A Microscopic Insight into Effect of Particle Gradation on Dynamic Compaction of Dry Sand

A Microscopic Insight into Effect of Particle Gradation on Dynamic Compaction of Dry 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 Research Article A Microscopic Insight into Effect of Particle Gradation on Dynamic Compaction of Dry Sand Yuqi Li, Liangchen Xu, Fu’an Yang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3416318/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 20 Jan, 2024 Read the published version in Multiscale and Multidisciplinary Modeling, Experiments and Design → Version 1 posted 7 You are reading this latest preprint version Abstract Particle gradation is an important feature of granular materials, which has a significant influence on the mechanical properties of soil. Several dynamic compaction (DC) tests for mono-sized dry sand samples and a well-graded dry sand sample were modeled using discrete element method. The effect of particle gradation on crater depth was analyzed as well as coordination number, porosity and contact stress from a microscopic view. It is indicated that the change rates of dynamic stress, coordination number and porosity of the well-graded sample were greater than the results from the mono-size samples. For the mono-sized samples and the well-graded sample, the differences of dynamic contact stress, coordination number and porosity became larger as the distance of measurement point from ground surface increased. The results also demonstrate from a microscopic view that the well-graded soil and the soil sample with small particle size were more prone to become dense under DC. This study at a grain level is helpful to understand the microscopic mechanism of DC and has certain guiding significance to the construction of DC. Dynamic compaction Discrete element method Particle gradation Coordination number Porosity Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 1 Introduction When the geological conditions of a construction site are poor, it is usually necessary to carry out foundation treatment on the site. Dynamic compaction (DC), first created by Menard in 1969 (Menard and Broise 1975 ), is a widely used foundation treatment method. This method can improve the bearing capacity of foundation and reduce post-construction settlement by converting the gravitational potential energy of a lifted tamper into kinetic energy and acting on the ground in the form of impact load. It is suitable for a wide variety of soil types, especially sandy soils and coarse granular soils; however, caution is required while used in saturated soils and clayey soils (Lukas et al. 1980; Miao et al. 2006 ; Ghassemi et al. 2010 ). Numerous studies on DC have been carried out through analytical solution, field and laboratory test and numerical simulation (Chow et al. 1992 ; Feng et al. 2000 ; Hu et al. 2001 ; Gu and Lee 2002 ; Wang et al. 2017 ; Li et al. 2018 ) due to its wide application in ground treatment. Few analytical solutions of DC to date have been deduced because of complexity of impact contact stress and heterogeneity of foundation soil. Aiming at the limitation of linear spring dashpot dynamic models applied to relatively soft ground, an improved analytical model was proposed and verified through experimental data (Thilakasiri et al. 1996 ). For a liquefiable soil and soft soil interbedded foundation in highway engineering practice, the in situ tests for evaluating effectiveness of the lower tamping energy were performed and the formulation of predicting the improvement depth of DC was proposed (Miao et al. 2006 ). As for field and laboratory tests, many researchers had done plenty of outstanding work (Mayne et al. 1984 ; Takada and Oshima 1994 ; Feng et al. 2000 ; Hu et al. 2001 ; Hwang and Tu 2006 ; Jia et al. 2009 ). Mayne et al. ( 1984 ) studied the relationship between the size of the craters, ground vibration levels and depth of influence and the level of energy per blow according to the data from more than 120 field tests. Hwang and Tu ( 2006 ) analyzed the effects of tamping energy and isolation trench on ground vibration based on the ground vibration data of an industrial site during DC. Through laboratory DC tests, the influence of pounder base shape on the efficiency of DC (Feng et al. 2000 ) and the variation of microstructural variation of soils (Hu et al. 2001 ; Jia et al. 2009 ) were analyzed. Furthermore, the centrifuge test of DC was also used to simulate an actual foundation treatment project (Takada and Oshima 1994 ). Apart from those investigations mentioned above, a lot of studies were focused on the numerical modeling of DC (Poran and Rodriguez 1992 ; Chow et al. 1992 ; Pan and Selby 2002 ; Gu and Lee 2002 ; Lee and Gu 2004 ; Ghassemi et al. 2010 ; Xie et al. 2013 ; Wang et al. 2017 ; Li et al. 2018 ). Some methods for estimating the degree and depth of improvement resulting from DC were proposed (Chow et al. 1992 ; Lee and Gu 2004 ; Wang et al. 2017 ) based on the results of numerical simulation. Also, a fully coupled hydro-mechanical finite element code for evaluating DC of saturated granular soil was developed by Ghassemi et al. ( 2010 ) and the simulation results were verified by the data from a real field case of DC treatment in a highway. In addition, micro mechanism of DC was studied using the discrete element method (Li et al. 2018 ; Ma et al. 2014 ; Jiang et al. 2017 ), in which the improvement effect and the influence depth of DC were analyzed based on the porosity change of the foundation soil (Ma et al. 2014 ; Jiang et al. 2017 ), and a centrifuge test of dry sand under DC (Takada and Oshima 1994 ) was modeled in order to investigate the microscopic characteristics of sand under DC at a particle level (Li et al. 2018 , 2021 ). These foregoing studies provided many data for design and construction of DC, and the influence of gradation on the behavior of soil has also attracted interest of a few researchers (Lu et al. 2013 ; Luo et al. 2014 ; Fang et al. 2016 ; Gu et al. 2017 ). The effects of the initial particle size and moisture on the compressive behavior of dense sand under high strain rates were investigated by Lu et al. ( 2013 ) and Luo et al. ( 2014 ). An analysis approach to model the mechanical behavior of dry sand under static and dynamic loadings was put forward and applied in the domain of projectile penetration into the sand particulate system, and the effect of particle size on the projectile penetration depth and ballistic instability was further discussed (Fang et al. 2016 ). By studying the effect of particle size distribution on the small strain shear stiffness of granular soils, Gu et al. ( 2017 ) explored the fundamental mechanism controlling this small strain shear stiffness. However, they seldom studied the influence of particle gradation on DC, especially at a grain level. In this paper, in order to assess the effect of particle gradation on DC from a microscopic view, based on the discrete element method (Ma et al. 2014 ; Jiang et al. 2017 ; Li et al. 2018 ; Liu et al. 2020 ), the DC simulations of sand samples with different grading were conducted using PFC software (Itasca Consulting Group 2004 ), and the dynamic stress, coordination number, porosity and crater depth of soils were thoroughly investigated. 2 Principle of PFC PFC 2D (Particle Flow Code in 2 Dimensions) models the movement and interaction of rigid circular particles by the distinct element method (Itasca Consulting Group 2004 ). The calculations performed in the PFC alternate between the application of Newton’s second law to the particles and a force-displacement law at the contacts. Newton’s second law is used to determine the motion of each particle arising from the contact and body forces acting on it, while the force-displacement law is used to update the contact forces arising from the relative motion at each contact. In PFC 2D , the presence of walls requires only that the force-displacement law accounts for ball-wall contacts. Newton’s second law is not applied to walls since the wall motion is specified by users (Itasca Consulting Group 2004 ). The calculation cycle is illustrated in Fig. 1 . 