A bond-based peridynamic constitutive method for quasi-brittle materials

Abstract

The application provides a bond-based near-field dynamic constitutive method suitable for quasi-brittle material mechanical behavior, the model considers the characteristics that the tensile and compressive stress-strain curves of the quasi-brittle material have nonlinear hardening and strain softening stages, the Prototype Microelastic Brittle (PMB) material model of the traditional bond-based near-field dynamics is improved, and the construction of the constitutive model is completed through three parts of the construction of the constitutive force function, the determination of the parameters, and the numerical simulation and model verification. The method makes up for the defects that the traditional bond-based near-field dynamics can only reflect the linear elastic stage of the quasi-brittle material, and provides a new theoretical method for establishing the damage model of the quasi-brittle material.

Description

Technical Field

[0001] This invention relates to the field of materials modeling technology, specifically to a bond-based near-field dynamic constitutive method applicable to the mechanical behavior of quasi-brittle materials. Background Technology

[0002] Quasi-brittle materials are those that fail under external forces with only minor deformation, such as concrete, rock, ice, tough ceramics, and various fiber or particulate composites. At small scales, quasi-brittle materials obey plasticity theory (or strength theory), characterized by material strength or yield strength; at large scales, they obey linear elastic fracture mechanics, characterized by fracture energy. The failure of quasi-brittle materials refers to the process under load where microcracks initiate and propagate within the material, leading to macroscopic cracks and ultimately, gradual deterioration and failure.

[0003] To study the failure of quasi-brittle materials, various numerical methods have been developed, such as the finite element method (FEM), the cohesive zone element method (CZM), and the extended finite element method (XFEM). The FEM requires prior knowledge of the crack's location and size, which is often unavailable in practical engineering applications. To overcome this limitation, the CZM utilizes the cohesion law to allow arbitrary crack propagation between adjacent elements; however, this method places high demands on the mesh, sometimes requiring re-meshing. The XFEM eliminates mesh dependence by introducing a local expansion function with degrees of freedom into the element function of the traditional FEM, and does not require mesh re-meshing when simulating crack growth. However, it has limitations in simulating the propagation of multiple cracks with complex morphologies. All of these methods use traditional continuum mechanics theory to study crack initiation and propagation, describing the deformation and internal forces of the object through partial differential equations. However, the presence of cracks leads to spatial discontinuities, causing singularities in the partial differential equations and rendering them absent from the governing equations of traditional continuum mechanics. Peripheral dynamics, based on the concept of nonlocal interactions, uses integral equations to describe the information of material points in the spatial domain. It can realize crack initiation and propagation, as well as the interaction of multiple cracks, without the need for additional fracture criteria. Peripheral dynamics includes bond-based periphery dynamics, ordinary-state-based periphery dynamics, and non-ordinary-state-based periphery dynamics. Although bond-based periphery dynamics is limited by Poisson's ratio, restricting its application, its constitutive relations are simple, and the forces between material points are only related to the deformation of a single bond, making it more suitable for describing the deformation of quasi-brittle materials.

[0004] The differences compared to existing applications are as follows:

[0005] In comparison with the technology of patent CN110414167A "Method for Modeling Near-Field Dynamic Constitutive Force Functions of Quasi-brittle Materials"

[0006] 1. In patent CN110414167A, the constitutive force function is obtained by mathematically fitting the damage variable, but no specific fitting method is given. In contrast, this invention gives a specific method for constructing the constitutive force function in the form of an analytical expression.

[0007] 2. Patent CN110414167A states that a bond breaks when it exceeds a certain fixed value during compression. However, this invention believes that bond breakage during compression is caused by cumulative damage and does not set a fixed value, thus avoiding abrupt changes in bond state.

[0008] 3. Patent CN110414167A gives a specific analytical expression of the constitutive force function in the form of a piecewise function, but does not give a specific method for determining the relevant critical elongation. In contrast, after constructing the constitutive force function, this invention clearly defines the elongation of each stage in the linear and nonlinear phases as well as the damage function representing the bond state.

[0009] 4. Patent CN110414167A proposes a near-field dynamic constitutive force function suitable for quasi-brittle materials, but it has not been verified by numerical simulation. In contrast, this invention verifies the reliability of the constitutive method proposed by numerically simulating the crack propagation trend of pre-cracked rocks and comparing it with experimental results.

