# Modeling the Effects of Particle Shape on Damping Ratio of Dry Sand by Simple Shear Testing and Artificial Intelligence

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}= 0.97 and 0.99 for the damping ratio and shear modulus parameters, respectively. Keshavarz and Mehramiri [33] proposed models based on gene expression programming (GEP) to predict the normalized shear modulus and the damping ratio of sands as a function of the mean effective confining pressure, void ratio and shear strain percentage; their proposed models were validated using published experimental data from the literature. Their GEP-based models demonstrated acceptable accuracy, with relative error margins lower than ±6% and ±2% for the normalized shear modulus and damping ratio parameters, respectively. In another study performed by Akbulut et al. [34], a neuro-fuzzy network was developed to model the dynamic behaviors of sand–rubber mixtures under varying conditions. Three predictive systems were trained and tested, using experimental data, to predict the shear modulus and damping ratio parameters. The study found that the adaptive neuro-fuzzy inference system (ANFIS) was the most effective method for predicting the dynamic behaviors of the composite materials, and further investigations of this paradigm were encouraged.

## 2. Materials and Methods

#### 2.1. Test Sand

_{10}= 0.19 mm, D

_{30}= 0.245 mm, D

_{50}= 0.265 mm and D

_{60}= 0.275 mm. In view of these values, the uniformity (i.e., C

_{u}= D

_{60}/D

_{10}) and curvature (i.e., C

_{c}= D

_{30}

^{2}/(D

_{10}D

_{60})) coefficients were calculated as 1.447 and 1.149, respectively; these values indicate that the test sand is poorly graded (i.e., SP) based on the Unified Soil Classification System (USCS) [45].

_{s}= 2.65 (measured as per ASTM D854 [45]), along with maximum and minimum void ratios (measured as per ASTM D4253 [46] and ASTM D4254 [47]) of e

_{max}= 1.07 and e

_{min}= 0.76, respectively. In view of the e

_{max}and e

_{min}values, the test sand can be characterized as having relatively poor compactability. It should be mentioned that the mineralogical composition of the examined sand was dominated primarily by silica, along with traces of 2.3% magnesium silicate and 1.2% ferric oxide.

#### 2.2. Dynamic Simple Shear Apparatus and Testing Plan

_{max–in}= largest inner radius of the sand particle corners; R

_{min–out}= smallest outer radius of the sand particle; i = index of summation; and N = number of inscribed spheres.

#### 2.3. Artificial Neural Network (ANN)

#### 2.4. Support Vector Machine (SVM)

## 3. Results

#### 3.1. Results of Cyclic Tests

#### 3.2. Artificial Neural Network (ANN) Performance

^{2}).

_{m}and X

_{p}= actual and predicted values, respectively; and $\overline{{\mathrm{X}}_{\mathrm{m}}}$ and $\overline{{\mathrm{X}}_{\mathrm{p}}}$ = average of the actual and predicted values, respectively.

**Note:**Ideally, the model should have R

^{2}= 1 and MAE, MSE, RMSE, MSLE and RMSLE values of 0.

^{2}) for the proposed ANN model, trained using a Levenberg–Marquardt (LM) algorithm, to predict the damping ratio as a function of the sand particle shape, vertical stress, number of loading cycles and CSR. The results indicate that the ANN model performs well in predicting the damping ratio, as evidenced by the high R

^{2}value of 0.962 for both the training and testing datasets, indicating that 96.2% of the variations in the actual output variable (i.e., the damping ratio) is captured and explained by the proposed ANN model. The MAE, MSE and RMSE values for the training dataset were found to be higher than those obtained for testing, indicating that the model performs better on the testing dataset. This could be due to overfitting, which occurs when the model fits the training data too closely, leading to relatively poorer performance on new (unseen) datasets. The MSLE and RMSLE values were low for both the training and testing datasets, implying that the predictions were associated with low forecast errors. In conclusion, the results suggest that the ANN model trained using the LM algorithm is a promising approach for predicting the damping ratio of sand based on the sand particle shape, vertical stress, number of loading cycles and CSR. However, further research is needed to validate the model’s performance on larger (and/or more diverse) datasets.

