Rock-soil particle crushing process prediction method based on population theory

Abstract

The rock-soil particle crushing process prediction method based on the population theory comprises the following steps: deriving a population symbiosis mathematical model based on the population evolution theory; comparing the characteristics of population evolution and particle fragmentation evolution, analyzing similarities and differences of the properties of the population evolution and the particle fragmentation evolution, and extracting common points; coupling the single particle size group model and the Markov chain model, and proposing a multi-particle size group fragmentation prediction model based on a population theory; determining the physical significance and the calculation mode of the model parameter migration rate k, the comprehensive environmental influence coefficient rij, the crushing energy consumption mij ratio and the effective restriction coefficient beta of the crushing energy consumption mij ratio; model characteristics prove that multiple particle size groups are simplified, extreme particle groups are consistent, and particle groups with over-limit content are inclusive; determining analysis dimensions and crushing parameters of the analysis dimensions; designing and executing an experimental scheme; and analyzing and comparing an experimental result with a prediction result, and verifying the prediction effects of unicity crushing and multi-particle-size group crushing of the model.

Description

Technical Field

[0001] The invention relates to a prediction method, in particular to a prediction method for rock and soil particle crushing process based on population theory. Background Art

[0002] Granular soil in the rock and soil category mainly refers to granular materials such as gravel, crushed stone and sandy soil, which have the characteristics of little or no viscosity. Granular soil is widely used in civil engineering construction as an excellent filler, such as large earth-rock dams, port riprap foundations, crushed stone roadbeds, railway ballast, etc. Granular soil particles are easily broken under external loads because of their point-to-point contact, which affects the strength and deformation of granular soil and significantly affects the stability of rock and soil structures. Soil particle size distribution is one of the basic studies in soil mechanics and is the basis for soil classification, mechanical strength calculation, and targeted engineering applications. The essence of particle crushing is the change of soil particle size distribution (gradation). Constructing a granular soil gradation curve prediction model under particle crushing conditions helps to understand the evolution law of particle gradation. Existing studies have proposed many theoretical models for single particle size and specific conditions in the evolution law of particle crushing, which provide important references for understanding particle crushing behavior, but the comprehensive description of the crushing of multiple particle size groups is still insufficient, and the mutual influence between each particle size group is not considered.

[0003] The prediction method of rock and soil particle crushing needs to be combined with different types of rock and soil materials and different engineering conditions to provide a prediction tool with high accuracy and reliability. With the increase in infrastructure construction, especially railway, highway, subway and other projects, the demand for geotechnical engineering is increasing. In particular, the construction of high-speed railways and heavy-duty railways requires higher stability of roadbed soil. The prediction of rock and soil particle crushing can provide accurate technical support for roadbed construction, earthwork engineering, etc., reduce construction risks and improve engineering quality.

[0004] Existing research on the evolution of particle breakage provides an important theoretical reference for understanding particle breakage behavior. Some studies focus on the breakage of particles of a single particle size, establish a relationship model between particle survival probability and load, and propose a calculation method for particle breakage limit, which provides a theoretical basis for the study of the breakage of particles of multiple particle sizes. The study of particles of a single particle size has become a leading work for further analysis of complex particle systems because of its relatively simple laws.

[0005] In the study of the crushing of particles of multiple particle sizes, there are existing models that can describe the crushing evolution of particles of multiple particle sizes. For example, the probability density function proposed based on dynamic loads can describe the crushing process of particles under complex stress conditions; the gradation evolution model of particles of multiple particle sizes is simulated through random processes, revealing the dynamic change law of particle crushing. In addition, by assuming the probability relationship of particles of different particle sizes being crushed to smaller particle sizes, a model suitable for particles of multiple particle sizes is constructed, enriching the research framework of particle crushing.

[0006] In the study of the coupling between particle crushing and mechanical behavior, the relevant models further improved the interaction mechanism between particle crushing and soil deformation by introducing crushing parameters and correcting the critical state line of granular materials. In terms of describing the properties and anisotropy of particle crushing, the improved model based on the concepts of particle stress and sliding surface can reflect the stress-deformation characteristics of the material crushing process.

[0007] In general, existing research has made certain progress in the research directions of single-size particles, multi-size groups of particles and mechanical coupling, providing an important basis for the construction of theoretical models and engineering applications of particle crushing.

[0008] Existing studies on the evolution of particle crushing have provided important theoretical references for understanding particle crushing behavior, but there are also many shortcomings. Most studies focus on the crushing laws of particles of a single size. Although this research direction is relatively simple and easy, and provides a leading reference for the crushing research of particles of multiple size groups, its scope of application is relatively limited, and it is difficult to fully reflect the true crushing evolution laws of particles under complex stress states. The survival probability model proposed based on the Weibull distribution and the crushing limit calculation method established using the Markov chain have certain theoretical significance, but they are limited to the analysis of particles of a single size and are not adaptable enough to particles of multiple size groups. [1-2] .

[0009] In addition, although there has been some progress in the study of particle crushing of multi-size groups, there are still problems such as high complexity of model construction, unreasonable parameter selection and insufficient actual prediction accuracy. Taking the proposed dynamic crushing probability density function as an example, this model can describe the crushing evolution law of multi-size coarse-grained soil to a certain extent, but it does not consider the influence between and within particle groups. [3] The multi-size group breakage model based on Markov chain has great theoretical limitations because the Weibull parameters used are derived from the breakage behavior of particles in a single size group and do not truly reflect the breakage characteristics of particles in multiple size groups. [4] .