[0010] In comparison with the technology of patent CN116895351A "BPD Finite Element Coupling Method and System for Predicting Failure of Quasi-brittle Materials"

[0011] 1. Patent CN116895351A mainly focuses on the coupling of near-field dynamics and finite element method, while this invention focuses on the constitutive method construction of quasi-brittle materials.

[0012] 2. In patent CN116895351A, the bond damage model is expressed as a quadratic function in each stage. However, this invention uses logarithmic and exponential functions in the attenuation stages of tension and compression, respectively. At the same time, the bond damage model is a continuous function, which reflects the process of the bond from integrity to cumulative damage until fracture.

[0013] 3. In patent CN116895351A, the elongation at break is determined by the ratio to the critical elongation, which involves two unknowns. However, the present invention is derived by calculating the fracture energy, which eliminates the unknowns and reduces the time required for model calibration. Summary of the Invention

[0014] To address the aforementioned technical problems, this invention proposes a bond-based near-field dynamic constitutive method applicable to the mechanical behavior of quasi-brittle materials, thereby overcoming the shortcomings of the traditional PMB model, which can only reflect elastic mechanical behavior.

[0015] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0016] A constitutive method for bond-based near-field dynamics applicable to the mechanical behavior of quasi-brittle materials is characterized by the following specific steps;

[0017] 1) Analyze the mechanical behavior of quasi-brittle materials;

[0018] The mechanical behavior of quasi-brittle materials can be divided into three stages: linear elastic stage, nonlinear strengthening stage, and strain softening stage. In the linear elastic stage, there is no crack initiation and no damage. In the nonlinear strengthening stage, microcracks initiate and propagate stably. In the strain softening stage, macrocracks begin to form and damage becomes localized.

[0019] 2) Construction of constitutive force functions;

[0020] Assuming that the variation trend of the constitutive force function in the near-field dynamics is consistent with the stress-strain curve, and referring to the analytical expression of the full stress-strain curve, the constitutive force function adopts a piecewise function representation method, which makes it more flexible while controlling unknown parameters.

[0021] 3) Parameter determination;

[0022] This includes determining the critical elongation at each stage, the fracture energy, the scalar function representing the bond state, and the material point damage function;

[0023] 4) Numerical simulation and model validation;

[0024] During numerical simulation, the model dimensions are determined, and material mechanical parameters, including mass, density, compressive strength, tensile strength, and elastic modulus, are input. The distance between material points and the neighborhood radius in the near-field dynamics are determined, the coordinates of the material points are generated, the time step is determined, points within the neighborhood range of each material point are searched and initialized, initial boundary conditions are applied to the numerical model, and the force, displacement, velocity, and acceleration of each material point are iteratively calculated according to the proposed constitutive model. The calculation results are output and a damage contour map is plotted. The experimental results are compared and analyzed with the numerical simulation results to verify the reliability of the model.

[0025] As a further improvement of the present invention, in step 2) the stretching stage, when the bond is stretched to the tensile elastic elongation s et Previously, it maintained linear growth, and the analytical expression of the function was consistent with the PMB model, with the tensile elastic elongation s et and tensile ultimate elongation s tThe nonlinear strengthening stage of the corresponding stress-strain curve causes the near-field force to increase in the form of a quadratic function, at the tensile limit elongation s. t The near-field force reaches its maximum value, and its analytical expression is f = As. 2 +Bs+c. Since the constitutive force function is a continuous function, substituting in two critical conditions:

[0026]

[0027] The value of the unknown that satisfies the conditions can be found:

[0028]

[0029] The constitutive force function expression corresponding to the nonlinear strengthening part is obtained as follows: The constitutive force function is attenuated using a logarithmic function in the strain softening stage corresponding to the stress-strain curve. When the elongation reaches s0, the near-field force decays to zero, and the bond breaks. The analytical expression is set as f = Aln(Bs + C). Substituting the two critical conditions:

[0030]

[0031] Find the value of the unknown that satisfies the conditions:

[0032]

[0033] The analytical expression obtained by rearranging is: The constitutive force function during the stretching stage is written as:

[0034]

[0035] For the compression stage of this model, the constitutive force functions of the elastic deformation and nonlinear strengthening portions of the compressive stress-strain curve remain consistent with those of the tensile stage. Since the compressive strength of quasi-brittle materials is much greater than their tensile strength, the bonds maintain a certain load-bearing capacity under compression; that is, the near-field force does not decay to zero with increasing elongation. Therefore, an exponential function is used for the decay function. Assume the analytical expression is f = Ase. Bs+C Similarly to the tensile part, to ensure that the constitutive force function is a continuous function, we substitute the critical condition. The values ​​of the unknowns that satisfy the conditions are as follows:

[0036]

[0037] The constitutive force function of the compression part is expressed as follows:

[0038]

[0039] As a further improvement of the present invention, in step 3) parameter determination, the constitutive model applicable to quasi-brittle materials involves five critical elongation rates, namely the compressive limit elongation s. c , compressive elongation s ec , tensile elastic elongation s et , tensile ultimate elongation s t The tensile elongation at break s0, where the ultimate tensile elongation s t and compressive limit elongation s c Each is determined by tensile strength σ t and compressive strength σ c The ratio to the elastic modulus E is determined and expressed as:

[0040]

[0041] Stretch elongation s et and compressive elongation s ec Determined by the ratio of the limiting elongation to the limiting elongation:

[0042] s et =αs t ,s ec =βs c

[0043] The values ​​of α and β were adjusted by comparing experimental and simulation results. It is worth noting that the values ​​of α and β are between 0 and 1.

[0044] s0 is the tensile elongation at fracture, which is determined by the fracture energy. Based on macroscopic characteristics, the strain softening stage is the main stage for unstable crack propagation and the formation of macroscopic damage. Therefore, when determining the work done by a single bond fracture, only the nonlinear decay stage is considered, i.e., the tensile ultimate elongation s. t The work done by a single bond during fracture in the region between the tensile elongation at break (s0) and the tensile fracture elongation (s0) is expressed as follows:

[0045]

[0046] Thus, the fracture energy G in the three-dimensional case is obtained. f

[0047]

[0048] Similarly, the fracture energy G in the two-dimensional case f

[0049]

[0050] Since the elongation is typically on the order of 10 -3 ~10 -4 ,so This term is approximately equal to 1. The tensile elongation at break s0 under different conditions is summarized as follows:

[0051]

[0052] Combined with tensile elastic elongation s et and compressive elongation s ec The definition reveals that the value of tensile elongation at break s0 is only related to the near-field size δ and the fracture energy G. f Elastic modulus E, tensile strength σ t The parameters are related to the scaling parameters. Among them, the only unknown parameter is the scaling parameter. The other parameters are material property parameters and near-field dynamics numerical simulation parameters, which greatly saves the time for calibrating the model in subsequent simulations.

[0053] The scalar function μ(t,ξ) representing the bond state is defined as a continuous function varying from 0 to 1. When the bond elongation s exceeds the tensile fracture elongation s0, the bond breaks. This reflects the process of the bond from intact to damaged and finally broken, avoiding the phenomenon of sudden bond breakage in the traditional PMB model. Its expression is as follows:

[0054]

[0055] Function representing material point damage Defined as the ratio of the number of broken bonds to the total number of bonds, the expression is as follows:

[0056]

[0057] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0058] The constitutive model proposed in this paper can accurately describe the nonlinear mechanical behavior of quasi-brittle materials, overcoming the limitation of the traditional PMB model, which can only reflect the linear elastic characteristics of quasi-brittle materials. By deriving the tensile elongation at fracture energy, the introduction of unknown parameters is reduced, significantly shortening the model calibration time. Attached Figure Description

[0059] Figure 1 This is a simplified flowchart of the invention;

[0060] Figure 2 This is a schematic diagram of the near-field dynamics of two material points before and after deformation;

[0061] Figure 3 This is a schematic diagram of the constitutive force function of PMB material;

[0062] Figure 4 It is a stress-strain curve diagram of a quasi-brittle material during the tensile stage;

[0063] Figure 5 This is a schematic diagram of an improved constitutive model for quasi-brittle materials;

[0064] Figure 6 This is a flowchart of the near-field dynamics numerical simulation;

[0065] Figure 7 This is a schematic diagram of a prefabricated single-crack model;

[0066] Figure 8 It is a diagram of the crack propagation of a prefabricated 60° oblique crack rock;

[0067] Figure 9 It is a diagram of the propagation mechanism of precast single-cracked rock;

[0068] Figure 10 This is a comparison of the compression propagation results of precast inclined crack rock samples at different angles. Detailed Implementation

[0069] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments:

