

[0110]

[0111] In the formula, L ij =2r is the bond length, ε 11 ε 22 ε 12 and ε 21 These are the strain tensor components in the global coordinate system.

[0112] Substituting equation (7) into equation (6) yields

[0113]

[0114] The elastic constant can be obtained from the formula Based on strain energy density calculations

[0115]

[0116] The generalized Hooke's law in tensor form is expressed as follows:

[0117]

[0118] Considering that the shear strains are equal ε 12 =ε 21 and introduce and γ 12 =ε 12 +ε 21 The above formula can be rewritten as follows:

[0119]

[0120] The generalized Hooke's law expression commonly used in engineering By comparison, we obtain the following 6 equations.

[0121]

[0122] It is worth noting that Equation (12) is a variant of the strain energy equivalence condition, which is the most essential equivalence condition that has been adopted by many scholars.

[0123] In ARLSM, nine spring stiffness parameters need to be determined. Currently, there are only six governing equations. However, the number of spring stiffness parameters exceeds the number of governing equations, resulting in an infinite number of solutions. Therefore, three more governing equations are needed for boundary value solutions. Here, the condition of mutual equality of shear stresses is added as the boundary value solution condition.

[0124] σ 12 =C 1211 ε 11 +C 1222 ε 22 +(C 1212 +C 1221 )ε 12 =C 2111 ε 11 +C 2122 ε 22 +(C 2112 +C 2121 )ε 12 =σ 21 (13)

[0125] Considering that equation (13) holds true for any strain, we can obtain three equations.

[0126]

[0127] Substituting equation (9) into equations (12) and (14) respectively, we obtain nine linearly independent equations.

[0128]

[0129] In the formula, the 6 equations marked with (*1) are obtained based on the commonly used strain energy equivalence condition, and the 3 equations marked with (*2) are obtained based on the supplementary shear stress equivalence condition.

[0130] Substituting the contents of Table 1 into equation (15), we can solve the system of equations to obtain the formulas for calculating the spring stiffness of the key in ARLSM, as shown in equations (1), (2) and (3).

[0131] The S3 loop solves the simulation model with configured parameters, and obtains the deformation and fracture development and evolution process.

[0132] Specifically, the core steps of the iterative solution include the Law of Motion and the Force-Displacement Law, such as... Figure 4 As shown; the deformation and fracture development and evolution process includes quasi-static problem simulation and dynamic problem simulation.

[0133] During the ARLSM iterative solution, lattice particles move according to the Law of Motion and interact through bond forces within the framework of the Force-Displacement Law. In the Law of Motion phase, the velocities and displacements of each lattice particle are updated using the current time step and the forces and torques calculated in the previous loop. In the Force-Displacement Law phase, the bond forces are updated based on the current state of the lattice particles, and the bond state (intact, broken, or sheared) is updated according to the failure criterion. As the simulation progresses, the model state advances over time through a series of computational loops.

[0134] The updates of the lattice particle velocities and displacements are achieved by equations (16) and (17), respectively. For the quasi-static problem, ARLSM uses a locally adaptive damping scheme to obtain the static solution. The locally adaptive damping acts on each lattice particle and is proportional to the unbalanced force, as shown in equation (16).

[0135]

[0136]

[0137] In the formula, and Let be the displacements of lattice particle i at times t and t+Δt, respectively. and Let be the velocities of lattice particle i at times t-Δt / 2 and t+Δt / 2, respectively. m is the resultant force (including applied external force) acting on lattice particle i. i Let be the mass of lattice particle i, and Δt be the time step. Furthermore, α is the dimensionless local damping coefficient, independent of mechanical properties and boundary conditions; it is typically set to 0.7 in static analysis and 0 or a small value in dynamic analysis.

