The impact of the fine-grained parameters of the anti-rotation model on the macroscopic mechanical properties of Fujian standard sand

The impact of the fine-grained parameters of the anti-rotation model on the macroscopic mechanical properties of Fujian standard sand | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article The impact of the fine-grained parameters of the anti-rotation model on the macroscopic mechanical properties of Fujian standard sand Hongshuai Liu, Bowen Ding, Dongtao Zhang, Liyun Li This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3880913/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Based on PFC3D software, the mechanical response of model meso parameters to macro parameters of the Fujian sand triaxial test was analyzed. Combined with the results of indoor tests, the range of model parameters was determined. The simulation was conducted under the confining pressures of 100kpa, 300kpa and 500kpa respectively. After the macro parameters were obtained, the influence degree of each micro parameter on the macro parameters was analyzed through orthogonal test design to conduct sensitivity analysis on the contact modulus, stiffness ratio, friction coefficient, and anti-rotation coefficient of the sample. The results show that the elastic modulus is positively correlated with the contact modulus, friction coefficient, and anti-rotation coefficient, and negatively correlated with the stiffness ratio. The peak friction angle is negatively correlated with the contact modulus and stiffness ratio and positively correlated with the friction coefficient and anti-rotation coefficient. With the increase of the contact modulus and friction coefficient, the strain softening degree of the sample will increase, and the stiffness ratio and anti-rotation coefficient have no obvious effect on the strain softening degree. The contact modulus has little influence on the stable value of the mechanical coordination number of the sample. In contrast, the friction coefficient, stiffness ratio, and anti-rotation coefficient have greatly influenced the stable value of the mechanical coordination number. Compared with the orthogonal test design, the contact modulus has the greatest impact on the elastic modulus, and the stiffness ratio has the least impact on the elastic modulus. The friction coefficient has the greatest effect on the peak friction angle, and the stiffness ratio has the least effect on the peak friction angle. Based on the orthogonal experimental design results, a set of parameters is obtained. Finally, the PFC calibration results of Fujian sand are obtained by comparing the experimental results obtained from the predicted microscopic parameters with the laboratory tests, which provides a reference for future discrete element simulation laboratory tests. Physical sciences/Engineering/Civil engineering Physical sciences/Mathematics and computing/Computational science Fujian sand macro and meso parameters orthogonal experimental design sensitivity analysis anti-rotation coefficient Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Figure 15 Figure 16 Figure 17 Figure 18 Figure 19 1 Introduction The calibration of granular parameters of sand is one of the important methods to study the mechanical properties of sand, and it is an important means to establish the correlation model between the macroscopic mechanical properties of sand and the microscopic parameters. By calibrating mesoscopic parameters, the macroscopic mechanical properties of sand can be predicted more accurately, which is of great significance for engineering practice and material design. The mesoscopic parameter calibration of sand reveals the relationship between the microstructure and macroscopic mechanical properties of sand[ 1 , 2 ]. The high-precision parameter calibration of the mesoscopic model of sand can establish a more accurate mechanical model of sand and provide a scientific basis for the behaviour prediction of soil engineering and the risk assessment of geological disasters[ 3 – 5 ]. Cundall and Strac [ 3 ]proposed the discrete element method for the first time in 1979, which represented the mechanical relationship between particles through the contact model. Common discrete element analysis software included FLAC, PFC, UDEC, Itasca3DEC, etc. Among them, Particle Flow Code (PFC) has high-precision particle simulation capability and visual particle parameter interface, which can simulate high-precision particle coupling problems [ 6 ]. In general, the physical and mechanical parameters of particle flow modes [ 7 ] cannot be directly associated with a series of microscopic structural parameters of particles. Therefore, in the simulation of engineering problems, the relation and connection between the micromechanical parameters of the PFC model and the macroscopic mechanical parameters of soil can be established using matching calculation or numerical simulation tests [ 8 , 9 ]. Due to the limitations of experimental methods, it is difficult for traditional test methods to restore actual engineering conditions and obtain accurate microstructure [ 10 ]. At the same time, the macro-mesoscale parameter calibration of the contact model in numerical simulation is the key to the successful application of the discrete element method [ 11 ]. It is mainly divided into the Hertz model [ 12 ]、linear model [ 13 ]、anti-rotation model [ 14 ], and so on. Wu et al. [ 15 ] proposed an improved Hertz-Mindlin rolling resistance model to represent the stiffness degradation response between particles effectively during particle shear. At the same time, the effect of particle shape was considered according to rolling resistance to reasonably simulate the real mechanical behaviour of granular soil under small and large strains. Zhou et al., by calibrating parameters of different shapes of wet gravel particles, concluded that the JKR model can simulate the screening process better than the Hertz-Mindlin model [ 16 ]. Compared with the Hertz-Mindlin model, the JKR model considers rigid and inelastic contact. Ehab Sabi et al. adopted a simple linear contact model. They used irregular shapes to simulate the behaviour of sand particles [ 17 ] to obtain the energy dissipation mechanism related to friction and slide between sand particles. Because the Hertzian and linear models ignore the large Angle torsion between particles due to the elastic and linear hypotheses, respectively, some simulation errors are introduced, and the model needs to be improved. Iwashita and Oda [ 18 ] found through shear tests that the rotation between particles greatly influences the mechanical properties of particles and, based on this, proposed a discrete element model considering the rotation resistance. Jiang et al. [ 19 ] proposed the definitions of pure sliding and pure rolling and used rolling and sliding components to describe contact displacement. A new rolling resistance model is developed. Then, some scholars improved the efficiency and accuracy of parameter calibration through parameter sensitivity analysis combined with computer algorithms and experimental design. Therefore, the anti-rotation model has advantages in simulating the dilatancy of dense sand by introducing the anti-rotation characteristics between particles and considering the rotation and deformation behaviours between particles. Liu Run's team and others from Tianjin University obtained the macro-mesoscopic parameter quantization relationship in the anti-rotating linear contact model, considering the coupling effect of various factors [ 8 ]. They proposed a method for rapid calibration of mesoscopic parameters. The particle anti-rotation occlusion phenomenon in sand has become an important branch of the research on the calibration of sand mesoscale parameters, and the choice of contact model and the precision of particle flow simulation will affect the calibration of sand mesoscale parameters. Based on the PFC3D discrete element simulation experiment, N. Barnett [ 20 ] conducted a sensitivity analysis on friction coefficient, stiffness ratio, porosity, and particle size distribution and then analyzed the degree of influence of different parameters on macroscopic mechanical properties by designing orthogonal experiments. Among them, the calibration methods of the discrete element mesoscale model parameters are mainly divided into trial and error methods [ 21 – 24 ] and optimization methods [ 25 , 27 ]. Among them, the trial and error method requires a lot of trial calculation and has low accuracy. In the orthogonal experiment method of the optimization method, meso parameters are regarded as independent variables, macro parameters are dependent variables, and different independent variables are combined orthogonally at different levels. Macro variables are screened based on the influence of meso parameters on macro variables [ 26 ] to determine which meso parameters can be adjusted to fit the laboratory test results quickly [ 28 – 29 ]. The orthogonal test method makes clear the law of the influence of microscopic parameters on macroscopic variables and Narrows the range of trial calculation to a great extent. Therefore, based on the discrete element software PFC3D, this paper introduced a reasonable anti-torsional moment between particles to ensure the accuracy of spherical particles in simulating the mechanical properties of irregular particles and improve particle simulation efficiency. The anti-rotational linear contact model was selected, an orthogonal test determined the controlled mesoscopic parameters, the trial range was narrowed, and the research results under different parameters were compared. Based on the correlation between the macroscopic mechanical parameters of the particle material unit and the microscopic parameters of the particle, the quantitative solution of the relationship between the macroscopic and microscopic parameters was further given through the PFC3D simulation and the sensitivity analysis of the results. 2 Numerical analysis of discrete element triaxial compression 2.1 Laboratory triaxial experiment The instrument used in this experiment is, the confining pressure range is, the experimental soil is Fujian standard sand, the height of the triaxial sample is 100mm, the diameter is 49.9mm, and the confining pressure of the sample is controlled at 100kPa, 300kPa and 500kPa respectively for consolidation undrained experiment. The physical property indexes are shown in Table 1 , and the gradation curve is shown in Fig. 1 . The particle gradation used in this experiment is the black part in the figure. Table 1 Basic physical indexes of experimental soil samples. Basic physical index of Fujian standard sand 1 ~ 0.25mm index Gs e min e max d 50 (mm) d 10 (mm) d 60 / d 10 Relative density Sand 2.64 0.69 1.01 0.175 0.115 1.65 70% According to the results obtained from the triaxial test, the deviational stress of the sample under different confining pressures of 100kPa, 300kPa and 500kPa is 260.65kPa, 980.65kPa and 1634.34kP. According to the formula of shear strength, the friction Angle is 30.93°. 2.2 PFC3D numerical simulation 2.2.1 Anti-rotation linear model The anti-rotation model is a new model proposed by Iwashita and Oda [ 18 ] after summiting predecessors' conclusions. It adds the anti-rotation coefficient based on the linear model; that is, it gives the particles the ability to resist rotation. The linear model can be imagined as a pair of springs with normal and tangential stiffness. On this basis, the anti-rotational torque, the anti-rotational linear contact model, is added. As shown in Fig. 2 . Its mechanical behaviour is similar to the linear contact model, and the anti-rotational moment is increased, whose value is equal to the product of the normal force, the anti-rotational coefficient and the effective contact radius. The relation between torque and Angle is described by introducing anti-rotation stiffness, which is defined as: \({M}^{r}={k}_{r}\theta\) (1) \({k}_{r}={k}_{s}{R}^{2}\) (2) \(\frac{1}{R}=\frac{1}{{R}_{1}}+\frac{1}{{R}_{2}}\) (3) In the above formula, \({k}_{r}\) is the rolling stiffness of the particle, R is the effective contact radius, and \({k}_{s}\) is the tangential stiffness. 