#### 3.3. Support Vector Machine (SVM)

^{2}= 0.973 and 0.892 for the training and testing datasets, respectively. Like the proposed ANN model, these results suggest that the SVM model performs well in predicting the damping ratio of sand based on the selected input parameters (the sand particle shape, vertical stress, number of loading cycles and CSR conditions).

## 4. Discussion

#### 4.1. Effects of Void Ratio Changes on Damping Ratio

_{min}and e

_{max}) variations against the number of cycles based on the R, S and ρ parameters are provided in Figure 18, Figure 19 and Figure 20, respectively. Note that the circle diameters in these figures represent the magnitude of the void ratio parameter. The findings demonstrate that with an increase in the number of loading cycles, and thus a consequent rounding of the sand grains, the void ratio between the grains reduces. This can be attributed to the fact that more rounded and spherical particles possess a higher packing capability, which leads to a reduction in the volume of voids (and an increase in the contact levels achieved between the particles). This increase in inter-particle interaction leads to higher energy dissipation during cyclic loading, which is reflected in the higher damping ratio (with an increasing number of cycles).

#### 4.2. Importance of Input Parameters and Sensitivity Analysis

#### 4.3. Limitations and Scope for Future Works

## 5. Summary and Conclusions

- The shape of the sand particles changes during cyclic loading, becoming progressively more rounded and spherical with an increasing number of loading cycles, resulting in an increase in the damping ratio.
- The damping ratio was found to decrease as the number of loading cycles increased. This can be attributed to the fact that cyclic loading rearranges the sand particles, prompting an increase in the packing capability (and hence a decrease in the volume of voids) of the samples. This is followed by an increase in the number of contact points between the particles, thereby leading to higher energy dissipation during cyclic loading.
- Compared to the SVM, the proposed ANN model, trained using LM algorithms, was found to produce more promising results in predicting the damping ratio of the dry sand as a function of the particle shape parameters, vertical stress, number of loading cycles and CSR. This was supported by the model’s high R
^{2}value of 0.962 for both the training and testing datasets. - Based on the sensitivity analysis results, vertical stress was found to be the most important parameter affecting the damping ratio, while the effects/importance of the CSR were relatively small. That is, increasing the vertical stress resulted in an increase in the damping ratio, while the effects of increasing the CSR on the damping ratio were fairly small. This is because vertical stress plays a major role in controlling the contact forces between the sand particles (and hence the energy dissipation) during cyclic loading.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Ashmawy, A.K.; Salgado, R.; Guha, S.; Drnevich, V.P. Soil damping and its use in dynamic analyses. In Proceedings of the International Conferences on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics, St. Louis, MO, USA, 2–7 April 1995; Volume 9. [Google Scholar]