3 Numerical simulation 3.1 Constitutive model In this paper, a hysteretic damping contact model was selected to simulate the dynamic characteristics of soil during DC. As shown in Fig. 2 (Itasca Consulting Group 2004 ), the normal stiffness of the model is different at the loading and unloading stages, and the normal stiffness on loading is smaller than that on unloading in the hysteretic damping model. The normal stiffnesses on loading, k n_load , and on unloading, k n_unload , used in the hysteretic damping model are calculated using the following equations: $${k_{{\text{n\_load}}}}=\frac{{2{R_{\text{h}}}{k_0}}}{{1+{R_{\text{h}}}}},\begin{array}{*{20}{c}} {}&{} \end{array}{k_{{\text{n\_unload}}}}=\frac{{2{k_0}}}{{1+{R_{\text{h}}}}}$$ 1 where k 0 is the initial normal stiffness, k n , of particles under static loading, which was determined by numerical biaxial test; and R h is the ratio of normal stiffness on loading, k n_load , to that on unloading, k un_load (0.05 < R h <1.0). The dynamic hysteretic effect of soil is remarkable when R h is close to 0, and R h was determined to be 0.75 in this work according to a previous study (Jia et al. 2015 ). 3.2 Modeling An axisymmetric numerical model for the sand sample adopted by Takada & Oshima centrifuge test (Takada and Oshima 1994 ) was established using PFC 2D . The tamper was generated by 40 overlapping balls using the clump command in PFC 2D . The discrete element simulation of dry sand centrifugal test under DC has been carried out by Li et al. ( 2018 ), and the same microscopic parameters were adopted in this paper, as listed in Table 1 . The generation method of the sample is the same to the literature (Li et al. 2018 ), and those particles, greater than 0.3 mm in diameter, were chosen to generate balls according to the proportion of particles in different range of particle size, as is shown in Figs. 3 and 4 and Table 2 (S1). A vertical acceleration of 50 g was applied to each particle so that a centrifugal field can be obtained, thus a centrifuge test can be simulated. The time step was determined to be 1.0×10 − 6 s. Table 1 Microscopic parameters of numerical model Ball Wall Tamper Particle density /(kg·m − 3 ) 2650 -- 1912 Normal stiffness k n /(N·m − 1 ) 1.2e7 1e10 1e8 Shear stiffness k s /(N·m − 1 ) 8e6 1e10 1e8 Friction coefficient 0.7 0 0.1 Table 2 Parameters of specimens for biaxial test S1(Sample 1) S2(Sample 2) S3(Sample 3) S4(Sample 4) Particle diameter /(mm) 0.3-2.0 2.0 1.0 0.3 Model dimensions* /(m×m) 0.04×0.02 0.12×0.06 0.06×0.03 0.02×0.01 Particle number 2392 2016 2016 2489 Normal stiffness k n /(N·m − 1 ) 1.2e7 8.2e6 8.0e6 8.5e6 Shear stiffness k s /(N·m − 1 ) 8.0e6 5.0e6 5.0e6 5.0e6 * Model dimensions are denoted as height×width. In order to analyze the effect of grain gradation on the DC of dry sand, the models with different grain size were established. According to the field test compared with Takada & Oshima’s centrifuge model test (Takada and Oshima 1994 ), the compression modulus of 5.05 MPa for the sand can be obtained (Li et al. 2018 , 2021 ). The Poisson’s ratio of dry sand was assumed to be 0.3. The microscopic parameters for the samples with different grading were obtained by calibrating the responses of biaxial test to achieve the results of the field test. The calibrated microscopic parameters by biaxial test are listed in Table 2 , wherein sample 1 (denoted as S1) was generated by the gradation curve, and the particle diameters for mono-sized samples 2, 3 and 4 (denoted as S2, S3 and S4) are 2.0 mm, 1.0 mm and 0.3 mm, respectively. The porosity, Poisson's ratio and friction coefficient of these models are all 0.12, 0.3 and 0.7, respectively, which are the same to those in the reference (Li et al. 2018 ). The numbers of particles generated by S1, S2, S3 and S4 in DC simulations were 57636, 5539, 21975 and 242800, respectively. 3.3 Layout of measurement circles 54 measurement circles, 0.012 m in radius, were arranged in numerical models, and the specific layout and numbering are shown in Fig. 5 . The measurement circles 8, 6 and 3, which correspond to 1.7 m, 2.9 m and 4.7 m below the ground surface, were selected to investigate the influence of grain size on the micro characteristics of soil under DC. 4 Result analysis 4.1 Propagation of Dynamic Contact Stress According to the data attained from the measurement circles 8, 6 and 3, the time histories of vertical contact stress can be plotted, as shown in Fig. 6. It is indicated in Figs. 6(a) to 6(c) that the time histories of contact stress with different particle gradation under dynamic load were all single peak curves. The closer the measurement circle was to the tamping point, the earlier the stress wave arrived, and the earlier the stress reached the peak value. The vertical stresses reached to the peak value at 0.2 s (Li et al. 2018 ) and attenuated at about 0.3 s. As the depth of test points increased, the difference of stress peak between the mono-sized samples (S2, S3 and S4) and the sample (S1) generated by the gradation curve became small since the influence of DC became small and gravity became large. It can also be seen from Figs. 6(a)-(c) that the peak values of the vertical stress for the mono-sized samples were smaller than those for the sample generated by the gradation curve. Also, the smaller the particle size was, the smaller the stress peak value was (Wu et al. 2017 ). With the increase of the depth of test points, the influence of grain size on the peak of vertical stress became small, as shown in Fig. 6(c). Therefore, the dynamic stress response for the sample (S1) generated by the gradation curve was larger under the same dynamic load, i.e., the variation of stress increase and attenuation was greater within the same time, which demonstrates that the better the particle gradation was, the better the effect of force transfer was. When choosing the foundation treatment parameters of DC, attention should be paid to particle size distribution and particle size of site soil. 4.2 Coordination number The time histories of average coordination number of particles are shown in Fig. 7 , which was in accord with the results of the indoor DC model test analysis (Jia et al. 2009 ). Under impact load, the increase of the coordination number of particles indicates that the soil particles became dense. It was pointed out in the literature (Li et al. 2018 ) that the curve of particle coordination number included three stages, that was, rising, decreasing and stabilizing, and that the coordination numbers of particles at the stable stage were greater than the initial values. Figure 7 displays that the coordination numbers of particles for the sample (S1) generated by the particle size distribution were smaller than those of mono-sized samples, and the difference became large with the increase of the depth of test points. This conclusion also further highlights the one drawn by Ueda et al. ( 2012 ) through numerical tests that the standard deviation of particle size was large, i.e., the gradation was good, and the coordinate number was small. Since the initial porosity of the model was constant, the better the gradation was, the more uneven the particle size was. The coordination number increased because the small-size particles more sufficiently contacted with those particles with large size after DC. As to the mono-sized specimen, since the particle was assumed to be a disk and their size was the same, the sample was difficult to become dense under DC. With the increase of depth, the impact effect of DC on soil became weak, and hence the change of coordination number also became very small. Therefore, the poorly graded soil is more difficult to be treated, especially under the assumption of 2D disk, which is consistent with engineering practice. 