A method for predicting crack propagation in brittle materials

Abstract

The application discloses a brittle material crack propagation prediction method, comprising the following steps: a brittle material entity model is discretized into a series of space material points through space grid multi-level division; parameters of the brittle material are acquired and taken as input of a prediction model; a prototype micro-brittle model constitutive relation is constructed based on space coordinates, the size of a near-field domain and a dual domain is set, and a key force action range is determined; initial boundary conditions are determined, and an external force is applied by using a progressive loading; physical and mechanical information such as material point position, velocity, acceleration and the like is updated in real time by using an explicit iteration method, and a critical elongation bond breaking criterion is used to describe damage cracking; displacement results and damage results at different times are output, the position and time of damage and cracking are recorded, and crack propagation paths at different times are acquired. The application solves the false stress wave problem caused by variable near-field domain size, and can effectively predict and analyze the mechanical behavior and damage problem of the brittle material under the action of a load.

Description

TECHNICAL FIELD

[0001] The present application relates to a brittle material crack propagation prediction method, in particular to a full-process prediction method for predicting and analyzing the whole process from the generation of a mesoscopic crack to the propagation and destruction of a brittle material under the action of a load based on a dual-domain near-field dynamics method, and belongs to the technical field of material structure destruction simulation. BACKGROUND

[0002] Impact destruction and fracture research of materials has always been of great guiding significance in engineering application fields. Whether in aerospace or in mechanical manufacturing, civil construction fields, as long as the safety and reliability of materials are involved, the fracture analysis of materials or engineering structures cannot be avoided.

[0003] The destruction of brittle materials is due to the generation of many microscopic defects in the material under the action of external stress and environment alone or together. These microscopic defects cannot be observed by the naked eye, but with the continuous influence of external forces or environmental factors, these microscopic defects continue to expand and merge with each other, which will cause the appearance of visible cracks in the material and eventually lead to material failure and complete destruction. Crack is an important form of destruction, and the appearance of cracks often has the following characteristics: on the one hand, the appearance of macroscopic cracks can easily lead to low-stress brittle fracture of materials, i.e. although the stress is far below the yield strength, the material will suddenly break, which is related to the change of stress field distribution in the material due to the appearance of cracks, and the stress concentration in some areas due to the appearance of cracks; on the other hand, the appearance of cracks itself is a material discontinuity problem, and in the traditional mechanics theory, such as the theory of elasticity, the continuity assumption is made when the theory is established, so the traditional mechanics theory has many complex problems in studying the material discontinuity problem of cracks.

[0004] Research on the cracking of brittle materials can be divided into two categories according to different research methods: physical test and numerical simulation. Physical test can obtain some reliable data, but it needs to consume a lot of manpower, financial resources and time cost. Compared with the test method, the numerical simulation method is low in cost and high in efficiency, and has no special requirements for the shape and size of the calculation model, and the boundary conditions and load setting are more free, which breaks through the limitations of test equipment conditions and specimen preparation methods. The finite element method based on the continuous theory needs to preset the crack path and crack propagation criterion when solving the non-continuous problem, and has problems such as inaccurate description of fracture and insufficient calculation accuracy.

[0005] Peridynamics is a new numerical method to describe material properties based on nonlocal model. Its advantage is to avoid the singularity of traditional partial differential equation solution when facing discontinuity (crack) problem and the complexity of existing multi-scale algorithm. Therefore, it has a unique advantage in simulating the cracking problem of materials. As shown in Figure 1 , the material body occupying the spatial domain R is discretized into closely arranged material points, and the circular (spherical) region with the material point x as the center and δ as the radius is called the near-field domain H δ . The interaction force between any two material points x' (x' ∈ H δ ) in the near-field domain is called a bond. At any time t, the motion equation of any material point x in the material is:

[0006]

[0007] where f is the point pair force function between material points x and x'; ρ is the density of the material; b(x, t) is the external load density; u is the displacement vector field of the material point; ξ = x' - x, η = u' - u; the integral domain H x is the region occupied by the object in the reference configuration, and H x = {x' ∈ R | ‖x'-x‖≤δ}, δ is the radius of the nonlocal near-field domain of the material point.

