A method for establishing a microscopic geometric model of particulate-reinforced composite materials

Abstract

This invention relates to a method for establishing a microscopic geometric model of particle-reinforced composite materials, comprising: determining the three-dimensional spatial range of particle movement; determining the target volume and volume growth rate of different particles; determining the number of different particles within the three-dimensional spatial range; randomly selecting the initial position of each particle and determining the initial volume of each particle; using the LJ potential function to determine the interaction force between particles at the current moment; determining the current velocity and direction of each particle based on the interaction force between particles at the current moment; obtaining the volume of each particle at the next moment based on the current volume and volume growth rate of each particle; obtaining the position of each particle at the next moment based on the current position and the current velocity and direction of each particle; repeating the above steps until each particle is uniformly distributed within the three-dimensional spatial range and the target volume is reached. This invention can rapidly construct a geometric model that reflects the small reinforcing components and high particle filling ratio of the material.

Description

Technical Field

[0001] This invention belongs to the field of composite material technology, and specifically relates to a method for establishing a microscopic geometric model of particle-reinforced composite materials. Background Technology

[0002] To study the local properties and microstructure of high particle-filling ratio materials using numerical simulation, it is first necessary to construct a representative volumetric unit model of isotropic random microstructures. Currently, there are two commonly used methods for mesoscale modeling of particle-reinforced composite materials: sequential algorithms and concurrent algorithms. The main characteristic of sequential algorithms is that they gradually add non-overlapping particles randomly to the computational domain until a certain particle volume filling fraction is achieved. In geometric models obtained using this method, the particle volume is typically below 40% due to the fixed particle positions (Widom, 1966). The main characteristic of concurrent algorithms is that they set all particles to an initial state and then add them all to the computational domain, continuously rearranging them using an algorithm until the desired particle volume filling fraction is achieved. Common concurrent algorithms include Monte Carlo random methods and molecular dynamics methods. These methods produce models with particle filling ratios as high as 70% (Rintoul and Torquato, 1996), but particle overlap often occurs during modeling, requiring repeated iterations and resulting in a high number of calculations. Summary of the Invention

[0003] The purpose of this invention is to overcome the above-mentioned defects and provide a method for establishing a microscopic geometric model of particle-reinforced composite materials. This method solves the technical problem that existing microscopic modeling methods for particle-reinforced composite materials cannot achieve rapid modeling of composite materials with high particle filling ratios. This invention can quickly construct a geometric model that reflects the small reinforcing components of the material and has a high particle filling ratio (>50%).

[0004] To achieve the above-mentioned objectives, the present invention provides the following technical solution:

[0005] This invention relates to a method for establishing a microscopic geometric model of particle-reinforced composite materials. Specifically, it relates to the ability to construct a geometric model that reflects the minute reinforcing components and has a high particle filling ratio (>50%) in the study of the microscopic properties of particle-reinforced composite materials. By importing the constructed geometric model into finite element software and performing simulation analysis, the mechanical properties of the particle-reinforced composite material, the damage mode of the particle-matrix interface, etc., can be obtained, which helps to reveal the failure mechanism of particle-reinforced composite materials under environmental conditions. This method is applicable to the construction of microscopic geometric models of particle-reinforced composite materials of various material types, such as composite solid propellant models and concrete models, and belongs to the field of composite material technology.

[0006] A method for establishing a microscopic geometric model of particulate-reinforced composite materials includes:

[0007] S1 determines the three-dimensional spatial range of particle motion;

[0008] S2 determines the volume content of different particles based on the composite material formulation; the different particles refer to particles with different target volumes.

[0009] S3 determines the target volume and volume growth rate of different particles;

[0010] S4 determines the number of different particles within the three-dimensional space based on the three-dimensional spatial range, the target volume of different particles, and the volume content of different particles.

[0011] S5 randomly selects the initial position of each particle within the three-dimensional space according to the number of different particles in the three-dimensional space, and determines the initial volume of each particle.

[0012] S6 uses the LJ potential function to determine the interaction force between particles at the current moment based on the position of each particle at the current moment;

[0013] S7 determines the current velocity and direction of each particle based on the interaction forces between them at the current moment.