4.3 Porosity Figure 8 shows the time histories of porosity at the measurement circles 8, 6 and 3, respectively. As can be seen from Fig. 8, the porosities were almost unchanged at the beginning, then decreased rapidly, then rebounded and eventually became stable, and the porosities after the stabilization were smaller than the initial values. For the well-graded sample (S1), when the coarse particles were dominant, the fine particles would fill into the voids between the coarse particles. However, some isolated fine particles might be trapped in the narrow gaps between the coarse particles, thus causing the wedging effect (Kwan et al. 2013 ). Therefore, the final porosity for the sample (S1) generated by the particle size distribution was the largest and the porosity for the sample (S3) with the diameter of 1.0 mm was the smallest. As mentioned above and shown in Fig. 8, the variation of porosity was insignificant at the initial stage. The deeper the measurement point was, the longer the duration at this stage was, as the result of gradual deepening of stress wave transmission. So there was a certain time delay in the compaction process of soil with the increase of depth. After the initial stage, the porosity curves for the samples all experienced a process of rapid decrease, rebound and stabilization no matter what particle size distribution was, which indicates that soil samples all became denser. In addition, as the depth of measurement point increased, the peak value of vertical stress (see Fig. 6(c)) became small, and the valley values of porosity all decreased, i.e. they were greater, regardless of particle size distribution. Hence, the influence of DC on soil became small with the increase of depth. The porosity variation of the samples before and after tamping is listed in Table 3 . It can be seen from Table 3 that there was a satisfactory compaction effect at the measurement circle 8. At 4.7 m (Measurement circle 3) below ground surface, the change rate of porosity was small, especially for the soil specimen with poor gradation or larger particle size, which further illustrated the effective reinforcement depth of about 5.0 m under current energy level. Table 3 Change rate of porosity after dynamic compaction Measurement circle No. 8 (1.7m) Measurement circle No. 6 (2.9m) Measurement circle No. 3 (4.7m) S1 -39.80% -30.58% -7.72% S2 -47.66% -12.12% 3.03% S3 -56.65% -19.11% -5.99% S4 -48.53% -25.85% -16.78% 4.4 Crater Depth Figure 9 shows the relationship between the crater depth and the number of blow for specimens with different particle gradation, which was consistent with the previous researches (Takada N and Oshima1994; Gu and Lee 2002 ; Li et al. 2018 ). The relationship between the crater depth and the number of blow for the specimen generated by the gradation curve has been reported and validated with the results of centrifuge test and field test (Li et al. 2018 ). As is shown in Fig. 9 , the crater depth at the first blow was the largest, and with the increase of tamping time, the crater depth of each blow decreased as a result of the compaction of soil (Jia et al. 2009 ; Li et al. 2018 ). The settlement values for the mono-sized samples of 2.0 mm (S2) and 1.0 mm (S3) in diameter were almost the same as those for the sample generated by the gradation curve (S1), especially at the first three tamping. However, the settlement values for the mono-sized sample of 0.3 mm (S4) in diameter were larger. According to Table 3 , at 4.7 m below the ground surface, the change rate of porosity for the mono-sized sample of 0.3 mm in diameter was still 16.78%, thus the sample was compressed more closely, and the settlement values were greater. 5 Conclusions Due to few studies on the microscopic behavior of soil under DC, the models for DC of dry sand with different particle size distribution were established using the hysteretic damping model in PFC 2D . The investigation on the effect of particle gradation on soil stress, coordination number, porosity and crater depth at a particle level was helpful to understand the microscopic mechanism of soil compaction and guide the construction of DC. The following conclusions were drawn: (1) The final stress and porosity for the sample generated by the gradation curve were the largest, whereas its coordination number was the smallest. (2) The change rates of stress, coordination number and porosity of the sample generated by the gradation curve were greater than the results from the mono-size samples. Therefore, the better the particle gradation was, the better the effect of force transfer was and the better the compaction was. (3) The influence depth of the sample generated by the gradation curve on stress and porosity was large; however, its effect on the coordination number was relatively shallow. (4) For a mono-sized sample and a well-graded sample, the differences of stress, coordination number and porosity of soil increased with the increase of depth. Declarations Data Availability Statement All data, models and code that support the findings of this study are available from the corresponding author upon reasonable request. Competing Interests: On behalf of all authors, the corresponding author states that there are no competing interests to declare. 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Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 20 Jan, 2024 Read the published version in Multiscale and Multidisciplinary Modeling, Experiments and Design → Version 1 posted Editorial decision: Revision requested 13 Nov, 2023 Reviews received at journal 20 Oct, 2023 Reviewers agreed at journal 08 Oct, 2023 Reviewers invited by journal 08 Oct, 2023 Editor assigned by journal 06 Oct, 2023 Submission checks completed at journal 06 Oct, 2023 First submitted to journal 06 Oct, 2023 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. 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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-3416318","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":238427338,"identity":"ab9e601b-a5d2-47b8-a559-0f1089893bdf","order_by":0,"name":"Yuqi 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University","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Liangchen","middleName":"","lastName":"Xu","suffix":""},{"id":238427340,"identity":"833f2861-3a67-483c-8688-42444d6c24bf","order_by":2,"name":"Fu’an Yang","email":"","orcid":"","institution":"Shanghai University","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Fu’an","middleName":"","lastName":"Yang","suffix":""}],"badges":[],"createdAt":"2023-10-06 12:59:24","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3416318/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3416318/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s41939-023-00348-5","type":"published","date":"2024-01-20T15:01:43+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":44460303,"identity":"3676aff9-2004-4dbd-8047-baffee9f496e","added_by":"auto","created_at":"2023-10-11 18:05:53","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":26811,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eCalculation cycle in PFC\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-3416318/v1/6cc05a3931daafed4fad9a56.png"},{"id":44459890,"identity":"ef22c7e4-eb3a-44bd-beed-99e286908ab4","added_by":"auto","created_at":"2023-10-11 17:57:53","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":27185,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eHysteretic damping contact model\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-3416318/v1/9ac2d468f6c353cc7c913378.png"},{"id":44459891,"identity":"eb9bf709-b00d-4f85-bbfd-e5bad45d1667","added_by":"auto","created_at":"2023-10-11 17:57:53","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":20370,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eParticle grading curve for Sample 1\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-3416318/v1/e7734f5ac5b0001f65735099.png"},{"id":44460306,"identity":"16548f05-deec-4579-983c-7762801d175e","added_by":"auto","created_at":"2023-10-11 18:05:53","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":708842,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eDiscrete