2.2.2 Model building First, a triaxial model was established according to the experiment, and a cylinder was used to simulate the triaxial instrument wall to constrain the particles. The initial size of the container was 100mm high, and the diameter was 49.9mm. Then, experimental particles were generated. The set gradation of the generated particles and the actual gradation generated by the experiment are shown in Fig. 3 , where the horizontal axis represents the particle diameter and the vertical axis represents the volume fraction. As the particles used in this experiment are too small, the number of particles generated according to the actual particle size is too large, which is difficult for ordinary computers to calculate. To reduce the calculation amount, according to the size effect analysis of previous scholars in the simulation [ 31 ]. [ 32 ] When the ratio of model size to the average radius of the sample is greater than 40, the number of particles has little influence on the final simulation result, so the method of reducing the model size is adopted. After trial calculation, the model's size was finally determined to be 2.98mm in radius and 10mm in height. Then, a steel wall was generated on each bottom of the model to simulate the loading plate, and the calculation area was covered with the command generate plane. In order to prevent sample overflow during the experiment, the height of the cylinder barrel was increased by 1.4 times, and the loading speed was 5% strain rate /min. 2.2.3 Experimental procedure The experiment is divided into three steps: sample formation, consolidation and loading. After the sample is generated, the sample is consolidated, and a certain force is exerted on the wall through servo control. The consolidation stress here is set as the laboratory consolidation stress (100kPa, 300kPa, 500kPa) to make the particle system uniform. The model adopted in this paper is the anti-rotation linear contact model, and the parameters to be determined include the anti-rotation coefficient, Young's modulus, friction coefficient and stiffness ratio. 3 Sensitivity analysis of microscopic parameters 3.1 Contact modulus The contact modulus between particles differs from the elastic modulus, which refers to the relationship between the contact force between particles and the amount of overlap between particles. In this section, the influence of the contact modulus on the macro-parameters of the samples is studied by setting different contact modulus for the samples, and the contact modulus is set as 300MPa, 600MPa, 900MPa and 1200MPa, respectively. Other granular mesoscopic parameters are shown in Table 2. Table 2 Particle mesoscopic parameters Anti-rotation coefficient Friction coefficient Stiffness ratio 0.4 0.4 2 The influence of the contact modulus on the stress-strain curve of the sample is shown in Fig. 4 . It can be seen from Fig. 4 that with the continuous increase of the contact modulus, the initial elastic modulus also gradually increases because with the increase of the contact modulus, the tangential and normal stiffness of the particles will also increase. During the loading process, the contact force will also increase, and the elastic modulus will increase with the increase of the contact modulus under the same strain. The peak strength decreases gradually, and when the contact modulus increases to a certain value, the peak strength tends to be the same. Under the same confining pressure, the strain-softening degree of the sample decreases gradually with the increase of the contact modulus. The coordination number is a micrometric parameter describing the contact number around particles, also known as the average contact number. Thornton [32] proposed that particles with less than two contact numbers bear no stress in the specimen. Excluding these coordination numbers, that is, the mechanical coordination number describes the mechanical property expression expressed by the soil skeleton as follows: $${C}_{n}=\frac{2{N}_{c}-{N}_{s1}}{{N}_{s}-{N}_{s0}-{N}_{s1}}$$ 4 In the formula, \({\text{N}}_{\text{c}}\)is the total number of contact samples, \({N}_{s1}\)and\({N}_{s0}\) are the number of particles whose contact numbers equal 1 and 0, respectively, and \({N}_{s}\)is the total number of particles. Figure 5 describes the changes in mechanical coordination numbers under different confining pressures when the contact modulus is different. With strain development, the mechanical coordination numbers increase first and then gradually decrease and eventually become stable; under different confining pressures, the coordination numbers gradually decrease with the increase of contact modulus; meanwhile, the coordination numbers under coaxial strain increase with the increase of confining pressure. Compared with the stress-strain curve, it can be seen that the mechanical coordination number also begins to become stable at the corresponding strain point when the peak strength appears, indicating that the sample skeleton particles rearrange after the peak strength to obtain a more stable structure. The relationship between macroscopic mechanical parameters and the contact modulus of soil under different confining pressures is shown in Fig. 6–7. As the contact modulus increases, the elastic modulus linearly increases, and the peak friction Angle linearly decreases. The elastic modulus and friction Angle can be obtained quickly by changing the contact modulus of particles during the calibration of sand soil, and the calibration efficiency can be improved. 3.2 Friction coefficient The friction coefficient is a micrometric parameter to characterize the friction characteristics between particles. Under the confining pressure conditions of 100kPa, 300kPa and 500kPa, the friction coefficient is set as 0.2, 0.4, 0.6 and 0.8. Other microscopic parameters of particles are shown in Table 3. The influence of the friction coefficient on the macroscopic parameters of the sample is studied. Table 3 Particle mesoscopic parameters Anti-rotation coefficient Contact modulus(Emod) Stiffness ratio 0.4 0.3GPa 2 It can be seen from Fig. 8 that with the increasing friction coefficient, the peak strength also increases. This can be explained by the fact that with the increase of the friction coefficient, the occlusion between the particles increases, and the stress required for the particles to slide is also greater, so the peak strength is increased. As can be seen from the observed figure, the degree of strain softening increases with the increasing friction coefficient, and the slope of the curve decreases with the increase of confining pressure under the same friction coefficient. Figure 9 describes the relationship between the mechanical coordination number and the friction coefficient of the sample under the confining pressure of 100kPa, 300kPa and 500kPa. It can be seen from the figure that the mechanical coordination number firstly increases, then decreases and finally tends to be stable with the increase of strain. With the increase of the friction coefficient, the stable value of the mechanical coordination number gradually decreases, which is because the increase of the friction coefficient increases the biting force between the particles, making it more difficult for the particles to generate relative movement and then produce a larger contact force between the particles. The efficiency of the transmission force is higher, and the contact number is less. The relationship between macroscopic mechanical parameters and the friction coefficient of soil under different confining pressures is shown in Fig. 10–11. As the friction coefficient increases, the elastic modulus and the peak friction Angle show an increasing trend. When calibrating sand soil, the target elastic modulus can be quickly obtained by adjusting the friction coefficient to improve the calibration speed. 3.3 Stiffness ratio The stiffness ratio refers to the ratio of normal stiffness and tangential stiffness. Under the conditions of confining pressure of 100kPa, 300kPa and 500kPa, the stiffness ratio is set as 2,6,10,14, respectively. Other microscopic parameters of particles are shown in Table 4 to study the influence of stiffness ratio on the macroscopic parameters of samples. Table 4 Particle mesoscopic parameters Anti-rotation coefficient Contact modulus(Emod) Friction coefficient 0.4 0.3GPa 0.4 It can be seen from Fig. 12 that with the increasing stiffness ratio, the slope gradually decreases; that is, the initial elastic modulus gradually decreases. This is because when the stiffness comparison is small when the normal stiffness is similar to the tangential stiffness, the sample has higher tangential deformation resistance, smaller particle displacement, larger overlap, and greater elastic modulus due to the change of axial stress. The degree of strain softening decreases with the increase in stiffness ratio. Moreover, the peak strength becomes smaller; that is, the tangential stiffness gradually decreases, which reduces the tangential resistance to deformation of the material and causes the specimen to gradually change from shear failure to tangential failure. The stress-strain curves all show a downward trend after the peak strength, showing a strain softening trend, and under higher confining pressure, the strain softening trend is more obvious. Fig. 13 - 15 shows the changes in mechanical coordination number with stiffness ratio under confining pressure of 100kPa, 300kPa and 500kPa, and the trend of first increasing, then decreasing, and finally stabilizing with the development of shear strain, that is, the number of particle contacts changes from less to more and then to less. Samples under low confining pressure are more stable after reaching peak strength than those under high confining pressure. Fig. 14 - 15 shows the relationship between macroscopic coefficients and microscopic parameters of the samples. It can be seen from the figure that both the elastic modulus and friction Angle of the samples show a linear trend of decrease with the increase of the stiffness ratio, and the changes of the elastic modulus and friction Angle under low confining pressure are lower than those under high confining pressure. 3.4 Anti-rotation coefficient Under the confining pressure of 100kPa, 300kPa and 500kPa, the anti-rotation coefficients were set as 0.2, 0.4, 0.6 and 0.8, respectively. Other granular mesoscale parameters were shown in Table 5 to study the influence of anti-rotation coefficients on the macro-parameters of the samples. Table 5 Particle mesoscopic parameters Stiffness ratio Contact modulus(Emod) Friction coefficient 2 0.3GPa 0.4 The anti-rotation coefficient will significantly affect the peak strength of the sample. As can be seen from Fig. 16 , with the increase of the anti-rotation coefficient, the resistance to be overcome by the relative rotation of the particles becomes larger, and the peak strength of the sample gradually increases. However, it can be observed that the increased range is decreasing continuously; that is, the coefficient value has an effect range, within which the peak strength and residual strength will increase with the increase of the anti-rotation coefficient. The increased range will become smaller and smaller. At the same time, the stress-strain curve will remain unchanged after the threshold because the increase of the anti-rotation coefficient leads to relatively large particle rotation resistance and small particle rotation. The strain-softening tendency of the sample increases with the increase of the anti-rotation coefficient, and the strain-softening conforms to the properties of medium-dense sand. Figure 17 shows the relationship between the anti-rotation coefficient and the mechanical coordination number. It can be seen from the figure that with the increase of the anti-rotation coefficient, the stable value of the mechanical coordination number will decrease accordingly. As the rotational impedance makes it more difficult for the particles to rotate, the contact force between the particles increases, and the effective contact decreases. Fig. 18 - 19 shows the relationship between macroscopic coefficients and microscopic parameters of the sample. It can be seen from the figure that both the elastic modulus and friction Angle of the sample tend to increase with the increase of the stiffness ratio. 