- Luna, R.; Jadi, H. Determination of dynamic soil properties using geophysical methods. In Proceedings of the first international conference on the application of geophysical and NDT methodologies to transportation facilities and infrastructure, St. Louis, MO, USA, 11–15 December 2000; pp. 1–15. [Google Scholar]
- Ventura, C.E.; Finn, W.L.; Lord, J.F.; Fujita, N. Dynamic characteristics of a base isolated building from ambient vibration measurements and low level earthquake shaking. Soil Dyn. Earthq. Eng.
**2003**, 23, 313–322. [Google Scholar] [CrossRef] - Sahebzadeh, S.; Heidari, A.; Kamelnia, H.; Baghbani, A. Sustainability features of Iran’s vernacular architecture: A comparative study between the architecture of hot–arid and hot–arid–windy regions. Sustainability
**2017**, 9, 749. [Google Scholar] [CrossRef] [Green Version] - Seed, H.B.; Wong, R.T.; Idriss, I.M.; Tokimatsu, K. Moduli and damping factors for dynamic analyses of cohesionless soils. J. Geotech. Eng.
**1986**, 112, 1016–1032. [Google Scholar] [CrossRef] - Senetakis, K.; Payan, M. Small strain damping ratio of sands and silty sands subjected to flexural and torsional resonant column excitation. Soil Dyn. Earthq. Eng.
**2018**, 114, 448–459. [Google Scholar] [CrossRef] - Senetakis, K.; Anastasiadis, A.; Pitilakis, K. Normalized shear modulus reduction and damping ratio curves of quartz sand and rhyolitic crushed rock. Soils Found.
**2013**, 53, 879–893. [Google Scholar] [CrossRef] [Green Version] - Edinçliler, A.; Yildiz, O. Effects of processing type on shear modulus and damping ratio of waste tire-sand mixtures. Geosynth. Int.
**2022**, 29, 389–408. [Google Scholar] [CrossRef] - Akbarimehr, D.; Fakharian, K. Dynamic shear modulus and damping ratio of clay mixed with waste rubber using cyclic triaxial apparatus. Soil Dyn. Earthq. Eng.
**2021**, 140, 106435. [Google Scholar] [CrossRef] - Payan, M.; Senetakis, K.; Khoshghalb, A.; Khalili, N. Characterization of the small-strain dynamic behaviour of silty sands; contribution of silica non-plastic fines content. Soil Dyn. Earthq. Eng.
**2017**, 102, 232–240. [Google Scholar] [CrossRef] - Li, W.; Lang, L.; Wang, D.; Wu, Y.; Li, F. Investigation on the dynamic shear modulus and damping ratio of steel slag sand mixtures. Constr. Build. Mater.
**2018**, 162, 170–180. [Google Scholar] [CrossRef] - Wichtmann, T.; Triantafyllidis, T. Effect of uniformity coefficient on G/G max and damping ratio of uniform to well-graded quartz sands. J. Geotech. Geoenviron. Eng.
**2013**, 139, 59–72. [Google Scholar] [CrossRef] [Green Version] - Hardin, B.O.; Drnevich, V.P. Shear modulus and damping in soils: Measurement and parameter effects (terzaghi leture). J. Soil Mech. Found. Div.
**1972**, 98, 603–624. [Google Scholar] [CrossRef] - Kumar, S.S.; Krishna, A.M.; Dey, A. Parameters influencing dynamic soil properties: A review treatise. In Proceedings of the National Conference on Recent Advances in Civil Engineering, Bhiwani, India, 15–16 November 2013; pp. 1–10. [Google Scholar]
- Okur, D.V.; Ansal, A. Stiffness degradation of natural fine grained soils during cyclic loading. Soil Dyn. Earthq. Eng.