[0010] Some studies have made overly idealistic assumptions about particle breakage behavior, ignoring the randomness and complexity of particle breakage. Some models assume that particles of different sizes have an equal probability of breaking into smaller sizes. Although this assumption is reasonable in model construction, it fails to fully consider the anisotropy of particle breakage and the impact of complex stress conditions on breakage behavior. [5] Some constitutive models that consider the crushing effect, such as the MPZ model improved based on state parameters, can describe the stress-deformation characteristics of rockfill materials, but the description of particle crushing behavior still remains at the level of generalized plasticity theory, which is difficult to meet the prediction needs of complex particle crushing behavior in actual engineering.[6] .

[0011] In summary, current research still has much room for improvement in the diversity and randomness of particle breakage laws and the comprehensive description of the breakage laws of multiple particle size groups: first, most models have common problems such as too many simplified assumptions, numerous parameters and insufficient precision in practical applications; second, most models only describe the initial and final states of particle breakage, and do not consider the migration and replenishment processes between particle groups during the breakage process; third, although most models are based on mathematical statistics theory, the parameters contained in the models have only mathematical meanings but no physical meanings, and therefore cannot reflect the influence of actual engineering stress states on the breakage process; finally, most models do not consider the mutual constraints within particle size groups during the particle breakage process.

[0012] References: [1] MCDOWELL G R. On the yielding and plastic compression of sand [J]. Soils and Foundations, 2002, 42 (1): 139–145.

[0013] [2] Tong Chenxi, Zhang Sheng, Li Xi, et al. Study on the crushing law and crushing limit of granular materials with a single particle size group [J]. Rock and Soil Mechanics, 2015, 36(Suppl. 1): 260-264.

[0014] [3] Long Yao, Zhang Tongwen, Zhang Jiasheng, et al. Dynamic test and particle crushing model of red sandstone coarse-grained soil[J]. Vibration and Shock, 2023, 42(3): 270-279.

[0015] [4] Tong Chenxi, Zhang Sheng, Li Xi, et al. Study on the evolution law of rock material particle crushing based on Markov chain [J]. Chinese Journal of Geotechnical Engineering, 2015, 37(5): 870-877.

[0016] [5]OZKAN G,ORTOLEVA P J.Evolution of the gouge particleized distribution:A Markov model[J].Pure and AppliedGeophysics,2000,157(3):449–468.

[0017] [6] ZHANG Xiang-tao, GAO Yi-zhao, YU Yu-zhen, et al. Generalized plasticity model considering grain crushing and anisotropy for rockfill materials [J]. Journal of Central South University, 2022, 29(4): 1274-1288. Summary of the invention

[0018] The technical problem to be solved by the present invention is the limitations of the existing particle crushing model in describing the crushing evolution law of granular soil particles with multiple particle size groups, the applicability and accuracy of the crushing process of different soil types and multiple particle size groups under complex stress conditions, especially the insufficiency in considering the mutual influence between particle size groups and the influence of internal constraints of particle size groups. The existing particle crushing model usually cannot fully consider the influence of the interaction between different particle size groups and the internal constraints of particle size groups on the crushing process, resulting in the model having insufficient accuracy and limited applicability when describing the crushing evolution process of different soil types and multiple particle size groups under complex stress conditions. The technical solution is as follows:

[0019] The rock and soil particle crushing process prediction method based on population theory is characterized by comprising the following steps:

[0020] Step 1: Based on the population evolution theory, derive the mathematical model of population symbiosis;

[0021] Step 2: Compare the characteristics of population evolution and particle fragmentation evolution, analyze the similarities and differences between the two properties, and extract the common points;

[0022] Step 3: Couple the single particle size group model and the Markov chain model to propose a multi-particle size group breakage prediction model based on population theory;

[0023] Step 4: Clarify the physical meaning and calculation method of the model parameters migration rate, comprehensive environmental impact coefficient, fragmentation energy consumption ratio and its effective control coefficient;

[0024] Step 5: Model feature demonstration, simplification of multiple particle size groups, consistency of extreme particle groups, and inclusion of particle groups with content exceeding the limit;

[0025] Step 6: Determination of analysis dimension and its fragmentation parameters;

[0026] Step 7: Design and implementation of experimental plan;

[0027] Step 8: Analyze and compare the experimental results with the predicted results, and continue to verify the model's single-size crushing and multi-size group crushing prediction effects.

[0028] Beneficial Effects

[0029] 1Introducing population evolution theory to fully describe the particle crushing process

[0030] Compared with the traditional particle crushing model, the present invention constructs a particle crushing evolution model of multiple particle size groups based on population evolution theory, which can more comprehensively and accurately describe the dynamic evolution law of particles in the crushing process. The existing technology usually focuses on a single particle size group or a simplified crushing model, while the present invention takes into account the mutual influence between multiple particle size groups, which is more in line with the complexity of particle crushing in actual engineering.

[0031] 2 Consider the mutual influence and internal constraints between multiple particle size groups

[0032] The present invention takes into account the mutual competition and synergistic effects between different particle size groups and introduces an effective constraint coefficient, which can more realistically simulate the inherent correlation in the crushing process of particles of multiple particle size groups, overcoming the limitation of the prior art that the interaction between different groups of particles is not fully considered when dealing with the crushing of multiple particle size groups.

[0033] 3 Improved crushing energy consumption model

[0034] The present invention proposes a model based on the crushing energy consumption ratio, which can accurately quantify the relationship between energy consumption and particle size change during particle crushing. Compared with the simplified energy model in the prior art, the crushing energy model of the present invention has more physical significance and practical application value.

[0035] 4 Introduction of Markov chain to describe the fragmentation process

[0036] Traditional particle breakage models are usually unable to accurately describe the migration process between particles. The present invention introduces a Markov chain matrix, which can accurately describe the process of particles migrating from one particle size group to another during the breakage process, thereby improving the model's ability to predict the dynamic changes of particle breakage.