[0008] This method is similar to the meshless method, which needs to discretize the entity model into individual particles with material properties. When analyzing the interaction force between particles, the number of material points in the near-field domain increases with the radius of the near-field domain in a cubic relationship; in the simulation of the material under load, a large number of numerical iteration steps are required to constantly solve and update the calculation parameters of each material point. The low computational efficiency of the peridynamics algorithm has always been the primary reason for its limited application. In order to take advantage of the unique advantages of peridynamics in analyzing non-continuous problems and more effectively and efficiently guide future engineering practice, it is necessary to develop peridynamics models and optimize the derived algorithms.

[0009] In summary, it is necessary to establish a prediction method for crack propagation of brittle materials to accurately predict cracks while reducing the computational scale as much as possible. SUMMARY

[0010] The purpose of the present application is to overcome the problem of the existing technology that the calculation efficiency of crack development of a material simulated by near-field dynamics under load is not ideal, propose the concept of a dual field corresponding to a near-field in the theory of near-field dynamics, split the constitutive force between particles into two forces, expand the theory of near-field dynamics to the theory of dual-field near-field dynamics, and realize the non-uniform discretization of a solid model without the theoretical limitation that the size of the near-field is a constant value.

[0011] A prediction method for crack propagation of brittle materials, comprising the following steps:

[0012] Step one: discretize a solid model of a brittle material into a series of spatial material points through multi-level division of a spatial grid, and allow non-uniform discretization of the model;

[0013] Step two: obtain structural parameters, material performance parameters and external load parameters of the brittle material, and use these parameters as inputs of a pre-constructed crack path prediction model based on dual-field near-field dynamics;

[0014] Step three: based on spatial coordinates, construct a prototype micro-brittle model constitutive relationship considering the brittleness of the material, and set the size of the near-field and the dual-field to determine the range of key force action;

[0015] Step four: determine the initial boundary conditions of the solid model of the brittle material, and apply external force in a gradual loading manner;

[0016] Step five: perform dynamic calculation by using an explicit iteration method, update physical and mechanical information such as the position, velocity and acceleration of the material points in real time, and use the critical elongation bond breaking criterion to describe damage and cracking;

[0017] Step six: output displacement results and damage results at different times, record the position and time of damage and cracking, and obtain the crack propagation path at different times.

[0018] Further, the multi-level spatial grid division method of the solid model of the brittle material in step one is as follows:

[0019] A material point in a material contains a series of related points with certain physical information in its vicinity. For the convenience of numerical calculation, the material can be discretized into a finite number of cubes with a spacing of Δx. Each cube has the same size, is arranged uniformly, and occupies a certain volume. The physical parameters of the division unit are represented by the geometric center of the small cube. The concept of dual domain is proposed and applied to the control equation. The dual domain is derived from the near-field domain and is understood as a collection of points, and the elements of the collection are points that contain target particles in their own near-field domains. Near-field dynamics based on dual domain theory allows variable near-field domain size, so the detailed area of interest can be discretized with smaller particle spacing. Non-uniform division of different sizes of near-field domains is performed in the same material. The division is performed in stages. First, the space is discretized into uniform material points with the same size. Then, the key area is divided into two levels. The coordinates of the second-level division are obtained by using the coordinates of the first-level points and the geometric relationship after the second-level division. In a two-dimensional plane, for example, one coordinate point can derive four coordinate points, which replace the original first-level coordinate. The same operation is performed for the third-level division.

[0020] The force term in the control equation after integrating the near-field domain and the dual domain is also expanded into two terms as follows:

[0021]

[0022] After discretization, the integral term is considered as the summation of a certain area in numerical implementation. The discretization of the control equation is as follows:

[0023]

[0024] It should be noted that the micro-modulus c in the composition of the force term depends on the radius of the material point x and is half of the corresponding micro-modulus in the traditional near-field dynamics theory. Wherein, V x′ is the volume of the material point x'; Hx is the near-field range of the material point x, and the meanings of other physical quantities are the same as above.

[0025] Further, the prototype micro-brittle model constitutive relation and bond breakage criterion of the material brittleness feature in step three are as follows:

[0026] The bond elongation of the material point in near-field dynamics is:

[0027] s(η,ξ)=(‖ζ‖-‖ξ‖) / ‖ξ‖.

[0028] Where s is the bond elongation; η is the relative displacement between material points x and x'; ξ is the relative position between material points x and x' in the reference configuration, the vector ζ = η + ξ, ||ξ|| and ||ζ|| are the lengths of the bond before and after deformation, respectively, and s is greater than the critical elongation s cAt this time, the pair force of the material point disappears.

