Recent Geotechnical Research at BOKU (Springer Series in Geomechanics and Geoengineering) [2024 ed.] 3031521587, 9783031521584

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Springer Series in Geomechanics and Geoengineering

Wei Wu Yunteng Wang   Editors

Recent Geotechnical Research at BOKU

Springer Series in Geomechanics and Geoengineering Series Editor Wei Wu, University of Natural Resources and Life Sciences, Vienna, Austria

Geomechanics deals with the application of the principle of mechanics to geomaterials including experimental, analytical and numerical investigations into the mechanical, physical, hydraulic and thermal properties of geomaterials as multiphase media. Geoengineering covers a wide range of engineering disciplines related to geomaterials from traditional to emerging areas. The objective of the book series is to publish monographs, handbooks, workshop proceedings and textbooks. The book series is intended to cover both the state-ofthe-art and the recent developments in geomechanics and geoengineering. Besides researchers, the series provides valuable references for engineering practitioners and graduate students. Indexed by SCOPUS, EI Compendex, INSPEC, SCImago.

Wei Wu · Yunteng Wang Editors

Recent Geotechnical Research at BOKU

Editors Wei Wu Institute of Geotechnical Engineering University of Natural Resources and Life Sciences Vienna, Austria

Yunteng Wang Institute of Geotechnical Engineering University of Natural Resources and Life Sciences Vienna, Austria

ISSN 1866-8755 ISSN 1866-8763 (electronic) Springer Series in Geomechanics and Geoengineering ISBN 978-3-031-52158-4 ISBN 978-3-031-52159-1 (eBook) https://doi.org/10.1007/978-3-031-52159-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Preface

This book brings together 19 papers by the members of our institute, our visiting scholars and our project partners. It presents a snapshot of the research activities on geomechanics and geohazards between 2020 and 2023 at the Institute of Geotechnical Engineering, University of Natural Resources and Life Sciences, Vienna, Austria. The research in this period is characterized by two research projects funded by the European Commission, i.e., HERCULES (towards geoHazards rEsilient infRastruCtUre under changing cLimatES) and FRAMED (Fracture Across Scales and Materials, Processes and Disciplines), and two projects funded by the Austrian Science Fund (FWF), i.e., MultiCBPR (Multiscale modelling of compaction bands in porous rocks) and STRECROPA (Experimental and numerical modelling of stress redistribution due to construction of cross passages of shallow tunnels). All four projects aim to advance and disseminate knowledge on geomechanics, i.e., fracture and strain localization, and geohazards, i.e., rock avalanche, landslide and debris flow. Our research includes constitutive modeling, numerical simulations, laboratory experiments and artificial intelligence techniques, e.g., hypoplasticity, geotechnical centrifuge, SPH and the phase-field approach, machine learning and data-driven intelligent surrogate models. Geomechanics and geohazards remain the hot research topics in our institute. We benefit from the staff exchange with partners in Europe, USA and China. We thank the European Commission and Austrian Science Fund for the financial support to the following projects: • HERCULES (towards geoHazards rEsilient infRastruCtUre under changing cLimatES), Grant agreement ID: 778360, Marie Skłodowska-Curie Research and Innovation Staff Exchange (RISE) within Horizon 2020. • FRAMED (Fracture Across Scales and Materials, Processes and Disciplines), Grant agreement ID: 734485, Marie Skłodowska-Curie Research and Innovation Staff Exchange (RISE) within Horizon 2020. • MultiCBPR (Multiscale modeling of compaction bands in porous rocks), Grant Nr. M3340-N, FWF Lise Meitner project.

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• STRECROPA (Experimental and numerical modelling of stress redistribution due to construction of cross passages of shallow tunnels), Grant Nr. P34257-N, FWF Stand-Alone project. All members of our institute and our project partners deserve our heartfelt thanks. Vienna, Austria February 2024

Wei Wu Yunteng Wang

Contents

A Simple Hypoplastic Model for Sand Under Cyclic Loading . . . . . . . . . . Mohammad-Javad Alipour and Wei Wu

1

Failure Mode and Mechanism of the Consequent Slope in Xuan’en County . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yi Du, Chao Cheng, Jiaqiang Zou, Jian Wang, and Echuan Yan

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Reliability Analysis of Slope Stability with Intelligent Surrogate Models: A Case Study in the Three Gorges Reservoir . . . . . . . . . . . . . . . . . Carlotta Guardiani, Enrico Soranzo, and Wei Wu

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Experimental Study on the Permeability of Hydrophobic Powders Treated Loess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yuqi He, Wei Wu, Hongjian Liao, and Xiaohua Liu

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Modelling and Assessment of Debris Flow Impact on Infrastructure in the Carpathians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Olena Ivanik

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On the Performance of CAES Pile in Overconsolidated Soils: A Numerical Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xuan Kang, Wei Wu, and Shun Wang

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Experimental and Numerical Analysis of Fluid-Injection Unloading Rock Failure Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miaomiao Kou, Yu Wang, Xinrong Liu, and Wei Wu

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A Visco-Hypoplastic Model with Solid Hardness Degradation for Granular Soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Haoyong Qian, Wei Wu, Xiuli Du, and Chengshun Xu Prediction of Tunnelling-Induced Settlement Trough by Artificial Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Enrico Soranzo, Christoph Pock, Carlotta Guardiani, Yunteng Wang, and Wei Wu vii

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Machine Learning Prediction of Bleeding of Bored Concrete Piles Based on Centrifuge Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Enrico Soranzo, Carlotta Guardiani, Yunteng Wang, and Wei Wu Stability Evaluation of Huangtupo Riverside Slump I Landslide Based on Soil-Water Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Xuexue Su, Carlotta Guardiani, Huiming Tang, Pengju An, and Wei Wu Effect of Different Factors on Dynamic Shear Modulus of Compacted Loess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Haiman Wang, Wankui Ni, and Kangze Yuan Unified Description of Viscous Behaviors of Clay and Sand with a Visco-Hypoplastic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Shun Wang, Xiao Xu, Xuan Kang, Guofang Xu, and Wei Wu Experimental and Numerical Investigation on Mechanical Behaviour of Gravel Soils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Shun Wang, Xuan Kang, Guofang Xu, Hongguang Bian, and Wei Wu A Basic Hypoplastic Model with Fabric Evolution . . . . . . . . . . . . . . . . . . . . 225 Yadong Wang and Wei Wu Phase-field Modeling of Brittle Failure in Rockslides . . . . . . . . . . . . . . . . . . 241 Yunteng Wang, Shun Wang, Enrico Soranzo, Xiaoping Zhou, and Wei Wu Triggering Mechanism and Mitigation Strategies of Freeze-Thaw Landslides for Engineering in Cold Regions: A Review . . . . . . . . . . . . . . . . 265 Xi Xu, Xiuli Du, and Wei Wu SPH Modeling of Water-Soil Coupling Dynamic Problems . . . . . . . . . . . . 283 Chengwei Zhu, Chong Peng, and Wei Wu Assessing Slope Stability Based on Measured Data Coupled with PSO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Jiaqiang Zou, Wei Zhang, and Aihua Liu

A Simple Hypoplastic Model for Sand Under Cyclic Loading Mohammad-Javad Alipour and Wei Wu

Abstract We propose a hypoplastic model for cyclic loading with the help of the intergranular strain concept. Based on a simple hypoplastic model, we try to recommend a straightforward framework in terms of equation and number of parameters. A comparison of the experimental results and the model’s predictions reveal the qualification of the model in predicting the experimental results as well as the model with the same approach.

1 Introduction The hypoplastic constitutive models are broadly employed in geotechnical studies and research projects. Modeling the seismic and vibration loading necessitates the improvement of the hypoplastic constitutive models for cyclic loading [1–7]. In the reference hypoplastic model, for the purpose of simplification, the history of loading is neglected and the model functions on the basis of current stress and strain increment. As a result, in a loop of loading, unloading, and reloading the reference model produce identical results for primary loading and reloading on the basis of the same Cauchy stress and stretching tensor. However, the experimental results imply the evolution in the granular soil stiffness in a loading cycle [3, 4]. Recommending a new hypoplastic model for cyclic loading necessitates including the history of loading in the new models. [6, 8]. The pioneering work by Niemunis and Herle [6] established the intergranular strain concept for the purpose of improving the hypoplastic model for cyclic loading. The intergranular strain can be interpreted as a notion that represents the deformation of the intergranular interface layers[6]. The intergranular strain is a state variable that captures the stretching tensor direction in the sequel of M.-J. Alipour · W. Wu (B) Institute of Geotechnical Engineering, University of Natural Resources and Life Sciences, Vienna, Austria e-mail: [email protected] M.-J. Alipour e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 W. Wu and Y. Wang (eds.), Recent Geotechnical Research at BOKU, Springer Series in Geomechanics and Geoengineering, https://doi.org/10.1007/978-3-031-52159-1_1

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loading steps. A sharp change in the strain path can be recognized using the intergranular strain. Consequently, the model’s stiffness according to the divergence of the strain rate from the intergranular strain can be modified. The intergranular strain concept has been utilized successfully in various studies for the purpose of cyclic loading modeling [2–5]. Reviewing the recent constitutive models for cyclic loading indicates the absence of a straightforward model with regard to formulation and application. In this work, we present a framework on the basis of a simple hypoplastic model [9] using the intergranular strain concept.

2 The Reference Model In this work, the hypoplastic model proposed by [9] is used as the reference model. ˚ is In the preliminary version of the reference model [10], the Jaumann stress rate (.T) a function of the Cauchy stress (.T) and stretching tensor (.D). Thereafter, the effect of void ratio (.e) with the use of the critical state concept was considered. The general form of the model is expressed as a tensor-valued function H such that: .

T˚ = H(T, D, e).

(1)

The model is proposed with two main parts that account for the soil’s linear and nonlinear behavior. With the help of representation theorem for two symmetric tensors [11] (where in this case, symmetric tensors are .T and .D), a tensor valued function is defined as: ˚ = L(T) : D + N(T)||D||Ie , .T (2) where .L : D and .N||D|| are linear and nonlinear in√ .D respectively, and term .||D|| is the norm of stretching tensor is defined as .||D|| = D : D. The fourth order tensor .L and second order tensor .N in Einstein abbreviation notation are defined as: L i jkl = c1 Tnn δik δ jl + c2 Ti j Tkl /Tnn .

Ni j = (c3 Tik Tk j + c4 Til∗ Tl∗j )/Tnn

(3)

Where .δi j is Kronecker delta, .ci (i = 1, ..., 4) are dimensionless constant, and .T∗ is defined as .Ti∗j = Ti j − Tnn δi j /3. The scalar functions . Ie consider the critical state and volume change effects inside the model: I = (q1 + q2 exp (q3 Tnn ) − 1)

. e

ecr − e +1 ecr − emin

(4)

Where .emin and .qi (i = 1, 2, 3) are constant parameters of the model, and .ecr is the critical state void ratio corresponding to .T.

A Simple Hypoplastic Model for Sand Under Cyclic Loading

3

3 Hypoplastic Model for Cyclic Loading The experimental study reveals the incrementally linear behavior of the granular soil at the beginning of loading. In this work, the intergranular strain concept is used to consider the linear behavior of the granular soil after a sudden change in the strain path. The predicted stiffness gradually returns to nonlinear with persisting the strain path [6].

3.1 Intergranular Strain Concept The intergranular strain concept proposed by [6] to considers the deformation of the interface layers and the evolution of the chain forces due to the change in loading conditions. The rate of intergranular strain .δ˙ is deified as: ] ⎧[ →− → ( )0.5 − → ⎨ I −− δ δ ||δ|| :D δ :D>0 R ˙= .δ − → ⎩ D δ :D≤0

(5)

the term R is the radius of the linear surface and the maximum value of the inter− → granular strain. The term . δ is the direction of the inter-granular strain and is defined as: { δ/||δ|| for δ /= 0 − → . δ = (6) 0 for δ = 0 According to Eq. 5, the value and direction of the .δ changes with a change in .D direction. The value of .δ changes from R to zero and then to R, while the direction of .δ changes toward the new D direction.

3.2 Cyclic Loading Functions With the use of an intergranular tensor, the functions . f L and . f N are suggested as T˚ = f L L(T) : D + f N N(T)||D||Ie

.

(7)

Where . f L and . f N are the functions responsible for the linear and nonlinear part of the model respectively. The functions alter the model’s stiffness with the help of the direction and magnitude of the .δ. The linear multiplier function (. f L ) is defined as: f = m + 0.5ρ χ (1 − m)(1 + cos θ )

. L

(8)

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Fig. 1 The scalar value of function . f L with respect to .θ for various values of .ρ

The term .ρ refers to the normalized value of .δ which is equal to .||δ||/R, and .χ is a constant parameter. The term .m is a constant parameter and implies the maximum multiplier of the linear stiffness. The term .cos θ has come to be used to refer to the − → − → deviation of the strain rate direction (. D ) from the intergranular strain direction (. δ ) − → − → which define as .cos θ = D : δ . In addition, instead of two multiplier parameters [6], we simplified the model using only one multiplier parameter. Figure 1 presents the scalar values of the . f L concerning .θ and .ρ. Maximum . f L happens at a sharp change in loading direction (.θ = 180°), and gradually decreases with the adjustment → − → − of . δ to . D . The function (. f N ) is defined as: f = ρ χ ⟨cos θ ⟩

. N

(9)

where .⟨ ⟩ are Macaulay brackets (.⟨x⟩ = x if .x ≥ 0, and .⟨x⟩ = 0 if .x < 0). As long as .θ ≥ 90◦ , . f N is equal to zero and deactivate the .N(T)||D||Ie part from the model. → − → − With the adjustment of . δ to . D and when .θ < 90◦ , the nonlinear part will activate and impact the predictions.

4 Model Performance In the present work, the experimental data of monotonic and non-monotonic loading tests on Karlsruhe fine sand is utilized [3, 4, 12]. The reference constitutive model is calibrated with the help of conventional drained and undrained triaxial test results (for the method of calibration see [10]). Table 1 presents the parameters of the model.

A Simple Hypoplastic Model for Sand Under Cyclic Loading

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Table 1 The parameters of the model .c1

.c2

-51.78

.−469.15

.c3 .−426.64

.c4

.q1

279.13 1.00

.q2

.q3

.emin

.m

.χ

R

.−0.31

.10−4

0.40

8.0

12

.10−4

Fig. 2 Predictions and experimental data of monotonic drained tests on Karlsruhe fine sand: a Deviatoric stress q versus axial strain .ε1 and b volumetric strain .εv versus axial strain .ε1 . Solid lines are the predictions of the model. Experimental data after [3, 4]

In this work, we use the critical sate line recommended by [3]: e = ec0 exp [−(3 p/ h s )n ]

. cr

(10)

where .ec0 = 1.054, .h s = 4000 MPa and .n = 0.27. The term . p is mean effective stress and defined as . p = Tnn /3. Figure 2 provides the prediction and experimental data of conventional triaxial tests on samples with diverse relative densities and effective confining stress. Comparing the experimental results and prediction of the model shows the model’s capability in predicting the soil strength, volume change, and dilatancy in the monotonic drained triaxial test. Figure 3 presents the experimental data and the prediction of the model for the triaxial undrained tests. The predictions imply the acceptable performance of the model in predicting phase transformations and the shear strength of the samples with different void ratios and confining pressure. In subsequent, we present the predictions of the Karlsruhe fine sand under the cyclic loading conditions using our hypoplastic model.

4.1 Predictions for Cyclic Loading The reference model predicts the stiffness of cohesionless soil under monotonic loading reasonably. Nevertheless, hypoplastic models in general are not very successful in predicting the behavior of the soil under cyclic loading. The newly presented

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Fig. 3 Predictions and experimental data of monotonic undrained triaxial test on Karlsruhe fine sand: a Deviatoric stress q versus mean effective stress p and b deviatoric stress q versus axial strain .ε1 . Solid lines are the predictions of the model. Experimental data after [3, 4]

Fig. 4 Predictions and experimental data of triaxial cyclic drained tests on Karlsruhe fine sand: a, b Deviatoric stress q versus axial strain .ε1 and c, d volumetric strain .εv versus axial strain .ε1 . Solid lines are the predictions of the model. Experimental data after [3, 4]

model provides a criterion after a change in the loading direction that deactivates the nonlinear and multiplies the linear part of the model. Figure 4 presents the experimental data and the predictions of the extended model for the triaxial cyclic drained test. The compression triaxial drained tests were unloaded to .q = 0 and reloaded

A Simple Hypoplastic Model for Sand Under Cyclic Loading

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between constant strain increments. The model predicts an increase in stiffness after the change of loading direction. Since the size of the linear surface is independent of the material’s history, the model cannot perfectly predict the reloading steps in Fig. 4. Such a shortcoming can be seen in the performance of other models as well [4, 5]. Under a cyclic loading condition, the results of the triaxial undrained test exhibit an increase and accumulation of pore water pressure and a decrease in mean effective stress. In an undrained cyclic loading test with constant shear stress amplitude, the stress path goes through several cycles and reaches the butter-fly-shaped zone. The reference model predicts an extensive pore water pressure compared with the experimental results which causes the stress path to reach the butterfly-shaped loop after a few cycles. Such a shortcoming can be better realized in the predictions of the undrained cyclic tests with minor shear stress and shear strain amplitude. Thanks to considering a linear surface in the model, a greater stiffness is predicted after a change in the loading direction. Consequently, as Fig. 5 shows, the modified model can predict a more accurate number of cycles prior to liquefaction for a medium-dense

Fig. 5 Prediction and experimental data of triaxial cyclic undrained test with constant deviatoric stress amplitude (q.amp = 40 kPa) on a medium dense sample of Karlsruhe fine sand: a, c Deviatoric stress q versus mean effective stress p and b, d deviatoric stress q versus axial strain.ε1 . Experimental data after [3, 4]

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Fig. 6 Prediction and experimental data of triaxial cyclic undrained test with constant axial strain amp amplitude (.ε1 = 6 × 10−4 ) on a medium dense sample of Karlsruhe fine sand: a, c Deviatoric stress q versus mean effective stress p and b, d deviatoric stress q versus axial strain.ε1 . Experimental data after [3, 4]

sample under cyclic loading. The experimental result of an undrained triaxial cyclic test with small strain amplitude cycles is provided with model prediction in Fig. 6. The medium-dense sample with initial anisotropic stress is sheared under a small strain amplitude. The model predicts fairly well the stress path and the extremum points of shear stress. As earlier discussed, the reference model does not discriminate between loading and reloading. In predicting isotropic and oedometer cyclic loading tests, this shortcoming brings about the prediction of extensive volume change and inaccurate stiffness [6]. Considering the intergranular strain in the model can solve the drawback to a significant degree [4, 5, 8]. Figure 7 provides model predictions and experimental results of three oedometer cyclic loading tests. Figure 8 shows the prediction of the model and the experimental result of a drained isotropic test with ten stress-controlled cycles (p.ampl = 750 kPa). As long as the model is independent of the loading history, slight ratcheting and overshooting in predictions are unavoidable.

A Simple Hypoplastic Model for Sand Under Cyclic Loading

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Fig. 7 Predictions and experimental data of the oedometer cyclic tests on samples of Karlsruhe fine sand: Void ratio e versus mean effective stress p. Solid lines are the predictions of the model. Experimental data after [3, 4]

Fig. 8 Predictions and experimental data of an isotropic cyclic test with constant mean effective stress amplitude (p.ampl = 750 kPa): Void ratio e versus mean effective stress p. Solid line is the prediction of the model. Experimental data after [3, 4]

5 Conclusion In this work, we employ an intergranular strain concept to propose a simple hypoplastic model for cyclic loading. With the use of intergranular strain, we recommend two multiplier functions responsible for the linear and nonlinear parts of the hypoplastic model. The number of multiplier parameters decreased to one parameter. The inner product of the stretching tensor and intergranular strain is utilized to simplify the equations. The recommended model can provide fairly decent predictions for drained and undrained cyclic loading tests. The intergranular strain provides a linear surface with a constant radius for all conditions independent of the mechanical history. As a result, the proposed model and models with the same strategy may show slight shortcomings such as ratcheting and overshooting in the predictions. Further work needs to be done to establish a framework in which the entire mechanical history is considered in the model.

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Acknowledgements The authors wish to acknowledge the financial support from the Otto Pregl Foundation of Fundamental Geotechnical Research in Vienna, and EU Horizon 2020 RISE project - FRAMED (Grant No. 734485) and EU Horizon 2020 RISE project - HERCULES (Grant No. 778360).

References 1. Poblete, M., Fuentes, W., Triantafyllidis, T.: On the simulation of multidimensional cyclic loading with intergranular strain. Acta Geotech. 11, 1263–1285 (2016) 2. Fuentes, W., Wichtmann, T., Gil, M., Lascarro, C.: ISA-Hypoplasticity accounting for cyclic mobility effects for liquefaction analysis. Acta Geotech. 15, 1513–1531 (2020) 3. Wichtmann, T., Triantafyllidis, T.: An experimental database for the development, calibration and verification of constitutive models for sand with focus to cyclic loading: part i—tests with monotonic loading and stress cycles. Acta Geotech. 11, 739–761 (2015) 4. Wichtmann, T., Triantafyllidis, T.: An experimental database for the development, calibration and verification of constitutive models for sand with focus to cyclic loading: part II—tests with strain cycles and combined loading. Acta Geotech. 11, 763–774 (2015) 5. Wichtmann, T., Fuentes, W., Triantafyllidis, T.: Inspection of three sophisticated constitutive models based on monotonic and cyclic tests on fine sand: Hypoplasticity vs. sanisand vs. ISA. Soil Dyn. Earthquake Eng. 124, 172–183 (2019) 6. Niemunis, A., Herle, I.: Hypoplastic model for cohesionless soils with elastic strain range. Mech. Cohesive-Friction. Mater. 2, 279–299 (1997) 7. von Wolffersdorff, P.-A.: A hypoplastic relation for granular materials with a predefined limit state surface. Mech. Cohesive-Friction. Mater. 1(3), 251–271 (1996) 8. Bode, M., Fellin, W., Mašín, D., Medicus, G., Ostermann, A.: An intergranular strain concept for material models formulated as rate equations. Int. J. Numer. Anal. Meth. Geomech. 44, 1003–1018 (2020) 9. Wu, W., Bauer, E., Kolymbas, D.: Hypoplastic constitutive model with critical state for granular materials. Mech. Mater. 23, 45–69 (1996) 10. Wu, W., Bauer, E.: A simple hypoplastic constitutive model for sand. Int. J. Numer. Anal. Meth. Geomech. 18, 833–862 (1994) 11. Wang, C.C.: A new representation theorem for isotropic functions: An answer to Professor G. F. Smith’s criticism of my papers on representations for isotropic functions. Arch. Ration. Mech. Anal. 36, 198–223 (1970) 12. Wichtmann, T.: Homepage.

Failure Mode and Mechanism of the Consequent Slope in Xuan’en County Yi Du, Chao Cheng, Jiaqiang Zou, Jian Wang, and Echuan Yan

Abstract The deformation degree and failure mode of the slope is always different under the influence of different factors. In this paper, the geological characteristics of the consequent slope failure happened in Xuan’en County are statistically analyzed, the sensitivity of several factors to the deformation degree of the slope is orthogonally studied. Further, the influence pattern of the main controlling factors is also discussed by response surface method. The results show that slope dip and strata dip are the main contributing factors, and the difference in their combination forms leads to the difference in slope failure modes. When the strata dip of the rock is greater than the slope dip, the slope shows slipping-bending failure under gravity pressure, otherwise, it is mostly slipping-rupture failure which develops gradually from the toe to the top of the slope.

Y. Du PipeChina Central China Company, Wuhan 430024, China e-mail: [email protected] Y. Du · J. Zou Institute of Geotechnical Engineering, University of Natural Resources and Life Sciences, Vienna, Feistmantelstrasse 4, 1180 Vienna, Austria e-mail: [email protected] C. Cheng · J. Wang · E. Yan (B) Faculty of Engineering, China University of Geosciences, Wuhan 430074, China e-mail: [email protected] C. Cheng e-mail: [email protected] J. Wang e-mail: [email protected] J. Zou College of Water Conservancy and Civil Engineering, South China Agricultural University, Guangzhou 510642, China © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 W. Wu and Y. Wang (eds.), Recent Geotechnical Research at BOKU, Springer Series in Geomechanics and Geoengineering, https://doi.org/10.1007/978-3-031-52159-1_2

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1 Introduction Since the inclination of rock strata in a consequent slope is similar to that of slope direction, sliding along a plane or soft interlayer is one of the consequent slope typical deformation modes. Xuan’en County is located in the southwest of Hubei Province, China. The sandy mudstone slope is widely exposed in this area. With the increase of engineering activities, the deformation and failure of the slope has been paid more and more attention. At present, many researchers have studied the failure modes of the consequent slope. Gao [1] studied four types of failure mode of the consequent slope by discrete element method (DEM), taking slope dip and strata dip as variables. Xiao [2] investigated failure modes of consequent rock slopes under complicated working conditions by DEM depending on three parameters. Zheng [3, 4] revealed the consequent slope failure mechanism of interbedded structure through model tests. Hu and Zhang [5, 6] analyzed the failure mode of bedding landslides from the perspective of progressive failure by considering the strain-softening characteristics of slip-zone soil. Chi [7] studied the influence of rock mass structural plane parameters on slope failure mode based on DEM simulations. So far, most of the existing results take the structural plane strength as the factor affecting the slope failure mode. Few studies have analyzed the slope failure mode and mechanism from the perspective of multi-factors statistics in a region. As such, on the basis of systematically extracting the controlling factors of consequent slope deformation, it is of great significance to quantitatively analyze the interaction pattern of each factor. In this paper, the consequent slope in Xuan’en County is orthogonally studied coupled with DEM. To this end, the parametric study of multiple contributing factors corresponding to consequent slope deformation is carried out and the main controlling factors are determined. What’s more, the quantitative relationship between the control factors and the deformation degree of different parts of the slope is fitted by the response surface method. Thus, the influence pattern of the interaction of controlling factors on the deformation and failure mode of the consequent slope is revealed. Finally, combined with typical numerical models in response surface design, the failure mechanism of the slope is also analyzed.

2 Sensitivity of Factors Orthogonal experimental design is a design method to study multiple factors and levels. As shown in (Table 1), the slope height, slope dip, strata dip, strata thickness, friction angle and cohesion of the rock strata are selected as factors in this experiment, and each factor contains four levels. The orthogonal table suitable for this experiment is L16(.46 ). Each group of experiments will be simulated by 3DEC, and the experimental target is the total displacement in each slope model. As shown in Fig. 1, the model used in the numerical simulation will be changed according to the differences in the levels of experimental factors in each group. In

Failure Mode and Mechanism of the Consequent Slope in Xuan’en County

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Table 1 Experimental factors and level design Factor Level A-Slope height (m) B-Slope dip (.◦ ) C-Strata dip (.◦ ) D-Strata thickness (m) E-Strata c (MPa) F-Strata .φ (.◦ )

15 30 15 0.5 1 35

40 47 30 1.2

65 63 45 1.8 5 40

90 80 60 2.5

Fig. 1 Numerical model of the slope Table 2 Physical and mechanical parameters of rock mass Strata types Density(g/cm.3 ) Elastic Tensile modulus strength (GPa) (MPa)

Poisson’s ratio

Sandy mudstone Structural plane

2.37

3.4

3.0

Tensile strength (MPa)

Cohesion (MPa)

Layer Joint

0.05 0.6

0.27 1

Friction angle Normal (.◦ ) stiffness (GN/m) 14 2.6 25 8.3

0.17 Tangential stiffness (GN/m) 1.7 6.2

addition, the non-variable parameters are obtained from laboratory experiments (see Table 2). In order to simplify the simulation, the whole slope body is assumed to be composed of homogeneous elastoplastic materials in the numerical model, without considering the effect of groundwater and other external forces. Figure 2 shows the change curves of total displacement acquired from all 16 different groups of numerical models based on the orthogonal experiment (see Table 3). It is clear that after 10,000 steps, each curve starts to divide, and at 20,000 steps, the degree of dispersion is relatively large, indicating two states of sharply rising and leveling off. As such, since it can better reflect the state of stability of the model, at

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Fig. 2 Total displacement of the experimental models Table 3 Orthogonal table L16(.46 ) No. Slope Slope dip height 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 R

1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 0.52

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 0.54

Strata dip

Strata thickness

Strata .φ

Strata c

Total displacement

1 2 3 4 2 1 4 3 3 4 1 2 4 3 2 1 0.35

1 2 3 4 3 4 1 2 4 3 2 1 2 1 4 3 0.16

1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 0.22

1 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2 0.07

0.0039 0.34 1.02 1.15 0.0128 0.008 0.0198 0.38 0.0012 0.089 0.0037 0.57 0.039 0.13 0.2 0.11

Failure Mode and Mechanism of the Consequent Slope in Xuan’en County

15

the point of 20,000 steps was therefore selected as the response index for the orthogonal experiment. According to the range analysis and calculation results showed in Table 3, the order of sensitivity of all factors is ranked from high to low, where slope dip is more sensitive than slope height, following strata dip, strata friction angle, strata thickness and finally strata cohesion. Hence, it can be concluded that the slope dip is the main contributing factor affecting slope deformation, while slope height and strata dip are the secondary controlling factors.

3 Analysis of the Interaction Between Main Control Factors 3.1 Response Surface Test and Results Based on the response surface design principle, the interaction of parameter uncertainties among the main control factors on the consequent slope deformation is analyzed. Response surface design is a test method to fit the response surface function between variable parameters and analysis target by least square method [8, 9]. Firstly, the response surface model (see Fig. 3) is constructed based on the Box-Behnken method. This method is suitable for a test with 2–5 factors. Three levels are selected in the range of each factor and coded with .(−1, 0, 1). 0 is the center of the test, .−1 and 1 are the relative small and large values of the cubic point. The function expression of the model is as follows:

.

yˆ = A +

k ∑ i=1

Bi xi +

∑∑ i

j

Ci j xi x j +

k ∑

Di i xi2

(1)

i=1

where . yˆ is the dependent variable; .xi j (.i, j=1,2,.. . .,n) is a random variable; . A, Bi , Ci j , Dii (.i, j=1,2,.· · · , n) is the undetermined factor.

Fig. 3 Distribution map of Box-Behnken design

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Table 4 Response surface test variables and levels Level Factor .−1 0 A- slope height B- slope dip C- strata dip

15 30 15

53 55 38

Table 5 Experimental factors and level design Test Slope height Slope dip 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0

0

.−1

.−1

1 0 0 1 0 .−1 .−1 1 1 .−1 0 0 0 0 0

1

1 .−1

1 0 1 0 0 0 .−1 1 0 0 0 0 .−1

90 80 60

Strata dip

Displacement at the top

Displacement at the toe

0 0 0 1 1 .−1 .−1 .−1 1 1 0 0 0 0 0 0 .−1

0.5800 0.0500 0.0050 0.6050 0.1290 0.0017 0.0036 0.0038 0.0170 0.0496 0.0930 0.1200 0.5800 0.5800 0.5800 0.5800 0.0017

0.9200 0.0080 0.0277 0.0030 0.7700 0.0108 0.0090 0.0100 0.0041 0.0630 0.0260 0.5120 0.9200 0.9200 0.9200 0.9200 0.0040

Then the model and the significance of each factor are verified using ANOVA. Among them, P denotes the probability of significance, and when P is less than 0.05, it indicates that the corresponding event is significant [10, 11]. In this test, slope height, slope dip, and strata dip were taken as independent variables (see Table 4). And the maximum horizontal displacement of slope top and toe at the 20,000th step of each simulation test (the location of monitoring point in Fig. 2) was taken as the response object. The test results are shown in Table 5.

Failure Mode and Mechanism of the Consequent Slope in Xuan’en County

17

3.2 Response Law of Displacement at the Top of Slope Through variance analysis, the second-order polynomial that can be fitted the relationship between the parameters and the horizontal displacement at the slope top is: . yˆ

= 0.58 − 0.0052 A − 0.062B + 0.099C − 0.04 AB + 0.0087AC − 0.12BC − 0.34 A2 − 0.17B 2 − 0.22C 2

where A, B, C can be found in Table 4. As can be seen from Table 6, when . P < 0.05 for the response surface model, it suggests a noteworthy difference, indicating a high degree of significance of the model. The determination coefficient . R 2 .= 0.93 .> 0.8 points out that the response surface model is 93% consistent with the test index. In the first term of ANOVA, the response of top displacement to strata dip is significant, and the interaction between slope dip and strata dip is also obvious. According to the response results, the interaction between AB and AC, the maximum horizontal displacement at the top of the slope appeared near the intermediate level of each factor. Its variation pattern is basically unaffected by the interaction of the controlling factors. As shown in Fig. 4, when the slope dip is small, the displacement increases with the increase of the strata dip, and the increase is about 0.6 m. When the slope is large enough and becomes larger than the maximum strata dip, the increase of the displacement can decrease to 0.15 m. Note that when the strata dip is larger than the slope dip, the deformation at the top of the slope is often more obvious.

Table 6 Variance analysis of response surface model Squares df Square Source Model A B C AB AC BC 2 .A 2 .B 2 .C 2 . R =0.93

1.08 0.0002 0.030 0.078 0.0062 0.0003 0.057 0.49 0.13 0.21

9 1 1 1 1 1 1 1 1 1

0.12 0.0002 0.030 0.078 0.0062 0.0003 0.057 0.49 0.13 0.21 2 . R Ad j .= 0.85

Value

P

10.10 0.018 2.55 6.57 0.53 0.025 4.81 41.01 10.64 17.51

0.0030 0.8966 0.1542 0.0373 0.4918 0.8779 0.0643 0.0004 0.0138 0.0041

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(a) The response surface BC

(b) Contour map between BC

Fig. 4 Total displacement of the experimental models

3.3 Response Law of Displacement at the Toe of Slope The second-order polynomial that can be fitted the relationship between the variables and the maximum horizontal displacement at the slope toe is: . yˆ

= 0.92 − 0.051A − 0.16B + 0.10C − 0.13AB + 0.015AC + 0.19BC − 0.48A2 − 0.30B 2 − 0.42C 2

where A, B, C can be found in Table 4. It can be seen from Table 7 that the response of the displacement at toe to slope dip is compelling (P .= 0.0003), and the interaction between slope dip and strata dip is also obvious. As shown in Fig. 5, in the interaction

Table 7 Variance analysis of response surface model Source Squares df Square Model A B C AB AC BC 2 .A 2 .B 2 .C 2 . R =0.96

2.83 0.021 0.20 0.081 0.063 0.0008 0.15 0.95 0.38 0.75

9 1 1 1 1 1 1 1 1 1

0.31 0.021 0.20 0.081 0.063 0.0008 0.15 0.95 0.38 0.75 2 . R Ad j .= 0.92

Value

P

20.58 1.35 13.34 5.31 4.12 0.055 9.49 62.22 24.93 49.11

0.0003 0.2833 0.0082 0.0546 0.0819 0.8211 0.0178 .< 0.0001 0.0016 0.0002

Failure Mode and Mechanism of the Consequent Slope in Xuan’en County

(a) The response surface BC

19

(b) Contour map between BC

Fig. 5 The action pattern of horizontal displacement response at the toe of slope

between AB and AC, the variation of horizontal displacement at slope toe similar to that at slope top. In the interaction between BC, when the slope dip is small, the displacement first increases and then decreases with the increase of strata dip. When the slope is large enough and becomes larger than the maximum strata dip, the displacement continues to rise with the increase of strata dip, the maximum displacement can be up to 0.8m. Note also that when the strata dip is smaller than the slope dip, the largest deformation of the slope tends to occur at the slope toe.

3.4 Discussion The test group 9 in Table 5 is a typical test model with the slope dip larger than strata dip. As shown in Fig. 6, the deformation process of the model (10k to 30k steps) shows a three-stage deformation mode of slipping-bending: (a) Firstly, the slope top tensile fracture stage. In this stage, the lateral displacement of the slope top is small, and the local tensile stress zone is come into being under gravity. (b) Then, bending uplift stage. With the development of deformation, the lateral displacement of the slope top continues to increase, and the tensile stress in slope also increases. In addition, a bulge deformation at the slope toe can be observed, and the slope hence shows a bending trend. (c) Finally, shear-band penetration stage. The middle and top of the slope are in a state of tension, but the increase of the total displacement of the slope becomes slow. At this time, a penetrating shear band begins to form inside the slope, and when it is completely penetrated through the slope, that is when the failure occurs. The actual failure mode of such slope in Xuan’en County is shown in Fig. 8. In the test model with a strata dip smaller than slope dip (group 12), as shown in Fig. 7, the deformation process of the model presents the failure mode of slipping-

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Fig. 6 Deformation characteristics of group 9 model

Fig. 7 Deformation characteristics of group 12 model

Fig. 8 Typical failure model of consequent slope in Xuan’en County

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Failure Mode and Mechanism of the Consequent Slope in Xuan’en County

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rupture: (a) Initially, the stress unloading stage. The local deformation occurs first at the slope toe, resulting in the readjustment of the rock mass stress after tensile fracture, and the shear strain in the slope body also appears. (b) Further, the fracture expansion stage. At this stage, the slope deformation gradually develops from the toe to the top, and the number of tensile fractures also increases with the expansion of the deformation area. (c) Eventually, the shear failure stage. When the tensile fracture deformation has developed to the slope top, instability failure occurs in the whole slope, and the maximum displacement occurs at the slope toe. Meanwhile, there may be interlayer dislocation in the deformation process, which is proven by the shear strain appearing in different rock layers in the simulation results. The failure mode of such slope is shown in Fig. 8.

3.5 Conclusions There are many factors affecting the state of stability of the consequent slope in Xuan’en County. In terms of slope geometry structure, slope dip mainly controls the deformation degree, while slope height and strata dip are the secondary controlling factors. According to ANOVA in response surface design, the interaction between slope dip and strata dip greatly affects the failure mode of the slope. When the strata dip is greater than the slope dip, the slope deformation is mainly concentrated at the top, and the deformation degree increases with the increase of the strata dip, showing the slipping-bending failure mode. However, when the strata dip is smaller than the slope dip, the displacement at the slope toe is more obvious than that at the top, and the deformation degree increases with the increase of the slope dip, leading to the slipping-rupture failure mode. Acknowledgements This work was financially supported by the National Natural Science Foundation of China (41807264), the International Training Program for Outstanding Young Scientific Researchers in Colleges and Universities of Guangdong Province (Grant no. HT202002314-4). These supports are gratefully acknowledged.

References 1. Gao, Y.T., Xiao, S., Wu, S.C. et al.: Numerical simulation of the deformation and failure characteristics of consequent rock slopes and their stability. Chinese J. Eng.(in Chinese) 37(11), 1403–1409 (2015) 2. Xiao, S., Wu, S.C., Gao, Y.T. et al.: Numerical simulation of failure modes and stability of consequent rock slopes. In: ISRM International Symposium-8th Asian Rock Mechanics Symposium (2014) 3. Zheng, Y., Chen, C., Liu, T., et al.: Slope failure mechanisms in dipping interbedded sandstone and mudstone revealed by model testing and distinct-element analysis. Bull. Eng. Geol. Env. 77(1), 49–68 (2018)

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4. Weng, M.C., Chen, T.C., Tsai, S.J.: Modeling scale effects on consequent slope deformation by centrifuge model tests and the discrete element method. Landslides 14(3), 981–993 (2017) 5. Hu, Q.J., Shi, R.D., Zheng, L.N., et al.: Progressive failure mechanism of a large bedding slope with a strain-softening interface. Bull. Eng. Geol. Env. 77(1), 69–85 (2018) 6. Zhang, S.L., Zhu, Z.H., Qi, S.C., et al.: Deformation process and mechanism analyses for a planar sliding in the Mayanpo massive bedding rock slope at the Xiangjiaba Hydropower Station. Landslides 15(10), 2061–2073 (2018) 7. Chi, E.A., Tao, T.J., Zhao, M.S., et al.: Failure mode analysis of bedding rock slope affected by rock mass structural plane. Appl. Mech. Mater. 602, 594–597 (2014). Trans Tech Publications Ltd 8. Mason, R.L., Gunst, R.F., Hess, J.L.: Statistical Design and Analysis of Experiments: With Applications to Engineering and Science, 2nd edn. Wiley 9. Du, Y., Yan, E.C., Gao, X., et al.: Identification of the main control factors and failure modes for the failure of Baiyuzui landslide control project. Geotech. Geol. Eng. 39(5), 3499–3516 (2021) 10. Freire, L., Carmezim, M.J., Ferreira, M.G.S., et al.: The passive behaviour of aisi 316 in alkaline media and the effect of pH: A combined electrochemical and analytical study. Electrochim. Acta 55(21), 6174 (2010) 11. Ranjbari, E., Hadjmohammadi, M.R.: Optimization of magnetic stirring assisted dispersive liquid-liquid microextraction of rhodamine B and rhodamine 6G by response surface methodology: application in water samples, soft drink, and cosmetic products. Talanta 139, 216–225 (2015)

Reliability Analysis of Slope Stability with Intelligent Surrogate Models: A Case Study in the Three Gorges Reservoir Carlotta Guardiani, Enrico Soranzo, and Wei Wu

Abstract Since the first impoundment of the Three Gorges Dam in China, the yearly fluctuations of the reservoir water level have reactivated several landslides. In this study, the stability of the Huangtupo landslide, one of the largest in the Three Gorges Reservoir region, is investigated with a probabilistic approach. The inherent uncertainty of the shear strength and the hydraulic conductivity is taken into account with the random variable method. Despite the increasing application of probabilistic methods to geotechnical problems, these are rarely applied to complex case studies because of the large computational effort required. To tackle this challenge, surrogate models based on machine learning are used to predict the factor of safety for different levels of the reservoir in a short time. Four models are trained and tested on the data produced with numerical simulations. The best performing model is selected to estimate the probability of failure of one section of the Huangtupo landslide. The results show that surrogate models based on machine learning can predict the factor of safety with good accuracy and improve the computational efficiency of fully probabilistic methods. This study may encourage the application of probabilistic slope stability analysis to complex landslides.

1 Introduction Since the Vajont landslide in 1963, the stability of slopes on reservoir banks has raised great interest in the engineering geology research community. Fluctuations in water levels caused by reservoir operations have a significant impact on the activation C. Guardiani (B) · E. Soranzo · W. Wu Institute of Geotechnical Engineering, University of Natural Resources and Life Sciences, Feistmantelstrasse 4, 1180 Vienna, Austria e-mail: [email protected] E. Soranzo e-mail: [email protected] W. Wu e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 W. Wu and Y. Wang (eds.), Recent Geotechnical Research at BOKU, Springer Series in Geomechanics and Geoengineering, https://doi.org/10.1007/978-3-031-52159-1_3

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of landslides, especially during rapid drawdown. With the construction of the Three Gorges Dam in China, the water level of the Yangtze River was raised initially from 69 to 139m a.s.l. From 2010, after the first impoundment, the water level varies every year from 145m to 175m a.s.l. More than 4000 landslides, including rock avalanches and debris flows, have been mapped along the Yangtze River [1]. The Huangtupo landslide is the largest landslide by volume in the Three Gorges Reservoir region and is located in Badong county (Hubei Province, China). Since 2012, the China University of Geosciences has operated a large underground experimental facility to promote research on landslide hazard and monitor the activity of the Huangtupo landslide. The experimental station consists of a main tunnel and smaller adits, which allowed the direct observation of the sliding zones in one part of the landslide called “Riverside Slump I“ (RSI). When establishing a geotechnical model, the choice of input values for soil properties is often based on limited site investigations and experimental testing. Probabilistic methods can incorporate the inherent uncertainties of soil properties in the analysis and design of geotechnical structures. In the last 20 years, they have been increasingly applied in geotechnical engineering and, in particular, in slope stability [2, 3]. In general, probabilistic methods describe the performance of a system as a function of random variables. This function is also named limit state function and separates the failure from the safety region in the probability space. In slope stability, the factor of safety is the performance function and the soil properties are modeled as random variables. Reliability-based design classifies probabilistic methods into three levels of sophistication: semiprobabilistic (implemented in Eurocode 0 [4]), analytical or approximate (e.g. First-Order and Second-Order Reliability Method) and numerical or fully probabilistic methods. The last category includes Monte Carlo simulations, which are often applied due to their conceptual simplicity, but, are computationally intensive. Surrogate models (also called “metamodels”) can reduce the calculation time by approximating the relationship between random variables (the soil properties) and the performance of the system (in this case, the factor of safety). These models can be mathematical functions or models based on artificial intelligence and have been applied to several geotechnical problems including slope stability, tunneling and shallow foundations [5–7]. However, to the authors’ knowledge, they have rarely been used to solve real case studies [8]. A considerable amount of literature has been published on the stability of Huangtupo. However, only a few studies adopted a probabilistic approach [9–12]. Based on past laboratory test results, Xue et al. [13] applied the random variable method and non stationary random fields to model the shear strength parameters of the two sliding zones in RSI. Liao et al. [14] conducted a reliability analysis of RSI under the effect of drying-wetting cycles due to the fluctuations of the reservoir. To take into account more possible failure modes, Liao et al. [15] estimated the failure probability related to a system that includes both shallow and deep sliding surfaces. In this study, we assessed the stability of a longitudinal section of RSI with a probabilistic approach based on Monte Carlo simulations and surrogate models.

Reliability Analysis of Slope Stability with Intelligent Surrogate …

25

With this approach, the factor of safety can be predicted for different levels of the reservoir. The case study and the deterministic model are described first, followed by an introduction to the probabilistic approach adopted with surrogate models.

2 Materials and Methods 2.1 Case Study The Huangtupo landslide has an area of 1.35.×10.6 m.2 and a volume equal to 70 .×10.6 m.3 . The site is located about 70 km upstream of the Three Gorges Dam in Badong county (Hubei Province) on the southern bank of the Yangtze River. It is a complex landslide consisting of four smaller slides, “Substation Landslide” and “Garden Spot Landslide” in the upper part, while “Riverside Slump I” and “Riverside Slump II” have the toe partially submerged by the Yangtze River. RSI has a volume of 23.×10.6 m.3 and an average thickness of 69.4m. It exhibits the largest movement with approximately 15mm a.−1 . GPS monitoring showed that the deformation velocity becomes larger with decreasing distance from the water level, thus indicating the great influence of the reservoir on the sliding mass [16]. The geological formation Middle Triassic Badong T.2 b dominates the area of the Huangtupo landslide, in particular the second and third members (.T2 b2 and .T2 b3 ), which consist of limestone and marlstone and come in succession from the top to the toe of RSI. The bedrock is composed of clastic and carbonate rocks, whereas the landslide body is made of loose rock and soil debris. Tang et al. [17] found out through Uranium-Thorium dating that two independent sliding zones with different calcium carbonate ages exist within RSI. The deep sliding zone is 100 ka, while the shallow sliding zone is younger, with 40 ka. Wang et al. [18] proposed a new model for the RSI based on the monitoring data obtained with inclinometers, in which RSI incorporates two independent sliding masses. The deep sliding zone has a grayishgreen color and is contains of silty clay and gravel. The material of the shallow one is brownish-yellow and includes finer particles and less gravel. In the main tunnel of the large test site, which is used also for the execution of in-situ experiments, the sliding zone is exposed in several sections and exhibits a thickness of 10–30 cm.

2.2 Probabilistic Approach Probabilistic methods take into account the uncertainties of a system by coupling deterministic analyses (from physically-based models) and sampling methods to achieve an estimate of the probability of failure (. pf ). For this purpose, Monte Carlo simulations are conceptually straightforward:

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• a sample data set of random variables is generated with sampling techniques; • for each sample unit, the limit state function is evaluated by replacing the values of the parameters selected as random variables in the deterministic model; • . pf is obtained as the ratio between the number of failures and sample size. In slope stability, failure occurs when the factor of safety (FS) is lower than unity and the probability of failure can be defined as

.

pf =

N 1 ∑ Nfailure I [F S < 1] = N i=1 Ntrial

(1)

where. I [·] = 1 is an indicator function of slope failure. The accuracy of the estimated pf depends on the sample size [19] as Eq. 2 shows:

.

/ COV pf =

.

(1 − pf ) . Ntrial pf

(2)

Since the Monte Carlo method generates a sample based on pseudo-random numbers, this procedure can be time consuming, especially for small probabilities of failure [20]. Two strategies are available to reduce the computational effort: either by reducing the sample size with variance reduction techniques or by using surrogate models as substitutes for physically-based numerical models. The former can be achieved with the Latin Hypercube Sampling (LHS) method [21], which divides the probability space into equal intervals and pseudo-random numbers can be generated with fewer simulations compared to the direct Monte Carlo method. With the latter strategy, a surrogate models replaces the deterministic model. In the problem herein analysed, the surrogate model approximates the relationship between the soil properties, which are modelled as random variables (e.g. the shear strength parameters), and the factor of safety. In this study, the random variables are .c' , .ϕ ' and .ks for the two sliding zones and the two soil layers of the landslide body apart from the bedrock, for a total of 12 random variables. The following assumptions are made: the random variables follow a lognormal distribution with a coefficient of variation (COV) equal to 15% and zero correlation. Once the surrogate model has been established (Sect. 2.2.2), the probability of failure can be obtained for samples of arbitrary dimensions. A more detailed description of the procedure can be found in Guardiani et al. [5].

2.2.1

Deterministic Model

The probabilistic method underlies the formulation of a deterministic model, which comprises a transient seepage analysis in a time span of one year and a slope stability analysis with the limit equilibrium method by Morgenstern-Price. The analyses are carried out with the SEEP/W and SLOPE/W modules available in the commercial

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400 Color

Name Bedrock

Elevation (m)

300

Dense soil and rock debris Loose soil and rock debris Sliding zone soil 1 Sliding zone soil 2

200

100

0 0

100

200

300

400

500

600

700

800

900

1.000

Distance (m)

Fig. 1 Deterministic model of RSI in GeoStudio Table 1 Material parameters ' .c (kPa) Material Bedrock Loose soil and rock debris Dense soil and rock debris Upper sliding zone Lower sliding zone

.ϕ

' (.◦ )

.ks

(m s.−1 )

380 31.8

44.2 21.5

5.8.×10−7 2.2.×10−5

33.2

23.2

8.1.×10−6

26.2 30.4

15.1 20.8

8.8.×10−6 2.8.×10−6

software GeoStudio [22]. The model shown in Fig. 1 is modified after Yang [23] and the soil properties of each layer, whose values are taken from Liao et al. [15], are shown in Table 1. In the seepage analysis, the boundary conditions are a constant water head of 265 m on the upstream side and a water total head varying according to the periodic fluctuations of the reservoir (blue line in Fig. 2). The latter boundary condition is a function of the reservoir water level over time and is obtained with a piecewise linear regression, which fits the monitoring data of the reservoir level between 2011 and 2019. In the time span of the simulation, five stages can be identified: slow drawdown, rapid drawdown, slow rise, rapid rise and a period with an approximately constant water level. The initial condition is a steady state seepage with a reservoir water level equal to 175 m a.s.l. One analysis per week is conducted, with a total of 52 simulations.

2.2.2

Machine Learning-Based Surrogate Models

A surrogate model based on machine learning (ML) is applied to increase the computational efficiency of fully probabilistic methods. Five ML algorithms are tested:

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linear regression (LIR), k-nearest neighbour (KNN), random forest (RF), gradient boosting (GB) and extreme gradient boosting (XGB). The last four models can be applied both for classification and regression problems. KNN [24] aims to predict the output from a k-sample of data closest in distance to a new point to be evaluated. RF [25] and GB [26] are two ensemble models based on decision trees. The main difference lies in the way trees are built and aggregated. In RF, decision trees are built in parallel (bagging) and trained on subsets of the data. Finally, the predictions of all trees are averaged. In GB, trees are built sequentially to improve the prediction in an iterative manner. XGB [27] is an advanced version of gradient boosting. In fact, GB uses gradient descent to find the minumum of the loss function, while XGB computes second-order gradients and applies an advanced regularization, limiting overfitting and improving generalization. These models are applied to solve a multi-output regression problem by fitting one regressor per target. The objective is to predict FS for each week during the analysis time for a total of 52 targets. Multi-output regression is implemented in the Python package Scikit-learn [28]. A sample of 1000 units is generated with LHS, of which 70% and 30% are assigned the training and test set, respectively. Fivefold cross-validation is applied to the training set to avoid overfitting. This procedure divides the training set into five folds, whereas four are considered for training and the remaining one for validation. Repeating this procedure for all combinations yields a mean performance score for all folds. For each model, there are hyperparameters that need to be tuned manually, like the number of neighbours for k-nearest neighbour or the number of estimators for random forest and gradient boosting. The Bayes search implemented in the scikit-optimize library [29] has been selected for this task. This algorithm constructs a probabilistic model of the objective function to be optimized and uses the information acquired from samples of the hyperparameter values to find a promising configuration based on current observations. At each iteration, the probability model improves and focuses the search in regions of the hyperparameter space that are likely to increase the model’s performance.

3 Results and Discussion In all time steps, the critical slip surface partially coincides with the shallow sliding surface (Fig. 2). Figure 3 shows the FS obtained with the limit equilibrium analysis by setting the mean values of the soil properties in Table 1. In the steady-state seepage analysis, when the reservoir is at its maximum level of 175m a.s.l., FS is equal to 1.22 and the sliding mass is almost submerged (except for a few slices on top). After 126 days of slow drawdown (with a rate of 0.11m d.−1 ), FS decreases to a value of 1.15. During this time (January to April), which also coincides with the dry season, the Three Gorges Dam supplies water. Subsequently, a phase of rapid drawdown takes place until day 161 (the descending rate is equal to 0.36m d.−1 ) and the FS further diminishes, reaching a minimum value of 1.11. This period (May and June) precedes the flooding season, during which intense and prolonged precipitation occurs. In

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29

400

300

Elevation (m)

-500

200

500

1.250

100 2.00 0

0 0

100

200

300

400

500

600

700

800

900

1.000

Distance (m)

Fig. 2 Results of seepage modelling in GeoStudio and critical slip surface after 161 days (pore water pressure labels are in kPa) 1.4

0.3

1.3

1.2

0.2

1.1

0.1

0.0

0

100

200 Time (days)

300

1.0

200 180 160 140 120

Reservoir water level (m a.s.l.)

Failure probability Factor of safety Reservoir water level

Factor of safety (-)

Failure probability (-)

0.4

100

Fig. 3 Variation over time of . pf (obtained with LHS), FS and reservoir water level

summer, FS increases until day 245 gaining a value of 1.13. This time (from June to September) serves for flood control and the reservoir level increases at a low rate of 0.05m d.−1 . A peak in FS is observed after a phase of rapid rise (0.5m d.−1 ) after 287 days from the beginning of the analysis, with a maximum value of 1.22. A rough estimate of . pf is obtained from the 1000 simulations carried out with GeoStudio to generate the training data for the surrogate model. The probability of failure mirrors the variation of the FS, reaching a maximum of 26% after rapid drawdown and a minimum of 5% after the rapid rise of the water level in autumn. Clearly, rapid drawdown is the most critical phase for the stability of RSI. Two factors play a major role in the decrease of the factor of safety: the reduction of the hydrostatic pressure on the slope, which has a stabilizing effect, and the slow dissipation of the pore water pressure, which occurs at a lower rate compared to the velocity of the reservoir. In this model, the response of the water level adjusts with very little delay to the variation of the reservoir’s water level.