[0037] 5Correlation analysis between fractal dimension and stress state

[0038] The present invention provides a more accurate method for characterizing particle crushing by fitting the relationship between fractal dimension and stress. Compared with the rough estimation in the prior art, the present invention can more accurately describe the distribution characteristics of particles after crushing, especially the behavior of particles under high stress conditions.

[0039] 6. High applicability of verification experiments and engineering applications

[0040] Verified by dynamic triaxial tests, one-dimensional compression tests and annular shear strain tests, the present invention has high prediction accuracy under different stress paths and soil conditions, can accurately describe the changes in the particle crushing process, and has stronger engineering applicability and higher prediction accuracy compared to the prior art.

[0041] 7 Consider the inclusiveness of extreme particle groups and over-limit particle groups

[0042] The model of the present invention can effectively describe the crushing behavior of extreme particle groups and over-limit particle groups, while the prior art mostly focuses on the description of conventional particle size groups and ignores the influence of these special particle groups. The present invention fills this gap and enhances the comprehensiveness and practical applicability of the model.

[0043] In summary, the present invention has obvious technical advantages over the prior art in terms of description of particle crushing process, prediction accuracy, comprehensiveness of the model and practical applicability. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 is a flow chart of the method of the present invention;

[0045] Figure 2 It is a schematic diagram of the gradation curve of the prior art test;

[0046] Figure 3 is the crushing energy E B and crushing parameter B r Schematic diagram of the relationship curve;

[0047] Figure 4 It is a schematic diagram of the test loading method of the present invention;

[0048] Figure 5 This is a schematic diagram of the state of the red sandstone 31.5-45mm particles from the initial single particle size to 10000 times of dynamic crushing in the present invention;

[0049] Figure 6 This is a schematic diagram of the crushing state of calcareous sand from a single particle size state to an axial stress of 1.6 MPa;

[0050] Figure 7 It is a schematic diagram of the crushing state of calcareous sand from a single particle size state to an axial stress of 51.2 MPa;

[0051] Figure 8 It is a schematic diagram of the crushing state of petroleum coke particles in one-dimensional compression test from load 7.5MPa to 10MPa;

[0052] Fig. 9 Schematic diagram of the crushing state of calcareous sand particles in the shear strain test from shear strain 285% to 1180%. DETAILED DESCRIPTION

[0053] The rock and soil particle crushing process prediction method based on population theory is characterized by comprising the following steps:

[0054] Step 1: Based on the population evolution theory, derive the mathematical model of population symbiosis;

[0055] (1) Theoretical basis

[0056] aBased on population evolution theory, it is assumed that there is competition and symbiotic relationship between populations, and the scale growth conforms to the Logistic model.

[0057] b The growth rate of population A is affected by population B, and vice versa.

[0058] c introduces the interaction coefficient ω AB _ and ω BA , characterizing the inhibitory or promoting effects between populations.

[0059] (2) Mathematical derivation

[0060] Based on the Logistic model, the following symbiosis model is established:

[0061]

[0062] In the formula, E A 、E B Respectively represent the size of two populations A and B in a certain period of time. A 、N B Respectively represent the maximum size of populations A and B when facing an unchanged environment within a certain period of time. BA is the influence of population B on population A, ω AB is the influence of population A on population B. A is the growth rate of population A, r B is the growth rate of population B, dE A (t) / dt represents the size of population A in time period t; dE B (t) / dt represents the size of population B in time period t.

[0063] (3) Key features

[0064] When AB >0,ω BA >0: Populations A and B have a symbiotic relationship.

[0065] When AB <0,ω BA <0: Populations A and B are in a competitive relationship.

[0066] Step 2: Compare the characteristics of population evolution and particle fragmentation evolution, analyze the similarities and differences between the two properties, and extract the common points;

[0067] 1. Similarities and differences between population evolution and particle fragmentation

[0068] (1) Similarities

[0069] a Both involve dynamic interactions between multiple components (between populations / between particle size groups);

[0070] b. Scale changes are affected by internal and external environments (limited resources / fragmented energy consumption);

[0071] c all follow a certain growth pattern (Logistic law / fragmentation-migration process)

[0072] (2) Differences

[0073] a Population evolution involves biological behavior, while particle breakage is mainly based on physical and mechanical effects;

[0074] b The population has the ability to self-repair, while particle breakage is irreversible.

[0075] 2. Common points extraction

[0076] (1) Dynamic changes depend on limited resources and migration mechanisms;

[0077] (2) Scale changes show nonlinear behavior and tend to be stable.

[0078] Step 3: Couple the single particle size group model and the Markov chain model to propose a single particle size group model and a Markov chain model;

[0079] Select i pseudo-limit particle sizes of rock and soil particles, from small to large, d1, d2, ..., d i According to the organizational population theory, for a certain particle size group i, we have:

[0080]

[0081] In the formula, x ii ' is the final content of a certain group of particles, %; x ii is the initial content of a certain particle group, %; x jj is the initial content of each particle group, %; k jj is the migration rate of each particle group from crushing to particle size group i, including the migration rate of particle size group i from crushing to this group, dimensionless; N ii , N jj The final contents of i and j particle groups, respectively, %; m ij is the crushing energy consumption ratio between this particle group and other particle groups, dimensionless; β is the effective constraint coefficient, which characterizes the constraint relationship within the multi-particle size group, dimensionless.