[0029]

[0030] Therefore, the bond damage factor can describe the fracture of the material point, and the damage of any local material point x in the material can be defined as a local damage function at the level of the material point pair. The local damage function ranges from 0 to 1. When the local damage is 1, all interactions related to the point are eliminated, and when the local damage is 0, it means that all interactions are complete.

[0031]

[0032] Further, in step five, when performing explicit dynamics calculation, the acceleration and velocity are approximated by finite difference form of displacement. Using the finite difference approximation of the second derivative of the displacement and velocity of the material point k at the given initial time, the velocity and displacement of the next time step are obtained:

[0033]

[0034]

[0035] Where Δt is the time step, and n is the calculation step number; and and respectively, are the displacement of the material point at t+Δt and t. In order to ensure the stability of the integration process, the time micro-step Δt needs to be as small as possible.

[0036] Preferably, in actual simulation, in order to ensure the accuracy requirement, the present application adopts the volume correction coefficient and surface correction term to reduce the numerical error for the particles in the near-field domain but not all in the near-field domain, and the material points of the entity boundary of the particles themselves.

[0037] Compared with the prior art, the present application has the following obvious and substantial characteristics and advantages:

[0038] 1. The present application gives the specific implementation of the near-field dynamics model based on the dual domain, expands and perfects the near-field dynamics model and numerical solution system, and introduces the near-field dynamics to a wider field for engineering application.

[0039] 2. The present application adopts the constitutive relation of material brittleness characteristics and the bond fracture criterion, inherits the properties of natural description of structure damage and direct penetration of crack propagation of near-field dynamics, and adds the volume correction coefficient and surface correction term to effectively improve the calculation accuracy of structure deformation.

[0040] 3. This invention proposes the idea of ​​multi-level spatial division, which can efficiently and accurately solve the problems of static and dynamic deformation, elastic wave propagation and brittle fracture of structures. Attached Figure Description

[0041] Figure 1 This is a diagram of the near-field dynamics model of the present invention;

[0042] Figure 2 This is a schematic diagram illustrating the implementation of the multi-level space partitioning algorithm of the present invention;

[0043] Figure 3 The constitutive relations and local damage factors of the prototype micro-brittle model used in this invention;

[0044] Figure 4 This is a schematic diagram of the local damage function of near-field dynamics in this invention;

[0045] Figure 5 This is a schematic diagram of the modeling and solution process for the dual-domain peri-field dynamics of this invention;

[0046] Figure 6 A schematic diagram of the solid model provided in an embodiment of the present invention;

[0047] Figure 7 This is a schematic diagram illustrating the progressive application of boundary conditions to a solid model provided in an embodiment of the present invention.

[0048] Figure 8 This is a schematic diagram comparing the damage distribution during the damage cracking process based on the calculation results provided in this embodiment of the invention. Detailed Implementation

[0049] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0050] A method for predicting crack propagation in brittle materials includes the following steps:

[0051] Step 1: Discretize the brittle material solid model into a series of spatial material points through multi-level spatial mesh partitioning, allowing for non-uniform discretization of the model;

[0052] Step 2: Obtain the structural parameters, material property parameters, and external load parameters of the brittle material, and use these parameters as inputs to a pre-built near-field dynamic crack path prediction model based on the dual domain;

[0053] Step 3: Based on spatial coordinates, construct the constitutive relation of the prototype micro-brittle model that considers the brittle characteristics of the material, and set the size of the near-field domain and the dual domain to determine the range of bond force action;

[0054] Step 4: Determine the initial boundary conditions of the brittle material solid model and apply external forces using a progressive loading method;

[0055] Step 5: Perform dynamic calculations using an explicit iterative method, updating physical and mechanical information such as the position, velocity, and acceleration of material points in real time, while using the critical elongation bond breakage criterion to describe damage and cracking.

[0056] Step 6: Output the displacement and damage results at different times, and record the location and time of the damage fracture to obtain the crack propagation path at different times.