[0138] The bond force update is achieved by equation (18). The normal component of the force vector of bond ij. and tangential components Calculated as

[0139]

[0140] In the formula, Let be the normal spring stiffness of key ij. Let be the tangential spring stiffness of key ij. Let the normal and tangential coupling spring stiffness of key ij be, such as Figure 3 As shown; Let be the normal component of the deformed vector of key ij. Let be the tangential component of the deformation vector of key ij, as shown in equation (5).

[0141] It is worth noting that ARLSM considers the real-time interactions between lattice particles and bonds by solving the complete dynamic equations, thereby simulating the deformation and fracture evolution of a continuum. The dynamic equations can be expressed as follows:

[0142]

[0143] In the formula, u is the displacement vector, M is the diagonal mass matrix, C is the damping matrix, K is the stiffness matrix, and F(t) is the resultant external force vector.

[0144] As an explicit time-stepping scheme, the second-order accurate Verlet algorithm is used to solve the above equations of motion. This solution is conditionally stable; the time step to maintain computational stability must not exceed a critical value related to the minimum characteristic period of the entire system. ARLSM employs a general scheme to estimate the critical time step. ARLSM is a collection of lattice particles and springs; assuming the degrees of freedom in the ARLSM are uncoupled, the critical time step can be estimated as follows:

[0145]

[0146] In the formula, m i K is the mass of lattice particle i. ij is the global stiffness matrix of bonds ij surrounding lattice particle i. It is important to note that, to ensure absolute stability of the calculation, the final critical time step needs to be multiplied by a safety factor (0.8 in ARLSM).

[0147] S4 determines the termination conditions of the deformation and fracture development process, obtaining displacement, stress, and crack distribution, such as... Figure 7 As shown.

[0148] Fracture simulation is one of the most important advantages of lattice models. Fracture calculation and discrimination in ARLSM are performed at the force-displacement law stage. The failure criterion is a necessary basis for fracture calculation and discrimination. The inventors, based on the nonlinear criterion for triaxial strength of inherently anisotropic rocks proposed by Mahendra Singh et al., added a tensile stress intercept to obtain a modified nonlinear criterion for triaxial strength of inherently anisotropic rocks, or simply the modified nonlinear strength criterion. Figure 5 As shown, ARLSM uses a modified nonlinear strength criterion as the basis for fracture calculation and judgment.

[0149] ARLSM defines the principal stresses of the key (compressive stress is specified as negative in the model, and...) as negative. The discrimination of bond fracture state and fracture mode is achieved by comparing the parameters with those of the modified nonlinear strength criterion. The discriminant is defined as follows:

[0150]

[0151]

[0152] In the formula, σ tβThe uniaxial tensile strength (UTS) of anisotropic rocks when the plane of anisotropy forms an angle β with the direction of the maximum principal stress, and satisfies the following relationship: The uniaxial compressive strength (UCS) of anisotropic rocks when the plane of anisotropy forms an angle β with the direction of maximum principal stress; c β0 and φ β0 The Mohr-Coulomb shear strength parameter is obtained by conducting triaxial strength tests on rock samples under low confining pressure (confining pressure approaching 0); σ crt The critical confining pressure for anisotropic rocks is statistically almost equal to 1.25 times the maximum UCS obtained by applying loads parallel or perpendicular to the anisotropic plane.

[0153] when When the key is pulled off, it breaks; otherwise, when At that time, the key was cut off.

[0154] In ARLSM, the stress of bond ij is taken as the average of the sum of the local stresses of the two lattice particles i and j connected to it, i.e.

[0155]

[0156] The local stress of lattice particle i is calculated as follows:

[0157]

[0158] In the formula, Let N be the effective volume of lattice particle i, and N be the number of bonds connected around lattice particle i. i Let x be the coordinates of lattice particle i. ij Let F be the coordinates of the bonds ij surrounding lattice particle i. ij Let be the force vector of the bonds ij surrounding lattice particle i.

[0159] Specifically, when the iterative solution is a quasi-static problem simulation, the termination condition of the deformation and fracture development process is determined to obtain the displacement, stress, and crack distribution, including:

[0160] The kinetic energy or unbalanced force during the deformation and fracture development process is used as the termination condition. When the termination condition is less than or equal to the target value, the simulation ends and the displacement, stress and crack distribution are obtained; otherwise, step S3 continues.