element model\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-3416318/v1/d4439fc3cb600b2b8c1951e9.png"},{"id":44460304,"identity":"584ae70b-cd8c-42c7-b42a-bd14066f1461","added_by":"auto","created_at":"2023-10-11 18:05:53","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":120073,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eLayout and numbering of measurement circles\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-3416318/v1/859c89d2952f028f83bdf6bd.png"},{"id":44459892,"identity":"13dfb09f-0204-4064-8594-68da8234c1be","added_by":"auto","created_at":"2023-10-11 17:57:53","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":77903,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eTime histories of vertical stress at different depth\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-3416318/v1/d3cabbefa2ee6270a72239de.png"},{"id":44459894,"identity":"ed387c15-398b-4322-b79d-384974f72f12","added_by":"auto","created_at":"2023-10-11 17:57:53","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":75740,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eTime histories of coordination number at different depth\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-3416318/v1/411793eb6b4ef10ab3ad0d65.png"},{"id":44460305,"identity":"5bd94950-09df-4609-b989-bc4472ab9169","added_by":"auto","created_at":"2023-10-11 18:05:53","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":72974,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eTime histories of porosity at different depth\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-3416318/v1/a2e2d66c2f3bf2c1668e2201.png"},{"id":44459897,"identity":"3d9d3902-59e4-486b-b143-bc2e1097a2a7","added_by":"auto","created_at":"2023-10-11 17:57:53","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":28709,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eVariation of crater depth with blow count\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-3416318/v1/1123fce9b571187b5c53dd93.png"},{"id":49978937,"identity":"287e44b7-676c-4bb1-911c-fcc0b660185b","added_by":"auto","created_at":"2024-01-22 15:10:16","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1651276,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3416318/v1/64bb84b2-7289-4edf-b932-9cae672d7795.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A Microscopic Insight into Effect of Particle Gradation on Dynamic Compaction of Dry Sand","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003eWhen the geological conditions of a construction site are poor, it is usually necessary to carry out foundation treatment on the site. Dynamic compaction (DC), first created by Menard in 1969 (Menard and Broise \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e1975\u003c/span\u003e), is a widely used foundation treatment method. This method can improve the bearing capacity of foundation and reduce post-construction settlement by converting the gravitational potential energy of a lifted tamper into kinetic energy and acting on the ground in the form of impact load. It is suitable for a wide variety of soil types, especially sandy soils and coarse granular soils; however, caution is required while used in saturated soils and clayey soils (Lukas \u003cem\u003eet al.\u003c/em\u003e 1980; Miao et al. \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2006\u003c/span\u003e; Ghassemi et al. \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2010\u003c/span\u003e). Numerous studies on DC have been carried out through analytical solution, field and laboratory test and numerical simulation (Chow et al. \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1992\u003c/span\u003e; Feng et al. \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2000\u003c/span\u003e; Hu et al. \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2001\u003c/span\u003e; Gu and Lee \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2002\u003c/span\u003e; Wang et al. \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Li et al. \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2018\u003c/span\u003e) due to its wide application in ground treatment.\u003c/p\u003e \u003cp\u003eFew analytical solutions of DC to date have been deduced because of complexity of impact contact stress and heterogeneity of foundation soil. Aiming at the limitation of linear spring dashpot dynamic models applied to relatively soft ground, an improved analytical model was proposed and verified through experimental data (Thilakasiri et al. \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e1996\u003c/span\u003e). For a liquefiable soil and soft soil interbedded foundation in highway engineering practice, the in situ tests for evaluating effectiveness of the lower tamping energy were performed and the formulation of predicting the improvement depth of DC was proposed (Miao et al. \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2006\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eAs for field and laboratory tests, many researchers had done plenty of outstanding work (Mayne et al. \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e1984\u003c/span\u003e; Takada and Oshima \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e1994\u003c/span\u003e; Feng et al. \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2000\u003c/span\u003e; Hu et al. \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2001\u003c/span\u003e; Hwang and Tu \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2006\u003c/span\u003e; Jia et al. \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2009\u003c/span\u003e). Mayne et al. (\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e1984\u003c/span\u003e) studied the relationship between the size of the craters, ground vibration levels and depth of influence and the level of energy per blow according to the data from more than 120 field tests. Hwang and Tu (\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2006\u003c/span\u003e) analyzed the effects of tamping energy and isolation trench on ground vibration based on the ground vibration data of an industrial site during DC. Through laboratory DC tests, the influence of pounder base shape on the efficiency of DC (Feng et al. \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2000\u003c/span\u003e) and the variation of microstructural variation of soils (Hu et al. \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2001\u003c/span\u003e; Jia et al. \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2009\u003c/span\u003e) were analyzed. Furthermore, the centrifuge test of DC was also used to simulate an actual foundation treatment project (Takada and Oshima \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e1994\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eApart from those investigations mentioned above, a lot of studies were focused on the numerical modeling of DC (Poran and Rodriguez \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e1992\u003c/span\u003e; Chow et al. \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1992\u003c/span\u003e; Pan and Selby \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e2002\u003c/span\u003e; Gu and Lee \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2002\u003c/span\u003e; Lee and Gu \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2004\u003c/span\u003e; Ghassemi et al. \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2010\u003c/span\u003e; Xie et al. \u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; Wang et al. \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Li et al. \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). Some methods for estimating the degree and depth of improvement resulting from DC were proposed (Chow et al. \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1992\u003c/span\u003e; Lee and Gu \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2004\u003c/span\u003e; Wang et al. \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e2017\u003c/span\u003e) based on the results of numerical simulation. Also, a fully coupled