[0057] Furthermore, the method for multi-level spatial mesh generation of the brittle material solid model in step one is as follows:

[0058] In a material, a point in the material contains a series of related points with certain physical information within its surrounding region. For ease of numerical calculation, the material can be discretized into a finite number of cubes with a spacing of Δx. Each cube is of equal size, uniformly arranged, and occupies a certain volume. The physical parameters of the partitioning unit are characterized by the geometric center of the smaller cube. The concept of a dual domain is proposed and applied to the governing equations. The dual domain is derived from the near-field domain and can be understood as a set of points, the elements of which are all points whose near-field domains contain the target particle. Near-field dynamics based on dual domain theory allows for variable near-field domain sizes, thus enabling the partitioning of the region of interest with smaller particle discretization spacing, and allowing for non-uniform partitioning of near-field domains of different sizes within the same material volume. The partitioning is performed hierarchically. First, the space is discretized into uniform, equal-sized points. Then, a second-level partition is performed in the critical region. The coordinates of the second-level partition are derived by inversely using the coordinates of the first-level points and the geometric relationship after the second-level partition. For example, in a two-dimensional plane, one coordinate point can derive four coordinate points, replacing the original first-level coordinates. The third-level partition operates similarly. The partitioning idea is as follows: Figure 2 As shown.

[0059] The force term in the governing equations after integrating the near-field and dual domains is also expanded to two terms, as follows:

[0060]

[0061] After the model is discretized, the integral term is considered as a summation over a certain region in the numerical implementation. The discretization of the governing equations is given below:

[0062]

[0063] It is important to note that the micromodulus c in the force term depends on the radius of the material point x, and is half of the corresponding micromodulus in traditional peri-field dynamics theory. Wherein, V x′ H is the volume of the material point x′; Hx is the near-field range of the material point x, and the meanings of other physical quantities are the same as above.

[0064] Furthermore, the prototype micro-brittle model constitutive relation and bond fracture criterion for the brittle characteristics of the material in step three are as follows:

[0065] The bond elongation of point bonds in near-field dynamics is:

[0066] s(η,ξ)=(‖ζ‖-‖ξ‖) / ‖ξ‖.

[0067] Where s is the bond elongation; η is the relative displacement between material points x and x′; ξ is the relative position between material points x and x′ in the reference configuration, the vector ζ = η + ξ, ||ξ|| and ||ζ|| are the bond lengths before and after deformation, respectively, and s is greater than the critical elongation s c When the point-to-point forces of the material disappear, the constitutive relation of the prototype microbrittle model is as follows: Figure 3 As shown.

[0068]

[0069] Therefore, the bond damage factor can describe the fracture of material point pairs. The damage at any local material point x in the material can be defined at the level of material point pairs using a local damage function. Indicates. For example... Figure 4 As shown, local damage function The range of local damage is 0 to 1. When the local damage is 1, all interactions associated with that point are eliminated, while when the local damage is 0, all interactions are intact.

[0070]

[0071] Furthermore, in step five, during explicit dynamic calculations, acceleration and velocity are approximated using finite difference forms of displacement. By using the finite difference approximation of the second derivative of the material point k with respect to the displacement and velocity at a given initial moment, the velocity and displacement at the next time step are obtained:

[0072]

[0073]

[0074] Where Δt is the time step and n is the number of computation steps; The velocities of the material point at time t+Δt and time t are respectively. Let t+Δt and t be the displacements of the material point, respectively. To ensure the stability of the integration process, the time step Δt should be as small as possible.

[0075] Preferably, in actual simulations, in order to ensure accuracy requirements, the present invention adopts volume correction coefficients and surface correction terms for particles that are in the near field but not entirely in the near field, as well as for material points of the particles at the solid boundary, to reduce numerical errors. Specific Implementation

[0077] like Figure 5 As shown, this embodiment takes the dynamic deformation and cracking failure of a 200mm×200mm concrete slab under a combination of shear and tensile loads as an example, and uses the method of this invention for near-field dynamic modeling and analysis; Young's modulus is E=30GPa, Poisson's ratio is υ=0.2, and mass density is ρ=2265kg / m³. 3 Type I energy release rate G Ιc =110J / m 2 Type II energy release rate G IIc =10G Ic A uniformly distributed load p with a resultant force of 5 kN is applied horizontally to the upper left half and lower right half of the thin plate. Then, a velocity load of v = 10 mm / s is applied to the upper and lower boundaries of the thin plate. The specific implementation includes the following steps:

[0078] S1: By multi-level spatial meshing, the brittle material solid model is discretized into a series of spatial material points, allowing for non-uniform discretization. In this embodiment, a solid model is established with an outer contour dimension of 200mm × 200mm, and the same material parameters are assigned to the material. The solid model is as follows: Figure 6 As shown. The size of the material points meets the calculation accuracy requirements, with Δx set to 1.25mm. Specifically, the entire structural solid model is first uniformly orthogonally divided into meshless particles, resulting in 6360 material points with a spacing of 1.25mm. Based on the multi-level spatial partitioning concept, the 1320 material points in the dense area are further divided into two levels, resulting in 5280 fine-particle material points. The solid model is thus discretized into 10320 material points. If a uniform Δx = 1.25mm is used, the model is divided into 25440 material points.