4 Orthogonal experimental design Orthogonal experiments were designed to conduct sensitivity analysis on the influence degree of the contact modulus, stiffness ratio, friction coefficient and anti-rotation coefficient of the microscopic parameters on the elastic modulus and peak friction Angle. The four factors and four levels of the design are shown in Table 6 . Under the confining pressure of 300kPa, 16 tests were conducted, respectively, as shown in Table 7 , and the results of the orthogonal experiments are shown in Table 8 . and the results of the orthogonal experiments are shown in Table 8 . The dependent variables are the elastic modulus and peak friction Angle of each group of experiments. Compared with the test results, the contact modulus and friction coefficient have the greatest influence on the elastic modulus, followed by the anti-rotation coefficient, and the stiffness ratio is the least. The results of orthogonal experiments analyzed by IBM SPSS are shown in Table 8 . The dependent variables are the elastic modulus and peak friction Angle of each group of experiments. Compared with the test results, the contact modulus and friction coefficient have the greatest influence on the elastic modulus, followed by the anti-rotation coefficient, and the stiffness ratio is the least. The order of the degree of influence on the elastic modulus is contact modulus > friction coefficient > anti-rotation coefficient > stiffness ratio. The friction coefficient and anti-rotation coefficient have the greatest influence on the peak friction Angle, and the contact modulus and stiffness ratio have almost no influence on the peak friction Angle. The order of the degree of influence on the peak friction Angle is friction coefficient > anti-rotation coefficient > contact modulus > stiffness ratio. The results of orthogonal experimental design and numerical analysis are consistent. Table 6 Four factors and four levels table Class number Emod, GPa Fric Rr_fric Krtio level 1 0.3 0.2 0.2 2 level 2 0.6 0.4 0.4 6 level 3 0.9 0.6 0.6 10 level 4 1.2 0.8 0.8 14 Table 7 Four-factor four-level orthogonal test table Number Emod,GPa Fric Rr_fric Krtio Elastic modulus, \({E}_{S}\) Peak friction angle,φ(°) 1 0.3 0.2 0.2 2 1.43101E7 29.33 2 0.3 0.4 0.4 6 2.47028E7 32.71 3 0.3 0.6 0.6 10 2.63206E7 34.53 4 0.3 0.8 0.8 14 1.99343E7 35.23 5 0.6 0.4 0.6 2 3.815E7 34.24 6 0.6 0.2 0.8 6 2.47645E7 30.27 7 0.6 0.8 0.2 10 3.93115E7 32.98 8 0.6 0.6 0.4 14 3.97037E7 34.01 9 0.9 0.6 0.8 2 5.0593E7 36.97 10 0.9 0.8 0.6 6 4.28102E7 36.92 11 0.9 0.2 0.4 10 3.68731E7 29.94 12 0.9 0.4 0.2 14 3.52881E7 30.94 13 1.2 0.8 0.4 2 7.983E7 35.88 14 1.2 0.6 0.2 6 4.77067E7 31.97 15 1.2 0.4 0.8 10 3.95994E7 33.86 16 1.2 0.2 0.6 14 2.6755E7 30.06 Table 8 Orthogonal experiment results table Test of intersubjective effects Source Class III sum of squares Degree of freedom Mean square F Significance Modified model \({E}_{S}\) 3.340E + 15 a 12 2.78E + 14 4.814 0.111 φ(°) 87.143 b 12 7.262 3.070 0.193 intercept \({E}_{S}\) 2.15E + 16 1 2.15E + 16 372.094 0.000 φ(°) 17544.539 1 17544.539 7416.237 0.000 emod \({E}_{S}\) 1.60E + 15 3 5.32E + 14 9.209 0.050 φ(°) 1.795 3 0.598 0.253 0.856 kratio \({E}_{S}\) 1.73E + 14 3 5.77E + 13 0.998 0.501 φ(°) 1.264 3 0.421 0.178 0.905 fric \({E}_{S}\) 8.46E + 14 3 2.82E + 14 4.879 0.113 φ(°) 60.155 3 20.052 8.476 0.056 rr_fric \({E}_{S}\) 5.59E + 14 3 1.86E + 14 3.220 0.181 φ(°) 14.783 3 4.928 2.083 0.281 error \({E}_{S}\) 1.73E + 14 3 5.78E + 13 φ(°) 7.097 3 2.366 total \({E}_{S}\) 2.50E + 16 16 φ(°) 17638.779 16 Revised total \({E}_{S}\) 3.51E + 15 15 φ(°) 94.240 15 a. R square = .951(Adjusted R square = .753) b. R square = .925(Adjusted R square = .623) 5 Conclusion Triaxial tests were carried out using the discrete element method under different confining pressures of 100kPa, 300kPa and 500kPa by introducing the anti-rotational contact model and reducing the particle size. The value range of model parameters was determined, and the influence of each mesoscopic parameter on the macro-parameters was analyzed through orthogonal test design. The sensitivity analysis of stiffness ratio, friction coefficient and anti-rotation coefficient shows the following conclusions: 1. The test shows that the elastic modulus of Fujian standard sand is positively correlated with the contact modulus, friction coefficient and anti-rotation coefficient and negatively correlated with the stiffness ratio. 2. The peak friction Angle is negatively correlated with the contact modulus and stiffness ratio and positively correlated with the friction and anti-rotation coefficients. 3. The contact modulus has little influence on the stability value of the mechanical digit of the sample. In contrast, the friction coefficient, stiffness ratio and anti-rotation coefficient greatly influence the stability value of the mechanical coordination number. It is necessary to set an appropriate anti-rotation coefficient value to improve the accuracy of parameter calibration. 4. Compared with the orthogonal experimental design, it can be concluded that the contact modulus has the greatest influence on the elastic modulus, and the stiffness ratio has the least influence on the elastic modulus. The friction coefficient has the greatest influence on the peak friction Angle, and the stiffness ratio has the least influence on the peak friction Angle. The sensitivity analysis of sand micromechanics parameters can provide a scientific basis for engineering design and construction and ensure the safety and stability of the project. Therefore, the sensitivity analysis of sand micromechanical parameters should be studied more. However, the sensitivity analysis of the mechanical properties of sand is not systematic at the present stage. This paper introduces the anti-rotation coefficient into a linear model, which is closer to the actual sand flow pattern than the non-rotation model. Improving the precision of sand micro parameter calibration and developing a high-fidelity precision particle flow model will become the focus of future research.. Declarations Author Contribution All the authors of this research paper made important contributions to the study. Specific contributions are as follows:H.S. Liu: Responsible for experimental design and data collection and analysis.B.W. Ding: Writing the thesisD.T. Zhang: Provided experimental equipment and technical support.L.Y. Li: Participated in experimental data analysis and result interpretation.We hereby declare that all authors involved in this study have reviewed and approved the final submitted version of the paper and agree to make a public statement about their contribution to this research. Acknowledgement This study was supported by the National Natural Science Foundation of China (Grant No.52278384) and the Science Research Project of Hebei Education Department (Grant No. ZD2020157); The authors would like to acknowledge the support by the HBU Innovation Team for Multi-disaster Prevention in Transportation Geotechnics (Grant No. IT2023C04). References S. Y. He, et al., The study on loess liquefaction in China: a systematic review, Natural Hazards. 103 (2) (2020) 1639-1669. D. Su, Z. Yang, A. 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Zhao, et al., Calibrating microparameters of DEM models by using CEM, DE, EFO, MFO, SSO algorithms and the optimal hyperparameters, Computational Particle Mechanics. (2023) 1-14. A. Fakhimi, Application of slightly overlapped circular particles assembly in numerical simulation of rocks with high friction angles, Engineering Geology. 74 (1-2) (2004) 129-138. S. Luc, F. V. Donzé, A DEM model for soft and hard rocks: role of grain interlocking on strength, Journal of the Mechanics and Physics of Solids. 61.2 (2013): 352-369. Z. H. Xu,W. Y. Wang,P. Lin, et al., A parameter calibration method for PFC simulation: Development and a case study of limestone, Geomech. Eng, 22 (1) (2020) 97-108. Yao L M, **ao Z M, Liu J B, et al. An optimized CFD-DEM method for fluid-particle coupling dynamics analysis[J]. International Journal of Mechanical Sciences, 2020, 174: 105503. L. Yu,C. Li,H. Zhang, et al., The uplift resistance of submarine pipelines buried in medium dense sand, Ocean Engineering. 266 (2022) 112732. S. Ji, J. Karlovšek, Optimized differential evolution algorithm for solving DEM material calibration problem, Engineering with Computers. 39 (3) (2023) 2001-2016. C. Yadong, Y. Yan, S. H. E. Yuexin, Method for determining mesoscopic parameters of sand in three-dimensional particle flow code numerical modeling, Chinese Journal of Geotechnical Engineering. 35 (zk2) (2013) 88-93. Y. X. Yang,J. H. Hu,J. F. Wang, Numerical Simulation of Meso-mechanism of Liquefaction in Saturated Stratified Silty Sands, Advanced Materials Research. 601 (2013) 222-226. X. T. Yin,Y. N. Zheng,S. K. Ma, Study of inner scale ratio of rock and soil material based on numerical tests of particle flow code, Rock and Soil Mechanics. 32 (04) (2011) 1211-1215. H. T. Liu,X. H. Cheng, Discrete element analysis for size effects of coarse-grained soils, Rock and Soil Mechanics. 30 (S1) (2009) 287-292. C. Thornton, Numerical simulations of deviatoric shear deformation of granular media, Géotechnique. 50 (1) (2000) 43–53. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3880913","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":271219248,"identity":"2d9867a6-136d-47a1-87f3-982a67a80351","order_by":0,"name":"Hongshuai Liu","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA3UlEQVRIiWNgGAWjYBACPmYGNgYGA5v6/e09bCABxgZCWtjAWgrSGDfwnCFWCxh9OMy4QSKHWC3sPGYPfxgcZjaXfHvsMQ+DjeyGA8zPHuB3GI+5MY9BOpvl7Lx0Yx6GNOMNB9jMDQhoMZNmMLDmYbidYybNw3A4ccMBHjYJQlokfxgwSzDcPAPS8p84LRI8Bs4GBjd4QFoOEKOFrUyaxyAtQbInx9xwjkGy8czDbGZ4tfDzH94m+eOPTQI/+xmzB28q7GT7jjc/w6sFDYCCipkE9aNgFIyCUTAKsAMAYm47QXQCiZAAAAAASUVORK5CYII=","orcid":"","institution":"Hebei University","correspondingAuthor":true,"prefix":"","firstName":"Hongshuai","middleName":"","lastName":"Liu","suffix":""},{"id":271219249,"identity":"dae2cf3b-dfc9-4b60-8ae4-14e55847e0ed","order_by":1,"name":"Bowen Ding","email":"","orcid":"","institution":"Hebei University","correspondingAuthor":false,"prefix":"","firstName":"Bowen","middleName":"","lastName":"Ding","suffix":""},{"id":271219250,"identity":"3f216164-b761-4ca8-929d-144593ea14eb","order_by":2,"name":"Dongtao Zhang","email":"","orcid":"","institution":"Hebei University","correspondingAuthor":false,"prefix":"","firstName":"Dongtao","middleName":"","lastName":"Zhang","suffix":""},{"id":271219251,"identity":"3abce592-dbae-4c21-973e-60570ba72863","order_by":3,"name":"Liyun Li","email":"","orcid":"","institution":"Beijing University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Liyun","middleName":"","lastName":"Li","suffix":""}],"badges":[],"createdAt":"2024-01-20 07:14:16","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-3880913/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3880913/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":50723671,"identity":"ecd9f3a0-978f-430b-98c6-a0c44b9774b0","added_by":"auto","created_at":"2024-02-06 10:30:10","extension":"jpg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":13578,"visible":true,"origin":"","legend":"\u003cp\u003eFujian sand grain grading curve\u003c/p\u003e","description":"","filename":"1.