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Table 2 Performance score of five-fold cross-validation and test Model Cross-validation Test 2 2 . R (–) . R (–) LIR KNN RF GB XGB

± 0.052 ± 0.039 .0.929 ± 0.020 .0.894 ± 0.024 .0.944 ± 0.020

RMSE (–)

.0.736

.0.737

.0.055

.0.619

.0.639

.0.065

.0.915

.0.032

.0.884

.0.038

.0.935

.0.028

The results of the training and validation phase are shown in Table 2 in terms of coefficient of determination (. R 2 ) and root mean square error (RMSE). KNN exhibits a poor performance compared to LIR, with a. R 2 for the cross-validation equal to 62% against 74% for LIR. This can be due to the high number of independent variables, for which a given observation can have a very few or none neighbors [30]. Among the decision trees ensemble models, XGB has the highest accuracy as expressed both by . R 2 and RMSE, followed by RF and GB. After the validation, the surrogate models are used to predict FS on a new larger sample of dimension 10.5 , with the same characteristics (mean value, COV and correlation) of the smaller sample produced with LHS. Figure 4 compares . pf estimated with LHS and the results of GeoStudio (green line) and the values obtained with the surrogate models. The discrepancy between the results obtained with XGB and GeoStudio is in the range 0.6–2.1%. Feature importance is applied to XGB to highlight the most useful variables in the implementation of the surrogate model. The importance score, which goes from 0 to 1, is calculated for each predictor. Figure 5 shows that the effective friction angle of the two soil layers yield the largest importance score for all time steps. The importance score of the friction angle of the dense soil layer (orange line in Fig. 5), which underlies the loose soil layer, follows the variation of the water level and lies within the range 0.33–0.53. For the friction angle of the loose soil layer (violet line), the importance score has the opposite behavior: it

0.30 Failure probability (-)

Fig. 4 Calculated . pf with GeoStudio (blue) and predicted . pf with the surrogate models

LHS MC LIR MC KNN MC RF MC GBR MC XGB

0.25 0.20 0.15 0.10 0.05 0.00

0

50

100

150 200 250 Time (days)

300

350

Reliability Analysis of Slope Stability with Intelligent Surrogate … 0.6 Feature importance (-)

Fig. 5 Results of the feature importance analysis: #1—layer of dense soil and rock debris; #2—loose soil and rock debris; #3—upper sliding zone; #4—lower sliding zone

31

0.5 0.4 0.3 coh1 phi1 ks1

0.2

coh2 phi2 ks2

coh3 phi3 ks3

coh4 phi4 ks4

0.1 0.0

0

10

20 30 Time (weeks)

40

50

increases up to 0.6 after 154 days, before . pf reaches its maximum value. The results of the feature importance analysis suggest that efforts in experimental investigations should be concentrated on these two parameters.

4 Conclusions This study was undertaken to perform a probabilistic analysis of the Huangtupo landslide and to evaluate the applicability of surrogate models on a complex case study. The results show that surrogate models can be successfully used to enhance the computational efficiency of fully probabilistic methods (e.g. Monte Carlo simulations) without compromising the accuracy of the estimated failure probability. Among the five models tested in a multi-output regression framework, the extreme gradient boosting emerged as the most accurate predictor of the factor of safety over time. The present study suggests a role for machine learning-based surrogate models to solve probabilistic slope stability analysis for real case studies with reasonable computational effort. Acknowledgements Financial support for this research is provided by the Otto Pregl Foundation for Geotechnical Fundamental Research in Vienna and by the project “HERCULES” within the European Union’s Horizon 2020 Marie Sklodowska-Curie Action Research and Innovation Staff Exchange (RISE) programme (Grant No. 778360).

References 1. Tang, H., Wasowski, J., Juang, C.H.: Geohazards in the three Gorges Reservoir Area, China Lessons learned from decades of research. Eng. Geol. 261, 105267 (2019). https://doi.org/10. 1016/j.enggeo.2019.105267 2. Griffiths, D.V., Fenton, G.A.: Probabilistic slope stability analysis by finite elements. J. Geotech. Geoenviron. Eng. 130(5), 507–518 (2004)

32

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3. Low, B.K., Lacasse, S., Nadim, F.: Slope reliability analysis accounting for spatial variation. Georisk: Assess. Manag. Risk Eng. Syst. Geohazards 1(4), 177–189 (2007). https://doi.org/ 10.1080/17499510701772089 4. CEN (European Committee for Standardization): Eurocode - Basis of structural design. Standard, Brussels (2002) 5. Guardiani, C., Soranzo, E., Wu, W.: Time-dependent reliability analysis of unsaturated slopes under rapid drawdown with intelligent surrogate models. Acta Geotechnica 17(4), 1071–1096 (2022). https://doi.org/10.1007/s11440-021-01364-w 6. Soranzo, E., Guardiani, C., Wu, W.: A soft computing approach to tunnel face stability in a probabilistic framework. Acta Geotechnica 17(4), 1219–1238 (2022). https://doi.org/10.1007/ s11440-021-01240-7 7. He, X., Wang, F., Li, W., Sheng, D.: Deep learning for efficient stochastic analysis with spatial variability. Acta Geotechnica 17(4), 1031–1051 (2022). https://doi.org/10.1007/s11440-02101335-1 8. Wang, L., Wu, C., Tang, L., Zhang, W., Lacasse, S., Liu, H., Gao, L.: Efficient reliability analysis of earth dam slope stability using extreme gradient boosting method. Acta Geotechnica 15(11), 3135–3150 (2020). https://doi.org/10.1007/s11440-020-00962-4 9. Cojean, R., Caï, Y.J.: Analysis and modeling of slope stability in the Three-Gorges Dam reservoir (China) - The case of Huangtupo landslide. J. Mountain Sci. 8(2), 166–175 (2011). https://doi.org/10.1007/s11629-011-2100-0 10. Sun, G., Zheng, H., Tang, H., Dai, F.: Huangtupo landslide stability under water level fluctuations of the Three Gorges reservoir. Landslides 13(5), 1167–1179 (2016). https://doi.org/10. 1007/s10346-015-0637-7 11. Li, X., Zhao, C., Hölter, R., Datcheva, M., Alimardani Lavasan, A.: Modelling of a large landslide problem under water level fluctuation-model calibration and verification. Geosciences 9(2), 89 (2019). https://doi.org/10.3390/geosciences9020089 12. Wang, J., Wang, S., Su, A., Xiang, W., Xiong, C., Blum, P.: Simulating landslide-induced tsunamis in the Yangtze River at the Three Gorges in China. Acta Geotechnica 16(8), 2487– 2503 (2021). https://doi.org/10.1007/s11440-020-01131-3 13. Xue, Y., Miao, F., Wu, Y., Li, L., Meng, J.: Application of uncertain models of sliding zone on stability analysis for reservoir landslide considering the uncertainty of shear strength parameters. Eng. with Comput. (2021). https://doi.org/10.1007/s00366-021-01446-z 14. Liao, K., Wu, Y., Miao, F., Li, L., Xue, Y.: Effect of weakening of sliding zone soils in hydrofluctuation belt on long-term reliability of reservoir landslides. Bullet. Eng. Geol. Environ. 80(5), 3801–3815 (2021). https://doi.org/10.1007/s10064-021-02167-9 15. Liao, K., Wu, Y., Miao, F.: System reliability analysis of landslides subjected to fluctuation of reservoir water level: a case study in the Three Gorges Reservoir area, China. Bullet. Eng. Geol. Environ. 81(6), 225 (2022). https://doi.org/10.1007/s10064-022-02698-9 16. Wang, J., Su, A., Xiang, W., Yeh, H.F., Xiong, C., Zou, Z., Zhong, C., Liu, Q.: New data and interpretations of the shallow and deep deformation of Huangtupo No. 1 riverside sliding mass during seasonal rainfall and water level fluctuation. Landslides 13(4), 795–804 (2016). https:// doi.org/10.1007/s10346-016-0712-8 17. Tang, H., Li, C., Hu, X., Su, A., Wang, L., Wu, Y., Criss, R., Xiong, C., Li, Y.: Evolution characteristics of the Huangtupo landslide based on in situ tunneling and monitoring. Landslides 12(3), 511–521 (2015). https://doi.org/10.1007/s10346-014-0500-2 18. Wang, J., Su, A., Xiang, W., Yeh, H.F., Xiong, C., Zou, Z., Zhong, C., Liu, J., Cao, S.: Threedimensional analyses of the sliding surface distribution in the Huangtupo No. 1 riverside sliding mass in the Three Gorges Reservoir area of China. Landslides 15(7), 1425–1435 (2018). https:// doi.org/10.1007/s10346-018-1003-3 19. Ang, A.H.S., Tang, W.H.: Probability Concepts in Engineering: Emphasis on Applications to Civil and Environmental Engineering, 2nd edn. Wiley, New York (2006) 20. Jiang, S.H., Huang, J.S.: Efficient slope reliability analysis at low-probability levels in spatially variable soils. Comput. Geotech. 75, 18–27 (2016). https://doi.org/10.1016/j.compgeo.2016. 01.016

Reliability Analysis of Slope Stability with Intelligent Surrogate …

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21. Vorechovský, M., Novák, D.: Correlation control in small-sample Monte Carlo type simulations I: a simulated annealing approach. Prob. Eng. Mech. 24(3), 452–462 (2009). https://doi.org/ 10.1016/j.probengmech.2009.01.004 22. GEO-SLOPE International Ltd.: Geostudio (2018). https://www.geoslope.com/ 23. Yang, J.: Research of recurrence mechanism of Huangtupo landslide under water level variation and rainfall in Badong county of Three Gorges Reservoir area. Master’s thesis, China University of Geosciences (2012). (in Chinese) 24. Goldberger, J., Hinton, G.E., Roweis, S., Salakhutdinov, R.R.: Neighbourhood components analysis. In: Advances in Neural Information Processing Systems, vol. 17. MIT Press (2004) 25. Breiman, L.: Random forests. Mach. Learn. 45(1), 5–32 (2001). https://doi.org/10.1023/A: 1010933404324 26. Friedman, J.H.:. Greedy function approximation: a gradient boosting machine. Ann. Stat. 1189– 1232 (2001) 27. Chen, T., Guestrin, C.: XGBoost: a scalable tree boosting system. In: Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ’16, pp. 785–794, New York, NY, USA (2016). Association for Computing Machinery. https:// doi.org/10.1145/2939672.2939785 28. Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V. and Vanderplas, J., Passos, A., Cournapeau, D., Brucher, M., Perrot, M., Duchesnay, E.: Scikit-learn: machine learning in python. J. Mach. Learn. Res. 12(85), 2825–2830 (2011). https://jmlr.csail.mit.edu/papers/v12/pedregosa11a.html 29. Head, T., Kumar, M., Nahrstaedt, H., Louppe, G., Shcherbatyi, I.: scikit-optimize/scikitoptimize (2020) 30. James, G., Witten, D., Hastie, T., Tibshirani, R.: An Introduction to Statistical Learning: with Applications in R. Springer, New York, 1st edn. (2013). Corr. 7th printing 2017 edition edition, 2013. https://doi.org/10.1007/978-1-4614-7138-7

Experimental Study on the Permeability of Hydrophobic Powders Treated Loess Yuqi He, Wei Wu, Hongjian Liao, and Xiaohua Liu

Abstract Natural loess soils generally possess loose structures and strong water sensitivity, which leads to various geological disasters upon wetting. This study aims to improve the anti-permeability and stability of loess under wetting conditions by using a novel hydrophobic power. We first treated loess specimens with hydrophobic powers at various concentrations, and a series of triaxial permeability tests were performed to study their permeability behaviour. Furthermore, soil-water contact angle tests and scanning electron microscope tests were carried out to investigate the mechanism of the hydrophobic powder reducing the permeability of loess from both micro and macro perspectives.

1 Introduction Loess soils in situ usually possess honeycomb-type meta-stable structures, which enable them to show higher strength and hydraulic conductivity [1]. However, those structures can be rapidly destructured and cause significant reductions in shear strength or collapse upon wetting. Collapse behaviours of loess will contribute to various geological problems to constructions constructed on loess soils. Furthermore, such collapse behaviours are closely related to their permeability; therefore, Y. He · H. Liao (B) Department of Civil Engineering, Xi’an Jiaotong University, Xi’an 710049, Shaanxi, P.R. China e-mail: [email protected] Y. He e-mail: [email protected] W. Wu Institute of Geotechnical Engineering, University of Natural Resources and Life Sciences, Feistmantelstrasse 4, 1180 Vienna, Austria e-mail: [email protected] X. Liu Guangzhou Metro Design and Research Institute Co., Ltd, Guangzhou, China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 W. Wu and Y. Wang (eds.), Recent Geotechnical Research at BOKU, Springer Series in Geomechanics and Geoengineering, https://doi.org/10.1007/978-3-031-52159-1_4

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reducing the permeability of loess soils and enhancing their hydraulic sensitivity is of great significance for preventing and controlling loess geological disasters. The factors affecting soil permeability can be classified into two categories: mechanical variables and physico-chemical variables [2]. The former includes size, shape, and preferred arrangement of soil particles. These variables determine the geometrical properties of the pore system of loess. The physico-chemical variables can be taken as the surface charge density, valency of the adsorbed cations, and viscosity of soil particles. Such variables may induce changes in the surface chemistry of soils. Over the past few decades, mechanical reinforcement [3] and chemical additives treatment [4] methods were primarily used to reduce the permeability of loess. The generally used mechanical reinforcement method, e.g., grouting, compaction, and plant reinforcement, can reduce the compressibility of soils and improve their stability. The mechanical additives treatment method refers to mixing some physical and chemical materials with soils [5]; as a result, those additives and soils undergo a series of biological or chemical reactions, resulting in a reduction in pore size and meta-stable structure or change in the surface properties of soils. The traditional additives include lime, cement, curing agent, chemical solution, etc. In recent years, nanomaterials, environment-friendly materials, and microbial grouting reinforcement technology have been the preferred choices to improve the stability of loess. Ng and Coo [6] mixed gamma-aluminum powder (.γ − Al2 O3 ) and nanocopper oxide (CuO) with clays to investigate its hydraulic conductivity, the results show that the hydraulic conductivity of clays decreased 30% and 40%, respectively. Furthermore, they conducted that the reduction in hydraulic conductivity is caused by the pores in clays being clogged under the effect of nanomaterials. Hydrophobic materials are environment-friendly and have been used to reduce the hydraulic conductivity of sands by earlier studies. Haquie and Hart [7] mixed Siloxane, a waterrepellent material, with Kaolinite soil to improve its ani-permeability. They found that 0.5–20% concentrations (by weight) of Siloxane can produce greater adherence between Kaolinite particles, resulting in a reduction in volume loss and moisture absorption upon wetting and drying cycles. Based on this, Saulick et al. [8] first modified the surface chemistry of sands by silanising sand grains and silica powders with an organosilane; then, the silica powders were adhered to the sands and improving the hydrophobicity of sands. As a result, the soil-water contact angle of sands was increased by 10–20%. In the studies mentioned above, hydrophobic materials were mainly used in granular soils, whereas their applications in loess soils are rarely evaluated. Furthermore, the macro- and micro-mechanism of hydrophobic materials reducing the permeability of loess have not been clearly investigated. For this reason, this study uses a novel hydrophobic material, silicone hydrophobic 60+ powder (SHP), to improve the anipermeability of loess soils. The optimal concentration of hydrophobic powder is first determined, and then the anti-permeability improvement mechanism is revealed from the perspectives of surface chemistry and the internal structure of loess.

Experimental Study on the Permeability of Hydrophobic …

37

2 Materials and Methods In this section, we first investigate the physical properties of the loess specimen and SHP, and then the lab testing procedures, including triaxial permeability tests, soil-water contact angle tests, and scanning electron microscope (SEM) tests, are introduced.

2.1 Materials SHP is a fine, free-flowing powder for fast and convenient easy mixing with excellent dispersion and workability. The finely dispersed silicon-based additives provide good storage stability and release the active ingredient upon wetting. The main compositions of SHP are silicone and resin, and its bulk density is 0.6 .g/cm3 . The tested loess specimens were retrieved from the southern Chinese Loess Plateau at a depth below ground level of 3 m. The loess is normally consolidated and has an initial void ratio of .e = 0.92. Figure 1 shows the Grain-size distribution curves of SHP and the tested loess, and the physical properties of the tested loess are shown in Table 1.

100

Percentage finer / %

80

SHP Loess soil

60 40 20 0 0.1

1

10

Particle size / µm

100

1000

Fig. 1 Grain-size distribution curves Table 1 Physical properties of the tested loess Variables Bulk density Water content Plastic limit Values

1.58 .g/cm3

12.35%

13.01%

Liquid limit

Specific gravity

29.90%

2.69

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2.2 Methods First, triaxial permeability tests were carried out to investigate the optimal SHP concentration and the influence of permeability time and SHP concentrations on the permeability coefficient. The loess specimens were crushed, dried, and passed through a 1 mm standard sieve, then mixed them evenly with SHP at concentrations of 0, 0.5, 1, and 2% (by weight). The water content was controlled at 15%, and then the well-mixed wet soil specimens were kept inside an impervious container for at least 48 h in a humidity- and temperature-controlled room. Thereafter, the specimens were compacted to the required dimensions of 39 mm diameter and 76 mm length by means of wet compaction. All the specimens were saturated before the permeability test. Followed by a consolidation process, and the consolidation pressures were 50, 100, and 200 kPa. The total permeability time is adopted as 130 h, and the experimental data were collected every 10 h. Furthermore, soil-water contact angle and surface free energy is closely related to the permeability of soils [9]. Therefore, we further conducted soil-water contact angle tests on the SHP treated loess to study the effect of SHP concentration on the soil-water contact angle and the permeability improvement mechanism from a micro perspective. The loess specimens were mixed with SHP at concentrations of 0, 1, 3, 5, 10, 15, and 20% and kept inside an impervious container for 48 h. After that, the specimens were compacted into a small cylinder shape with a 10 mm diameter and 3 mm length in a compression mould. All the specimens have the same water content and bulk density, and a total of 3 groups of parallel tests were performed at a constant temperature of .20◦ . Finally, SEM tests were conducted on SHP treated loess under various consolidation conditions to analyze their microstructure and void distribution characters. The SEM samples were trimmed from the loess specimens before and after triaxial permeability tests, then the surface samples were gold-plated, and a total of 24 group tests were carried out.

3 Experimental Results and Discussion 3.1 The Permeability Coefficient of SHP Treated Loess In this section, the relationship between the permeability time and the permeability coefficient of SHP treated loess, as well as the optimal SHP concentration, were investigated through triaxial permeability tests.

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3.1.1

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The Relationship Between the Permeability Time and the Permeability Coefficient of SHP Treated Loess

Figure 2 shows the relationship between the permeability time and the permeability coefficient of pure loess (0% SHP concentration) at various confining pressure levels. It can be seen that the permeability coefficient gradually decreases to a stable value with increasing permeability time, which follows a power function relationship. The reasons for the reduction in permeability can be attributed to the cemented bonding inside the soil disintegrating under seepage pressure. As a result, the tiny particles are separated from the soil aggregates and slowly migrate to the bottom of the specimen, which blocks some seepage channels and further reduces the permeability coefficient until the seepage becomes stable. The relationship between the permeability time and the permeability coefficient of SHP treated loess is shown in Fig. 3. It can be seen that the permeability coefficient is significantly decreased, whereas the general evolution law of the permeability coefficient remains unchanged. Such evolution law can be described by a power function as follows: −b .k = k 0 t (1) where .k0 refers to the initial permeability coefficient, .b controls the decrease rate of the permeability coefficient, and .t is the permeability time. Figures 2 and 3 show that Eq. (1) can reasonably predict the permeability coefficient of SHP treated loess at various confining pressures. Once the initial permeability coefficient .k0 and parameter .b are determined, the ultimate permeability coefficient can be obtained. 16 14 12 k / (10-2cm h-1 )

Fig. 2 The relationship between permeability coefficient and time of pure loess

10 8 6

c

= 50 kPa, k = 0.179t -0.10, R2 = 0.95

c

= 100 kPa, k = 0.076t -0.22, R2 = 0.98

c

= 200 kPa, k = 0.232t -0.77, R2 = 0.97

4 2 0

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c

k / (10-2cm h-1 )

5

12 10

= 100 kPa, k = 0.009t

-0.11

, R = 0.97

= 200 kPa, k = 0.018t c

-0.13

, R2 = 0.93

c

4

= 50 kPa, k = 0.071t -0.19, R2 = 0.96 2

3 2

k / (10-3cm h-1 )

6

1

8

c

= 50 kPa, k = 0.015t -0.28, R2 = 0.96

c

= 100 kPa, k = 0.009t -0.27, R2 = 0.98

c

= 200 kPa, k = 0.020t -0.82, R2 = 0.88

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(a)

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6 c

10

= 50 kPa, k = 0.099t -0.98, R2 = 0.97 5

= 100 kPa, k = 0.059t -0.93, R2 = 0.96 c c

8

= 200 kPa, k = 0.008t -0.71, R2 = 0.92

6 4 2

k / (10-3cm h-1 )

k / (10-3cm h-1 )

t/h

4

c

= 50 kPa, k = 0.011t -0.38, R2 = 0.88

c

= 100 kPa, k = 0.003t -0.22, R2 = 0.98

c

= 200 kPa, k = 0.008t -0.71, R2 = 0.92

3 2 1

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60

t/h

(c)

80

100

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140

0

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(d)

Fig. 3 The relationship between permeability coefficient and time of a 0.5% SHP, b 1% SHP, c 2% SHP, and d 3% SHP treated loess

3.1.2

The Effect of SHP Concentration on the Permeability Coefficient of Loess

To quantitatively describe the effect of SHP concentration on the permeability coefficient of SHP treated loess, the average permeability coefficient is used, which can be calculated as follows: i=1 1∑ .k a = ki (2) n n where .ka is the average permeability coefficient, n refers to the number of test data, and .ki means the permeability coefficient at .ith hours. Figure 4 shows the relationship between the average permeability coefficient and the SHP concentration. Take the specimen at a confining pressure of 50 kPa as an example; the average permeability coefficient of the specimen without SHP mixed is 0.12 cm/h, whereas the average permeability coefficient is reduced by 73%, 95%, 98%, and 98% after added SHP at a concentration of 0.5%, 1%, 2%, and 3%, respec-

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13

Fig. 4 The effect of SHP concentrations on average permeability coefficient

c

12

c

ka / (10-2cm h-1 )

= 50 kPa

= 100 kPa c = 200 kPa

11 4 3 2 1 0 -1 -0.5

0.0

0.5

1.0 1.5 2.0 2.5 SHP concentration / %

Table 2 Average permeability coefficient of SHP treated loess 0%-.kaa 0.5%-.ka 1%-.ka Confining (.g/cm3 ) (.g/cm3 ) pressure (kPa) (.g/cm3 ) 50 100 200 a

. .m%

0.120 0.032 0.012

0.032 0.023 0.011

0.00500 0.00320 0.00096

3.0

2%-.ka (.g/cm3 )

3%-.ka (.g/cm3 )

0.00250 0.00180 0.00057

0.0024 0.0013 0.00029

3.5

− ka refers to the average permeability coefficient of loess treated with .m% SHP

tively. Detailed test results are shown in Table 2. We can conduct that SHP can significantly reduce the permeability of the loess. In the case of SHP concentration less than 1%, the average permeability coefficient decreases sharply with increasing SHP concentrations. However, such decreasing trend disappears, and the average permeability coefficient tends to be stable when SHP concentration is greater than 1%, which indicates that the optimal SHP concentration is 1–2%.

3.2 Soil-Water Contact Properties of SHP Treated Loess Figure 5a and b show the results of soil-water contact angle tests on the pure loess (0% SHP), and 10% SHP treated loess, respectively. As can be seen from the figures, the soil-water contact angle of the pure loess specimen (0% SHP) is .θw = 25.5◦ . In this case, the water drops on the specimen will be instantly absorbed by the loess. However, the soil-water contact angle is increased to .θw = 122.1◦ after we added 10% SHP into the loess specimen. When the water drops on the specimen, it will stay on the loess surface for a short time and slowly infiltrate into it. According to the test results, we can conclude that SHP significantly affects the soil-water contact angle. In addition, we found that in the case of .θw ≤ 90◦ , the water infiltrates the loess body relatively fast, whereas the infiltration rate is reduced for .θw > 90◦ .

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Fig. 5 Soil-water contact angle of a 0% SHP and b 10% SHP treated loess 180

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Experiments Curve fitting

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Experiments Curve fitting

60 m-1

140 120

40

w

100 tot sg

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20

60 40 20

0

5

10 15 SHP concentration / %

20

0

25

0

5

10 15 SHP concentration / %

20

25

(b)

(a)

Fig. 6 The relationship between SHP concentration and a soil-water contact angle and b surface free energy

Figure 6a shows the relationship between the soil-water contact angle and the SHP concentration. Error bars indicate .± standard deviation. The result shows that the soil-water contact angle of the loess specimen increases with increasing the SHP concentration. Such increasing trend slows down when SHP concentration is greater than 10%; thereafter, the soil-water contact angle becomes stable. The observations are consistent with the earlier study [10]. In this study, a critical soil-water contact of SHP treated loess is obtained as .θw = 133◦ . Figure 6b shows the relationship between soil surface free energy and SHP concentrations of SHP treated loess. The total surface free energy of loess .γsgtot consists p of dispersive .γsgd and polar parts .γsg according to: γ tot = γsgd + γsgp

. sg

(3)

p

The dispersive part.γsgd and polar part.γsg can be further calculated based on OWRK method [11] using water and diiodomethane: / / p p γsgd γlgd + γsg γlg = 0.5γlgtot (1 + cosθi )

.

(4)

Experimental Study on the Permeability of Hydrophobic …

43

p

where .γlgd and .γlg refers to dispersive and polar parts of liquids, .γlgtot is the total surface free energy of liquids, and .θi means the contact angle between soils and liquids. It can be seen that the surface free energy decreases sharply with increasing SHP concentration when SHP concentration is less than 10%. After that, it reaches a stable value of 21 mN/m. Moreover, the maximum surface free energy of 71.8 mN/m is observed in a pure loess specimen. According to the test results, we can conduct that the soil-water contact angle of loess is increased after mixing with SHP. As a result, the soil surface free energy is decreased and, in turn, results in a significant reduction in permeability. Therefore, the loess soil becomes more stable upon wetting.

3.3 Microstructure of SHP Treated Loess Figure 7 shows the microstructure of 2% SHP treated loess before and after triaxial permeability tests. It can be observed that SHP powers adhere to the surface of some loess particles and wrap some loess aggregates before the permeability test. In addition, finer SHP particles will fill in the inter-granular pores of loess as a bonding material, which changes the original inter-contacts into cemented contacts and cohesion of some loess particles. Moreover, some SHP powders will fall in the spaced pores of loess, decreasing the loess porosity and increasing the tortuosity of the seepage channel. Thus, the permeability coefficient is reduced. However, the loess particles are more closely arranged after the loess specimen undergoes a triaxial permeability test at a confining pressure of 50 kPa. The edges and corners of loess particles become more blurred, and a layer of bonding can be observed on the loess surface, indicating that a chemical reaction occurred between the SHP and loess upon wetting. This chemical reaction results in a hydrophobic film coated on the loess surface and causes a reduction of hydraulic conductivity.

Fig. 7 Microstructure of SHP treated loess

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4 Conclusions In this paper, an organic silicon hydrophobic powder is used to improve the antipermeability of loess. A serious of lab tests, including triaxial permeability tests, soil-water contact angle tests, and SEM tests, are carried out. Furthermore, the permeability improvement mechanism from the micro and macro perspectives is investigated. Based on the test results, the main conclusions are summarized as follows: (1) Mixing SHP in loess soil can significantly reduce its permeability coefficient. The permeability coefficient of loess decreases rapidly with the increase of SHP concentration at the onset of a permeability test and then tends to be a stable value. The optimal SHP concentration is 1–2%, and the permeability of loess decreases by 90% in this case. (2) The permeability coefficient of SHP treated loess decreases with increasing the confining pressure. In addition, with the increase of the permeability time, the permeability coefficient of SHP treated loess gradually decreases to a stable value, which follows a power function relationship. (3) The macro- and micro-mechanisms of SHP reducing the permeability of loess mainly include two aspects: the changing in surface chemistry and the variation of internal structure. The former can be explained that SHP increases the roughness of the loess surface; meanwhile, under the water repellency effect of the silicon, the soil-water contact angle is increased, causing a lower surface free energy. The variation of internal microstructure is attributed to the fact that SHP can be adhered to the surface of loess particles and falls in the inter-granular pores and spaced pores in a loess soil, which reduces the porosity and blocks the seepage channel of loess. As a result, the permeability of loess is significantly reduced. Acknowledgements The authors wish to acknowledge the financial support from the National Natural Science Foundation of China (Grant No. 51879212, 41630639), Key Research and Development Program of Shaanxi (Grant No. 2019KWZ-09), Qin Chuangyuan “scientist+engineer” Team Construction Project of Shaanxi (2023KXJ-178), the EU Horizon 2020RISE project—HERCULES (Grant No. 778360), and the Austrian Science Fund (FWF) for Lise Meitner Project—MultiCBPR (Grant No. M 3340-N).

References 1. Li, P., Vanapalli, S., Li, T.: Review of collapse triggering mechanism of collapsible soils due to wetting. J. Rock Mech. Geotech. Eng. 8(2), 256–274 (2016) 2. Mesri, G., Olson, R.E.: Mechanisms controlling the permeability of clays. Clays Clay Miner. 19(3), 151–158 (1971) 3. Valizade, N., Tabarsa, A.: Laboratory investigation of plant root reinforcement on the mechanical behaviour and collapse potential of loess soil. Eur. J. Environ. Civ. Eng. 26(4), 1475–1491 (2022) 4. Xu, P., Zhang, Q., Qian, H., Qu, W.: Effect of sodium chloride concentration on saturated permeability of remolded loess. Minerals 10(2), 199 (2020)

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5. Tabarsa, A., Latifi, N., Meehan, C.L., Manahiloh, K.N.: Laboratory investigation and field evaluation of loess improvement using nanoclay-A sustainable material for construction. Constr. Build. Mater. 158, 454–463 (2018) 6. Ng, C.W.W., Coo, J.L.: Hydraulic conductivity of clay mixed with nanomaterials. Can. Geotech. J. 52(6), 808–811 (2015) 7. Haquie, A., Hart, M.L.: Inducing hydrophobicity to improve long term engineering performance of Kaolinite Clay. In: The International Congress on Environmental Geotechnics, Singapore, pp. 481–488 (2018) 8. Saulick, Y., Lourenço, S.D., Baudet, B.A.: Optimising the hydrophobicity of sands by silanisation and powder coating. Géotechnique 71(3), 250–259 (2021) 9. Doerr, S.H., Shakesby, R.A., Walsh, R.: Soil water repellency: its causes, characteristics and hydro-geomorphological significance. Earth Sci. Rev. 51(1–4), 33–65 (2000) 10. Leelamanie, D.A.L., Karube, J., Yoshida, A.: Characterizing water repellency indices: contact angle and water drop penetration time of hydrophobized sand. Soil Scie. Plant Nutr. 54(2), 179–187 (2008) 11. Gao, Y., Guo, R., Fan, R., Liu, Z., Kong, W., Zhang, P., Du, F.P.: Wettability of pear leaves from three regions characterized at different stages after flowering using the OWRK method. Pest Manag. Sci. 74(8), 1804–1809 (2018)

Modelling and Assessment of Debris Flow Impact on Infrastructure in the Carpathians Olena Ivanik

Abstract Debris flows are water-gravitational processes which are characterized by specific geological, geomorphological and climatic conditions. Formation of the debris flow is characterized by non-stationary and avalanche movements, as well as a multi-phase formation process. The general technique and modelling of debris flow impact on infrastructure in the Carpathians have been proposed. They are based on the algorithms of debris flow loadings calculations. These algorithms are applying the fundamental hydrodynamic laws and empirical data. A calculating module for the simulating debris flow impact on infrastructure in the Carpathians has been developed. In order to show the capabilities of the developed model, the results of its application to the model site test are analyzed, and finally, the application to a plausible debris-flow scenario, taken from a case study, is discussed. The module is the instrument of debris flow hazard assessment and aims towards the identification of potential debris flow risks. The developed process-based model predicts the motion of a mass movement and could be applied for a better understanding of the vulnerability of mountainous areas and design appropriate protective measures to withstand the impact of potential debris flows in the Carpathians.

1 Introduction Debris flows are water-gravitational processes which are characterized by specific geological, geomorphological and climatic conditions. The formation of a debris flow is characterized by non-stationary and avalanche movements. Debris flows are multi-phase flows consisting of different materials and it is extremely challenging to predict the formation, dynamic and impact on infrastructure, which will facilitate mitigation as well as evaluation of risks. Debris and mudflow hazards belong to the O. Ivanik (B) Taras Shevchenko National University of Kyiv, 90 Vasylkivska str., Kyiv 03022, Ukraine e-mail: [email protected]; [email protected] CRPG, CNRS, Université de Lorraine, 15 rue Notre Dame des Pauvres, 54500 Vandoeuvre-lès-Nancy, France © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 W. Wu and Y. Wang (eds.), Recent Geotechnical Research at BOKU, Springer Series in Geomechanics and Geoengineering, https://doi.org/10.1007/978-3-031-52159-1_5

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temporal category and have definite frequency [1]. Debris and mudflows are some of the main geological hazards in the Carpathian Mountains. Field and analytic research proved that debris flows occur every 20–25 years and are exceptionally rare. These flows are unpredictable and have high velocities ranging from 2 to 10 m/s. There are 400 debris flows basins in this region. The area of the basins ranges from 0.1 to 50 km2 . Debris flows are a widespread means of sediment transport in the Carpathians. In recent years, the economic losses from landslides and debris flows have exceeded dozens of millions of hryvnias. Many facilities periodically damaged by landslides and debris flows include villages, cities, railway lines, farmlands, highways and pipelines. In addition to the destruction from mass movements, debris flow deposits reduce the flood conveyance in the stream channels. Increasing human occupation of attractive mountain areas over the past decades has increased the need to better understand the debris flow processes and to predict their impact. The lack of quantitative information concerning debris flow processes complicates development of techniques for forecasting debris flow hazards. Such lack of information is directly related to the unpredictable and short-term flow formation, the poor accuracy and imperfect method of debris flow parameter measurements, etc. That is why it is necessary to develop adequate techniques of debris flow forecasting coupled with an assessment of the specific geological structure of the study area, analysis of the tectonic movements, and specific features of the debris flow formations.

2 Materials and Methods 2.1 Main Approaches to Debris Flow Modelling Debris flow prediction and evaluation of factors for debris flow formation are quite difficult tasks, coping with which is impossible without a quantitative theory of debris flow formation and impact based on complex mathematical models describing different stages of the debris flow process. The development of such a theory and its practical application are related to fundamental challenges, stipulated by complexity and multifactor debris flow phenomena, and a great number of changeable parameters. The results of theoretical and experimental dynamic studies of water flows, which are similar to fluid streams, are widely used for debris flow modelling. Many researchers [8, 9] offered a range of methods to determine discharges of flows, volumes and types of river floods, equation of hydrographs, analytical research of parameters of run-offs, such as run-off coefficient, duration of flood, etc. Also, the principle methods and research procedures of common factors of river run-off, which may also be used in the estimation of debris flow hazard have been determined. In order to create mathematical models of fluid streams, methods of suspended stream hydraulics are usually used, which provide a better option for studying debris flow movement and dynamics. One of the first methods of calculating the velocity of

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fluids was suggested by Wang [11]. Later a method of calculating flow velocity as a fluid using the Chezy formula has been applied by many researchers. Debris flow velocity, slope and roughness of the channel, specific gravity of sediment and runoff mass, hydraulic radius of the stream, etc. are the main parameters for this method. To determine the dynamic parameters of debris flows, Fleischman [2] offered to use the equation of Newton for viscous fluid movement on a flat surface with a constant slope. The main principles and correlations characterizing the movement of water flow with solid ingredients have been used for the description of fluid stream movements and to significantly simplify calculations, although their applicability to assess the debris flow hazards issue is still disputable. Numerous models devoted to assessing the main features of debris flow phenomena have been proposed by many scientists. Thus, O’Brien [5] has designed a mudflow model for watershed channels based on the Bingham model. Takahashi and Tsujimoto [10] proposed a 2D finite volume model for debris flows based on a dilatant fluid model coupled with Coulomb flow resistance. O’Brien et al. [6] conceived a 2D finite difference model. O’Brien et al. conceived a 2D finite difference model (FLO-2D) for routing nonNewtonian flood flows on alluvial fans, based on the Saint–Venant equations. Based on the integral method, Papa and Pianese [7] have developed the debris flow model which helps to calculate the impact of cross-sectional velocity, density and pressure on debris flow modelling. At the same time, in order to study specific characteristics of formation and dynamics of viscous-plastic flows or solid particle–viscous fluid mixtures, we must use the results of the theory of visco-plastic media application and constitutive modelling. A suitable constitutive model which can capture the solid-like and fluidlike behavior of a solid–fluid mixture describes the development of pore water pressure (or effective stresses) in the initiation stage and determine the residual effective stresses. Wu et al. [3, 12, 13] have developed a constitutive model of debris materials based on a framework where a static portion for the frictional behavior and a dynamic portion for the viscous behavior are combined. The frictional behavior is described by a hypoplastic model with critical state for granular materials. The viscous behavior is described by the tensor form of a modified Bagnold’s theory for solid–fluid suspension. Summarizing the existing methods for debris flows modelling and calculations, it should be noted that despite the great number and diversity of approaches, none of the proposed methods could be universal and applied for the evaluation of debris flows formations and dynamics in all areas. Several features of debris flow phenomena still constitute an open issue, such as the rheological behavior, real velocity, mechanisms of sediment transport etc. The majority of the characterized models describe in detail the various stages of debris flow formation and its effects. However, peculiarities of geological and geomorphological structure of debris flow hazard areas, hydrometeorological characteristics as well as neotectonic and hydrogeological regimes dictate the need for the development of proper debris flow hazard prediction tools and determination of hydrodynamic debris flow parameters, which are peculiar exactly to certain areas.

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In this framework, based on the above-mentioned, these investigations propose the proper complex for the Carpathians.

2.2 The Algorithm for Calculating the Debris Flow Impact on Infrastructure in the Carpathians The below algorithm is based on empirical data from the Carpathian Mountains and fundamental laws of hydrodynamics, in particular, the Bernoulli equation. Using this equation, we can obtain the illustration of the impingement hydrodynamic pressure. Assuming that motion of an ideal fluid is adiabatic and stationary, while mass forces are conservative, the following formula is executed on the flow lines. v2 + ∏ + P = const, that is called the Bernoulli integral. 2 For homogenous uncompressed liquid ρ = const and that’s why p v2 + ∏ + = const 2 ρ

(1)

If mass forces are gravity forces (∏ = gz), then v2 p +z+ = const 2g ρg

(2)

2

p v where 2g —velocity height; z—elevation; ρg —piezometric height. If we have two cross sections of stream 1 and 2 and indicate the corresponding values of pressure, velocity and height for each of these cross sections via p1 , v1, z1 , p2 , v2, z2 , , we’ll have another form for the Bernoulli integral:

v2 2 v1 2 p p + z1 + 1 = + z2 + 2 2g ρg 2g ρg

(3)

Choosing the position of the second cross-section on the wall of a fixed obstacle (for example, road, bridge, etc. (Fig. 1)) we will have the obvious equations: v2 = 0, z1 = z2 . Hence p p v1 2 + 1 = 2 2g ρg ρg

(4)

( ) p2 − p1 v2 p = = γ γ 2g

(5)

or

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Fig. 1 Cross section of debris flow channel along the road: 1—bridge; 2—debris flow channel; 3—debris flow formation

Considering the scenario of flow collision with an obstacle, the last formula may be rewritten in the form: v2 p = Kα , γ 2g

(6)

where the K —coefficient, considering the characteristics of bodies, is guessed to be equal to 2; α—the evaluation velocity coefficient by the quantity of motion is guessed to be equal to 1–1.33. It should be noted that in the Bernoulli distribution, we assume that the flow acts like an ideal uncompressed liquid, although in reality, it is a mixture with a great difference in densities of solid and liquid parts. Now we’ll have a formula for hydrostatic pressure. According to the hydrostatic Pascal’s paradox, we have the following formula for the main vector of fluid-pressureinduced force on the wall of the obstacle with surface area S: R = γ z c S,

(7)

where z c – is the vertical coordinate of the gravity centre C of area S. Hence, for the pressure we’ll have: p= or

R = γ zc S

(8)

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O. Ivanik

p H = zc = , γ 2

(9)

where H – flow depth. Thus, making expressions for shock hydrodynamic and hydrostatic pressures, we obtain an expression for the full pressure of the stream: p=γ

H v2 γ + γ Kα = 2 2g g

(

) gH v2 . + Kα 2 2

(10)

Assuming that g ≈ 10 in this formula, we’ll have: p=γ

( ) v2 v2 H + γ Kα = 0.1γ 5H + K α 2 2g 2

(11)

By K = 2 and α = 1 we’ll obtain the formula: ( ) P total = 0, 1 γc 5H0 + vc 2

(12)

where P total (T/m2 )—total pressure; γc (T/m3 )—average density of the debris flow; H0 (m)—depth of debris flow; vc (m/s)—velocity of debris flow. For quantitative prediction of the force effect of the debris flow impact on various infrastructures, it is necessary to determine a range of debris flow parameters, which is quite a complicated task. Geological and geomorphological data, parameters of river valleys and flow channels, characteristics of potential debris flow formation and others have been defined by previous investigations and included in regional databases and maps. It also includes data of long-term hydrometeorological observations, in particular for defining the rainfall regime, frequency of extreme situations, density of debris flows, etc. Significant difficulties arise in determining the main dynamic parameters of the debris flow such as velocity, as well as the depth associated with the channels and valleys, rainfall, runoff, flow rates etc. Calculated formulas for the maximum flow discharge are very diverse and based on different principles. Most of them take into account the area of the watershed, intensity of rain and run-off, as well as less significant factors. In the formula of Sokolovskyi [9] flow discharge Qc (m3 /sec) and form of hydrograph have been considered: Qc =

0.28α Ht Fν fh , t

(13)

where Ht (mm)—rainfall, determined as a result of average rainfall intensity a and its duration; α—run-off coefficient; (h)—time of flood raise; Fv (km2 )—watershed area; —coefficient of hydrograph form. The run-off coefficient depends on rainfall, soil moisture and infiltration regime. The intensity of infiltration is one of the main factors of the substrate surface, which is

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determined by the nature of soil and vegetation. During the early stages, infiltration intensity could be quite high, but after filling pores with water, it decreases and asymptotically approaches the established value of filtration. According to our investigation in the Carpathian region the run-off coefficient in the dry period varies from 0 to 0.2, in the wet period it increases to 0.4–0.5, and in particularly rainy periods it reaches 0.7. The effectiveness of the formula of D. Sokolovskyi in the calculations module is decreased by an inaccurate determination of the value and the f h coefficient. We can use the last empirical formula for this: fh =

12 , 4+ν

(14)

where v (m/sec)—the maximum velocity at the cross section of a flow, it requires data of flow velocity. In general, according to the effect of run-off, the rainfalls are divided into several types: (1) rainfalls—short and intense rains lasting no more than 2–3 h and with average intensity a ≥ 10–20 mm/h; (2) rain showers lasting from a few hours to several days with average intensity a ≥ 2–10 mm/h; (3) widespread rain, as a rule, with an intensity of 3–5 days and more with a small average intensity a < 2 mm/h. The maximum debris flow effect is executed by rainfalls of the first type, although the probability of debris flow occurrence due to other types is also high. It should be noted that the overall duration of the rains is not sufficient for their characteristics regarding the drainage effect, but it is desirable to take into account the duration of their effective part, within which the rain intensity exceeds infiltration intensity. It is clear that with the increase in rainfall duration, its intensity decreases (Fig. 2). For the Eastern European area, this formula is followed (calculated on the data of long-term meteorological observations over 28 meteorological stations in this region): a=

5 1 + 0, 06t

(15)

It is necessary to note that the choice of the method for calculating the debris flows velocity is of key importance. It is known that velocities depend on channel shape, its inequalities and depth of flow. Usually, the velocity is determined by the well-known Chezy formula. However, due to the variability of the coefficients of channel inequalities, and also the fact that the Chezy formula reflects a uniform motion, sometimes we have a low accuracy of calculations. Morphometric elements of the channel, in particular depth and slope, included in the Chezy formula, vary not only from the velocity. The exact velocity is often a more stable value than its constituents. Nevertheless, the formula of maximum velocity has been proposed: Vmax = 17, 0 J 0,40 h 0,50 cp

(16)

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Fig. 2 Diagram of the dependence of rainfall intensity on its duration (for Eastern Europe): 1— average intensity of rainfall; 2—rainfall intensity according to the calculation formula of M. Protodyakonov [8]

where Vmax – maximum velocity in the cross-section; J – channel slope; h cp – average depth on the effective cross-section at maximum accumulation. Protodiakonov [9], considering the dependence of run-off velocity V m/sec on channel slope I ‰, discharge Q m3 /sec, as well as the surface irregularities and channel cross-section, has proposed to determine the flow velocity according to the Chezy formula with a coefficient by Manning: 1

3

V = k1 Q 4 I 8 ; 1 3 1 k1 = 0, 075( ) 4 k 2 n

(17)

Here I —average channel slope, ‰; n1 —coefficient of bottom irregularities; k = √R —coefficient depending on the form of the channel cross-section; R—hydraulic ω radius, m; ω—area of the effective cross-section, m2 . These formulas are used in the calculation module as a controlling comparison of calculation results obtained by other methods, which are based mainly on generalized empirical data. It is important to note that the maximum level of flow velocities simultaneously reflect rainfall intensity (flow discharges) and valley features (longitudinal and crossing profiles, terrace existence etc.), which in its turn reflect the time of valley formation and potential erosion processes. Trends of velocity may be defined for a region by long-term observations and used as integral characteristics for predictive assessment of flood and debris flow hazards.

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Analysis of hydrological data of the Carpathian region allowed classifying flow channels by length. Channels are proposed to be divided into three groups by the channel length: (1) L 1 < 1 km; (2) 1 ≤ L 2 < 5 km; (3) L 3 ≥ 5 km. On the base of available data, the following empirical dependences on rainfall duration have been determined the change of intensity and critical velocities: √ for each√group. We consider √ v1 = 5 T ; v2 = 3 4 T ; v3 = 2 6 T [4]. Thus, setting rainfall intensity—Ht , its duration in hours—T , area– Fv , soil ability to absorb water—α coefficient and channel length for this district—L, we can calculate the velocity and discharge of debris flows—correspondingly vc and Qc. On the other hand, these values are connected with the area of the debris flow channel with the following ratio: Q c = vc · S,

(18)

where S—the area of the cross-section of the debris flow channel. Then S=

Qc vc

(19)

Let’s think that the channel cross section has the form of a trapezium with a, b bases and H0 height. Then. S=

a+b · H0 2

(20)

Assuming that the channel width on the bottom (a value) and the angles of the channel slope (φ1 , φ2 ) are known, geometrically we have: b = a + H0 (ctgφ1 + ctgφ2 )

(21)

] [ H0 S= a+ (ctgφ1 + ctgφ2 ) · H0 2

(22)

and

We receive an equation for finding out H0 debris flow depth according to the set values of slope angles and flow width on the bottom as well as by the earlier calculated debris flow discharge and velocity: H02 (ctgφ1 + ctgφ2 ) + 2a H0 − 2 Hence

Qc =0 vc

(23)

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H0 =

−a +

/ a 2 + 2 Qvcc (ctgφ1 + ctgφ2 ) ctgφ1 + ctgφ2

(24)

Thus, by setting debris flows density and using the developed mathematical model, we can calculate a complete unstable (dynamic) debris flow pressure on the infrastructure. The hydrodynamic force affecting the obstacle is calculated according to the formula: F = P total · S. The presented algorithm, based on empirical data from the Carpathian region and the fundamental laws of hydrodynamics, demonstrates the necessity of choosing calculation formulas and quantities, the determination of which is possible under specific situations and geological and geomorphological conditions. Based on the proposed algorithm, the software module was created. This module allows us to model the impact of debris flows on the infrastructure in the Carpathian region.

3 Case Study and Model Application The proposed method was tested in the study area of the Ukrainian Carpathians. Analytical and field research was held in the Latoritsa river basin in the Carpathian Mountains. The research area is extended from 48º30’ to 48º 50’ N latitude and 22º 55’ to 23º 15’ E longitude with the altitude range from 94 to 1656 m a.s.l. (Fig. 3). It covers an area of approximately 804 km2 . Several existing and potentially dangerous areas have been found there. One of them is situated in the Abranka river basin where heavy rains caused flash flood and debris flows (Fig. 4). The debris flows caused damage to houses, infrastructures and agricultural fields (Fig. 5). From the geological point of view, the study area is situated in the Eastern Carpathians and the Transcarpathian depression. It covers the Duklyanskaya, Magurskaya, Porculetskaya and Penninskaya structural-facial zones. These zones are the regional sheets with the changeable configuration and different amplitudes of thrust. The asymmetry of the mountain ridge with inclined south-west slopes and steep north-east slopes is the main relief peculiarity of the area. There are two structural layers of geology. The lower layer consists of the carbonaceousterrigenous and terrigenous Mesosoic-Cenosoic (mostly flysch) formations. Coeval deposits have a different facial structure. Their main characteristics are the matter composition of the flysch deposits, their colouring, calcification, texture features, presence of organic matter and fossils, reference horizons and thicknesses. These characteristics of flysch deposits are the most important for the hazardous exogenic processes formation because different types of flysch are very diverse in their mechanical properties and reaction on destructive processes. The upper layer consists of the Neogene-Quaternary sedimentary deposits, volcanic, and volcanic-polymict deposits with flat bedding. Quaternary deposits are presented by alluvial, deluvial, lacustrine and glacial genetic types. The territory is prone to be affected by debris flows due

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Fig. 3 Map of the study area and digital elevation model of Latoritsa river basin, Ukrainian Carpathians

to the geological, geomorphological and climatic conditions. Therefore, the main conditions of the debris flow formation are as follows: (a) presence of rock destruction products which could be a solid matter of the mudflow and debris flow; (b) sufficient rainfall excess for removal of unconsolidated material; (c) rugged surface of relief that provides for simultaneous movement of large quantities of water/soil mass at high velocity levels. All these parameters are included in the method and calculations.