[0079] S2: Obtain the structural parameters, material property parameters, and external load parameters of the brittle material, and use these parameters as input to a pre-constructed near-field dynamic crack path prediction model based on the dual domain. Determine the near-field domain H associated with the point based on the initial position coordinates of the material point and the near-field radius. x and dual domain H' x , represented as:

[0080] Hx ={x'∈R|‖x'-x‖≤δ}

[0081] H' x ={x'∈R|x∈H x'}

[0082] Where x and x′ are the coordinates of the material points in the initial configuration.

[0083]

[0084] Then, all the geometric parameters and attribute parameters from the previous steps are input into the dual-domain peri-field dynamics model as input items.

[0085] S3: Based on spatial coordinates, construct the constitutive relation of a prototype micro-brittle model considering the brittle characteristics of materials, and determine the range of bond force action by setting the sizes of the near-field and dual domains. The constitutive relation of the prototype micro-brittle model of brittle materials is as follows:

[0086]

[0087] Among them, f xx′ For the constitutive force component of bond xx′, f x′x This is the constitutive force component of the dual bond x′x; both constitutive forces are functions of their respective bond elongations s, where s is greater than the critical elongation s. c When , it indicates that the point-to-point force of the material point disappears; μ is the local damage function; η is the relative displacement between material points x and x′; ξ is the relative position between material points x and x′ in the reference configuration, and the vector ζ = η + ξ, where ||ξ|| and ||ζ|| are the lengths of the bond before and after deformation, respectively.

[0088] Based on the material point size and the characteristic scale of the loaded material structure, the near-field ranges of point association and bond association are generated. Specifically, based on the material point size Δx and the characteristic scale of the loaded material structure, the radius of the near-field range is determined to be δ = 3.015Δx.

[0089] S4: Determine the initial boundary conditions for the brittle material solid model and apply external force using a progressive loading method. In this embodiment, the initial conditions are set as follows: the initial displacement and initial velocity of all nodes are 0; a time-varying uniform velocity boundary and loading method are used, with the application time being the first 2000 time steps, Δt = 0.1 μs. Figure 7 As shown.

[0090] S5: An explicit iterative method is used for dynamic calculations, updating the physical and mechanical information such as the position, velocity, and acceleration of the material points in real time. Simultaneously, the critical elongation bond breakage criterion is used to describe damage and cracking. The simplest explicit time integral is applied, and the position, velocity, and acceleration of the material points are updated in real time through iterative calculations. Specifically, for the dynamic deformation and cracking failure problem in this embodiment, a difference algorithm using an explicit Verlet velocity scheme is used to achieve time stepping, obtaining the velocity and displacement at time t+Δt, i.e.:

[0091]

[0092]

[0093] Where Δt is the time step and n is the number of computation steps; The velocities of the material point at time t+Δt and time t are respectively. These are the displacements of the material point at time t+Δt and time t, respectively.

[0094] S6: Outputs displacement and damage results at different times, and records the location and time of fracture to obtain the crack propagation path at different times. After obtaining the explicit dynamic calculation results, the critical elongation fracture criterion is used to describe the damage and cracking, and the displacement and damage contour plot results at different times are output, such as... Figure 8 The method records the location and time when the damage approaches the critical value of 0.5, calculates the time, location and speed of cracking, and verifies the accuracy of the method with experiments or other numerical methods. It also verifies the efficiency of the dual-domain peri-field dynamics model with a uniformly partitioned peri-field dynamics model. In practice, the efficiency is improved by 119.49% in this problem, thus enabling the prediction and analysis of crack propagation in brittle materials.