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3880913/v1/e641df4352da8fc2730522bd.jpg"},{"id":50724001,"identity":"8e8d6e54-b404-49d5-93ee-45eea7b4e16d","added_by":"auto","created_at":"2024-02-06 10:38:10","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":15100,"visible":true,"origin":"","legend":"\u003cp\u003eContact model\u003c/p\u003e","description":"","filename":"2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3880913/v1/8ca987864a3e5131cac98c31.jpg"},{"id":50723665,"identity":"90220273-e90e-4f12-9866-0d1ee49b5836","added_by":"auto","created_at":"2024-02-06 10:30:10","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":43967,"visible":true,"origin":"","legend":"\u003cp\u003eParticle size distribution of numerical experiment\u003c/p\u003e","description":"","filename":"3.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3880913/v1/d0f6bcc452a98addfbc5f0bb.jpg"},{"id":50724002,"identity":"f066b53e-4f24-47ca-8376-98dbee5a28a5","added_by":"auto","created_at":"2024-02-06 10:38:10","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":44739,"visible":true,"origin":"","legend":"\u003cp\u003eStress-strain curve under different contact modulus\u003c/p\u003e","description":"","filename":"4.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3880913/v1/63081ce157db4e86f6b82933.jpg"},{"id":50723666,"identity":"a479d3ff-57bf-4120-9dfc-cc5ccaa009d7","added_by":"auto","created_at":"2024-02-06 10:30:10","extension":"jpg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":61182,"visible":true,"origin":"","legend":"\u003cp\u003eVariation of mechanical coordination number under different contact modulus\u003c/p\u003e","description":"","filename":"5.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3880913/v1/c5381325169b848c8456942a.jpg"},{"id":50723669,"identity":"78993d23-9af9-4fb1-bb50-fd208d62a1c7","added_by":"auto","created_at":"2024-02-06 10:30:10","extension":"jpg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":13696,"visible":true,"origin":"","legend":"\u003cp\u003eInfluence of contact modulus on elastic modulus under different confining pressures\u003c/p\u003e","description":"","filename":"6.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3880913/v1/dca836e8aecacbb4bf5feb4d.jpg"},{"id":50723674,"identity":"08dee020-9508-4b9b-8abf-abf5db8527f3","added_by":"auto","created_at":"2024-02-06 10:30:10","extension":"jpg","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":14251,"visible":true,"origin":"","legend":"\u003cp\u003eInfluence of contact modulus on peak friction Angle under different confining pressures\u003c/p\u003e","description":"","filename":"7.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3880913/v1/427c038287024da482b2b4b6.jpg"},{"id":50724010,"identity":"64d194d8-d16a-42f1-b3ae-4199a5bc1d81","added_by":"auto","created_at":"2024-02-06 10:38:10","extension":"jpg","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":52430,"visible":true,"origin":"","legend":"\u003cp\u003eStress-strain curves under different friction coefficients\u003c/p\u003e","description":"","filename":"8.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3880913/v1/3f423e4c387880408f1c3929.jpg"},{"id":50724006,"identity":"606e59b6-919c-491d-bee1-5e147df41c77","added_by":"auto","created_at":"2024-02-06 10:38:10","extension":"jpg","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":59470,"visible":true,"origin":"","legend":"\u003cp\u003eVariation of mechanical coordination number under different friction coefficients\u003c/p\u003e","description":"","filename":"9.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3880913/v1/a6cc12e3a210d2e57920fec8.jpg"},{"id":50724274,"identity":"fe6606be-30f1-49e4-91fe-79e5c1cb47ac","added_by":"auto","created_at":"2024-02-06 10:46:10","extension":"jpg","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":11982,"visible":true,"origin":"","legend":"\u003cp\u003eRelationship between friction coefficient and elastic modulus under different confining pressures\u003c/p\u003e","description":"","filename":"10.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3880913/v1/9c4d552ffa9959cf9661edf1.jpg"},{"id":50724003,"identity":"ec4edacf-d1e5-4be1-8c20-0a27ca7edd06","added_by":"auto","created_at":"2024-02-06 10:38:10","extension":"jpg","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":11991,"visible":true,"origin":"","legend":"\u003cp\u003eRelationship between friction coefficient and peak friction Angle under different confining pressures\u003c/p\u003e","description":"","filename":"11.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3880913/v1/72c7f9f3a3582cfc76678e15.jpg"},{"id":50724008,"identity":"2b01d845-f695-4307-91a5-3a8f6651c6b8","added_by":"auto","created_at":"2024-02-06 10:38:10","extension":"jpg","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":54376,"visible":true,"origin":"","legend":"\u003cp\u003eStress-strain curves under different stiffness ratios\u003c/p\u003e","description":"","filename":"12.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3880913/v1/bb242aabc96b627b82ef7572.jpg"},{"id":50724913,"identity":"966758b9-5e46-4188-ab2b-96f7dcb0c278","added_by":"auto","created_at":"2024-02-06 10:54:10","extension":"jpg","order_by":13,"title":"Figure 13","display":"","copyAsset":false,"role":"figure","size":60827,"visible":true,"origin":"","legend":"\u003cp\u003eVariation of mechanical coordination number under different stiffness ratio\u003c/p\u003e","description":"","filename":"13.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3880913/v1/c0248b09a07679f7d356392d.jpg"},{"id":50723680,"identity":"68ac7f58-5e08-4442-970e-d93d14710312","added_by":"auto","created_at":"2024-02-06 10:30:10","extension":"jpg","order_by":14,"title":"Figure 14","display":"","copyAsset":false,"role":"figure","size":14176,"visible":true,"origin":"","legend":"\u003cp\u003eRelation between stiffness ratio and elastic modulus under different confining pressures\u003c/p\u003e","description":"","filename":"14.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3880913/v1/3d9abb0709a60c9c1723fc96.jpg"},{"id":50723677,"identity":"3180e1ca-e147-407d-a977-1bcfffbf206a","added_by":"auto","created_at":"2024-02-06 10:30:10","extension":"jpg","order_by":15,"title":"Figure 15","display":"","copyAsset":false,"role":"figure","size":15194,"visible":true,"origin":"","legend":"\u003cp\u003eRelation between stiffness ratio and peak friction Angle under different confining pressures\u003c/p\u003e","description":"","filename":"15.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3880913/v1/d7a96fac662374a7dff13f94.jpg"},{"id":50724011,"identity":"d53599f0-60a0-455a-a068-d38547bf773d","added_by":"auto","created_at":"2024-02-06 10:38:10","extension":"jpg","order_by":16,"title":"Figure 16","display":"","copyAsset":false,"role":"figure","size":57457,"visible":true,"origin":"","legend":"\u003cp\u003eStress-strain curves under different anti rotation coefficient\u003c/p\u003e","description":"","filename":"16.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3880913/v1/263d62a7bc35bababe7e4b9f.jpg"},{"id":50724281,"identity":"78c1bb6d-34a1-4d75-a027-a2b5a04eb114","added_by":"auto","created_at":"2024-02-06 10:46:10","extension":"jpg","order_by":17,"title":"Figure 17","display":"","copyAsset":false,"role":"figure","size":56984,"visible":true,"origin":"","legend":"\u003cp\u003eVariation of mechanical coordination number under different anti-rotation coefficient\u003c/p\u003e","description":"","filename":"17.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3880913/v1/ab78a86c8755d0398edf2e83.jpg"},{"id":50724005,"identity":"ed405466-9834-4d12-a101-8cfe5c5a98c8","added_by":"auto","created_at":"2024-02-06 10:38:10","extension":"jpg","order_by":18,"title":"Figure 18","display":"","copyAsset":false,"role":"figure","size":12594,"visible":true,"origin":"","legend":"\u003cp\u003eCoefficient and elastic modulus under different confining pressures\u003c/p\u003e","description":"","filename":"18.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3880913/v1/bc4332604abe205f2380e9ef.jpg"},{"id":50724275,"identity":"f973f6cf-ed44-422e-9035-10dd1e7f70ef","added_by":"auto","created_at":"2024-02-06 10:46:10","extension":"jpg","order_by":19,"title":"Figure 19","display":"","copyAsset":false,"role":"figure","size":10833,"visible":true,"origin":"","legend":"\u003cp\u003eCoefficient and peak friction angle under different confining pressures\u003c/p\u003e","description":"","filename":"19.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3880913/v1/91a772f91c7ca60926928826.jpg"},{"id":56405769,"identity":"c3d0ee13-22fa-4ac0-a2d3-d66fd093ecce","added_by":"auto","created_at":"2024-05-13 18:24:28","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1178977,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3880913/v1/d02f5141-dc73-45c2-8878-131d5bd8b9b2.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"The impact of the fine-grained parameters of the anti-rotation model on the macroscopic mechanical properties of Fujian standard sand","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003eThe calibration of granular parameters of sand is one of the important methods to study the mechanical properties of sand, and it is an important means to establish the correlation model between the macroscopic mechanical properties of sand and the microscopic parameters. By calibrating mesoscopic parameters, the macroscopic mechanical properties of sand can be predicted more accurately, which is of great significance for engineering practice and material design. The mesoscopic parameter calibration of sand reveals the relationship between the microstructure and macroscopic mechanical properties of sand[\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e, \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. The high-precision parameter calibration of the mesoscopic model of sand can establish a more accurate mechanical model of sand and provide a scientific basis for the behaviour prediction of soil engineering and the risk assessment of geological disasters[\u003cspan additionalcitationids=\"CR4\" citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]. Cundall and Strac [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e]proposed the discrete element method for the first time in 1979, which represented the mechanical relationship between particles through the contact model. Common discrete element analysis software included FLAC, PFC, UDEC, Itasca3DEC, etc. Among them, Particle Flow Code (PFC) has high-precision particle simulation capability and visual particle parameter interface, which can simulate high-precision particle coupling problems [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e]. In general, the physical and mechanical parameters of particle flow modes [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e] cannot be directly associated with a series of microscopic structural parameters of particles. Therefore, in the simulation of engineering problems, the relation and connection between the micromechanical parameters of the PFC model and the macroscopic mechanical parameters of soil can be established using matching calculation or numerical simulation tests [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eDue to the limitations of experimental methods, it is difficult for traditional test methods to restore actual engineering conditions and obtain accurate microstructure [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]. At the same time, the macro-mesoscale parameter calibration of the contact model in numerical simulation is the key to the successful application of the discrete element method [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e]. It is mainly divided into the Hertz model [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e]、linear model [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e]、anti-rotation model [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e], and so on. Wu et al. [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e] proposed an improved Hertz-Mindlin rolling resistance model to represent the stiffness degradation response between particles effectively during particle shear. At the same time, the effect of particle shape was considered according to rolling resistance to reasonably simulate the real mechanical behaviour of granular soil under small and large strains. Zhou et al., by calibrating parameters of different shapes of wet gravel particles, concluded that the JKR model can simulate the screening process better than the Hertz-Mindlin model [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]. Compared with the Hertz-Mindlin model, the JKR model considers rigid and inelastic contact. Ehab Sabi et al. adopted a simple linear contact model. They used irregular shapes to simulate the behaviour of sand particles [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e] to obtain the energy dissipation mechanism related to friction and slide between sand particles. Because the Hertzian and linear models ignore the large Angle torsion between particles due to the elastic and linear hypotheses, respectively, some simulation errors are introduced, and the model needs to be improved. Iwashita and Oda [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e] found through shear tests that the rotation between particles greatly influences the mechanical properties of particles and, based on this, proposed a discrete element model considering the rotation resistance. Jiang et al. [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e] proposed the definitions of pure sliding and pure rolling and used rolling and sliding components to describe contact displacement. A new rolling resistance model is developed. Then, some scholars improved the efficiency and accuracy of parameter calibration through parameter sensitivity analysis combined with computer algorithms and experimental design. Therefore, the anti-rotation model has advantages in simulating the dilatancy of dense sand by introducing the anti-rotation characteristics between particles and considering the rotation and deformation behaviours between particles. Liu Run's team and others from Tianjin University obtained the macro-mesoscopic parameter quantization relationship in the anti-rotating linear contact model, considering the coupling effect of various factors [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e]. They proposed a method for rapid calibration of mesoscopic parameters. The particle anti-rotation occlusion phenomenon in sand has become an important branch of the research on the calibration of sand mesoscale parameters, and the choice of contact model and the precision of particle flow simulation will affect the calibration of sand mesoscale parameters. Based on the PFC3D discrete element simulation experiment, N. Barnett [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e] conducted a sensitivity analysis on friction coefficient, stiffness ratio, porosity, and particle size distribution and then analyzed the degree of influence of different parameters on macroscopic mechanical properties by designing orthogonal experiments. Among them, the calibration methods of the discrete element mesoscale model parameters are mainly divided into trial and error methods [\u003cspan additionalcitationids=\"CR22 CR23\" citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e] and optimization methods [\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e, \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e]. Among them, the trial and error method requires a lot of trial calculation and has low accuracy. In the orthogonal experiment method of the optimization method, meso parameters are regarded as independent variables, macro parameters are dependent variables, and different independent variables are combined orthogonally at different levels. Macro variables are screened based on the influence of meso parameters on macro variables [\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e] to determine which meso parameters can be adjusted to fit the laboratory test results quickly [\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e]. The orthogonal test method makes clear the law of the influence of microscopic parameters on macroscopic variables and Narrows the range of trial calculation to a great extent.\u003c/p\u003e \u003cp\u003eTherefore, based on the discrete element software PFC3D, this paper introduced a reasonable anti-torsional moment between particles to ensure the accuracy of spherical particles in simulating the mechanical properties of irregular particles and improve particle simulation efficiency. The anti-rotational linear contact model was selected, an orthogonal test determined the controlled mesoscopic parameters, the trial range was narrowed, and the research results under different parameters were compared. Based on the correlation between the macroscopic mechanical parameters of the particle material unit and the microscopic parameters of the particle, the quantitative solution of the relationship between the macroscopic and microscopic parameters was further given through the PFC3D simulation and the sensitivity analysis of the results.\u003c/p\u003e"},{"header":"2 Numerical analysis of discrete element triaxial compression","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 Laboratory triaxial experiment\u003c/h2\u003e \u003cp\u003eThe instrument used in this experiment is, the confining pressure range is, the experimental soil is Fujian standard sand, the height of the triaxial sample is 100mm, the diameter is 49.9mm, and the confining pressure of the sample is controlled at 100kPa, 300kPa and 500kPa respectively for consolidation undrained experiment. The physical property indexes are shown in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, and the gradation curve is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. The particle gradation used in this experiment is the black part in the figure.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eBasic physical indexes of experimental soil samples.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"8\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colspan=\"8\" nameend=\"c8\" namest=\"c1\"\u003e \u003cp\u003eBasic physical index of Fujian standard sand 1\u0026thinsp;~\u0026thinsp;0.25mm\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eindex\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGs\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cem\u003ee\u003c/em\u003e\u003csub\u003emin\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cem\u003ee\u003c/em\u003e\u003csub\u003emax\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u003cem\u003ed\u003c/em\u003e\u003csub\u003e50\u003c/sub\u003e (mm)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u003cem\u003ed\u003c/em\u003e\u003csub\u003e10\u003c/sub\u003e (mm)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e\u003cem\u003ed\u003c/em\u003e\u003csub\u003e60\u003c/sub\u003e/ \u003cem\u003ed\u003c/em\u003e\u003csub\u003e10\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003eRelative density\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSand\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e2.64\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.69\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.175\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.115\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e1.65\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e70%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAccording to the results obtained from the triaxial test, the deviational stress of the sample under different confining pressures of 100kPa, 300kPa and 500kPa is 260.65kPa, 980.65kPa and 1634.34kP. According to the formula of shear strength, the friction Angle is 30.93\u0026deg;.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2 PFC3D numerical simulation\u003c/h2\u003e \u003cdiv id=\"Sec5\" class=\"Section3\"\u003e \u003ch2\u003e2.2.1 Anti-rotation linear model\u003c/h2\u003e \u003cp\u003eThe anti-rotation model is a new model proposed by Iwashita and Oda [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e] after summiting predecessors' conclusions. It adds the anti-rotation coefficient based on the linear model; that is, it gives the particles the ability to resist rotation. The linear model can be imagined as a pair of springs with normal and tangential stiffness. On this basis, the anti-rotational torque, the anti-rotational linear contact model, is added. As shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIts mechanical behaviour is similar to the linear contact model, and the anti-rotational moment is increased, whose value is equal to the product of the normal force, the anti-rotational coefficient and the effective contact radius. The relation between torque and Angle is described by introducing anti-rotation stiffness, which is defined as:\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"No\" id=\"Taba\" border=\"1\"\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({M}^{r}={k}_{r}\\theta\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e(1)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{r}={k}_{s}{R}^{2}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e(2)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{1}{R}=\\frac{1}{{R}_{1}}+\\frac{1}{{R}_{2}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e(3)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eIn the above formula, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{r}\\)\u003c/span\u003e\u003c/span\u003e is the rolling stiffness of the particle, R is the effective contact radius, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{s}\\)\u003c/span\u003e\u003c/span\u003e is the tangential stiffness.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section3\"\u003e \u003ch2\u003e2.2.2 Model building\u003c/h2\u003e \u003cp\u003eFirst, a triaxial model was established according to the experiment, and a cylinder was used to simulate the triaxial instrument wall to constrain the particles. The initial size of the container was 100mm high, and the diameter was 49.9mm. Then, experimental particles were generated. The set gradation of the generated particles and the actual gradation generated by the experiment are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, where the horizontal axis represents the particle diameter and the vertical axis represents the volume fraction. As the particles used in this experiment are too small, the number of particles generated according to the actual particle size is too large, which is difficult for ordinary computers to calculate. To reduce the calculation amount, according to the size effect analysis of previous scholars in the simulation [\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e]. [\u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e] When the ratio of model size to the average radius of the sample is greater than 40, the number of particles has little influence on the final simulation result, so the method of reducing the model size is adopted. After trial calculation, the model's size was finally determined to be 2.98mm in radius and 10mm in height. Then, a steel wall was generated on each bottom of the model to simulate the loading plate, and the calculation area was covered with the command generate plane. In order to prevent sample overflow during the experiment, the height of the cylinder barrel was increased by 1.4 times, and the loading speed was 5% strain rate /min.