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Fig. 4 Digital Elevation Model and debris flow inventory of Abranka catchment (Ukrainian Carpathians)

Fig. 5 Consequences of debris flows activity in Abranka village, Transcarpathian region, Ukraine

4 Results As a result of using the calculation module “Debris flows”, created on the base of the developed algorithm, the differentiation of indicators of the debris flow hazard within Abranka River and its tributary was determined. They depend on the morphometric

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parameters of debris flow channels, lithology and mechanical properties of potential debris flow material, catchment area and hydrometeorological factors (Tables 1, 2 and 3). With the using of GIS modelling and field observations, it was determined that the watershed area of Abranka is 5.54 km2 , the length of the channel is 4.5 km, the width of the channel is about 3 m, the right slope is 21º, the left slope is 15º. The density of the flow, which was characterized by the content of detrital material represented by sand and pebble deposits, is about 1.5 T/m3 . According to the analysis of the impact of the duration of rainfall on such indicators of debris flow hazard as the total pressure and hydrodynamic force (Table 1) it has been confirmed that for the Abranka river and its tributaries, a general tendency is decreasing of this parameters with the increasing of rainfall time. This is explained by the decrease in the intensity of rainfall over time under the condition of water saturation of the soil, which corresponds to the maximum value of the runoff coefficient. For the Abranka river, the value of the total pressure of debris flow varies from 3.91 to 3.39 T/m2 , and the hydrodynamic force varies from 98.67 to 68.18 T. For tributaries we have lower values of these parameters (total pressure from 3.73 to 1.27 T/m2, hydrodynamic force from 6.59 to 1.22 T). The analysis of the soil moisture factor showed that the highest values of flow rates and their force impact are characteristic of full soil moisture conditions (Table 2, Fig. 6). These features, in turn, are determined by the values of the runoff coefficient, which, as is known, depends not only on precipitation and the nature of the soil, but more on the previous moisture of the soil and on the saturation conditions of the territory as a whole, as well as on the nature of the vegetation cover. The inclusion of natural parameters in the modelling made it possible to determine such an important parameter of the debris flow process as the runoff coefficient, the calculation of which in the mountainous areas of the Carpathians is a problematic issue due to the complexity of the landscape, orographic situation and changes in the physical and mechanical properties of sediments. The runoff coefficient calculated in this way is 0.35. The study of the impact of the debris flow density on the debris flow hazards shows that with the increase of solid material, the force parameters increase (Table 3, Fig. 7). The density of the debris flow (its volumetric weight) affects the transport capacity of the flow. In addition, if the debris flow is characterized by significant viscosity and density, it is transportable capacity is increased. The following conclusions can be made based on the calculations and analysis of the impact of various combinations of debris flow hazard factors. Morphometric and morphological characteristics of debris flow channels have an important influence on the flow formation, its velocity and depth. The maximum levels of the average flow velocities depend on the intensity of rainfall and characteristics of the valleys, which in turn reflect the time of formation of the valleys, erosion processes and tectonic regime. Critical velocities are in the short, high-order tributaries with significant slopes and relatively narrow valleys. They are quickly reached during short-term rainfalls, and, conversely, in long streams (rivers) with terraced valleys, the critical velocities are possible only during long-time rains. At the same

The width of the channel at the bottom, m

3

3

3

Duration of rainfall, min

40

50

60

92

92

92

Elevation, m

4500

4500

4500

The length of the channel projection, m

15

15

15

Left slope (degree)

21

21

21

Right slope (degree)

3,48

3,61

3,76

Debris flow velocity, m/s

Table 1 The impact of changes in the duration of rainfall on debris flow hazards in Abranka catchment

2,09

2,22

2,39

The depth of the debris flow, m

3,39

3,62

3,91

Total pressure of debris flow, T/m2

68,18

80,9

98,67

Hydrodynamic force, T

60 O. Ivanik

Type of Soils

2

Dry

Saturated

Oversaturated

Dry

Saturated

Oversaturated

River and its tributaries

1

River Abranka

River Abranka

River Abranka

River Abranka, 1st right tributary

River Abranka, 1st right tributary

River Abranka, 1st right tributary

0,45

0,45

0,45

5,54

5,54

5,54

3

Catchment area, km2

1

1

1

3

3

3

4

The width of the channel at the bottom, m

78

78

78

92

92

92

5

Elevation, m

478

478

478

4500

4500

4500

6

The length of the channel projection, m

65

65

65

15

15

15

7

Left slope (degree)

Table 2 The impact of changes in soil moisture on debris flow hazards in Abranka river basin

70

70

70

21

21

21

8

Right slope (degree)

4,36

3,66

2,68

3,76

3,16

2,31

9

Debris flow velocity, m/s

1,18

0,79

0,36

2,39

1,75

0,97

10

The depth of the debris flow, m

3,73

2,6

1,34

3,91

2,81

1,53

11

Total pressure of debris flow, T/ m2

6,59

2,73

0,55

(continued)

98,67

42,22

8,95

12

Hydrodynamic force, T

Modelling and Assessment of Debris Flow Impact on Infrastructure … 61

Type of Soils

Dry

Saturated

Oversaturated

Dry

Saturated

River and its tributaries

River Abranka, 2nd right tributary

River Abranka, 2nd right tributary

River Abranka, 2nd right tributary

River Abranka, 3d right tributary

River Abranka, 3d right tributary

Table 2 (continued)

0,64

0,64

0,36

0,36

0,36

Catchment area, km2

0,8

0,8

0,5

0,5

0,5

The width of the channel at the bottom, m

145

145

129

129

129

Elevation, m

749

749

664

664

664

The length of the channel projection, m

70

70

65

65

65

Left slope (degree)

76

76

70

70

70

Right slope (degree)

4,26

3,11

4,39

3,7

2,7

Debris flow velocity, m/s

1,12

0,52

1,33

0,94

0,47

The depth of the debris flow, m

3,56

1,85

3,89

2,75

1,44

Total pressure of debris flow, T/ m2

4,57

0,93

5,45

2,29

0,47

(continued)

Hydrodynamic force, T

62 O. Ivanik

Type of Soils

Oversaturated

Dry

Saturated

Oversaturated

Dry

River and its tributaries

River Abranka, 3d right tributary

River Abranka, 4th right tributary

River Abranka, 4th right tributary

River Abranka, 4th right tributary

River Abranka, 5th right tributary

Table 2 (continued)

0,36

0,68

0,68

0,68

0,64

Catchment area, km2

3

3,5

3,5

3,5

0,8

The width of the channel at the bottom, m

137

133

133

133

145

Elevation, m

689

1238

1238

1238

749

The length of the channel projection, m

27

22

22

22

70

Left slope (degree)

31

15

15

15

76

Right slope (degree)

2,72

4,14

3,48

2,54

5,06

Debris flow velocity, m/s

0,1

0,54

0,36

0,16

1,65

The depth of the debris flow, m

1,19

2,97

2,09

0,09

5,09

Total pressure of debris flow, T/ m2

0,38

8,37

3,48

0,71

(continued)

10,98

Hydrodynamic force, T

Modelling and Assessment of Debris Flow Impact on Infrastructure … 63

Type of Soils

Saturated

Oversaturated

Dry

Saturated

Oversaturated

River and its tributaries

River Abranka, 5th right tributary

River Abranka, 5th right tributary

River Abranka, 6th right tributary

River Abranka, 6th right tributary

River Abranka, 6th right tributary

Table 2 (continued)

0,5

0,5

0,5

0,36

0,36

Catchment area, km2

3,5

3,5

3,5

3

3

The width of the channel at the bottom, m

137

137

137

137

137

Elevation, m

802

802

802

689

689

The length of the channel projection, m

21

21

21

27

27

Left slope (degree)

20

20

20

31

31

Right slope (degree)

4,44

3,82

2,8

4,43

3,72

Debris flow velocity, m/s

0,39

0,25

0,11

0,4

0,24

The depth of the debris flow, m

3,4

2,38

1,25

3,22

2,26

Total pressure of debris flow, T/ m2

6,38

2,66

0,55

4,48

1,87

(continued)

Hydrodynamic force, T

64 O. Ivanik

Type of Soils

Dry

Saturated

Oversaturated

Dry

Saturated

River and its tributaries

River Abranka, 7th right tributary

River Abranka, 7th right tributary

River Abranka, 7th right tributary

River Abranka, 1st left tributary

River Abranka, 1st left tributary

Table 2 (continued)

3,3

3,3

0,2

0,2

0,2

Catchment area, km2

4

4

2,5

2,5

2,5

The width of the channel at the bottom, m

306

306

46

46

46

Elevation, m

2473

2473

522

522

522

The length of the channel projection, m

12

12

30

30

30

Left slope (degree)

10

10

34

34

34

Right slope (degree)

5,44

3,98

2,83

2,38

1,74

Debris flow velocity, m/s

0,63

0,35

0,39

0,25

0,11

The depth of the debris flow, m

4,95

2,63

1,49

1,03

0,53

Total pressure of debris flow, T/ m2

(continued)

25,67

5,34

1,8

0,74

0,15

Hydrodynamic force, T

Modelling and Assessment of Debris Flow Impact on Infrastructure … 65

Type of Soils

Oversaturated

Dry

Saturated

Oversaturated

Dry

River and its tributaries

River Abranka, 1st left tributary

River Abranka, 2nd left tributary

River Abranka, 2nd left tributary

River Abranka, 2nd left tributary

River Abranka, 3d left tributary

Table 2 (continued)

0,7

2,8

2,8

2,8

3,3

Catchment area, km2

2

3

3

3

4

The width of the channel at the bottom, m

156

276

276

276

306

Elevation, m

1359

2911

2911

2911

2473

The length of the channel projection, m

22

15

15

15

12

Left slope (degree)

15

13

13

13

10

Right slope (degree)

2,63

5,62

4,73

3,46

6,47

Debris flow velocity, m/s

0,24

1,13

0,81

0,42

0,97

The depth of the debris flow, m

1,21

5,59

3,96

2,11

7

Total pressure of debris flow, T/ m2

0,79

(continued)

47,56

20,04

4,17

61,04

Hydrodynamic force, T

66 O. Ivanik

Type of Soils

Saturated

Oversaturated

River and its tributaries

River Abranka, 3d left tributary

River Abranka, 3d left tributary

Table 2 (continued)

0,7

0,7

Catchment area, km2

2

2

The width of the channel at the bottom, m

156

156

Elevation, m

1359

1359

The length of the channel projection, m

22

22

Left slope (degree)

15

15

Right slope (degree)

4,27

3,59

Debris flow velocity, m/s

0,68

0,48

The depth of the debris flow, m

3,25

2,29

Total pressure of debris flow, T/ m2

9,09

3,82

Hydrodynamic force, T

Modelling and Assessment of Debris Flow Impact on Infrastructure … 67

Flow density, t/ m3

2

1,2

1,5

1,8

2

1,2

1,5

River and its tributaries

1

River Abranka

River Abranka

River Abranka

River Abranka

River Abranka, 1st right tributary

River Abranka, 1st right tributary

0,45

0,45

5,54

5,54

5,54

5,54

3

Catchment area, km2

1

1

3

3

3

3

4

The width of the channel at the bottom, m

78

78

92

92

92

92

5

Elevation, m

478

478

4500

4500

4500

4500

6

The length of the channel projection, m

65

65

15

15

15

15

7

Left slope (degree)

Table 3 The impact of debris flow density on debris flow hazards in Abranka river basin

70

70

21

21

21

21

8

Right slope (degree)

4,36

4,36

3,76

3,76

3,76

3,76

9

Debris flow velocity, m/s

1,18

1,18

2,39

2,39

2,39

2,39

10

The depth of the debris flow, m

3,73

2,99

5,21

4,69

3,91

3,13

11

Total pressure of debris flow, T/ m2

6,59

5,27

(continued)

131,56

118,4

98,67

78,93

12

Hydrodynamic force, T

68 O. Ivanik

Flow density, t/ m3

1,8

2

1,2

1,5

1,8

River and its tributaries

River Abranka, 1st right tributary

River Abranka, 1st right tributary

River Abranka, 2nd right tributary

River Abranka, 2nd right tributary

River Abranka, 2nd right tributary

Table 3 (continued)

0,36

0,36

0,36

0,45

0,45

Catchment area, km2

0,5

0,5

0,5

1

1

The width of the channel at the bottom, m

129

129

129

78

78

Elevation, m

664

664

664

478

478

The length of the channel projection, m

65

65

65

65

65

Left slope (degree)

70

70

70

70

70

Right slope (degree)

4,39

4,39

4,39

4,36

4,36

Debris flow velocity, m/s

1,33

1,33

1,33

1,18

1,18

The depth of the debris flow, m

4,67

3,89

3,11

4,98

4,48

Total pressure of debris flow, T/ m2

6,54

5,45

4,36

8,79

7,91

(continued)

Hydrodynamic force, T

Modelling and Assessment of Debris Flow Impact on Infrastructure … 69

Flow density, t/ m3

2

1,2

1,5

1,8

2

River and its tributaries

River Abranka, 2nd right tributary

River Abranka, 3d right tributary

River Abranka, 3d right tributary

River Abranka, 3d right tributary

River Abranka, 3d right tributary

Table 3 (continued)

0,64

0,64

0,64

0,64

0,36

Catchment area, km2

0,8

0,8

0,8

0,8

0,5

The width of the channel at the bottom, m

145

145

145

145

129

Elevation, m

749

749

749

749

664

The length of the channel projection, m

70

70

70

70

65

Left slope (degree)

76

76

76

76

70

Right slope (degree)

5,06

5,06

5,06

5,06

4,39

Debris flow velocity, m/s

1,65

1,65

1,65

1,65

1,33

The depth of the debris flow, m

6,78

6,1

5,09

4,07

5,19

Total pressure of debris flow, T/ m2

14,65

13,78

10,98

8,79

7,27

(continued)

Hydrodynamic force, T

70 O. Ivanik

Flow density, t/ m3

1,2

1,5

1,8

2

1,2

River and its tributaries

River Abranka, 3d right tributary

River Abranka, 3d right tributary

River Abranka, 3d right tributary

River Abranka, 3d right tributary

River Abranka, 5th right tributary

Table 3 (continued)

0,36

0,68

0,68

0,68

0,68

Catchment area, km2

3

3,5

3,5

3,5

3,5

The width of the channel at the bottom, m

137

133

133

133

133

Elevation, m

689

1238

1238

1238

1238

The length of the channel projection, m

27

22

22

22

22

Left slope (degree)

31

15

15

15

15

Right slope (degree)

4,43

4,14

4,14

4,14

4,14

Debris flow velocity, m/s

0,38

0,54

0,54

0,54

0,54

The depth of the debris flow, m

2,58

3,96

3,57

2,97

2,38

Total pressure of debris flow, T/ m2

3,58

11,13

10,02

8,37

6,68

(continued)

Hydrodynamic force, T

Modelling and Assessment of Debris Flow Impact on Infrastructure … 71

Flow density, t/ m3

1,5

1,8

2

1,2

1,5

River and its tributaries

River Abranka, 5th right tributary

River Abranka, 5th right tributary

River Abranka, 5th right tributary

River Abranka, 6th right tributary

River Abranka, 6th right tributary

Table 3 (continued)

0,5

0,5

0,36

0,36

0,36

Catchment area, km2

3,5

3,5

3

3

3

The width of the channel at the bottom, m

137

137

137

137

137

Elevation, m

802

802

689

689

689

The length of the channel projection, m

21

21

27

27

27

Left slope (degree)

20

20

31

31

31

Right slope (degree)

4,44

4,44

4,43

4,43

4,43

Debris flow velocity, m/s

0,39

0,39

0,38

0,38

0,38

The depth of the debris flow, m

3,4

2,72

4,3

3,87

3,22

Total pressure of debris flow, T/ m2

6,38

5,11

5,97

5,38

4,48

(continued)

Hydrodynamic force, T

72 O. Ivanik

Flow density, t/ m3

1,8

2

1,2

1,5

1,8

River and its tributaries

River Abranka, 6th right tributary

River Abranka, 6th right tributary

River Abranka, 7th right tributary

River Abranka, 7th right tributary

River Abranka, 7th right tributary

Table 3 (continued)

0,2

0,2

0,2

0,5

0,5

Catchment area, km2

2,5

2,5

2,5

3,5

3,5

The width of the channel at the bottom, m

46

46

46

137

137

Elevation, m

522

522

522

802

802

The length of the channel projection, m

30

30

30

21

21

Left slope (degree)

34

34

34

20

20

Right slope (degree)

2,83

2,83

2,83

4,44

4,44

Debris flow velocity, m/s

0,39

0,39

0,39

0,39

0,39

The depth of the debris flow, m

1,79

1,49

1,19

4,53

4,08

Total pressure of debris flow, T/ m2

2,16

1,8

1,44

8,51

7,66

(continued)

Hydrodynamic force, T

Modelling and Assessment of Debris Flow Impact on Infrastructure … 73

Flow density, t/ m3

2

1,2

1,5

1,8

2

River and its tributaries

River Abranka, 7th right tributary

River Abranka, 1st left tributary

River Abranka, 1st left tributary

River Abranka, 1st left tributary

River Abranka, 2nd left tributary

Table 3 (continued)

3,3

3,3

3,3

3,3

0,2

Catchment area, km2

4

4

4

4

2,5

The width of the channel at the bottom, m

306

306

306

306

46

Elevation, m

2473

2473

2473

2473

522

The length of the channel projection, m

12

12

12

12

30

Left slope (degree)

10

10

10

10

34

Right slope (degree)

6,47

6,47

6,47

6,47

2,83

Debris flow velocity, m/s

0,97

0,97

0,97

0,97

0,39

The depth of the debris flow, m

9,34

8,4

7

5,6

1,99

Total pressure of debris flow, T/ m2

81,39

73,25

61,04

48,84

2,4

(continued)

Hydrodynamic force, T

74 O. Ivanik

Flow density, t/ m3

1,2

1,5

1,8

2

1,2

River and its tributaries

River Abranka, 2nd left tributary

River Abranka, 2nd left tributary

River Abranka, 2nd left tributary

River Abranka, 3d left tributary

River Abranka, 3d left tributary

Table 3 (continued)

0,7

2,8

2,8

2,8

2,8

Catchment area, km2

2

3

3

3

3

The width of the channel at the bottom, m

156

276

276

276

276

Elevation, m

1359

2911

2911

2911

2911

The length of the channel projection, m

22

15

15

15

15

Left slope (degree)

15

13

13

13

13

Right slope (degree)

4,27

5,62

5,62

5,62

5,62

Debris flow velocity, m/s

0,68

1,13

1,13

1,13

1,13

The depth of the debris flow, m

2,6

7,45

6,71

5,59

4,47

Total pressure of debris flow, T/ m2

7,27

63,41

57,07

47,56

38,04

(continued)

Hydrodynamic force, T

Modelling and Assessment of Debris Flow Impact on Infrastructure … 75

Flow density, t/ m3

1,5

1,8

2

River and its tributaries

River Abranka, 3d left tributary

River Abranka, 3d left tributary

River Abranka, 3d left tributary

Table 3 (continued)

0,7

0,7

0,7

Catchment area, km2

2

2

2

The width of the channel at the bottom, m

156

156

156

Elevation, m

1359

1359

1359

The length of the channel projection, m

22

22

22

Left slope (degree)

15

15

15

Right slope (degree)

4,27

4,27

4,27

Debris flow velocity, m/s

0,68

0,68

0,68

The depth of the debris flow, m

4,33

3,89

3,25

Total pressure of debris flow, T/ m2

12,12

10,91

9,09

Hydrodynamic force, T

76 O. Ivanik

Fig. 6 The impact of changes in soil moisture on debris flow hazards in Abranka river and its tributaries

Modelling and Assessment of Debris Flow Impact on Infrastructure … 77

Fig. 7 The impact of debris flow density, total pressure and hydrodynamic force on debris flow hazards in Abranka river and its tributaries

78 O. Ivanik

Modelling and Assessment of Debris Flow Impact on Infrastructure …

79

time, in valleys of different lengths, velocities can stabilize reaching a certain level at any duration of precipitation. Thus the main channel of the Abranka river has a significant potential for debris flow hazard only in case of a significant amount of precipitation and the presence of debris. The hazard is possible only with flow depths more than 2.5 m and in fact similar to critical flood events in the Carpathians.

5 Discussion and Conclusions Peculiarities of the geological and geomorphological structure of the landslide-prone area, its hydrometeorological characteristics, as well as neotectonic and hydrogeological regimes dictate the need to develop appropriate approaches to the landslide hazard prediction and determine the parameters of landslides that are specific to a certain region. In this regard, the development of mathematical models of debris flows phenomena requires the solution of a number of tasks related to the selection of determining ratios, the construction of an algorithm, and the creation of such a debris flows risk assessment complex that would meet the conditions and needs of a specific region. At the same time, it should be noted that the assessment of some parameters of debris flows basins and processes is problematic. Thus, the maximum levels of flow velocities simultaneously reflect the intensity of precipitation (flow rates) and the features of the valleys (longitudinal and transverse profiles, terracing, etc.), which in turn reflect the time of formation of the valleys and the erosion rates due to neotectonic movements. Trends in velocities reaching critical and maximum levels in channels of various orders, depending on the duration of rainfalls, could be defined for the region based on long-term observations and used as an integral characteristic for predictive assessment of flood and debris flow hazards. Unfortunately, empirical data on the real debris flows velocities that occurred in different-order streams of the Carpathian region are not enough to establish unambiguous dependencies, such data should be accumulated in the future with the appropriate monitoring. Nevertheless, the attempt to generalize the available data within the Carpathian model site turned out to be quite promising. Therefore, the presented algorithm, based on empirical data on the Carpathian region and fundamental laws of hydrodynamics, demonstrates the need to choose calculation formulas and values, the determination of which is possible in specific situations and corresponding geological and geomorphological conditions. On the basis of the proposed algorithm, a software module was created for calculating the debris flow impact on the infrastructure. Acknowledgements This research has been done in the framework of projects on risk assessment of landslide hazards and impact on communities, funded by the JESH and PAUSE programmes.

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References 1. Cannon, S.H., Gartner J.E., Rupert, M.G, Michael, J.A.: A method for the rapid assessment of the probability of post-wildfire debris flow from recently burned basins in the intermountain west. U.S.A. Geophys. Res. Abstr. 8(02030), 125–129 (2006) 2. Fleishmann, S.M.: Mudflows. Leningral Hydrometeorological Press, Leningrad (1978) 3. Guo, X., Peng, C., Wu, W., et al.: A hypoplastic constitutive model for debris materials. Acta Geotech. 11, 1217–1229 (2016). https://doi.org/10.1007/s11440-016-0494-0 4. Ivanik, O.M.: Spatial analysis and forecast of the water-gravitational processes based on GIS in Carpathian region. Geoinformatics 4, 52–58 (2008) 5. O’Brien, J.S.: Physical processes, rheology and modeling of mudflows. Ph.D thesis. Colorado State University. Fort Collins. Colorado (1986) 6. O’Brien, J.S., Julien, P.Y., Fullerton, W.T.: Two-dimensional water flood and mudflow simulation. J. Hydraul. Eng. ASCE 119(2), 244–261 (1993) 7. Papa, M., Pianese, D.: Influence of cross-sectional velocity, density and pressure gradients on debris flow modelling. Phys. Chem. Earth 27(36), 1545–1550 (2002) 8. Protodyakonov, M.M.: Flow Maximal Discharge Assessment in Small Basins. Leningral Hydrometeorological Press (1960) 9. Sokolovskiy, D.l.: River FLOWINg (Basic Theory and Calculations). Leningral Hydrometeorological Press (1968) 10. Takahashi, T., Tsujimoto, H.: Delineation of the debris flow hazardous zone by a numerical simulation method. In: Proceedings International Syrup on Erosion, Debris Flow and Disaster Prevention. Tsukuba, Japan, pp. 457–462 (1985) 11. Wang, F.: Die Schlammstrom Abwehr Bauweise. Lpz. 142 (1902) 12. Wang, X., Wu, W.: An update hypoplastic constitutive model, its implementation and application. In: Wan, R., Alsaleh, M., Labuz, J. (eds.) Bifurcations, Instabilities and Degradations in Geomaterials, pp. 133–143. Springer, Berlin (2011) 13. Wu, W.: On high-order hypoplastic models for granular materials. J. Eng. Math. 56, 23–34 (2006)

On the Performance of CAES Pile in Overconsolidated Soils: A Numerical Study Xuan Kang, Wei Wu, and Shun Wang

Abstract In this paper, we investigate the performance of a compressed air energy storage (CAES) pile in overconsolidated soils through finite element analyses. A hypoplastic constitutive model is used to account for the overconsolidation effects of the soil surrounding a CAES pile in a plane-strain model. The numerical results show that the normal force on the interface between the outer surface of the pile and the surrounding soil is a function of internal air pressure. The pressurization can influence the stress state of the surrounding soil only when the internal pressure is larger than a threshold value. Moreover, the overconsolidation effect increases the stiffness of the surrounding soil and reduces its compressibility. In turn, the change in soil behaviour influences the mechanical response of the CAES pile.

1 Introduction Inevitable intermittency of solar and wind energy resources and their mismatch with the energy demand cycle are among the main factors that impose a significant burden on the electric grid system and hinder the maximum exploitation of renewable energy; thus, viable energy storage systems are critically needed to address such an intermittency challenge [1]. Compressed air energy storage (CAES) is a competitive energy storage option that allows for the economical storage of energy X. Kang · W. Wu (B) Institute of Geotechnical Engineering, University of Natural Resources and Life Sciences, Vienna, 1180 Vienna Feistmantelstrasse 4,, Austria e-mail: [email protected] X. Kang e-mail: [email protected] S. Wang State Key Laboratory of Water Resources Engineering and Management, Wuhan University, 299 Bayi Road, Wuhan, China e-mail: [email protected]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 W. Wu and Y. Wang (eds.), Recent Geotechnical Research at BOKU, Springer Series in Geomechanics and Geoengineering, https://doi.org/10.1007/978-3-031-52159-1_6

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using excess energy during off-peak hours and later releasing the air with heat provided by fossil fuels to drive turbines and generate electricity for peak periods (i.e., diabatic concept). A major cost factor in CAES is the provision of storage space. Unlike the largescale commercial CAES currently in practice, which requires large caverns in the underground, small- and micro-scale CAES do not need immense underground storage and are, therefore, free from the geological limitations of rock caverns [2]. Other advantages of small- and micro-scale CAES include longer life span, possibly lower cost than batteries, and potentially easier integration with existing heat and cooling sources [3, 4]. In addition, larger markets are available for the micro-scale CAES, such as residential complexes, hotel facilities, hospitals, shopping malls, and farms. Purposely constructed caverns and containers are mainly for large energy plants and are rather expensive, while pile foundations under residential buildings provide attractive storage space for small facilities. In this case, closed-ended steel piles can provide space where pressurized air is recurrently stored. Although there have been several studies to explore the feasibility of CAES using steel piles [5, 6], research on the mechanical behaviours of CAES pile system is still in its infancy because the loading conditions are rather unique. The main difference between the loading condition of a conventional pile and that of a pile used for CAES is the additional loading from the compressed air pressure inside the pile. Therefore, the effect of the additional loading due to the CAES on the behaviour of the piles needs to be considered. In addition to the unique loading condition, the consolidation history of the soil also influences significantly the performance of CAES piles. Accurate characterization of the performance of CAES pile in OC clays is of great importance for the operation of CAES pile system. While the effect of overconsolidation on the mechanical behaviour of clays is well documented in the literature, there is no report on the subject of CAES pile system, and thus all the existing researches take the CAES pile system as a steel pile surrounded by normally consolidated soil [1, 7, 8]. In this paper, we investigate the effects of consolidation history on the performance of a CAES pile system in overconsolidated soils. To do this, numerical simulations with a single CAES pile are carried out by using a novel hypoplastic model. To account for the overconsolidation effect, the hypoplastic model is enhanced by introducing a structure tensor presenting consolidation history. A plane strain analysis was carried out to show the effects of consolidation history on the mechanical response of a single CAES pile installed in overconsolidated soil. The deformation behaviour and soil-pile interaction are investigated under different scenarios.

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2 Hypoplastic Constitutive Model 2.1 Hypoplastic Framework In this work, a novel hypoplastic framework proposed by Wang and Wu [9] is used to simulate the overconsolidated soil in the CAES pile system. The model includes a structure tensor to account for the history dependence. To gain perspective, let us consider the following hypoplastic framework: T˚ = L(T + S) : D + N(T − S)||D||

.

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where .T˚ is the Jaumann stress rate tensor; .T is the Cauchy stress tensor; .D is the strain rate (stretching) tensor. The tensorial functions .L and .N are of the .4th and .2nd order, respectively. .||D|| stands for the Euclidean norm of the stretching tensor; .S is the structure tensor representing the consolidation history. A simple formulation is adopted for the structure tensor: ( ) 1 T .S = αln R

(2)

where.α is a model parameter; . R is the stress ratio representing the overconsolidation defined as: pe+ .R = (3) pe in which . pe+ and . pe are the pre-consolidation pressure in the Modified Camclay model and the Hvorslev equivalent pressure, respectively. The formulations of . pe+ and . pe are given as follows: .

[ q 2] ) , pe+ = p 1 + ( Mp

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where the inclination . M corresponds to the stress ratio at the critical state by . M = 6sinφc /(3 − sinφc ) with .φc being the critical state friction angle; . N denotes the logarithmic value of the specific volume at the reference stress . pr = 1 kPa; .e is the void ratio of the soil; the index .λ∗ is the slope of the normal compression line in the double logarithmic .ln(1 + e) − ln p plane. Bear in mind, however, . R has a different physical meaning from the conventional overconsolidation ratio OCR; and a smaller value of . R corresponds to a larger OCR.

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2.2 Hypoplastic Model for OC Soils With the hypoplastic framework incorporating stress history, a simple hypoplastic constitutive model for overconsolidated soils was proposed by Wang and Wu [10]: .

[ ] ˇ ˇ + f v (trD)Tˇ + a 2 tr(TD) Tˇ + a(Tˆ + Tˆ ∗ )||D|| T˚ = f s (tr T)D tr Tˇ

(5)

where .Tˇ = T + S and .Tˆ = T − S are adopted for simplicity. The multipliers . f s and . f v account for the stiffness and volumetric response of the soil, respectively, which are defined as: √ 2 3 1 2 r (3 + a . fs = − , f = − − 3a) (6) v i 3ri λ∗ 2 3 where .ri is the ratio of the bulk modulus in isotropic compression and the shear modulus in undrained shear test on isotropic consolidated sample. It has similar physical meaning to the Poisson’s ratio; The material constant .a corresponds to the limit stress at the critical state. It reads: √ 3η(3 − sinφc ) .a = (7) √ 2 2sinφc where the factor .η is adopted to incorporate the Matsuoka-Nakai failure criterion, to give: 2I1 / .η = / (8) 3 (I1 I2 − I3 )(I12 − 3I2 )/(11 I2 − 9I3 ) − (I12 − 3I2 ) in which . I1 , . I2 , and . I3 are stress invariants. The constitutive model (5) possesses a simple formulation and is characterized by the following features: (1) history dependence can be considered by the structure tensor; (2) Matsuoka-Nakai failure surface is included by a simple formulation, and (3) loading and unloading behaviour of clays can be described without adopting additional loading criterion. Those advantages make this model a stand-alone tool for modelling overconsolidated soils. The numerical implementation of this model in finite element codes can be found in the literature [11].

3 Plane Strain Analysis of CAES Pile System 3.1 FE Model and Simulation Scenarios Under the increase of the inner pressure, a pile expands in both radial and axial directions, similar to a pressurized vessel. However, since pile length is typically much greater than its diameter, at least by a few orders of magnitude, a plane-strain

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P Z

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analysis could be a feasible approach to understanding the fundamental behaviour in the horizontal direction. Numerical analysis of CAES pile is performed using the commercial finite element software Abaqus standard. In this simulation, we consider a single pile for simplicity. The pile is close-end, circular in cross-section, and vertically preinstalled in the ground at the depth of 20 m below the ground surface, and the pile has an outer diameter of 500 mm and a wall thickness of 15 mm. A series of plane strain analyses are performed to study the influence of the inner pile pressure on the increase of contact pressure applied by the soil on the pile. In the simulations, steel pile is considered as an elastic medium, while the surrounding soil is modelled using a novel hypoplastic constitutive model, which incorporates the consolidation effect. A contact interaction is used to model the pile-soil interface with the pile-soil frictional stress being realized. The plane-strain model is shown in Fig. 1. The computational domain is a circle of 10 m in diameter. This size is large enough, compared with the size of the CAES pile (.d = 0.5 m), to avoid any boundary effects. An initial pressure of 100 kPa is applied on all nodes, provided that the plane strain model is a section buried below the ground at a depth of 5.1 m. During the simulation, the CAES pile is linearly pressurized from the atmospheric pressure of. p0 = 0.1 MPa to the maximum pressure . pmax = 10, 000 p0 , which is much higher than the service pressure of the CAES pile [7]. The computation is performed to find the actual pressure, . pint , applied to the surrounding soil. The analyses consider different stress histories with OCR = 1, 2, 4, and 8. This can be realized by setting different initial void ratios for a specific initial stress state. The material parameters used in the analyses are listed in Table 1. It is assumed that the computational domain is totally saturated conditions, so the generation of excess pore water pressure and the underground fluid flow are not considered.

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Table 1 Material parameters used in the finite element simulations Mechanical parameters Steel pile Yound’s modulus: . E, MPa Poission’s ratio: .ν Mass density: .ρ, kg/m.3 Critical state friction angle: .φc , .[◦ ] Compression index: .λ∗ Specific volume: . N Modulus ratio: .ri OCR parameter: .α Interface friction: .φ

OC soil

210000 0.3 7800 –

– – 1600 25.1 0.46 0.540 1.3 0.6 0.35

0.35

3.2 Numerical Results The relationship between the applied internal pressure and the actual interface stress is shown in Fig. 2. The contact stress acting on the interface between the outer surface of the pile and the surrounding soil remains as . p0 until the internal air pressure becomes larger than 100. p0 . For the CAES pile installed in normally consolidated soil, the interface stress increases from . p0 to 3.32. p0 with the internal air pressure increasing from . p0 to 10,000. p0 . This implies that pressurization of the CAES pile to 10 Mpa does not influence the stress state of the surrounding soil. With increasing the OCR from 1 to 8, the contact stress increases dramatically from 3.32. p0 to 8.37. p0 . The contour plot of the radial stress and strain at the end of the simulation for the CAES pile installed in the normally consolidated soil is shown in Fig. 3. The radial strain in the soil occurs as a result of the pressurization on the pile face, as well as the Poisson effects due to the constraint along the out-of-plane direction. The influence zone of the pressurization reaches up to 1.7 m, which is much larger than

Fig. 2 Soil-pile interface stress response with air pressure inside the CAES pile installed in overconsolidated soil

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the diameter (0.5 m) of the CAES pile. It is observed that the strain gives rise to a certain stress increase in the soil. The predicted radial stresses within the soil with normal consolidation are shown in Fig. 4a for all various internal air pressure. It shows that with the pressurization of the pile, radial stresses in the soil are compressive with the maximum stress occurred at the pile-soil contact. The compressional pressure reduces with distance away from the pile face. For pressurizing pressure of 1000. p0 , the stress drops to 1.0 . p0 at 1.11 m away from the pile face. However, with increasing the pressurizing pressure to 10,000. p0 , the influence zone expands up to 1.7 m. The radial strains in the soil with respect to pile pressurization are shown in Fig. 4b. The soil surrounding the pile exhibits compressional radial strains within a distance equal to more than two times the pile diameter with the maximum compressional strain occurring at the interface. Presumably, the maximum compressional strain increases 10 times with a 10 times increase in the internal air pressure. The mechanical responses of the CAES pile system in overconsolidated soil are shown in Fig. 5. It can be observed that with the increase of the OCR, the maximum

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radial stress at the pile-soil interface increase. In contrast, the maximum radial strain decreases with increasing the OCR. This is because, the stiffness of the soil increase with the OCR, leading to the reduction of the compressibility of the soil. Moreover, the influence zone of the pressurizing pressure also increases with the increase of the OCR. For normally consolidated soil, the influence zone is about 1.7 m when the pressurizing pressure reaches 10,000. p0 , whereas the influence zone expands to the boundary of the computational domain, which is more than 5 m. The maximum radial strain at the pile-soil interface decrease with the OCR. It reaches 0.1 when the pressurizing pressure reaches 10,000. p0 for normally consolidated soil. However, this value decreases to 0.06 for OCR = 8. The above results suggest that consolidation history indeed influences the mechanical response of the CAES pile. Most of the natural deposit soils exhibit a certain degree of OC behaviour, especially for the superficial layer of soil. Without considering the OCR effect, the prediction of the interaction between the CAES pile and the surrounding soil might be underestimated in the current feasibility assessments. Hence, it is recommended to include the consolidation effect when evaluating the performance of the CAES pile.

4 Conclusion Pressurization of the CAES pile can result in a variety of mechanical responses. It can in turn influence the serviceability of CAES piles. In addition, pore water pressure change, preconsolidated pressure and shear strength of soil can be influenced by the operation of CAES piles. This paper numerically examines the effect of consolidation history on the mechanical performance of a single CAES pile. The pressurizationinduced lateral stress and strain of the CAES pile under different OCRs are examined. Some concluding remarks from this paper are summarized as follows. • The numerical results reveal that the normal force on the interface between the outer surface of the pile and the surrounding soil is a function of internal air pressure. The pressurization can influence the stress state of the surrounding soil only when the internal pressure is larger than a threshold value. • The OCR effect increased the stiffness of the surrounding soil and reduces its compressibility. The change in soil behaviour in turn influences the mechanical response of the CAES pile. Specifically, the maximum contact stress at the pilesoil interface increases with the increase of the OCR. However, the influence zone of the pressurizing pressure shrinks with the increase of the OCR. Although this paper reveals some insights into the mechanical response of CAES piles in OC soils, some behaviours, such as the long-term behaviour of CAES piles under cyclic loading, and degradation of soil-pile interface properties, cannot be evaluated by the current numerical analysis. In addition, it is unclear how

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pressurization-induced temperature affects the operation of the CAES pile. Quantitative assessments of the impact of thermally induced interface effects and the resulted pile capacity changes using constitutive models considering thermal effects are required. Acknowledgements This work was funded by the National Natural Science Foundation of China (No.42241109), the Fundamental Research Funds for the Central Universities (No.2042023kfyq03), the Austrian Science Fund (No.ESP-342, P-35921), the European Commission Horizon Europe Marie Skłodowska-Curie Actions Staff Exchanges Project - LOC3G (Grant No. 101129729), and the Otto Pregl Foundation of Fundamental Geotechnical Research in Vienna.

References 1. Ko, J., Kim, S., Kim, S., Seo, H.: Utilizing building foundations as micro-scale compressed air energy storage vessel: Numerical study for mechanical feasibility. J. Energy Storage 28, 101225 (2020) 2. Zhang, L., Ahmari, S., Sternberg, B., Budhu, M.: Feasibility study of compressed air energy storage using steel pipe piles. In: Hryciw, R.D., Athanasopoulos-Zekkos, A., Yesiller, N. (eds.) GeoCongress 2012 State Art Pract, pp. 4272–4279. Geotech. Eng. American Society of Civil Engineers, Oakland, CA, USA (2012) 3. Budt, M., Wolf, D., Span, R., Yan, J.: A review on compressed air energy storage: basic principles, past milestones and recent developments. Appl. Energy 170, 250–268 (2016) 4. Rogers, A., Henderson, A., Wang, X., Negnevitsky, M.: Compressed air energy storage: thermodynamic and economic review. IEEE PES Gen. Meet. | Conf. Expo. MD, USA, IEEE, National Harbor 2014, 1–5 (2014). https://doi.org/10.1109/PESGM.2014.6939098 5. Kersten, M.S.: Thermal Properties of Soils, Bulletin 28, Engineering Experiment Station. University of Minnesota, Minneapolis, Minn, USA (1949) 6. Salomone, L.A., Kovacs, W.D., Kusuda, T.: Thermal performance of fine-grained soils. J. Geotech. Eng. 110(3), 359–374 (1984) 7. Kim, S., Kim, S., Seo, H., Jung, J.: Mechanical behaviour of a pile used for small-scale compressed air energy storage. Geo-Chicago 2016, 135–143 (2016) 8. Kim, S., Ko, J., Kim, S., Seo, H., Tummalapudi, M.: Investigation of a small-scale compressed air energy storage pile as a foundation system. Geotech. Front. 2017, 103–112 (2017) 9. Wang, S., Wu, W.: A simple hypoplastic constitutive model for overconsolidated clays. Acta Geotech. 16, 21–29 (2020) 10. Wang, S., Wu, W.: Validation of a simple hypoplastic constitutive model for overconsolidated clays. Acta Geotech. 16, 31–41 (2020) 11. Wang, S., Wu, W., Peng, C., He, X.Z., Cui, D.S.: Numerical integration and FE implementation of a hypoplastic constitutive model. Acta Geotech. 13(6), 1265–1281 (2018)

Experimental and Numerical Analysis of Fluid-Injection Unloading Rock Failure Process Miaomiao Kou, Yu Wang, Xinrong Liu, and Wei Wu

Abstract The aim of this study is to investigate the mechanical responses and failure characteristics of fissured rock-like materials subjected to fluid injection unloading, using a combination of laboratory experiments and numerical simulations. A novel configuration of fissured rock-like specimens containing a single fluid-injection borehole has been fabricated in the laboratory to allow for simultaneous application of mechanical loads and internal fluid pressure. Advanced techniques, such as 3-D X-ray computed tomography and 3-D digital image reconstruction, have been utilized to examine the effects of initial unloading states on the morphological characteristics of internal fracture networks under coupled hydro-mechanical unloading conditions. Furthermore, a series of numerical simulations have been conducted to deepen our understanding of the fluid-injection unloading failure mechanisms. Keywords Fluid-injection · Unloading failure · Crack initiation and propagation · Initial unloading states · Computed tomography and reconstruction · Numerical simulations

M. Kou (B) · Y. Wang School of Civil Engineering, Qingdao University of Technology, Qingdao 266033, P.R. China e-mail: [email protected] M. Kou · X. Liu Key Laboratory of New Technology for Construction of Cities in Mountain Area, Ministry of Education, Chongqing University, Chongqing 400045, China W. Wu Institute of Geotechnical Engineering, University of Natural Resources and Life Sciences, Vienna Feistmantelstrasse 4, 1180, Vienna, Austria e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 W. Wu and Y. Wang (eds.), Recent Geotechnical Research at BOKU, Springer Series in Geomechanics and Geoengineering, https://doi.org/10.1007/978-3-031-52159-1_7

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1 Introduction Rock engineering projects constructed below the groundwater table often face complex and varied rock structures that are exposed to external triaxial loads [1, 2] and internal hydraulic pressures [3, 4]. The presence of multiscale fractures can significantly influence the mechanical deformation and transport properties of these rock formations. Hydro-mechanical coupling plays a pivotal role in the dynamic evolution of rock mass failure and permeability, directly impacting the stability of fluid-saturated strata [5]. In water-rich strata, such as karst and subsea regions, the presence of cracks can lead to water inrush, a common occurrence in engineering disasters [6]. Similarly, the initiation and propagation of preexisting fractures due to fluid injection under hydraulic pressures below the minimum in situ stress can enhance permeability and induce seismic activity during geothermal reservoir stimulation in geothermal engineering [7]. Therefore, understanding the mechanics behind crack initiation and propagation via pressurized fluid injection is crucial for the construction of hydraulic tunnels and the success of enhanced geothermal systems. This knowledge also assists scientists and engineers in comprehending tunnel excavation and reservoir stimulation mechanisms. The fracture mechanism of fluid-pressurized fissures in rock and rock-like materials is influenced by both the hydro-mechanical stress state and the stress paths during hydromechanical loading and unloading [1, 8–11]. In recent decades, numerous researchers have delved into the mechanical behaviors and fracture mechanisms of both intact and fissured rock and rock-like materials, including sandstone, granite, salt rock, red sandstone, coal, resin, gypsum, and PMMA. They have conducted a range of experiments, such as triaxial unloading tests [12] and true triaxial unloading experiments [13]. These studies have focused on understanding the impact of loading and unloading stress paths, unloading rates, and confining pressures on two key aspects: (i) deformation and strength characteristics [14] and (ii) fracture evolution and ultimate failure modes [15, 16]. The resulting stress-strain curves during loading and unloading have been instrumental in analyzing the elastic modulus, strength parameters, and energy changes in rock and rock-like materials under various unloading conditions. Advanced monitoring techniques, including Acoustic Emission (AE) [13], Digital Image Correlation (DIC) [17], Computed Tomography (CT) [18], and Digital Volume Correlation (DVC) [19], have been employed to gather experimental data that describe various crack types, modes of crack coalescence, and ultimate failure patterns. Furthermore, some researchers have shifted their focus to the fracture mechanism of rock masses induced by fluid injection under hydro-mechanical loading conditions. Most of these investigations center around hydraulic fracturing, which involves initiating and propagating new cracks under higher fluid injection pressures than the minimum principal stress. Despite extensive laboratory experiments exploring the fracture behavior of rock and rock-like materials under unloading or coupled hydro-mechanical loading conditions, there has been relatively limited investigation into the coupling effects of hydro-mechanical unloading on the initiation and propagation of fluid-pressurized pre-existing cracks in rock and rock-like materials.

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A deeper understanding of the coupled hydro-mechanical unloading effects holds potential benefits for both underground construction in water-rich strata and deep energy exploitation. The primary objective of this article is to explore the mechanical properties, failure behavior, and permeability evolution of rock-like specimens with fissures under coupled hydro-mechanical pre-peak unloading conditions. To achieve this, we conduct triaxial-injection unloading tests on ISRM standard rock-like specimens. Our tests aim to replicate fluid pressurization on rock fractures, similar to geophysical experiments carried out in previous studies. Unlike traditional triaxial-injection tests that disregard fluid seepage effects in porous rock matrices, our approach involves the use of a specially designed drilling borehole to apply internal fluid pressurization on preexisting fissure surfaces. Throughout these tests, both the injection fluid pressurization and the unloading of confining pressure work in tandem to facilitate the propagation and coalescence of new cracks originating from the tips of preexisting fissures. We employ hydro-mechanical measurements, permeability assessments, and 3-D X-ray computed tomography along with 3-D digital reconstruction techniques to examine the following key aspects: (i) the interplay of coupled hydro-mechanical responses, (ii) the progression and merging of cracks, (iii) the three-dimensional internal fracture morphology, (iv) the spatiotemporal evolution of permeability, and (v) three-dimensional numerical simulations of mechanisms governing the evolution of the internal fracture network. The results derived from our laboratory experiments and numerical simulations offer valuable insights into the mechanical responses and failure mechanisms induced by fluid injection during unloading. These findings hold practical significance for underground engineering excavation projects, particularly in the context of deep water subsea tunnels. The structure of this article is organized as follows. Laboratory testing results are presented in Sect. 2. The three-dimensional numerical simulations of fluid-injection unloading process are performed in Sect. 3. Finally, conclusions are drawn in Sect. 4.

2 Experimental Observations 2.1 Fissured Rock-Like Specimen Preparation The artificial rock-like material is composed of silicon sands, high-strength cement (specifically, 42.5 Poland cement), water, a water-reducing agent, and rubber powder, with a specific mass ratio.The geometry of rock-like samples is manufactured in accordance with the standards set by the International Society for Rock Mechanics and Rock Engineering (ISRM). Each rock-like specimen has a height of 100.0 mm and a diameter of 50.0 mm. Within the specimen, a single fissure measuring 10.0 mm in length, inclined at an angle of .α, is incorporated, along with a fluid-injection borehole with a diameter of 0.8 mm, as illustrated in Fig. 1a

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Fig. 1 Layout of fluid-injection unloading laboratory tests: a geometric and boundary conditions and b triaxial MTS testing apparatus

The preparation of the specimens involved several key steps. Initially, a steel mold, comprising a thin steel plate, a steel column, and a swivel plate, was designed and assembled. The mixed cement grout was subsequently poured into the mold, which was then secured on a vibrating table for 5 min to eliminate any entrapped air within the grout. After curing for 15 hrs, the mold components, including the thin steel plate, steel column, and swivel plate, were disassembled after a 24-hour period. The cast fissured rock-like specimens were then submerged in distilled water for seven days, followed by an additional 21-day curing period in a controlled environment at 20.◦ C, with a relative humidity of 95.% and atmospheric pressure.

2.2 Experimental Setup and Procedure The rock servo-controlled triaxial facility employed in this study, the Rock 600-50 HT PLUS, boasts a robust loading system with a capacity of 1000 kN, a confining pressure system capable of achieving pressures of up to 60 MPa, and a water flow pressure system with a maximum pore pressure of 60 MPa. Additionally, the facility is equipped with an automated data collection system that includes axial and radial strain LVDTs, boasting an accuracy of 1.0 mm, as depicted in Fig. 1. All tests conducted in this study were executed utilizing this state-of-the-art facility. To examine the internal fracture networks post-testing, we employ a cutting-edge X-ray CT scanning system, the SIEMENS-SOMATOM scope X-ray CT scanner. The X-ray CT tests are performed with the following parameters: a source voltage of 130.0 kV, a source current of 80 .μA, an object-to-source distance of 200 mm, a camera-to-source distance of 500 mm, slice intervals of 0.5 mm, and a rotation angle of 0.18.◦ .

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Fig. 2 The schematic diagram of fluid-injection unloading stress path in laboratory experiments

To conduct laboratory fluid-injection unloading tests, we followed the following loading procedures: Initially, we applied uniform stresses of .σ1 , .σ2 , and .σ3 at a controlled rate of 1.0 MPa per minute to the fissured rock-like specimens until they reached 8.0 MPa. Subsequently, we introduced internal hydraulic pressure to the surfaces of prefabricated fissures via a fluid-injection borehole within the specimen, maintaining an internal hydraulic pressure of . pw = 2.0 MPa. While sustaining a confining pressure of .σ2 = σ3 at 8.0 MPa, we progressively increased the axial stress .σ1 at a consistent loading rate of 0.01 mm per minute until the sample reached the ultimate failure. A schematic representation of the entire fluid-injection unloading stress path is provided in Fig. 2. In addition, in these fluid-injection unloading tests, the initial two loading procedures mirror the same steps as those employed in the coupled hydro-mechanical loading tests. In the third stage, we set the initial confining pressure, denoted as .σ2 = σ3 , at 8.0 MPa. Meanwhile, we increment the axial stress, .σ1 , at a rate of 0.01 mm/min, represented by line “bc” in Fig. 2. Once .σ1 reaches the initial unloading stress state, which corresponds to point c in Fig. 2, we maintain .σ1 at a constant value, while the confining pressure (.σ2 = σ3 ) decreases at a rate of 1.0 MPa/min, represented by line “cd” in Fig. 2. Moreover, this study explores three different initial unloading stress states: 70.%, 80.%, and 90.%. In addition, we hypothesize that the injected fluid go through the failed rock-like samples satisfying the known Darcy’s law as [8] 2q Pout μL ( ) .k = (1) 2 A Pin2 − Pout in which, .q denotes the fluid flow rate; .μ represents viscosity of the injected fluid; . L is the sample length of 100.0 mm; . A is the associated cross-sectional area, and . Pin and . Pout are the inlet and outlet measuring pressures of fluids, respectively.

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2.3 Failure Characteristic Analysis We adopt the methodology of crack classification [12, 15, 16] to describe the crack propagation and coalescence patterns of rock-like samples subjected to hydromechanical loads. There are seven various types of cracks occurring in the failed samples, involving wing/anti-wing cracks, secondary shear cracks, oblique/coplanar secondary cracks, out-of-plane tension and shear cracks. The ultimate failure patterns of three different kind of fissured rock specimens, including S-45-8-2-90.%, S-45-8-2-80.% and S-45-8-2-70.%, in the fluid-injection unloading tests are depicted in Figs. 3, 4 and 5, respectively. When the initial unloading stress state is 90.%σpeak , wing cracks are initiated from the fissure tips and propagate along the maximum principal stress direction, as shown in Fig. 3. The shear crack propagation dominates the ultimate fluid-injection unloading failure patterns.

Fig. 3 Ultimate failure characteristics of the fissured rocks specimen-I in the fluid-injection unloading test: a image of failed sample; b crack growth paths; c X-ray CT image of a failed sample; d 3-D fracture networks by digital reconstruction technique; e horizontal cross section at different hight . Z

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Fig. 4 Ultimate failure characteristics of the fissured rocks specimen-II in the fluid-injection unloading test: a image of failed sample; b crack growth paths; c X-ray CT image of a failed sample; d 3-D fracture networks by digital reconstruction technique; e horizontal cross section at different hight . Z

When the initial unloading stress state is 80.%σpeak , we can observe the occurrence of wing and anti-wing cracks, and the ultimate crack coalescence patterns due to fluidinjection and the unloading effects, which belongs to the shear crack coalescence mode as illustrated in Fig. 4. When the initial unloading stress state is 70.%σpeak , wing cracks coalesce with the far shear cracks, and secondary cracks link with the far tensile cracks, which belongs to the mixed tension-shear failure mode (see Fig. 5). We also find that the ultimate failure mode transforms from the pure shear failure mode to the mixed tension-shear failure mode, as the initial unloading stress state decreases. In order to investigate the internal fracture morphology of the specimens after failure, the fissured specimens are subjected to X-ray CT scanning. Using Avizo software, the internal fracture networks of the failed fissured rock-like specimens are reconstructed under coupled hydro-mechanical loading and unloading conditions.

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Fig. 5 Ultimate failure characteristics of the fissured rocks specimen-III in the fluid-injection unloading test: a image of failed sample; b crack growth paths; c X-ray CT image of a failed sample; d 3-D fracture networks by digital reconstruction technique; e horizontal cross section at different hight . Z

We depict the 3D internal fracture morphologies by digital reconstruction techniques in the hydro-mechanical unloading tests with . Pw = 2.0 MPa and different unloading stress points in Figs. 3, 4 and 5, respectively. It can be found from these figures that the secondary cracks are inhibited as the initial unloading state increases from 70.%σpeak to 90.%σpeak . The X-ray CT scanning system is used to measure the crack area and aperture extent in the failed fissured rock-like specimens subjected to internal hydraulic pressures and mechanical loads during pre-peak unloading tests. These measurements are presented in Figs. 3, 4 and 5, respectively, for the three different kinds of samples. To quantitatively assess the internal fracture morphology in the fissured rock-like specimens that have failed under coupled hydro-mechanical unloading conditions, we employ fractal theory. This theory enables us to describe the internal fracture networks with varying spatial distributions. Fractal theory is not only useful for

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quantitatively characterizing the distribution and geometric morphology of cracks on surfaces in 2-D cases but can also be applied to study the distribution and geometric morphology of crack surfaces in 3-D cases, as established by Kou (2021) [8]. The box-counting fractal dimension can be calculated as follows: D=

.

log Nε ln Nε = ln ε log (1/ε)

(2)

in which .Nε is the number of squares covering cracks, and .ε is the characteristic length of the uniform squares.

2.4 Mechanical Responses In Fig. 6a–c, we present the comprehensive stress-strain curves of three different types of rock samples. These curves are derived from purely mechanical loading and unloading conditions, each with varying initial unloading stress states. The included curves encompass.σd ∼ ∈a ,.σd ∼ ∈r and.σd ∼ ∈v . It is noted that Fig. 6 also highlights the characteristic complete stress-strain curves for each of these curve types. The initial stage of deformation in the fissured rock-like specimens is characterized by a linear elastic relationship between axial strain and deviatoric stress. In this phase, the preexisting fissures and voids gradually close under the applied hydrostatic stresses, resulting in a slight reduction in permeability. The specimens remain in this stage as long as the confining pressure remains constant and axial stress continues to increase, up to the point of reaching the initial unloading state. Upon the onset of unloading of the confining pressure, the second stage commences, marked by nonlinear deformation. It is during this phase that cracks are initiated and propagate within the specimens, contributing to an increase in permeability. After achieving a peak value, the specimens enter the third stage: post-peak drop. In this stage, unstable crack propagation and coalescence occur, leading to a second surge in permeability

3 Numerical Simulations 3.1 Numerical Approach This section presents a 3-D numerical simulation of jointed rock masses using FLAC3D. The numerical specimen’s dimensions and mechanics parameters are identical to those of the laboratory experiments described above. To model brittle fracture behavior accurately, we modified FLAC3D’s original elastic-plastic model into a novel elastic-brittle model. In this model, the strength of elements drops sharply to an extremely low residual strength after reaching peak strength.

100 Fig. 6 Mechanical responses and permeability evolution, i.e., stress-strain-permeability curves, of the fissured rock specimens in the fluid-injection unloading tests with .σ3c = 8.0 MPa, . Pw = 2.0 MPa, and different initial unloading states: a 90.% .σpeak , b 80.% .σpeak , and c 70.% .σpeak

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To implement this modification, we placed our focus on two critical analysis aspects: stress analysis and failure analysis. We chose the Mohr-Coulomb model as the failure criterion for stress analysis. In the realm of failure analysis, we carefully assessed the type of failure experienced by each element, whether it was shear failure, tension failure, or a combination of both. When an element experienced failure, we systematically assigned substantially reduced values for strength and stiffness to every post-failure element, in line with degradation principles. Over successive iterations, this process led to comprehensive macro-damage throughout the entire specimen. The primary objective of this approach was to transition the original constitutive relation into a brittle failure model following the peak value. The framework of damage mechanics guided our approach to failure analysis, where we adapted the properties of post-failure elements according to principles of degradation. If an element had undergone shear failure without any prior or subsequent tension failure, we reduced its tension and cohesive strengths to 30.% of their initial values. However, for elements presently experiencing or having previously undergone tension failure, their tension and cohesive strengths were adjusted to only 5.% of their initial values. Regardless of the type of failure, both the shear modulus and bulk modulus of post-failure elements were uniformly reduced relative to their initial values. Notably, the friction angle remained constant throughout the process.”

3.2 Numerical Validation The reduction magnitude of mechanics parameters mentioned above is based on the synchronous evolution theory of tension and shear damages [15, 16] and the residual strength theory of rocks. However, obtaining an appropriate value for the residual strength requires significant and thorough exploration. Despite this, the straightforward principles of this new model make it highly efficient in solving 3-D and numerous element issues, and adjustments can be made to the reduction magnitude to make it effective for specific types of rocks. Several scholars have used FLAC3D to simulate the crack propagation process of real rocks using the original elastic-plastic model [20]. However, these simulations tend to result in large-scale, irregular plastic zones around or wrapping the pre-cracks, which is inconsistent with experimental results. In reality, the 2-D crack propagation locus in experiments is threadlike, while the 3-D crack propagation locus is a curved surface (as shown later in this paper). Therefore, it is clear that the elastic-plastic model is not effective on brittle materials, especially rocks. To demonstrate the effectiveness and correctness, we simulate the fluid-injection unloading test on the fissured rock-like sample of “S-45-8-2-90.%”. For demonstrating the numerical convergence of our numerical model, we simulate three numerical samples with different FE mesh sizes, i.e., .h e = 1.8 mm, .h e = 1.5 mm and .h e = 1.2 mm. We compare the predicted mechanical responses, i.e., deviatoric stress .σ1 − σ3 versus axial strain .ε1 curves, against the laboratory testing data, as illustrated in

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Fig. 7 Comparison of the simulated stress-strain curves against the experimental data in fluid-injection loading test

Fig. 7. We can observe that the numerical responses become close to the experimental observation as .h e decreases. Figure 8a shows the crack initiation and propagation process during the fluidinjection unloading test. The injection causes stress redistribution and concentration around preexisting fissure tips, leading to the initiation of wing cracks that propagate toward the direction of maximum principal stress. In three-dimensional cases (as shown in Fig. 8), the wing crack surfaces have a wrapping shape. Moreover, the ultimate failure mode in this test is dominated by the propagation of far shear crack surfaces. Our numerical model accurately predicts the failure patterns and propagation crack surfaces observed in the laboratory experiment, indicating its effectiveness and correctness. We also present the mechanical responses in Fig. 8b, including the deviatoric stress versus axial strain curves and volumetric strain versus axial strain curves.