[0082] Considering that multiple particle size groups are not simply linear combinations of single particle sizes, we must consider that the existence of other particle groups can increase or decrease this particle group. jj The comprehensive environmental impact coefficient of each particle group:

[0083]

[0084] Combining the above two equations, we can get:

[0085]

[0086] In the formula, x ii ' is the final content of a certain group of particles, %; x ii is the initial content of a certain particle group, %; x jj is the initial content of each particle group, %; k ii is the migration rate of particle group i to its own particle size group, dimensionless; k ij is the migration rate of particle group j to particle size group i, dimensionless; r ij is the comprehensive environmental impact coefficient of particle size group j during the process of particle size group j being crushed into particle size group i, dimensionless; r ii It is the comprehensive environmental impact coefficient of particle size group i when particle size group i is crushed into its own particle size group. It is dimensionless.

[0087] According to existing literature (Tong Chenxi, Zhang Sheng, Li Xi, et al. Study on the evolution law of rock material particle crushing based on Markov chain [J]. Journal of Geotechnical Engineering, 2015, 37(5): 870-877.), the crushing of multiple particle groups should satisfy the one-step transfer Markov chain matrix, and the one-step transfer matrix is ​​used to describe the migration behavior between multiple particle size groups:

[0088]

[0089] Among them, p ij is the transition probability from particle size group j to i

[0090] Coupling the above formulas, we can get the broken mobility matrix P:

[0091]

[0092] In the formula, r ij is the comprehensive environmental impact coefficient of particle size group j during the process of particle size group j being crushed into particle size group i, dimensionless; r ii k is the comprehensive environmental impact coefficient of particle size group i when particle size group i is crushed into its own particle size group, dimensionless; ii is the migration rate of particle group i to its own particle size group, dimensionless; k ij is the migration rate of particle group j to particle size group i, dimensionless.

[0093] Step 4: Clarify the model parameters migration rate k and comprehensive environmental impact coefficient r jj , crushing energy consumption ratio m ij The physical meaning of its effective constraint coefficient and its calculation method;

[0094] For different particle groups, the migration rate is different, which can be determined according to the crushing law of a single particle size group: for a certain particle group, its limiting particle size is D i , its initial percentage is 1, and the percentage of each particle size group is obtained after one crushing. Then the percentage of each particle size group is the migration rate obtained by crushing the particle size group. i -d j , the migration rate k is:

[0095]

[0096] And ∑k ij =1

[0097] In the formula, k ij is the migration rate of particle group j to particle size group i, dimensionless; F di is smaller than a certain particle size d i The mass percentage of F is dimensionless; dj is smaller than a certain particle size d j The mass percentage of D max is the maximum particle size, mm; d i is the size of particle size group i, mm, d j is the size of the j-particle size group, mm; α is the fractal dimension, dimensionless.

[0098] There is a final gradation for particle crushing. Under this gradation, large and small particles will form a self-similar distribution, and the corresponding gradation curves before loading, during loading, and at the end of loading can be expressed using a mass percentage function.

[0099]

[0100] Where, d m is the minimum particle size of the sample, mm; d M is the maximum particle size of the sample; F0(d), F(d) and F u (d) are the mass percentages before loading, during loading, and at the end of loading, respectively, dimensionless; α is the fractal dimension, dimensionless.

[0101] Corrected Hardin crushing parameter B r There is a certain relationship between it and crushing energy consumption:

[0102]

[0103] Where a, b and c are the crushing parameter coefficients; E B is the crushing energy consumption, kPa; B r To modify the Hardin crushing parameters.

[0104]

[0105] Crushing energy ratio m ji for:

[0106]

[0107] The effective constraint coefficient β measures the internal constraint relationship of multiple actual particle groups based on the crushing energy consumption. After experimental crushing under certain working conditions, the vector of the actual content of each particle size group is:

[0108] X ij T =[x 1j ,x 2j …,x ij ]

[0109] In the formula, x ij After the experimental crushing, the particle size groups d1, d2, ..., d i The percentage of X ij T It is the transposed vector of the percentage of each particle size group after experimental crushing.

[0110] For the breakage of multiple particle size groups, the corresponding predicted calculated value is obtained from the breakage matrix of multiple particle size groups:

[0111] X ij ' T =[x 1j ',x 2j ',…,x ij ']

[0112] In the formula, x ij ' is the predicted value of the percentage of each particle size group calculated after particle crushing; X ij ’T It is the transposed matrix of the predicted percentage of each particle size group after particle crushing.

[0113] The following objective function is constructed by the least squares method:

[0114] X”=[x 1j '-x 1j ] 2 +[x 2i '-x 2j ] 2 +……+[x ij '-x ij ]2

[0115] Where X” is a function of the effective restriction coefficient β. By finding the minimum value of X” and deriving the function, the corresponding effective restriction coefficient β can be obtained.

[0116] Step 5: Model feature demonstration, simplification of multiple particle size groups, consistency of extreme particle groups, and inclusion of particle groups with content exceeding the limit;

[0117] 1. Simplification of multi-particle group models

[0118] The simplification of the multi-particle group model means that when only one particle group exists and the content of other particle groups is 0, the single particle size crushing law should be satisfied. At this time, for the multi-particle group model, since there is only a single particle size, the effective constraint coefficient between different particle groups is 0, then: β = 0, x ii ≠0, x jj =0,k jj ≠0.

[0119] For a single particle size group i:

[0120]

[0121] For other particle size groups, i≠j:

[0122]

[0123] In the formula, x ii ' is the final content of a certain group of particles, %; x ii is the initial content of a certain particle group, %; x jj ' is the final content of other granule groups, %; x jj is the initial content of other particle groups, %; k jj is the migration rate of each particle group from crushing to particle size group i, including the migration rate of particle size group i from crushing to this group, dimensionless; k ij is the migration rate of particle group j to particle size group i, dimensionless; N ii , N jj The final contents of i and j particle groups, %; m ij is the crushing energy consumption ratio between this particle group and other particle groups, dimensionless; β is the effective constraint coefficient, which characterizes the constraint relationship within the multi-particle size group, dimensionless.