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section3\"\u003e \u003ch2\u003e2.2.3 Experimental procedure\u003c/h2\u003e \u003cp\u003eThe experiment is divided into three steps: sample formation, consolidation and loading. After the sample is generated, the sample is consolidated, and a certain force is exerted on the wall through servo control. The consolidation stress here is set as the laboratory consolidation stress (100kPa, 300kPa, 500kPa) to make the particle system uniform. The model adopted in this paper is the anti-rotation linear contact model, and the parameters to be determined include the anti-rotation coefficient, Young's modulus, friction coefficient and stiffness ratio.\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e"},{"header":"3 Sensitivity analysis of microscopic parameters","content":"\u003cdiv id=\"Sec9\"\u003e\n \u003ch2\u003e3.1 Contact modulus\u003c/h2\u003e\n \u003cp\u003eThe contact modulus between particles differs from the elastic modulus, which refers to the relationship between the contact force between particles and the amount of overlap between particles. In this section, the influence of the contact modulus on the macro-parameters of the samples is studied by setting different contact modulus for the samples, and the contact modulus is set as 300MPa, 600MPa, 900MPa and 1200MPa, respectively. Other granular mesoscopic parameters are shown in Table\u0026nbsp;2.\u003c/p\u003e\n \u003cdiv\u003e\n \u003ctable id=\"Tab2\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv\u003eTable 2\u003c/div\u003e\n \u003cdiv\u003e\n \u003cp\u003eParticle mesoscopic parameters\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eAnti-rotation coefficient\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eFriction coefficient\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eStiffness ratio\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003c/div\u003e\n \u003cp\u003eThe influence of the contact modulus on the stress-strain curve of the sample is shown in \u003cstrong\u003eFig.\u0026nbsp;4\u003c/strong\u003e. It can be seen from \u003cstrong\u003eFig.\u0026nbsp;4\u003c/strong\u003e that with the continuous increase of the contact modulus, the initial elastic modulus also gradually increases because with the increase of the contact modulus, the tangential and normal stiffness of the particles will also increase. During the loading process, the contact force will also increase, and the elastic modulus will increase with the increase of the contact modulus under the same strain. The peak strength decreases gradually, and when the contact modulus increases to a certain value, the peak strength tends to be the same. Under the same confining pressure, the strain-softening degree of the sample decreases gradually with the increase of the contact modulus.\u003c/p\u003e\n \u003cp\u003eThe coordination number is a micrometric parameter describing the contact number around particles, also known as the average contact number. Thornton [32] proposed that particles with less than two contact numbers bear no stress in the specimen. Excluding these coordination numbers, that is, the mechanical coordination number describes the mechanical property expression expressed by the soil skeleton as follows:\u003c/p\u003e\n \u003cdiv id=\"Equ1\"\u003e\n \u003cdiv id=\"FileID_Equ1\" name=\"EquationSource\"\u003e$${C}_{n}=\\frac{2{N}_{c}-{N}_{s1}}{{N}_{s}-{N}_{s0}-{N}_{s1}}$$\u003c/div\u003e\n \u003cdiv\u003e4\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003eIn the formula, \\({\\text{N}}_{\\text{c}}\\)is the total number of contact samples, \\({N}_{s1}\\)and\\({N}_{s0}\\) are the number of particles whose contact numbers equal 1 and 0, respectively, and \\({N}_{s}\\)is the total number of particles.\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003eFigure\u0026nbsp;5\u003c/strong\u003e describes the changes in mechanical coordination numbers under different confining pressures when the contact modulus is different. With strain development, the mechanical coordination numbers increase first and then gradually decrease and eventually become stable; under different confining pressures, the coordination numbers gradually decrease with the increase of contact modulus; meanwhile, the coordination numbers under coaxial strain increase with the increase of confining pressure. Compared with the stress-strain curve, it can be seen that the mechanical coordination number also begins to become stable at the corresponding strain point when the peak strength appears, indicating that the sample skeleton particles rearrange after the peak strength to obtain a more stable structure.\u003c/p\u003e\n \u003cp\u003eThe relationship between macroscopic mechanical parameters and the contact modulus of soil under different confining pressures is shown in Fig.\u0026nbsp;6\u0026ndash;7. As the contact modulus increases, the elastic modulus linearly increases, and the peak friction Angle linearly decreases. The elastic modulus and friction Angle can be obtained quickly by changing the contact modulus of particles during the calibration of sand soil, and the calibration efficiency can be improved.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec10\"\u003e\n \u003ch2\u003e3.2 Friction coefficient\u003c/h2\u003e\n \u003cp\u003eThe friction coefficient is a micrometric parameter to characterize the friction characteristics between particles. Under the confining pressure conditions of 100kPa, 300kPa and 500kPa, the friction coefficient is set as 0.2, 0.4, 0.6 and 0.8. Other microscopic parameters of particles are shown in Table\u0026nbsp;3. The influence of the friction coefficient on the macroscopic parameters of the sample is studied.\u003c/p\u003e\n \u003cdiv\u003e\n \u003ctable id=\"Tab3\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv\u003eTable 3\u003c/div\u003e\n \u003cdiv\u003e\n \u003cp\u003eParticle mesoscopic parameters\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eAnti-rotation coefficient\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eContact modulus(Emod)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eStiffness ratio\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.3GPa\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003c/div\u003e\n \u003cp\u003eIt can be seen from \u003cstrong\u003eFig.\u0026nbsp;8\u003c/strong\u003e that with the increasing friction coefficient, the peak strength also increases. This can be explained by the fact that with the increase of the friction coefficient, the occlusion between the particles increases, and the stress required for the particles to slide is also greater, so the peak strength is increased. As can be seen from the observed figure, the degree of strain softening increases with the increasing friction coefficient, and the slope of the curve decreases with the increase of confining pressure under the same friction coefficient.\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003eFigure\u0026nbsp;9\u003c/strong\u003e describes the relationship between the mechanical coordination number and the friction coefficient of the sample under the confining pressure of 100kPa, 300kPa and 500kPa. It can be seen from the figure that the mechanical coordination number firstly increases, then decreases and finally tends to be stable with the increase of strain. With the increase of the friction coefficient, the stable value of the mechanical coordination number gradually decreases, which is because the increase of the friction coefficient increases the biting force between the particles, making it more difficult for the particles to generate relative movement and then produce a larger contact force between the particles. The efficiency of the transmission force is higher, and the contact number is less.\u003c/p\u003e\n \u003cp\u003eThe relationship between macroscopic mechanical parameters and the friction coefficient of soil under different confining pressures is shown in Fig.\u0026nbsp;10\u0026ndash;11. As the friction coefficient increases, the elastic modulus and the peak friction Angle show an increasing trend. When calibrating sand soil, the target elastic modulus can be quickly obtained by adjusting the friction coefficient to improve the calibration speed.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec11\"\u003e\n \u003ch2\u003e3.3 Stiffness ratio\u003c/h2\u003e\n \u003cp\u003eThe stiffness ratio refers to the ratio of normal stiffness and tangential stiffness. Under the conditions of confining pressure of 100kPa, 300kPa and 500kPa, the stiffness ratio is set as 2,6,10,14, respectively. Other microscopic parameters of particles are shown in Table\u0026nbsp;4 to study the influence of stiffness ratio on the macroscopic parameters of samples.\u003c/p\u003e\n \u003cdiv\u003e\n \u003ctable id=\"Tab4\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv\u003eTable 4\u003c/div\u003e\n \u003cdiv\u003e\n \u003cp\u003eParticle mesoscopic parameters\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eAnti-rotation coefficient\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eContact modulus(Emod)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eFriction coefficient\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.3GPa\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003c/div\u003e\n \u003cp\u003eIt can be seen from \u003cstrong\u003eFig.\u0026nbsp;12\u003c/strong\u003e that with the increasing stiffness ratio, the slope gradually decreases; that is, the initial elastic modulus gradually decreases. This is because when the stiffness comparison is small when the normal stiffness is similar to the tangential stiffness, the sample has higher tangential deformation resistance, smaller particle displacement, larger overlap, and greater elastic modulus due to the change of axial stress. The degree of strain softening decreases with the increase in stiffness ratio. Moreover, the peak strength becomes smaller; that is, the tangential stiffness gradually decreases, which reduces the tangential resistance to deformation of the material and causes the specimen to gradually change from shear failure to tangential failure. The stress-strain curves all show a downward trend after the peak strength, showing a strain softening trend, and under higher confining pressure, the strain softening trend is more obvious.\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003eFig. 13\u003c/strong\u003e-\u003cstrong\u003e15\u003c/strong\u003e shows the changes in mechanical coordination number with stiffness ratio under confining pressure of 100kPa, 300kPa and 500kPa, and the trend of first increasing, then decreasing, and finally stabilizing with the development of shear strain, that is, the number of particle contacts changes from less to more and then to less. Samples under low confining pressure are more stable after reaching peak strength than those under high confining pressure.\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003eFig. 14\u003c/strong\u003e-\u003cstrong\u003e15\u003c/strong\u003e shows the relationship between macroscopic coefficients and microscopic parameters of the samples. It can be seen from the figure that both the elastic modulus and friction Angle of the samples show a linear trend of decrease with the increase of the stiffness ratio, and the changes of the elastic modulus and friction Angle under low confining pressure are lower than those under high confining pressure.