3.3 Numerical Studies To investigate the effect of .σ2 = σ3 on the mechanical responses and failure characteristics of fissured rock-like specimens under fluid-injection unloading, we conduct numerical simulations on three samples, each featuring a single preexisting fissure inclined at .α = 45◦ . A constant fluid-injection hydraulic pressure of . pw = 2.0 MPa is applied to the preexisting fissure surfaces. The initial unloading stress state is set at 80.% of.σpeak . All other geometric and boundary conditions remain consistent with the descriptions provided earlier. In addition, the mechanical and hydraulic parameters remain unchanged, as previously stated.

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Fig. 8 a Crack initiation and propagation process, and b mechanical responses of the fissured rock specimen with .α = 45◦ in the fluid-injection loading test with .σ3c = 8.0 MPa and . Pw = 2.0 MPa

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Fig. 9 Numerical investigation on effect of the initial confining pressure on fluid-injection unloading rock failure patterns: a .σ3c = 6.0 MPa; b.σ3c = 8.0 MPa; and c .σ3c = 10.0 MPa, with the same internal hydraulic pressure of 2.0 MPa and initial unloading stress states 80.%σpeak

Figure 9 illustrates the 3-D ultimate failure patterns of the fissured rock-like specimens derived from fluid-injection unloading tests through FLAC 3D numerical simulations. For a .σ3 of 6.0 MPa, the fluid-injection unloading exhibited a mixed tension-shear failure pattern, as displayed in Fig. 9a. However, when .σ3 increased to 8.0 MPa or 10.0 MPa, the ultimate failure patterns shifted to shear-dominated modes, characterized by the propagation of shear cracks, as presented in Fig. 9b, c. These numerical findings highlight the transformation in crack propagation patterns from mixed tension-shear mode to pure shear mode with an increase in confining pressure from 6.0 MPa to 10.0 MPa.

4 Closure In this study, we conduct a series of laboratory experiments and numerical simulations to investigate the mechanical responses and failure characteristics of fissured rock-like materials under fluid-injection unloading conditions. To achieve this, we introduce an innovative configuration of fissured rock-like specimens featuring a single fluid-injection borehole, allowing us to study coupled hydro-mechanical loading-unloading scenarios. We employ advanced laboratory experimental techniques to scrutinize the internal fracture characteristics of these specimens. Our find-

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ings reveal a transformation in the ultimate failure mode, shifting from pure shear failure to mixed tension-shear failure as the initial unloading stress state decreases, while keeping the internal hydraulic and confining pressures constant. Furthermore, we observe a shift in the ultimate crack propagation patterns, transitioning from the mixed tension-shear mode to pure shear mode as the confining pressure increases, all while the initial unloading stress state and hydraulic pressure remain unchanged. These transformations in mechanical responses provide profound insights into the observed phenomena. Acknowledgements This research is financially supported by open research fund from Key Laboratory of New Technology for Construction of Cities in Mountain Area, Ministry of Education in Chongqing University (Grant No. LNTCCMA-20230109), National Natural Science Foundation of China (Grant No. 42107178), Shandong Provincial Natural Science Foundation (Grant. No. ZR2021QE162).

References 1. Feng, X.T., Haimson, B., Li, X., Chang, C., Ma, X., Zhang, X., Suzuki, K.: ISRM suggested method: determining deformation and failure characteristics of rocks subjected to true triaxial compression. Rock Mech. Rock Eng. 52, 2011–2020 (2019) 2. Zhao, Y., Bi, J., Wang, C., Liu, P.: Effect of unloading rate on the mechanical behavior and fracture characteristics of sandstones under complex triaxial stress conditions. Rock Mech. Rock Eng. 54(9), 4851–4866 (2021) 3. Maleki, S., Fiorotto, V.: Hydraulic brittle fracture in a rock mass. Rock Mech. Rock Eng. 54(9), 5041–5056 (2021) 4. Martino, J.B., Chandler, N.A.: Excavation-induced damage studies at the underground research laboratory. Int. J. Rock Mech. Mining Sci. 41(8), 1413–1426 (2004) 5. Zuo, L., Xu, L., Baudet, B.A., Gao, C., Huang, C.: The structure degradation of a silty loess induced by long-term water seepage. Eng. Geol. 272, 105634 (2020) 6. Li, S., Liu, R., Zhang, Q., Zhang, X.: Protection against water or mud inrush in tunnels by grouting: a review. J. Rock Mech. Geotech. Eng. 8(5), 753–766 (2016) 7. Zang, A., Zimmermann, G., Hofmann, H., Stephansson, O., Min, K.B., Kim, K.Y.: How to reduce fluid-injection-induced seismicity. Rock Mech. Rock Eng. 52, 475–493 (2019) 8. Kou, M.M., Liu, X.R., Wang, Z.Q., Nowruzpour, M.: Mechanical properties, failure behaviors and permeability evolutions of fissured rock-like materials under coupled hydro-mechanical unloading. Eng. Fract. Mech. 254, 107929 (2021) 9. Olsson, R., Barton, N.: An improved model for hydromechanical coupling during shearing of rock joints. Int. J. Rock Mech. Mining Sci. 38(3), 317–329 (2001) 10. Preisig, G., Eberhardt, E., Smithyman, M., Preh, A., Bonzanigo, L.: Hydromechanical rock mass fatigue in deep-seated landslides accompanying seasonal variations in pore pressures. Rock Mech. Rock Eng. 49, 2333–2351 (2016) 11. Qiu, S.L., Feng, X.T., Xiao, J.Q., Zhang, C.Q.: An experimental study on the pre-peak unloading damage evolution of marble. Rock Mech. Rock Eng. 47, 401–419 (2014) 12. Zhou, X.P., Zhang, J.Z., Wong, L.N.Y.: Experimental study on the growth, coalescence and wrapping behaviors of 3D cross-embedded flaws under uniaxial compression. Rock Mech. Rock Eng. 51, 1379–1400 (2018) 13. He, M.C., Miao, J.L., Feng, J.L.: Rock burst process of limestone and its acoustic emission characteristics under true-triaxial unloading conditions. Int. J. Rock Mech. Mining Sci. 47(2), 286–298 (2010)

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14. Meng, L., Li, T., Xu, J., Chen, G., Ma, H., Yin, H.: Deformation and failure mechanism of phyllite under the effects of THM coupling and unloading. J. Mountain Sci. 9(6) (2012) 15. Wong, L.N.Y., Einstein, H.H.: Crack coalescence in molded gypsum and Carrara marble: part 1. Macroscopic observations and interpretation. Rock Mech. Rock Eng. 42, 475–511 (2009) 16. Zhou, X.P., Cheng, H., Feng, Y.F.: An experimental study of crack coalescence behaviour in rock-like materials containing multiple flaws under uniaxial compression. Rock Mech. Rock Eng. 47, 1961–1986 (2014) 17. Wang, P., Guo, X., Sang, Y., Shao, L., Yin, Z., Wang, Y.: Measurement of local and volumetric deformation in geotechnical triaxial testing using 3D-digital image correlation and a subpixel edge detection algorithm. Acta Geotech. 15, 2891–2904 (2020) 18. Yun, T.S., Jeong, Y.J., Kim, K.Y., Min, K.B.: Evaluation of rock anisotropy using 3D X-ray computed tomography. Eng. Geol. 163, 11–19 (2013) 19. Heap, M.J., Baud, P., McBeck, J.A., Renard, F., Carbillet, L., Hall, S.A.: Imaging strain localisation in porous andesite using digital volume correlation. J. Volcanol. Geother. Res. 404, 107038 (2020) 20. Zhang, Z., Gao, H.: Simulating fracture propagation in rock and concrete by an augmented virtual internal bond method. Int. J. Numer. Anal. Methods Geomech. 36(4), 459–482 (2012)

A Visco-Hypoplastic Model with Solid Hardness Degradation for Granular Soil Haoyong Qian, Wei Wu, Xiuli Du, and Chengshun Xu

Abstract In order to investigate creep failure behaviour of granular soil, a viscohypoplastic constitutive model accounting for the solid hardness degradation and time-dependent behaviour is proposed. The visco-hypoplastic constitutive model is divided into inviscid part and viscous part, respectively. The inviscid component, incorporating the solid hardness degradation, is validated by comparing with experimental results of various void ratio and confining pressure. The viscous component is assumed to be the rate form of the power law equation related to the strain acceleration. Comparison between numerical and experimental results illustrates that the model is capable of simulating creep behaviour of granular soils. Moreover, the creep deformation mechanism between inviscid and viscous stresses is explained in view of proposed visco-hypoplastic model.

1 Introduction The creep behaviour of granular soils plays an essential role in the design of geotechnical engineering.Creep behaviour of silt and clay, such as long-term settlement issue, has been explored extensively in the last few years [1]. The reason of long-term settlement is due to the primary and secondary creep phase of the silt and clay. Opposite H. Qian · X. Du · C. Xu (B) Key Laboratory of Urban Security and Disaster Engineering of the Ministry of Education, Beijing University of Technology, Beijing, China e-mail: [email protected] H. Qian e-mail: [email protected] X. Du e-mail: [email protected] W. Wu Institute of Geotechnical Engineering, University of Natural Resources and Life Sciences, Vienna Feistmantelstrasse 4, 1180, Vienna, Austria e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 W. Wu and Y. Wang (eds.), Recent Geotechnical Research at BOKU, Springer Series in Geomechanics and Geoengineering, https://doi.org/10.1007/978-3-031-52159-1_8

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to clayed soil, the time-dependent behaviour about the granular soil have rarely been studied. This is attributed to that granular soils tend to be regarded little viscous property in geotechnical material. However, unbound gravel presents obvious creep behaviour according to experiments [2]. Creep is one of most important time-dependent property of granular materials since the creep failure occurs under constant stress which is below to the peak strength [3]. Recently, there are some researches focus on the viscous behaviour of various geomaterials. Gravelly soil presents different rate-dependent behaviours under large and medium sample test which exhibited various viscous properties, such as Isotach and non-Isotach behaviour [4]. A series of drained Triaxial compression tests on Toyoura sand was investigated under different stress level throughout shearing. It was reported that the evolution of strain was closely related to the stress level which is an notable issue in engineering [5]. Recently, the hypoplastic model has been widely used for granular material [8, 9]. With the framework of hypoplasticity, Wu [10] proposed a workable viscous equation, which was linear dependence in relatively low strain rate and quadratic dependence in relatively high strain rate. Based on the framework of visco-hypoplastic model, similar viscous equation of Wu [10] was applied to the frozen soil and it successfully presented 3 classical creep stages for the first time, including primary, secondary and tertiary phases [7]. Rheology models, such as Herschel-Bulikley model [11], was combined with hypoplastic model to study the viscous behaviour of soils, which demonstrated the rate-dependent behaviour and creep behaviour [12]. Despite these researches, however, the viscous part is too complex and has too many parameters. Moreover, the creep failure time has rarely investigated systematically. Investigation of creep failure time is crucial both in numerical and theoretical researches [3]. Therefore, a visco-hypoplastic model and a formulation between creep failure time and the minimum strain rate have been developed for the creep properties of granular soil in the following. In this work, a novel visco-hypoplastic constitutive model is proposed to evaluate creep behaviour of granular soil. Highlight of proposed novel model is that the viscous part is the rate form of a power law equation which is illustrated to be efficient to describe the viscous property of granular soils. Solid hardness degradation is incorporated into visco-hypoplastic model to simulate the evolution of void ratio. The performance of the visco-hypoplastic model was verified by comparisons with experimental results. Based on the numerical results, a formulation between creep failure time and minimum strain rate is proposed.

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2 Critical Void Ratios and Solid Hardness 2.1 Isotropic Compression of Granular Material A relationship, between the void ratio and the mean effective stress p, is utilized to simulate the isotropic compression behaviour of granular material. Bauer [13] employed an exponential function and a slightly rectified formulation with wide application is proposed: ξ .e = e0 + λ( p/ h s ) (1) where .e0 , .λ and .ξ are constants. .h s is the constant which is related to the solid hardness. Experimental results demonstrate that.h s is closely related to the stress level when the grain crushing dominates the mechanical behaviour. Under the condition of high stress level, the void ratio in Eq. (1) evolves to zero which can be ascribed to grain plastification and crushing. Herein, Eq. (1) is reasonable except for the condition that the stress level approaches infinity.

2.2 Pressure Dependence of Critical Void Ratios Granular materials present different densities as the void ratio ranges. Monotonic shear loading evolves to a critical state which is corresponding to a constant void ratio, i.e. the critical void ratio. Experimental results denote that a unique function is proved to exist for a sand between the critical void ratio .ec and stress level. As proved by experimental data, the critical void ratio is dependent strongly on the stress level. Though various formulations for .ec are proposed in previous researche [12], the following relationship is suggested: e = ec0 + λ( p/ h s )ξ

. c

(2)

where .ec0 , .λ and .ξ are constants calibrated in an .e − logp curve at critical state. The terms can be calibrated based on an isotropic compression test [13].

2.3 Pressure Dependence of Critical Void Ratios Friction sliding and the rearrangement of granular particles contributed to creep behaviour under relative low stresses, while the particle crushing played a key role under relative high stresses. For poor particle grading, the generated fragments with crushing are filled in the void space which causes creep behaviour. The compressibility of granular materials is higher for the poorly-graded material than for the well-graded material. The higher compressibility of the poorly-graded material can

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be explained by the degradation of the stiffness of the solid material due to particle crushing and consequently to a rearrangement of particles into a denser state. In view of the effect of disintegration of granular material, the solid hardness .h s is assumed to be dependent of particle grading and history as .h st . The generalized relationship is derived: ξ .e = e0 + λ( p/ h st ) (3) where .h st ranges within .h st ≤ h sg . The maximum .h sg represents the solid hardness of initial grading characteristics. Moreover, the degradation is regarded to be timedependent, i.e. .h st (t) varies with time. The rate of the void ratio is derived: p .e ˙ = ξ λ( )ξ h st

[

] p˙ h˙ st − p h st

(4)

where .e, ˙ h˙ st and p˙ represent the time derivation of .e, h st and p. Under the constant density, the volumetric strain rate is shown: e˙ = (1 + e)˙εv

.

(5)

Substituting Eq. (4) into Eq. (5): ε˙ =

. v

p ξλ ( )ξ 1 + e h st

[

] p˙ h˙ st − p h st

(6)

In particular, during the creep stage the stress rate is equal to zero, i.e. . p˙ = 0. When particle crushing and rearrangement takes place, the evolution of the hardness degradation is calculated: ˙ st = 1 + e (− h st )ξ h st ε˙ v (7) .h ξλ p According to the time-dependent void ratio, considering the degradation of the solid hardness during creep stage, Eq. (1) can be evolved to: e = ec0 + λ(

. c

p ξ ) . h st

(8)

3 Hypoplastic Constitutive Model 3.1 Framework of Hypoplastic Modelling In this paper, a basic assumption is made, as in previous work [7, 10], that the stress in granular material can be decomposed into two components with the inviscid and the viscous stresses. The formulation of the stress is given as follows:

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T = Ts + Tv

.

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(9)

where.Ts and Tv are the inviscid and viscous stresses, denoting the time-independent and time-dependent behaviour of soil, respectively. In order to derive a concrete formulation for granular soils, some basic requirements should be applied to Eq. (1). At first, the Eq. (1) should capture the major time-independent mechanical behaviour of granular soils. Secondly, the equation should reflect the effect of the strain acceleration. At last, the equation should reveal the complete behaviour of creep, i.e. primary, secondary and tertiary creep phases using a unified formulation. In this work, the enhanced hypoplastic constitutive model based on Wang and Wu [14] and rate form of a power law equation are adopted to calibrate the new constitutive equation. The rate expression of the new equation is shown as follows: ˙ = T˙ s (D) + T˙ v (D, D) ˙ .T (10) ˙ T˙ s and T˙ v are the Cauchy, time-independent inviscid and time-dependent where .T, viscous stress rate, respectively. The viscous stress rate is the function of both .D and ˙ while the inviscid stress rate is only dependent on strain rate .D. .D

3.2 Inviscid Component of Hypoplastic Equation The inviscid component of the new constitutive model is a nonlinear rate independent theory that has been used to many mechanical problems. In the framework of hypoplastic model, the constitutive model can be written in two parts, the linear and nonlinear behavior respectively. The formulation proposed by Wu and Bauer [15] is adopted as follows: ˚ = L(T, D) + N(T) ||D|| .T (11) where .L and N represent linear and nonlinear√ terms respectively, .T is Cauchy stress tensor and .D is the stretching tensor. .||D|| = trD2 stands for the Euclidean norm. The Jaumann stress rate .T˚ is composed of the rate Cauchy stress tensor .T˙ and the spin tensor .W: ˚ = T˙ + TW − WT .T (12) Based on the framework of Eq. (3), a specific hypoplastic equation for inviscid part of granular soils is proposed by Wang and Wu [14], which composes 3 linear and 1 nonlinear part as below: (trTD) T˚ = C1 (trT)D + C2 (trD)T + C3 T + C4 (trT + trT∗ ) ||trD|| trT

.

(13)

where .Ci (i = 1, 2, 3, 4) are dimensionless material constants. .trT∗ is the deviatoric stress in Eq. (13) defined by .T∗ = T − 1/3(trT)1.

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The stress-strain curve given by Eq. (13) usually is too stiff in monotonic loading tests. In order to rectify the shortcoming, a function of stiffness is used to improve the accuracy of the Eq. (13): I =

. se

exp[β(ec − e)] (1 + r )2

(14)

where .β is a material parameter, .r represents the stress ratio .||T|| /trT and .ec is the critical void ratio. In view of Eqs. (13) and (14), rectified constitutive equation is illustrated: (trTD) T˚ = Ise [C1 (trT)D + C2 (trD)T + C3 T + C4 (trT + trT∗ ) ||trD||] trT

.

(15)

The influence of void ratio is discussed in lots of soil mechanical tests. During an oedometer experiment, the compressibility of loose granular soils is higher than that of dense granular soils. During a triaxial experiment, the dilatancy of granular soils is much dependent of the initial void ratio. Generally speaking, the granular soil with various initial void ratios is regarded as different materials. The constants of soils should be different if the initial void ratio is not included. It is convenient to take the void ratio into account, and the constants of soils remain effective when the initial void ratio ranges from loose to dense granular soils. In order to solve this problem, the improved constitutive equation proposed Wu [16] is adopted: ˚ = H(T, D, e) T

.

(16)

where .e is void ratio. A critical state is described by stress rate and void ratio with continuing deformation of soil: T˙ = 0, e˙ = 0

.

(17)

Correspondingly, the friction angle and the void ratio at critical state is defined as φ and ec . Based on proposed critical void ratio, slight modification is adopted to change the maximum void ratio .emax into the critical void ratio .ec :

. c

.

Dc =

e − emin ec − emin

(18)

where . Dc is redefined relative density and .emin is the minimum void ratio. The rate of the void ratio can be calculated during the continuing deformation is as follows: e˙ = (1 + e)trtrD

.

(19)

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Fig. 1 Void ratio function versus to void ratio

The void ratio function . Ie is defined as follows: I = Dcα

. e

(20)

where .α is a dimensionless index. As demonstrated in Fig. 1, Eq. (20) satisfies the following boundary conditions: I|

. e e=e c

=1

Ie |e=emin = 0

(21)

Following the above assumptions, the extended specific hypoplastic constitutive equation with critical state representing the inviscid stress reads: (trTD) T + C4 (T + T∗ ) ||D|| Ie ] T˚ = Ise [C1 (trT)D + C2 (trD)T + C3 trT

.

(22)

The constants,.Ci (i = 1, 2, 3, 4), in Eq. (22) are first calculated at critical state. Ie = 1. The detailed process of identification is referred to Wu and Bauer [17].

3.3 Viscous Component of Hypoplastic Equation The viscous component of hypoplastic equation formulated in rate term has been developed independently of the inviscid theories. One alternative choice is try to apply non-Newtonian fluid model and the rate form of viscous stress is the function ˙ of both .D and D: ˙ v = H(D,D) ˙ .T (23) Many viscous model proposed in previous literatures are in terms of the stress and strain rate relationship. For instance, Chen[6] adopted power law Eq. (24) to analyze viscous property of granular materials. Results indicated that power law equation between the stress and strain rate was suitable for the granular materials.

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τ = k γ˙ n

. v

(24)

Where .k s viscosity coefficient which has the dimension of .Nsn /mn and .n is liquidity index. However, rate form of the viscous stress is expected if inviscid and viscous models are combined [10, 12]. It is noted to regard that similar rate forms have been adopted successfully for frozen soil and granular material to illustrate the creep behaviour of frozen soil and granular materials in the previous literatures [7, 12]. In view of the application of rate form of viscous stress, the differential form of Eq. (24) is derived: n−1 .τ˙v = kn γ˙ γ¨ (25) where .γ˙ and γ¨ enote shear strain rate and acceleration, respectively. In order to fulfill 3-dimention problem, Eq. (25) is extended to be the viscous component of the hypoplastic model as follows: ˚ T˚ v = kn||D||n−1 D

.

(26)

where .D is the Jaumann stretching-rate tensor: ˚ =D ˙ + DW − WD D

.

(27)

3.4 The Comprehensive Visco-Hypoplastic Model To describe the inviscid and viscous behaviour of granular soil in a unified way, Eq. (22) and 26 is combined to form the comprehensive new model as follows: ˚ .T

= Ise [C1 (trT)D + C2 (trD)T + C3

(trTD) ˚ T + C4 (T + T∗ ) ||D|| Ie ] + kn||D||n−1 D trT

(28) The stress rate is vanishing as the strain rate is zero and it causes a problem to simulate the stress relaxation. On the contrary, the strain acceleration can be solved as the stress rate is vanishing. Therefore, the creep rate can be obtained based on the ordinary differential Eq. (28) during creep stage. The property of visco-hypoplastic model will be demonstrated in the following.

4 Model Simulation The model performance was demonstrated by simulating some experiments. At first, the new proposed model was verified by some drained triaxial tests under constant strain rate condition on various confining pressure and density sand. In addition,

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some creep tests were simulated to demonstrate the viscous performance of granular soils, including stepwise strain rate test on Toyoura sand and Chiba gravelly soil.

4.1 Triaxial Compression Test Drained triaxial tests with constant strain rate on Tacheng gravelly sand performed by Jiang and Liu [18] were adopted for simulation. The parameters used in the proposed model for simulation are shown in Table 2. In the drained triaxial tests simulation, the second time derivative of the strain is equal to zero because of the constant strain rate. Therefore, the constitutive model 28 returns to the model 22. In the drained triaxial tests, three confining pressures (0.4MPa, 0.8 MPa and 1.6 MPa) and four different initial void ratio (0.181, 0.225, 0.263and 0.291) were employed for simulating the tests (Table 1). Together with the drained triaxial tests results, the simulated constant strain rate tests on Tacheng gravelly sand are presented in Fig. 2. Figure 2A shows that the simulated stress-strain results are in agreement with experimental results with various densities. With the increasing of the densities, stress-strain curves vary from contractive to dilatant. The critical stresses gradually converge to a certain value under the same confining pressure [19]. In addition, the critical stress is independent of initial void ratio, which is consistent with experimental results [18]. Figure 2B demonstrates that the stress-void ratio simulations and experiments are consistent. Before the failure occurs, i.e. the inflection point of the curves, the volumetric strain deformation decreases with the decreasing of the void ratio. Figure 2C presents that the simulated stress-strain results are in line with experimental results with various confining pressures excepted that the initial tangent modulus of simulation is underestimated in high confining pressure sample. The peak stresses and critical stresses increasing with the increasing of the confining pressures. These properties are also validated by other experimental results [19]. A series of stress-strain curve combining different confining pressures and void ratios can be successfully reproduced by proposed model. The simulation results vary from contractive to dilatant when the conditions range from low density and high stress level to high density and low stress level.

Table 1 Material parameters for calculating the triaxial tests Para. .C1 .C 2 .C 3 .C 4 .e .λ Value

.−121

.−75

.−648

.−258

0.25

.−0.002

.ξ

.h st

.α

.β

0.51

101

0.12

10

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0.3

p =0.4MPa

e =0.291 0

i

Void ratio,e

q= 1- 3 (MPa)

2.4 1.8

e 0=0.225 e 0=0.181 e =0.263

1.2

0

0.6

e 0=0.263 e =0.225 0

0.2

experimental numerical

e =0.291 0

e 0=0.181

pi =0.4MPa

0.15

0 0

(a)

0.25

3

6

9

12

0

15

0.5

(b)

Axial strain(%)

1

1.5

2

2.5

Stress ratio q/p

8 e =0.217

q= 1- 3 (MPa)

0

6 pi =1.6MPa 4 pi =0.8MPa

2

pi =0.4MPa

0 0

(c)

3

6

9

12

15

Axial strain(%)

Fig. 2 Comparison of numerical and experimental results for drained triaxial tests on Tacheng gravelly sand: A Stress-strain relationship under various densities (. pi =0.4MPa); B Stress-void ratio relationship under various density (. pi =0.4MPa); C Stress-strain relationship under various confining pressures (.e0 = 0.217)

4.2 Creep Simulation In order to further validate the ability of proposed model, a series of unconfined compression creep tests carried out by Enomoto [20] are simulated. Contrary to conventional constitutive model, an accelerated loading phase is assumed to reach the prescribed deviatoric stress, where the initial inviscid and viscous stress are calculated. Therefore, the initial stress state including initial inviscid stresses and initial viscous stress is obtained. Enomoto [20] conducted a series of drained unconfined creep tests based on moist compacted Miho sand. The specimens of Miho sand, with the optimum water content, are 10cm in diameter and 20cm in height which was conducted by wet material of ten layers. The confining pressure of each specimen is zero. The character parameters of the specimen included: the maximum diameter . Dmax = 2mm, mean diameter . D50 = 0.161mm, the uniformity coefficient .Uc = 34.7, fine content . FC = 0.212, the optimum water content .wopt = 0.161, maximum dry density

A Visco-Hypoplastic Model with Solid Hardness Degradation for Granular Soil Fig. 3 Deviatoric stress-strain curve of Miho sand in unconfined compression test

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Deviatoric stress (kPa)

50

q/q

max

=0.99

q/q =0.94 max q/q =0.88 max q/qmax=0.83

40

q/qmax=0.72

30

Creep

20 Experimental Numerical

10

Vertical strain rate: D 0=0.0625%/min

0 0

Table 2 Material parameters for creep simulation .C 2 .C 3 .C 4 .e .λ Para. .C1 Value .−74

30

.−825 .−246

0.67

.−0.1

.ξ

0.5

.h st

0.726 100

1 1.5 Vertical strain(%)

2

2.5

.α

.β

.k

.n

1.0

10

30

0.1

ρ = 1.644g/cm3 . Each sample applied in the creep tests possessed the same initial void ratio. In the simulation, different deviatoric stress levels in drained condition were selected to validate the proposed model. In view of the Miho sand, a constant strain rate drained test is simulated to obtain the material parameters. As illustrated in Fig. 3, the numerical and experimental results are in good agreement. The material parameters used in the simulation are presented in Table 2. As shown in Fig. 3, Five point are chosen to be the beginning point for the creep test which correspond to q/q.max =0.72,0.83,0.88,0.94 and 0.99, respectively. Both liquidity index .n and the viscosity coefficient .k an be acquired by fitting the creep experimental results. During the accelerated loading phase, same initial vertical strain rate D=0.000625/min is obtained. During the monotonic loading phase, deviatoric stress gradually increases until the prescribed stress with constant vertical strain rate D=0.000625/min. Additionally, various creep strain will be accumulated, since the magnitude of the creep strain is affected by the initial stress level [21]. Figure 4 illustrates the simulation results and experimental results about the vertical strain rate-creep time and vertical strain-creep time relationships. As demonstrated in Fig. 4, both the vertical strain rate-creep time and vertical strain-creep time relationships are consistent with the experimental results. The peak stress q.max is calculated from the stress strain relationship in monotonic loading test with the constant strain rate D .= 0.000625/min. In Fig. 4A, the vertical strain rate-time relationships under different creep stress level can be presented as straight lines, except for creep stresses over a critical value, such as q/q.max = 0.88, 0.94 and 0.99. During creep phase, the creep strain rates

. d max

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q/q max=0.99

q/q max=0.88

10-2

10-4

10

(a)

2

q/q max=0.94

-6

101

q/q max=0.83

q/q max=0.72

Vertical strain(%)

Vertical strain rate (%)

100

Experimental Numerical

1.5

q/q max =0.94

q/q max =0.88

1 q/q max =0.99

q/q max =0.83

0.5

q/q max =0.72

Numerical Experimental 102

103

Creep time (s)

104

105

0

(b)

0

2000

4000

6000

8000

10000

Creep time(s)

Fig. 4 Comparison of the numerical and experimental results for the drained unconfined compression tests on Miho sand: A Vertical strain rate versus to creep time; B Vertical strain versus to creep time

decay more slowly when the stress is higher, i.e., the slope of vertical strain rate-time relationship is smaller, which is related to the constant .n. Moreover, the secondary phase will begin earlier which is caused by a higher creep stress level. The relationships between vertical strain rate and time are expressed as descending curves in a concave-downwards shape. In particularly, 3 creep phases are demonstrated for q/q.max =0.88,0.94 and 0.99, .e., primary phase (the strain rate decreases), secondary phase (the strain rate keep constant) and tertiary phase (the strain rate increases), successively. During the primary phase, the vertical strain rate decreases monotonically and reaches the inflection point, i.e. the minimum creep strain rate. It is worth noting that the corresponding time is defined as creep failure time .t f . After inflection point, the creep strain rate begins to increase rapidly until the sample present creep rupture. The time corresponding to the minimum creep strain rate, which means the ˙ = 0, is also regarded as the creep failure time. strain acceleration equals to null .D As illustrated in Fig. 4B, the vertical strain increases a little at the onset of the creep when the deviatoric stress is lower than a certain value, such as q/q.max =0.72 and 0.83. Only primary phase and secondary phase are presented and the tertiary phase will hardly occur. On the contrary, three creep phases occur for q/qmax=0.88 and 0.94. Since q/q.max = 0.99 is almost closed to the peak stress, the vertical strain begins from secondary creep phase instead of the primary creep phase. Figure 5 shows minimum strain rate versus to stress level of experimental and numerical results. The minimum strain rate, corresponding to the secondary phase, increases with increasing stress level. As shown in Fig. 5, the numerical results of minimum strain rate are consistent with the experimental results with various stress level. The fitting curve of numerical results is also shown in Fig. 5 and it demonstrates that the relationship between minimums strain rate and stress level is linear. Moreover, the comparison of numerical and experimental results illustrates the accuracy and validity of proposed model. As shown in Fig. 6, the numerical inviscid and viscous stresses under various stress level develop with the time. Because q/q.max =0.99 is very closed to peak stress, the

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Minimum strain rate(%/s)

10-2 Fitting curve of numerical results 10-3

y=1E (10.98*x-13.83) R 2=0.99

10

-4

10-5

Experimental Numerical

10-6 0.7

0.75

0.8

0.85

0.9

0.95

1

Stress level q/q max Fig. 5 Minimum strain rate versus to stress level 14

70 q/q max =0.88

65 q/q max =0.83 q/q max =0.99

60

q/q max =0.72

q/q max =0.94

10

q/q max =0.88 q/q max =0.83

8

q/q max =0.72

6 4 2

55 0 (a)

q/q max =0.99

12 Viscous stress(kPa)

Inviscid stress(kPa)

q/q max =0.94

2000

4000 6000 Creep time(min)

8000

0

10000 (b)

2000

4000 6000 Creep time(s)

8000

10000

Fig. 6 Numerical results with different stress level: A Inviscid stress versus to time; B Viscous stress versus to time

inviscid stress begins from secondary phase which means the stress decreases monotonically until creep failure occurs as illustrated in Fig. 6A. Regarding to q/q.max =0.88 and 0.94, it is obvious to identify 3 classical creep phases. Corresponding to the change of the 3 creep phases, the inviscid stress increases during the primary phase, and evolves to the inflection point, i.e. the maximum inviscid stress representing the second stage where the inviscid stress keeps constant. Additionally, the inviscid stress decreases until the sample occurs rupture in the tertiary stage. As illustrated in Fig. 7, normalized inviscid and viscous stress difference under various stress levels develops with creep time. Regarding to q/q.max =0.88 and 0.94, it is obvious to identify 3 classical creep phases. Normalized inviscid stress difference, which is the difference between the current stress to the initial stress divided by the maximum difference, increases monotonically and reaches the maximum value, i.e., the inflection point .I1 and .I2 . After inflection point, normalized inviscid stress difference begins to decrease rapidly until the sample present creep rupture, which means that the inviscid stress rate evolves from positive to negative. Normalized

120 1.5

Normalized stress difference

Fig. 7 Normalized inviscid and viscous stress difference versus to creep time

H. Qian et al.

I1

1

I2

Inviscid component 0.5 q/q max =

0 -0.5

0.99 0.94 0.88 0.83 0.72

Viscous component

-1 -1.5 100

101

102

103

104

Creep time(s) Fig. 8 Normalized inviscid and viscous stress difference versus to creep time Stress rate(kPa/s)

1 0.5

10-3

Inviscid

0 -0.5

Viscous q/q max =

-1 102

103

0.94 0.88 0.83 0.72

104

Creep time(s)

viscous stress difference illustrates in a converse manner. The opposite change of two kind of stresses keep the total stress remain constant for the whole process in creep test. Figure 8 presents different behaviour of the inviscid and viscous stresses rate with various stress level. As shown in Fig. 8, for q/q.max =0.88 and 0.94, the positive inviscid stress rate keeps decreasing until negative which means the sample occurs creep failure in the test. The evolution tendency of inviscid stress rate is in accordance with that of normalized inviscid stress difference in Fig. 7. On the contrary, the viscous stress rate presents in a converse manner. The time inflection point of two stress rate is becomes longer when the stress level changes from q/q.max =0.94 to q/q.max =0.88. According to Eq. (26), the viscous stress rate is depended on strain rate and strain acceleration. As the value of strain acceleration is zero, the viscous stress rate is equal to zero and the strain rate is close to the minimum. For q/q.max =0.72 and 0.83, the inviscid stress rate is approach to zero infinitely and will not exceed zero. Therefore, the sample will not occur creep failure in relatively low stress level.

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Acknowledgements The authors wish to acknowledge the financial support from the National Outstanding Youth Science Fund Project of the National Natural Science Foundation of China (Grant No. 51722801), EU Horizon 2020 RISE project–HERCULES (Grant No. 778360), OeAD WTZ project (Grant No. CN14/2021), Otto Pregl Foundation of Fundamental Geotechnical Research in Vienna and China Scholarship Council (No. 202106540024).

References 1. Di Prisco, C., Flessati, L., Frigerio, G., Castellanza, R., Caruso, M., Galli, A., et al.: Experimental investigation of the time-dependent response of unreinforced and reinforced tunnel faces in cohesive soils. Acta Geotech. 13, 651–670 (2018) 2. Oldecop, L.A., Alonso, E.: Theoretical investigation of the time-dependent behaviour of rockfill. Géotechnique 57, 289–301 (2007) 3. Wang, J.F., Xia, Z.Q.: DEM study of creep and stress relaxation behaviors of dense sand. Comput. Geotech. 134, 104–142 (2021) 4. Yin, Z.Y., Jin, Y.F., Shen, J.S., Hicher, P.Y.: Optimization techniques for identifying soil parameters in geotechnical engineering: comparative study and enhancement. Int. J. Numer. Anal. Methods Geomech. 42, 70–94 (2018) 5. Murayama, S., Michihiro, K., Sakagami, T.J.S.: foundations,: Creep characteristics of sands. Soils Found. 24, 1–15 (1984) 6. Chen, Y.M., Liu, H.L., Zhou, Y.D.: Analysis on flow characteristics of liquefied and postliquefied sand. Chin. J. Geotech. Eng. 28, 1139–1143 (2006) 7. Xu, G.F., Wu, W., Qi, J.: Modeling the viscous behavior of frozen soil with hypoplasticity. Int. J. Numer. Anal. Methods Geomech. 40, 2061–2075 (2016) 8. Qian, H.Y., Wu, W., Xu, C.S., Liao, D., Du, X.L.: An extended hypoplastic constitutive model considering particle breakage for granular material. Comput. Geotech. 159, 105503 (2023) 9. Qian, H.Y., Wu, W., Du, X.L., Xu, C.S.: A hypoplastic constitutive model for granular materials with particle breakage. Int. J. Geomech. 23, 04023065 (2023) 10. Wu, W.: On high-order hypoplastic models for granular materials. J. Eng. Math. 56, 23–34 (2006) 11. Atapattu, D., Chhabra, R., Uhlherr, P.: Creeping sphere motion in Herschel-Bulkley fluids: flow field and drag. J. Non-Newtonian Fluid Mech. 59, 245–265 (1995) 12. Wang, S., Wu, W., Yin, Z.Y., Peng, C., He, X.Z.: Modelling the time-dependent behaviour of granular material with hypoplasticity. Int. J. Numer. Anal. Methods Geomech. 42, 1331–1345 (2018) 13. Bauer, E.: Calibration of a comprehensive hypoplastic model for granular materials. Soils Found. 36, 13–26 (1996) 14. Wu, W., Lin, J., Wang, X.: A basic hypoplastic constitutive model for sand. Acta Geotech. 12, 1373–1382 (2017) 15. Wu, W., Kolymbas, D.: Numerical testing of the stability criterion for hypoplastic constitutive equations. Mech. Mater. 9, 245–253 (1990) 16. Wu, W., Bauer, E., Kolymbas, D.: Hypoplastic constitutive model with critical state for granular materials. Mech. Mater. 23, 45–69 (1996) 17. Wu, W., Bauer, E.: A simple hypoplastic constitutive model for sand. Int. J. Numer. Anal. Methods Geomech. 18, 833–862 (1994) 18. Jiang, J.S., Liu, H.L., Cheng, Z.L., Ding, H.S., Zuo, Y.Z.: Influence of density and confining pressure on mechanical properties for coarse-grained soils. J. Yangtze River Sci. Res. Inst. 26, 46–50 (2009) 19. Verdugo, R., Ishihara, K.: The steady state of sandy soils. Soils Found. 36, 81–91 (1996)

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20. Enomoto, T., Koseki, J., Tatsuoka, F., Sato, T.: Creep failure of sands exhibiting various viscosity types and its simulation. Soils Found. 55, 1346–1363 (2015) 21. Karimpour, H., Lade, P.V.: Creep behavior in virginia beach sand. Can. Geotech. J. 50, 1159– 1178 (2013)

Prediction of Tunnelling-Induced Settlement Trough by Artificial Neural Networks Enrico Soranzo, Christoph Pock, Carlotta Guardiani, Yunteng Wang, and Wei Wu

Abstract Tunnelling-induced settlement is usually estimated based on field data. However, the data are representative of the local study area only depending on such parameters as geology setting and tunnel geometry. Moreover, the number of training data samples is also limited. In this study, surrogate models are developed to account for the variation of the tunnel parameters, so that they are representative of many types of conditions. The data is generated with numerical simulations by employing the Hardening Soil Model and considering various stress reduction factors. Exploiting their pattern recognition capabilities, single and multi-output artificial neural networks are trained to predict the maximum settlement and the trough width. The networks employ only 10 features and return very accurate predictions with a coefficient of determination generally higher than 90%. The network architecture, activation functions and weight initialisers are optimised by grid search. The relative importance of the various features is also studied. A computer script is provided to predict the settlement and trough width with custom input data based on the trained networks.

Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-3-031-52159-1_9. E. Soranzo (B) · C. Pock · C. Guardiani · Y. Wang · W. Wu Department of Civil Engineering and Natural Hazards Institute of Geotechnical Engineering, University of Natural Resources and Life Sciences, Vienna, Feistmantelstraße 4, 1180 Vienna, Austria e-mail: [email protected] C. Guardiani e-mail: [email protected] Y. Wang e-mail: [email protected] W. Wu e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 W. Wu and Y. Wang (eds.), Recent Geotechnical Research at BOKU, Springer Series in Geomechanics and Geoengineering, https://doi.org/10.1007/978-3-031-52159-1_9

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1 Introduction Shallow tunnels in the urban environment lead inevitably to settlement. These must be carefully controlled to avoid damage to surface structures [1] and repair cost claims [2, 3]. In the past, tunnelling-induced settlement has been analysed using two conventional approaches, namely analytical methods and the finite element method [4]. Analytical methods [5–9] are based on simplifications concerning the tunnel geometry, soil layers, constitutive model, boundary and initial conditions. The simplest method consists in a pseudoelastic analysis and calculates the maximum surface settlement .smax (Fig. 1) as γ R2 , .smax = kλ (1) E where . R is the tunnel radius, . E is the Young’s modulus of elasticity of the soil, .k is an empirical factor, depending on the soil stress and tunnel geometry, and .λ is the stress reduction factor that depends on the construction method and workmanship experience. O’Reilly and New [10] postulated that the transverse settlement trough be expressed by the Gaussian curve as s(y, z) = smax e

.

2

− 2(Ky z)2

,

(2)

where . y and .z are the horizontal and vertical distance from the tunnel axis, respectively and . K is the trough width parameter (. K = 0.4 to .0.6 for stiff clay and sandy clay, and . K = 0.25 to .0.45 for less stiff sands and gravels [4, 11]). The product .i = K z in Eq. 2 defines the trough width that corresponds to the distance between the inflection point of the settlement trough and the tunnel axis. In practice, the total

Fig. 1 Idealised transverse settlement trough

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trough width is taken as .6i. The volume of the settlement trough .Vs is calculated based on Eq. 2 as √ . Vs = 2πismax . (3) Vs is approximately equal to the ground loss .Vt , i.e. the volume of the soil excavated in excess of the designed excavation (Fig. 1). Methods exist to estimate the ground loss [12]. In practice, the ground loss ratio varies between .0.5% to .2% of the area of the excavated cross-section for TBM and NATM tunnelling, respectively [13]. Hence, the maximum settlement can be expressed as

.

s

. max

Vt . =√ 2πi

(4)

Analytical methods have the merit of simplicity and are used in the preliminary design phase. Au contraire, numerical methods are generally adopted for detailed design. Numerical methods such as the Finite Element (FEM) [14] or Finite Difference Method (FDM) [15] can account for, inter alia, soil heterogeneity, non-linear material behaviour, in situ initial and boundary conditions and time-dependent effects such as construction stages, concrete hardening and creep, and water seepage. Although tunnel excavation is a three-dimensional problem, especially in the zone close to the tunnel face, full 3D numerical analyses are complex, computational intensive and time-consuming. Hence, two-dimensional simulations are often preferred. The threedimensional effects are accounted for with various methods [16–18], among which the .λ-method [19], sometimes referred to as the .β-method, is the most frequently used [20, 21]. It consists in the reduction of the in situ stress and the resulting partial convergence of the analysed cross-section that simulates the effect of the excavation ahead of the tunnel face. Past studies have been devoted to calibrating the stress reduction factor .λ based on the results of three-dimensional simulations [20, 22– 25]. Guidelines exist on the choice of the stress reduction factor depending on the soil strength, anisotropy and support delay length for various tunnel depths [26, 27]. Generally accepted, empirical values are 50% for NATM [28] and 20% for TBM [20] tunnelling, respectively. Other than analytical or numerical methods, an approach that has recently gained momentum is the settlement prediction with machine learning. Machine learning is a branch of artificial intelligence (AI) and computer science that deals with the use of data to imitate the human learning process, by gradually improving the predictive accuracy through algorithms [29]. Machine learning has been widely applied to geotechnical engineering in general [30–38] and tunnelling in particular [39–45]. The rationale behind the adoption of machine learning approaches is that analytical formulae do not simultaneously take into consideration all the relevant factors [46], whereas numerical analysis is time-consuming. A selection of the relevant literature is presented in the following. From the early studies of [46, 47], tunnelling-induced settlement has focussed on tunnels constructed with both the NATM and TBM. The vast majority of data employed for the machine learning predictions has been acquired with field monitoring (FM). Nontheless, [48]

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studied the minimum amount of numerical simulations required to calibrate an accurate prediction for a given set of input parameters (the so-called “features” in machine learning parlance). The predictions are generally formulated as regression problems. References [49, 50] defined settlement ranges and framed the prediction as a classification problem by defining settlement intervals. Other authors [51, 52] focussed on the relative importance of the various features involved in the settlement prediction. The number of features employed vary from as litte as three [53] up to 24 [50]. Typical features include tunnel geometry, soil strength and deformation parameters, TBM advance rate and operational parameters. Various algorithms have been employed, the most frequent ones being the Artificial Neural Network (ANN) [54], Support Vector Machine [55] and Random Forests [56]. Feature selection has been carried out either with the Random Forests or with dedicated algorithms, such as Boruta [57] and SHAP [58]. Although most algorithms pertain to traditional machine learning, [53] used the Imperialist Competitive Algorithm [59], an evolutionary optimisation method, and [60] ventured into the field of Reinforcement Learning by using the Deep-Q-Network [61]. Other algorithms used are Decision Tree [62], Gradient Boosting [63], K-Nearest Neighbours [64], Linear Regression [65], Multivariate Adaptive Regression Splines [66], Vector Auto-Regressor [67] and Extreme Gradient Boosting [68]. In general, the algorithms are trained ex post. However, some authors explore the possibility of updating the prediction as the data become available [60, 69]. The relative performance of the various algorithms has been discussed by [70]. Most studies predicted the maximum settlement .smax only, but some also provided the width of the settlement trough .i. Although the predictive capacity achieved with these studies is high, the data used for the regression is collected during project execution. Also, the data is representative of the local study area only. Hence, some features tend to remain unchanged (tunnel geometry) while others may vary within certain ranges (geology). The number of training data samples is often limited (some 100 data points usually). Therefore numerical simulations can help generate additional data and control the variation of features. The numerical simulations may cover a broad range of tunnel features from geological condition to tunnelling geometry. One variable that has received little attention so far is the stress reduction factor. This is especially interesting during the design phase, where engineers design various tunnel cross-sections. Based on simplified calculations [25], the stress reduction factor .λ is determined and used in the numerical analysis to simulate the excavation ahead of the tunnel face and calculate the maximum settlement. The proposed technique, which does not depend on the tunnelling method (NATM/TBM), returns the maximum settlement and trough width with only 10 features and considers the stress reduction factor. Hence, it provides a quick crosscheck for tunnel designers. The accuracy of the settlement estimation is guaranteed by the sophisticated constitutive model used (Hardening Soil Model by [71, 72]). A Python script [73] is referenced in the code availability section that returns .smax and .i for user input data. This paper is organised as follows. Section 2 presents the numerical calculations and their statistics. The results are used to train the artificial neural networks.

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Section 3 shows the results of the machine learning predictions. These are discussed in Sect. 4. Finally, Sect. 5 concludes the manuscript.

2 Methods The data for the artificial neural networks is acquired via numerical simulations carried out with FLAC3D [74], an explicit Lagrangian finite-volume program for engineering mechanics computation, widely adopted in geotechnics.

2.1 Mesh Grid, Boundary and Initial Conditions A mesh grid with 8011 nodes and 3866 elements is generated (Fig. 2). For reasons of symmetry, only half of the model is sufficient for the settlement calculation. Hexahedrical elements with eight vertices and six quadrilateral faces are used. A circular tunnel cross-section is assumed for both NATM and TBM tunnels. The tunnel diameter . D is selected within the range 5 to 15 m. The cover to diameter ratio .C/D is comprised between 0.5 and 5. A random number generator is used to create the data points. The different geometric combinations are visually summarised in Fig. 3. Although certain combinations are similar to each other, the soil types (Sect. 2.2) assigned to the data points are different. The upper edge of the mesh simulates the terrain surface and is free to deform. The displacement of the side edges is fixed in horizontal direction. The bottom edge

Z

X

Fig. 2 Example of a FLAC3D model mesh grid generated to calculate the maximum settlement

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C/D (-)

Soil

D (m)

Fig. 3 Overburden to diameter ratio versus diameter of the ten selected soil parameter combinations

is fixed in both directions. The distance between the tunnel crown and the soil surface (overburden) and the tunnel diameter are variable. The distance between the tunnel bench and vertical edges, and between the tunnel invert and the bottom edge are denoted by . X and . Z , respectively. The side and bottom boundaries of the mesh must be far enough from the tunnel so that their effect can be neglected. A ratio . X/R = 20 is ensured in the simulations, as recommended by [75] and a ratio. Z /R = 6 is applied based on recently published guidelines [76, 77]. The initial stress state is determined by .σv' = γ ' z vertically and by .σh' = K 0 σv' horizontally, where the earth pressure at rest coefficient is defined according to [78] as ' . K 0 = 1 − sin ϕ , (5) where .ϕ ' is the effective friction angle.

2.2 Soil Constitutive Model and Parameters Complex material laws have been developed to describe the non-linear stressdeformation behaviour of the soil. Simple models like the linear elastic model with Mohr-Coulomb failure criterion require only a few input parameters, but oversimplify the mechanics of soils. In practice, there is a trade-off between the complexity, the number of parameters and the accuracy of the constitutive model [79]. In this work, the Hardening Soil Model (HSM) is used [71]. The HSM is an elasto-plastic material law with stress- and loading historydependent stiffness featuring both a shear and a cap yield surface. The relation between the vertical strain .ε1 and the deviatoric stress .q is hyperbolic and the soil

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Fig. 4 Hyperbolic stress-strain relation in primary loading for a standard drained triaxial test [81]

shows a decreasing stiffness under loading as irreversible plastic deformations occur (Fig. 4). When the maximum deviatoric stress q =

. f

6 sin ϕ (σ3 + c cot ϕ) 3 − sin ϕ

(6)

is reached, defined according to the Mohr-Coulomb failure criterion, shear plastic flow occurs. The secant and unloading/reloading stiffnesses are formulated as follows: ) ( σ3 + c cot ϕ m ref . E 50 = E 50 (7) pref + c cot ϕ ( .

ref E ur = E ur

σ3 + c cot ϕ pref + c cot ϕ

)m (8)

The oedometric stiffness is given by: ( .

ref E oed = E oed

σ3 + c cot ϕ pref + c cot ϕ

)m (9)

ref The parameters . E 50 and . E ur and the exponent for the stress dependence .m can be determined with a triaxial test, . E oed with an oedometer test. Hardening occurs both through plastic shear and volumetric stress. The yield surface is shown in Fig. 5. The material parameters are chosen from the published literature. To create a comprehensive dataset, a wide range of material parameters is considered. Hence, 10 different soils are defined according to Table 1.

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Fig. 5 Representation of the yield functions of the Hardening Soil model in principal stress space for cohesionless soils [81]

Table 1 Material parameters of the Hardening Soil Model for different soil types considered in this study No.

Source

Soil type

.ρ

.c (kPa)

(kg/m.3 )

.ϕ (.◦ )

.ν (–)

.ψ (.◦ )

ref . E 50

ref . E oed

ref . E ur

(MPa)

(MPa)

(MPa)

.m (–)

1

[80]

Gr

2100

0.0

42.0

0.2

2.0

90.0

90.0

270.0

0.40

2

[81]

Sa

1900

0.0

35.0

0.2

5.0

45.0

45.0

180.0

0.55

3

[82, 83] Si

2000

10.0

25.2

0.2

0.0

4.4

3.4

14.0

0.70

4

[84]

Cl

1950

32.8

23.0

0.2

0.0

9.5

12.0

30.0

1.00

5

[85]

Cl, si

2019

10.4

12.5

0.2

0.0

12.0

12.0

36.2

1.00

6

[86]

Si, cl

2200

30.0

26.0

0.2

0.0

37.6

37.6

150.4

0.30

7

[87]

Cl

2200

33.6

17.5

0.2

1.6

3.4

3.6

12.0

0.70

8

[88]

Cl

1800

5.1

20.8

0.2

0.0

2.1

1.6

6.2

1.00

9

[89]

Gr

1900

40.0

36.0

0.2

6.0

94.2

94.2

282.7

0.50

10

[90]

Sa, si

2000

50.0

39.0

0.2

0.0

75.0

60.0

150.0

0.75

2.3 Lining Constitutive Model and Parameters The lining properties are taken from [91, 92] and vary depending on the simulated construction method (Table 2). For NATM tunnelling, the lining consists of shotcrete, whereas for TBM tunnelling, precast concrete segments are considered. The tunnel lining is modelled as a linear-elastic material with no failure criterion by using shell elements. The Young’s modulus . E SpC .= 15 GPa is considered for the hardened shotcrete. The Young’s modulus of the segmental lining . E cm .= 35 GPa is representative of the concrete strength class C40/50. Based on experience and confirmed by preliminary studies, given that its stiffness is some orders of magnitude higher than that of the ground, the lining has a very limited impact on the ground settlement.

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Table 2 Material parameters of the concrete/shotcrete considered in this study Unit weight .γ Young’s modulus Poisson’s ratio .ν Thickness .d (cm) (kN/m.3 ) . E cm |. E SpC (GPa) (–) Hardened shotcrete Segmental lining

25

15

0.2

35

25

35

0.2

40

2.4 Stress Reduction Due to Tunnelling As previously mentioned, tunnel construction is a three-dimensional process and various approaches exist to account for the stress redistribution and soil deformation in two-dimensional calculations [79]. With the .λ-method [19], the in situ stress is reduced by a factor .λ according to .

p = (1 − λ) p0 ,

(10)

where . p and . p0 are the reduced and in situ stresses, respectively. Due to the different construction methods, different stress reduction factors .λ are considered, namely the interval 0.15 to 0.5 for NATM and 0.1 for TBM tunnelling.

2.5 Numerical Analysis Workflow The workflow of the numerical simulations is as follows: 1. The geometry parameters . D (tunnel diameter), .C (overburden) and the soil constitutive parameters are chosen 2. The mesh grid is created 3. The constitutive parameters are assigned to the soil and the initial and boundary conditions are applied 4. The tunnel elements are deactivated and a pressure corresponding to the in situ stess is applied on the tunnel boundary (Sect. 2.1) 5. The pressure . p0 on the cavity is reduced by .λ up to . p (Eq. 10) 6. The lining is installed 7. The maximum settlement .smax and the width of the trough .i are calculated.

2.6 Dataset Description According to [48] more than 75 simulations are necessary to train machine learning models and achieve tunnel settlement prediction errors below 6.5%. In this study, a

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Table 3 Descriptive statistics of the dataset obtained with the FLAC3D models. Feature (stress reduction factor, Hardening soil model material parameters and geometry) and labels (maximum settlement and trough width) .λ (-)

.ρ

(kg/m.3 )

.c (kPa) .ϕ (.◦ )

ref . E 50

ref . E oed

ref . E ur

(MPa)

(MPa)

(MPa)

.C (m)

. D (m)

.C/D

.smax

(–)

(mm)

.i (m)

Values 281

281

281

281

281

281

281

281

281

281

281

281

Mean

1993

15.4

29.6

38.6

38.3

122.9

28.4

10.2

2.8

30

21.7

St.Dev. 0.11

95

16

8.9

35.5

34.9

107.3

16.9

3

1.4

34

10.7

Min.