**2007**, 27, 843–854. [Google Scholar] [CrossRef] - Wichtmann, T.; Triantafyllidis, T. Influence of a cyclic and dynamic loading history on dynamic properties of dry sand, part I: Cyclic and dynamic torsional prestraining. Soil Dyn. Earthq. Eng.
**2004**, 24, 127–147. [Google Scholar] [CrossRef] - Jafarzadeh, F.; Sadeghi, H. Experimental study on dynamic properties of sand with emphasis on the degree of saturation. Soil Dyn. Earthq. Eng.
**2012**, 32, 26–41. [Google Scholar] [CrossRef] - Wu, S.; Gray, D.H.; Richart, F.E., Jr. Capillary effects on dynamic modulus of sands and silts. J. Geotech. Eng.
**1984**, 110, 1188–1203. [Google Scholar] [CrossRef] - Clayton, C.R.; Priest, J.A.; Best, A.I. The effects of disseminated methane hydrate on the dynamic stiffness and damping of a sand. Geotechnique
**2005**, 55, 423–434. [Google Scholar] [CrossRef] - Baghbani, A.; Choudhury, T.; Costa, S.; Reiner, J. Application of artificial intelligence in geotechnical engineering: A state-of-the-art review. Earth Sci. Rev.
**2022**, 228, 103991. [Google Scholar] [CrossRef] - Bayat, M.; Ghalandarzadeh, A. Stiffness degradation and damping ratio of sand-gravel mixtures under saturated state. Int. J. Civ. Eng.
**2018**, 16, 1261–1277. [Google Scholar] [CrossRef] - Ling, X.Z.; Zhang, F.; Li, Q.L.; An, L.S.; Wang, J.H. Dynamic shear modulus and damping ratio of frozen compacted sand subjected to freeze–thaw cycle under multi-stage cyclic loading. Soil Dyn. Earthq. Eng.
**2015**, 76, 111–121. [Google Scholar] [CrossRef] - Chen, G.; Zhou, Z.; Sun, T.; Wu, Q.; Xu, L.; Khoshnevisan, S.; Ling, D. Shear modulus and damping ratio of sand–gravel mixtures over a wide strain range. J. Earthq. Eng.
**2019**, 23, 1407–1440. [Google Scholar] [CrossRef] - Wichtmann, T.; Hernández, M.N.; Triantafyllidis, T. On the influence of a non-cohesive fines content on small strain stiffness, modulus degradation and damping of quartz sand. Soil Dyn. Earthq. Eng.
**2015**, 69, 103–114. [Google Scholar] [CrossRef] - Tong, L.; Wang, Y.H. DEM simulations of shear modulus and damping ratio of sand with emphasis on the effects of particle number, particle shape, and aging. Acta Geotech.
**2015**, 10, 117–130. [Google Scholar] [CrossRef] - Jafarian, Y.; Javdanian, H.; Haddad, A. Dynamic properties of calcareous and siliceous sands under isotropic and anisotropic stress conditions. Soils Found.
**2018**, 58, 172–184. [Google Scholar] [CrossRef] - Baghbani, A.; Costa, S.; O’Kelly, B.C.; Soltani, A.; Barzegar, M. Experimental study on cyclic simple shear behaviour of predominantly dilative silica sand. Int. J. Geotech. Eng.
**2022**, 23, 91–105. [Google Scholar] [CrossRef] - Baghbani, A.; Choudhury, T.; Samui, P.; Costa, S. Prediction of secant shear modulus and damping ratio for an extremely dilative silica sand based on machine learning techniques. Soil Dyn. Earthq. Eng.
**2023**, 165, 107708. [Google Scholar] [CrossRef] - Nguyen, M.D.; Baghbani, A.; Alnedawi, A.; Ullah, S.; Kafle, B.; Thomas, M.; Moon, E.M.; Milne, N.A. Experimental Study on the Suitability of Aluminium-Based Water Treatment Sludge as a Next Generation Sustainable Soil Replacement for Road Construction. Available online: https://ssrn.com/abstract=4331275 (accessed on 25 March 2023).
- Baghbani, A.; Baumgartl, T.; Filipovic, V. Effects of Wetting and Drying Cycles on Strength of Latrobe Valley Brown Coal. In Proceedings of the Copernicus Meetings, Vienna, Austria, 23–28 April 2023. [Google Scholar]
- Baghbani, A.; Nguyen, M.D.; Alnedawi, A.; Milne, N.; Baumgartl, T.; Abuel-Naga, H. Improving Soil Stability with Alum Sludge: An AI-Enabled Approach for Accurate Prediction of California Bearing Ratio. Preprints