[0124] It can be seen that the multi-particle size group model is consistent with the crushing model of single particle group particles, proving that the multi-particle group model can be unified.

[0125] 2. Consistency of extreme particle groups

[0126] Extreme particle groups refer to the smallest particle group that cannot be broken any further and the largest particle group that cannot be replenished during the breaking process. The population model of particle breaking should have commonalities and be applicable to extreme particle groups.

[0127] For the minimum particle size group x 11 Due to the lower limit of particle crushing, this particle group can no longer be crushed. 11 , the survival rate of the particles migrating to their own group is 1, and the fragmentation rate of particles migrating to other groups is 0. The increment of this group can only be supplemented by other groups. Its migration rate k 11 =1, k 1j =0(j≠1).

[0128]

[0129] In the formula, x 11 ' is the final content of the smallest particle size group, %; x 11 is the initial content of the smallest particle size group, %; x jj is the initial content of other particle groups, %; k jj is the migration rate of each particle group from crushing to particle size group i, including the migration rate of particle size group i from crushing to this group, dimensionless; k 1j is the migration rate of the j-size group to the smallest size group, dimensionless; N 11 is the final content of the smallest particle size group, %; N jj is the final content of j-granule group, %; m ij is the crushing energy consumption ratio between this particle group and other particle groups, dimensionless; β is the effective constraint coefficient, which characterizes the constraint relationship within the multi-particle size group, dimensionless.

[0130] For the largest particle size group, since this particle group can only be broken into smaller particle groups, and the survival rate of other particle groups migrating to this particle group is 0, the increment of this particle group can only be caused by its own breakage, and this increment is a non-positive value. That is, the migration rate k of other particle size groups to the largest particle size group is timax =0,t≠i max .

[0131]

[0132] In the formula, x imax ' is the final content of the largest particle size group, %; x imax is the initial content of the largest particle size group, %; x jj is the initial content of other particle groups, %; k jj is the migration rate of each particle group from crushing to particle size group i, including the migration rate of particle size group i from crushing to this group, dimensionless; k imax is the migration rate of the largest particle size group from the original particle size group, dimensionless; N imax is the final content of the largest particle size group, %; Nii , N jj The final contents of i and j particle groups, respectively, %; m ij is the crushing energy consumption ratio between this particle group and other particle groups, dimensionless; β is the effective constraint coefficient, which characterizes the constraint relationship within the multi-particle size group, dimensionless.

[0133] It can be seen that the crushing model can judge the crushing amount of extreme particle groups, that is, the minimum particle size group and the maximum particle size group.

[0134] 3. Inclusiveness of the group of particles with excessive content

[0135] The actual crushing process is an uncertain process. The content of a certain particle group may increase or decrease. When the content of a particle group increases, it is possible that the content of the particle group exceeds the limit content. In this case, the instantaneous content exceeding is acceptable, but the overall trend is still stable. It can be described by a stable equilibrium point. Under certain conditions, the particle crushing will reach stability, that is, satisfying the equation group:

[0136]

[0137] In the formula, g1, g2, ..., g i is the instantaneous growth rate of each particle size group, %; x 11 ,x 22 ,…,x ii is the initial content of each particle size group, %; N 11 ,N 22 ,…,N ii ,N jj is the final content of each particle size group, %; m ij F is the crushing energy consumption ratio of this particle group to other particle groups, dimensionless; β is the effective constraint coefficient, which characterizes the constraint relationship within the multi-size group, dimensionless. i (x 11 ,x 22 ,…,x ii ) is a function of the initial content of each particle size group; x ii ' is the final content of a certain particle size group.

[0138] By solving this equation, the corresponding equilibrium point can be obtained.

[0139] Step 6: Determination of analysis dimension and its fragmentation parameters;

[0140] (1) Determination of fractal dimension

[0141] Fractal dimension calculation formula:

[0142]

[0143] Where M(d) is the mass of particles with a diameter smaller than d, g; M max The mass corresponding to the maximum particle size, g; d max is the maximum particle size, mm; d is a certain particle size, mm; α is the fractal dimension.

[0144] The calculation formula of fractal dimension can be obtained:

[0145]

[0146] There are experimental gradation curves in other literature Figure 2

[0147] The fitting formula of fractal dimension α and compressive stress σ is as follows:

[0148] α=e·ln(σ)+f

[0149] Among them, e = 0.268, f = 1.3653, correlation coefficient R 2 =0.973.

[0150] (2) Crushing parameter coefficients a, b and c, see Figure 3 shown.

[0151] According to the existing experimental data, the crushing energy consumption E B and crushing parameter B r The relationship curve, such as Figure 2 The crushing parameter coefficients a, b and c can be fitted using formula (11). It can be obtained that: a = 0.9813, b = 0.2665, c = 0.1412, and the correlation coefficient R 2 =0.9694.

[0152] Step 7: Design and implementation of experimental plan;

[0153] This test uses the TAJ-2000 large-scale dynamic and static triaxial tester as the main experimental instrument, and uses red sandstone coarse-grained soil as the test material, which is divided into three single-size groups for testing. The loading method adopts graded loading, dynamically controlling the number, frequency and amplitude of loading to simulate actual working conditions. The porosity is strictly controlled during the sample preparation process to ensure the representativeness of the sample and the reliability of the test results.