0.1

1800

0

12.5

2.1

1.6

6.2

2.5

5

0.5

1.6

3.4

25%

0.1

1900

0

23

9.5

12

30

14.2

7.5

1.7

9.6

13.8

50%

0.13

2000

10

26

37.6

37.6

150

26.5

10.4

2.8

18.5

19.8

75%

0.28

2100

32.8

36

90

90

270

36.5

12.8

4

34.3

27.7

Max.

0.7

2200

50

42

94.2

94.2

282.7

75

15

5

229.3

53.5

0.19

total of 281 different combinations of geometry, soil and stress reduction parameters are considered to obtain the settlement and trough width. The descriptive statistics of the variables are listed in Table 3. The whole dataset is provided in a public repository as indicated in the code availability section. The stress reduction factor.λ is comprised in the interval 0.10 to 0.50, the soil density .ρ is between 1800 and 2200 kg/m.3 . The cohesion and friction angle are in the range 0 to 50 kPa and 12.5.◦ to 42.◦ , respectively. The elasticity moduli are between 2 and 280 MPa. The geometric parameters have been discussed in Sect. 2.1. The overburden to diameter ratio.C/D is a derived feature that is introduced to enhance model performance [93]. The calculated settlement.smax varies from a negligible value of 2 mm to a possibly critical one of 230 mm, the trough width .i from about 3 to 54 m. The correlation matrix of the numerical analysis data is shown in Fig. 6. It can be seen that the settlement is primarily negatively correlated to soil cohesion .c and positively correlated to the overburden .C. The maximum settlement and trough width are moderately correlated, a fact that suggests the adoption of a multi-output regression (Sect. 2.8.6).

2.7 Data Preprocessing In this work, the machine learning models are implemented with the programming language Python version 3.8.5. 70% of the data is used to train the ANN and 30% is assigned to the test set. This splitting task is performed by the function train_test_split of the Scikit-learn library [94]. Different units of measurement determine scale differences among the features that can hamper ANN performance, due to numerical instability. Hence, the features . X are scaled with the StandardScaler of the Scikit-learn library [94] according to

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Fig. 6 Heatmap of the correlation matrix of the dataset used to train the ANNs

Fig. 7 Schematic representation of the working principle of an artificial neural network

.

X' =

X −μ , σ

where .μ and .σ are the mean values and the standard deviations of the features.

(11)

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2.8 Artificial Neural Networks The underlying idea of the ANN is to artificially mimic biological intelligence [54]. In practice, the labels . y (.smax and .i, in our case) are approximated by a function of ref ref ref , . E oed , . E ur , .C, . D and .C/D) with a neural network the features . X (.λ, .ρ, .c, .ϕ, . E 50 (Fig. 7). Neurons represent the input features . X , the output labels . y and one or more interposed hidden layers. The output function of each neuron . j is expressed as ( .

yˆj = f

n ∑

) wi, j · xi + b j ,

(12)

i=1

where .wi, j are the weights and .b j the bias term. Various activation functions . f (xi ) can be selected for the neurons to impart non-linearity to the network.

2.8.1

Activation Functions

Two activation functions are considered in this study, namely the REctified Linear Unit (RELU) and the Gaussian Error Linear Unit (GELU) (Fig. 8). Let .z j = wi, j · xi + b j , the REctified Linear Unit (RELU) activation function is defined according to Eq. 13. .a j = f (z j ) = max(0, z j ) (13) Although RELU units are a common choice for traditional neural networks, they are affected by the problem of “dying RELUs”, arising when the activation function constantly returns .a j = 0 [95]. This problem is solved by GELU [96] by considering negative output values for .z j ≥ 0 according to 3

output activation function

2.5

2

1.5 GeLU ReLU

1

0.5

0 -3

-2

-1

0

1

2

-0.5

input of the activation function

Fig. 8 Visual comparison of the GELU and RELU activation functions

3

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a j = f (z j ) =

.

2.8.2

)] [ ( zj zj 1 + erf √ 2 2

135

(14)

Backpropagation

The weights and biases of the network are optimised by minimising the mean squared error .MSE (backpropagation), defined in Eq. 15, where . yi and . yˆi are the observed and predicted labels, respectively. n ∑ (yi − yˆi )2 .MSE = n i=1

(15)

The .MSE is the “cost function” of the ANN and is minimised using the method for stochastic optimisation Adam [97], with an initial step size (“learning rate”) of 0.001 and the parameters.β1 and.β2 controlling the decay rate of 0.9 and 0.999, respectively.

2.8.3

Weight Initialisation

The MSE is minimised by backpropagating its gradient from the labels to the features. As the gradient comes closer to the features it gets smaller. With very small gradients, the weights are hardly updated at all and no learning occurs. Proper weight initialisation overcomes this problem by keeping the variance of the activation functions and the backpropagated gradients as constant as possible across the layers. Hence, the variance of the layer outputs must be equal to the variance of the input [98]. In this paper, four weight initialisation approaches are considered, namely the GlorotUniform and GlorotNormal [99], the HeUniform and HeNormal [100]. The GlorotUniform initialises the weights from a uniform distribution with limits .W according to / 6 .W = ± (16) n j + n j+1 so that their variance assumes the value σ2 =

.

2 n j + n j+1

(17)

where .n j and .n j+1 refer to the input and output neurons of a layer. The weights of the GlorotNormal are unbounded with mean value 0. The HeUniform initialisation strategy considers only the input nodes: / .

W =±

6 nj

(18)

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σ2 =

.

2 nj

(19)

The weights of the HeNormal are unbounded with mean value 0.

2.8.4

Weight Regularisation

Weight regularisation is one of the strategies commonly adopted to overcome overfitting, which happens when the model fits the training set too well, resulting in a low performance on the test set. The underlying idea is to limit the degrees of freedom of neural networks so that the weights do not grow indefinitely. In this study, L2-regularisation is considered, in which the expression

.

n λ ∑ 2 w 2n i=1 i

(20)

is added to the cost function. The parameter .λ specifies how much the model should be penalised. In L2-regularisation, large weights are penalised more heavily [98].

2.8.5

Hyperparameter Tuning and Cross-Validation

The ANN is both parametric (the weights .wi, j and biases .b j are determined by the algorithm) and hyperparametric, the number of layers and the number of neurons in each layer (the ANN architecture), the activation function and initialiser being set by the user. The optimal number of layers is searched within the interval .[1, 4] and the number of neurons is optimised among .10, 20, ..., 50 neurons per layer. The hyperparameters are tuned with a five-fold cross-validation grid search [94] that searches the best combination within the defined subset of hyperparameters. This procedure is computationally expensive, but it is effective when the hyperparameters have discrete values. For a given set of hyperparameters, the model is trained on four folds and tested on the remaining one. At each iteration, the mean cross-validated score . R 2 is computed and the hyperparameters providing the highest . R 2 are selected.

2.8.6

Network Output Labels

The deep-learning libraries TensorFlow [101] and Keras [102] are used to implement the ANNs. Three artificial neural networks are defined with different number of output labels. ANN 1 has two labels, namely the maximum settlement .smax and the trough width .i. ANN 2 and ANN 3 have each one label, namely .smax and .i, respectively. ANN 1 is a multi-output regression, in which both labels are dependent variables. The weights of ANN 1 are shared among the labels except for the output

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layer. This type of regression is preferable if the labels are correlated. As previously stated (Sect. 2.6), a moderate correlation between .smax and .i justifies this approach. On the other hand, since ANN 2 and ANN 3 have only one label, their weights are better optimised to fit the underlying data than ANN 1. However, this comes at the additional computational cost due to the separate networks.

2.8.7

Training and Performance Metrics

Once the network architecture is defined, its training can begin. Early stopping, an inexpensive way to avoid strong overfitting [103], is implemented by monitoring the ANN performance to ensure that the MSE of the test set decreases within every 50 steps (patience) and by interrupting the training as soon as this criterion is no longer fulfilled. Two performance metrics are computed for the training and test sets, namely the coefficient of determination (. R 2 ) and the Mean Absolute Percentage Error (MAPE). They are defined as ∑n ˆ i )2 i=1 (yi − y 2 . R = 1 − ∑n (21) ¯ )2 i=1 (yi − y | n | 1 ∑ || yi − yˆi || .MAPE = , n i=1 | yi |

(22)

where . y¯ is the mean value of the observed labels.

2.9 Feature Importance Because of their complexity, ANN are often regarded as black box models [98]. However, some methods exist to provide interpretability, such as the SHapley Additive exPlanations (SHAP) by [58]. This method is based on the use of Shapley values from the cooperative game theory. In the machine learning framework, the features represent the “players” that form coalitions whose objective is the reproduction of the labels (the “game”). An advantage of SHAP over other feature importance strategies, e.g. permutation importance, is that all combinations of features are examined to determine the importance of one feature.

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3 Results A workstation equipped with an Intel Core i7-4810MQ CPU at 2.80 GHz with 4 cores and 8 logical processors is used for the calculations. The network architecture optimisation lasts about 8 h. The training and test of the ANN takes about a minute. The optimisation of the architecture of the three networks is summarised in Table 4. Table 5 provides an overview of the performance of the networks with regard to the maximum settlement and trough width for the training and test sets. The maximum settlement predicted with the ANN against the one computed with the FDM in FLAC are shown in Figs. 9 and 11 for the training and test sets, where the bisector represents a perfect prediction and the dashed lines .±20% deviations. The trough width predictions are shown in Figs. 10 and 12. The prediction is very accurate for all networks as manifested by the coefficients of determination . R 2 in the order of magnitude of 90%. Appreciable deviations occur on very few samples. Figures 14 and 15 shows the SHAP values of ANN 1 to 3. The stiffness, stress reduction and tunnel diameter have the largest impact on the maximum settlement. The overburden, diameter and the overburden to diameter ratio have the largest impact on the trough width.

Table 4 Optimal architectures of the artificial neural networks for the prediction of the maximum settlement and trough width Predicted Hidden layers Neurons per Activation Initialiser ANN labels layer function ANN 1 ANN 2 ANN 3

.smax , .i .smax .i

4 4 1

40 50 10

GELU GELU GELU

HeNormal GlorotUniform GlorotUniform

Table 5 Performance of the artificial neural networks for the prediction of the maximum settlement and trough width ANN Performance Predicted labels metrics .smax .i Training Test Training Test ANN 1 ANN 2 ANN 3

.R

2

MAPE 2 .R MAPE 2 .R MAPE

0.998 5.8% 0.993 8.7% – –

0.897 13.4% 0.912 11.0% – –

0.993 4.1% – – 0.993 4.2%

0.978 8.1% – – 0.989 6.3%

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(a) Training dataset

139

(b) Test dataset

Fig. 9 Prediction results of the maximum settlement .smax with ANN 1

(a) Training dataset

(b) Test dataset

Fig. 10 Prediction results of the trough width .i with ANN 1

4 Discussion Numerical calculations rely on an idealisation of natural conditions. In this work, a homogeneous soil without groundwater is considered. A circular tunnel geometry is assumed both for the lower values of the stress reduction factor.λ (representative of the TBM excavations) and for the higher ones (NATM tunnelling). The HSM is chosen in this study as the material law, an acceptable compromise between computational accuracy and effort. However, some authors found that the HSM delivers wider settlement troughs than field observations [79].

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(a) Training dataset

(b) Test dataset

Fig. 11 Prediction results of the maximum settlement .smax with ANN 2

(a) Training dataset

(b) Test dataset

Fig. 12 Prediction results of the trough width .i with ANN 3

The advantage of this study is the consideration of the 3D effects by the .λ-method, where the values of the stress reduction factors are based on the method of [25].

4.1 Dataset Size The dataset consists of 280 different combinations of stress reduction, soil and geometry parameters. In Machine Learning applications, it is crucial to have broad and representative datasets. Although an even larger and more varied database could

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increase the network performance, especially for the test datasets, the stress reduction factors encompass both TBM and NATM tunnelling and the soil and geometry parameters cover a large range of values that can be considered realistic for shallow tunnels.

4.2 Network Architecture, Activation Functions and Initialisers The performance of an ANN is highly dependent on its architecture. In this work, the ANN architectures are systematically optimised by grid search. It is evident that the optimal architectures of ANN 1 and 2 differ from ANN 3 (Table 4). The architecture of the multi-output ANN 1 is very similar to the single-output ANN 2 with four hidden layers in both cases. This deep architecture indicates that the relationship between the features and the maximum settlement is complex. This is confirmed by Figs. 13a and 14 where no feature has a prevalent impact on .smax . On the contrary, the trough width demands a shallower architecture. In ANN 3, the best results are achieved with a shallow network with only one hidden layer and 10 neurons. This indicates that the relationships between the features and the trough width is more straightforward. This conclusion is also supported by the results of the feature importance, whereby the overburden .C plays a lead role (Fig. 15), as reported in previous studies that prevalently linked .i to the overburden [4].

(a) Maximum settlement

max

(b) Trough width

Fig. 13 Feature importance for the maximum settlement .smax and trough width .i with ANN 1

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Fig. 14 Feature importance for the maximum settlement .smax with ANN 2

Fig. 15 Feature importance for the trough width .i with ANN 3

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143

Activation Functions

In this work, the RELU and GELU activation functions are considered the network optimisation. As shown in Table 4, the GELU activation function leads to the best model performance for all the three networks. Due to its curved form and the fact that it can assume negative values, the GELU could better approximate the results. Given the dataset of 280 data points, the adoption of the GELU activation function, although more computationally intensive, delivers timely results.

4.2.2

Weight Initialisation

The choice of the weight initialisation method has no significant influence on the performance of the three networks. Nonetheless, Glorot initialisers are chosen both in ANN 2 and 3, which might indicate that they outperform He initialisers. Since the Glorot was specifically developed for the sigmoid function, whereas the He initialiser targeted RELU and GELU functions, the results do not entirely match the expectations. One possible explanation is that the networks considered contain only a few hidden layers and the vanishing gradient problem is not very pronounced.

4.3 Network Performance The performance of the networks on both the training and test sets is very good (Table 5). Overfitting is successfully avoided during ANN training, as manifested by the high test . R 2 . A stronger adjustment during the training would improve the training performance, but in return the test performance would deteriorate. Thus, a balance must always be struck between the generalisation ability of the networks and training fitness. Very accurate results were achieved in this work both with the multi and singleoutput networks. A multi-output ANN is subject to the assumption that the two output variables are dependent. The correlation between .smax and .i of 0.42 is moderate (Fig. 6). Due to the very similar performance of the networks, no recommendation can be made as to whether single or multi-output architectures are better suited.

4.4 Feature Importance The feature importance provides an overview of the impact of the individual features on the results. Figures 13b and 15 show the large difference between the multi and single-output networks with regard to the trough width .i. By embedding the maximum settlement as a network label, the importance of the features become more

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even. Hence, the single-output neural networks are better suited for assessing the relative importance of the input features. The most important input features with regard to the maximum settlement are the moduli of elasticity, stress reduction factor and tunnel diameter. These features are in line with expectations. The modulus of elasticity determines the stiffness of the soil. With higher values, less settlement is to be expected. The stress reduction factor and the diameter influence the affected area of stress redistribution in the soil. With larger values, a larger soil volume is affected by the stress redistribution. With regard to the trough width, the most important input features for both ANN 1 and 3 are the geometry parameters, namely the overburden, the tunnel diameter and their ratio. In agreement with the field data collected by [10, 11], the overburden has the major impact on the results (Fig. 15).

4.5 Python Script for Custom Data A Python script is developed to aid the practitioner in estimating tunnelling-induced ground settlement and ensure that this study is not an end in itself. This script loads the previously trained ANNs, takes the user input data, scale them according to the mean and standard deviation of the training data of the artificial neural networks and returns the maximum settlement and trough width.

4.6 Limitations and Possible Improvements Obviously, the performance of the machine learning algorithms depends on the number of the training data. The algorithms can be steadily enhanced along with augmented training data. However, when expanding the dataset of numerical results, care must be taken to use broadly distributed soil parameter values. The inclusion of geometry combinations beyond the range of this study would also prove beneficial. In this work, the tunnel diameter and overburden ratio are kept within certain limits, i.e. tunnels of small diameters and large depth are not considered. Finally, the scope of the prediction could be widened by considering the influence of groundwater and soil stratification, whereby the position of the layers and the groundwater level could serve as additional features.

5 Conclusions Inspired by the recent developments of machine learning, a method for the prediction of tunnel-induced settlements is developed in this work based on a database of numerical calculations and by taking the stress reduction factor into consideration. The Hardening Soil Model is adopted in the calculations.

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A very good agreement between the predicted and calculated settlements is achieved both on the training and test sets. Both multi-output (.smax and .i) and singleoutput (.smax or .i) regressions are considered. Since their results are comparable, both can be adopted for settlement predictions. However, the impact of the features on the results is more evident with single-output networks. The most important features for the maximum settlement prediction are the elastic modulus, tunnel diameter and stess reduction factor. With regard to the trough width, the overburden, diameter and their ratio are the most relevant. The artificial neural networks are provided together with the Python script. Acknowledgements Financial support for this research is provided by the Otto Pregl Foundation for Geotechnical Fundamental Research and by the Austrian Science Fund (FWF): P 34257 StandAlone Project.

6 Supplementary Material The source codes are available for download at the link: https://github.com/soranz84/220627_Tun_Sett/.

References 1. Houlsby, G.T., Burd, H.J., Augarde, C.E.: Analysis of tunnel-induced settlement damage to surface structures. In: Proceedings of the 12th European Conference on Soil Mechanics and Foundation Engineering, pp. 31–44 (1999) 2. Crossrail. D12—Ground settlement. Published online (2008). https://bit.ly/3nPerZc 3. High Speed Two. Guide to ground settlement. Published online (2021). https://bit.ly/3yVWqi8 4. Leca, E., New, B.: Settlements induced by tunneling in soft ground. Tunn. Undergr. Space Technol. 22, 119–149, 03 2007. https://doi.org/10.1016/j.tust.2006.11.001 5. Wayne Clough, G., Schmidt, B.: Design and Performance of Excavations and Tunnels in Soft Clay, volume 20 of Developments in Geotechnical Engineering, Chapter 8, pp. 567–634. Elsevier (1981). https://doi.org/10.1016/B978-0-444-41784-8.50011-3 6. Dormieux, L., De Buhan, P., Leca, E.: Estimation par une méthode variationnelle en élasticité des déformations lors du creusement d’un tunnel: application au calcul du tassement de surface. Revue Française de Géotechnique 59, 15–32 (1992). https://doi.org/10.1051/geotech/ 1992059015 7. Kerry Rowe, R., Lee, K.M.: Subsidence owing to tunnelling. II. Evaluation of a prediction technique. Canadian Geotech. J. 29(6), 941–954 (1992). https://doi.org/10.1139/t92-105 8. Sagaseta, C.: Analysis of undrained soil deformation due to ground loss. Géotechnique 37(3), 301–320 (1987). https://doi.org/10.1680/geot.1987.37.3.301 9. Yi, X., Kerry Rowe, R., Lee, K.M.: Observed and calculated pore pressures and deformations induced by an earth balance shield. Canadian Geotech. J. 30(3), 476–490 (1993). https://doi. org/10.1139/t93-041 10. O’Reilly, M.P., New, B.M.: Settlements above tunnels in the United Kingdom—their magnitude and prediction. In: Tunnelling 82, pp. 173–181, London, UK (1982). Institution of Mining and Metallurgy. Proceedings of the 3rd International Symposium

146

E. Soranzo et al.

11. Mair, R., Taylor, R.: Bored tunnelling in the urban environment. In: Proceedings of the 14th International Conference on Soil Mechanics and Foundation Engineering, pp. 2353–2385, Rotterdam, Netherlands, Sept. 1997. Balkema 12. Lee, K.M., Kerry Rowe, R., Yee Lo, K.: Subsidence owing to tunnelling. I. Estimating the gap parameter. Canadian Geotech. J. 29(6), 929–940 (1992). https://doi.org/10.1139/t92-104 13. Mair, R.: General report on settlement effects of bored tunnels. In: Mair, R., Taylor, R.N. (eds.), Proceedings of the 2nd International Symposium on Geotechnical Aspects of Underground Construction in Soft Grounds, Geotechnical Aspects of Underground Construction in Soft Ground, pp. 43–53, Rotterdam, Netherlands (1996). Balkema 14. Clough, R.W.: The finite element method in plane stress analysis. In: Proceedings of 2nd ASCE Conference on Electronic Computation, Pittsburgh, USA, Sept. 1960 15. Collatz, L.: Numerische Behandlung von Differentialgleichungen. Springer (1955). https:// doi.org/10.1007/978-3-662-22248-5 16. Powell, D.B., Sigl, O., Beveridge, J.P.: Heathrow express-design and performance of platform tunnels at terminal 4. In: Tunnelling, vol. 97, pp. 565–593, London, UK, Sept. 1996. Institution of Mining and Metallurgy 17. Rowe, R.K., Lo, K.Y., Kack, G.J.: A method of estimating surface settlement above tunnels constructed in soft ground. Canadian Geotech. J. 20(1), 11–22 (1983). https://doi.org/10. 1139/t83-002 18. Swoboda, G.: Finite element analysis of the new Austrian tunnelling method (natm). In: Proceedings of the 3rd International Conference on Numerical Methods in Geomechanics, Aachen, vol. 2, pp. 581–586. Balkema, Rotterdam, the Netherlands (1979) 19. Panet, M., Guenot, A.: Analysis of convergence behing the face of a tunnel. In: Tunnelling, vol. 82, pp. 197–204, London, UK, June 1982. Institution of Mining and Metallurgy. Proceedings of the 3rd International Symposium 20. Möller, S.C., Vermeer, P.A.: On numerical simulation of tunnel installation. Tunn. Undergr. Space Technol. 23(4), 461–475 (2008). ISSN 0886-7798. https://doi.org/10.1016/j.tust.2007. 08.004 21. Mohyla, M., Vojtasik, K., Hrubesova, E., Stolarik, M., Nedoma, J., Pinka, M.: Approach for optimisation of tunnel lining design. Appl. Sci. 10(19) (2020). https://doi.org/10.3390/ app10196705 22. Do, N.A., Dias, D.: A comparison of 2D and 3D numerical simulations of tunnelling in soft soils. Environ. Earth Sci. 76(102) (2017). https://doi.org/10.1007/s12665-017-6425-z 23. Hoek, E., Carranza-Torres, C., Diederichs, M., Corkum, B.: Integration of geotechnical and structural design in tunnelling. In: Proceedings University of Minnesota 56th Annual Geotechnical Engineering Conference, pp. 1–53, Jan. 2008 24. Karakus, M.: Appraising the methods accounting for 3D tunnelling effects in 2D plane strain FE analysis. Tunn. Underg. Space Technol. 22(1), 47–56 (2007). https://doi.org/10.1016/j. tust.2006.01.004 25. Vlachopoulos, N., Diederichs, M.: Improved longitudinal displacement profiles for convergence confinement analysis of deep tunnels. Rock Mech. Rock Eng. 42, 131–146 (2009). https://doi.org/10.1007/s00603-009-0176-4 26. Negro, A. Jr.: Design of shallow tunnels in soft ground. Ph.D. thesis, Department of Civil Engineering, University of Alberta, Edmonton, Canada (1988) 27. Kim, H.-J., Eisenstein, Z., Chae, B.-G., Jeong, C.-H.: Estimates of stress reduction factors for the tunnel design. Tunn. Underg. Space Technol. 21(3), 451 (2006). ISSN 0886-7798. https:// doi.org/10.1016/j.tust.2005.12.089. Safety in the Underground Space—Proceedings of the ITA-AITES 2006 World Tunnel Congress and 32nd ITA General Assembly 28. Muir Wood, A.M.: The circular tunnel in elastic ground. Géotechnique 25(1), 115–127 (1975). https://doi.org/10.1680/geot.1975.25.1.115 29. International Business Machines. Machine learning (2020). https://www.ibm.com/cloud/ learn/machine-learning 30. Achmet, Z., Di Sarno, L.: State-of-the-art review of machine learning applications to geotechnical earthquake engineering problems. In: EASD Procedia EURODYN, pp. 3424–3437, Jan. 2020. https://doi.org/10.47964/1120.9281.18886

Prediction of Tunnelling-Induced Settlement Trough by Artificial Neural Networks

147

31. Guardiani, C., Soranzo, E., Wei, W.: Time-dependent reliability analysis of unsaturated slopes under rapid drawdown with intelligent surrogate models. Acta Geotechnica 17(4), 1071–1096 (2022). https://doi.org/10.1007/s11440-021-01364-w 32. Jong, S.C., Ong, D.E.L., Oh, E.: State-of-the-art review of geotechnical-driven artificial intelligence techniques in underground soil-structure interaction. Tunn. Underg. Space Technol. 113, 103946 (2021). https://doi.org/10.1016/j.tust.2021.103946 33. Ebid, A.M.: 35 years of (AI) in geotechnical engineering: state of the art. Geotech. Geol. Eng. 39, 637–690 (2021). https://doi.org/10.1007/s10706-020-01536-7 34. Moayedi, H., Mosallanezhad, M., Rashid, A.S.A., Jusoh, W.A.W., Muazu, M.A.: A systematic review and meta-analysis of artificial neural network application in geotechnical engineering: theory and applications. Neural Comput. Appl. 32(2), 495–518 (2020). https://doi.org/10. 1007/s00521-019-04109-9 35. Shahin, M.A.: State-of-the-art review of some artificial intelligence applications in pile foundations. Geosci. Front. 7(1), 33–44 (2016). https://doi.org/10.1016/j.gsf.2014.10.002. Special Issue: Progress of Machine Learning in Geosciences 36. Soranzo, E., Guardiani, C., Saif, A., Wei, W.: A reinforcement learning approach to the location of the non-circular critical slip surface of slopes. Comput. Geosci. 166, 105182 (2022). https:// doi.org/10.1016/j.cageo.2022.105182 37. Zhang, W., Zhang, R., Wu, C., Goh, A.T.C., Lacasse, S., Liu, Z., Liu, H.: State-of-the-art review of soft computing applications in underground excavations. Geosci. Front. 11(4), 1095–1106 (2020a). ISSN 1674-9871. https://doi.org/10.1016/j.gsf.2019.12.003 38. Zhang, W., Li, H., Li, Y., Liu, H., Chen, Y., Ding, X.: Application of deep learning algorithms in geotechnical engineering: a short critical review. Artif. Intell. Rev. 54, 5633–5673 (2021). https://doi.org/10.1007/s10462-021-09967-1 39. Marcher, T., Erharter, G.H., Winkler, M.: Machine learning in tunnelling—capabilities and challenges. Geomech. Tunn. 13(2), 191–198 (2020). https://doi.org/10.1002/geot.202000001 40. Shreyas, S.K., Dey, A.: Application of soft computing techniques in tunnelling and underground excavations: state of the art and future prospects. Innov. Infrastruct. Sol. 4, 46 (2019). https://doi.org/10.1007/s41062-019-0234-z 41. Shahrour, I., Zhang, W.: Use of soft computing techniques for tunneling optimization of tunnel boring machines. Underg. Space 6(3), 233–239 (2021). https://doi.org/10.1016/j.undsp.2019. 12.001 42. Soranzo, E., Guardiani, C., Wei, W.: A soft computing approach to tunnel face stability in a probabilistic framework. Acta Geotechnica 17(4), 1219–1238 (2022). https://doi.org/10. 1007/s11440-021-01240-7 43. Soranzo, E., Guardiani, C., Wei, W.: The application of reinforcement learning to NATM tunnel design. Underground Space. In Press (2022). https://doi.org/10.1016/j.undsp.2022.01. 005 44. Wang, X., Lu, H., Wei, X., Wei, G., Behbahani, S.A., Iseley, T.: Application of artificial neural network in tunnel engineering: a systematic review. IEEE Access 8(9125920), 119527– 119543 (2020). https://doi.org/10.1109/ACCESS.2020.3004995 45. Zhang, W., Li, Y., Wu, C., Li, H., Goh, A.T.C., Liu, H.: Prediction of lining response for twintunnel construction in anisotropic clays using machine learning techniques. Underg. Space 7(1), 122–133 (2022). https://doi.org/10.1016/j.undsp.2020.02.007 46. Kim, C.Y., Bae, G.J., Hong, S.W., Park, C.H., Moon, H.K., Shin, H.S.: Neural network based prediction of ground surface settlements due to tunnelling. Comput. Geotech. 28(6), 517–547 (2001). ISSN 0266-352X. https://doi.org/10.1016/S0266-352X(01)00011-8 47. Shi, J., Ortigao, J.A.R., Bai, J.: Modular neural networks for predicting settlements during tunneling. J. Geotech. Geoenviron. Eng. 124(5), 389–395 (1998). https://doi.org/10.1061/ (ASCE)1090-0241(1998)124:5(389) 48. Saadallah, A., Egorov, A., Cao, B.-T., Freitag, S., Morik, K., Meschke, G.: Active learning for accurate settlement prediction using numerical simulations in mechanized tunneling. In: Procedia CIRP, vol. 81, pp. 1052–1058 (2019). ISSN 2212–8271. https://doi.org/10.1016/ j.procir.2019.03.250. 52nd CIRP Conference on Manufacturing Systems (CMS), Ljubljana, Slovenia, 12–14 June (2019)

148

E. Soranzo et al.

49. Dindarloo, S.R., Siami-Irdemoosa, E.: Maximum surface settlement based classification of shallow tunnels in soft ground. Tunn. Underg. Space Technol. 49, 320–327 (2015). https:// doi.org/10.1016/j.tust.2015.04.021 50. Kim, D., Kwon, K., Pham, K., Ju-Young, O., Choi, H.: Surface settlement prediction for urban tunneling using machine learning algorithms with Bayesian optimization. Autom. Const. 140, 104331 (2022). https://doi.org/10.1016/j.autcon.2022.104331 51. Kannangara, K.K.P.M., Zhou, W., Ding, Z., Hong, Z.: Investigation of feature contribution to shield tunneling-induced settlement using Shapley additive explanations method. J. Rock Mech. Geotech. Eng. (2022). https://doi.org/10.1016/j.jrmge.2022.01.002 52. Zhang, P.: A novel feature selection method based on global sensitivity analysis with application in machine learning-based prediction model. Appl. Soft Comput. 85, 105859 (2019). https://doi.org/10.1016/j.asoc.2019.105859 53. Mohammad Reza Moghaddasi and Majid Noorian-Bidgoli: ICA-ANN, ANN and multiple regression models for prediction of surface settlement caused by tunneling. Tunn. Underg. Space Technol. 79, 197–209 (2018). https://doi.org/10.1016/j.tust.2018.04.016 54. Rosenblatt, F.: The perceptron: a probabilistic model for information storage and organization in the brain. Psychol. Rev. 65(6), 386–408 (1958). https://doi.org/10.1037/h0042519 55. Boser, B.E., Guyon, I.M., Vapnik, V.N.: A training algorithm for optimal margin classifiers. In: Proceedings of the 5th Annual Workshop on Computational Learning Theory, COLT ’92, New York, US, pp. 144–152 (1992). Association for Computing Machinery. ISBN 089791497X. https://doi.org/10.1145/130385.130401 56. Ho, T.K.: Random decision forests. In: Proceedings of 3rd International Conference on Document Analysis and Recognition, pp. 278–282 (1995). https://doi.org/10.1109/ICDAR.1995. 598994 57. Kursa, M.B., Rudnicki, W.R.: Feature selection with the boruta package. J. Stat. Soft. 36(11), 1–13 (2010). https://doi.org/10.18637/jss.v036.i11 58. Lundberg, S.M., Lee, S.-I.: A unified approach to interpreting model predictions. In: Proceedings of the 31st International Conference on Neural Information Processing Systems, NIPS’17, pp. 4768–4777, Red Hook, New York, US, Dec. 2017. Curran Associates Inc. ISBN 9781510860964 59. Atashpaz-Gargari, E., Lucas, C.: Imperialist competitive algorithm: an algorithm for optimization inspired by imperialistic competition. In: 2007 IEEE Congress on Evolutionary Computation, pp. 4661–4667 (2007). https://doi.org/10.1109/CEC.2007.4425083 60. Zhang, P., Li, H., Ha, Q.P., Yin, Z.-Y., Chen, R.-P.: Reinforcement learning based optimizer for improvement of predicting tunneling-induced ground responses. Adv. Eng. Inf. 45, 101097 (2020b). https://doi.org/10.1016/j.aei.2020.101097 61. Mnih, V., Kavukcuoglu, K., Silver, D., Rusu, A.A., Veness, J., Bellemare, M.G., Graves, A., Riedmiller, M., Fidjeland, A.K., Ostrovski, G., Petersen, S., Beattie, C., Sadik, A., Antonoglou, I., King, H., Kumaran, D., Wierstra, D., Legg, S., Hassabis, D.: Human-level control through deep reinforcement learning. Nature 518(7540), 529–533 (2015). https://doi. org/10.1038/nature14236 62. Breiman, L., Friedman, J.H., Olshen, R.A., Stone, C.J.: Classification and Regression Trees, Routledge, London, UK (1984). ISBN ISBN 9780412048418. https://doi.org/10.1201/ 9781315139470 63. Friedman, J.: Stochastic gradient boosting. Comput. Stat. Data Anal. 38, 367–378, Feb. 2002. https://doi.org/10.1016/S0167-9473(01)00065-2 64. Fix, E., Hodges, J.L.: Discriminatory analysis. Nonparametric discrimination: consistency properties. Int. Stat. Rev. 57(3), 238–247 (1989). https://doi.org/10.2307/1403797 65. Legendre, A.M.: Nouvelles méthodes pour la détermination des orbites des comètes. F. Didot, Paris, France (1805) 66. Friedman, J.H.: Multivariate adaptive regression splines. Ann. Stat. 19(1), 1–67 (1991). https:// doi.org/10.1214/aos/1176347963 67. Mann, H.B., Wald, A.: On the statistical treatment of linear stochastic difference equations. Econometrica 11(3/4), 173–220 (1943). https://doi.org/10.2307/1905674

Prediction of Tunnelling-Induced Settlement Trough by Artificial Neural Networks

149

68. Chen, T., Guestrin, C.: Xgboost: a scalable tree boosting system. In: Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 785–794, Aug. 2016. https://doi.org/10.1145/2939672.2939785 69. Yan, K., Dai, Y., Xu, M., Mo, Y.: Tunnel surface settlement forecasting with ensemble learning. Sustainability 12(1) (2020). https://doi.org/10.3390/su12010232 70. Tang, L., Na, S.H.: Comparison of machine learning methods for ground settlement prediction with different tunneling datasets. J. Rock Mech. Geotech. Eng. 13(6), 1274–1289 (2021). https://doi.org/10.1016/j.jrmge.2021.08.006 71. Schanz, T., Vermeer, P.A., Bonnier, P.G.: The Hardening Soil Model: Formulation and Verification, pp. 281–296. Routledge, London, UK (1999). https://doi.org/10.1201/ 9781315138206-27 72. Cheng, Z., Detournay, C.: Plastic hardening model I: Implementation in FLAC3D. In: Proceedings of the 4th Itasca Symposium on Applied Numerical Modeling, Mar. 2016 73. Van Rossum, G., Drake, F.L.: Python 3 Reference Manual. CreateSpace, Scotts Valley, CA (2009). 1441412697 74. Itasca Consulting Group: Fast Lagrangian Analysis of Continua in 3 Dimensions. Itasca Consulting Group, Minneapolis (2017) 75. Vlachopoulos, N., Diederichs, M.S.: Appropriate uses and practical limitations of 2D numerical analysis of tunnels and tunnel support response. Geotech. Geolog. Eng. 32(2), 469–488 (2014) 76. Do, N.-A., Dias, D., Oreste, P., Djeran-Maigre, I.: 2D tunnel numerical investigation: the influence of the simplified excavation method on tunnel behaviour. Geotech. Geolog. Eng. 32(1), 43–58 (2014). https://doi.org/10.1007/s10706-013-9690-y 77. Janin, J.P., Dias, D., Emeriault, F., Kastner, R., Le Bissonnais, H., Guilloux, A.: Numerical back-analysis of the southern toulon tunnel measurements: a comparison of 3D and 2D approaches. Eng. Geol. 195, 42–52 (2015). https://doi.org/10.1016/j.enggeo.2015.04.028 78. Jaky, J.: Pressure in silos. In: Proceedings of the 2nd International Conference on Soil Mechanics and Foundation Engineering, NIPS’17, pp. 103–107 (1948) 79. Meschke, G., Alsahly, A., Dang, T.S., Freitag, S., Gall, V.E., Nini´c, J., Stascheit, J., Zhou, M.: Numerische Simulation im Tunnelbau [Numerical Simulation in Tunnelling], pp. 171–232. Ernst & Sohn, Berlin, Germany (2014). In German 80. Pramthawee, P., Jongpradist, P., Kongkitkul, W.: Evaluation of hardening soil model on numerical simulation of behaviors of high rockfill dams. Songklanakarin J. Sci. Technol. 33(3), 325–334 (2011) 81. Bentley Systems. The Hardening Soil model (Isotropic hardening), pdf 6, pp. 67–81. Bentley, Exton, US (2022). https://bit.ly/3IpOKrG 82. Yanbin, F., He, S., Zhang, S., Yang, Y.: Parameter analysis on hardening soil model of soft soil for foundation pits based on shear rates in Shenzhen Bay, China. Adv. Mater. Sci. Eng. 2020, 7810918 (2020). https://doi.org/10.1155/2020/7810918 83. Obrzud, A., Truty, R.F.A.: The hardening soil model—a practical guidebook. Z Soil. PC 100701 report. Zace Services, Préverenges, Switzerland (2018). https://bit.ly/3RglyaA 84. Surarak, C., Likitlersuang, S., Wanatowski, D., Balasubramaniam, A., Oh, E., Guan, H.: Stiffness and strength parameters for hardening soil model of soft and stiff Bangkok clays. Soils Found. 52(4), 682–697 (2012). ISSN 0038-0806. https://doi.org/10.1016/j.sandf.2012. 07.009 85. Momeni, E., Maizir, H., Gofar, N., Nazir, R.: Comparative study on prediction of axial bearing capacity of driven piles in granular materials. Jurnal Teknologi (Sciences and Engineering) 61, 15–20 (2013). https://doi.org/10.11113/jt.v61.1777 86. Scharinger, F., Schweiger, H., Galavi, V.: FE-analysis of deep excavations in lacustrine clay with different constitutive models. In: Proceedings of the 16th International Conference on Soil Mechanics and Geotechnical Engineering, pp. 971–974. Millpress (2005) 87. Saleh, S., Yunus, N.Z.M., Ahmad, K., Said, K.N.M.: Numerical simulation with hardening soil model parameters of marine clay obtained from conventional tests. SN Appl. Sci. 3, 156 (2021). https://doi.org/10.1007/s42452-020-04115-w

150

E. Soranzo et al.

88. Klai, M., Bouassida, M., Seifeddine, T.: Numerical modelling of tunis soft clay. Geotech. Eng. J. SEAGS & AGSSEA 46, 12 (2015) 89. Sharifzadeh, M., Kolivand, F., Ghorbani, M., Yasrobi, S.: Design of sequential excavation method for large span urban tunnels in soft ground—niayesh tunnel. Tunn. Underg. Space Technol. 35, 178–188 (2013). https://doi.org/10.1016/j.tust.2013.01.002 90. Govindasamy, D., Ismail, M.A.M., Zaki, F., Abidin, M.H.Z.: Calibration of stiffness parameters for hardening soil model in residual soil from kenny hill formation. In: Inaugural SEGRM Symposium on: Geo-engineering in Tropical Region, June 2019. https://doi.org/10.7186/ bgsm67201915 91. European Committee for Standardization: Eurocode 1: Actions on structures—Part 1–1: General actions—Densities, self-weight, imposed loads for buildings. European Committee for Standardization, Bruxelles, Belgium (2002) 92. European Committee for Standardization: EN 1992-1-1: Eurocode 2: Design of concrete structures—Part 1–1: General rules and rules for buildings. European Committee for Standardization, Bruxells, Belgium (2004) 93. Linoff, G.S., Berry, M.J.A.: Data Mining Techniques. Wiley Publishing, Indianapolis, USA. (2011)978-1-118-08745-9 94. Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., Vanderplas, J., Passos, A., Cournapeau, D., Brucher, M., Perrot, M., Duchesnay, É.: Scikit-learn: machine learning in python. J. Mach. Learn. Res. 12(85), 2825–2830 (2011) 95. Lu, L.: Dying ReLU and initialization: theory and numerical examples. Commun. Comput. Phys. 28(5), 1671–1706 (2020). https://doi.org/10.4208/cicp.oa-2020-0165 96. Hendrycks, D., Gimpel, K.: Bridging nonlinearities and stochastic regularizers with Gaussian Error Linear Units (2016). arxiv:1606.08415. https://doi.org/10.48550/arXiv.1606.08415 97. Kingma, D.P., Ba, J.L.: Adam: a method for stochastic optimization. In: Proceedings of the International Conference on Learning Representations, San Diego, USA (2015) 98. Géron, A.: Hands-on Machine Learning with Scikit-Learn, Keras & TensorFlow: Concepts, Tools, and Techniques to Build Intelligent Systems. O’Reilly, Sebastopol, USA (2019) 99. Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. J. Mach. Learn. Res.—Proc. Track 9, 249–256, Jan. 2010 100. He, K., Zhang, X., Ren, S., Sun, J.: Delving deep into rectifiers: surpassing human-level performance on imagenet classification. In: 2015 IEEE International Conference on Computer Vision (ICCV), pp. 1026–1034 (2015). https://doi.org/10.1109/ICCV.2015.123 101. Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I.J., Harp, A., Irving, G., Isard, M., Jia, Y., Józefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mané, D., Monga, R., Moore, S., Murray, D.G., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P.A., Vanhoucke, V., Vasudevan, V., Viégas, F.B., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016). arxiv:abs/1603.04467. https://doi.org/10.48550/arXiv.1603.04467 102. Chollet, F. et al.: Keras. Published online (2015). https://github.com/fchollet/keras 103. Bengio, Y.: Practical Recommendations for Gradient-Based Training of Deep Architectures, pp. 437–478. Springer, Berlin, Heidelberg (2012). ISBN 978-3-642-35289-8. https://doi.org/ 10.1007/978-3-642-35289-8_26

Machine Learning Prediction of Bleeding of Bored Concrete Piles Based on Centrifuge Tests Enrico Soranzo, Carlotta Guardiani, Yunteng Wang, and Wei Wu

Abstract Bleeding in fresh concrete refers to the rise of excessive water as a result of sedimentation and consolidation of the aggregates. Various machine learning models are applied in this study to predict the amount of bleeding water based on centrifuge model tests on concrete bored piles. The soil type, its consolidation, concrete age and exposition class, pile geometry and amount of mixing water are the model test variables. Together with derived features at the prototype scale and geometric ratios, the degree of dewatering is predicted with three algorithms, namely the linear and decision tree regressions and artificial neural networks. The predictions match the observed data with a coefficient of determination up to 0.882. Given that the data are retrieved from centrifuge model tests, a reasonable agreement with field measurements is expected.

1 Introduction Bored piles are cylindrical concrete elements that are installed into the ground to transfer high structural loads to load bearing soil layers [1]. They are installed by casing and drilling out the soil, inserting the reinforcement cages, pouring the concrete and withdrawing the casing (Fig. 1). Bored piles have various applications in E. Soranzo (B) · C. Guardiani · Y. Wang · W. Wu University of Natural Resources and Life Sciences, Vienna, Austria e-mail: [email protected] C. Guardiani e-mail: [email protected] Y. Wang e-mail: [email protected] W. Wu e-mail: [email protected] Department of Civil Engineering and Natural Hazards, Institute of Geotechnical Engineering, Feistmantelstraße 4, 1180 Vienna, Austria © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 W. Wu and Y. Wang (eds.), Recent Geotechnical Research at BOKU, Springer Series in Geomechanics and Geoengineering, https://doi.org/10.1007/978-3-031-52159-1_10

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soft soil

soft soil

soft soil

soft soil

load bearing soil

load bearing soil

load bearing soil

load bearing soil

Drilling and casing insertion

Steel reinforcement insertion

Concrete pouring

Casing retraction

Fig. 1 Stages of the construction process of bored piles

geotechnical engineering, such as supporting deep excavations, especially close to existing buildings, as well as stabilising and retaining slopes, often in combination with ground anchors or soil nails [2]. Bleeding of fresh concrete is a segregation process that occurs during hardening and affects bored concrete piles. Due to the different unit weights of the concrete constituents, the water pressure can exceed the hydrostatic pressure and determine a vertical water flow through the concrete structure. Although this phenomenon has only a negligible effect on the compressive strength of concrete, it affects its durability [3]. Few studies investigated the factors affecting bleeding in bored concrete piles, such as the water to cement ratio and hardening temperature [4, 5], concrete composition [6, 7], presence of very soft soil layers (undrained shear strength .cu < 25 kPa, tip resistance .qc < 500 kPa) at shallow depth, the effect of the groundwater table with large hydraulic gradients and the layered soil stratigraphy [8]. The bleeding potential is higher for higher fresh concrete columns and a certain yield height for bleeding to occur has been observed [3, 9]. A relationship has been also established between the specific surface of the admixtures and the amount of bleeding water [10]. Bleeding is more pronounced in large diameter bored piles and can be minimised with the use of higher-grade concrete and by limiting the dosage of retarders [11, 12]. Concrete guidelines recommend maximising the reinforcement spacing, avoiding multiple reinforcement layers and controlling the concrete rheology and workability [13, 14]. Available methods explain the causes and mechanisms of concrete bleeding in bored piles [3, 8, 9, 11, 12, 15], propose some mitigation measures [10, 14], but fail to predict both the exact circumstances under which bleeding occurs and the amount of bleeding water. Although numerical simulations of bleeding in piles have been put forward [13] under the framework of computational fluid dynamics (CFD) [16] and by considering the concrete as a non-Newtonian frictional plastic-viscous fluid, they remain to this day without follow-up.

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The objective of this study is to predict the relative amount of bleeding water based on the results of centrifuge tests [17] with machine learning. The centrifuge model features are integrated with derived features to enhance the model performance [18]. Different feature sets are compared and critically evaluated based on their importance. Three algorithms with increasing complexity are selected, namely the linear regression, decision tree and artificial neural network.

2 Methodology 2.1 Data Acquisition and Exploratory Analysis The dataset underlying this study is collected with the centrifuge tests described in detail in the companion paper [17] and briefly summarised in the following. The beam geotechnical centrifuge has a diameter of 3.0 m, maximum acceleration of 200g and maximum model weight of 90 kg. The model piles with various pile lengths are simulated with two concentric polyethylene (PE) pipes. One pipe accommodates the concrete, the other the model soil. The inner pipe is gently extracted after the fresh concrete is poured. Five model soils are considered (Table 1). They are either pluviated (NC) or compacted with the Proctor hammer (OC). 122 tests are carried out. In 20 of these, the volume of the cylindrical annulus between the outer and inner pipes is filled with the model soils; the remaining 102 are carried out with fresh concrete only. The tests vary based on the geometry of the model piles, fresh concrete age and exposure class, radial acceleration, soil type and preparation. The volume of the bleeding water .Vf is measured after each test. The degree of dewatering . D is the output variable (label), defined according to Eq. 1. .

.

D=

Vf Vm

(1)

Vm is the total volume of the mixing water provided at the beginning of the test.

Table 1 Classification results for the model soils No. Symbol .dmax .C U (-) (-) (mm) (-) S1 S2 S3 S4 S5

cGr mGr Sa, fgr fmSa Si, sa

32 16 4 100 8

1.3 1.3 2.5 2.6 11.1

.C C

.d10

.d50

(-)

(mm)

(mm)

1.0 1.0 1.3 1.1 1.7

19.0 9.30 0.63 0.123 0.004

20.4 14.8 1.4 0.028 0.035

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The exploratory data analysis and machine learning models outlined in the following are developed with the Python programming language [19]. The descriptive statistics of the continuous features are listed in Table 2. The model pile diameter . Dmod and length . L mod are comprised in the intervals 70–260 mm and 19–46.8 cm, respectively. The concrete age .Ca is on average 3.7 h and the test duration .T is generally one hour. Although a few tests last up to six hours, it is shown in the companion paper that the pore water pressure stabilises after about one hour [17]. Radial accelerations . N up to 120 g are considered with a mean loading velocity .v of 618 m/h. The mean volume of the mixing water .Vw,mod is 1133 ml or approximately 1/6 of the mean model pile volume .Vmod . The observed degree of dewatering . D ranges from 4 to 60%. In addition to these data, the model soil types (. St ) and preparation methods (. Sp ), and the concrete exposure classes B9, B10, B11 (.Ce ), defined according to the concrete standards [20, 21], are considered as one-hot encoded features. The correlation matrix of the centrifuge model test data is shown in Fig. 2. It can be seen that the degree of dewatering . D is primarily negatively correlated to the lateral area of the model piles . Alat,mod and positively correlated to the radial acceleration . N .

2.2 Derived Data The geometric features . Dmod , . L mod , . Alat,mod and .Vmod refer to the model piles. To generalise our predictions to the field, however, we introduced the geometric features . Dprot , . L prot , . A lat,prot and . Vprot at the prototype scale based on the scaling laws for centrifuge testing [22]. . Dprot = N · Dmod (2) L prot = N · L mod

(3)

Alat,prot = N 2 · Alat,mod

(4)

Vprot = N 3 · Vmod

(5)

.

.

.

We also introduce the length to diameter ratio . RL and the water to pile volume . RW to enhance the model performance [18]. .

.

L prot N · L mod L mod = = Dprot N · Dprot Dmod

(6)

Vw,prot N 3 · Vw,mod Vw,mod = 3 = Vw,prot N · Vw,mod Vw,mod

(7)

RL =

RW =

Values Mean St.Dev. Min. 25% 50% 75% 100%

122 152.9 71.0 70 100 125 200 260

Model pile diameter . Dmod (mm)

122 34.97 9.76 19 20.33 39.95 41.2 46.8

Model pile length . L mod (cm)

122 1527.1 555.2 631.5 1188.33 1551.9 1675.8 2940.5

Model pile lateral area . Alat,mod (cm.2 )

Table 2 Descriptive statistics of the dataset

122 6383.2 4191.0 1404.7 2970.8 5215.5 10087.7 14702.7

Model pile volume . Vmod (cm.3 ) (h)

(h) 122 1.31 1.12 1 1 1 1 6

.T

.C a

122 3.68 1.11 2.92 3.08 3.34 3.83 8.08

Test duration

Concrete age

122 41.43 32.84 0 0 50 50 120

(g)

.N

122 617.9 463.74 75 150 750 750 1500

(m/h)

.v

Radial Loading acceleration velocity

122 1133.2 716.83 217 581.3 846 1866 2525

Model mixing water . Vw,mod (ml)

122 17.6 12.13 4 8 14.5 24 60

Degree of dewatering .D (%)

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Fig. 2 Correlation matrix of the centrifuge model test data

2.3 Feature Sets Some of the features introduced refer exclusively to the model tests (. Dmod , . L mod , Alat,mod , .Vmod , .T , . N , .v, .Vw,mod ), some others to the prototype piles (. Dprot , . L prot , . A lat,prot , . Vprot , . Vw,prot ), whereas the ratios . RL and . RW apply to both. The features describe the pile geometry (. Dmod , . L mod , . Alat,mod , .Vmod , . Dprot , . L prot , . Alat,prot , .Vprot , . Vw,prot , . RL ) and water volumes (. Vw,mod , . Vw,prot ), the soil type (. St ), its consolidation (. Sp ), the concrete age (.Ca ) and exposition class (.Ce ). To assess the effect of the different groups of features, we consider six different feature sets (Table 3). F0 is the baseline set that considers the features of the model tests. F1 considers also the features derived in Sect. 2.2. Its purpose is to assess the effect of the derived parameters on the predictive capability of the model. By considering only prototype features, F2 can be used to make predictions in the field. F3 considers only the soil and geometric features, thus weighing the importance of the concrete parameters. F4 considers only the concrete and the geometry, thus assessing the effect of the soil parameters. Finally, F5 comprises only the soil and concrete features and appraises the impact of the pile geometry on the results. .

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Table 3 Feature sets of the predictive models considered No. Description Variables considered F0

Only model features

F1

All features

. St , . Sp , . Dmod , . L mod , . Alat,mod , . Vmod , .C e , .C a , . T , . N , .v, . Vw,mod , . RL , . RW

. St , . Sp , . Dmod , . L mod , . Alat,mod , . Vmod , .C e , .C a , . T , . N , .v, . Dprot , . L prot , . Alat,prot , . Vprot , . Vw,mod , . Vw,prot , . RL , . RW

F2

Only prototype features

. St , . Sp , .C a , .C e , . Dprot , . L prot , . Alat,prot , . Vprot ,

F3

Only soil and geometry

F4

Only concrete and geometry

F5

Only soil and concrete

. Vw,mod , . Vw,prot , . RL , . RW . St , . Sp , . Dprot , . L prot , . Alat,prot , . Vprot , . Vw,mod , . Vw,prot , . RL , . RW .C a , .C e , . Dprot , . L prot , . Alat,prot , . Vprot , . Vw,mod , . Vw,prot , . RL , . RW . St , . Sp , .C e , .C a

2.4 Feature Selection Alongside testing different feature sets, we also evaluate the feature importance with the Boruta package [23]. Originally developed for R [24], Boruta creates several randomly shuffled shadow attributes to establish the baseline performance of a model. To determine whether a feature is only randomly correlated with the label or carries significant information, Boruta runs the hypothesis test that the difference between the mean value of each feature and of the label is equal to zero. Features that fail to reject this hypothesis are discarded. As uninformative features are iteratively removed, the importance of the remaining features improves.

2.5 Prediction Algorithms Three predictive algorithms are used in this study, namely the linear [25] and decision tree [26] regressions, and the Artificial Neural Network (ANN) [27]. The objective of the linear regression is to find a hyperplane that best fits the data points according to Eq. 9. n ∑ .y ˆ = β0 + βi X i (8) i=1

The coefficients in Eq. 9 are found by minimising the residual sum of squares RSS between the predicted and observed values, calculated as follows RSS =

.

n ∑ ( yˆi − yi )2 i=1

(9)

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Fig. 3 Visual representation of the working principle of a linear regression in a one feature input space

Fig. 4 Visual representation of the working principle of a decision tree regression with a depth of three as a flowchart

where . yi and . yˆi are the observed and predicted labels, respectively (Fig. 3). Decision tree regression segments the domain of the features. In each segment, predictions are made based on their mean value. The name “decision tree” comes from the set of splitting rules used which resembles a tree. The optimal splitting rules are found by minimising the mean squared error (MSE) defined in Sect. 2.6. The decision tree regression generally outperforms the linear regression, but underperforms the ANN. It is, however, a straightforward model that delivers a clear decision path and can be potentially be applied in the practice, especially if the depth of the resulting tree is small (Fig. 4). The underlying idea of the ANN is to artificially mimic biological intelligence [27]. In practice, the label . y is approximated by a function of the features . X with a neural network. Neurons represent the input features . X , the output label . y and one or more interposed hidden layers. The output function of each neuron . j is expressed as in Eq. 10. ( n ) ∑ . yˆj = f wi, j · xi + b j (10) i=1

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Fig. 5 Visual representation of the working principle of an artificial neural network with two input features, two hidden layers (with two and three neurons, respectively) and one output label

where .wi, j are the weights and .b j the bias term. Various activation functions . f (xi ) can be selected for the neurons. Let .z j = wi, j · xi + b j , the REctified Linear Unit (RELU) activation function is used in this study according to Eq. 11. a j = f (z j ) = max(0, z j )

.