[0070] As a preferred embodiment of the present invention, the present invention provides a bond-based near-field dynamic constitutive method applicable to the mechanical behavior of quasi-brittle materials, such as... Figure 1 As shown, it includes the following steps:

[0071] Step 1: Analysis of the mechanical behavior of quasi-brittle materials

[0072] Common quasi-brittle materials exhibit crack initiation and propagation under load, and their stress-strain curves, whether under tension or compression, display both linear and nonlinear segments. Taking the tensile stage as an example, its stress-strain curve is as follows: Figure 4 As shown. The mechanical behavior of quasi-brittle materials can be divided into three stages: (1) OA: linear elastic stage, in which no cracks initiate and no damage occurs; (2) AB: nonlinear strengthening stage, in which microcracks initiate and propagate stably; (3) BC: strain softening stage, in which macroscopic cracks begin to form and damage becomes localized. The traditional PMB model (such as...) Figure 3 The figure (shown) describes a simple linear relationship between near-field force and elongation, which only reflects the characteristics of the linear elastic stage and cannot accurately reflect the mechanical properties of quasi-brittle materials.

[0073] Step 2: Constructing Constitutive Force Functions

[0074] like Figure 2 As shown, in the near-field dynamics theory of bond groups, the macroscopic continuum is divided into a large number of matter points, and any matter point x exists only in its finite "near-field domain H". x"Within this near-field region, there is interaction with other matter points x' through bonds, resulting in an interaction force f, while there is no interaction with matter points outside this near-field region. The near-field dynamics equations can be expressed as:"

[0075]

[0076] In the formula: H x ={x'||x'-x|≤δ} is the near-field region, representing a circular region centered on particle x with radius δ. b represents the body force density field, and f(x'-x,u'-u,t) is the paired force density function, representing the force vector per unit volume squared exerted by particle x' on particle x at time t. Figure 3 As shown, traditional bond-based near-field dynamics typically employs prototype microelastic brittle (PMB) material, and its expression is as follows:

[0077]

[0078] In the formula, s=(ξ+η|-|ξ|) / |ξ| is the elongation of the bond, and c is a constant representing the micromodulus or bond stiffness. Its value is derived by equating the strain energy density of the PD model at material point x with the strain energy density of traditional continuum mechanics. The values ​​of the micromodulus under different conditions are as follows:

[0079]

[0080] In the formula, E is the elastic modulus, h is the thickness, and A is the cross-sectional area. The derivation of the above formula yields a fixed value for the Poisson's ratio in the near-field dynamics of the bond basis, namely ν = 1 / 3 in the case of plane stress and plane strain, and ν = 1 / 4 in the three-dimensional case.

[0081] The traditional PMB model describes a simple linear relationship between near-field force and elongation. As long as the elongation does not reach the critical elongation, the near-field force and elongation maintain a linear relationship; however, once the elongation reaches the critical elongation, the near-field force suddenly drops to zero. This model is often used for modeling brittle-elastic materials, reflecting only the characteristics of the linear elastic stage. However, for brittle materials such as rock and concrete, this model cannot accurately reflect their mechanical properties.

[0082] Considering that the constitutive force function in peridynamics essentially describes the relationship between force and deformation, this paper assumes that the variation trend of the constitutive force function in peridynamics is consistent with the stress-strain curve. The traditional PMB model is improved to make it suitable for describing the mechanical behavior of quasi-brittle materials. Referring to the analytical expression of the full stress-strain curve, the constitutive force function adopts a piecewise function representation, making it more flexible while controlling unknown parameters.

[0083] Taking the stretching stage as an example, when the bond is stretched to the tensile elastic elongation s et Previously, it maintained linear growth, and the analytical expression of the function was consistent with the PMB model, with the tensile elastic elongation s et and tensile ultimate elongation s t The nonlinear strengthening stage of the corresponding stress-strain curve causes the near-field force to increase in the form of a quadratic function, at the tensile limit elongation s. t The near-field force reaches its maximum value, and its analytical expression is f = As. 2 +Bs+c. Since the constitutive force function is a continuous function, substituting in two critical conditions:

[0084]

[0085] The value of the unknown that satisfies the conditions can be found:

[0086]

[0087] After simplification, the analytical expression of the constitutive force function corresponding to the nonlinear strengthening part is obtained as follows: Considering that bond stretching is the primary cause of material failure, the constitutive force function is attenuated using a logarithmic function during the strain softening stage of the stress-strain curve. When the elongation reaches s0, the near-field force decays to zero, and the bond breaks. The analytical expression is set as f = Aln(Bs + C), and substituting the two critical conditions:

[0088]

[0089] The value of the unknown that satisfies the conditions can be found:

[0090]

[0091] The analytical expression can be obtained by rearranging the equation as follows: The constitutive force function during the stretching stage can be written as:

[0092]

[0093] For the compression stage of this model, the constitutive force functions of the elastic deformation and nonlinear strengthening portions of the compressive stress-strain curve remain consistent with those of the tensile stage. Since the compressive strength of quasi-brittle materials is much greater than their tensile strength, the bonds maintain a certain load-bearing capacity under compression; that is, the near-field force does not decay to zero with increasing elongation. Therefore, an exponential function is used for the decay function. Assume the analytical expression is f = Ase. Bs+C Similarly to the tensile part, to ensure that the constitutive force function is a continuous function, we substitute the critical condition. The values ​​of the unknowns that satisfy the conditions are as follows:

[0094]

[0095] The constitutive force function of the compression portion can be expressed as follows:

[0096]

[0097] The improved constitutive force function diagram is shown below. Figure 5 As shown.

[0098] Step 3: Parameter Determination

[0099] The constitutive model constructed for quasi-brittle materials involves five critical elongations, namely the compressive limit elongation s. c , compressive elongation s ec , tensile elastic elongation s et , tensile ultimate elongation s t The tensile elongation at break is s0. Where the ultimate tensile elongation is s... t and compressive limit elongation s c Each is determined by tensile strength σ t and compressive strength σ c The ratio of the elastic modulus E to the elastic modulus can be determined and expressed as:

[0100]

[0101] Stretch elongation s et and compressive elongation s ec Determined by the ratio of the limiting elongation to the limiting elongation:

[0102] s et =αs t ,s ec =βs c

[0103] The values ​​of α and β can be adjusted by comparing experimental and simulation results. It is worth noting that the values ​​of α and β are between 0 and 1.

[0104] s0 represents the tensile elongation at fracture, its value determined by the fracture energy. Based on macroscopic characteristics, the strain softening stage is the primary stage for unstable crack propagation and the formation of macroscopic damage. Therefore, when determining the work done in single-bond fracture, only the nonlinear decay stage, i.e., the tensile ultimate elongation s0, is considered. t The region between the tensile elongation at break (s0) and the tensile fracture elongation (s0). The work done by a single bond fracture is expressed as follows:

[0105]

[0106] Thus, the fracture energy G in the three-dimensional case is obtained. f

[0107]

[0108] Similarly, the fracture energy G in the two-dimensional case f

[0109]

[0110] Since the elongation is typically on the order of 10 -3 ~10 -4 ,so This term is approximately equal to 1. The tensile elongation at break s0 under different conditions can be summarized as follows:

[0111]

[0112] Combined with tensile elastic elongation s et and compressive elongation s ec Given the definition of and the value of the micromodulus c, the tensile elongation at break s0 can be further expressed as:

[0113]

[0114] It can be observed that the value of the tensile elongation at break s0 is only related to the near-field size δ and the fracture energy G. f Elastic modulus E, tensile strength σ t The parameters are related to the scaling parameters. Among them, the only unknown parameter is the scaling parameter. The other parameters are material property parameters and near-field dynamics numerical simulation parameters, which greatly saves the time for calibrating the model in subsequent simulations.

[0115] The scalar function μ(t,ξ) representing the bond state is defined as a continuous function varying from 0 to 1. When the bond elongation s exceeds the tensile fracture elongation s0, the bond fractures, reflecting the process of the bond from intact to damaged and finally fractured. This avoids the sudden bond fracture phenomenon in the traditional PMB model. Its expression is as follows:

[0116]

[0117] Function representing material point damage Defined as the ratio of the number of broken bonds to the total number of bonds, the expression is as follows:

[0118]

[0119] Step 3: Numerical Simulation and Model Validation

[0120] During numerical simulation, the model dimensions are determined, and material mechanical parameters, including mass, density, compressive strength, tensile strength, and elastic modulus, are input. The spacing between material points and the neighborhood radius in the near-field dynamics are determined, and the coordinates of the material points are generated. The time step is determined, and points within the neighborhood of each material point are searched and initialized. Initial boundary conditions are applied to the numerical model, and the force, displacement, velocity, and acceleration of each material point are iteratively calculated according to the proposed constitutive model. The calculation results are output, and a damage contour map is plotted. The experimental procedure is as follows: Figure 6 As shown, the experimental results are compared and analyzed with the numerical simulation results to verify the reliability of the model.