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec12\"\u003e\n \u003ch2\u003e3.4 Anti-rotation coefficient\u003c/h2\u003e\n \u003cp\u003eUnder the confining pressure of 100kPa, 300kPa and 500kPa, the anti-rotation coefficients were set as 0.2, 0.4, 0.6 and 0.8, respectively. Other granular mesoscale parameters were shown in Table\u0026nbsp;5 to study the influence of anti-rotation coefficients on the macro-parameters of the samples.\u003c/p\u003e\n \u003cdiv\u003e\n \u003ctable id=\"Tab5\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv\u003eTable 5\u003c/div\u003e\n \u003cdiv\u003e\n \u003cp\u003eParticle mesoscopic parameters\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eStiffness ratio\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eContact modulus(Emod)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eFriction coefficient\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.3GPa\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003c/div\u003e\n \u003cp\u003eThe anti-rotation coefficient will significantly affect the peak strength of the sample. As can be seen from \u003cstrong\u003eFig.\u0026nbsp;16\u003c/strong\u003e, with the increase of the anti-rotation coefficient, the resistance to be overcome by the relative rotation of the particles becomes larger, and the peak strength of the sample gradually increases. However, it can be observed that the increased range is decreasing continuously; that is, the coefficient value has an effect range, within which the peak strength and residual strength will increase with the increase of the anti-rotation coefficient. The increased range will become smaller and smaller. At the same time, the stress-strain curve will remain unchanged after the threshold because the increase of the anti-rotation coefficient leads to relatively large particle rotation resistance and small particle rotation. The strain-softening tendency of the sample increases with the increase of the anti-rotation coefficient, and the strain-softening conforms to the properties of medium-dense sand.\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003eFigure\u0026nbsp;17\u003c/strong\u003e shows the relationship between the anti-rotation coefficient and the mechanical coordination number. It can be seen from the figure that with the increase of the anti-rotation coefficient, the stable value of the mechanical coordination number will decrease accordingly. As the rotational impedance makes it more difficult for the particles to rotate, the contact force between the particles increases, and the effective contact decreases.\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003eFig. 18\u003c/strong\u003e-\u003cstrong\u003e19\u003c/strong\u003e shows the relationship between macroscopic coefficients and microscopic parameters of the sample. It can be seen from the figure that both the elastic modulus and friction Angle of the sample tend to increase with the increase of the stiffness ratio.\u003c/p\u003e\n\u003c/div\u003e"},{"header":"4 Orthogonal experimental design","content":"\u003cp\u003eOrthogonal experiments were designed to conduct sensitivity analysis on the influence degree of the contact modulus, stiffness ratio, friction coefficient and anti-rotation coefficient of the microscopic parameters on the elastic modulus and peak friction Angle. The four factors and four levels of the design are shown in Table\u0026nbsp;\u003cspan refid=\"Tab6\" class=\"InternalRef\"\u003e6\u003c/span\u003e. Under the confining pressure of 300kPa, 16 tests were conducted, respectively, as shown in Table\u0026nbsp;\u003cspan refid=\"Tab7\" class=\"InternalRef\"\u003e7\u003c/span\u003e, and the results of the orthogonal experiments are shown in Table\u0026nbsp;\u003cspan refid=\"Tab8\" class=\"InternalRef\"\u003e8\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eand the results of the orthogonal experiments are shown in Table\u0026nbsp;\u003cspan refid=\"Tab8\" class=\"InternalRef\"\u003e8\u003c/span\u003e. The dependent variables are the elastic modulus and peak friction Angle of each group of experiments. Compared with the test results, the contact modulus and friction coefficient have the greatest influence on the elastic modulus, followed by the anti-rotation coefficient, and the stiffness ratio is the least. The results of orthogonal experiments analyzed by IBM SPSS are shown in Table\u0026nbsp;\u003cspan refid=\"Tab8\" class=\"InternalRef\"\u003e8\u003c/span\u003e. The dependent variables are the elastic modulus and peak friction Angle of each group of experiments. Compared with the test results, the contact modulus and friction coefficient have the greatest influence on the elastic modulus, followed by the anti-rotation coefficient, and the stiffness ratio is the least. The order of the degree of influence on the elastic modulus is contact modulus\u0026thinsp;\u0026gt;\u0026thinsp;friction coefficient\u0026thinsp;\u0026gt;\u0026thinsp;anti-rotation coefficient\u0026thinsp;\u0026gt;\u0026thinsp;stiffness ratio. The friction coefficient and anti-rotation coefficient have the greatest influence on the peak friction Angle, and the contact modulus and stiffness ratio have almost no influence on the peak friction Angle. The order of the degree of influence on the peak friction Angle is friction coefficient\u0026thinsp;\u0026gt;\u0026thinsp;anti-rotation coefficient\u0026thinsp;\u0026gt;\u0026thinsp;contact modulus\u0026thinsp;\u0026gt;\u0026thinsp;stiffness ratio. The results of orthogonal experimental design and numerical analysis are consistent.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab6\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 6\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eFour factors and four levels table\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eClass number\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eEmod, GPa\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eFric\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eRr_fric\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eKrtio\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003elevel 1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003elevel 2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003elevel 3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003elevel 4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab7\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 7\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eFour-factor four-level orthogonal test table\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"7\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNumber\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eEmod,GPa\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eFric\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eRr_fric\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eKrtio\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eElastic modulus, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({E}_{S}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003ePeak\u003c/p\u003e \u003cp\u003efriction angle,φ(\u0026deg;)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e1.43101E7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e29.33\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e2.47028E7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e32.71\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e2.63206E7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e34.53\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e1.99343E7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e35.23\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e3.815E7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e34.24\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e2.47645E7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e30.27\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e3.93115E7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e32.98\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e3.97037E7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e34.01\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e5.0593E7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e36.97\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e4.28102E7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e36.92\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e3.68731E7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e29.94\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e3.52881E7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e30.94\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e13\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e7.983E7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e35.88\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e4.77067E7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e31.97\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e3.95994E7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e33.86\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e2.6755E7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e30.06\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab8\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 8\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eOrthogonal experiment results table\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"7\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colspan=\"7\" nameend=\"c7\" namest=\"c1\"\u003e \u003cp\u003eTest of intersubjective effects\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e \u003cp\u003eSource\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eClass III sum of squares\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eDegree of freedom\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eMean square\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eF\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003eSignificance\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eModified model\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({E}_{S}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e3.340E\u0026thinsp;+\u0026thinsp;15\u003csup\u003ea\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e2.78E\u0026thinsp;+\u0026thinsp;14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e4.814\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.111\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eφ(\u0026deg;)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e87.143\u003csup\u003eb\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e7.262\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e3.070\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.193\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eintercept\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({E}_{S}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2.15E\u0026thinsp;+\u0026thinsp;16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e2.15E\u0026thinsp;+\u0026thinsp;16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e372.094\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eφ(\u0026deg;)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e17544.539\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e17544.539\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e7416.237\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eemod\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({E}_{S}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.60E\u0026thinsp;+\u0026thinsp;15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e5.32E\u0026thinsp;+\u0026thinsp;14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e9.209\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.050\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eφ(\u0026deg;)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