**2023**, 2023030197. [Google Scholar] - Cabalar, A.F.; Cevik, A. Modelling damping ratio and shear modulus of sand–mica mixtures using neural networks. Eng. Geol.
**2009**, 104, 31–40. [Google Scholar] [CrossRef] - Keshavarz, A.; Mehramiri, M. New Gene Expression Programming models for normalized shear modulus and damping ratio of sands. Eng. Appl. Artif. Intell.
**2015**, 45, 464–472. [Google Scholar] [CrossRef] - Akbulut, S.; Hasiloglu, A.S.; Pamukcu, S. Data generation for shear modulus and damping ratio in reinforced sands using adaptive neuro-fuzzy inference system. Soil Dyn. Earthq. Eng.
**2004**, 24, 805–814. [Google Scholar] [CrossRef] - Manafi Khajeh Pasha, S.; Hazarika, H.; Yoshimoto, N. An Artificial Intelligence Approach for Modeling Shear Modulus and Damping Ratio of Tire Derived Geomaterials. In Advances in Computer Methods and Geomechanics: IACMAG Symposium 2019; Springer: Singapore, 2020; Volume 2, pp. 591–606. [Google Scholar]
- Abdolrasol, M.G.; Hussain, S.S.; Ustun, T.S.; Sarker, M.R.; Hannan, M.A.; Mohamed, R.; Ali, J.A.; Mekhilef, S.; Milad, A. Artificial neural networks based optimization techniques: A review. Electronics
**2021**, 10, 2689. [Google Scholar] [CrossRef] - Baghbani, A.; Daghistani, F.; Baghbani, H.; Kiany, K. Predicting the Strength of Recycled Glass Powder-Based Geopolymers for Improving Mechanical Behavior of Clay Soils Using Artificial Intelligence; EasyChair: Manchester, UK, 2023. [Google Scholar]
- Baghbani, A.; Daghistani, F.; Baghbani, H.; Kiany, K.; Bazaz, J.B. Artificial Intelligence-Based Prediction of Geotechnical Impacts of Polyethylene Bottles and Polypropylene on Clayey Soil; EasyChair: Manchester, UK, 2023. [Google Scholar]
- Baghbani, A.; Daghistani, F.; Kiany, K.; Shalchiyan, M.M. AI-Based Prediction of Strength and Tensile Properties of Expansive Soil Stabilized with Recycled Ash and Natural Fibers; EasyChair: Manchester, UK, 2023. [Google Scholar]
- Baghbani, A.; Daghistani, F.; Naga, H.A.; Costa, S. Development of a Support Vector Machine (SVM) and a Classification and Regression Tree (CART) to Predict the Shear Strength of Sand Rubber Mixtures. In Proceedings of the 8th International Symposium on Geotechnical Safety and Risk (ISGSR), Newcastle, Australia, 14–16 December 2022. [Google Scholar]
- Baghbani, A.; Costa, S.; Choundhury, T.; Faradonbeh, R.S. Prediction of Parallel Desiccation Cracks of Clays Using a Classification and Regression Tree (CART) Technique. In Proceedings of the 8th International Symposium on Geotechnical Safety and Risk (ISGSR), Newcastle, Australia, 14–16 December 2022. [Google Scholar]
- Baghbani, A.; Baghbani, H.; Shalchiyan, M.M.; Kiany, K. Utilizing artificial intelligence and finite element method to simulate the effects of new tunnels on existing tunnel deformation. J. Comput. Cogn. Eng.
**2022**, 2022, 1–10. [Google Scholar] - Shahin, M.A. Artificial intelligence in geotechnical engineering: Applications, modeling aspects, and future directions. In Metaheuristics in Water, Geotechnical and Transport Engineering; Elseiver: Amsterdam, The Netherlands, 2013; p. 169204. [Google Scholar]
- Ebid, A.M. 35 Years of (AI) in geotechnical engineering: State of the art. Geotech. Geol. Eng.