[0154] Step 8: Analyze and compare the experimental results with the predicted results, and continue to verify the model's single-size crushing and multi-size group crushing prediction effects.

[0155] 1. Verification of the crushing evolution of a single particle size group

[0156] The experimental results were compared with the model predictions to verify the model's ability to describe the breakup behavior of a single particle size group.

[0157] 2. Verification of the crushing evolution of multiple particle size groups

[0158] Verify the model's ability to describe complex interactions among multiple particle size groups.

[0159] 3. Model suitability assessment

[0160] Summarize the prediction accuracy of the model under different soil types and stress paths and confirm its practical value.

[0161] 1. Model Construction

[0162] The organizational population model is based on three assumptions: (1) The organizational population is composed of many similar organizations, and the size of the population is the sum of the production capacity of many similar organizations. The increase in population size is due to two aspects: population size and production capacity. (2) Within a certain period of time, environmental resources are fixed, and the size of the population cannot change infinitely. As the size changes, the growth rate of the population continues to decline until it stabilizes. Within a certain period of time, facing an unchanged environment, the size of each population has a maximum value N. i (3) In the organizational network, different organizational populations have different characteristics and different growth rates of organizational population size. The population size growth conforms to the Logistic model:

[0163]

[0164] There is a competitive relationship between populations. The growth of population B will inhibit the growth of population A, and the size of population A will decrease. Similarly, due to the inhibitory effect of population A, the size of population B will also decrease. The symbiotic mathematical model of populations A and B is:

[0165]

[0166] In the formula, E A 、E B Respectively represent the size of two populations A and B in a certain period of time. A 、N B Respectively represent the maximum size of populations A and B when facing an unchanged environment within a certain period of time. BA is the influence of population B on population A, ω AB is the influence of population A on population B. A is the growth rate of population A, r B is the growth rate of population B, dE A (t) / dt represents the size of population A in time period t; dE B (t) / dt represents the size of population B in time period t.

[0167] The crushing of multiple particle groups is that different particle size groups are crushed under certain space and external conditions, and the crushed particles migrate to other particle groups, causing the content of other particle size groups to increase or decrease. After the particles are crushed, the final gradation will be reached, and each particle size group will reach the limit value. Select i pseudo-limit particle sizes of rock and soil particles, from small to large, d1, d2, ..., d i According to the organizational population theory, for a certain particle size group, we have:

[0168]

[0169] In the formula, x ii ' is the final content of a certain group of particles, %; x ii is the initial content of a certain particle group, %; x jj is the initial content of each particle group, %; k jj is the migration rate of each particle group from crushing to particle size group i, including the migration rate of particle size group i from crushing to this group, dimensionless; N ii , N jj The final contents of i and j particle groups, respectively, %; m ij is the crushing energy consumption ratio between this particle group and other particle groups, dimensionless; β is the effective constraint coefficient, which characterizes the constraint relationship within the multi-particle size group, dimensionless.

[0170] Considering that multiple particle size groups are not simply linear combinations of single particle sizes, we must consider that the existence of other particle groups can increase or decrease this particle group. ij The comprehensive environmental impact coefficient of each particle group:

[0171]

[0172] Combining equations (3) and (4), we can get equation (5):

[0173]

[0174] In the formula, x ii ' is the final content of a certain group of particles, %; x ii is the initial content of a certain particle group, %; x jj is the initial content of each particle group, %; k ii is the migration rate of particle group i to its own particle size group, dimensionless; k ij is the migration rate of particle group j to particle size group i, dimensionless; r ij is the comprehensive environmental impact coefficient of particle size group j during the process of particle size group j being crushed into particle size group i, dimensionless; r ii It is the comprehensive environmental impact coefficient of particle size group i when particle size group i is crushed into its own particle size group. It is dimensionless.

[0175] The multi-particle group crushing should satisfy the Markov chain matrix of one-step transfer:

[0176]

[0177] In the formula, r ij is the comprehensive environmental impact coefficient of particle size group j during the process of particle size group j being crushed into particle size group i, dimensionless; r ii k is the comprehensive environmental impact coefficient of particle size group i when particle size group i is crushed into its own particle size group, dimensionless; ii is the migration rate of particle group i to its own particle size group, dimensionless; k ij is the migration rate of particle group j to particle size group i, dimensionless.

[0178] 2. Clarify model parameters

[0179] The migration rate is a quantitative representation of the survival of a certain size group and its fragmentation into other size groups. The fragmentation rate only describes the total amount of fragmentation of this size group, but does not give a quantitative description of the amount of fragmentation of this size group into other size groups. The migration rate refines the fragmentation rate and clearly indicates the amount of fragmentation of this size group into a certain size group.

[0180] For different particle groups, the migration rate is different, which can be determined according to the crushing law of a single particle size group: for a certain particle group, its limiting particle size is D i , its initial percentage is 1, and the percentage of each particle size group is obtained after one crushing. Then the percentage of each particle size group is the migration rate obtained by crushing the particle size group. i -d j , the migration rate k is:

[0181]

[0182] And Σk ij =1 (8)

[0183] In the formula, k ij is the migration rate of particle group j to particle size group i, dimensionless; F di is smaller than a certain particle size d i The mass percentage of F is dimensionless; dj is smaller than a certain particle size d j The mass percentage of D max is the maximum particle size, mm; d i is the size of particle size group i, mm, d j is the size of the j-particle size group, mm; α is the fractal dimension, dimensionless.

[0184] There is a final gradation for particle crushing. Under this gradation, large and small particles will form a self-similar distribution, and the corresponding gradation curves before loading, during loading, and at the end of loading can be expressed using a mass percentage function.