.795\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.598\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.253\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.856\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003ekratio\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({E}_{S}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.73E\u0026thinsp;+\u0026thinsp;14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e5.77E\u0026thinsp;+\u0026thinsp;13\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.998\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.501\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eφ(\u0026deg;)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.264\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.421\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.178\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.905\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003efric\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({E}_{S}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e8.46E\u0026thinsp;+\u0026thinsp;14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e2.82E\u0026thinsp;+\u0026thinsp;14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e4.879\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.113\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eφ(\u0026deg;)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e60.155\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e20.052\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e8.476\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.056\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003err_fric\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({E}_{S}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e5.59E\u0026thinsp;+\u0026thinsp;14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1.86E\u0026thinsp;+\u0026thinsp;14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e3.220\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.181\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eφ(\u0026deg;)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e14.783\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e4.928\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e2.083\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.281\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eerror\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({E}_{S}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.73E\u0026thinsp;+\u0026thinsp;14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e5.78E\u0026thinsp;+\u0026thinsp;13\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eφ(\u0026deg;)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e7.097\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e2.366\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003etotal\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({E}_{S}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2.50E\u0026thinsp;+\u0026thinsp;16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eφ(\u0026deg;)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e17638.779\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eRevised total\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({E}_{S}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e3.51E\u0026thinsp;+\u0026thinsp;15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eφ(\u0026deg;)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e94.240\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"7\" nameend=\"c7\" namest=\"c1\"\u003e \u003cp\u003ea. R square\u0026thinsp;=\u0026thinsp;.951(Adjusted R square\u0026thinsp;=\u0026thinsp;.753)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"7\" nameend=\"c7\" namest=\"c1\"\u003e \u003cp\u003eb. R square\u0026thinsp;=\u0026thinsp;.925(Adjusted R square\u0026thinsp;=\u0026thinsp;.623)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e"},{"header":"5 Conclusion","content":"\u003cp\u003eTriaxial tests were carried out using the discrete element method under different confining pressures of 100kPa, 300kPa and 500kPa by introducing the anti-rotational contact model and reducing the particle size. The value range of model parameters was determined, and the influence of each mesoscopic parameter on the macro-parameters was analyzed through orthogonal test design. The sensitivity analysis of stiffness ratio, friction coefficient and anti-rotation coefficient shows the following conclusions:\u003c/p\u003e\n\u003cp\u003e1. The test shows that the elastic modulus of Fujian standard sand is positively correlated with the contact modulus, friction coefficient and anti-rotation coefficient and negatively correlated with the stiffness ratio.\u003c/p\u003e\n\u003cp\u003e2. The peak friction Angle is negatively correlated with the contact modulus and stiffness ratio and positively correlated with the friction and anti-rotation coefficients.\u003c/p\u003e\n\u003cp\u003e3. The contact modulus has little influence on the stability value of the mechanical digit of the sample. In contrast, the friction coefficient, stiffness ratio and anti-rotation coefficient greatly influence the stability value of the mechanical coordination number. It is necessary to set an appropriate anti-rotation coefficient value to improve the accuracy of parameter calibration.\u003c/p\u003e\n\u003cp\u003e4. Compared with the orthogonal experimental design, it can be concluded that the contact modulus has the greatest influence on the elastic modulus, and the stiffness ratio has the least influence on the elastic modulus. The friction coefficient has the greatest influence on the peak friction Angle, and the stiffness ratio has the least influence on the peak friction Angle.\u003c/p\u003e\n\u003cp\u003eThe sensitivity analysis of sand micromechanics parameters can provide a scientific basis for engineering design and construction and ensure the safety and stability of the project. Therefore, the sensitivity analysis of sand micromechanical parameters should be studied more. However, the sensitivity analysis of the mechanical properties of sand is not systematic at the present stage. This paper introduces the anti-rotation coefficient into a linear model, which is closer to the actual sand flow pattern than the non-rotation model. Improving the precision of sand micro parameter calibration and developing a high-fidelity precision particle flow model will become the focus of future research..\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eAll the authors of this research paper made important contributions to the study. Specific contributions are as follows:H.S. Liu: Responsible for experimental design and data collection and analysis.B.W. Ding: Writing the thesisD.T. Zhang: Provided experimental equipment and technical support.L.Y. Li: Participated in experimental data analysis and result interpretation.We hereby declare that all authors involved in this study have reviewed and approved the final submitted version of the paper and agree to make a public statement about their contribution to this research.\u003c/p\u003e\u003ch2\u003eAcknowledgement\u003c/h2\u003e \u003cp\u003eThis study was supported by the National Natural Science Foundation of China (Grant No.52278384) and the Science Research Project of Hebei Education Department (Grant No. ZD2020157); The authors would like to acknowledge the support by the HBU Innovation Team for Multi-disaster Prevention in Transportation Geotechnics (Grant No. IT2023C04).\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eS. Y. He, et al., The study on loess liquefaction in China: a systematic review, Natural Hazards. 103 (2) (2020) 1639-1669.\u003c/li\u003e\n\u003cli\u003eD. Su, Z. Yang, A. 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Ma, Study of inner scale ratio of rock and soil material based on numerical tests of particle flow code, Rock and Soil Mechanics. 32 (04) (2011) 1211-1215.\u003c/li\u003e\n\u003cli\u003eH. T. Liu,X. H. Cheng, Discrete element analysis for size effects of coarse-grained soils, Rock and Soil Mechanics. 30 (S1) (2009) 287-292.\u003c/li\u003e\n\u003cli\u003eC. Thornton, Numerical simulations of deviatoric shear deformation of granular media, G\u0026eacute;otechnique. 50 (1) (2000) 43\u0026ndash;53.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":true,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Fujian sand, macro and meso parameters, orthogonal experimental design, sensitivity analysis, anti-rotation coefficient","lastPublishedDoi":"10.21203/rs.3.rs-3880913/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3880913/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eBased on PFC3D software, the mechanical response of model meso parameters to macro parameters of the Fujian sand triaxial test was analyzed. Combined with the results of indoor tests, the range of model parameters was determined. The simulation was conducted under the confining pressures of 100kpa, 300kpa and 500kpa respectively. After the macro parameters were obtained, the influence degree of each micro parameter on the macro parameters was analyzed through orthogonal test design to conduct sensitivity analysis on the contact modulus, stiffness ratio, friction coefficient, and anti-rotation coefficient of the sample. The results show that the elastic modulus is positively correlated with the contact modulus, friction coefficient, and anti-rotation coefficient, and negatively correlated with the stiffness ratio. The peak friction angle is negatively correlated with the contact modulus and stiffness ratio and positively correlated with the friction coefficient and anti-rotation coefficient. With the increase of the contact modulus and friction coefficient, the strain softening degree of the sample will increase, and the stiffness ratio and anti-rotation coefficient have no obvious effect on the strain softening degree. The contact modulus has little influence on the stable value of the mechanical coordination number of the sample. In contrast, the friction coefficient, stiffness ratio, and anti-rotation coefficient have greatly influenced the stable value of the mechanical coordination number. Compared with the orthogonal test design, the contact modulus has the greatest impact on the elastic modulus, and the stiffness ratio has the least impact on the elastic modulus. The friction coefficient has the greatest effect on the peak friction angle, and the stiffness ratio has the least effect on the peak friction angle. Based on the orthogonal experimental design results, a set of parameters is obtained. Finally, the PFC calibration results of Fujian sand are obtained by comparing the experimental results obtained from the predicted microscopic parameters with the laboratory tests, which provides a reference for future discrete element simulation laboratory tests.\u003c/p\u003e","manuscriptTitle":"The impact of the fine-grained parameters of the anti-rotation model on the macroscopic mechanical properties of Fujian standard sand","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-02-06 10:30:05","doi":"10.21203/rs.3.rs-3880913/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"476bdc89-b91b-4e44-8670-7a4ecb6b542f","owner":[],"postedDate":"February 6th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":28588357,"name":"Physical sciences/Engineering/Civil engineering"},{"id":28588358,"name":"Physical sciences/Mathematics and computing/Computational science"}],"tags":[],"updatedAt":"2024-05-13T18:24:15+00:00","versionOfRecord":[],"versionCreatedAt":"2024-02-06 10:30:05","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-3880913","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3880913","identity":"rs-3880913","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}