(11)

The weights and biases of the network are updated by minimising the MSE (backpropagation). The MSE is minimised using Adam, a method for stochastic optimisation [28], with a step size (“learning rate”) of 0.001 (Fig. 5).

2.6 Workflow To ensure the replicability of the results the pseudorandom number generator is initialised with a constant seed. 70% of the data (85 data points) is used to train the models and 30% of the data (37 data points) is assigned to validation. This splitting task is performed by the function train_test_split of the Scikit-learn library [29]. Different units of measurement determine scale differences among the features that do not affect the linear or decision tree regression, but can hamper ANN performance, due to numerical instability. Hence, for the ANN, the features . X are scaled with the StandardScaler of the Scikit-learn library [29] according to Eq. 12. .

X' =

X −μ σ

μ and .σ are the mean values and the standard deviations of the features.

.

(12)

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The linear regression is a parametric method that determines the values of the coefficients .βi of Eq. 9 by minimising RSS, whereas the decision tree regression is hyperparametric. Hyperparameters are not learned by the algorithm, but set by the user. For the decision tree regression, we consider two hyperparameters, namely the depth of the decision trees and the splitter. The depth of the decision tree is the length of the longest path from the root to a leaf. We consider the minimum and maximum depths of two and seven, respectively. The splitter is one of the two strategies used to choose the split at each node, namely “best split” (the algorithm considers all the features and chooses the best split at each node) and “best random split” (only a random subset of features is considered). The ANN is both parametric (the weights .wi, j and biases .b j are determined by the algorithm) and hyperparametric, the number of neurons in each layer (the ANN architecture) and the batch size (the chunk of data fed to the ANN at each substep) being set by the user. The ANN has two hidden layers and the number of neurons is optimised among 20, 30 and 40 neurons per layer. The batch size is optimised among the values of 1, 2 and 4. The number of epochs, i.e. the process of passing the entire dataset forward and backward through the ANN, is set to 250. The hyperparameters are tuned with a five-fold cross-validation grid search [29] that searches the best combination within the defined subset of hyperparameters. This procedure is computationally expensive, but it is effective when the hyperparameters have discrete values. For a given set of hyperparameters, the model is trained on four folds and tested on the remaining one. At each iteration, the mean cross-validated score . R 2 is computed and the hyperparameters providing the highest . R 2 are selected. Once the tuning of the hyperparameters is completed, the performance metrics are computed for the training and validation sets. Three performance metrics are considered, namely the coefficient of determination (. R 2 ), the Root Mean Squared Error (RMSE) and the Mean Absolute Error (MAE). They are computed as shown in Eqs. 13–15. ∑n ˆ i )2 i=1 (yi − y 2 . R = 1 − ∑n (13) ¯ )2 i=1 (yi − y ⎡ | n |∑ (yi − yˆi )2 .RMSE = √ n i=1 ∑n MAE =

.

where . y¯ is the mean observed label.

i=1

| | | yi − yˆi | n

(14)

(15)

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3 Results A workstation equipped with an Intel Core i7-4810MQ CPU at 2.80 Ghz with 4 cores and 8 logical processors is used for the calculations. The hyperparameter tuning, training and test complete in a matter of seconds for the linear and decision tree regressions, and in a few minutes for the ANN. Figures 6, 7, 8, 9, 10 and 11 show the prediction results of the degree of dewatering ˆ on the training (a) and test (b) sets for the linear regressions with the feature sets F0 .D to F5. The abscissae represent the observed values, the ordinate the predictions. The nearer the scatter points are to the bisector, the better is the prediction. The dashed lines describe .±30% deviations from the bisector. The prediction with the feature set F0 (model features only) is very accurate (Fig. 6). Nonetheless, it can be further improved by considering the derived features of the F1 set (Fig. 7). The prediction with the prototype features F2 only (Fig. 8) is more accurate than that with the model

(a) Training dataset

(b) Test dataset

Fig. 6 Prediction results of the degree of dewatering . D (%) with the linear regression and feature set F0

(a) Training dataset

(b) Test dataset

Fig. 7 Prediction results of the degree of dewatering . D (%) with the linear regression and feature set F1

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(a) Training dataset

(b) Test dataset

Fig. 8 Prediction results of the degree of dewatering . D (%) with the linear regression and feature set F2

(a) Training dataset

(b) Test dataset

Fig. 9 Prediction results of the degree of dewatering . D (%) with the linear regression and feature set F3

(a) Training dataset

(b) Test dataset

Fig. 10 Prediction results of the degree of dewatering . D (%) with the linear regression and feature set F4

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(b) Test dataset

Fig. 11 Prediction results of the degree of dewatering . D (%) with the linear regression and feature set F5

features (F0). By neglecting the concrete features .Ce and .Ca (F3), the prediction performance sensibly decreases on the training set, but slightly increases on the test set (Fig. 9). As shown in the following (Fig. 12), the concrete age .Ca is an important feature. By ignoring the soil features . St and . Sp the feature set F4 has an intermediate performance on the training set, although the performance on the test set is still high (Fig. 10). Finally, by discarding the geometric features in F5 (. Dprot , . L prot , . Alat,prot , . Vprot , . Vw,prot , . RL and . RW ), the performance drops dramatically both on the training set and on the test set. The coefficients .βi of the linear regression for the feature sets F2 to F5 are summarised by Eqs. 16 in which the label . D is given in percentage, the features S1 to S5 of the soil types, the consolidation state (. N C, . OC) and the concrete exposure class are binary variables (e.g. . S1 = 1 if the pile is bored in cGr, S1 = 0 in another soil; NC = 1 if the soil is normally consolidated; B9 = 1 if the concrete exposure class is B9), the concrete age .Ca is measured in hours and the geometric features . Dprot , . L prot , . A lat,prot , . Vprot , . Vw,prot are in metres. The ratios . RL and . RW are dimensionless. .

D = 28.19% + 4.05 · S1 + 0.64 · S2 + 6.95 · S3 + 3.18 · S4 + 16.10 · S5 +20.36 · NC + 10.56 · OC − 4.00 · B9 + 5.73 · B10 − 1.73 · B11 − 0.84 · Ca −0.22 · Dprot + 1.45 · L prot − 0.03 · Alat,prot −0.01 · Vprot + 0.08 · Vw,prot − 3.08·RL − 99.30 · RW (16a)

.D

= 22.43% + 1.30 · S1 − 1.55 · S2 + 7.72 · S3 + 4.86 · S4 + 14.958 · S5

+ 18.59 · NC + 8.7 · OC − 0.90 · Dprot + 1.09 · L prot − 0.004 · Alat,prot − 0.01 · Vprot + 0.072 · Vw,prot − 3.444 · RL − 70.17 · RW (16b)

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Fig. 12 Feature importance results obtained with the Boruta package on the feature set F2. Features ranking and indication on whether each feature has be retained, tentatively retained or discarded

.

D = 25.86% − 0.26 · B9 + 3.74 · B10 − 3.48 · B11 − 1.16 · Ca +0.45 · Dprot +3.31 · L prot − 0.09 · Alat,prot − 0.01 · Vprot + 0.13 · Vw,prot −2.915 · RL − 185.62 · RW (16c)

.

D = 22.02% + 7.11 · S1 + 0.80 · S2 + 5.31 · S3 − 3.05 · S4 + 24.02 · S5 + 21.59 · NC + 12.60 · OC − 4.54 · B9 + 6.87 · B10 − 2.33 · B11 − 1.46 · Ca (16d)

From this point, we only consider feature set F2 since it returns the best prediction with the prototype features. Figure 12 shows the feature importance results obtained with the Boruta package on the feature set F2 (all the prototype features). The features are ranked from the most (.Ca ) to the least (S2 and S4) important. The results indicate that the features .Ca , . L prot , . RL , . RW and NC must be retained, . Alat,prot and .Vw,prot can be tentatively retained and the remaining features are to be discarded. Most of the features to be retained are geometric features such as the pile length . L prot , the length to diameter ratio . RL and the ratio of mixing water to pile volume . RW . Hence, the derived features . RL and . RW play a pivotal role. Although the geometric features are

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(b) Test dataset

Fig. 13 Prediction results of the degree of dewatering . D (%) with the decision tree regression and feature set F2

(a) Training dataset

(b) Test dataset

Fig. 14 Prediction results of the degree of dewatering . D (%) with the artificial neural network and feature set F2

overrepresented among the important features, all three groups of features are listed, such as the concrete age (.Ca ) and soil consolidation (NC). Figure 13 shows the results obtained with the decision tree regression and the feature set F2. The prediction is very accurate, outperforming the linear regression both on the training on testing set. The optimised depth of the tree is six and the random best splitter is selected. Figure 14 shows the results obtained with the ANN and the feature set F2. The prediction outperforms both the linear regression and the decision tree regression on the training and testing sets. Figure 15 shows the convergence of the ANN. The MSE on the training set decreases monotonically, whereas the MSE on the test set decreases rapidly for the first 10 epochs, then fluctuates. The optimised ANN has 30 neurons in the two hidden layers and a batch size of two. Figure 16 compares the results of the three algorithms on the feature set F2. The performance metrics are summarised in Table 4.

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Fig. 15 Error minimisation for the train and test sets of the artificial neural network with the feature set F2

(a) Training dataset

(b) Test dataset

Fig. 16 Comparison of the prediction results with the linear regression, decision tree regression and artificial neural network and feature set F2

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Table 4 Performance metrics of algorithms and feature sets considered Feature set Training dataset Test dataset R.2 RMSE MAE R.2 RMSE (-) (%) (%) (-) (%) Linear regression 0.847 F0 F1 0.903 F2 0.831 F3 0.796 F4 0.622 F5 0.603 Decision tree 0.882 F2 Artificial neural network F2 0.931

MAE (%)

4.454 3.542 4.679 5.139 6.994 7.162

3.410 2.624 3.809 4.152 5.496 5.945

0.821 0.915 0.849 0.854 0.818 0.417

5.698 3.927 5.234 5.152 5.746 10.289

4.327 3.304 4.034 4.489 5.136 7.620

3.900

2.637

0.875

4.770

3.975

2.987

2.035

0.882

4.633

3.656

4 Discussion and Conclusions This study derives predictive models for the bleeding water of bored concrete piles from centrifuge model tests. The model test data are post-processed by deriving four new features by making use of the centrifuge scaling laws and by introducing two geometric ratios. The predictive capability of six feature sets is assessed, namely the centrifuge model (F0), centrifuge model and prototype pile (F1), prototype pile (F2), soil and prototype pile geometry (F3), concrete and prototype pile geometry (F4) and soil and concrete features (F5). The prediction with the feature set F2 shows a good agreement with the observed data. As confirmed by the feature importance, features belonging to all groups (soil, geometry and concrete) must be retained, primarily the concrete age .Ca , prototype pile length . L prot , geometric ratios . RL and . RW as well as soil consolidation NC. Although a good predictive performance is reached with the linear regression (. R 2 = 0.849), further improvements are achieved with the decision tree regression (. R 2 = 0.875) and the artificial neural network (. R 2 = 0.882). Given that the data are retrieved from centrifuge model tests, an adequately simplified method of analysing full-scale geotechnical structures [22], we can expect a reasonable agreement of our predictive models with field measurements. Acknowledgements Financial support for this research is provided by the Otto Pregl Foundation for Geotechnical Fundamental Research.

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References 1. Bauer Spezialtiefbau. Bauer bored piles. https://www.bauertech.co.uk/export/shared/ documents/pdf/bst/print/905_015_2_BAUER-Bored-Piles_en.pdf, Nov 2018. Accessed: 2022-04-12 2. Keller UK. Bored piles and pile walls. https://www.keller.co.uk/sites/keller-uk/files/2019-09/ E_Brochure_210BoredPilesUK.pdf, Sept 2019. Accessed: 2022-04-12 3. Loh, C.-K., Tan, T.-S., Yong, T.-S., Wee, T.-H.: An experimental study on bleeding and channelling of cement paste and mortar. Adv. Cem. Res. 10(1), 1–16 (1998). https://doi.org/10. 1680/adcr.1998.10.1.1 4. Josserand, L., Coussy, O., de Larrard, F.: Bleeding of concrete as an ageing consolidation process. Cem. Concr. Res. 36(9), 1603–1608 (2006). https://doi.org/10.1016/j.cemconres.2004. 10.006 5. Topçu, ˙IB., Elgün, V.B.: Influence of concrete properties on bleeding and evaporation. Cem. Concr. Res. 34(2), 275–281 (2004). https://doi.org/10.1016/j.cemconres.2003.07.004 6. Niederegger, C., Bruschek, C., Koppi, M., Schröcker, H.P., Wagner, D.: Verbesserung von frisch und festbetoneigenschaften durch minimierung der haufwerk-sporosität von bindemitteln mittels approximation der fuller- kurve durch mischen von kornfraktionen [improvement of fresh and hardened concrete properties by minimizing the porosity of binders by approximation of the fuller curve by mixing grain fractions]. Beton 5, 220–222 (2007). https://doi.org/10.1016/ j.cemconres.2004.10.006 7. Blask, O.: Zur Rheologie von polymermodifizierten Bindemittelleimen und Mörtelsystemen [On the Rheology of Polimer-modified Binder Glues and Mortar Systems]. PhD thesis, Universität-Gesamthochschule Siegen (2002) 8. Beute, J.C.: Problems with cast-in-situ concrete piles: a study on the possible causes of excessive bleeding. Master’s thesis, TU Delft (2020) 9. Micheals, A.S., Bolger, J.C.: Settling rates and sediments volumes of flocculated kaolin suspensions. Ind. Eng. Chem. Fundam. 1(1), 24–33 (1962). https://doi.org/10.1021/i160001a004 10. Tazky, M., Hela, R.: High-performance concretes intended for deep foundations of constructions. Civil Eng. Archit. 8, 46–54 (2020). https://doi.org/10.13189/cea.2020.080202 11. Kog, Y.C.: Bleeding and channelling in bored piles. Mag. Concr. Res. 61(10), 759–765 (2009). https://doi.org/10.1680/macr.2008.61.10.759 12. Kog, Y.C.: Integrity problem of large-diameter bored piles. J. Geotech. Geoenviron. Eng. 135(2), 237–245 (2009). https://doi.org/10.1061/(ASCE)1090-0241(2009)135:2(237) 13. Larisch, M., Qin, Z., Alehossein, H.: Performance control tests and numerical simulations for concrete in deep foundations. Concr. Aust. 39(4), 26–34 (2013) 14. EFFC/DFI Concrete Task Group. Guide to Tremie Concrete for Deep Foundations. EFFC/DFI Concrete Task Group, Faversham, United Kingdom (2018) 15. Thasnanipan, N., Boonyarak, T., Aye, Z.Z.: Concrete bleeding in bored pile construction in Bangkok soil. In: Proceedings of PILE 2017, Bali, Indonesia, Sept 2017 16. Harlow, F.H.: Fluid dynamics in group t-3 los alamos national laboratory: (la-ur-03-3852). J. Comput. Phys. 195(2), 414–433 (2004). ISSN 0021-9991. https://doi.org/10.1016/j.jcp.2003. 09.031 17. Soranzo, E., Brückl, J., Wu, W.: Centrifuge model test on bleeding of fresh concrete in bored piles. Springer Series in Geomechanics and Geoengineering, page accepted (2022) 18. Linoff, G.S., Berry, M.J.A.: Data Mining Techniques. Wiley Publishing, Indianapolis, U.S.A. (2011). ISBN 978-1-118-08745-9 19. Van Rossum, G., Drake, F.L.: Python 3 Reference Manual. CreateSpace, Scotts Valley, CA (2009). ISBN 1441412697 20. Austrian Standards. ÖNORM B 4710-1: Concrete – Specification, performance, production, use and conformity – Part 1: Rules for the implementation of ÖNORM EN 206 for normal and heavy concrete. Austrian Standards, Vienna, Austria (2018)

Machine Learning Prediction of Bleeding of Bored Concrete Piles …

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21. European Committee for Standardization. Concrete - Part 1: Specification, perfor-mance, production and conformity, EN 206-1. European Committee for Standardization, Bruxelles, Belgium (2000) 22. Kutter, B.L.: Dynamic centrifuge modeling of geotechnical structures, vol. 1336. Transportation Research Record, Washington, U.S.A (1992) 23. Kursa, M.B., Rudnicki, W.R.: Feature selection with the Boruta package. J. Stat. Softw. 36(11), 1–13 (2010). https://doi.org/10.18637/jss.v036.i11 24. R Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria (2021). https://www.R-project.org/ 25. Legendre, A.M.: Nouvelles méthodes pour la détermination des orbites des comètes. F. Didot, Paris, France (1805) 26. Breiman, L., Friedman, J.H., Olshen, R.A., Stone, C.J.: Classification and Regression Trees. Routledge, London, United Kingdom (1984). ISBN 9780412048418. https://doi.org/10.1201/ 9781315139470 27. Rosenblatt, F.: The perceptron: a probabilistic model for information storage and organization in the brain. Psychol. Rev. 65(6), 386–408 (1958) 28. Kingma, D.P., Ba, J.L.: Adam: a method for stochastic optimization. In: 3rd International Conference on Learning Representations, ICLR 2015 - Conference Track Proceedings, San Diego, USA (2015) 29. Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., Vanderplas, J., Passos, A., Cournapeau, D., Brucher, M., Perrot, M., Duchesnay, É.: Scikit-learn: machine learning in python. J. Mach. Learn. Res. 12(85), 2825–2830 (2011)

Stability Evaluation of Huangtupo Riverside Slump I Landslide Based on Soil-Water Interaction Xuexue Su, Carlotta Guardiani, Huiming Tang, Pengju An, and Wei Wu

Abstract The impoundment of Three Gorges Reservoir in China has adversely influenced the stability of the landslide by periodic water level fluctuation, which has been a worldwide issue. The hydrologic action has induced changes in the seepage field and deterioration in sliding zone soil. Taking the Huangtupo landslide as a case, this study experientially investigates the strength weakening laws of the sliding zone soil, and a numerical simulation by Geostudio is also performed to evaluate the landslide stability. It shows that: (1) the deterioration in sliding zone soil caused by soil–water interaction can be characterized with an exponential function model. (2) the seepage field, displacement, and stability are closely related to the scheduling of reservoir water level. In the rapid decline period, the factor of safety is also the lowest, and the displacement increases significantly. (3) combined with the weakening model of shear strength, the landslide behaves as a step-like deformation and has the potential to move longer distances in the long-term hydrologic changes.

X. Su · H. Tang (B) Faculty of Engineering, China University of Geosciences, Wuhan 430074, China e-mail: [email protected] X. Su e-mail: [email protected] C. Guardiani · W. Wu Institute of Geotechnical Engineering, University of Natural Resources and Life Sciences, Feistmantelstrasse 4, 1180 Vienna, Austria e-mail: [email protected] W. Wu e-mail: [email protected] P. An Institute of Rock Mechanics, Ningbo University, Ningbo, China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 W. Wu and Y. Wang (eds.), Recent Geotechnical Research at BOKU, Springer Series in Geomechanics and Geoengineering, https://doi.org/10.1007/978-3-031-52159-1_11

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1 Introduction After the construction of the largest water conservancy project, the Three Gorges Dam has created the well-known Three Gorges Reservoir area (TGRA), where nearly 500 km of geohazard-prone reservoir bank, and more than 5300 landslides in the reservoir area have been founded [1]. And the reservoir water level fluctuates periodically between 145 and 175m, causing a hydro-fluctuation belt and a suite of environmental problems, such as hydrological alterations, soil erosion, etc., among which, the frequent occurrence of geohazards are the foremost issues. It documented that more than 2000 huge landslides were triggered or reactivated by the influences of water level fluctuation, two extrema cases are (1) the Qianjiangping landslide, with 24 death toll and 346 damaged houses [2], (2) the Huangtupo landslide, causing the relocation in Badong Town [3]. Clearly, the wide-distributed reservoir landslides have given rise to heavy casualties and irreparable harm to the natural environment and social economy, so it has been an imperative problem [4]. The hydro-fluctuation belt, with a water level difference of 30 m, would periodically change the groundwater seepage field when the water level rises and drops [5]. And the sliding zone, which plays an important role in controlling the movement rate, the movement pattern, and the deformation characteristics of the landslides, will certainly experience alternating wetting–drying processes. However, the repeated wetting–drying cycles cause irreversible deterioration on rock/soil properties, such as the microstructure, mineral composition, and mechanical strength. Therefore, two alterations, i.e., the scheduled seepage field and deteriorated soil properties, will further cause a less stable state of the sliding zone, consequently affecting the landslide stability. Generally, with the influence of hydro alterations, the sliding zone soil is definitely impacted by soil–water effects, including mechanical, physical, and chemical interactions [6]. The mechanical interactions, including the hydrostatic and hydrodynamic seepage pressures, are mostly associated with the water level fluctuation, as there is a dynamic process in groundwater level, seepage field, and pore water pressure, and the landslide stability is dynamically changed [7]. The physical interaction is primarily relevant to the softening and saturation by water immersion and the moisture change, inducing the weakening of shear strength by the cyclic soil– water interaction [8]. While the chemical interactions refer to a long-term process controlled by the pore water chemistry, it would also gradually affect the soil properties in the sliding zone, thus contributing a potential major threat to strength deterioration [9]. Obviously, the soil–water interaction has complex and comprehensive impacts, which have been less focused on. This paper focuses on the complex Huangtupo landslide, evaluating the deformation and stability of Riverside Slump I based on the strength weakening laws from the triaxial test of sliding zone soil. It provides a reference to the evaluation on the long-term stability of reservoir landslides under periodical water level fluctuations.

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2 Materials 2.1 Geological Setting The Huangtupo landslide, located in Badong County of China, is known as the volumetrically largest reservoir landslide in the Three Gorges reservoir area. It is developed in the strata of the Middle Triassic Badong Formation (T2 b2 and T2 b3 ), and mainly comprises mudstone, pelitic siltstone, argillaceous limestone, and some weak layers. The overall slope is shaped gently in the middle area but steeply in the upper and underpart riverside areas, thus forming a complex landslide of multiple slumps. There are four slumps for this huge landslide, i.e., Riverside Slump I, Riverside Slump II, the Substation landslide, and the Garden Spot landslide (Fig. 1). The Riverside Slump I is in the foreside and submerged in the river at an elevation of 50–90 m.a.s.l., above which is Garden Spot landslide. Divided by Sandaogou, the Riverside Slump II is adjacent to the Riverside Slump I and lies in the foreside, above which is the Substation landslide, thus the elevation of the crown reaches to 600 m.a.s.l. After the impoundment of Three Gorges Reservoir, the water level fluctuation has significantly influenced the stability of the riverside slumps, despite some stabilization work applied, this huge landslide is still active with a creep rate of 15 mm/a [10]. Meanwhile, the geochemical characteristics of the Huangtupo landslide, including the chemical water environments and soil mineral composition, would certainly influence the evolutionary process of the sliding zone and movement rate of the landslide [11]. Therefore, it delivers the necessity to evaluate the landslide stability considering the soil–water interaction.

2.2 Experimental Works The sliding zone soil was collected from the 5# exploratory tunnel in the Riverside Slump I, see Fig. 1. The undisturbed soil from the sliding zone is mainly silty clay with some gravels. The silty clay is mostly sandy and silty in plastic to semi-solid state and the gravels are rounded to subangular. The basic physical properties are displayed in Fig. 2, determined according to Chinese standard GB/T 20,123–2019 [12]. In this work, the triaxial specimens are 39.1 mm in diameter and 80 mm in height. For a natural state, the density and water content were controlled to 1.95 g/cm3 and 12% respectively. There are 28 specimens prepared before the test, four specimens are the normal group with no wetting–drying test, and the other 24 specimens are divided into six groups for 1, 2, 3, 5, 7, and 10 wetting–drying cycles. And the submerged solution is groundwater, which is collected from the Huangtupo landslide.

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Fig. 1 Huangtupo landslide and A-A’ profile in Riverside Slump I (1 limestone; 2 argillaceous limestone; 3 dense soil and rock debris; 4 sliding mass of Garden spot landslide; 5 loose soil and rock debris; 6 cataclastie; 7 sliding zones with possible slip surfaces; 8 rock-soil interfaces; 9 main sliding surfaces.)

Fig. 2 The basic physical properties and grain size distribution of the sliding zone soil

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Table 1 Basic physical parameters of the sliding zone soil Physical properties

Value

Physical properties

Natural water content (ω/%)

11.96

Density (g/cm3 )

Degree of saturation (Sr/%)

70.92

Void ratio (e) Plasticity index (I p )

0.49

Atterberg limit (%)

11.32

Value Natural (ρ)

2.18

Dry (ρ d )

1.95

Liquid limit (ωl )

26

Plastic limit (ωp )

14.68

The consolidated undrained test (CU) was employed here. The triaxial testing procedures follow the Chinese standard GB/T 20,123–2019 (China, 2019). Specifically, (1) saturate the specimens for more than 12 h. (2) take out the specimens carefully from the saturator and put them into a rubber membrane. (3) consolidate the samples for 24 h with the confining stress of 100, 200, 300, and 400 kPa, respectively. (4) perform the shear process under the shear rate of 0.8 mm/min. (5) obtain the peak shear strength under different confining pressures. Note that the particular stress at the strain of 15% might be treated as the shear strength if no peak stress had been experienced (Table 1).

3 Methodology 3.1 Weakening Model of the Sliding Zone Soil The regression analysis method was performed to calculate the cohesion and friction angle of sliding zone soil undergoing wetting–drying cycles. Based on the principle of effective stress and the Mohr–Coulomb strength theory, the effective parameters, cohesion (c’) and friction angle (ϕ’), were calculated by Eqs. 1 and 2. The Principle of Effective Stress is, σ = σ + u

(1)

And, the Mohr–Coulomb Strength Theory is, ⎧ 1 1 ⎪ σ = (σ1 + σ3 ) + (σ1 − σ3 ) cos 2α ⎪ ⎪ ⎪ 2 2 ⎪ ⎨ 1 τ = (σ1 − σ3 ) sin 2α ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎩ α = 45◦ + 1 ϕ 2

(2)

Subsequently, the cohesion and friction are obtained on the least square method and displayed in Fig. 3. The strength parameters show a decreasing tendency with increasing wetting–drying cycles, c tends to decrease with an average rate of 2.81 kPa

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Fig. 3 The weaking laws of cohesion and friction angle

per cycle, ϕ appears to averagely decline 0.38° per cycle. This decreasing tendency is greatly related to the degradation by soil–water interactions in the wetting–drying alterations. Based on the existing experimental results and theoretical analysis, the weakening of strength parameters under the wetting–drying cycles can be well characterized by an exponential function: f (n) = a + be−nd

(3)

where f (n) signifies the shear strength parameters; n denotes the number of wetting– drying cycles; a signifies the residual weakening coefficient; b demotes the weakening proportionality coefficient; d is the weakening law coefficient. The weakening function is performed to fit the variation of cohesion and friction angle, i.e., c(n) and ϕ(n), with the increasing wetting–drying cycles, And the results of the fitting are also displayed in Fig. 3. And the deterioration coefficient (η), the ratio of shear strength parameter and the initial one, was also calculated.

3.2 Numerical Simulation Based on the geographical analysis of Riverside slump I, a two-dimensional finite element is established to simulate the seepage field and sigma field of the landslide, considering the impact of reservoir water level fluctuations. According to the Three Gorges Reservoir actual operating situation, the reservoir water level fluctuates between 145 and 175 m. To study the transient seepage field with a fluctuating water level, the actual reservoir water level is taken as the foundation of the calculation, which is as follows (Fig. 4):

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Fig. 4 The scheduling curve of reservoir water level for one year

⎧ 145 + 0.5t, t ∈ [0, 60] ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 175, t ∈ (60, 120] H (t) = 175 − (t − 120)/9, t ∈ (120, 210] ⎪ ⎪ ⎪ 165 − (t − 210)/3, t ∈ (210, 270] ⎪ ⎪ ⎪ ⎩ 145, t ∈ (270, 365]

(4)

As seen in Fig. 5, the established model, with a height of 357 m and a distance of 943m, comprises quadrilateral cells and trilateral transitional cells. It is also partitioned into 4283 grid cells and 4096 nodes. Then, the boundary condition is determined. In the SEEP field, the foreside of the landslide suffered from significant impacts by the fluctuating water level, so the right boundary is determined as a changing water total head, which is scheduled with the reservoir water level. The left boundary, also the groundwater level at the trailing water, is determined as a constant water head boundary with an elevation of 270 m. The lower boundary is confining boundary, which is regarded as waterproof. In the SIGMA field, the left and right boundary is subjected to horizontal restraints, while the lower boundary is subjected to horizontal and vertical restraints. Meanwhile, Table 2 shows the simulated physical and mechanical parameters of rock mass and sliding zone soil, which are obtained based on the Investigation Report on the Huangtupo landslide. Thus, the transient seepage field and displacement field were calculated based on the strength reduction method and limit equilibrium method.

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Fig. 5 Computation model of the Huangtupo landslide Riverside slump I

Table 2 The calculation parameters of the finite element Parameters

Bedrock Sliding mass Sliding zone

Elastic modulus (E/MPa)

Poisson’s ratio (ν)

Friction angle (ϕ/°)

Cohesion (c/kPa)

Density (ρ/kN/ m3 )

Saturated permeability (m/d)

3674

0.26

44.2

380

26

0.1

Shallow

672

0.31

25

80

21

2.35

Deep

2178

0.29

34

190

23

1.88

Shallow

28.6

0.34

18.8

25

19.9

0.76

Deep

36.8

0.35

22.2

62.9

21.3

0.24

4 Results 4.1 The Transient Seepage Field Based on the SEEP/W module in the Geostudio 2021.3, the seepage field of the landslide is simulated. The initial condition is the steady seepage field when the water level is 145 m. The pore water pressure and saturation line in different stages during the water level fluctuation is calculated and displayed in Fig. 6. (1) Storage period (0–60 days): as seen in Fig. 6a, the saturation line rises in the foreside of the sliding mass and declines in the back part, displaying a concave shape when the water level rises from 145 to 175 m. It shows the reservoir water level tends to supplement the groundwater level because there is a water head difference between external reservoir water and internal groundwater, and the concave shape also indicates a certain lag response in the seepage field

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Fig. 6 Saturation line at different period a 145–175 m b 175 m c 175–145 m d 145 m

that reservoir water flows to the landslide mass. And the pore water pressure is increasing with the rise of water level. (2) High water level (60–120 days): as seen in Fig. 6b, although the water level maintains stable at 175 m, the saturation line continues to rise and appears to be gentle gradually. The areas that are affected by the saturation line extend to the lower sliding mass, and the saturated areas become larger with the rising saturation lines. It is also caused by the lag effect between reservoir water and groundwater; thus, the groundwater level would still adjust with time to reduce the water head differences, and gradually reach a smooth state. The pore water pressure tends to be steady in the largest value in this period. (3) Downwards period (120–270 days): the decline of water is of different rates, the slow decline period (120–180 days) and the rapid decline period (180–270 days). From Fig. 6c, the saturation line drops in the front and rear edges of the sliding mass, displaying a convex shape. It indicates that the supplement is from groundwater to the reservoir water, thus the saturation line in the sliding mass appears to decline, and the decline rate of groundwater in the rapid decline period is greater than that in the slow period. And for the pore water pressure, it decreases with decline of groundwater level. (4) Low water level (270–365 days): the saturation lines still decline due to the lag effect of the seepage field, when in 290 d and 310 d, the drop of saturation lines is still significant, while in 310–365 d, the leading edge of lines is essentially same, the back edges also decline gradually. It illustrates that the groundwater still supplements the reservoir water, and the back edge lay behind the leading edge. And the pore water pressure appears to be unchanged.

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4.2 The Displacement Field Based on the SIGMA/W module in the Geostudio, the stress field of the landslide is simulated. The initial condition is the steady stress field of gravity, which is assigned as the beginning of the simulation. The displacement and the maximum shear strain of the landslide in different stages during the water level fluctuation is calculated and displayed in Fig. 7. (1) Storage period (0–60 days): as seen in Fig. 7a, the maximum X-displacement occurred at the front of the landslide, and the significant deformation extend from the leading edge to the middle part of the shallow mass. With the rise of the reservoir water, the leading edge of the landslide body is subjected to increasing hydrostatic and dynamic water pressure, resulting in the deformation of the leading edge to the trailing edge of the landslide body. Meanwhile, the shear stress at the shear outlet in the shallow landslide zone gradually increases, while only slight deformation occurs in the deep landslide zone. (2) High water level (60–120 days): in the trailing of landslide, the groundwater level gradually rises, accompanied by the formation of a stable groundwater seepage field. Consequently, the rising groundwater level weakens the strength of geotechnical materials and decreases effective stress, the maximum shear strain at the leading edge slightly increases. During the stable operation period, the landslide experiences relatively small overall deformation, with the maximum horizontal displacement occurring at the front edge of the shallow landslide, seen as Fig. 7b. (3) Downwards period (120–270 days): see Fig. 7c, in the slow decline period (120– 210 days), the horizontal displacement still increases in the slow decline period,

Fig. 7 X-displacement at different times a 145–175 m b 175 m c 175–145 m d 145 m

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the deformation mainly concentrated at the leading edge of the shallow landslide, and gradually extend to the middle and trailing of the landslide, showing notable traction movement. And the shear strain area of the shallow landslide zone gradually expands, and the shear strain of the deep sliding zone begins to extend. In the rapid decline period (210–270 days), the shear strain continues to increase, accompanied with the significant increase in the horizontal displacement. (4) Low water level (270–365 days): as shown in Fig. 7d, the water level stays at 145m in this period, the groundwater level decreased and stabilized, and there was no significant change in the shear strain. Meanwhile, the horizontal displacement of the leading edge and middle part slightly decreased or remained unchanged, while the displacement in the trailing continues to increase. Thus, the deformation process of the landslide is highly related to the schedule of water level fluctuation, and an irreversible progression in stress change and displacement increase. Generally, the shear strain primarily occurs in the leading edge and cause the increase of horizontal displacement, then the shear strain area tends to extend and induces the deformation in the middle and trailing part, thus the displacement becoming continually increase.

4.3 The Variation of Stability Additionally, by the computation of SIGMA/W, the displacement of the landslide can be seen in Fig. 8a. The monitor points M1, M2, and M3, see Fig. 5, are in the upper, middle, and lower part of the landslide, which are also corresponded to the site of the GPS monitoring of G02, G07, and G11. The displacement calculated by the simulation can well fit the monitor data, implying the reliability of the simulation results. Need to add that some exceptions of M3 and M1 exist, the simulated displacement appears to decrease in the storage period, however, the monitor data still increase when the water level rises, which may be attributed to the tensile cracks developed at the front of the landslide, providing some deformation space which can hardly be measured by GPS. Based on the simulation, the factor of safety, F s , of the shallow sliding zone and deep sliding zone are also computed by the SLOPE/W. It appears from Fig. 8b that F s is variable with the changing water level: in the storage period, F s tends to increase with the increasing reservoir water level; in the high-water level, F s starts to decrease significantly; in the downward period, it still declines, however, the decline rates are different. F s of a rapid decline period is much higher than that of a slow decline period. In the low water level, F s increases again. It indicates that when the water level maintains at a low water level or rises, the stability would increase, while in a high-water level or decline period, the stability of landslide decreases, and the rapid downward rates would pose to greater drop to the factor of safety, so the most unsafe state is in the rapid decline period [13]. And it indicated from the F s that the deep

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Fig. 8 The variation trend comparison diagram of the factor of safety

sliding zone is safer than the shallow sliding zone, which can also be evidenced by their displacement and maximum shear stress in Fig. 7.

4.4 The Long-Term Evaluation of Huangtupo Landslide According to the deterioration coefficient and weakening model of the sliding zone soil, the landslide stability and displacements of 10 years are computed by the SLOPE/W and SIGMA/W of the Geostudio software. From Fig. 9, it shows that the displacement tends to increase with step-like behavior when the landslide experiences periodic hydraulic action, including the seepage field change caused by the water level change and the strength deterioration induced by the soil–water interaction. And the displacement of different parts appears to increase with a differed rate, averagely 21.47, 9.68, and 5.17 mm/a for the lower, middle, and upper parts of Huangtupo landslide. Thus, the creep rate of the landslide is 5–30 mm/a, which is corresponded to the monitoring data. For the factor of safety, it decreases with the hydrological year for deep sliding zone and shallow sliding zone. The deep sliding zone is quite safe with a F s approaching 1.1, while the shallow sliding zone is less safe, it may move a longer distance in the future. So, some monitoring equipment, including GPS and InSAR, and also some grouted rubble revetment is required to be constructed in the lower part of Huangtupo landslide.

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Fig. 9 The long-term displacement a and stability b of Huangtupo landslide

5 Concluding Remarks (1) Huangtupo landslide is a large and complex landslide undergoing periodic reservoir water level fluctuation, causing the strength deterioration by the soil–water coupling. An exponential function model is performed to fit the variation of c and ϕ with good-of-fitness. It also indicates the weakening effect of soil–water interaction. (2) When the reservoir water level changes, the seepage field of the landslide tends to change accordingly, thus the saturation line shows a concave shape in the storage period and a convex shape when the water level declines. It implies the water head differences between groundwater level and reservoir water level impact the direction of water flow, and the lag effect is also influenced a lot. (3) The displacement of landslide is variable in different periods of water level fluctuation, it appears to increase in high water level and the downwards period, whereas in the rapid decline period, the displacement dramatically increased. This trend is also shown in the variation of Fs, meaning that when the reservoir water declines rapidly, the landslide is in a most unsafe state. (4) Combined with the weakening model of strength deterioration, the displacement and stability analysis of the Huangtupo landslide have been obtained, the landslide behaves as a step-like deformation. Although this large landslide is in a stable state, it remains active, especially for the shallow sliding zone, with a creep rate of 5-30 mm/a. So, some engineering prevention measures, and monitoring are still required to be conducted in Huangtupo landslide for long-term stability. Acknowledgements This research was supported by Otto Pregl Foundation of Fundamental Geotechnical Research in Vienna, the National Major Scientific Instruments and Equipment Development Projects of China (No. 41827808).

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References 1. Tang, H.M., Janusz, W., C. Hsein J.: Geohazards in the three Gorges Reservoir Area, China– Lessons learned from decades of research. Eng. Geol. 261,105267 (2019). https://doi.org/10. 1016/j.enggeo.2019.105267 2. Wang, F., Zhang, Y., Huo, Z., Peng, X., Wang, S., Yamasaki, S.: Mechanism for the rapid motion of the Qianjiangping landslide during reactivation by the first impoundment of the Three Gorges Dam reservoir, China. Landslides 5, 379–386 (2008). https://doi.org/10.1007/ s10346-008-0130-7 3. Gong, W., Juang, C.H., Wasowski, J.: Geohazards and human settlements: Lessons learned from multiple relocation events in Badong, China—Engineering geologist’s perspective. Eng. Geol. 10, 106051 (2021). https://doi.org/10.1016/j.enggeo.2021.106051 4. Wang, L.Q., Xiao, T., Liu, S.L., Zhang, W.G., Yang, B.B., Chen, L.C.: Quantification of model uncertainty and variability for landslide displacement prediction based on Monte Carlo simulation. Gondwana Res. (2023). https://doi.org/10.1016/j.gr.2023.03.006 5. Zou, Z., Luo, T., Zhang, S., Duan, H., Li, S., Wang, Jinge, Deng, Y., Wang, J.: A novel method to evaluate the timedependent stability of reservoir landslides: exemplified by Outang landslide in the three Gorges Reservoir. Landslides (2023). https://doi.org/10.1007/s10346-023-02056-0 6. Wu, Y., Miao, F., Li, L., Xie, Y., Chang, B.: Time-varying reliability analysis of Huangtupo Riverside No.2 Landslide in the Three Gorges Reservoir based on water-soil coupling. Eng. Geol. 10 226, 267–276 (2017). https://doi.org/10.1016/j.enggeo.2017.06.016 7. Wu, Q., Liu, Y., Tang, H., Kang, J., Wang, L., Li, C., Wang, D., Liu, Z.: Experimental study of the influence of wetting and drying cycles on the strength of intact rock samples from a red stratum in the Three Gorges Reservoir area. Eng. Geol. 314, 107013 (2023). https://doi.org/10. 1016/j.enggeo.2023.107013 8. Miao, F., Wu, Y., Li, L., Tang, H., Xiong, F.: Weakening laws of slip zone soils during wetting– drying cycles based on fractal theory: a case study in the Three Gorges Reservoir (China). Acta Geotech. (2019). https://doi.org/10.1007/s11440-019-00894-8 9. Wang, J., Wang, S., Su, A., Xiang, W., Xiong, C., Blum, P.: Simulating landslide-induced tsunamis in the Yangtze River at the Three Gorges in China. Acta Geotech. 16, 2487–2503 (2021). https://doi.org/10.1007/s11440-020-01131-3 10. Wang, S., Wu, W., Wang, J., Yin, Z., Cui, D., Xiang, W.: Residual-state creep of clastic soil in a reactivated slow-moving landslide in the Three Gorges Reservoir Region, China. Landslides 8(15), 2413–2422 (2018). https://doi.org/10.1007/s10346-018-1043-8 11. Miao, H., Wang, G.: Effects of clay content on the shear behaviors of sliding zone soil originating from muddy interlayers in the Three Gorges Reservoir, China. Eng. Geol. 294, 106380 (2021). https://doi.org/10.1016/j.enggeo.2021.106380 12. China, M.O.C.O.T.P.R.O.: Standard for soil test Methods (GB/T 50123–2019). China Planning Press, Beijing (2019) 13. Hu, X., Tang, H., Li, C., Sun, R.: Stability of Huangtupo riverside slumping mass II# under water level fluctuation of Three Gorges Reservoir. J. Earth Sci. 23, 326–334 (2012). https:// doi.org/10.1007/s12583-012-0259-0

Effect of Different Factors on Dynamic Shear Modulus of Compacted Loess Haiman Wang, Wankui Ni, and Kangze Yuan

Abstract Loess dynamic parameters are extensively affected by earthquakes. Accurate determination of loess dynamic parameters plays a significance to solve the geotechnical engineering problems related to the earthquake. The dynamic shear modulus (DSM) is one of the most relevant dynamic parameters. The GDS high system of dynamic triaxial test imported from the UK was adopted to test the DSM of compacted loess with different water contents, dry densities, confining pressures, consolidation stress ratios, and loading frequencies. Experimental results presented the DSM decreased with the increase of water content, but increased with the increase of dry density, confining pressure, consolidation stress ratio, and loading frequency. The initial DSM had a nonlinear relationship with water content and loading frequency, and a linear relationship with dry density, confining pressure, and consolidation stress ratio. Water content, dry density, confining pressure, consolidation stress ratio, and loading frequency were independent variables, linear regression analysis was carried out on the dependent variable (initial DSM), and the influenced degree of each factor on the DSM and the regression equation were obtained. The innovative contribution of this study is that provides a basis for seismic response and seismic stability analysis of the loess field.

H. Wang · W. Ni (B) College of Geological Engineering and Geomatics, Chang’an University, No.126 Yanta Road, Xi’an, Shaanxi, P.R. China e-mail: [email protected] H. Wang e-mail: [email protected] K. Yuan Department of Civil and Environmental Engineering, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milan, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 W. Wu and Y. Wang (eds.), Recent Geotechnical Research at BOKU, Springer Series in Geomechanics and Geoengineering, https://doi.org/10.1007/978-3-031-52159-1_12

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1 Introduction Loess is special sediment formed during the Quaternary Period under arid and semiarid climate conditions [1]. Loess deposits cover 10% of the world’s continents, including Asia, Africa, Europe, and northern France [2–4]. In northwestern China, loess is extremely important and covers 6.4 .× 104.km.2 , primarily in the Shanxi, Shaanxi, Gansu, and Ningxia regions. An aeolian deposit with an open and metastable structure has been described as the most distinctive characteristic of loess. Therefore, loess foundations are very vulnerable to seismic-hazard, such as ground subsidence or landslide during an earthquake [5, 6]. The M8 earthquake of 1654 in Tianshi south, Gansu, triggered a large amount of loess landslides. In 1718, the M7.5 earthquake in Tongwei County, Gansu Province caused over 300 massive loess landslides, resulting in the destruction of a large number of buildings. In 1920, the Haiyuan earthquake in Ningxia caused 467 large-scale loess landslides, covering an area of 4000–5000 .km.2 . The M8 Gansu Gulang earthquake in 1927 also had a great impact on buildings and people’s safety [7, 8]. Dynamic loads caused by earthquakes are the main cause of these engineering problems in loess foundations. DSM is one of the key elements in the analysis of foundation deformation under dynamic loads. Significant research aimed at understanding the influence on DSM under different factors has been carried out. Wan et al. [9] found that DSM increases with increasing confining pressure by conducting experimental studies on soft soils with free vibrating columns. Feng et al. [10] added the consolidation ratio factor on the DSM depending on confining pressure, and found that the aeolian soils DSM increases with the confining pressure and consolidation ratio. The different confining pressure durations were separated into two phases by SAS et al. [11], who explored the variation of DSM. Kong et al. [12] measured the DSM through a series of tests and demonstrated that the DSM values were influenced by the pore fluid. Bedr et al. [13] adopted a similar test method and combined with the normalized model to discuss the influence of different factors on DSM. For non-cohesive soil, Dyka et al. [14] conducted that the particle size distribution had a major impact on the DSM. Through the above literatures that the current research on DSM is mainly focused on a soft foundation. However, few studies have focused on loess foundations due to the characteristics of loess such as high porosity and collapsibility. Therefore, it is necessary to conduct research regarding loess fields [15]. Deng et al. [16] performed dynamic triaxial tests on loess under different confining pressures and analyzed the variation rule of DSM. Liu et al. [17] investigated cyclic triaxial tests, proving that shear modulus decreases with the increase of the porosity ratio, but it had little impact on the shear modulus at the higher shear strain level. The consolidated-drained triaxial tests were performed by Wang et al. [18], who exhibited that the dynamic characteristics of saturated remolded loess were closely related to the consolidation confining pressure and the initial stress state. Although scholars have tried their best to supplement the studies on DSM in the loess region. However, there are few explanations on the DSM of loess in different initial states, and the influence of different initial states on the degree of DSM is even less.

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Based on the above considerations, the GDS high system of dynamic triaxial test imported from the UK is adopted in this paper to test the DSM of compacted loess with different water content, dry density, confining pressure, consolidation stress ratio, and loading frequency. Meanwhile, Linear regression analysis is used to analyze the initial DSM under different factors to explore the influence of different factors on the DSM. The most important contribution of this study is that it has a full understanding of the DSM and provides some guidance for future analysis of landslides.

2 Materials and Methods 2.1 Original Loess Samples The samples investigated were loess from a foundation pit of the Loess Plateau in Yan’an City , China. Loess samples were taken at a depth of 7.0–7.5 m and were defined as Malan loess [17]. ASTM standard [19] was used for testing the basic physical properties of the original loess (Table 1). Mineralogical compositions were quantitatively calculated from the XRD pattern of the original loess (Fig. 1).

Table 1 Physical properties of loess Sample measurements In situ density (g/cm.3 ) Natural water content (%) Specific gravity Plastic limit .wP (%) Liquid limit .wL (%) Optimal water content w(%) Maximum dry density (g/cm.3 ) Quartz (%) Feldspar (%) Calcite (%) Chlorite (%) Kaolinite (%) Illite (%)

Value 1.35–1.42 13.0 2.71 16.1 28.9 14.1 1.74 45.2 21.0 15.5 8.0 5.8 4.5

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Fig. 1 XRD pattern of the original loess sample

2.2 Test Instrument The soil tests of dynamic property are performed on the GDS high system of dynamic triaxial test imported from the UK, this test system can make axial load of the arbitrary waveform applied directly to the specimen through the up and down motion of loading piston. Instruments include: brake unit, triaxial pressure chamber, confining pressure controller, back pressure controller, high-speed data acquisition, and control card (IEEE card). The three-axis pressure chamber structure has an axial brake and is connected with the three-axis pressure chamber base. The base contains all hydraulic joints and pore pressure and confining pressure sensors leading to the pressure chamber, and the soil sample is installed in the triaxial pressure chamber. The confining pressure controller has a chamber volume of 200,000 .mm.3 , and a tube at the top is connected to a triaxial pressure chamber, which is pressurized through piston movement, and the maximum pressure is up to 3 MPa. The maximum volume of the backpressure controller is 200,000 .mm.3 and the maximum pressure is up to 2 MPa. It is connected with the drain valve of the triaxial pressure chamber, which can not only apply back pressure but also serve as the drainage channel of the experimental device. The volume variation of the soil sample can be measured by the volume variation of the backpressure controller during consolidation.

2.3 Preparation and Installation of Loess Samples 2.3.1

Sample Preparation

After collection, the test samples were prepared as follows. Firstly, all aggregates of the original loess were crushed. The obtained material was passed through a 2 mm sieve and then oven-dried at 105.◦ C for 8 h. Afterward, a quantity of deionized water

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was gradually added to the sample using a spray bottle until the target water content is reached. Then tightly sealed with plastic film and placed in a humidity chamber for approximately 2 days at room temperature to homogenize the humidity. To ensure the uniformity of the samples, static compaction was performed on the samples three times. Finally, the triaxial samples with the size of .Φ70 mm × h140 mm under different water content and dry density were obtained.

2.3.2

Sample Installation

The rubber membrane was covered in the mold bearing cylinder, and the rubber membrane was closely attached to the inner wall of the bearing barrel with the sucking ball. Put the sample into the bearing cylinder, removed the sucking ball, at this time the sample and rubber membrane are close. Then, the top and bottom of the sample were placed on the permeable stones, and the filter paper should be placed between the permeable stone and the sample. Before installing the sample, the loading base shall be moved to the initial position, and then the rubber membrane was fixed to the sample cap and the base with the help of the O ring. After the confining pressure chamber was bolted to the base, opened the oil inlet valve and oil pump of the confining pressure chamber. When oil spilled out of the confining pressure chamber, first closed the oil inlet valve of the confining pressure chamber, then closed the oil pump, and finally closed the confining pressure chamber. Moving the loading base upward so that the contact pressure between the sample cap and the loading head was 0.0 kN.

2.4 Dynamic Triaxial Experiment For the dynamic triaxial tests on compacted loess, a stress-controlled loading method was employed, with a sine wave created by a servo system selected for step-by-step loading. The influence of different initial conditions on DSM of the compacted loess was explored. The water content of the compacted loess was controlled as 6, 10, 12, 14 and 18%, the dry densities were 1.6, 1.7 and 1.8 g/.cm.3 , the confining pressures were 100, 200, 300 and 400 kPa, consolidation stress ratios were 1.0, 1.45, 1.56, 1.69, 1.83, 2.0, 2.2, and the loading frequencies were 0.1, 0.5, 1, 5 and 10 Hz respectively. Before the dynamic load was applied, the sample was consolidated under uniform pressure for 1 h. After consolidation was completed, the drain valve was closed and the dynamic load on the specimen was increased step by step without drainage, from 5 to 20 steps, and dynamic triaxial tests were carried out with constant dynamic load levels in each layer.

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2.5 Statistical Analysis Linear regression analysis is often used to find out the extent and direction of the influence of one or more variables on another, and to understand how the dependent variable (y) changes relative to changes in the independent variable (.xi ) [20]. A multiple linear regression analysis models as follows: y = β0 + β1 x1 + · · · + βk xk + ξ

.

(1)

where, .x1 · · · xk are non-random variables; y is the random dependent variable; β · · · βk are the regression coefficients; .ξ is the residual, i.e., the prediction error. If the experiment makes n observations of y and x, and gets n groups of observations .yi , .x1i · · · xki (.i = 1, 2, .. . . , n), it satisfies the following relationship: . 0

y = β0 + β1 x1i + · · · + βk xki + ξi

. i

(2)

The matrix can be expressed as: ⎡ ⎤ 1 y1 ⎢1 ⎢ y2 ⎥ ⎢ ⎢ ⎥ .y = ⎢ . ⎥ , X = ⎢ . ⎣ .. ⎣ .. ⎦ yn 1 ⎡

x11 x12 .. . x1n

⎡ ⎤ ⎡ ⎤ ⎤ β0 ξ1 xk1 ⎢ β1 ⎥ ⎢ ξ2 ⎥ xk2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ .. ⎥ , β = ⎢ .. ⎥ , ξ = ⎢ .. ⎥ ⎣ . ⎦ ⎣ . ⎦ . ⎦ · · · xkn βk ξn ··· ··· .. .

(3)

Therefore, the model can be written as: y = Xβ + ξ

.

(4)

If X is full rank, the least square estimation of the linear regression model parameters is: ˆ = (X X T )−1 X T y .β (5) Thus, the estimated value of y is: .

yˆ = X βˆ

(6)

ˆ therefore, the least square estimate of The residual vector is.e = y − yˆ = y − X β; the random error variance .σˆ 2 is .

σˆ 2 =

eT e n−k −1

(7)

After the estimated values of the regression model parameters are obtained, a significance test of the regression equation and regression coefficient is needed. (1) The significance test of regression equation:

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/ SSR k .F = ∼ F(k, n − k − 1) SSE/(n − k − 1) where .SSR =

n ∑

(ˆyi − y¯ )2 is the sum of regression squares; .SSE =

i=1

(8) n ∑

(yi − yˆ i )2 is

i=1

the sum of residuals squares. For a given significance level p, the rejection domain of the test is .F > Fp (k, n − k − 1). (2) The significance test of regression coefficient: / SSEj k ∼ F(k, n − k − 1) .F = SSE/(n − k − 1)

(9)

where.SSEj is the sum of residuals squares without.xj . For a given significance level p, the rejection domain of the test is .F > Fp (1, n − k − 1). It can also use test statistics: t =

. j

βˆj √ ∼ t(n − k − 1) σˆ cjj

(10)

where, for a given significance level p, the rejection domain of the test is .|tj | > t(p/2) (n − k − 1). Linear regression analysis enables checking the difference between predicted values, values from real soil experiments, and observation results [21]. In addition, the correlation and suitability of the regression model results and the direct experimental results were assessed. In this study, correlation and regression analyses were performed using Statistical Product and Service Solutions (SPSS) software as statistical analysis, and the initial DSM was estimated from the different factors.