hydro-mechanical finite element code for evaluating DC of saturated granular soil was developed by Ghassemi et al. (\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2010\u003c/span\u003e) and the simulation results were verified by the data from a real field case of DC treatment in a highway. In addition, micro mechanism of DC was studied using the discrete element method (Li et al. \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2018\u003c/span\u003e; Ma et al. \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2014\u003c/span\u003e; Jiang et al. \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2017\u003c/span\u003e), in which the improvement effect and the influence depth of DC were analyzed based on the porosity change of the foundation soil (Ma et al. \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2014\u003c/span\u003e; Jiang et al. \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2017\u003c/span\u003e), and a centrifuge test of dry sand under DC (Takada and Oshima \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e1994\u003c/span\u003e) was modeled in order to investigate the microscopic characteristics of sand under DC at a particle level (Li et al. \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2018\u003c/span\u003e, \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2021\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eThese foregoing studies provided many data for design and construction of DC, and the influence of gradation on the behavior of soil has also attracted interest of a few researchers (Lu et al. \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; Luo et al. \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2014\u003c/span\u003e; Fang et al. \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2016\u003c/span\u003e; Gu et al. \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2017\u003c/span\u003e). The effects of the initial particle size and moisture on the compressive behavior of dense sand under high strain rates were investigated by Lu et al. (\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2013\u003c/span\u003e) and Luo et al. (\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2014\u003c/span\u003e). An analysis approach to model the mechanical behavior of dry sand under static and dynamic loadings was put forward and applied in the domain of projectile penetration into the sand particulate system, and the effect of particle size on the projectile penetration depth and ballistic instability was further discussed (Fang et al. \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2016\u003c/span\u003e). By studying the effect of particle size distribution on the small strain shear stiffness of granular soils, Gu et al. (\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2017\u003c/span\u003e) explored the fundamental mechanism controlling this small strain shear stiffness. However, they seldom studied the influence of particle gradation on DC, especially at a grain level. In this paper, in order to assess the effect of particle gradation on DC from a microscopic view, based on the discrete element method (Ma et al. \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2014\u003c/span\u003e; Jiang et al. \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Li et al. \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2018\u003c/span\u003e; Liu et al. \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2020\u003c/span\u003e), the DC simulations of sand samples with different grading were conducted using PFC software (Itasca Consulting Group \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2004\u003c/span\u003e), and the dynamic stress, coordination number, porosity and crater depth of soils were thoroughly investigated.\u003c/p\u003e"},{"header":"2 Principle of PFC","content":"\u003cp\u003ePFC\u003csup\u003e2D\u003c/sup\u003e (Particle Flow Code in 2 Dimensions) models the movement and interaction of rigid circular particles by the distinct element method (Itasca Consulting Group \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2004\u003c/span\u003e). The calculations performed in the PFC alternate between the application of Newton\u0026rsquo;s second law to the particles and a force-displacement law at the contacts. Newton\u0026rsquo;s second law is used to determine the motion of each particle arising from the contact and body forces acting on it, while the force-displacement law is used to update the contact forces arising from the relative motion at each contact. In PFC\u003csup\u003e2D\u003c/sup\u003e, the presence of walls requires only that the force-displacement law accounts for ball-wall contacts. Newton\u0026rsquo;s second law is not applied to walls since the wall motion is specified by users (Itasca Consulting Group \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2004\u003c/span\u003e). The calculation cycle is illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e"},{"header":"3 Numerical simulation","content":"\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e\n \u003ch2\u003e3.1 Constitutive model\u003c/h2\u003e\n \u003cp\u003eIn this paper, a hysteretic damping contact model was selected to simulate the dynamic characteristics of soil during DC. As shown in Fig. 2 (Itasca Consulting Group \u003cspan class=\"CitationRef\"\u003e2004\u003c/span\u003e), the normal stiffness of the model is different at the loading and unloading stages, and the normal stiffness on loading is smaller than that on unloading in the hysteretic damping model. The normal stiffnesses on loading, \u003cem\u003ek\u003c/em\u003e\u003csub\u003en_load\u003c/sub\u003e, and on unloading, \u003cem\u003ek\u003c/em\u003e\u003csub\u003en_unload\u003c/sub\u003e, used in the hysteretic damping model are calculated using the following equations:\u003c/p\u003e\n \u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e$${k_{{\\text{n\\_load}}}}=\\frac{{2{R_{\\text{h}}}{k_0}}}{{1+{R_{\\text{h}}}}},\\begin{array}{*{20}{c}} {}\u0026amp;{} \\end{array}{k_{{\\text{n\\_unload}}}}=\\frac{{2{k_0}}}{{1+{R_{\\text{h}}}}}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003ewhere \u003cem\u003ek\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e is the initial normal stiffness, \u003cem\u003ek\u003c/em\u003e\u003csub\u003en\u003c/sub\u003e, of particles under static loading, which was determined by numerical biaxial test; and \u003cem\u003eR\u003c/em\u003e\u003csub\u003eh\u003c/sub\u003e is the ratio of normal stiffness on loading, \u003cem\u003ek\u003c/em\u003e\u003csub\u003en_load\u003c/sub\u003e, to that on unloading, \u003cem\u003ek\u003c/em\u003e\u003csub\u003eun_load\u003c/sub\u003e (0.05\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eR\u003c/em\u003e\u003csub\u003eh\u003c/sub\u003e\u0026lt;1.0). The dynamic hysteretic effect of soil is remarkable when \u003cem\u003eR\u003c/em\u003e\u003csub\u003eh\u003c/sub\u003e is close to 0, and \u003cem\u003eR\u003c/em\u003e\u003csub\u003eh\u003c/sub\u003e was determined to be 0.75 in this work according to a previous study (Jia et al. \u003cspan class=\"CitationRef\"\u003e2015\u003c/span\u003e).\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec5\" class=\"Section2\"\u003e\n \u003ch2\u003e3.2 Modeling\u003c/h2\u003e\n \u003cp\u003eAn axisymmetric numerical model for the sand sample adopted by Takada \u0026amp; Oshima centrifuge test (Takada and Oshima \u003cspan class=\"CitationRef\"\u003e1994\u003c/span\u003e) was established using PFC\u003csup\u003e2D\u003c/sup\u003e. The tamper was generated by 40 overlapping balls using the clump command in PFC\u003csup\u003e2D\u003c/sup\u003e. The discrete element simulation of dry sand centrifugal test under DC has been carried out by Li et al. (\u003cspan class=\"CitationRef\"\u003e2018\u003c/span\u003e), and the same microscopic parameters were adopted in this paper, as listed in Table \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e. The generation method of the sample is the same to the literature (Li et al. \u003cspan class=\"CitationRef\"\u003e2018\u003c/span\u003e), and those particles, greater than 0.3 mm in diameter, were chosen to generate balls according to the proportion of particles in different range of particle size, as is shown in Figs. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e and \u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e and Table \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e (S1). A vertical acceleration of 50 \u003cem\u003eg\u003c/em\u003e was applied to each particle so that a centrifugal field can be obtained, thus a centrifuge test can be simulated. The time step was determined to be 1.0\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;6\u003c/sup\u003e s.\u003c/p\u003e\n \u003cp\u003e\u003c/p\u003e\u0026nbsp;\u003ctable id=\"Tab1\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eMicroscopic parameters of numerical model\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\u0026nbsp;\u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eBall\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eWall\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eTamper\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\u003eParticle density /(kg\u0026middot;m\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2650\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e--\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1912\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eNormal stiffness \u003cem\u003ek\u003c/em\u003e\u003csub\u003en\u003c/sub\u003e /(N\u0026middot;m\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.2e7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1e10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1e8\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eShear stiffness \u003cem\u003ek\u003c/em\u003e\u003csub\u003es\u003c/sub\u003e /(N\u0026middot;m\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e8e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1e10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1e8\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eFriction coefficient\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.1\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003cp\u003e\u003c/p\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e\u003c/p\u003e\u0026nbsp;\u003ctable id=\"Tab2\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eParameters of specimens for biaxial test\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\u0026nbsp;\u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eS1(Sample 1)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eS2(Sample 2)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eS3(Sample 3)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eS4(Sample 4)\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\u003eParticle diameter /(mm)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.3-2.0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2.0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.3\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eModel dimensions* /(m\u0026times;m)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.04\u0026times;0.02\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.12\u0026times;0.06\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.06\u0026times;0.03\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.02\u0026times;0.01\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eParticle number\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2392\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2016\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2016\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2489\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eNormal stiffness \u003cem\u003ek\u003c/em\u003e\u003csub\u003en\u003c/sub\u003e /(N\u0026middot;m\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.2e7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e8.2e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e8.0e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e8.5e6\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eShear stiffness \u003cem\u003ek\u003c/em\u003e\u003csub\u003es\u003c/sub\u003e /(N\u0026middot;m\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e8.0e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e5.0e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e5.0e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e5.0e6\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003cp\u003e\u003c/p\u003e\n \u003cp\u003e* Model dimensions are denoted as height\u0026times;width.\u003c/p\u003e\n \u003cp\u003eIn order to analyze the effect of grain gradation on the DC of dry sand, the models with different grain size were established. According to the field test compared with Takada \u0026amp; Oshima\u0026rsquo;s centrifuge model test (Takada and Oshima \u003cspan class=\"CitationRef\"\u003e1994\u003c/span\u003e), the compression modulus of 5.05 MPa for the sand can be obtained (Li et al. \u003cspan class=\"CitationRef\"\u003e2018\u003c/span\u003e, \u003cspan class=\"CitationRef\"\u003e2021\u003c/span\u003e). The Poisson\u0026rsquo;s ratio of dry sand was assumed to be 0.3. The microscopic parameters for the samples with different grading were obtained by calibrating the responses of biaxial test to achieve the results of the field test. The calibrated microscopic parameters by biaxial test are listed in Table \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e, wherein sample 1 (denoted as S1) was generated by the gradation curve, and the particle diameters for mono-sized samples 2, 3 and 4 (denoted as S2, S3 and S4) are 2.0 mm, 1.0 mm and 0.3 mm, respectively. The porosity, Poisson\u0026apos;s ratio and friction coefficient of these models are all 0.12, 0.3 and 0.7, respectively, which are the same to those in the reference (Li et al. \u003cspan class=\"CitationRef\"\u003e2018\u003c/span\u003e). The numbers of particles generated by S1, S2, S3 and S4 in DC simulations were 57636, 5539, 21975 and 242800, respectively.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec6\" class=\"Section2\"\u003e\n \u003ch2\u003e3.3 Layout of measurement circles\u003c/h2\u003e\n \u003cp\u003e54 measurement circles, 0.012 m in radius, were arranged in numerical models, and the specific layout and numbering are shown in Fig. \u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e. The measurement circles 8, 6 and 3, which correspond to 1.7 m, 2.9 m and 4.7 m below the ground surface, were selected to investigate the influence of grain size on the micro characteristics of soil under DC.\u003c/p\u003e\n\u003c/div\u003e"},{"header":"4 Result analysis","content":"\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e\n \u003ch2\u003e4.1 Propagation of Dynamic Contact Stress\u003c/h2\u003e\n \u003cp\u003eAccording to the data attained from the measurement circles 8, 6 and 3, the time histories of vertical contact stress can be plotted, as shown in Fig. 6. It is indicated in Figs. 6(a) to 6(c) that the time histories of contact stress with different particle gradation under dynamic load were all single peak curves. The closer the measurement circle was to the tamping point, the earlier the stress wave arrived, and the earlier the stress reached the peak value. The vertical stresses reached to the peak value at 0.2 s (Li et al. \u003cspan class=\"CitationRef\"\u003e2018\u003c/span\u003e) and attenuated at about 0.3 s. As the depth of test points increased, the difference of stress peak between the mono-sized samples (S2, S3 and S4) and the sample (S1) generated by the gradation curve became small since the influence of DC became small and gravity became large. It can also be seen from Figs. 6(a)-(c) that the peak values of the vertical stress for the mono-sized samples were smaller than those for the sample generated by the gradation curve. Also, the smaller the particle size was, the smaller the stress peak value was (Wu et al. \u003cspan class=\"CitationRef\"\u003e2017\u003c/span\u003e). With the increase of the depth of test points, the influence of grain size on the peak of vertical stress became small, as shown in Fig. 6(c). Therefore, the dynamic stress response for the sample (S1) generated by the gradation curve was larger under the same dynamic load, i.e., the variation of stress increase and attenuation was greater within the same time, which demonstrates that the better the particle gradation was, the better the effect of force transfer was. When choosing the foundation treatment parameters of DC, attention should be paid to particle size distribution and particle size of site soil.