[0014] S8 calculates the volume of each particle at the next moment based on the current volume and volume growth rate of each particle.

[0015] The position of each particle at the next moment is obtained based on the current position, velocity, and direction of each particle at the current moment.

[0016] S9 Repeat steps S6 to S8 until each particle is evenly distributed in three-dimensional space and the target volume is reached.

[0017] Furthermore, step S1 also includes determining periodic boundary conditions;

[0018] In step S8, under periodic boundary conditions, the position of each particle at the next moment is obtained based on the current position, velocity, and direction of each particle at the current moment.

[0019] Furthermore, the periodic boundary condition is:

[0020] If a particle is removed from one boundary during its movement, it is moved into the other boundary in that direction.

[0021] Furthermore, in step S3, the volume growth rate is the radius growth rate.

[0022] Furthermore, in step S3, the radius growth rate is set as follows:

[0023] Set the radius growth rate of all particles to be equal;

[0024] or,

[0025] Calculate the radius growth rate v of each particle based on its target volume:

[0026] v = R / t, where R is the particle diameter corresponding to the target volume, and t is the time from the initial moment.

[0027] Furthermore, in step S5, the initial positions of each particle are randomly selected within the three-dimensional space and are not repeated, thus determining that the initial volume of each particle is 0.

[0028] Furthermore, in step S6, the method for calculating the interaction force between particles using the LJ potential function based on the current position of each particle includes:

[0029] Set the cutoff radius between any particle and all other particles based on the target volume size of different particles;

[0030] Based on the cutoff radius between a certain particle and all other particles, and the current position of each particle, the LJ potential function is used to calculate the interaction force between the particle and other particles within the cutoff radius of the particle at the current moment.

[0031] Furthermore, in step S7, the method for determining the velocity and direction of each particle at the current moment based on the interaction forces between particles includes:

[0032] S7.1 Based on the forces exerted on this particle by other particles, calculate the acceleration of the particle under the resultant force of all the forces exerted by the other particles.

[0033] S7.2 Based on the acceleration obtained by the particle under the resultant force of all other particles, determine the current velocity and direction of the particle based on the acceleration and the velocity and direction of the particle at the previous moment.

[0034] Furthermore, the microscopic geometric model of the particle-reinforced composite material is represented by the three-dimensional coordinate matrix of the centroid of each particle and the corresponding particle radius matrix.

[0035] Compared with the prior art, the present invention has at least one of the following advantages:

[0036] (1) This invention creatively proposes a method for constructing a micro-geometric model of a particle-matrix composite material with a high particle volume filling ratio. It introduces a potential function (LJ potential function) to describe the interaction between particles, which can ensure that each pair of particles always maintains a certain distance and reduce the phenomenon of overlap. This greatly reduces the process of repeated checking and iteration in previous modeling methods to avoid particle overlap, and the model can be obtained more quickly, shortening the modeling time.

[0037] (2) In this invention, the initial particle volume is set to 0. Although the initial spatial coordinates of the particles are random, the initial positions of the particles have almost no possibility of overlapping, which can avoid the problem of subsequent calculations not converging due to the distortion of the initial state of the model.

[0038] (3) The present invention provides two ways to set the particle volume growth rate, which can be selected according to actual needs and has wide applicability. Attached Figure Description

[0039] Figure 1 This is a flowchart of a method for establishing a microscopic geometric model of a particle-reinforced composite material according to the present invention;

[0040] Figure 2 A schematic diagram of the LJ potential function;

[0041] Figure 3 This is a schematic diagram of the HTPB mesoscopic finite element model established in the embodiments of the present invention;

[0042] Figure 4 The stress-strain curves obtained from the finite element model in this embodiment of the invention are shown below.

[0043] Figure 5 The stress cloud diagrams obtained from the finite element model in this embodiment of the invention are shown in (a) and (b) respectively. (c) is the stress cloud diagram when the strain is 0%, (b) is the stress cloud diagram when the strain is 17%, and (c) is the stress cloud diagram when the strain is 50%. Detailed Implementation

[0044] The features and advantages of the present invention will become clearer and more explicit from the following detailed description.