[0161] When the iterative solution is used for dynamic problem simulation, termination conditions are determined for the deformation and fracture development process to obtain displacement, stress, and crack distribution, including:

[0162] The simulation physical time during the deformation and fracture development process is used as the termination condition. When the termination condition is greater than or equal to the target value, the simulation ends and the displacement, stress and crack distribution are obtained; otherwise, continue to step S3.

[0163] This invention presents a simulation method for regular lattice spring models of anisotropic material deformation and fracture, offering significant advantages. First, the inventors developed a locally regular lattice spring model. Building upon the simultaneous use of normal and tangential springs, they innovatively introduced coupled normal and tangential springs to alleviate the Poisson's ratio limitation and improve simulation accuracy. Simultaneously, they derived explicit expressions for accurately calculating the stiffness parameters of each spring using engineering elastic constants. This allows the invented regular lattice spring model to be easily and directly used for the deformation and fracture calculation and analysis of anisotropic continuums. It is important to emphasize that the focus of this invention is to construct a unified framework for the simulation modeling and calculation of regular lattice spring models of anisotropic material deformation and fracture, offering significant advantages.

[0164] This invention can accurately characterize the elastic deformation and fracture failure behavior of anisotropic materials. The modeling is as simple as the finite element model (FEM) and has the significant advantages of the discrete element bond model for simulating fracture. It provides a new method, a new tool and a new option for the calculation and analysis of deformation and fracture of anisotropic materials.

[0165] The above description is merely a preferred embodiment of the simulation method for a regular lattice spring model of anisotropic material deformation and fracture according to the present invention. Of course, it should not be construed as limiting the scope of the present invention. Those skilled in the art can understand that all or part of the process of implementing the above embodiments and equivalent changes made in accordance with the claims of the present invention still fall within the scope of the invention.

Claims

1. A simulation method for a regular lattice spring model of anisotropic material deformation and fracture, characterized in that, Includes the following steps: The continuum is discretized into a set of regular lattice particles and bonds, and the set is used to construct a simulation model; Calculate and configure the parameters of the lattice particles and bonds in the simulation model; The simulation model with configured parameters is solved iteratively to obtain the deformation and fracture development and evolution process; The termination conditions of the deformation and fracture development process are determined to obtain the displacement, stress, and crack distribution. In the regular lattice spring model, the lattice particles have the same size, shape, and mass, and are connected by massless bonds to generate interactions. In the regular lattice spring model, each bond consists of three springs: a normal spring, a tangential spring, and a normal-tangential coupling spring. Each spring has its own independent stiffness. In the regular lattice spring model, the bonds have strength, and they break when the force exceeds a set strength. In the step of calculating and configuring the parameters of the lattice particles and bonds of the simulation model, the parameters include physical parameters and boundary conditions; The physical parameters include the density and damping of lattice particles, and the spring stiffness and strength of bonds; The boundary conditions include the physical forces and velocities of lattice particles, and the internal forces and deformations of bonds.