3 Result and Discussion 3.1 Loess DSM Statistics Under Different Factors 3.1.1

Water Content

Figure 2 shows the variation between DSM and dynamic shear strain of compacted loess with different water content, which presents that the DSM decreases with the increase of dynamic shear strain. When the water content is 6%, the DSM decreases most obviously. When the water content is greater than 6%, the dynamic elastic modulus decreasing rate of compacted loess is basically the same. The DSM decreases with the increase of water content. There are two main reasons for this. One is that water smoothes the connections between particles. With the increase of water content, the internal friction between particles decreases. On the other hand, the bound water on the surface of loess particles thickens with the increase of water content, which will lead to the decrease of cohesion between particles. Interestingly, when the water content increases from 10 to 12%, the DSM decreases obviously, indi-

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Fig. 2 Water content on DSM curve

cating that the water content has an obvious effect on the DSM of the compacted loess. Moreover, with the increase of water content, there exists a threshold value that causes the cohesion between loess particles to suddenly decrease.

3.1.2

Dry Density

Figure 3 indicates the variation between DSM and dynamic shear strain of compacted loess with different dry densities. The compacted loess DSM decreases with the increase of dynamic shear strain under different dry densities. The DSM increases with the increase of dry density. With the increase of dry density, the loess structure becomes denser, the cementation between loess particles becomes stronger, and the resistance to shear deformation becomes stronger. The larger the dynamic shear

Fig. 3 Dry density on DSM curve

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Fig. 4 Confining pressure on DSM curve

stress required for the loess to produce the same dynamic shear strain, the larger the corresponding DSM.

3.1.3

Confining Pressure

The DSM of compacted loess under different confining pressures decreases with the increase of dynamic shear strain (Fig. 4). When the dynamic shear strain is the same, the DSM increases with the increase of confining pressure. With the increase of the confining pressure on the compacted loess, the cementation of loess particles becomes stronger and the loess structure becomes more denser, which makes the movement among loess particles more difficult. Meanwhile, the larger confining pressure enhances the cohesion and the ability of loess to resist deformation, which leads to the increase of DSM.

3.1.4

Consolidation Stress Ratio

The different consolidation stress ratio reflects the different loess initial stress state before the application of dynamic load. Figure 5 demonstrates that the DSM of compacted loess under different consolidation stress ratios decreases with the increase of dynamic shear strain. When the dynamic shear strain is less than 0.1%, the DSM increases with the increase of the consolidation stress ratio. However, when the dynamic shear strain is greater than 0.1%, the DSM does not increase significantly. The reason is that the initial principal stress of the sample increases with the increase of the consolidation stress ratio. With the increase of the principal stress, the compaction coefficient of the loess increases significantly, which leads to the increase of the compact DSM between the particles. However, with the continuous action of dynamic shear stress, the dynamic shear strain gradually increases, the loess structure

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Fig. 5 Consolidation stress ratio on DSM curve

becomes looser, the cohesive force between loess particles decreases, and the effect of consolidation stress ratio on the DSM of loess is obviously weakened.

3.1.5

Load Frequency

The DSM of compacted loess increases with the increase of loading frequency (Fig. 6). With the increase of loading frequency, the number of vibrations of loess under dynamic stress increases in unit time, and the stress cannot be completely transferred and distributed through adjacent loess particles, which shows that soil strength increases, leading to the increase of DSM. When the loading frequency is 0.5.∼10 Hz, the DSM decreases slowly with the decrease of loading frequency.

Fig. 6 Load frequency on DSM curve

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When the loading frequency is less than 0.5 Hz, the DSM decreases obviously with the decrease of loading frequency.

3.1.6

Relationship Between Dynamic Shear Stress and Dynamic Shear Strain

By studying DSM (.G d ) and dynamic shear strain (.γd ) under different factors. It can be found that .G d and .lgγd basically show a linear reduction law, so Eq. (11) is used to fit the experimental data, and the fitting results are shown in Table 2. .

G d = algγd + b

(11)

Where, .G d is the DSM (MPa), .γd is the dynamic shear strain (%), and a and b are fitting parameters.

3.2 Influence of Different Factors on Initial DSM of Compacted Loess The initial DSM represents the level of dynamic shear strength of loess under the initial state, and the initial DSM of loess under different factors,.G 0 , can be calculated (Fig. 7). According to the change law of the initial DSM, the fitting equations and parameters were shown in Table 3. The dry density, confining pressure, and consolidation stress ratio have a linear relationship with the initial dynamic shear strength, while the water content and loading frequency have a nonlinear relationship with the initial dynamic shear strength (Fig. 7). With the increase of water content, the friction between loess particles decreases, which leads to the decrease of the initial DSM. Initial DSM increases with the increase of dry density. The larger the dry density, the stronger the cementation between particles, the denser the loess structure, and the greater the friction between particles. Initial DSM increases with the increase of the confining pressure and consolidation stress ratio. The increase of the confining pressure and consolidation stress ratio reflects the increase of the vertical stress, which makes the movement between soil particles more difficult. The confining pressure enhances the resistance of loess to deformation, which is also one of the reasons for the increase of initial DSM. With the increase of loading frequency, the time of stress acting on loess becomes shorter, and the dynamic shear strain of loess structure under the same dynamic stress becomes smaller, which leads to the increase of initial DSM. Therefore, the water content is negatively correlated with the initial DSM. Meanwhile, the dry density, confining pressure, consolidation stress ratio, and load frequency is positively correlated with the initial DSM.

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Table 2 .G d and .γd formula parameters Factors Factor levels a .−16.3

Water content (%) 6 10 12 14 18 Dry density 1.6 (g/.cm.3 ) 1.7 1.8 Confining 100 pressure (kPa) 200 300 400 Consolidation 1.0 stress ratio 1.45 1.56 1.69 1.83 2.0 2.2 Load frequency 0.1 (Hz) 0.5 1 5 10

.−11.1 .−11.1 .−11.5 .−10.0 .−11.1 .−9.5 .−6.6 .−7.9 .−11.1 .−16.8 .−17.6 .−8.9 .−11.1 .−10.6 .−11.1 .−17.5 .−19.0 .−23.9 .−6.7 .−12.1 .−11.1 .−11.1 .−10.0

.R

60.3 59.8 33.1 19.3 12.2 59.8

0.9578 0.9713 0.9608 0.9649 0.9840 0.9713

76.4 92.2 45.0

0.9462 0.9144 0.9275

59.8 60.8 71.0 42.5

0.9713 0.9596 0.9601 0.9686

40.5 45.2 59.8 46.6 45.9 48.9 48.2

0.9623 0.9885 0.9713 0.9888 0.9837 0.9882 0.9057

46.5 59.8 72.0 89.0

0.9031 0.9713 0.9807 0.9761

Table 3 Fitting equations under different factors Factors Fitting equation Water content, .w (%) Dry density, .ρd (.g/cm.3 ) Confining pressure, .σ3 (kPa) Consolidation stress ratio, .kc Load frequency, f (Hz)

= = 70.76 + 28.41ρd .G 0 = 69.46 + 0.19σ3 .G 0 = 71.19 + 2.35kc f .G 0 = 140.22 − 66.62 .× 0.34 .G 0 .G 0

2

b

35.76 − 260.56 .× 0.88w

.R

2

0.8192 0.9994 0.8897 0.9029 0.8241

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Fig. 7 Variation of initial DSM of different factors

3.3 Linear Regression Analysis on the DSM Water content, dry density, confining pressure, consolidation stress ratio, and loading frequency were taken as independent variables, and the results of linear regression analysis on the dependent variables (initial DSM) were shown in Table 4. The F value is 16.241, and the P value is less than 0.001, indicating that the independent variable has a significant effect on the dependent variable. The .R2 value of the linear

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Table 4 Statistical results of initial DSM using linear regression analysis of loess samples B SD Beta t-value P Dependent variables .−34.728

(Constant) (%) 3 .ρd (.g/cm. ) .σ3 (kPa) .kc f (Hz)

.−7.811

.w

43.551 0.169 70.223 4.416

106.23 1.365 62.241 0.055 13.829 1.391

–

.−0.327

.−0.577

.−5.722

0.071 0.308 0.510 0.320

0.700 3.054 5.078 3.174

0.748 0.000 0.493 0.007 0.000 0.005

regression analysis model for the initial DSM is 0.819, indicating that the explanatory power of different factors for the initial DSM is 0.976. The regression coefficient and significance test show that water content (B .= −7.811, t .= −5.722, P .< 0.001) has a significant negative effect on the initial DSM of the dependent variable. Confining pressure (B .= 0.169, t .= 3.054, P .< 0.001), consolidation stress ratio (B .= 70.223, t .= 5.078, P .< 0.001), and load frequency (B .= 4.416, t .= 3.174, P .= 0.005.< 0.01) have a significant positive effect on the initial DSM of the dependent variable. However, dry density (B .= 43.551, t .= 0.700, P .= 0.493 .> 0.05) has no significant effect on the initial DSM of the dependent variable. By comparing standardized coefficients, the influence degree of each factor on the initial DSM is as following: consolidation stress ratio .> load frequency .> confining pressure .> water content .> dry density. The regression equation between different factors and initial DSM is finally obtained, as follows: .

G 0 = −7.811w + 0.169σ3 + 70.223kc + 4.416f + 34.728

(12)

4 Conclusions The GDS high system of dynamic triaxial test imported from the UK was adopted in this study to test the DSM of compacted loess with different water content, dry density, confining pressure, consolidation stress ratio, and loading frequency. The conclusions drawn from this study are as follows: (1) The DSM decreases with the increase of water content, but increases with the increase of dry density, confining pressure, consolidation stress ratio, and loading frequency. (2) The initial DSM increases non-linearly with the increase of loading frequency, and increases linearly with the increase of dry density, confining pressure, and consolidation stress ratio. However, the initial DSM decreases non-linearly with the increase of water content.

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(3) Through linear regression analysis, water content has a significant negative effect on the initial DSM of the dependent variable. Confining pressure, consolidation stress ratio, and load frequency have a significant positive effect on the initial DSM of the dependent variable. However, dry density has no significant effect on the initial DSM of the dependent variable. Meanwhile, the influence degree of each factor on the initial DSM is as following: consolidation stress ratio .> load frequency .> confining pressure .> water content .> dry density. Acknowledgements The authors gratefully acknowledge the Key Program of the National Natural Science Foundation of China (Grant no. 41931285).

References 1. Pye, K.: The nature, origin and accumulation of loess. Quat. Sci. Rev. 14, 653–667 (1995) 2. Smalley, I., Rogers, C.: Loess: the yellow earth. Geol. Today 12, 186–193 (1996) 3. Rost, K.T.: Late Holocene loess deposits and dust accumulation in the alpine meadow belt of the Wutai Shan, China. Quat. Int. 76–77, 85–92 (2001) 4. Yu, B., Liu, G., Liu, Q., Huang, C., et al.: Seasonal variation of deep soil moisture under different land uses on the semi-arid Loess Plateau of China. J. Soils Sed. 19, 1179–1189 (2019) 5. Karam, J.P., Cui, Y.J., Tang, A.M., et al.: Experimental study on the cyclic resistance of a natural loess from Northern France. Soils Found. 49, 421–429 (2009) 6. Guo, T., Prakash, S.: Liquefaction of silts and silt-clay mixtures. J. Geotech. Geoenviron. Eng. 125, 706–710 (1999) 7. Yang, X.P., Feng, X.J., Huang, X.N.: The late quaternary activity characteristics of the LixianLuojiabu fault: a discussion on the seismogenic mechanism of the Lixian M8 earthquake in 1654. Chinese J. Geophys. (in Chinese) 58(2), 504–519 (2015) 8. Li, C.L., Wu, J., Gao, M.T.: Study on interaction between major earthquakes around Ordos Block. Chinese J. Geophys. (in Chinese) 61(12), 4862–4872 (2018) 9. Wan, X., Yang, D.: Experimental study on dynamic shear modulus and damping ratio of soft soils in Binhu new district of Hefei. Appl. Mech. Mater. 405–408, 1957–1960 (2013) 10. Feng, S.Y., Li, X.Y., Wang, S.Y.: Experimental investigation on the dynamic shear modulus and damping ratio of aeolian soils. International Conference on Power Engineering and Energy, Environment, pp. 534–539 (2016) 11. Sas, W., Gabry, K., Szyma´nski, A.: Effect of time on dynamic shear modulus of selected cohesive soil of one section of express way no. S2 in Warsaw. Acta Geophys. 63, 398–413 (2015) 12. Kong, G., Li, H., Yang, G., et al.: Investigation on shear modulus and damping ratio of transparent soils with different pore fluids. Granular Matter 20, 8 (2018) 13. Bedr, S., Mezouar, N., Verrucci, L., et al.: Investigation on shear modulus and damping ratio of Algiers marls under cyclic and dynamic loading conditions. Bull. Eng. Geol. Environ. 78, 2473–2493 (2019) 14. Dyka, I., Srokosz, P.E., Bujko, M.: Influence of grain size distribution on dynamic shear modulus of sands. Open Eng. 7, 317–329 (2017) 15. Cheng, X., Li, X., Nie, J., et al.: Research on the dynamic parameters of loess. Geotech. Geol. Eng. 37, 77–93 (2019) 16. Deng, Y., Mu, H., Peng, J., et al.: Experiment research on dynamic characteristics of ground fissure belt loess in Xi ’an area. Appl. Mech. Mater. 387, 138–146 (2013) 17. Liu, X., Fan, W., Wang, J. et al.: Experimental study on shear modulus and damping properties of loess. Fundam. Soil Behav. 320–328 (2018)

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18. Wang, H., Sun, P., Liu, E., et al.: Dynamic properties of Tianshui saturated remolded loess: a laboratory study. Eng. Geol. 272, 105570 (2020) 19. ASTM: Annual Book of ASTM Standards. ASTM International, West Conshohocken, Pa (2006) 20. Ebrahimi, M., Sarikhani, M., Safari, S., et al.: Estimating the soil respiration under different land uses using artificial neural network and linear regression models. CATENA 174, 371–382 (2019) 21. Kim, K., Kim, M., Lee, M., et al.: Regression equations for estimating landslide-triggering factors using soil characteristics. Appl. Sci. 10, 3560 (2020)

Unified Description of Viscous Behaviors of Clay and Sand with a Visco-Hypoplastic Model Shun Wang, Xiao Xu, Xuan Kang, Guofang Xu, and Wei Wu

Abstract Time-dependent deformations are commonly observed in geotechnical engineering. This paper presents a unified visco-hypoplastic model to describe viscous behaviors of clay and sand by introducing a new viscous flow rule. The proposed model is capable of capturing a wide range of viscous behaviors, such as isotach behavior of clay and non-isotach behavior of sand in constant rate of strain tests. Moreover, relaxation and creep tests can be simulated as long as the corresponding conditions are prescribed. Keywords Viscous · Hypoplasticity · CRS · Relaxation · Creep

1 Introduction Time-dependent deformations of materials are ubiquitous in nature. For geotechnical engineering, numerous structures, such as tunnels, embankments, and foundation pits [1–4], show time-dependent deformations, which may cause economic losses during S. Wang (B) · X. Xu State Key Laboratory of Water Resources Engineering and Management, Wuhan University, 299 Bayi Road, Wuhan 430072, PR China e-mail: [email protected] X. Xu e-mail: [email protected] X. Kang · W. Wu Institute of Geotechnical Engineering, University of Natural Resources and Life Sciences, Vienna, Feistmantelstrasse 4, 1180 Vienna, Austria e-mail: [email protected] W. Wu e-mail: [email protected] G. Xu State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Xiaohongshan str. 2, Wuhan 430071, China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 W. Wu and Y. Wang (eds.), Recent Geotechnical Research at BOKU, Springer Series in Geomechanics and Geoengineering, https://doi.org/10.1007/978-3-031-52159-1_13

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their service period. Therefore, accurate description of the time-dependent behaviors of geomaterials is of great importance for geotechnical practices. During the past decades, much effort has been devoted to the description of rheological characteristics through mathematic models within the framework of hypoplasticity [5–9]. It was reported that for clay, referred to as isotach material, stepwise change in strain rate has a permanent influence on stress-strain relation, while for sand, referred to as non-isotach material, the influence is temporary [10, 11]. Although some achievements have been made, previous models can describe either the isotach behavior of clay or the non-isotach behavior of sand, while attempts on unified description of time-dependent behaviors of clay and sand with a single constitutive model is less satisfactory. In this paper, a visco-hypoplasticity constitutive model is developed to describe rate effects of clay and sand in a unified way. This is realized by introducing a new viscous flow rule into an existing visco-hypoplastic model [6]. The viscous flow rule contains a rate-sensitivity [12] to describe the so-called isotach and non-isotach behaviors. In addition, the unified model is able to simulate relaxation and creep behaviors as well.

2 The Visco-Hypoplastic Model 2.1 Model Framework The development of our model is based on a visco-hypoplastic model developed by Niemunis [6]. This model is able to describe viscous behaviors of clays such as rate dependence, relaxation and creep. According to Niemunis, the constitutive framework of the visco-hypoplastic model is expressed as: .

T˚ = f b L : (D − Dvis )

(1)

˚ is the Jaumann stress rate tensor, . f b is the barotropy factor, .L is the fourthwhere .T order tensorial function, .D is the strain rate tensor and .Dvis is the so-called viscous strain rate tensor. (1) The Jaumann stress rate tensor .T˚ is defined in terms of the material timederivative of the Cauchy stress rate tensor .T˙ and the spin tensor .W: .

˚ = T˙ + TW − WT T

(2)

(2) Based on the hypoplastic model proposed by von Wolffersdorff [13], .L reads: L = F 2 I + a 2 Tˆ ⊗ Tˆ

.

(3)

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where .I is a fourth-order unit tensor, .Tˆ is the normalized Cauchy stress tensor defined as .Tˆ = T/trT and .a is associated with the critical friction angle .ϕc : √ 3(3 − sinϕc ) .a = √ 2 2sinϕc

(4)

Considering the Matsuoka-Nakai failure criterion, . F in Eq. (3) is defined as: / .

F=

1 2 2 − tan2 ψ 1 tan ψ + − √ tanψ √ 8 2 + 2tanψcos3θ 2 2

with: tanψ =

.

√ 3||Tˆ ∗ ||

√ tr(Tˆ ∗3 ) cos3θ = − 6 ˆ ∗2 )]3/2 [tr(T

.

(5)

(6) (7)

ˆ − 1/3I where .Tˆ ∗ is the deviatoric normalized stress tensor defined as .Tˆ ∗ = T with .I being the second-order unit tensor. (3) .Dvis , also known as the creep rate, is described analogously to Norton’s law [14] using the overconsolidation ratio (.OCR): vis

D

.

)1/Iv ( 1 → = −Dr B OCR

(8)

where . Dr is a reference creep rate, . Iv is the viscosity index of Leinenkugel [15], → = B/||B|| is the direction of the creep rate, defined as: and .B ˆ + Tˆ ∗ ) + Tˆ : T ˆ Tˆ ∗ − Tˆ Tˆ : Tˆ ∗ ( F )2 (T →= B = a B ˆ + Tˆ ∗ ) + Tˆ : T ˆ Tˆ ∗ − Tˆ Tˆ : Tˆ ∗ || ||B|| ||( Fa )2 (T

.

(9)

The calculation of .OCR will be presented in Sect. 2.2. (4) To obtained the barotropy factor. f b , we consider an oedometer test with confined radial deformation (.ε˙ 1 /= 0, .ε˙ 2 = ε˙ 3 = 0). An oedometer test has the following expressions of stress, stress rate and strain rate: ⎤ ⎡ ⎤ ⎡ ⎤ σ˙ 1 0 0 ε˙ 1 0 0 σ1 0 0 ˚ = T˙ = ⎣ 0 σ˙ 3 0 ⎦ , D = ⎣ 0 0 0⎦ .T = ⎣ 0 σ3 0 ⎦ , T 0 0 σ3 0 0 σ˙ 3 0 0 0 ⎡

With the help of an oedometer test, the barotropy factor . f b is obtained as:

(10)

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f =−

. b

[1 +

trT + 2K 0 )]κ 0

a 2 /(1

(11)

where . K 0 is the earth pressure coefficient calculated by: .

K0 =

−2 − a 2 +

√

36 + 36a 2 + a 4 16

(12)

and .κ 0 is the oedometric swell index.

2.2 Unified Visco-Hypoplastic Model Combing the basic visco-hypoplastic model proposed by Niemunis [6] and the elastic-viscoplastic model proposed by Yuan [12], we proposed a new viscous strain rate tensor .Dvis by replacing . Dr in Eq. (8) with a state variable . Ra , and thus .Dvis can be formulated as follows: )1/Iv ( 1 → Dvis = −Ra B OCR

.

(13)

wherein . Ra is subject to the following temporal evolution: .

R˙ a = [ f (˙ε) − Ra ]m t

(14)

where . f (˙ε) is activation function and .m t is transient coefficient. It should be noticed that .ε˙ here refers to .ε˙ 1 in Eq. (10). The formulations of . f (˙ε ) and .m t are given as follows: ) ( )( λ−κ ε˙ −β . f (˙ ε) = (15) ε˙ λ ε˙ r e f ( mt =

.

) vis ε˙ λ −1 + ε˙ α κn

(16)

where .λ and .κ are, respectively, compression and swelling indexes in the modified Cam-Clay model [16, 17], .ε˙ r e f is a reference strain rate, the exponent .−β represents the rate-sensitivity of . f (˙ε), .α is a material constant, .ε˙ vis is viscous strain rate associated with viscous strain rate tensor by .ε˙ vis = Dvis (1, 1) and .n is porosity connected with void ratio by .n = e/(1 + e). For an oedometer test, .OCR is defined as the ratio of the reference effective stress ' ' ' ' .σp and the vertical effective stress .σv , i.e., .OCR = σp /σv . By virtue of integration of differential equation (1), the vertical effective stress can be achieved owing to ' .σv = T(1, 1). For one thing, .ε ˙ vis is subject to Eq. (13); for another, it should comply with the following equation:

Unified Description of Viscous Behaviors of Clay and Sand …

ε˙ vis = (λ − κ)n

.

205

σ˙ p'

(17)

σp'

Based on Eq. (17), we can derive the formulation of the rate of the reference effective stress .σ˙ p' as follows: ε˙ vis .σ ˙ p' = σp' (18) (λ − κ)n

3 Model Performance 3.1 Parameters To show the performance of the proposed model, some oedometer tests, e.g., constant rate of strain (CRS) tests, creep and relaxation tests, were carried out. The parameters shown in Table 1 will be used for the numerical simulations.

3.2 Simulation of CRS Tests The CRS test is simulated by carrying out a one-dimensional consolidation test with constant and stepwise changed rate of strain. In the simulation, .ε˙ 0 = 0.5 %/h is prescribed as a reference strain rate, and several strain rates, i.e., .0.1˙ε 0 , .10˙ε 0 and .100˙ ε 0 , are involved during the test. Figures 1 and 2 illustrate the results of CRS tests with different strain rates. The unified visco-hypoplastic model describes parallel compression lines with .β = 0.02, as shown in Fig. 1a. The faster strain rate generates higher effective stress level at the same void ratio. For a CRS test with stepwise changes in strain rate, as shown in Fig. 1b, .β = 0.02 can capture the salient isotach behavior of clay [11, 18]. On the other hand, with .β = 0, Fig. 2a predicts a unique compression line independent from the strain rates. In Fig. 2b, each change in strain rate leads to a temporary jump in effective stress, which indicates that .β = 0 can capture the salient non-isotach behavior of sand [11, 19].

Table 1 Parameters used for simulation of oedometer tests Parameters .ϕc Value

.30◦

. Iv

.λ

.κ

.κ 0

.α

. Ra0

.e0

' .σv0

0.5

0.76

0.05

0.04

0.02

0. %/h

1.49

100. kPa 200. kPa 0.5. %/h

' .σp0

.ε˙r e f

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1.2 1.2

1.2 1.2

1

Void ratio, e (log scale)

0.8 0.8

0.8 0.8

0.6 0.6

0.6 0.6

0.4 0.4

0.4 0.4

0.2 0.2 102 102

CRS CRS test test Stepwise changed strain rates

1

1

Void ratio, e (log scale)

1

Void ratio, e (log scale)

1.6 1.6 1.4 1.4

CRSCRS test test Different Different strainstrain ratesrates

Void ratio, e (log scale)

1.6 1.6 1.4 1.4

0.2 0.2 102 102

103 103

103 103

(b) stepwise changed strain rates

(a) different strain rates

Fig. 1 Demonstration of predicted rate effects on vertical effective stress-void ratio curves for CRS tests with .β = 0.02 1.6

CRS test Different strain rates

1.4 1.2

1.6 1.2 1

Void ratio, e (log scale)

Void ratio, e (log scale)

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

CRS test Stepwise changed strain rates

1.4

102

103

(a) different strain rates

0.2

102

103

(b) stepwise changed strain rates

Fig. 2 Demonstration of predicted rate effects on vertical effective stress-void ratio curves for CRS tests with .β = 0

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1

1.5 1.45

Relaxation Independent from

0.98

1.4

0.96

Void ratio, e (log scale)

1.35

0.94

1.3

1.25

0.92

1.2

0.9

1.15

0.88

1.1 1.05 1

Relaxation e=1.05 100

200

300

(a) initial state

400 500 600

0.86 0.84 10-1

100

101

Elapsed time, t: h (log scale)

102

(b) relaxation

Fig. 3 Demonstration of model predictions on vertical effective stress-elapsed time curves for relaxation tests with different values of .β

3.3 Simulation of Relaxation and Creep Tests The relaxation test is simulated by carrying out a one-dimensional consolidation test with constant strain (.ε˙ = 0). As time goes by, the vertical effective stress continuously decreases-it relaxes. As a result of .ε˙ = 0, we have . f (˙ε) = 0 and . R˙ a /Ra = −m t according to Eqs. (14) and (15). The proposed model can simulate relaxation as long as . Ra > 0. Thus, . Ra0 ought to be a positive value but close to zero. ' Figure 3 illustrates relaxation behavior from an initial state: .σvc = 300 kPa, .e = ' in Fig. 3b, the vertical effective stress decreases linearly 1.05. Normalized by .σvc with the elapse of time after a transition. Remarkably, relaxation is independent from rate-sensitivity parameter .β, since .ε˙ = 0 during stress relaxation. The creep test is simulated by carrying out a one-dimensional consolidation test with constant effective stress (.σ˙ v' = 0). As time goes by, the strain continuously increases-it creeps. In other words, the void ratio gets smaller over time. When simulating creep, we have .D = Dvis according to Eq. (1). Therefore, the proposed model can simulate creep behaviour if .D equals to .Dvis . Figure 4 illustrates creep behavior from the same initial state as relaxation test shown in Fig. 3. Normalized by.ec in Fig. 4b, the void ratio decreases linearly with the elapse of time after a transition. For different values of .β, the model gains different creep curves.

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1.5

Creep Dependent from

1.45 0.995

1.4

Void ratio, e (log scale)

1.35 0.99

1.3

1.25

0.985

1.2

1.15

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1.1 1.05 1

0.00 0.01 =0.02

0.975 Creep 100

200

300

(a) initial state

400 500 600

0.97 100

101

102

103

Elapsed time, t: h (log scale)

(b) creep

Fig. 4 Demonstration of model predictions on void ratio-elapsed time curves for creep tests with different values of .β

4 Conclusions This paper presents a unified visco-hypoplastic model for describing the viscous behaviors of clay and sand. By virtue of a key parameter, the rate-sensitivity .β, the proposed model can specially characterize the isotach behavior of clay (when .β > 0) and the non-isotach behavior of sand (when.β = 0) in CRS tests. Moreover, the salient behavior of relaxation and creep can also be captured by this model. It should be noted that the dissipation of pore water pressure should be taken into account during consolidation in practical engineering. Therefore, our future work will focus on the simulation of long-term consolidation of soil with different thicknesses within the framework of visco-hypoplasticity. Acknowledgements This work was funded by the National Natural Science Foundation of China (No.42241109, 52178372), the Fundamental Research Funds for the Central Universities (No.2042023kfyq03), and the Austrian Science Fund (No.ESP-342, P-35921).

References 1. Arora, K., Gutierrez, M., Hedayat, A., et al.: Time-dependent behavior of the tunnels in squeezing ground: an experimental study. Rock Mech. Rock Eng. 54(4), 1755–1777 (2021)

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2. Cattoni, E., Miriano, C., Boco, L., et al.: Time-dependent ground movements induced by shield tunneling in soft clay: a parametric study. Acta Geotechnica 11(6), 1385–1399 (2016) 3. Karstunen, M., Yin, Z.Y.: Modelling time-dependent behaviour of Murro test embankment. Geotechnique 60(10), 735–749 (2010) 4. Zhang, Z., Xu, R., Wu, X., et al.: ANN-based dynamic prediction of daily ground settlement of foundation pit considering time-dependent influence factors. Appl. Sci. (2076-3417). 12(13):N.PAG (2022) 5. Niemunis, A., Krieg, S.: Viscous behaviour of soil under oedometric conditions. Can. Geotech. J. 33(1), 159–168 (1996) 6. Niemunis, A.: Extended hypoplastic models for soils. Habilitation thesis. Ruhr-University. Bochum (2003) 7. Niemunis, A., Grandas Tavera, C.E., Prada Sarmiento, L.F.: Anisotropic visco-hypoplasticity. Acta Geotechnica 4(4), 293–314 (2009) 8. Wang, S., Wu, W., Peng, C., et al.: Modelling the time-dependent behaviour of granular material with hypoplasticity. Int. J. Numer. Anal. Methods Geomech. 42(12), 1331–1345 (2018) 9. Gajári, G., Kisgyörgy, L., Ádány, S., et al.: A visco-hypoplastic constitutive model for rolled asphalt. Periodica Polytechnica: Civil Eng. 65(3), 798–809 (2021) 10. Yin, Z.Y., Yin, J.H., Huang, H.W.: Rate-dependent and long-term yield stress and strength of soft Wenzhou marine clay: experiments and modeling. Marine Georesour. Geotechnol. 33(1), 79–91 (2015) 11. Augustesen, A., Liingard, M., Lade, P.V.: Evaluation of time-dependent behavior of soils. Int. J. Geomech. 4(3), 137–156 (2004) 12. Yuan, Y., Whittle, A.J.: A novel elasto-viscoplastic formulation for compression behaviour of clays. Geotechnique 68(12), 1044–1055 (2018) 13. von Wolffersdorff, P.A.: A hypoplastic relation for granular materials with a predefined limit state surface. Mech. Cohesive-frictional Mater. 1(3), 251–271 (1996) 14. Norton, F.: The Creep of Steel at High Temperatures. Mc Graw Hill Book Company Inc., New York (1929) 15. Leinenkugel, H.: Deformations- und Festigkeitsverhalten bindiger Erdstoffe. Experi mentelle Ergebnisse und ihre physikalische Deutung. Ph.D. thesis, Insitut für Boden und Felsmechanik, Universität Karlsruhe, Heft 66 (1978) 16. Roscoe, K.H., Thurairajah, A., Schofield, A.N.: Yielding of clays in states wetter than critical. Geotechnique 13(3), 211–240 (1963) 17. Roscoe, K.H., Burland, J.B.: On the generalised stress-strain behaviour of an ideal wet clay. In: Heyman, J., Leckie, F.A. (eds.) Engineering Plasticity, pp. 535–609. Cambridge University Press, Cambridge (1968) 18. Suklje, L.: The analysis of the consolidation process by the isotaches method. In: Proceedings of the 4th International Conference on Soil Mechanics and Foundation Engineering, pp. 201–206. Butterworths Scientific, London, UK (1957) 19. Tatsuoka, F., Ishihara, M., Di Benedetto, H., et al.: Time-dependent shear deformation characteristics of geomaterials and their simulation. Soils Found. 42(2), 103–129 (2002)

Experimental and Numerical Investigation on Mechanical Behaviour of Gravel Soils Shun Wang, Xuan Kang, Guofang Xu, Hongguang Bian, and Wei Wu

Abstract Gravel soil is a distinctive geological material comprising a blend of soil and gravel fragments. Unlike pure soil, the mechanical behaviour of gravel soils depends largely on the properties of the gravel particles, including their sizes, contents, spatial distributions, and shapes. This study aims to investigate the impact of gravel content and stress level on the mechanical properties of gravel soils. Firstly, undrained triaxial compression tests are conducted to determine the shear strength of the soil matrix with fine particles (.d < 7.5 mm). Then, a series of finite element simulation of plane strain tests using a clay hypoplastic constitutive model is carried out to assess the shear strength and deformation behaviour of gravel soils with varying contents of gravel (.d > 7.5 mm). The numerical simulation indicates that the presence of gravel particles enhances shear strength while reducing the potential for deformation in gravel soil. Keywords Gravel soil · Large-scale triaxial test · Numerical simulation · Hypoplastic model

S. Wang · H. Bian State Key Laboratory of Water Resources Engineering and Management, Wuhan University, Wuhan 430072, China e-mail: [email protected] H. Bian e-mail: [email protected] X. Kang (B) · W. Wu Institute of Geotechnical Engineering, University of Natural Resources and Life Sciences, Vienna, Feistmantelstrasse 4, 1180 Vienna, Austria e-mail: [email protected] W. Wu e-mail: [email protected] G. Xu State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Xiaohongshan str. 2, Wuhan 430071, China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 W. Wu and Y. Wang (eds.), Recent Geotechnical Research at BOKU, Springer Series in Geomechanics and Geoengineering, https://doi.org/10.1007/978-3-031-52159-1_14

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1 Introduction Gravel soil is ubiquitous in nature and can be classified as a type of soil-rock mixture. This geomaterial typically comprises gravel of varying sizes ranging up to 60 mm, as well as fine-grained materials like sand, silt, clay, and pore spaces [1, 2]. The complex nature of gravel soil, characterized by its inhomogeneous multiphase composition, poses significant challenges for both testing and modelling [1]. Understanding and studying the properties of gravel soil require intricate analysis and comprehensive approaches due to its intricate and heterogeneous nature. Over the past few decades, extensive effort has been dedicated to comprehending the shear strength characteristics of gravel soils. These investigations involve both in-situ tests [3–5] and laboratory experiments [6]. A comparison between in-situ tests and laboratory direct shear tests has revealed that the presence of gravel enhances the strength of the soil within the slip zone [5]. Moreover, it has been observed that the shear strength of gravel soil increases with a higher gravel content. Several studies have indicated that the internal friction angle exhibits a linear relationship with the gravel content [3, 4], while the cohesion of gravel soil decreases [6, 7]. However, it is important to note that in-situ tests are both costly and time-consuming, while laboratory experiments are limited to regular testing conditions [1]. Alternatively, numerical simulations, such as the finite-element method (FEM) and finite-difference method (FDM), are widely used to investigate the strength characteristics of complex geomaterials, such as gravel soils in landslide shear zones [1]. Much effort has been devoted to establishing numerical meso-structure models of gravel soil. Based on the Monte-Carlo random sampling principle, random mesostructure models are generated with round and regularly polygonal gravels [8, 9]. With the help of digital image processing (DIP), the actual meso-structure of geomaterials are established and translated into the vector format which can be imported into the FEM software [10]. In order to reduce costs and establish random mesostructure models more easily and quickly, the database with different shape gravels is set up using DIP. Recently, the computed tomography (CT) is used in conjunction with DIP to characterize the full kinematics of the particles within a sample [11, 12]. This paper aims to investigate the influence of gravel content and confining pressure on the mechanical response of gravel soil through numerical simulation. Firstly, large-scale undrained triaxial compression tests on reconstituted samples are carried out to study the basic mechanical response of soil matrix in gravel soils. A gravel generation method is proposed to generate numerical samples with varying contents of gravels with arbitrary shape. The mechanical behaviours of gravel soils are numerically studied by adopting a hypoplastic constitutive model through finite element simulations of plane strain tests.

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2 Large-Scale Triaxial Compression Test 2.1 Sample Preparation The soil from Linzi to Linyi highway (Shandong, China) was sampled to study the mechanical properties of gravel soils. The threshold size of soil/rock (.d S/RT ) is an important concept of gravel soil. Medley and Lindquist [13] and Xu et al. [12] suggest that 0.05 times engineering scale .(L c ) should be taken as the limit of the threshold size of soil/rock. For the triaxial tests in this work, . L c = 150 mm is the diameter of the sample, and thus .d S/RT = 7.5 mm can be regarded as the threshold size of soil/rock. Therefore, the soil sample with particles diameter less than 7.5 mm can be considered as the soil matrix in the triaxial tests. The grading curve obtained from the particle sieving test is depicted in Fig. 1. The dry density of the soil matrix is 1.95 g/cm.3 , and its natural water content is approximately 10.21%. The uniformity coefficient (.Cu ) is 35.17, and the curvature coefficient (.Cc ) is 1.53. The lithology of the soil matrix is silty clay, while the gravel is mainly from limestone. The sample preparation process follows the Standard for Geotechnical Testing Method (GB/T 50123-2019). The samples are prepared in five layers, compacted in a stratified manner, while maintaining the same density and water content as in their natural state. To accommodate the protruding edges and corners of certain blocks, rubber molds with a thickness of 0.8 mm are chosen for the test. For each sample, the vacuum saturation method and back pressure saturation method are employed until the B-value exceeds 0.95.

Fig. 1 Grain size distributions of soil matrix (with particle .d < 7.5 mm) in gravel soils

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2.2 Apparatus and Test Program The experiment adopts the advanced dynamic triaxial testing system developed by GDS at Wuhan University, as shown in Fig. 2. This testing system consists of a driving system, a saturated consolidation system, a measuring system, and a computer operation and data acquisition system. The maximum axial load capacity of the system is 60 kN and the displacement range is set at 100 mm. Both the confining pressure and back pressure controllers have a range of 2 MPa. The primary objective of this test is to determine the shear strength parameters of the soil matrix in the gravel soils. The shearing process is meticulously conducted under three distinct confining pressures: 100, 200, and 400 kPa, respectively. To ensure accurate and consistent results, all tests employ the strain-controlled method, with a controlled shear rate of 0.1 mm/min. The tests are designed to conclude when the maximum strain reaches 20%, which is sufficient to bring the samples to the critical state.

Fig. 2 GDS triaxial test system for gravel soil at Wuhan University

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3 Numerical Simulation of Plane Strain Tests 3.1 Gravel Generation Algorithm For the numerical simulation of plane strain tests, a Python script-based gravel generation algorithm is developed to create randomly shaped gravels. This algorithm consists of three steps, outlined as follows: (1) generate round gravels with different diameters. (2) gain the coordinate and radius of the round gravels. (3) generate random shape gravels based on the round gravels. Notably, the generation of a round gravel needs to follow two basic rules, as shown in Fig. 3. • The gravels can not overlap or touch each other. To this end, the distance of two particles should meet the following requirement: r + rn >

√

. m

(xm − xn )2 + (ym − yn )2

(1)

where .Ci (xi , yi ) and .ri .(i = m, n) are the coordinate of the circle center and the radius of the circle, respectively. • The gravels are not intersected or tangent to the boundary of the sample. This requirement can be meet through: .

L=

|A| > ri l

(2)

where . L is the / distance from the circle center to the boundary, .l is the length of the

boundary .l = (Q 2X − Q 1X )2 + (Q 2Y − Q 1Y )2 with . Q 1 , . Q 2 being the endpoints of the boundary. . A is the value of the second-order determinant, formulated as follows: | | X |Q − Q X QY − QY | 2 1 2 1 | | .A = (3) | xi − Q X yi − Q Y | 1 1

(a)

(b)

Fig. 3 Schematic illustration for generating gravels: The distance between a two gravels; b and a gravel to the boundary

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feature boundary

vertices connection

Fig. 4 The process for generating random shape gravel

The process of generating a single gravel is presented in Fig. 4. Starting from the reference point of each gravel center, the circle is initially divided into N parts, with each part corresponding to a center angle of 360.◦ /N. Subsequently, N vertices are generated, wherein the distance of each vertex from the circle’s center is determined by multiplying the radius reduction factor .α with the radius of the respective circle. For this study, the value of .α ranges from 0.3 to 1.0. By sequentially connecting these vertices, a random-shaped gravel is obtained.

3.2 Plane Strain Models In the numerical simulations, each sample has a dimension of 100 mm in width and 200 mm in height. In order to simulate the plane strain test, a series of plane strain models with different gravel contents are generated by adopting the aforementioned gravel generation algorithm. Then, the plane strain model with varying gravel contents are discretized into 100 .× 200 finite elements (four-node plane strain elements, CPE4), shown as follows ⎡

digital sample Am×n

.

a11 ⎢ a21 ⎢ =⎢ . ⎣ ..

a12 a22 .. .

··· ··· .. .

⎤ a1w a2w ⎥ ⎥ .. ⎥ . ⎦

ah1 ah2 · · · ahw

in which .w = 100 and .h = 200. In the above matrix, each component in the digital sample represents a soil element or a gravel element in the finite element simulation. The elements representing gravel can be found during the generation of the plane strain model. In the numerical simulation, the gravels are treated as elastic media, while the soil matrix is assumed to exhibit a plastic response. A simple step for generating the plane strain model with with various gravel contents is shown in Fig. 5. In this simulation, 6 plane strain models with gravel contents ranging from 14 to 39% are generated, as shown in Fig. 6. Note that, the interface behaviour between gravels and soil matrix is not considered in this simulation.

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Fig. 5 The process for generating plane stain model with various gravel contents in the finite element simulation

The numerical test is performed by applying a displacement boundary condition on the top surface of the numerical sample while keeping the confining pressure constant. An additional boundary condition at the bottom is applied to constrain the vertical displacement. In addition, the middle of the bottom boundary should be constrained to prevent the overall motion along the horizontal direction. The compression is conducted by applying a displacement boundary upon the upper boundary. Generally, a total of 30 mm of shearing displacement is required to bring the gravel sample to a critical state.

3.3 Constitutive Model To replicate the mechanical response of the soil matrix, the simulation incorporates a hypoplastic model proposed by Wang and Wu [14]. The formulation of the hypoplastic model is outlined as follows: .

[ ] ˇ ˇ + f v (trD)Tˇ + a 2 tr(TD) Tˇ + f u a(Tˆ + Tˆ ∗ )||D|| T˚ = f s (tr T)D tr Tˇ

(4)

where .T˚ is the Jaumann stress √ rate, T is the Cauchy stress tensor, .D is the strain rate (stretching) tensor, .||D|| = tr(D2 ) stands for the norm of the strain rate tensor. .Tˇ = T + S and .Tˆ = T − S with .S denoting the structure tensor for overconsolidation. It is expressed as: (1) T .S = αln (5) R

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(a) 14%

(b) 20%

(c) 24%

(d) 31%

(e) 36%

(f) 39%

Fig. 6 Artificial clastic samples with different mass gravel contents, gravel diameter .d ∈ [7.5 30] mm

in which . R stands for the degree of overconsolidation: .

( q 2 ) [ ln(1 + e) − N ] (0 < R ≼ 1) R = p + 2 exp M p λ∗

(6)

The stiffness factor . f s and the multipliers . f v and . f u are defined as: √ 2 3 1 . fs = − , f v = vi − (3 + a 2 − 3a), f u = ∗ 3vi λ 2 3

(

||B : D|| ||B|| ||D||

)μ (7)

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where .B denotes the flow rule of the model and .μ is a fitting parameters controls the undrained response of the soil. In the above equations, .a and . M correspond to the limit stress at critical state, giving: √ 3(3 − sinφc ) 6sinφc .a = , and M = (8) √ (3 − sinφc ) 2 2sinφc There are six parameters for the hypoplastic model: .φc , .vi , .λ∗ , . N , .α, and .μ. .φc is the critical state friction angle; .vi is the ratio of the bulk modulus in the isotropic compression and the shear modulus in the undrained shear test on an isotropic consolidated sample; .λ∗ is the slope of the isotropic normal compression line in the double logarithmic plane .ln(1 + e) − ln p; . N is the value of .ln(1 + e) at the isotropic normal compression line for . pr = 1 kPa; .α and .μ are fitting parameters controlling the magnitude of the structure tensor and undrained response of a soil, respectively.

4 Test Results 4.1 Triaxial Compression Test Figure 7 presents the results of the triaxial compression tests. The stress-strain curves and stress paths feature obvious strain softening behaviours in undrained condition. When the axial strain reaches 10%, the soil mass has reached the critical state with constant stress and continuous development of axial strain. Before conducting the numerical plane strain test, it is crucial to verify the material parameters used in the constitutive model. For this purpose, a numerical simulation of undrained triaxial consolidation tests with one axisymmetric element (CAX4P) is performed. The material parameters employed for the numerical simulation are provided in Table 1. The comparison between triaxial compression tests and numerical simulation is presented in Fig. 7 as well. The simulation effectively captures the strain softening behaviour of the soil matrix under undrained conditions. Additionally, it successfully replicates the variations in the stress path. However, it should be noted that the numerical simulation conducted with a confining pressure of 400 kPa deviates significantly from the experimental results. As a result, only the numerical results under confining pressures of 100 and 200 kPa are presented.

4.2 Numerical Plane Strain Tests Figure 8 shows the numerical results of the plane strain tests with different gravel content under confining pressure of 100 kPa. Note that, the numerical simulations with high gravel content show less numerical convergence; therefore, the stress-

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Deviatric stress, q [kPa]

400

(a)

model prediction 100 kPa 200 kPa 400 kPa

350 300 250 200 150 100 50 0 0

Deviatric stress, q [kPa]

450

5

10 Axail strain,εa [%]

20

model prediction 100 kPa 200 kPa 400 kPa

(b)

400

15

350 M = 1.53

300 250 200 150 100 50 0 0

50

100 150 200 250 300 350 400 450 [kPa]

Fig. 7 Comparison between triaxial compression tests and numerical simulation: a stress-strain curves and b stress paths Table 1 Material parameters used in the finite element simulations Material Mechanical parameters Values Gravel

Young’s modulus: . E, MPa Poisson’s ratio: .ν

3000 0.3

Soil matrix

Critical friction angle: .φc , /.◦ Compression index: .λ∗ Specific volume: . N Modulus ratio: .ri OCR parameter: .α Undrained control parameter: .μ

37.8 0.02 0.61 1.0 0.3 11

Experimental and Numerical Investigation on Mechanical Behaviour of Gravel Soils 400 Deviatoric stress, q [kPa]

Fig. 8 Numerical results of the plane strain tests with various gravel content ranging from 14 to 39% under confining pressure of 100 kPa: a stress-strain response, and b volumetric changes

(a)

350 300 250 200 150

14% 20% 24% 31% 36% 39%

100 50 0 0

0.02 0.04 0.06 0.08 0.1 Axial strain,εa [-]

0.01

0.12 0.14 0.16

14% 20% 24% 31% 36% 39%

(b) 0.005 Volumetric strain, εv [-]

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0 -0.005 -0.01 -0.015 -0.02 -0.025 -0.03 0

0.02 0.04 0.06 0.08 0.1 Axial strain,εa [-]

0.12 0.14 0.16

strain curves and volumetric changes of the these simulations are not complete. Generally, all stress-strain curves follow the same trend that only strain hardening and contractive volume changes are reproduced in the simulation simulation. Obviously, increasing the gravel content can enhance the shear strength of the gravel soil. Meanwhile, the compressibility of the gravel soil decreases by increasing the gravel content. Figure 9 shows numerical results of the plane strain tests with 20% gravel content under different confining pressures, together with the evolution process of the plastic strain zone. As shearing starts, the maximum localised strain usually occurs at the interface between gravel and soil matrix. Due to the highly elastic mismatch of gravel and soil matrix, all localised plastic strain zone propagate to bypass the gravel. It is interesting to show that the plastic strain is also affected by the stress level. Two localized strain paths are developed in the simulation under the confining pressure of 100 kPa, while the number of localized strain path convert to one for the simulation under 400 kPa confining pressure.

222 1400 Deviatoric stress, q [kPa]

Fig. 9 Numerical results of the plane strain tests with 20% gravel content under confining pressures of 100, 200 and 400 kPa: a stress-strain response, and b volumetric changes

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(a) 400kPa

1200 1000 800

200kPa

600 400

100kPa

200 0 0

0.01

0.02 0.04 0.06 0.08 0.1 Axial strain,εa [-]

0.12 0.14 0.16

(b)

Volumetric strain, εv [-]

0 -0.01 -0.02

100kPa

-0.03 200kPa

-0.04

400kPa

-0.05 -0.06 -0.07 0

0.02 0.04 0.06 0.08 0.1 Axial strain,εa [-]

0.12 0.14 0.16

5 Conclusions In this paper, the mechanical properties of gravel soils are investigated through finite element simulation. The influence of gravel content and confining pressure on the mechanical response of gravel soil is analyzed by using a hypoplastic constitutive model. The following conclusions are drawn: (1) A gravel generation method is proposed to generate numerical samples with different contents of gravel with random shape. This algorithm is adopted in the FEM simulation of plane strain tests for generating gravels with random shapes. (2) The existence of the gravel increases the shear strength while decrease the compressibility of gravel soil. In addition, the confining pressure can influence the development of plastic strain and eventually affect the localized shear path of the plane strain samples. Acknowledgements This work was funded by the National Natural Science Foundation of China (No.42241109, 52178372), the Fundamental Research Funds for the Central Universities (No.2042023kfyq03), and the Austrian Science Fund (No.ESP-342, P-35921).

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References 1. Wang, S., Wu, W., Cui, D.: On mechanical behavior of clastic soils: numerical simulations and constitutive modelling. Géotechnique 72, 706–721 (2022) 2. Xu, W., Zhang, H., Jie, Y., et al.: Generation of 3D random meso-structure of soil-rock mixture and its meso-structural mechanics based on numerical tests. J. Central South Univ. 22, 619–630 (2015) 3. Coli, N., Berry, P., Boldini, D.: In situ non-conventional shear tests for the mechanical characterisation of a bimrock. Int. J. Rock Mech. Min. Sci. 48, 95–102 (2011) 4. Xu, W.J., Qiang, X., Rui, R.H.: Study on the shear strength of soil–rock mixture by large scale direct shear test. Int. J. Rock Mech. Min. Sci. 48, 1235–1247 (2011) 5. Zou, Z., Zhang, Q., Xiong C., et al.: In situ shear test for revealing the mechanical properties of the gravelly slip zone soil. Sensors 20, 6531 (2020) 6. Zhang, Y., Lu, J., Han, W., et al.: Effects of moisture and stone content on the shear strength characteristics of soil-rock mixture. Materials 16, 567 (2023) 7. Zhang, Z.L., Xu, W.J., Xia, W., et al.: Large-scale in-situ test for mechanical characterization of soil-rock mixture used in an embankment dam. Int. J. Rock Mech. Min. Sci. 86, 317–322 (2016) 8. Li, X., Liao, Q.L., He, J.M.: In situ tests and a stochastic structural model of rock and soil aggregate in the Three Gorges reservoir area, China. Int. J. Rock Mech. Min. Sci. 41, 494 (2004) 9. Li, S.H., Wang, Y.N.: Stochastic model numerical simulation of uniaxial loading test for rock, soil blending by 3D-DEM. Chin. J. Geotech. Eng. 26(2), 172–7 (2004) [in Chinese with English abstract] 10. Xu, W.J., Yue, Z., Hu, R.: Study on the mesostructure and mesomechanical characteristics of the soil-rock mixture using digital image processing based finite element method. Int. J. Rock Mech. Min. Sci. 45, 749–762 (2008) 11. Jiang, J., Xiang, W., Rohn, J., et al.: Research on mechanical parameters of coarse-grained sliding soil based on CT scanning and numerical tests. Landslides 13, 1261–1272 (2016) 12. Xu, W.J., Zhang, H.Y.: Meso and macroscale mechanical behaviors of soil-rock mixtures. Acta Geotechnica 17, 3765–3782 (2022) 13. Medley, E., Lindquist, E.S.: The engineering significance of the scale-independence of some Franciscan melanges in California, USA. U.S. Symp. Rock Mech. 6, 907–914 (1995) 14. Wang, S., Wu, W.: Validation of a simple model for overconsolidated clay. Acta Geotechnica 16, 31–41 (2020, 2022)

A Basic Hypoplastic Model with Fabric Evolution Yadong Wang and Wei Wu

Abstract The fabric anisotropy has great impacts on the behaviour of granular soils but constitutive modelling considering fabric evolution still pose difficulties. In this paper, a hypoplastic constitutive model is developed within the realm of anisotropic critical state theory. A deviatoric fabric tensor is introduced into a critical state hypoplastic enhanced by Matsuoka-Nakai failure criterion. The critical state function and failure criterion are embedded with a fabric anisotropic variable indicating fabric evolution during loading. In such way, the impacts of fabric anisotropy on the dilatant and strength behaviours can be characterized, and the conditions of fabric anisotropy are concurrently satisfied along with the traditional conditions at the critical state. Moreover, the material parameters can be readily obtained from conventional undrained triaxial compression tests without resorting to any complicated calibration procedures. The comparison between numerical simulations and experimental findings indicate that the new model is robust and can reasonably captures the salient annisotropic behaviours of geomaterials at various scenarios of different bedding angles and intermediate principal stress ratios.

1 Introduction The observation of the experimental tests show that, the behaviour of the anisotropic granular materials associated with the characteristics of soil particles, material state, stress state, various shearing modes (with intermediate principal stress ratio .b) and loading direction, etc. cannot be properly described by an isotropic criterion [1]. In fact, isotropic failure surface generally overestimates the strength of the anisotropic soil specimen compare with the data from the experiments [2, 3]. Noticeably the Y. Wang · W. Wu (B) Institute of Geotechnical Engineering, University of Natural Resources and Life Sciences, Vienna, Feistmantelstrasse 4, 1180 Vienna, Austria e-mail: [email protected] Y. Wang e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 W. Wu and Y. Wang (eds.), Recent Geotechnical Research at BOKU, Springer Series in Geomechanics and Geoengineering, https://doi.org/10.1007/978-3-031-52159-1_15

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stress-strain-strength behaviour, dilative material response of the sands also changes with the different orientations of the depositional plane and also varies with b [4]. While there is abundant evidence of the impact of fabric anisotropy on granular soil behaviour as shown above, it remains a challenging task to take the anisotropic effects into account into the well-established continuum mechanics framework. Based on the framework of microstructure tensor [5], several attempts have been made to construct the generalized anisotropic failure criteria and constitutive models incorporating with the fabric effects [6–8]. For modelling the evolution of fabric tensor and maintaining the uniqueness of critical state line (CSL), the anisotropic critical state theory (ACST) for granular materials was proposed by Li and Dafalias based on the thermodynamic theories [8]. The main idea is a normalized fabric tensor evolves toward a critical state value, and the critical state direction of fabric tensor is coaxial with the loading direction. The dilatancy state parameter, obtained by the fabric anisotropic variable and critical parameters, varies with the loading state until the critical state. During the shearing, a relocate state line called dilatancy state line (DSL) replaced CSL is used to restrict the behaviour of the stress and plastic deformation. Thus the fabric evolution is essentially the DSL evolves to CSL. This flexible DSL not only guarantees the uniqueness of CSL, but also continuously updates the effects of anisotropy during the deformation process. Further more, ACST has been widely used in various constitutive models [9, 10], proving the great potential of this theory. The above anisotropic methods and constitutive models are flexible, but it is not easy to derivate the long term partial derivations from the constitutive formula including fabric tensor, with the condition of consistency of the yield function. As an alternative, hypoplastic model based on non-linear tensorial functions without resorting to the yield surface function and other ingredients associated with the plastic theory. A simple hypoplastic equation for granular materials was given by Wu et al. [11]. Based on the framework of that simple model, the extended models with critical state have been proposed [12, 13]. For anisotropy, a hypoplastic model combined with the ACST have been proposed by Yang et al. [10]. This pioneering paper presented two kinds of hypoplastic equation for ACST, one of which is an intergranular strain equation that can simulate experiments with good performance. In this paper, the main topic is to develop a simplified and robust constitutive model with ACST based on a basic hypoplastic model. The interpolation function for Matsuoka-Nakai limit condition (SMP) failure criterion is added to embed the effects from the intermediate principal stress. Then the anisotropic variable is incorporated into the dilatancy and strength of the constitutive equation as ACST to account for the fabric evolution. The calibration of each parameter and the numerical verification examples are given to show the effectiveness of the model. Among them, a set of elementary tests and fabric evolution under the monotonic loading stress paths, will be simulated to compare with the classical experiments tests. In the following, except where specified otherwise, effective stress is used.