[0045] The term “exemplary” as used herein means “serving as an example, embodiment, or illustration.” Any embodiment illustrated herein as “exemplary” is not necessarily to be construed as superior to or better than other embodiments. Although various aspects of embodiments are shown in the accompanying drawings, the drawings are not necessarily drawn to scale unless specifically indicated otherwise.

[0046] This invention proposes a molecular dynamics modeling method based on a potential function. First, some particles with zero volume are randomly arranged in the computational domain. Then, an interaction potential function is set for these particles, and then these particles are randomly assigned a certain motion speed and linear growth rate. During the motion of the particles, due to the interaction forces between each pair, their motion direction and speed are constantly changing. At the same time, with the increase of volume, the particles are constantly attracted and repelled. As the computation time increases, the particles gradually increase in size until the volume fraction of the particles meets the requirements of the relevant geometric model. The advantages of this modeling method are: (1) Due to the existence of the potential function, the repulsive force makes the distance between particles not infinitely small, so there is almost no particle overlap during the modeling process, which is conducive to improving computational efficiency; (2) The volume filling ratio of the particles in the model can be as high as 50% or more, which is conducive to characterizing the microstructure characteristics of high-volume particle-reinforced composite materials.

[0047] The implementation process of this invention, which calculates and controls the motion trajectory of each particle in the system using the interparticle interaction potential function and Newtonian mechanics principles in the model, is as follows: Figure 1 As shown.

[0048] First, a series of non-repeating coordinate points are randomly selected in geometric space as the initial coordinate points of the particles, and the initial volume of the particles is set to 0. Then, a volume growth rate (usually the radius growth rate) is set for each particle, and an initial spatial motion velocity is randomly assigned. The three-dimensional spatial range of the particle motion and periodic boundary conditions are set. If a particle is removed from the x-boundary during motion, it needs to be moved into the corresponding -x-boundary. The same applies to the y and z boundaries.

[0049] When particles move in space, they exhibit mutual attraction and repulsion. During this motion, the Lennard-Jones (LJ) potential function describes the interaction between particles, and its expression is:

[0050]

[0051] Where ε is the potential well depth, reflecting the strength of particle interactions; σ is the distance between particles when the interaction potential is equal to 0; r is the distance between two particles; r c To determine the cutoff radius, when the distance between particles exceeds r... cAt this time, the interparticle interactions are considered negligible. The first term (12th power term) of the LJ potential function represents the repulsive effect between particles, and the second term (6th power term) represents the attractive effect. When the particles are very close together, the first term is greater than the second term, and the particles will exhibit a significant repulsive phenomenon; conversely, if the distance between particles is very short, an attractive phenomenon will occur. A schematic diagram of the LJ potential function is shown below. Figure 2 As shown.

[0052] Differentiating equation (1) yields the interaction force between the two particles, i.e.

[0053]

[0054] Then, the acceleration of the particle is obtained through Newton's equations of motion, i.e.

[0055]

[0056] In the formula: a and m are the acceleration and mass of the particle, respectively, and t represents time.

[0057] The acceleration generated by the interaction between two particles can be calculated using equations (1) to (3), but usually a particle is surrounded by more than one adjacent particle. Assume the distance from particle i is less than or equal to the cutoff radius r. c The number of particles within the range is N. These N particles are the nearest neighbors of particle i. When calculating the acceleration of particle i, the effects of the N particles on particle i need to be summed.

[0058] As the particles move, their volume continuously increases at a predetermined rate. The size of the particle at the next moment is calculated based on this volume growth rate. A cutoff radius is set for each particle based on its target volume; that is, a cutoff radius is set between every two particles, and for each particle, a cutoff radius is set between it and all other particles. Alternatively, two methods can be used to set the particle volume growth rate: one is to set the radius growth rate of all particles to be the same, which allows small particles to quickly reach the target volume; the other is to calculate the radius growth rate (R / t, where R is the final particle radius and t is the running time) based on the different target volumes of each particle, which allows all particles to reach the required volume simultaneously.