[0121] Numerical simulation of uniaxial compression of pre-fabricated oblique crack rock specimens, such as... Figure 7 As shown, the model has geometric dimensions of 50mm × 100mm, a crack length of 28mm, and is subjected to displacement loading at a rate of 0.1m / s at the upper loading boundary, while the lower loading boundary remains fixed. Since the crack has a relatively small impact on thickness, it is simplified as a plane stress problem. The elastic modulus is 4.71GPa, and the density is 1810kg / m³. 3 The tensile strength is 1.5 MPa, and the compressive strength is 12.57 MPa. The time step is 4.0 × 10⁻⁶. -7 The particle spacing is 0.0005 m, the near-field radius is δ = 3Δx, the proportionality coefficient of elastic elongation is 0.6, and the fracture energy of the rock sample is 30 J / m. 2 . Figure 8 Damage contour plots of the crack propagation process in a rock specimen with a crack inclination angle of 60° are presented. The aeroid crack initially originates at the tip of the pre-existing crack. As deformation intensifies, the aeroid crack propagates in a curved path towards the end of the specimen. Figure 8 (b) The propagation direction is consistent with the direction of the maximum principal stress, and the airfoil crack is a tension crack. At a time step of 7000 steps ( Figure 8 (c) The anti-airfoil crack initiates in the opposite direction to the airfoil crack, while the secondary coplanar shear crack propagates in a direction nearly coplanar with the pre-existing crack. Both the anti-airfoil crack and the secondary coplanar shear crack initiate from both ends of the pre-existing crack, and both are shear cracks. As the load continues to be applied, the crack further propagates, reaching a point when the time step reaches 8000 steps. Figure 8 (d) The crack propagates to its final form, with pre-existing crack tips appearing in the most severely damaged area. The crack propagation mechanism is as follows: Figure 9 As shown.

[0122] The crack propagation trends under pre-fabricated oblique cracks at 30°, 45°, and 60° were simulated, and the final failure results were compared with experimental results. The comparison results are as follows: Figure 10As shown, the failure results of the rock specimens simulated using the improved quasi-brittle material constitutive model show a certain degree of similarity to the experimental results in terms of crack initiation location and propagation direction. Pre-cracks with different inclination angles result in different penetration patterns in the rock specimens. When the pre-crack inclination angle is low, the wing-shaped cracks gradually develop incompletely and merge with secondary coplanar shear cracks, thus forming macroscopic failure. However, regardless of the inclination angle of the pre-cracks, the most severely damaged areas of the rock specimens appear at both ends of the crack, consistent with the experimental results. There are also some differences between the near-field dynamic numerical simulation results and the experimental results, possibly due to the strong heterogeneity of the rock; under the same loading conditions, multiple experiments may also yield different results.

[0123] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any other way. Any modifications or equivalent changes made based on the technical essence of the present invention shall still fall within the scope of protection claimed by the present invention.