**2021**, 39, 637–690. [Google Scholar] [CrossRef] - ASTM D2487-17; Standard Practice for Classification of Soils for Engineering Purposes (Unified Soil Classification System). ASTM International: West Conshohocken, PA, USA, 2017.
- ASTM D854–14; Standard Test Methods for Specific Gravity of Soil Solids by Water Pycnometer. ASTM International: West Conshohocken, PA, USA, 2014.
- ASTMD4253 A; Standard Test Methods for Maximum Index Density and Unit Weight of Soils Using a Vibratory Table. ASTM International: West Conshohocken, PA, USA, 2006.
- ASTMD4254 A; Standard Test Methods for Minimum Index Density and Unit Weight of Soils and Calculation of Relative Density. ASTM International: West Conshohocken, PA, USA, 2006.
- Ladd, R.S. Specimen preparation and liquefaction of sands. J. Geotech. Eng. Div.
**1974**, 100, 1180–1184. [Google Scholar] [CrossRef] - Carraro, J.A.; Prezzi, M.; Salgado, R. Shear strength and stiffness of sands containing plastic or nonplastic fines. J. Geotech. Geoenviron. Eng.
**2009**, 135, 1167–1178. [Google Scholar] [CrossRef] - Krumbein, W.C.; Sloss, L.L. Stratigraphy and Sedimentation; LWW: Philadelphia, PA, USA, 1951; Volume 71, p. 401. [Google Scholar]
- Cho, G.C.; Dodds, J.; Santamarina, J.C. Closure to “particle shape effects on packing density, stiffness, and strength: Natural and crushed sands” by Gye-Chun Cho, Jake Dodds, and J. Carlos Santamarina. J. Geotech. Geoenviron. Eng.
**2007**, 133, 1474. [Google Scholar] [CrossRef] - Kay, L.E. From logical neurons to poetic embodiments of mind: Warren S. McCulloch’s project in neuroscience. Sci. Context
**2001**, 14, 591–614. [Google Scholar] [CrossRef] - Yadav, N.; Yadav, A.; Kumar, M.; Yadav, N.; Yadav, A.; Kumar, M. History of neural networks. In An Introduction to Neural Network Methods for Differential Equations; Springer: Dordrecht, The Netherlands, 2015; pp. 13–15. [Google Scholar]
- Werbos, P.J. The Roots of Backpropagation: From Ordered Derivatives to Neural Networks and Political Forecasting; John Wiley & Sons: Hoboken, NJ, USA, 1994. [Google Scholar]
- Azadeh, A.; Ghaderi, S.F.; Sohrabkhani, S. Forecasting electrical consumption by integration of neural network, time series and ANOVA. Appl. Math. Comput.
**2007**, 186, 1753–1761. [Google Scholar] [CrossRef] - Cruz, J.A.; Wishart, D.S. Applications of machine learning in cancer prediction and prognosis. Cancer Inform.
**2006**, 2, 117693510600200030. [Google Scholar] [CrossRef] - Pascanu, R.; Mikolov, T.; Bengio, Y. On the difficulty of training recurrent neural networks. In Proceedings of the International Conference on Machine Learning, Atlanta, GA, USA, 16–21 June 2013; pp. 1310–1318. [Google Scholar]
- Kecman, V. Learning and Soft Computing: Support Vector Machines, Neural Networks, and Fuzzy Logic Models; MIT Press: Cambridge, MA, USA, 2001. [Google Scholar]
- Vapnik, V.; Chapelle, O. Bounds on error expectation for support vector machines. Neural Comput.
**2000**, 12, 2013–2036. [Google Scholar] [CrossRef] [Green Version] - Boswell, D. Introduction to Support Vector Machines; Departement of Computer Science and Engineering, University of California: San Diego, CA, USA, 2002. [Google Scholar]
- Joshua, V.; Priyadharson, S.M.; Kannadasan, R. Exploration of machine learning approaches for paddy yield prediction in eastern part of Tamilnadu. Agronomy
**2021**, 11, 2068. [Google Scholar] [CrossRef]