[0059] When particles are small, due to the large geometric space and uneven spatial distribution, some particles are far apart, resulting in little or no interaction. As particle volume increases and distribution becomes more uniform, interparticle forces become more prominent. The forces acting on a particular particle can be obtained through statistical analysis and calculation. Based on the magnitude and direction of these forces, the particle's velocity and direction are determined. Combining this with the current velocity, direction, and position, the particle's position for the next moment is calculated. By continuously calculating and updating particle size and spatial position, particles can gradually grow and become evenly distributed, ultimately achieving the target volumetric filling ratio.

[0060] Because the potential function is characterized by the fact that the smaller the distance between two particles, the greater the repulsive force between them, and the particles will move in opposite directions. Therefore, the existence of repulsion ensures that particles always maintain a distance from each other, and particles almost never overlap in geometric space.

[0061] In summary, the present invention can be summarized into the following steps:

[0062] S1 determines the three-dimensional spatial range of particle motion and the periodic boundary conditions;

[0063] S2 determines the target volume of different particles and sets a volume growth rate for each particle;

[0064] S3 determines the number of different particles within a three-dimensional spatial range;

[0065] S4 randomly selects a series of non-repeating coordinate points in geometric space as the initial coordinate points of the particles.

[0066] S5 sets the initial volume of the particles to 0;

[0067] S6 randomly assigns an initial spatial velocity to the particles;

[0068] S7 sets the cutoff radius;

[0069] S8 uses the LJ potential function to describe the interaction force between particles. By statistics and calculation, the magnitude and direction of the resultant force of all the forces acting on a certain particle by all its adjacent particles can be obtained.

[0070] S9 calculates the size of the particle at the next moment based on the particle's volume growth rate;

[0071] S10 determines the particle's velocity and direction based on the magnitude and direction of the force applied.

[0072] S11 calculates the particle's position at the next moment based on its velocity, direction, and location.

[0073] S12 Repeat steps S8 to S11 until the particles are evenly distributed in space and the target volume filling ratio is achieved.

[0074] Example:

[0075] (1) Geometric model construction objectives

[0076] Taking a certain hydroxyl-terminated polybutadiene (HTPB) propellant as an example, the main solid filler of this propellant is ammonium perchlorate (AP) particles. The relevant parameters are: the total volume content of the particles is 66.25%, and there are 14 different particle sizes. Based on the different particle volume contents, when the size of the calculation area is 2×2×2mm... 3 At that time, the number of particles filled was approximately 90. In practical applications, the volume percentage (i.e., content) of AP particles is related to the HTPB propellant formulation, while the specific number is related to the size of the computational domain. When the volume percentage of AP particles is constant, the larger the computational domain, the more particles there are, and the greater the computational load. Therefore, a reasonable computational domain should be selected based on available computational resources.

[0077] (2) Geometric model construction steps

[0078] 1) Construct a three-dimensional space [-1 1; -1 1; -1 1] for particle expansion and motion, and set periodic boundary conditions for this modeled three-dimensional space;

[0079] 2) Establish a column vector of target particle size and arrange them in ascending order, and at the same time establish a corresponding column vector of particle radius growth rate;

[0080] 3) Based on the target volume of different particles, set different cutoff radii for each pair of particles to ensure that they do not overlap with other particles;

[0081] 4) Calculate the number of each type of particle based on the target particle size, content, and size of the three-dimensional space (computation area) to be filled;

[0082] 5) Allow the particles to expand and move freely in three-dimensional space until the volume of each particle reaches the target value and the total volume reaches the target. After calculation, the total volume target of 66.25% was finally achieved.

[0083] 6) By using the method of this invention for modeling, a series of three-dimensional coordinate matrices of the centroid of circular particles and corresponding particle radius matrices can be obtained.