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec9\" class=\"Section2\"\u003e\n \u003ch2\u003e4.2 Coordination number\u003c/h2\u003e\n \u003cp\u003eThe time histories of average coordination number of particles are shown in Fig. \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e, which was in accord with the results of the indoor DC model test analysis (Jia et al. \u003cspan class=\"CitationRef\"\u003e2009\u003c/span\u003e). Under impact load, the increase of the coordination number of particles indicates that the soil particles became dense. It was pointed out in the literature (Li et al. \u003cspan class=\"CitationRef\"\u003e2018\u003c/span\u003e) that the curve of particle coordination number included three stages, that was, rising, decreasing and stabilizing, and that the coordination numbers of particles at the stable stage were greater than the initial values. Figure \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e displays that the coordination numbers of particles for the sample (S1) generated by the particle size distribution were smaller than those of mono-sized samples, and the difference became large with the increase of the depth of test points. This conclusion also further highlights the one drawn by Ueda et al. (\u003cspan class=\"CitationRef\"\u003e2012\u003c/span\u003e) through numerical tests that the standard deviation of particle size was large, i.e., the gradation was good, and the coordinate number was small. Since the initial porosity of the model was constant, the better the gradation was, the more uneven the particle size was. The coordination number increased because the small-size particles more sufficiently contacted with those particles with large size after DC. As to the mono-sized specimen, since the particle was assumed to be a disk and their size was the same, the sample was difficult to become dense under DC. With the increase of depth, the impact effect of DC on soil became weak, and hence the change of coordination number also became very small. Therefore, the poorly graded soil is more difficult to be treated, especially under the assumption of 2D disk, which is consistent with engineering practice.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec10\" class=\"Section2\"\u003e\n \u003ch2\u003e4.3 Porosity\u003c/h2\u003e\n \u003cp\u003eFigure 8 shows the time histories of porosity at the measurement circles 8, 6 and 3, respectively. As can be seen from Fig. 8, the porosities were almost unchanged at the beginning, then decreased rapidly, then rebounded and eventually became stable, and the porosities after the stabilization were smaller than the initial values. For the well-graded sample (S1), when the coarse particles were dominant, the fine particles would fill into the voids between the coarse particles. However, some isolated fine particles might be trapped in the narrow gaps between the coarse particles, thus causing the wedging effect (Kwan et al. \u003cspan class=\"CitationRef\"\u003e2013\u003c/span\u003e). Therefore, the final porosity for the sample (S1) generated by the particle size distribution was the largest and the porosity for the sample (S3) with the diameter of 1.0 mm was the smallest.\u003c/p\u003e\n \u003cp\u003eAs mentioned above and shown in Fig.\u0026nbsp;8, the variation of porosity was insignificant at the initial stage. The deeper the measurement point was, the longer the duration at this stage was, as the result of gradual deepening of stress wave transmission. So there was a certain time delay in the compaction process of soil with the increase of depth. After the initial stage, the porosity curves for the samples all experienced a process of rapid decrease, rebound and stabilization no matter what particle size distribution was, which indicates that soil samples all became denser. In addition, as the depth of measurement point increased, the peak value of vertical stress (see Fig.\u0026nbsp;6(c)) became small, and the valley values of porosity all decreased, i.e. they were greater, regardless of particle size distribution. Hence, the influence of DC on soil became small with the increase of depth.\u003c/p\u003e\n \u003cp\u003eThe porosity variation of the samples before and after tamping is listed in Table \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e. It can be seen from Table \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e that there was a satisfactory compaction effect at the measurement circle 8. At 4.7 m (Measurement circle 3) below ground surface, the change rate of porosity was small, especially for the soil specimen with poor gradation or larger particle size, which further illustrated the effective reinforcement depth of about 5.0 m under current energy level.\u003c/p\u003e\n \u003cp\u003e\u003c/p\u003e\u0026nbsp;\u003ctable id=\"Tab3\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eChange rate of porosity after dynamic compaction\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\u0026nbsp;\u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eMeasurement circle No. 8 (1.7m)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eMeasurement circle No. 6 (2.9m)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eMeasurement circle No. 3 (4.7m)\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\u003e\u003cstrong\u003eS1\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-39.80%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-30.58%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-7.72%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eS2\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-47.66%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-12.12%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e3.03%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eS3\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-56.65%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-19.11%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-5.99%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eS4\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-48.53%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-25.85%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-16.78%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003cp\u003e\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec11\" class=\"Section2\"\u003e\n \u003ch2\u003e4.4 Crater Depth\u003c/h2\u003e\n \u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e9\u003c/span\u003e shows the relationship between the crater depth and the number of blow for specimens with different particle gradation, which was consistent with the previous researches (Takada N and Oshima1994; Gu and Lee \u003cspan class=\"CitationRef\"\u003e2002\u003c/span\u003e; Li et al. \u003cspan class=\"CitationRef\"\u003e2018\u003c/span\u003e). The relationship between the crater depth and the number of blow for the specimen generated by the gradation curve has been reported and validated with the results of centrifuge test and field test (Li et al. \u003cspan class=\"CitationRef\"\u003e2018\u003c/span\u003e). As is shown in Fig. \u003cspan class=\"InternalRef\"\u003e9\u003c/span\u003e, the crater depth at the first blow was the largest, and with the increase of tamping time, the crater depth of each blow decreased as a result of the compaction of soil (Jia et al. \u003cspan class=\"CitationRef\"\u003e2009\u003c/span\u003e; Li et al. \u003cspan class=\"CitationRef\"\u003e2018\u003c/span\u003e). The settlement values for the mono-sized samples of 2.0 mm (S2) and 1.0 mm (S3) in diameter were almost the same as those for the sample generated by the gradation curve (S1), especially at the first three tamping. However, the settlement values for the mono-sized sample of 0.3 mm (S4) in diameter were larger. According to Table \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e, at 4.7 m below the ground surface, the change rate of porosity for the mono-sized sample of 0.3 mm in diameter was still 16.78%, thus the sample was compressed more closely, and the settlement values were greater.