**Figure 5.**Illustration of the selection process for the 25 sand particles for each sample (not to scale).

**Figure 7.**Results of the cyclic tests for CSR = 0.3 and vertical stress = 150 kPa: (

**a**) shear stress–strain hysteresis loops; and (

**b**) volumetric strain by number of cycles.

**Figure 8.**Effects of increasing number of loading cycles on the damping ratio for different CSR conditions and vertical stresses of (

**a**) 50 kPa, (

**b**) 150 kPa and (

**c**) 250 kPa for the first reconstruction.

**Figure 9.**Effects of increasing number of loading cycles on the damping ratio for different CSR conditions and vertical stresses of (

**a**) 50 kPa, (

**b**) 150 kPa and (

**c**) 250 kPa for the second reconstruction.

**Figure 10.**Effects of increasing number of loading cycles on the damping ratio for different CSR conditions and vertical stresses of (

**a**) 50 kPa, (

**b**) 150 kPa and (

**c**) 250 kPa for the fifth reconstruction.

**Figure 11.**(

**a**) Three-dimensional image, (

**b**) two-dimensional cross-sectional area and (

**c**) calculation of the three shape parameters S, R and ρ for one particle before cyclic testing.

**Figure 12.**(

**a**) Three-dimensional image, (

**b**) two-dimensional cross-sectional area and (

**c**) calculation of the three shape parameters S, R and ρ for one particle after four cyclic tests.

**Figure 13.**Variations of the three shape descriptors (

**a**) before cycling, (

**b**) after 30 cycles and (

**c**) after 120 cycles. Note: S, R and ρ denote sphericity, roundness and regularity, respectively.

**Figure 14.**Effects of number of loading cycles, along with (

**a**) R, (

**b**) S and (

**c**) ρ, on the damping ratio.

**Figure 15.**Effects of number of loading cycles, along with (

**a**) vertical stress and (

**b**) CSR, on the damping ratio.

**Figure 16.**Performance of the best ANN model to predict the damping ratio for (

**a**) training and (

**b**) testing datasets.

**Figure 17.**Correlation performance of the best SVM model to predict the damping ratio for (

**a**) training and (

**b**) testing datasets.

**Figure 18.**Effects of cycling loading on (

**a**) e

_{min}and (

**b**) e

_{max}based on the roundness R parameter.

**Figure 19.**Effects of cycling loading on (

**a**) e

_{min}and (

**b**) e

_{max}based on the sphericity S parameter.

**Figure 20.**Effects of cycling loading on (

**a**) e

_{min}and (

**b**) e

_{max}based on the regularity ρ parameter.

Variable | Minimum | Maximum | Mean | SD |
---|---|---|---|---|

S (–) | 0.690 | 0.743 | 0.713 | 0.014 |

R (–) | 0.498 | 0.534 | 0.518 | 0.011 |

ρ (–) | 0.594 | 0.638 | 0.616 | 0.011 |

D (%) | 8.500 | 25.500 | 16.050 | 5.903 |

Variable | Minimum | Maximum | Mean | SD |
---|---|---|---|---|

S (–) | 0.733 | 0.793 | 0.760 | 0.017 |

R (–) | 0.529 | 0.572 | 0.557 | 0.014 |

ρ (–) | 0.631 | 0.683 | 0.658 | 0.014 |

D (%) | 7.200 | 21.400 | 13.392 | 4.784 |

Variable | Minimum | Maximum | Mean | SD |
---|---|---|---|---|

S (–) | 0.741 | 0.807 | 0.774 | 0.020 |

R (–) | 0.537 | 0.592 | 0.570 | 0.018 |

ρ (–) | 0.640 | 0.695 | 0.672 | 0.018 |

D (%) | 6.900 | 20.200 | 12.425 | 4.536 |

Variable | Observations | Minimum | Maximum | Mean | SD |
---|---|---|---|---|---|

Damping Ratio, D (%) | 29 | 6.900 | 25.500 | 13.355 | 5.370 |

Number of Cycles (–) | 29 | 0.000 | 120.000 | 56.897 | 51.555 |

Sphericity, S (–) | 29 | 0.702 | 0.807 | 0.753 | 0.028 |

Roundness, R (–) | 29 | 0.502 | 0.592 | 0.552 | 0.025 |

Regularity, ρ (–) | 29 | 0.612 | 0.695 | 0.653 | 0.025 |

Vertical Stress (kPa) | 29 | 50.000 | 250.000 | 143.103 | 84.223 |

CSR (%) | 29 | 0.200 | 0.500 | 0.348 | 0.112 |

Variable | Observations | Minimum | Maximum | Mean | SD |
---|---|---|---|---|---|