[0084] (3) Finite element model construction and calculation

[0085] The constructed geometric model was imported into the finite element simulation software ABAQUS using Python, and mesh generation was performed. The results are shown below. Figure 3As shown. After assigning material parameters and periodic boundary conditions to the finite element model, uniaxial tension is performed, resulting in... Figure 4 The stress-strain curve shown is as follows: Figure 4 The two curves in the middle are the results obtained by calculation using different material parameters. In addition, we can also obtain... Figure 5 The stress cloud diagram shown illustrates the dehydration phenomenon at the interface between the particles and the substrate.

[0086] The geometric model established by this invention generates a finite element model, which enables simulation studies of the mechanical and other properties of materials. The rapid and accurate establishment of the geometric model of this invention is beneficial to improving the speed and accuracy of material performance simulation results, and has profound significance for the micro-design and microstructure and performance research of materials.

[0087] The present invention has been described in detail above with reference to specific embodiments and exemplary examples; however, these descriptions should not be construed as limiting the present invention. Those skilled in the art will understand that various equivalent substitutions, modifications, or improvements can be made to the technical solutions and embodiments of the present invention without departing from the spirit and scope of the invention, and all such modifications and improvements fall within the scope of the present invention. The scope of protection of the present invention is defined by the appended claims.

[0088] The contents not described in detail in this specification are common knowledge to those skilled in the art.

Claims

1. A method for establishing a microscopic geometric model of a particle-reinforced composite material, characterized in that, include: S1 determines the three-dimensional spatial range of particle motion; S2 determines the volume content of different particles based on the composite material formulation; S3 determines the target volume and volume growth rate of different particles; S4 determines the number of different particles within the three-dimensional space based on the three-dimensional spatial range, the target volume of different particles, and the volume content of different particles. S5 randomly selects the initial position of each particle within the three-dimensional space according to the number of different particles in the three-dimensional space, and determines the initial volume of each particle. S6 uses the LJ potential function to determine the interaction force between particles at the current moment based on the position of each particle at the current moment; S7 determines the current velocity and direction of each particle based on the interaction forces between them at the current moment. S8 calculates the volume of each particle at the next moment based on the current volume and volume growth rate of each particle. The position of each particle at the next moment is obtained based on the current position, velocity, and direction of each particle at the current moment. S9 Repeat steps S6 to S8 until each particle is evenly distributed in three-dimensional space and the target volume is reached; In step S3, the radius growth rate is set as follows: Set the radius growth rate of all particles to be equal; or, Calculate the radius growth rate of each particle based on its target volume. v : v=R / t ,in, R The particle diameter corresponds to the target volume. t This is the time elapsed since the initial moment; In step S5, the initial positions of each particle are randomly selected in the three-dimensional space and are not repeated, so the initial volume of each particle is determined to be 0. In step S6, the method for calculating the interaction force between particles using the LJ potential function based on the current position of each particle includes: Set the cutoff radius between any particle and all other particles based on the target volume size of different particles; Based on the cutoff radius between a certain particle and all other particles and the position of each particle at the current moment, the LJ potential function is used to calculate the force exerted on the particle by other particles within the cutoff radius range of the particle at the current moment. Step S1 also includes determining periodic boundary conditions; In step S8, under periodic boundary conditions, the position of each particle at the next moment is obtained based on the current position, velocity, and direction of each particle at the current moment. The periodic boundary conditions are as follows: If a particle is removed from one boundary during its movement, it is moved into the other boundary in that direction. In step S3, the volume growth rate is the radius growth rate; In step S7, the method for determining the velocity and direction of each particle at the current moment based on the interaction forces between particles includes: S7.1 Based on the forces exerted on this particle by other particles, calculate the acceleration of the particle under the resultant force of all the forces exerted by the other particles. S7.2 Based on the acceleration obtained by the particle under the resultant force of all other particles, determine the current velocity and direction of the particle based on the acceleration and the velocity and direction of the particle at the previous moment.

2. The method for establishing a microscopic geometric model of a particle-reinforced composite material according to claim 1, characterized in that, The microscopic geometric model of the particle-reinforced composite material is represented by the three-dimensional coordinate matrix of the centroid of each particle and the corresponding particle radius matrix.

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