\u003c/p\u003e\n\u003c/div\u003e"},{"header":"5 Conclusions","content":"\u003cp\u003eDue to few studies on the microscopic behavior of soil under DC, the models for DC of dry sand with different particle size distribution were established using the hysteretic damping model in PFC\u003csup\u003e2D\u003c/sup\u003e. The investigation on the effect of particle gradation on soil stress, coordination number, porosity and crater depth at a particle level was helpful to understand the microscopic mechanism of soil compaction and guide the construction of DC. The following conclusions were drawn:\u003c/p\u003e \u003cp\u003e(1) The final stress and porosity for the sample generated by the gradation curve were the largest, whereas its coordination number was the smallest.\u003c/p\u003e \u003cp\u003e(2) The change rates of stress, coordination number and porosity of the sample generated by the gradation curve were greater than the results from the mono-size samples. Therefore, the better the particle gradation was, the better the effect of force transfer was and the better the compaction was.\u003c/p\u003e \u003cp\u003e(3) The influence depth of the sample generated by the gradation curve on stress and porosity was large; however, its effect on the coordination number was relatively shallow.\u003c/p\u003e \u003cp\u003e(4) For a mono-sized sample and a well-graded sample, the differences of stress, coordination number and porosity of soil increased with the increase of depth.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eData Availability Statement\u003c/h2\u003e \u003cp\u003eAll data, models and code that support the findings of this study are available from the corresponding author upon reasonable request.\u003c/p\u003e\u003cp\u003e\u003cstrong\u003eCompeting Interests:\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eOn behalf of all authors, the corresponding author states that there are no competing interests to declare.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eChow YK, Yong DM, Yong KY, Lee SL (1992) Dynamic compaction of loose sand deposits. \u003cem\u003eSoils and Foundations\u003c/em\u003e \u003cstrong\u003e32(4):\u0026nbsp;\u003c/strong\u003e93-106.\u003c/li\u003e\n \u003cli\u003eFang Q, Zhang J, Zhang Y, Liu J (2016) Mesoscopic investigation of the sand particulate system subjected to intense dynamic loadings. \u003cem\u003eInternational Journal of Impact Engineering\u003c/em\u003e \u003cstrong\u003e89:\u003c/strong\u003e 62-71.\u003c/li\u003e\n \u003cli\u003eFeng TW, Chen KH, Su YT, Shi YC.(2000) Laboratory investigation of efficiency of conical-based pounders for dynamic compaction. \u003cem\u003eGeotechnique\u003c/em\u003e \u003cstrong\u003e50(6):\u003c/strong\u003e 667-674.\u003c/li\u003e\n \u003cli\u003eGhassemi A, Pak A,\u0026nbsp;Shahir H (2010) Numerical study of the coupled hydro-mechanical effects in dynamic compaction of saturated granular soils. \u003cem\u003eComputers and Geotechnics\u003c/em\u003e \u003cstrong\u003e37:\u003c/strong\u003e 10-24.\u003c/li\u003e\n \u003cli\u003eGu Q, Lee FH (2002) Ground response to dynamic compaction of dry sand. \u003cem\u003eGeotechnique\u003c/em\u003e \u003cstrong\u003e52(7):\u003c/strong\u003e 481-493.\u003c/li\u003e\n \u003cli\u003eGu X, Lu L, Qian J (2017) Discrete element modeling of the effect of particle size distribution on the small strain stiffness of granular soils. \u003cem\u003eParticuology\u003c/em\u003e \u003cstrong\u003e32:\u003c/strong\u003e 21-29.\u003c/li\u003e\n \u003cli\u003eHu RL, Yeung MR, Lee CF, Wang SJ (2001) Mechanical behavior and microstructural variation of loess under dynamic compaction. \u003cem\u003eEngineering Geology\u003c/em\u003e \u003cstrong\u003e59:\u0026nbsp;\u003c/strong\u003e203-217.\u003c/li\u003e\n \u003cli\u003eHwang JH, Tu TY (2006) Ground vibration due to dynamic compaction. \u003cem\u003eSoil Dynamics and Earthquake Engineering\u003c/em\u003e \u003cstrong\u003e26:\u003c/strong\u003e 337-346.\u003c/li\u003e\n \u003cli\u003eItasca Consulting Group (2004) \u003cem\u003eParticle Flow Code in 2 Dimensions (Version 3.1)\u003c/em\u003e. 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Several dynamic compaction (DC) tests for mono-sized dry sand samples and a well-graded dry sand sample were modeled using discrete element method. The effect of particle gradation on crater depth was analyzed as well as coordination number, porosity and contact stress from a microscopic view. It is indicated that the change rates of dynamic stress, coordination number and porosity of the well-graded sample were greater than the results from the mono-size samples. For the mono-sized samples and the well-graded sample, the differences of dynamic contact stress, coordination number and porosity became larger as the distance of measurement point from ground surface increased. The results also demonstrate from a microscopic view that the well-graded soil and the soil sample with small particle size were more prone to become dense under DC. This study at a grain level is helpful to understand the microscopic mechanism of DC and has certain guiding significance to the construction of DC.\u003c/p\u003e","manuscriptTitle":"A Microscopic Insight into Effect of Particle Gradation on Dynamic Compaction of Dry Sand","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2023-10-11 17:57:48","doi":"10.21203/rs.3.rs-3416318/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2023-11-13T17:07:21+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2023-10-20T10:20:41+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"d774bcd0-a83d-45f7-9ae5-16a37d7479a3","date":"2023-10-08T18:38:42+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2023-10-08T18:18:21+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2023-10-07T02:02:28+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2023-10-07T02:02:28+00:00","index":"","fulltext":""},{"type":"submitted","content":"Multiscale and Multidisciplinary Modeling, Experiments and Design","date":"2023-10-06T12:57:43+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"multiscale-and-multidisciplinary-modeling-experiments-and-design","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"mmed","sideBox":"Learn more about [Multiscale and Multidisciplinary Modeling, Experiments and Design](https://link.springer.com/journal/41939)","snPcode":"41939","submissionUrl":"https://submission.nature.com/new-submission/41939/3","title":"Multiscale and Multidisciplinary Modeling, Experiments and Design","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"6d961454-f2bb-4066-92ce-ebc0625ed171","owner":[],"postedDate":"October 11th, 2023","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2024-01-22T15:07:25+00:00","versionOfRecord":{"articleIdentity":"rs-3416318","link":"https://doi.org/10.1007/s41939-023-00348-5","journal":{"identity":"multiscale-and-multidisciplinary-modeling-experiments-and-design","isVorOnly":false,"title":"Multiscale and Multidisciplinary Modeling, Experiments and Design"},"publishedOn":"2024-01-20 15:01:43","publishedOnDateReadable":"January 20th, 2024"},"versionCreatedAt":"2023-10-11 17:57:48","video":"","vorDoi":"10.1007/s41939-023-00348-5","vorDoiUrl":"https://doi.org/10.1007/s41939-023-00348-5","workflowStages":[]},"version":"v1","identity":"rs-3416318","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3416318","identity":"rs-3416318","version":["v1"]},"buildId":"ehx78VzkSd0WSzXnipQa-","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}