Damping Ratio, D (%) | 7 | 11.000 | 22.000 | 16.443 | 3.756 |

Number of Cycles (–) | 7 | 0.000 | 120.000 | 21.429 | 44.881 |

Sphericity, S (–) | 7 | 0.690 | 0.788 | 0.730 | 0.038 |

Roundness, R (–) | 7 | 0.498 | 0.575 | 0.533 | 0.029 |

Regularity, ρ (–) | 7 | 0.594 | 0.682 | 0.631 | 0.033 |

Vertical Stress (kPa) | 7 | 50.000 | 250.000 | 178.571 | 75.593 |

CSR (%) | 7 | 0.200 | 0.500 | 0.357 | 0.127 |

Variable | Number of Cycles | S | R | ρ | Vertical Stress | CSR | D |
---|---|---|---|---|---|---|---|

Number of Cycles | 1 | 0.705 | 0.715 | 0.727 | 0.000 | 0.000 | −0.256 |

Sphericity, S | 0.705 | — | 0.705 | 0.780 | 0.193 | 0.227 | −0.044 |

Roundness, R | 0.715 | 0.705 | — | 0.771 | 0.193 | 0.206 | −0.040 |

Regularity, ρ | 0.727 | 0.780 | 0.771 | — | 0.198 | 0.223 | −0.043 |

Vertical Stress | 0.000 | 0.193 | 0.193 | 0.198 | — | 0.324 | 0.798 |

CSR | 0.000 | 0.227 | 0.206 | 0.223 | 0.324 | — | 0.277 |

Damping Ratio, D | −0.256 | −0.044 | −0.040 | −0.043 | 0.798 | 0.277 | — |

**Table 7.**Overall performance of the best ANN model to predict the damping ratio for both training and testing datasets.

Metric | Training Dataset | Testing Dataset |
---|---|---|

MAE (%) | 0.887 | 0.551 |

MSE (%) | 1.056 | 0.460 |

RMSE (%) | 1.027 | 0.679 |

MSLE (%) | 0.007 | 0.001 |

RMSLE (%) | 0.084 | 0.037 |

R^{2} (–) | 0.962 | 0.962 |

**Table 8.**Overall performance of the best SVM model to predict the damping ratio for both training and testing datasets.

Metric | Training Dataset | Testing Dataset |
---|---|---|

MAE (%) | 0.716 | 0.831 |

MSE (%) | 0.761 | 1.302 |

RMSE (%) | 0.872 | 1.141 |

MSLE (%) | 0.006 | 0.003 |

RMSLE (%) | 0.079 | 0.057 |

R^{2} (–) | 0.973 | 0.892 |

Model | Input Parameters | |||||
---|---|---|---|---|---|---|

Number of Cycles | S | R | ρ | Vertical Stress | CSR | |

ANN | 3 | 5 | 2 | 4 | 1 | 6 |

SVM | 4 | 5 | 3 | 2 | 1 | 6 |

Total Score | 7 | 10 | 5 | 6 | 2 | 12 |

Ranking | 4 | 5 | 2 | 3 | 1 | 6 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Baghbani, A.; Costa, S.; Faradonbeh, R.S.; Soltani, A.; Baghbani, H.
Modeling the Effects of Particle Shape on Damping Ratio of Dry Sand by Simple Shear Testing and Artificial Intelligence. *Appl. Sci.* **2023**, *13*, 4363.
https://doi.org/10.3390/app13074363

**AMA Style**

Baghbani A, Costa S, Faradonbeh RS, Soltani A, Baghbani H.
Modeling the Effects of Particle Shape on Damping Ratio of Dry Sand by Simple Shear Testing and Artificial Intelligence. *Applied Sciences*. 2023; 13(7):4363.
https://doi.org/10.3390/app13074363

**Chicago/Turabian Style**

Baghbani, Abolfazl, Susanga Costa, Roohollah Shirani Faradonbeh, Amin Soltani, and Hasan Baghbani.
2023. "Modeling the Effects of Particle Shape on Damping Ratio of Dry Sand by Simple Shear Testing and Artificial Intelligence" *Applied Sciences* 13, no. 7: 4363.
https://doi.org/10.3390/app13074363