[0095] The embodiments of the present invention have been described above in conjunction with the accompanying drawings. However, the present invention is not limited to the above embodiments. Various changes can be made according to the purpose of the invention. Any changes, modifications, substitutions, combinations or simplifications made based on the spirit and principle of the technical solution of the present invention shall be equivalent substitutions. As long as they meet the purpose of the invention and do not deviate from the technical principle and inventive concept of the present invention, they shall fall within the protection scope of the present invention.

Claims

1. A method for predicting crack propagation in brittle materials, characterized in that, Includes the following steps: (1) By using the multi-level spatial grid partitioning method, the brittle material solid model is discretized into a series of spatial material points, allowing for non-uniform discretization of the model; (2) Obtain the structural parameters, material property parameters, and external load parameters of brittle materials, and use them as inputs to a pre-built near-field dynamic cracking path prediction model based on dual domain; (3) Based on spatial coordinates, construct the constitutive relation of the prototype micro-brittle model that considers the brittle characteristics of the material, set the size of the near field and dual field, and determine the range of bond force action; (4) Determine the initial boundary conditions of the brittle material solid model and apply external forces by progressive loading; (5) The explicit iterative method is used to perform dynamic calculations and update the physical and mechanical information of the material point position, velocity, and acceleration in real time. At the same time, the critical elongation bond breaking criterion is used to describe the damage and cracking of the material. (6) Output the displacement and damage results at different times, and record the location and time of damage and fracture to obtain the crack propagation path at different times; The discretization of the brittle material entity model in step (1) is as follows: a certain material point in the material contains a series of related material points with certain physical information in its vicinity. In order to facilitate numerical calculation, the material is discretized into a finite number of cubes with a spacing of Δx. Each cube is of equal size, uniformly arranged and occupies a certain volume. The physical parameters of the dividing unit are characterized by the geometric center of the small cube. The concept of dual domain is proposed and applied to the governing equation. The dual domain is derived from the near field domain and can be understood as a set of points. The elements of the set are all points in their own near field domain that contain the target particle. The near field dynamics based on the dual domain theory allows for variable near field domain size. Therefore, the detailed region of interest is divided with a smaller particle discretization spacing. Non-uniform division of near field domains of different sizes is carried out in the same material body. The division method is to proceed step by step. First, the space is discretized into uniform material points of equal size, and then a second-level division is carried out in the key region. The force term in the governing equations after integrating the near-field and dual domains is also expanded to two terms, as follows: After the model is discretized, the integral term is considered as a summation over a certain region in the numerical implementation. The discretization of the governing equations is given below: It should be noted that the micromodulus c in the force term depends on the radius of the material point x, and is half of the corresponding micromodulus in the traditional peri-field dynamics theory; where Vx′ is the volume of the material point x′; and Hx is the peri-field range of the material point x.

2. The method for predicting crack propagation in brittle materials according to claim 1, characterized in that, The prototype micro-brittle model constitutive relation and bond breakage criterion of the brittle characteristics in step (3) are as follows: The bond elongation of point bonds in near-field dynamics is: s(η,ξ)=(‖ζ‖-‖ξ‖) / ‖ξ‖. Where s is the bond elongation; η is the relative displacement between material points x and x′; ξ is the relative position between material points x and x′ in the reference configuration, the vector ζ = η + ξ, ||ξ|| and ||ζ|| are the bond lengths before and after deformation, respectively, and s is greater than the critical elongation s c When this occurs, it indicates that the point-to-point force between the material points disappears; Therefore, the bond damage factor describes the fracture of material point pairs. The damage at any local material point x in the material is defined at the level of material point pairs using a local damage function. express: Local damage function The range of local damage is indicated as 0 to 1.

3. The method for predicting crack propagation in brittle materials according to claim 1, characterized in that, In the explicit dynamics calculation in step (5), acceleration and velocity are approximated using the finite difference form of displacement; by using the finite difference approximation of the second derivative of the material point k with respect to the displacement and velocity at a given initial moment, the velocity and displacement at the next time step are obtained: Where Δt is the time step and n is the number of computation steps; The velocities of the material point at time t+Δt and time t are respectively. Let t+Δt and t be the displacements of the material point, respectively. To ensure the stability of the integration process, the time step Δt should be as small as possible.

4. The method for predicting crack propagation in brittle materials according to claim 3, characterized in that, In explicit dynamic calculations, to ensure accuracy, volume correction coefficients and surface correction terms are applied to reduce numerical errors for particles that are in the near field but not entirely in the near field, as well as for material points of the particles at the solid boundary.

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