2. The simulation method for a regular lattice spring model of anisotropic material deformation and fracture as described in claim 1, characterized in that, The spring stiffness of the key is precisely calculated by the relationship between spring stiffness and engineering elastic constant, as shown in equations (1), (2) and (3); The relationship between the spring stiffness and the engineering elastic constant is derived from the dual control conditions of strain energy equivalence and shear stress reciprocity, as shown in equations (12) and (14). The shear stress reciprocity condition is used as a boundary condition to determine the number of spring stiffness parameters that are greater than the number of control equations, as shown in equation (13). Specifically, the spring stiffness of the key in the ARLSM is precisely calculated using engineering elastic constants, and the calculation formula is as follows: (1) In the formula, , , , , and These are the engineering elastic constants in the constitutive model of an anisotropic continuum. For model thickness; When the constitutive model of an anisotropic continuum degenerates into the constitutive model of an isotropic continuum, equation (1) degenerates into... plane stress, (2) plane strain, (3) In the formula, and These are the engineering elastic modulus and Poisson's ratio in the constitutive model of an isotropic continuum, respectively. In ARLSM, the relationship between spring stiffness and engineering elastic constants is derived from the dual control conditions of strain energy equivalence and shear stress reciprocity. Shear stress reciprocity is a boundary condition used to determine the number of spring stiffness parameters that are greater than the number of control equations. The derivation process is as follows. Define connected lattice particles and The key is Then its deformation vector is determined by the lattice particles. and The relative displacement vector is calculated as follows: (4) In the formula, and They are lattice particles and The displacement vector; defining lattice particles point to The normal vector is The tangential vector is Then the key Normal components of the deformation vector and tangential components Calculated as (5) key It consists of three springs: a normal spring, a tangential spring, and a spring that couples the normal and tangential directions, and each spring has its own independent stiffness. , and ; Assuming lattice particles With the surroundings Each lattice particle is connected, lattice particles The average strain energy density around is calculated as follows (6) In the formula, lattice particles The effective volume, lattice particles The radius of the inscribed circle, For model thickness; when calculating strain energy density, lattice particles The strain energy of each surrounding bond is reduced to half to avoid overlap; Considering a small rotation problem, two lattice particles and The relative displacement between them is expressed as (7) In the formula, The length of the bond. , , and These are the strain tensor components in the global coordinate system; Substituting equation (7) into equation (6) yields (8) The elastic constant is given by the formula Based on strain energy density calculations (9) The generalized Hooke's law in tensor form is expressed as follows: (10) Considering shear strain reciprocity and introduce and The above formula can be rewritten as follows: (11) The generalized Hooke's law expression commonly used in engineering By comparison, we obtain the following 6 equations. (12) Equation (12) is a variant of the strain energy equivalence condition; In ARLSM, nine spring stiffness parameters need to be determined. Currently, there are only six governing equations. However, the number of spring stiffness parameters exceeds the number of governing equations, resulting in an infinite number of solutions. Therefore, three more governing equations are needed for boundary value solutions. Here, the condition of mutual equality of shear stresses is added as the boundary value solution condition. (13) Considering that equation (13) holds true for any strain, we obtain three equations. (14) Substituting equation (9) into equations (12) and (14) respectively, we obtain nine linearly independent equations. (15) In the formula, the 6 equations marked with (*1) are obtained based on the commonly used strain energy equivalence condition, and the 3 equations marked with (*2) are obtained based on the supplementary shear stress equivalence condition. Substituting the ARLSM key normal vector and key spring stiffness into equation (15), and solving the system of equations, we obtain the formulas for calculating the spring stiffness of the key in the ARLSM, as shown in equations (1), (2) and (3).