Claims

1. A bond-based near-field dynamic constitutive method applicable to the mechanical behavior of quasi-brittle materials, characterized in that, The specific steps are as follows; 1) Analyze the mechanical behavior of quasi-brittle materials; The mechanical behavior of quasi-brittle materials can be divided into three stages: linear elastic stage, nonlinear strengthening stage, and strain softening stage. In the linear elastic stage, there is no crack initiation and no damage. In the nonlinear strengthening stage, microcracks initiate and propagate stably. In the strain softening stage, macrocracks begin to form and damage becomes localized. 2) Construction of constitutive force functions; Assuming that the variation trend of the constitutive force function in the near-field dynamics is consistent with the stress-strain curve, and referring to the analytical expression of the full stress-strain curve, the constitutive force function adopts a piecewise function representation method, which makes it more flexible while controlling unknown parameters. Step 2) Tensioning stage: When the bond is stretched to its tensile elastic elongation... Previously, it maintained linear growth, and the functional expression was consistent with the PMB model, representing the tensile elastic elongation. and tensile ultimate elongation The nonlinear strengthening stage corresponding to the stress-strain curve causes the near-field force to increase in the form of a quadratic function, at the tensile limit elongation. The near-field force reaches its maximum value, and the analytical expression is set as follows: Since the constitutive force function is a continuous function, we can substitute two critical conditions: ； The value of the unknown that satisfies the conditions can be found: ； The constitutive force function expression corresponding to the nonlinear strengthening part is obtained as follows: Corresponding to the strain softening stage of the stress-strain curve, the constitutive force function is attenuated using a logarithmic function, and this decays when the elongation reaches... At this point, the near-field force decays to zero, the bond breaks, and the analytical expression is set as follows: Substitute the two critical conditions: ； Find the value of the unknown that satisfies the conditions: ； The analytical expression obtained by rearranging is: The constitutive force function during the stretching stage is written as: ； For the compression stage of this model, the constitutive force function corresponding to the elastic deformation and nonlinear strengthening parts of the compressive stress-strain curve remains consistent with that of the tensile stage. Since the compressive strength of quasi-brittle materials is much greater than their tensile strength, the bonds maintain a certain load-bearing capacity under compression; that is, the near-field force does not decay to zero with increasing elongation. Therefore, an exponential function is used for the decay function, assuming the analytical expression is... Similarly to the tensile part, to ensure that the constitutive force function is a continuous function, we substitute the critical condition. The values ​​of the unknowns that satisfy the conditions are as follows: ； For compression limit elongation, To express the compressible elastic elongation, the constitutive force function of the compressed portion is as follows: ； 3) Parameter determination; This includes determining the critical elongation at each stage, the fracture energy, the scalar function representing the bond state, and the material point damage function; 4) Numerical simulation and model validation; During numerical simulation, the model dimensions are determined, and material mechanical parameters, including mass, density, compressive strength, tensile strength, and elastic modulus, are input. The distance between material points and the neighborhood radius in the near-field dynamics are determined, the coordinates of the material points are generated, the time step is determined, points within the neighborhood range of each material point are searched and initialized, initial boundary conditions are applied to the numerical model, and the force, displacement, velocity, and acceleration of each material point are iteratively calculated according to the proposed constitutive model. The calculation results are output and a damage contour map is plotted. The experimental results are compared and analyzed with the numerical simulation results to verify the reliability of the model.

2. The bond-based near-field dynamic constitutive method for the mechanical behavior of quasi-brittle materials according to claim 1, characterized in that, In step 3) parameter determination, the constitutive model constructed for quasi-brittle materials involves five critical elongations, namely the compressive limit elongation. Compression elongation , tensile elastic elongation Elongation at the limit of tension Elongation at break of the bond Among them, the ultimate elongation at break and compression limit elongation Based on tensile strength and compressive strength With elastic modulus The ratio is determined and expressed as: ； Tensile elastic elongation and compressive elongation Determined by the ratio of the limiting elongation to the limiting elongation: ； in, , The value of is adjusted by comparing and simulating the experimental results. It is worth noting that... and The value is between 0 and 1; The tensile elongation at fracture is determined by the fracture energy. Based on macroscopic characteristics, the strain softening stage is the main stage of unstable crack propagation and macroscopic damage formation. Therefore, when determining the work done by a single bond fracture, only the nonlinear decay stage is considered, i.e., the tensile ultimate elongation. and tensile elongation at break The work done by breaking a single bond in the region between is expressed as follows: ； Thus, the fracture energy in the three-dimensional case is obtained. ; ； Similarly, fracture energy in the two-dimensional case ; ； Since the order of magnitude of elongation is usually... ,so This term is approximately equal to 1, representing the elongation at break under different conditions. Sorted as: ； Combined with tensile elastic elongation and compressive elongation The definition of tensile elongation at break was found The value is only related to the near-field range size. fracture energy Elastic modulus ,tensile strength The parameters are related to the scaling parameters. Among them, the only unknown parameter is the scaling parameter. The other parameters are material property parameters and near-field dynamics numerical simulation parameters, which greatly saves the time for calibrating the model in subsequent simulations. Scalar value function representing key state Defined as a continuous function varying from 0 to 1, when the bond elongation... Exceeding the tensile elongation at break When the bond breaks, it reflects the process from a intact bond to damage and finally breakage, avoiding the phenomenon of sudden bond breakage in the traditional PMB model. Its expression is as follows: ； Function representing material point damage Defined as the ratio of the number of broken bonds to the total number of bonds, the expression is as follows: 。

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