[0185]

[0186]

[0187] Where, d m is the minimum particle size of the sample, mm; d M is the maximum particle size of the sample; F0(d), F(d) and F u (d) are the mass percentages before loading, during loading, and at the end of loading, respectively, dimensionless; α is the fractal dimension, dimensionless.

[0188] Crushing parameter B r There is a certain relationship between it and crushing energy consumption:

[0189]

[0190] Where a, b and c are the crushing parameter coefficients; E B is the crushing energy consumption, kPa; B r To modify the Hardin crushing parameters.

[0191]

[0192] Crushing energy ratio m ji for:

[0193]

[0194] The effective constraint coefficient β measures the internal constraint relationship of multiple actual particle groups based on the crushing energy consumption. After experimental crushing under certain working conditions, the vector of the actual content of each particle size group is:

[0195] X ij T =[x 1j ,x 2j …,x ij ] (14)

[0196] In the formula, x ij After the experimental crushing, the particle size groups d1, d2, ..., d i The percentage of X ij T It is the transposed vector of the percentage of each particle size group after experimental crushing.

[0197] For the breakage of multiple particle size groups, the corresponding predicted calculated value is obtained from the breakage matrix of multiple particle size groups:

[0198] X ij ' T =[x 1j ',x 2j',…,x ij '] (15)

[0199] In the formula, x ij ' is the predicted value of the percentage of each particle size group calculated after particle crushing; X ij ’T It is the transposed matrix of the predicted percentage of each particle size group after particle crushing.

[0200] The following objective function is constructed by the least squares method:

[0201] X”=[x 1j '-x 1j ] 2 +[x 2i '-x 2j ] 2 +……+[x ij '-x ij ] 2 (16)

[0202] Where X” is a function of the effective restriction coefficient β. By finding the minimum value of X” and deriving the function, the corresponding effective restriction coefficient β can be obtained.

[0203] 3. Determination of model parameters

[0204] (1) Determination of fractal dimension

[0205] Fractal dimension calculation formula:

[0206]

[0207] Where M (d) is the mass of particles with a diameter smaller than d, g; M max The mass corresponding to the maximum particle size, g; d max is the maximum particle size, mm; d is a certain particle size, mm; α is the fractal dimension.

[0208] The calculation formula of fractal dimension can be obtained:

[0209]

[0210] There are experimental gradation curves in other literature Figure 2 ;

[0211] The fitting formula of fractal dimension α and compressive stress σ is as follows:

[0212] α=e·ln(σ)+f (19)

[0213] Among them, e = 0.268, f = 1.3653, correlation coefficient R 2=0.973.

[0214] (2) Crushing parameter coefficients a, b and c, see Figure 3 shown.

[0215] According to the existing experimental data, the crushing energy consumption E B and crushing parameter B r The relationship curve, such as Figure 2 The crushing parameter coefficients a, b and c can be fitted using formula (11). It can be obtained that: a = 0.9813, b = 0.2665, c = 0.1412, and the correlation coefficient R 2 =0.9694.

[0216] IV. Experimental Design and Implementation

[0217] (1) Test equipment

[0218] Considering the influence of particle size effect, it is necessary to use a dynamic triaxial tester with a large diameter specimen tube, model TAJ-2000 large dynamic and static triaxial testing instrument.

[0219] (2) Test materials

[0220] The red sandstone coarse-grained soil material is used to crush the red sandstone material mined back into three single particle sizes. They are D = 45 ~ 31.5mm, D = 25 ~ 16mm and D = 16 ~ 7.1mm. The physical properties of the material are as follows: natural moisture content 6.5%, soil particle specific gravity 2.52, plastic limit 12.37%, liquid limit 25.56%

[0221] (3) Experimental design

[0222] Experimental protocol

[0223]

[0224] (4) Loading method, see Figure 4 shown.

[0225] (5) Sample preparation method

[0226] The porosity control method was used to prepare the sample. The red sandstone coarse-grained soil was sampled with the natural moisture content and subjected to undrained triaxial test. The consolidation ratio of the sample was 1.0, the consolidation was 2h, and the designed porosity was 30%.

[0227] 5. Model Validity Verification

[0228] (1) Prediction of particle breakage evolution of a single size group

[0229] Dynamic triaxial data verification

[0230] Using existing literature data, a one-dimensional compression experiment was conducted on the crushability of 2.5 mm calcareous sand, with axial stress ranging from 0.8 to 51.2 MPa.

[0231] (2) Prediction of the crushing evolution of multiple particle size groups

[0232] Using existing literature data, this paper uses the crushing gradation of a single particle size under a load of 7.5MPa as the initial gradation and further crushes it under a load of 10MPa. The experimental values ​​and predicted values ​​representing the percentage of each particle size under different stress states are compared.

[0233] At the same time, the experimental values ​​of the percentage of each particle size in the crushed state of calcareous sand with shear strain ranging from 285% to 1180% are compared with the predicted values.

[0234] This model is suitable for predicting the crushing process of single and multi-size groups of particles. It effectively describes the crushing of particles under different soil types and stress paths, and takes into account the extreme particle groups and the particle groups exceeding the limit value. The model takes into account the mutual influence between different particle size groups, and also discusses the extreme particle groups and the particle groups exceeding the limit value. The model can effectively evaluate the crushing process of particles with high accuracy and strong prediction ability.

[0235] This paper proposes a new particle breakage model based on population evolution theory. The model not only considers the influence between particle size groups, but also introduces constraints within the particle size groups, such as migration rate, comprehensive environmental impact coefficient, crushing energy consumption ratio and effective constraint coefficient, to more accurately simulate the complex evolution of particle breakage. Through this innovation, the model can better describe the crushing behavior of single-size and multi-size group particles under different soil properties and stress states, and improve the applicability and prediction accuracy of the model in complex actual environments.