3. The simulation method for a regular lattice spring model of anisotropic material deformation and fracture as described in claim 1, characterized in that, The iterative solution steps include the laws of motion and the force-displacement law; The evolution of deformation and fracture includes quasi-static problem simulation and dynamic problem simulation; Specifically, the core steps of the iterative solution include the laws of motion and the force-displacement law; The deformation and fracture development and evolution process includes quasi-static problem simulation and dynamic problem simulation; During the ARLSM cyclic solution process, lattice particles move based on the laws of motion and interact with each other through bond forces within the framework of the force-displacement law. In the motion law phase, the velocity and displacement of each lattice particle are updated using the current time step and the force and torque calculated in the previous loop. In the force-displacement law stage, the bond forces are updated based on the current state of the lattice particles, and the bond state is updated according to the failure criterion. As the simulation progresses, the model state is advanced over time through a series of computational loops; The updates of the velocity and displacement of the lattice particles are achieved by Equations (16) and (17), respectively. For the quasi-static problem, ARLSM uses a local adaptive damping scheme to obtain the static solution. The local adaptive damping acts on each lattice particle and is proportional to the unbalanced force, as shown in Equation (16). (16) (17) In the formula, and They are lattice particles exist and Displacement at any moment and They are lattice particles exist and The speed of time For action on lattice particles The combined force of the above lattice particles quality For time step; in addition, It is a dimensionless local damping coefficient, which is independent of mechanical properties and boundary conditions. It is 0.7 in static analysis and 0 or a small value in dynamic analysis. Bond force update is achieved by equation (18); bond Normal component of the force vector and tangential components Calculated as (18) In the formula, For key The normal spring stiffness, For key Tangential spring stiffness, For key The normal and tangential coupling spring stiffness; For key The normal component of the deformed vector, For key The tangential component of the deformation vector is shown in equation (5); ARLSM simulates the deformation and fracture evolution of a continuum by solving the complete dynamic equations, taking into account the real-time interactions between lattice particles and bonds. The dynamic equations are expressed as follows: (19) In the formula, It is a displacement vector. The mass matrix is ​​a diagonal matrix. Here is the damping matrix. Here is the stiffness matrix. The resultant external force vector, For velocity vectors, It is the acceleration vector; As an explicit time-stepping scheme, the second-order precision Verlet algorithm is used to solve the above equations of motion. This solution is conditionally stable; the time step to maintain computational stability must not exceed a critical value related to the minimum characteristic period of the entire system. ARLSM employs a general scheme to estimate the critical time step. An ARLSM is a collection of lattice particles and springs; assuming the degrees of freedom in an ARLSM are uncoupled, the critical time step is estimated as follows: (20) In the formula, lattice particles quality It is a lattice particle Surround keys The global stiffness matrix; to ensure absolute stability of the calculation, the final critical time step needs to be multiplied by a safety factor.

4. The simulation method for the deformation and fracture of anisotropic materials using a regular lattice spring model as described in claim 3, characterized in that, The deformation and fracture development process adopts a modified nonlinear strength criterion as the basis for fracture calculation and discrimination; Specifically, ARLSM distinguishes the bond fracture state and fracture mode by comparing the principal stresses of the bond with the modified nonlinear strength criterion parameters. The discriminant is defined as follows: (21) (22) In the formula, For anisotropic rocks, the anisotropic plane is perpendicular to the direction of the maximum principal stress. Uniaxial tensile strength at an angle, and satisfying the relationship ; For anisotropic rocks, the anisotropic plane is perpendicular to the direction of the maximum principal stress. Uniaxial compressive strength at an angle; and The Mohr-Coulomb shear strength parameters are obtained by conducting triaxial strength tests on rock samples under low confining pressure. The critical confining pressure for anisotropic rocks is statistically equal to 1.25 times the maximum UCS obtained by applying loads parallel or perpendicular to the anisotropic plane. when When the key is pulled off, it breaks; otherwise, when At that time, the key was cut off; ARLSM middle mouse button The stress is taken as the stress of the two lattice particles connected to it. and The average value of the sum of local stresses, i.e. (23) Lattice particles The local stress is calculated as follows (24) In the formula, lattice particles The effective volume, lattice particles The number of surrounding connected keys, lattice particles coordinates lattice particles Surround keys coordinates lattice particles Surround keys The force vector; Specifically, when the iterative solution is a quasi-static problem simulation, the termination condition of the deformation and fracture development process is determined to obtain the displacement, stress, and crack distribution, including: The kinetic energy or unbalanced force during the deformation and fracture development process is used as the termination condition. When the termination condition is less than or equal to the target value, the simulation ends, and the displacement, stress and crack distribution are obtained. When the iterative solution is used for dynamic problem simulation, termination conditions are determined for the deformation and fracture development process to obtain displacement, stress, and crack distribution, including: The simulation physical time during the deformation and fracture development process is used as the termination criterion. When the termination criterion is greater than or equal to the target value, the simulation ends, and the displacement, stress and crack distribution are obtained.

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