[0236] The above shows and describes the basic principles, main features and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The above embodiments and descriptions only describe the principles of the present invention. The present invention may be subject to various changes and improvements without departing from the spirit and scope of the present invention. These changes and improvements fall within the scope of the present invention. The scope of protection claimed by the present invention is defined by the attached claims and their equivalents.

Claims

1. A prediction method for rock and soil particle crushing process based on population theory, which is characterized by: The steps include: Step 1: Based on population theory, derive the mathematical model of population symbiosis; Step 2: Compare the characteristics of population evolution and particle fragmentation evolution, analyze the similarities and differences between the two properties, and extract the common points; Step 3: Couple the single particle size group model and the Markov chain model to establish a multi-particle size group breakage prediction model based on population theory; Step 4: Clarify the physical meaning and calculation method of the model parameters migration rate, comprehensive environmental impact coefficient, fragmentation energy consumption ratio and its effective control coefficient; Step 5: Model feature demonstration, simplification of multiple particle size groups, consistency of extreme particle groups, and inclusion of particle groups with content exceeding the limit; Step 6: Determination of fractal dimension and its fragmentation parameters; Step 7: Design and implementation of experimental plan; Step 8: Analyze and compare the experimental results with the predicted results, and continue to verify the model's single-size crushing and multi-size group crushing prediction effects.

2. The method for predicting rock and soil particle crushing process based on population theory according to claim 1 is characterized by: The step 1 includes the following contents: Based on population theory, a mathematical model of population symbiosis is derived. By analyzing the co-evolution mechanism between populations, the mathematical expressions of key factors such as migration rate, resource competition, and environmental adaptability are studied, and a theoretical model reflecting the symbiotic relationship between multiple populations is constructed. This model takes the description of the laws of population change as its core, comprehensively considers the competition and cooperation effects between different populations, and lays the foundation for modeling the evolution of particle fragmentation.

3. The method for predicting rock and soil particle crushing process based on population theory according to claim 1 is characterized in that: step 2 comprises the following contents: By comparing the characteristics of population evolution and particle fragmentation evolution, we conducted in-depth analysis from the aspects of dynamic process, probability distribution and behavior pattern, and extracted the common characteristics and differences between the two in essence; focusing on analyzing the randomness and evolutionary laws in the particle fragmentation process, and establishing a corresponding relationship with the individual renewal and resource allocation characteristics in population theory, providing theoretical support for the scientific nature and operability of the prediction method.

4. The method for predicting rock and soil particle crushing process based on population theory according to claim 1 is characterized in that: step 3 comprises the following contents: A state transfer matrix is ​​constructed to describe the transformation relationship between different particle size groups and the migration behavior of particles during the crushing process. At the same time, the nonlinear evolution process of particle crushing is simulated by combining the dynamic characteristics of population evolution, thereby achieving accurate prediction of the crushing behavior of multiple particle size groups.

5. The method for predicting rock and soil particle crushing process based on population theory according to claim 1 is characterized in that: step 4 comprises the following contents: Clarify the physical meaning and calculation method of each key parameter in the model, including migration rate k, comprehensive environmental impact coefficient r ij , crushing energy consumption ratio m ij and its effective constraint coefficient β; the action mechanism and influence degree of these parameters in the particle crushing process are analyzed, and targeted calculation methods are proposed to improve the applicability and accuracy of the model.

6. The method for predicting rock and soil particle crushing process based on population theory according to claim 1 is characterized in that: step 5 comprises the following contents: The characteristics of the prediction model are comprehensively demonstrated, including the applicable scope of the simplification of multi-particle size groups, the theoretical verification of the consistency of extreme particle groups, and the inclusiveness and dynamic evolution trend of particle groups with excessive content limits during the crushing process; through the detailed analysis of the model characteristics, the stability and applicability of the prediction model under various complex working conditions are ensured.

7. The method for predicting rock and soil particle crushing process based on population theory according to claim 1 is characterized in that: step 6 comprises the following contents: Analyze the fractal dimension characteristics in the crushing process, clarify the crushing parameters in the model and their relationship with the dimension, and determine the scientific method of these parameters; combine with actual experimental data to verify the rationality and reliability of parameter selection, thereby improving the model's description accuracy of the particle crushing process.

8. The method for predicting rock and soil particle crushing process based on population theory according to claim 1 is characterized in that: said step 7 comprises the following contents: Design the experimental plan and formulate in detail the selection of experimental materials, loading methods, test parameters and measurement methods to ensure that the experimental conditions can meet the requirements of the model assumptions; during the experiment, systematically collect particle size distribution data and mechanical response data of particle crushing to provide a reference basis for model verification.

9. The method for predicting rock and soil particle crushing process based on population theory according to claim 1 is characterized by: The step 8 includes the following contents: The experimental results were deeply analyzed and compared with the model prediction results, and the prediction ability of the model was verified from multiple dimensions such as data fit, consistency of evolution trend and error analysis. The prediction effects of single particle size group crushing and multi-particle size group crushing were combined to ensure its applicability and stability under different particle size groups and working conditions.

10. A non-volatile storage medium, characterized in that: The non-volatile storage medium includes a stored program, wherein when the program is executed, the device where the non-volatile storage medium is located is controlled to execute the method according to claim 1.

11. An electronic device, characterized in that: It comprises a processor and a memory; the memory stores computer-readable instructions, and the processor is used to execute the computer-readable instructions, wherein the computer-readable instructions execute the method according to claim 1 when executed.

Images