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2 Hypoplastic Model 2.1 Framework of Hypoplastic Model The hypoplastic constitutive equation with critical state be can expressed by the following tensorial function .H such that [12]: ˚ = H(T, D, e) T

.

(1)

˚ is the Jaumann stress rate, .T the Cauchy stress, .D the strain tensor and .e is where .T the current void ratio. The Jaumann stress rate .T˚ is expressed as ˚ = T˙ + WT − TW T

.

(2)

where.W is the spin tensor and the superposed dot means material time differentiation. In this paper, we consider the following framework: T˚ = L : D + Ie N||D||

.

(3)

where .L and .N are isotropic tensorial functions, the symbol . : denotes an inner product between two tensors. .||D|| stands for the norm of the strain rate tensor, and . Ie is a density function to account for the effects of density on the stress-strain behaviour of the granular material. The density function has the value . Ie = 1 at the critical state, greater than 1 for a loose state, and less than 1 for a dense state. The above equation is further enhanced to give better performance in undrained tests [10]. Such that, the Eq. 3 is recast to: ˚ = L : D + ϖ Ie N||D|| .T (4) where .ϖ is the joint invariant of the normalized of strain rate and flow rule function of Eq. 3. The factor .ϖ can be defined as: ( ϖ =

.

|D : B| ||D||||B||

)u (5)

where .B = −L−1 : N is a second order tensor presenting the flow rule, .u is the non-physical coefficient.

2.2 The Basic Hypoplastic Model with Critical State With the above framework, a basic hypoplastic constitutive model, which includes the critical state of sand, was proposed by Wu et al. [14], it reads

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( ) tr (TD) T˚ = c1 (trT) D + c2 (trD) T + c3 T + Ie c4 T + T∗ ||D|| trT

.

(6)

in which .c1 , .c2 , .c3 and .c4 are dimensionless parameters related to some wellestablished material parameters, such as: the initial tangent modulus . E i , initial Poisson’s ratio .νi , the dilatancy angle .ψ, and the critical state friction angle .φc , the subscript .c denotes the critical state. .T∗ is the deviatoric part of the stress tensor .T, and tr represents the trace of a tensor. At the critical state, the constitutive model gives rise to vanished stress rate and volumetric change, such as: ˚ = 0, trD = 0 .T (7) To account for the effects of intermediate principal stress, the SMP limit surface can be embedded into the hypoplastic model through various approaches [16]. Analogously, the SMP failure surface can be incorporated into the basic model [14] in the following way: [ ] ( ) tr (TD) T˚ = It a 2 c1 (trT) D + a 2 c2 (trD) T + c3 T + Ie ac4 T + T∗ ||D|| trT

.

(8)

where . It is the interpolation function of stiffness; .a is an interpolation function to account for SMP failure criterion. In this paper, the following formulation as proposed by Bauer is used to represent the density function [13]: ) ( e − emin χ . Ie = (9) ec − emin where .χ is the non-physical constant, .emin is the minimum void ratio representing the densest state at isotropic consolidation condition. .ec is the critical void ratio. The location of CSL in the .e − p space can be expressed by the following equation proposed by Li et al. [15]: ( ' )ξ p .ec = e⎡ − λ (10) pa where .λ and .ξ are the granular material parameters, .e⎡ is the void ratio at the initial state, . p ' is the mean effective pressure and . pa = 101.325 kPa is the atmospheric pressure. Different to the formulation used in Von Wolfferdorf’s model [16], a generalized function of .a proposed by Yao et al. [17] is used: / 2I1 I12 − 3I2

a= √ 3 (I1 I2 − I 3)/(I1 I2 − 9I3 ) − 1

.

where . I1 , . I2 and . I3 are the stress invariants.

(11)

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Fig. 1 The comparison between failure and bounding surfaces of hypoplastic model

Figure 1 compares the failure surface (FS) and bounding surface (BS) of hypoplastic models (6) and (8). In this figure, the dashed and bold lines represent the original and modified surfaces, respectively. Clearly, with the interpolation factor.a, the radius of the original failure surface is changed from .ρ to .aρ. To the end, the stiffness factor . It , manipulating the incremental stiffness of the model follows the line of Von Wolfferdorf’s [16], as follows: I =

. t

(trT)2 T: T

(12)

3 Hypoplastic Model with Fabric Evolution As mentioned by Li and Dafalias [8], the fabric anisotropy variable . A provides a link to bridge the fabric anisotropy with some important constitutive elements, such as dilatancy and shear strength. To account for these effect, the hypoplastic model incorporating ACST can be assumed as: .

˚ = H (T, D, e, A) T

(13)

A = F : n = FnF : n

(14)

where . A is defined as follows: .

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where .n is a unit-norm deviatoric tensor which represents the loading direction, and F is the deviatoric fabric tensor with . F and .nF standing for its norm and direction, respectively. During fabric evolution, the direction of .F will evolve to .n, lead to . A = 1 at critical state. The expression of .F is given as [10] .

F = n + (Fi − n) exp(−c0 |∈q |)

.

(15)

where√.c0 is a non-negative constant manipulating the rate of fabric evolution, ∈ = 2/3||∈ ∗ || is the generalized shear strain with .∈ ∗ being the deviatoric part of the strain tensor. Furthermore, the loading directing .n coincide with the direction of the deviatoric stress in triaxial monotonic loading [8], such as .n = T∗ /||T∗ || is adopted in the proposed model. The initial .Fi under triaxial compression loading can be obtained: √ ⎡ ⎤ 0 −2/ 6 0√ .Fi = Fi nF = Fi ⎣ (16) 0 1/ 6 0√ ⎦ 0 0 1/ 6

. q

where . Fi denotes the initial fabric norm. Notably, if the sample has been rotated, .nF should be updated by a proper orthogonal transformation. Then, the following requirements must be concurrently satisfied at critical state: T˚ = 0, trD = 0, A = Ac = 1

.

(17)

Clearly, Eq. 17 is the governing condition for the explicit expression of Eq. 13. Thus in the following, the constitutive model (8) is incorporated with ACST by introducing . A (in the form of . A − 1) into the functions of . Ie and .a.

3.1 Fabric Related . Ie for Dilatancy and Strength As observed in experiments [4], the strength and dilatancy behaviour of the samples change with varying the bedding angle and shearing modes. Thus the new . Ie with the fabric anisotropy variable . A can be obtained by replacing the critical state void ratio .ec to the dilatant void ratio .ed through the following formulation: ( I =

. e

e − emin ed − emin

)χ (18)

in which the dilatant void ratio .ed can be obtained by combing the Eqs. 10 and 14, as follows: ( ' )ξ p .ed = e⎡ − λ + e A (A − 1) (19) pa It can be clearly seen that the density factor depends not only on the void ratio and stress state, but also on the loading direction, which controls the magnitude of fabric

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Fig. 2 The schematic diagram of . Ie related to . A

anisotropy variable. Let us consider the following typical parameters with . E i = 170, ν = 0.05, .ψ = 0◦ , .φc = 34◦ , .emin = 0.597, .e⎡ = 0.977, .λ = 0.019, .ξ = 0.7, .χ = 0.35, . Fi = 0.5, .e A = 0.1 and .trT = −1 to obtain the failure surface of model (8). The method to calculate failure surface related to . A in model (8) is same as the treating in paper [18]. Thus the the value of .n can be calculated by the stress state on failure surface. The variation of . Ie along the failure surface with different Lode angle .θ and bedding angle .α is shown in Fig. 2.

.

3.2 Fabric Related . a for Shear Strength With the fabric anisotropy variable . A in the new density function, the failure surface of model (8) also evolves with fabric. To reflect the effect of . A on the shear strength, the failure surface with . A can be expressed by the following formulation: .

f = H(T, φc , A)

(20)

where . f is the general expression of the anisotropic failure criterion. An explicit expression of the anisotropic failure criterion can be described by [6, 7]: .

f ∝ : fˆ − a Mg = 0

(21)

where . fˆ is the function of limit surface. .g is an interpolation function with the following general form [5]: .

g = g[g1 (A − 1), g2 (A − 1)2 , ..., gn (A − 1)n ]

where .gi (i = 1, 2, ..., n) are positive model constants.

(22)

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Fig. 3 The schematic diagram of .a related to . A

Fig. 4 The schematic diagram of failure surface related to . A in Eq. 8

The aforementioned method is a generic concept to incorporate fabric anisotropy. In this paper, to achieve the same end, a simplified exponential function of . A − 1 is chosen, as follows . g = exp(es (A − 1)) (23) where .es is a non-negative model constant. Thus, the interpolation function .a related to . A can be obtained as follows: / 2I1 I12 − 3I2 (24) .a = √ exp(es (A − 1)) 3 (I1 I2 − I 3)/(I1 I2 − 9I3 ) − 1 Following the same typical parameters in Fig. 2 with .es = 0.45, the variation of .a along the failure surface with different Lode angle .θ and bedding angle .α is shown in Fig. 3. While the related surfaces with .α = 0◦ , .α = 45◦ and .α = 90◦ , are shown in Fig. 4.

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4 Validation with Experimental Results In this section, the proposed model is validated with experimental results on Toyoura sands [4]. A series of monotonic triaxial compression tests and simple shear test with fixed intermediate principal stress coefficient .b, and fixed bedding angle .α are simulated.

4.1 Calibration of the Model Parameters The model contains 13 material constants that can be classified into three groups: (1) Five parameters of mechanical behaviour: . E i , .νi , .ψ, .φc , and .u; (2) Four parameters for critical state behaviour: .emin , .e⎡ , .λ and .ξ ; (3) Four additional parameters for fabric anisotropy, . Fini ,.e A , .es , .χ , and .c0 . The four material parameters . E i (initial Young’s modulus), .νi (initial Poisson’s ratio), .ψ = 0 (dilatant angle), .φc (critical state friction angle), for mechanical behaviour can be readily attained from triaxial compression tests. They will be used to calculate the dimensionless constant .c1 , .c2 , .c3 and .c4 . The parameters .emin , .e⎡ , .ξ , and .λ for critical state behaviour can be measured from an .e − p plot of critical states based on three triaxial isotropic compression tests; The parameters, initial fabric . Fini and its pace controller .es , strength related parameter .c0 , dilatancy related parameter .e A , for fabric anisotropy can be determined through undrained triaxial tests, and .χ can be determined through trial and error based on the density effect on strain softening or hardening. The parameter .u used for improving the undrained performance can be forced to zero in drained condition, while it may be set between 1.0 and 4.0 for undrained tests. In the following, a procedure to calibrate this model for Toyoura sand is outlined: • Basic parameters (1) Obtain the value of . E i , .ν, .ψ and .φc from experimental tests [2, 4] (2) The value of .c1 , .c2 , .c3 and .c4 can be obtained by Eq. 8 in TC test [11] .• Critical state parameters (3) .emin , .e⎡ , .λ and .ξ in the paper [15] .• Drained or undrained condition (4) Drained: .u = 0. Undrained: .u = 1 − 4 [10] .• Anisotropy related parameters (5) Get the value of . Fini from literature (or DEM simulate) [19] (6) Estimate the value range of .e A by TC test [20] (7) trial and error run performance: .e A , .es , .χ and .c0 [20] .

Thus, the parameters are given as followings: . E i = 264 mPa, .ν = 0.05, .ψ = 0◦ , ◦ .φc = 34 , .emin = 0.597, .e⎡ = 0.977, .λ = 0.019, .ξ = 0.7, .χ = 0.35, .u = 1.5, . Fi = 0.5, .e A = 0.1, .es = 0.4, .c0 = 8.6.

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4.2 Drained Compression Tests In this section, drained torsional shear tests on Toyoura sand [2] are simulated, with the conditions that the value of mean principal stress . p ' = 100 kPa and .b = 0.5 are fixed. The values of .φc = 37◦ , .es = 0.2 related to the peak strength is modified (by the same calibration procedure in Sect. 4.1), due to the slight difference in experimental samples preparation. Among them, the relationships between stress ratio and strains for .α = 0◦ and .α = 60◦ tests are shown in Fig. 5. The dependence of shear deformation characteristics and dilatancy behaviour on the direction of principal stress axes is shown in Fig. 6. The above simulations show the well performance of Eq. 8 in drained shearing tests under monotonic lading.

4.3 Undrained Tests with Fixed . b-Value and .α Conditions The comparison between experiments and simulations of conventional undrained triaxial compression (.b = 0, .α = 0◦ ) and extension (.b = 1, .α = 90◦ ) tests on Toyoura sand are presented in Fig. 7. The simulation is performed with different initial confining pressures varying from 50 to 500 kPa. It can be seen that the simulations

Fig. 5 Experimental data (a, c) and simulations (b, d) for drained shearing tests with fixed. p ' = 100 kPa and fixed .b = 0.5 (. Dr ≈ 81.5%) [2]

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Fig. 6 Experimental data (a, c) and simulations (b, d) for drained shearing tests with fixed. p ' = 100 kPa and fixed .b = 0.5 (. Dr ≈ 81.5%) [2]

agree well with the experiments. For the same initial confining pressure, the extension tests gives rise to higher excess pore water pressure than that of the compression tests. This shearing behaviour can be well captured by the simulation. The results also highlight the significant effects of .b-value and bedding angle .α on the dilatancy behaviour of sand. In addition, the different magnitude of the intermediate principal stress ◦ .b = 0 − 1 and constant bedding angle .α = 45 with similar relative density (.e = 0.849 − 0.861) are presented in Fig. 8. One may notice that the behaviour of the sand softer and the developed excess pore water pressure with the increased .b-value. These simulation results have good agreement with the tests [4] and shown the effectiveness of the Eq. 8.

4.4 Evolution of Fabric Variables Although in experimental tests, it is different to arrival at the critical state, the numerical simulations significantly show that the critical state is unique with the framework of ACST. The simulation follows the previous studying including TC tests

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Fig. 7 Experimental data (a, c) and simulations (b, d) for undrained TC (.b = 0 and .α = 0◦ , = 0.861 − 0.890) and TE (.b = 1 and .α = 90◦ , .e = 0.860 − 0.888) tests [4]

.e

(.b = 0, α = 0◦ − 45◦ ), normal torsional shearing (.b = 0.5, α = 15◦ − 75◦ ) and TE test (.b = 1, α = 60◦ − 90◦ ). The observation shows the agreement with the DEM simulation [21]. Among them, the relative angle between deviatoric fabric tensor .F and .n can be obtained by Eq. 14. The evolution of the relative angles keep decreasing until equal to zero under the monotonic shearing . In Fig. 9, .ed from Eq. 19 they all converge to the CSL with the huge difference of the response in the initial state. In addition, in Fig. 10, the vertical dash line at .γ = 12% − 15% means the normal strain stage within which the experimental tests and simulation carries out. As mentioned in paper [8], at the normal strain stage, the soil fabric exit a far distance away from the critical value in all the cases. Thus it is necessary to consider the fabric evolution compared with the cross-anisotropy. Notice that if . A < 0 at the initial state, the values of the norm . F reduce and then increase to the similar critical value. It is because the initial relative angle larger than 90.◦ and the direction of .F needs to converge to .n. Clearly, the fabric evolution of . A and . F are in agreement within the DEM observation presented by Li [21]. Compared with Yang’s model [10], the values of all parameters except . Fi in Eq. 8 can be obtained by drained/undrained TC (.b = 0 and ◦ ◦ .α = 0 ) or TE (.b = 1 and .α = 90 ) test, directly.

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Fig. 8 Experimental data (a, c) and simulations (b, d) for undrained shearing tests with fixed = 45◦ and constant .b = 0 − 1 (.e = 0.849 − 0.861) [4]

.α

Fig. 9 The evolution of .ed and eventually coincide with .ec

5 Conclusions In this paper, the fabric anisotropy variable . A proposed by ACST is embedded into a basic hypoplastic model to account for fabric evolution. In addition, the SMP failure criterion is introduced into the basic equation by interpolated function to reflect the effects on varies .b-value correctly,. Then the terms of the hypoplastic equation,

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Fig. 10 Evolution of the fabric anisotropy variable . A

which can affect the contractility and dilatancy of the model performance, associated with . A are redefined. Refer to the previous studying, the values of parameters are given by undrained triaxial compression test directly without complicated calibration procedures. Meanwhile, the undrained and drained simulation results show the good performance and robustness of the new model. An extension to the stress rotation with fabric anisotropy evolution is straightforward. Acknowledgements The authors wish to acknowledge the financial support from Austrian Science Fund (FWF) for Principal Investigator Project - HIME (Grant No. P 37175-N), OeAD WTZ project (Grant Nos. CN 04/2022, CN14/2021 and FR 03/2024), European Commission Horizon Europe Marie Skłodowska-Curie Actions Staff Exchanges Project - LOC3G (Grant No. 101129729) and Otto Pregl Foundation of Fundamental Geotechnical Research in Vienna.

References 1. Pradhan, T.B., Tatsuoka, F., Horii, N.: Simple shear testing on sand in a torsional shear apparatus. Soils Found. 28(2), 95–112 (1988) 2. Miura, K., Miura, S., Toki, S.: Deformation behavior of anisotropic dense sand under principal stress axes rotation. Soils Found. 26(1), 36–52 (1986) 3. Kirkgard, M.M., Lade, P.V.: Anisotropic three-dimensional behavior of a normally consolidated clay. Can. Geotech. J. 30(5), 848–858 (1993) 4. Yoshimine, M., Ishihara, K., Vargas, W.: Effects of principal stress direction and intermediate principal stress on undrained shear behavior of sand. Soils Found. 38(3), 179–188 (1998) 5. Pietruszczak, S., Mroz, Z.: Formulation of anisotropic failure criteria incorporating a microstructure tensor. Comput. Geotech. 26(2), 105–112 (2000) 6. Lade, P.V.: Failure criterion for cross-anisotropic soils. J. Geotech. Geoenviron. Eng. 134(1), 117–124 (2008) 7. Gao, Z., Zhao, J., Yao, Y.: A generalized anisotropic failure criterion for geomaterials. Int. J. Solids Struct. 47(22–23), 3166–3185 (2010) 8. Li, X.S., Dafalias, Y.F.: Anisotropic critical state theory: role of fabric. J. Eng. Mech. 138(3), 263–275 (2012)

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9. Tian, Y., Yao, Y.P.: Constitutive modeling of principal stress rotation by considering inherent and induced anisotropy of soils. Acta Geotechnica 13(6), 1299–1311 (2018) 10. Yang, Z., Liao, D., Xu, T.: A hypoplastic model for granular soils incorporating anisotropic critical state theory. Int. J. Numer. Anal. Methods Geomech. 44(6), 723–748 (2020) 11. Wu, W., Bauer, E.: A simple hypoplastic constitutive model for sand. Int. J. Numer. Anal. Methods Geomech. 18(12), 833–862 (1994) 12. Wu, W., Bauer, E., Kolymbas, D.: Hypoplastic constitutive model with critical state for granular materials. Mech. Mater. 23(1), 45–69 (1996) 13. Bauer, E.: Calibration of a comprehensive hypoplastic model for granular materials. Soils Found. 36(1), 13–26 (1996) 14. Wu, W., Lin, J., Wang, X.: A basic hypoplastic constitutive model for sand. Acta Geotechnica. 12(6), 1373–1382 (2017) 15. Li, X.S., Wang, Y.: Linear representation of steady-state line for sand. J. Geotech. Geoenviron. Eng. 124(12), 1215–1217 (1998) 16. Von Wolffersdorff, P.A.: A hypoplastic relation for granular materials with a predefined limit state surface. Mech. Cohesive-frictional Mater. Int. J. Exp. Model. Comput. Mater. Struct. 1(3), 251–271 (1996) 17. Yao, Y., Lu, D., Zhou, A., Zou, B.: Generalized non-linear strength theory and transformed stress space. Sci. China Ser. E: Technol. Sci. 47(6), 691–709 (2004) 18. Wu, W., Niemunis, A.: Failure criterion, flow rule and dissipation function derived from hypoplasticity. Mech. Cohesive-frictional Mater. Int. J. Exp. Model. Comput. Mater. Struct. 1(2), 145–163 (1996) 19. Dafalias, Y.F.: Must critical state theory be revisited to include fabric effects? Acta Geotechnica 11(3), 479–491 (2016) 20. Petalas, A.L., Dafalias, Y.F., Papadimitriou, A.G.: SANISAND-F: sand constitutive model with evolving fabric anisotropy. Int. J. Solids Struct. 188, 12–31 (2020) 21. Li, X., Li, X.S.: Micro-macro quantification of the internal structure of granular materials. J. Eng. Mech. 135(7), 641–656 (2009)

Phase-field Modeling of Brittle Failure in Rockslides Yunteng Wang, Shun Wang, Enrico Soranzo, Xiaoping Zhou, and Wei Wu

Abstract In this paper, we use a phase-field model with the mixed-mode fracture driving forces to study the brittle failure in rockslides. Our phase-field model is capable of predicting the crack initiation, propagation, and coalescence in rock/rock-like materials under different loading conditions. The phase-field model is validated against the experimental data in uniaxial compression tests.We proceed to simulate the brittle failure in rock slopes with three representative discontinuous structures, i.e., en-echelon joints, locking sections, and randomly distributed fissures. The numerical model provides insights into the failure mechanism of rockslides. Keywords Phase-field model · Mixed-mode fracture · Rock materials · Slope stability · Rockslides

Y. Wang · E. Soranzo · W. Wu (B) Institut für Geotechnik, Universität für Bodenkultur Wien, Feistmantelstraße 4, 1180 Vienna, Austria e-mail: [email protected] Y. Wang e-mail: [email protected] E. Soranzo e-mail: [email protected] S. Wang State Key Laboratory of Water Resources Engineering and Management, Wuhan University, Wuhan 430072, People’s Republic of China e-mail: [email protected] X. Zhou School of Civil Engineering, Chongqing University, Chongqing 40045, People’s Republic of China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 W. Wu and Y. Wang (eds.), Recent Geotechnical Research at BOKU, Springer Series in Geomechanics and Geoengineering, https://doi.org/10.1007/978-3-031-52159-1_16

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1 Introduction Rockslides are major geohazards in mountainous regions all over the world, e.g., Austria [1], Italy [2], China [3, 4], Greece [5] etc. Geological observations indicate that rockslides usually occur along surfaces formed by interactions among discontinuous structures in rock bridge zones [6–8], as shown in Fig. 1a–d. The failure behavior of rock bridge zones plays a critical role in rock slope stability, and better understanding of the brittle failure mechanism in rock bridge zones is vital to predicting rockslides. Due to the importance and complexities in rock sliding evolution, the mechanical behavior and failure characteristics of rock bridge zones have attracted much attention in the past. Obviously, the essence of triggering rockslides is the brittle failure behavior in rock bridge zones [8, 9]. Over the past decades, many efforts were devoted to understanding the brittle failure mechanism in rock bridge zones through in-situ observations [3, 8, 10–12] and laboratory experiments [9, 13–15, 17]. For example, [3] proposed a three-sections model, including sliding, tension cracking, shearing, to demonstrate the rock sliding mechanism in western China. According to the in-situ geological study, [12] constructed a detailed energy budget of a rock avalanche at Lake Coleridge, New Zealand [16]. Huang et al. [9] presented a physical experimental study on the brittle failure mechanism of the locking section in the large-scale rockslides [14]. Numerical models offer an alternative approach to study brittle failure mechanism in rock bridge zones. In recent years, discrete element method (DEM) [19] and extended finite element method (XFEM) [20] represent two major numerical methods for this purpose [18, 21–27]. For instance, Zhou and Chen [28] reported a XFEM with .J-integral calculation to study the step-path failure mechanism of rock slopes

Fig. 1 Photographs of rock slope instability in a US national park [8], b Xiaowan hydroelectric station, China [18], c, d Xinguokeng mining field, Fujian, China; and e schematic diagram of rock slope with multiscale discontinuous structures

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with non-persistent en-echelon joints. Chen et al. [29] utilized DEM to investigate the effect of critical tension crack depth on the triggering failure mechanism in rockslides. However, some drawbacks in XFEM and DEM have been noticed during their applications to brittle failure in rockslides. For example, XFEM cannot efficiently model interactions among multiple cracks and requires much computational effort in calculating the crack morphology. DEM cannot capture the crack growth paths properly, giving rise to rather crude prediction of potential sliding surfaces in rockslides. In addition, the microscopic parameters in DEM simulations often require the complicated calibration procedures. Phase-field theory is a promising computational approach for simulating brittle failure, where crack growth paths can be automatically determined. Based on Griffith’s theory [30], the sets of cracks is modeled by minimizing the total potential energy of solids. [31]. The sharp discontinuous structures are regularized as diffuse damage zones by the phase-field variable .d and inherent characteristic length .l [32, 33], as shown in Fig. 1e. The diffuse damage zones correspond to fracture progress zones (FPZ), which consist of main cracks and microscopic fissures/voids [34]. Furthermore, phase-field theory possesses some merits over other numerical methods, e.g., mesh-independence, and without ad hoc bifurcation criteria, in modeling crack initiation and propagation [35–37]. These advantages make it a promising approach to address some problems in solid fracture mechanics [38–42]. However, the existing phase-field models mainly focused on the tensile fracture and cannot properly describe the complex brittle failure behavior of rocks. More recently, some phase-field models were developed to simulate cracks in rocks by modifying fracture driving forces [43, 44]. This idea [44] was then widely adopted to simulate the mixed-mode cracks [45–56]. Moreover, double-phase-field models were proposed to differentiate the tensile and shear cracks by introducing two scalar phase-field variables [51, 52]. Zhuang et al. [57] reported laboratory experiments and phase-field simulations on the deformation and failure behavior of rocks. The Mohr-Coulomb failure criterion was also implemented to study rock fracture in uniaxial compression tests [47], Brazilian tests [54] and fluid-injection tests [49]. Despite these noteworthy works, however, the compressive-shear mixedmode fracture initiation and propagation in rocks cannot be modeled properly in phase-field simulations. In this paper, we present a phase-field model to study the triggering mechanism of brittle failure in rockslides based on our previous work [55]. The driving forces for mixed-mode fractures are derived by modifying Benzeggagh-Kenane failure criterion. To our knowledge, this work is the first application of phase-field model to predict the rock slope instabilities. This paper is organized as follows: our phase-field framework is briefly stated in Sect. 2. Numerical validation is presented in Sect. 3. In Sect. 4, the brittle failure mechanism of rock slopes is studied. Some conclusions are drawn in Sect. 5.

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2 Phase-Field Framework for Mixed-Mode Fracture 2.1 Geometry and Variational Approach to Rock Damage The key features of phase-field model for brittle failure in elasticity are briefly summarized. Let us consider the elastic domain .Ω ⊆ Rn containing the discontinuous structures .Γd . The computational domain .Ω is bounded by .∂Ω, which is divided into the prescribed displacement boundary .∂Ωu and the prescribed traction boundary.∂Ωt , i.e.,.∂Ω = ∂Ωu ∪ ∂Ωt and.∂Ωu ∩ ∂Ωt = ∅. The applied displacement field .u ¯ and the external surface traction .t¯ are prescribed on .∂Ωu and .∂Ωt , respectively, as shown in Fig. 2. In the phase-field theory [31–33], the sharp crack topology is described in the diffuse one (see Fig. 2) by introducing the crack surface density function .γ (d, ∇d), which reads γ (d, ∇d) =

.

1 2 l d + |∇d|2 2l 2

(1)

( ) where(.d )is phase-field variable representing rock damage, i.e., .d x = 0 for intact and .d x = 1 for complete failure. Griffith’s criterion in Linear Elastic Fracture Mechanics is generalized in the phase-field framework [31–33, 37], the total potential energy can be recast as { ( ( ) ) ( ( ) ) { E ε x ,d = ψl ε x , d dΩ + Gc γ (d, ∇d) dΩ Ω Ω { { . ( ) ( ) − b x · u x dΩ − t¯ · u (x) d S Ω

Fig. 2 Schematic illustration of diffuse crack topologies in phase-field framework

∂Ωt

(2)

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where .ψl is the stored elastic strain energy density after damage (degradation, .Gc ) represents the critical fracture energy release rate of solids and .b x denotes the body force density.

2.2 Mixed-Mode Fracture Driving Forces To simulated the mixed-mode fracture behaviors in rocks, a modified energy-based failure criterion based the Benzeggagh-Kenane semi-empirical failure criterion [58] was proposed in our previous work [55], where critical value of the equivalent mixedmode fracture energy release rate yields (

GT c

.

HI I = G I c + (G I I c − G I c ) H+ I + HI I

)m (3)

where .G I c and .G I I c are, respectively, the critical fracture energy release rates for the pure mode-I and pure mode-II fracture types; .m denotes the empirical parameter, where .m = 2.6 is adopted. Thus, in our phase-field model, the modified mixed-mode fracture driving force reads ( + ) HI HI I .Hm = + (4) η GI c GI I c Hm (G I c G I I c ) where .η = G H is a parameter for avoiding numerical instabilities. It + Tc( I I G I c +H I G I I c ) is worth noting that fracture driving forces in the classical phase-field models [33, 36] are the special cases of Eqs. 3 and 4 when .G I c = G I I c . Thus, our model is a generalized version and is available for the cases with .G I I c > G I c , which is capable of describing the complex failure characteristics in geological media. To satisfy the crack irreversibility principle, the fracture driving forces are stated as { ( ) 2} 1 sph + sph e sph f t ε :K :ε , Tr(ε) ≥ 0 ε , d = max .H I (5a) 2 2E ( ) − εsph , d = 0 .H I Tr(ε) < 0 (5b) } { ( ) 1 dev dev ε : K e : εdev .H I I ε , d = max (5c) 2

in which . K e represents the fourth-order tensor that characterizes the stiffness of any element, . f t denotes the uniaxial tensile strength of rocks, and . E is the Young’s modulus.

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2.3 Governing Equations In summary, the governing equations of our phase-field model for the mixed-mode fracture in the elastic, isotropic and homogeneous solids are: • Kinematic admissibility and compatibility: ( ) 1[ ( ) ( )] ∇ · u x +T ∇ · u x , ε x = ( ) 2sph ( ) ( ) x + εdev x , .ε x = ε ( ) ¯ .u x = u .

∀x ∈ Ω

(6)

∀x ∈ Ω

(7)

∀x ∈ ∂Ωu

(8)

• Static admissibility: ( ( ) ( )) ( ) ∇ · σ ε x ,d x + b x = 0 ( ( ) ( )) ( ) · n x = t¯ .σ ε x , d x .

∀x ∈ Ω

(9)

∀x ∈ ∂Ωt

(10)

• Mass conservation: ) ( + ( ) ( ( )) ( ) HI HI I η + l2 ∇ 2 d x − d x = 0 l .2 1 − d x + GI c GI I c ( ) ( ) .∇d x · n x = 0

∀x ∈ Ω

(11)

∀x ∈ ∂Ω (12)

• Constitutive equations: ( ( ) ) ∂ψl ε x , d ∂ψ I+0 ∂ψ I−0 ∂ψ I I 0 ( ) ( ) + g (d) ( ) + ( ) = g (d) σ u, d = ∂ε x ∂ε x ∂ε x ∂ε x . (

)

∀x ∈ Ω

(13)

= g (d) K e : ε sph+ + g (d) K e : ε dev + K e : ε sph−

in which .ε sph = εsph+ + εsph− , the symbol of .± represents the tensile/compression part of strain tensors, and .g (d) is the degradation function.

2.4 Solution Strategy In the computational domain .Ω, the continuous Galerkin finite elements is used to obtain the displacement field and phase-field variable for the quasi-static conditions.

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The numerical solver aims at finding the admissible displacement field .u and the admissible phase-field variable.d based on the principle of minimum potential energy, which reads { ( ) ( )} ( ( ) ( )) = arg min E u x , d x (14) . u x ,d x u∈Su &d∈Sd

( ) } { { ( ) } where .Su = u|u x = u¯ on ∂Ωu and .Sd = d x |∇d · n = 0 on ∂Ω are the admissible displacement field space and the admissible phase-field space, respectively. The nonlinear solver with the fully monolithic solution strategy is employed to solve the following system of governing equations. [ .

Kuu Kud Kdu Kdd

] [ u] R δu = δd Rd

][

(15)

where .Kuu , .Kdu , .Kud and .Kdd are components of global stiffness matrix, referring to [55]; .Ru = 0 and .Rd are two objectives of this nonlinear optimization problem in Eq. 14.

3 Numerical Validation of Laboratory Tests To show the performance of our model, several fissured rock-like samples in the uniaxial compression tests are simulated. There are two kinds of fissured samples with dimensions of .76 mm .× .152 mm, including one with a single preexisting fissure, and the other ones containing three preexisting fissures, as shown in Fig. 3a, b, respectively. The bottom boundary is fixed, i.e.,.u = 0, and a prescribed displacement fields of .u˙ y = 0.002 mm/s is applied on the top one. The geometries of samples are depicted in Fig. 3, and the details, involving fissure length .2a and fissure inclination angles .α, .β1 .β2 , .β3 , are summarized in Table 1. Following [59], the rock-like material properties are chosen as follows: Young’s modulus . E = 30 MPa, Possion’s ratio .ν = 0.3, mode-I fracture energy release rate 2 2 .G I c = 5.0 J/m. and mode-II fracture energy release rate .G I I c = 13.0 J/m. . The four

Table 1 Geometric details of the preexisting fissures Number of .2a .α No. fissures I II III IV

One Three Three Three

12.7 mm 12.7 mm 12.7 mm 12.7 mm

45.◦ – – –

.β1

.β2

.β3

– 45.◦ 120.◦ 90.◦

– 45.◦ 45.◦ 150.◦

– 45.◦ 45.◦ 30.◦

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Fig. 3 Geometry and boundary conditions of fissured rock-like specimens in the uniaxial compression tests: a single preexisting fissure and b three preexisting fissures

Fig. 4 Progressive failure process of fissured rock-like sample-I: a crack growth paths, b maximum principle stress, and c comparisons with [59]

fissured samples are discretized into irregular triangular elements with the minimum size of .h min = 0.5 mm and the maximum size of .h max = 6.0 mm. The characteristic length in the phase-field modelling is adopted as .l = 1.0 mm. The FE elements in these four fissured rock-like samples are 17,550; 20,802; 22,816; and 23,944, respectively. Numerical results of four fissured rock-like specimens subjected to the uniaxial compressive loads are plotted in Figs. 4, 5, 6 and 7. Figure 4 presents the progressive failure process of rock-like sample-I containing one single preexisting fissure with an inclination angle of .α = 45◦ . The evolutions of crack growth paths and maximum principal stress distributions are plotted in Fig. 4a, b, respectively. The tensile wing

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Fig. 5 Progressive failure process in the fissured rock-like sample-II: a crack growth paths, b maximum principle stress, and c comparisons with [59]

Fig. 6 Progressive failure process in the fissured rock-like specimen-III: a crack growth paths, b maximum principle stress, and c comparisons with [59]

cracks driven by the maximum principal stress and the compressive-shear secondary cracks can be successfully reproduced. For validation, the current phase-field modelling results are compared with the previous experimental data [59], as shown in Fig. 4c. The predicted fracture pattern agrees well with that observed in laboratory tests. The axial load—axial displacement curve obtained from simulations also show a reasonable agreement with that measured from experiments. Figure 5a shows the progressive failure process of the rock-like sample-II containing three preexisting fissures with the identical incline angles of 45.◦ . As observed in Fig. 5(a-1), (a-2), wing cracks are initiated from tips of the preexisting fissures, and propagate along the axial compression. With the increase of axial loading, wing cracks coalesce with secondary cracks in the rock bridge regions, as shown in Fig. 5(a-3), (a-4). The spatiotemporal distribution of .σ1 in Fig. 5b indicates that wing

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Fig. 7 Progressive failure process in the fissured rock-like specimen-IV: a crack growth paths, b maximum principle stress, and c comparisons with [59]

cracks are driven by tensile stress, while the initiation and propagation of secondary cracks are caused by the concentrations of shear stresses. The comparison between numerical and experimental results [59] of crack coalescence patterns with corresponding axial load - axial displacement curves implies the reliability of our model. Moreover, numerical results of sample-III with .(β1 , β2 , β3 ) = (120◦ , 45◦ , 45◦ ) and sample-IV with .(β1 , β2 , β3 ) = (90◦ , 15◦ , 30◦ ) are presented in Figs. 6 and 7, respectively. The stress distribution reveals the crack coalescence mechanism in rock bridge zones, which provides better understanding of brittle failure characteristics at different loading stages. A reasonable agreement between numerical simulation and laboratory tests [59] is achieved, which further demonstrates the capability of our model in simulating complex brittle failure behavior in rock/rock-like materials.

4 Simulations of Brittle Failure in Rockslides Geological and geophysical evidences indicate that the natural discontinuous structures in rock bridge regions play key roles in triggering failure initiation, forming the potential compound fracture surfaces and rockslides/rock avalanches [3, 6, 7, 60, 61]. In this section, numerical rock slope stability analysis with three different discontinuous structures are considered to study the triggering failure mechanism of rockslides. The progressive failure process of en-echelon joints and brittle failure mechanism of locking sections in rockslides are comprehensively studied by carrying out a series of simulations.

4.1 En-Echelon Joints in Rock Slopes The en-echelon joints are the typical planar non-persistent/intermittent discontinuous structures, which widely exist in rock slopes, and may cause step-like failure

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Fig. 8 Schematic diagram of a rock slope with en-echelon joints: a geometry of a rock slope (unit: mm) and b mesh discretization in numerical simulations

paths/surfaces in rockslides, e.g., the Aishihik River landslide [62] and the rock avalanche near Xiaowan hydroelectric station, China [18]. The step-path failure characteristics of en-echelon fissures in rock bridge zones attracts much attention due to its importance on understanding rockslides and rock avalanches [8, 18, 28, 63]. In the first numerical engineering application, our phase-field model is applied to analyze the progressive step-failure paths in rock slopes with en-echelon joints. Geometry of the rock slope containing the en-echelon joints with the length of 48.0 m and the height of 32.0 m is plotted in Fig. 8a. A set of en-echelon joints consists of four parallel fissures with the inclination angles .β and fissure lengths .l j summarizing in Fig. 8a. The rock bridges have different angles .γ and lengths .lb . The vertical displacement at the bottom boundary and horizontal displacement at the right boundary are prescribed. In all simulations, the gravitational force is gradually increased to trigger failure initiation of the rock slopes. The physical and mechanical parameters are given as follows: rock mass density 3 .ρ = 2, 640 kg/m. , Young’s modulus. E = 4.20 GPa, Poisson’s ratio.ν = 0.30, modeI fracture energy release rate .G I c = 5.0 J/m.2 and mode-II fracture energy release rate 2 .G I I c = 25.0 J/m. . The rock slope is discretized into triangular FE-meshes with the maximum size of .h max = 2.50 m and the minimum size of .h min = 0.15 m, resulting in 11,011 elements, as shown in Fig. 8b. The internal characteristic length is adopted as .l = 0.30 m, and the computational time step is employed as .Δt = 1.0 × 10−6 s. In addition, nine monitoring points are set in three rock bridge zones to present the brittle failure characteristics in rockslides. The numerical predictions of progressive step-path failure are depicted in Fig. 9. To reveal/the rock sliding initiation mechanism, the relevant displacement fields (i.e., .u = u 2x + u 2y ), maximum principal stress fields (i.e., .σ1 ) and maximum shear stress fields (i.e., .τmax ) obtained from the presented numerical simulations are shown in Figs. 10, 11 and 12 during the step-path failure sliding process. At the initial stage, due to the gravity, the effective displacement fields at the rock slope top are larger than those at the toe (see Fig. 10a). It can be observed from Figs. 11a and 12a that the major principal stress fields and the maximum shear stress fields are concentrated around the inner tip of Joint-I near the toe, which causes the mixed tensile-shear crack initiation and propagation towards the lower tip of Joint-II, as shown in Fig. 9a. With the increase of the gravity, the mixed tensile-shear crack

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Fig. 9 Numerical prediction of crack growth paths in rock slopes with en-echelon joints

Fig. 10 Numerical predicted displacement fields, i.e., .u = cess of rock slopes with en-echelon joints. (Unit: mm)

/

u 2x + u 2y , in the step-path failure pro-

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Fig. 11 Numerical predicted maximum principal stress fields in the step-path failure process. (Unit: MPa)

coalescence mode happens in the rock bridge-I between the Joint-I and Joint-II (Fig. 9b). Due to the linkage between two preexisting joints, it can be observed from Fig. 10b that the total displacement fields at the rock fall spalling part increase. At the same time, concentrations of the major principal stress fields (see Fig. 11b) and the maximum shear stress fields (Fig. 12b) at the upper tip of Joint-II lead to the initiation and propagation of the mixed-mode cracks in the rock bridge-II, as shown in Fig. 9b. After that, because of the stress redistribution (see Figs. 11c, d and 12c, d), the occurrence of similar cracking behaviors in the rock bridge-III induces the coalescence of Joint-III and Joint-IV, which results in the ultimate rock sliding failure surface. The large part of the total displacement fields move from the rock slope top to the toe, as shown in Fig. 10c. Although the mixed tensile-shear crack coalescence happens in the three rock bridge zones (see Figs. 11 and 12), the shear cracks dominate the linkage of preexisting joints in the rock slopes. Moreover, when the ultimate noncoplanar failure surface is formed, the rock masses slide along the noncoplanar failure surface, see Fig. 10d. In a rock slope, the rock bridge suffers from stress concentration, which prevents rock mass from sliding. This phenomenon is relevant to the propagation and coalescence of cracks between discontinuities. The in-situ and laboratory observations indicates that the accelerated deformation state of rock bridge zone can be regarded as a precursor for predicting rockslides [17, 70, 71]. Therefore, the vertical and horizontal stress states at the monitoring points, i.e., . A1 ∼ A3, . B1 ∼ B3 and .C1 ∼ C3,

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Fig. 12 Numerical predicted maximum shear stress fields in the step-path failure process. (Unit: MPa)

respectively pre-located in the rock bridge-I, rock bridge-II and rock bridge-III, are recorded during the rock sliding process, which are depicted in Fig. 13. The positions of dashed lines correspond to the failure states in Figs. 9, 10, 11 and 12. The variation of the measured stress states in Fig. 13 indicates the stress concentration dominates the crack initiation and propagation, and the stress redistribution leads to the crack initiation in the upper rock bridges. When the crack coalescence happens in the rock bridge zones, the sudden stress drops appear and the three rock bridges are cut through to produce rockslides.

4.2 Brittle Failure of Locking Section in Rockslides Geohazard evidence indicates that the locking section plays an important role in rock slope stability and may subsequently result in the catastrophic failure [8, 9, 64–68]. In this section, numerical simulations of rockslides with locking sections are carried out to study the triggering mechanism. Configuration of the numerical model with .48.0 m in length and .32.0 m in height is presented in Fig. 14a. The preexisting tensile crack with length of .12.0 m, and bottom crack with length of .15.0 m and inclination angle of .β = 30◦ both contribute

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Fig. 13 Spatiotemperal evolution of the stress fields during the rock slope step-path failure process: a–c vertical stress-time curves and d–f horizontal stress-time curves at the monitoring points plotted in Fig. 8b

to form the locking section, as shown in Fig. 14a. The boundary conditions are the same as those in Sect. 4.1. Rock material properties are adopted based on physical tests [9], including material density .ρ = 2, 700 kg/m.3 , Young’s modulus . E = 55.0 GPa, Poisson’s ratio 2 .ν = 0.13 and mode-I fracture energy release rate .G I c = 5.0 J/m. . To study the effect of difference between mode-I and mode-II fracture energy release rate on the rock sliding failure of locking section in rock slopes, four rock slope samples with

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Fig. 14 Sketch of the rock slope with locking section and its features: a geometry and b FE mesh-based discretization

the same geometric conditions and various ratios of .G I I c /G I c , i.e., .G I I c /G I c = 2.0, G I I c /G I c = 3.0, .G I I c /G I c = 4.0, and .G I I c /G I c = 6.0, are simulated. This rock slope is discretized into the irregular triangular meshes with the maximum size .h max = 2.0 m and the minimum size of .h min = 0.1 m, as shown in Fig. 14b. Furthermore, the internal characteristic length is employed as .l = 0.20 m, and the computational time step is adopted as .Δt = 1.0 × 10−7 s in all simulations. The progressive failure process of crack growth paths in the rock slope with .G I I c /G I c = 2.0 are plotted in Fig. 15. The relevant evolution of the maximum principal stress and maximum shear stress are presented in Fig. 16a, b, respectively. The shear cracks are first initiated from the tip of bottom fissure due to the concentration of maximum shear stress. With the increase of gravity loading, the trajectories of shear cracks emanating from the bottom fissure tip at the slope toe transfer from the direction of bottom fissure to the direction of gravity. This change is caused by the concentrated degree of maximum principal stress around crack tips, as shown in Figs. 15b and 16. Then, the concentration of maximum principal stress around tensile fissure tip at the rock slope top leads to the initiation and propagation of tensile cracks. Meanwhile, tensile failure damage occurs around the shear cracks initiated from bottom fissure, and shear damage happens around the tensile cracks emanating from the top tensile fissure. Finally, the ultimate rock sliding surface is created by the coalescence between the tensile and shear cracks in the locking section, which belongs to the mixed tensile-shear failure surface, as shown in Fig. 15d. The predicted failure surfaces of rock slopes with different material properties .G I I c /G I c are shown in Fig. 17. When the ratio .G I I c /G I c changes from 2 to 4, cracks initiated from the bottom fissure tip transfer from the shear cracks to the tensile ones in the region of rock slope toe, as shown in Fig. 17a–c. While, shear cracking trajectories emanating from the tensile fissure tips remain in the region of rock slope top. Thus, the final sliding failure surface is the product by the mixed tensileshear crack coalescence in the locking sections. However, when .G I I c /G I c = 6.0, .

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Fig. 15 Numerical prediction of crack growth paths in the locking region during the rock sliding process with .G I I c /G I c = 2.0

Fig. 16 Evolutions of the internally physical variables during rockslides: a maximum principal stress fields (unit: MPa) and b maximum shear stress fields (Unit: MPa)

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Fig. 17 Ultimated rock sliding surfaces for locking sections with different fracturing parameters .G I I c /G I c

The ultimate sliding failure surface is formed by the shear coalescence in the locking section (see Fig. 17d).

4.3 Multiple Discontinuities in Rockslides Field-scale evidence demonstrates the presence of rock bridge regions with multiple discontinuities, i.e., discrete fracture networks, in the realistic rock mass slopes [8, 69]. The discrete fracture network is a reliable tool for analyzing rock slope stability [8]. To study the triggering failure mechanism of rock bridge regions in rockslides, the discrete fracture network are randomly distributed in the numerical rock slope model with .48.0 m in length and .32.0 m in height, as shown in Fig. 18a. The mechanical properties of brittle rock materials and boundary conditions are the same in Sect. 4.1. The computational domain of rock slopes with multiple discontinuities is discretized into 20,632 irregular triangular elements (see Fig. 18b), where the internal characteristic length is .l = 0.5 m. Crack growth paths and the associated displacement field evolution obtained from the phase-field simulations are depicted in Figs. 19 and 20, respectively. The progressive failure process in Fig. 19a suggests the tensile and shear cracks are first initiated from the tips of fissures around the slope toes. With the increase of loads, tensile cracks propagate along the direction of maximum principal stress, while shear

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Fig. 18 a Geometry of the rock slope with randomly distributed fissures and b FE mesh-based discretization in simulations

Fig. 19 Numerical predicted crack growth paths of discrete fracture networks in rock slopes

cracks extend in the approximately parallel direction, as shown Fig. 19b. Then, Fig. 19c indicates crack coalescence happens in some rock bridge regions. Finally, the coalescence of cracks and the initial discrete fracture networks both contribute to the compound sliding failure surfaces formation in rock slopes (see Fig. 19d), which may result in sudden rockslides. From the evolutionary process of rockslides observed from Fig. 19, it can be speculated that the critical rock bridge zones contribute to the

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Fig. 20 Numerical prediction of the displacement fields, i.e., .u = multiple discontinuities. (Unit: mm)

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/ u 2x + u 2y in rock slopes with

sliding surfaces formation, while the non-critical rock bridges may be involved only in the fragmentation of the rock masses during movements, which is similar to the geological observations [8]. Figure 20a, b show the evolution of displacement fields in the sliding failure surfaces formation. The large deformation concentrates in the rock slope top region at the failure initiation and propagation stages. After the occurrence of crack coalescence and formation of discrete fragments at the toe, the concentration of large deformation transfers from top to the toe. When the ultimate compound failure surfaces happen, the sudden displacement field can be observed in Fig. 20d, which indicates the mass movements in rockslides. The progressive failure process observed from the numerical simulation uncover the brittle failure mechanism of rockslides, which can be summarized as follows: • The initiation of tensile and shear cracks occurs at the tips of discrete fracture networks around the rock slope toe. • The mixed tensile shear crack coalescence in the rock bridge regions makes the occurrence of initial fragments around the rock slope toe, which results in the large displacement fields around rock slope toe. • The coalescence of crack near rock slope toe and discrete fracture networks around rock slope top commonly formulate the compound rock sliding surface.

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5 Conclusions A phase-field model with modified mixed-mode fracture driving forces is applied to study the triggering brittle failure mechanism of rockslides. This model is capable of predicting pure tensile, pure shear, and tensile/compressive-shear mixed-mode cracks in geological media, which is validated through several numerical examples. The satisfactory agreement between the current numerical results and the reported experimental data demonstrates the reliability of our model. The developed model is further applied to study the triggering mechanism of brittle failure in rock slopes with representative discontinuous structures. Some findings are drawn as follows: • The shear crack coalescence dominates the step-path failure of en-echelon fissures in rock slopes, which is accompanied by the sudden stress drops. • The ratio.G I I c /G I c plays an important role in controlling the brittle fracture patterns in the locked section, where crack growth paths is a semi-circular arc instead of the straight-line failure paths. • For the rock slope with multiple randomly distributed fissures, the crack coalescence in critical rock bridge zones mainly contributes to the formation of rock sliding surfaces, while the non-critical rock bridges only affect the moving fragments in rockslides. Acknowledgements The authors wish to acknowledge the financial support from FWF Lise Meitner Project—MultiCBPR (Grant No. M 3340-N), Fundamental Research Funds for the Central Universities (Grant No. 2042023kfyq03), FWF Principal Investigator Project—HIME (Grant No. P 37175-N), OeAD WTZ project (Grant Nos. CN 04/2022, CN14/2021 and FR 03/2024), European Commission Horizon Europe Marie Skłodowska–Curie Actions Staff Exchanges Project—LOC3G (Grant No. 101129729), Otto Pregl Foundation of Fundamental Geotechnical Research in Vienna and National Key Research and Development Program of China (Grant No. 2023YFC2907200).

References 1. Rechberger, C., Zangerl, C.: Rock mass characterisation and distinct element modelling of a deep-seated rock slide influenced by glacier retreat. Eng. Geol. 300, 106584 (2022) 2. Tarchi, D., Antonello, G., Casagli, N., Farina, P., Fortuny-Guasch, J., Guerri, L., Leva, D.: On the use of ground-based SAR interferometry for slope failure early warning: the Cortenova rock slide (Italy). Landslides: Risk Analysis and Sustainable Disaster Management, pp. 337–342 (2005) 3. Huang, R.: Mechanisms of large-scale landslides in China. Bull. Eng. Geol. Environ. 71(1), 161–170 (2012) 4. Li, Y., Chen, J., Zhou, F., Bao, Y., Li, Z., Song, S., Cao, C.: Stability evaluation of rock slope based on discrete fracture network and discrete element model: a case study for the right bank of Yigong Zangbu Bridge. Acta Geotechnica 17(4), 1423–1441 (2021) 5. Christaras, B., Dimitriou, A., Dimopoulos, G., Pavlides, S., Marinos, P.: World heritage “Monasteries of Mount Athos” at rock slide risk, in Greece. Landslides: Risk Analysis and Sustainable Disaster Management, pp. 235–239 (2005)

262

Y. Wang et al.

6. Einstein, H.H., Veneziano, D., Baecher, G.B., O’reilly, K.J.: The effect of discontinuity persistence on rock slope stability. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 20(5), 227–236. Pergamon (1983, Oct) 7. Eberhardt, E., Stead, D., Coggan, J.S.: Numerical analysis of initiation and progressive failure in natural rock slopes-the 1991 Randa rockslide. Int. J. Rock Mech. Min. Sci. 41(1), 69–87 (2004) 8. Elmo, D., Stead, D., Yang, B., Marcato, G., Borgatti, L.: A new approach to characterise the impact of rock bridges in stability analysis. Rock Mech. Rock Eng. 55(5), 2551–2569 (2022) 9. Huang, R., Chen, G., Guo, F., Zhang, G., Zhang, Y.: Experimental study on the brittle failure of the locking section in a large-scale rock slide. Landslides 13, 583–588 (2016) 10. Spreafico, M.C., Franci, F., Bitelli, G., Borgatti, L., Ghirotti, M.: Intact rock bridge breakage and rock mass fragmentation upon failure: quantification using remote sensing techniques. Photogramm. Rec. 32(160), 513–536 (2017) 11. Elmo, D., Donati, D., Stead, D.: Challenges in the characterisation of intact rock bridges in rock slopes. Eng. Geol. 245, 81–96 (2018) 12. Davies, T.R., Reznichenko, N.V., McSaveney, M.J.: Energy budget for a rock avalanche: fate of fracture-surface energy. Landslides 17, 3–13 (2020) 13. Zhong, Z., Huang, D., Zhang, Y., Ma, G.: Experimental study on the effects of unloading normal stress on shear mechanical behaviour of sandstone containing a parallel fissure pair. Rock Mech. Rock Eng. 53, 1647–1663 (2020) 14. Lin, F., Wu, L.Z., Huang, R.Q., Zhang, H.: Formation and characteristics of the Xiaoba landslide in Fuquan, Guizhou, China. Landslides 15, 669–681 (2018) 15. Tang, P., Chen, G.Q., Huang, R.Q., Zhu, J.: Brittle failure of rockslides linked to the rock bridge length effect. Landslides 17(4), 793–803 (2020) 16. Lee, J., Davies, T., Bell, D.: Successive Holocene rock avalanches at Lake Coleridge, Canterbury, New Zealand. Landslides 6, 287–297 (2009) 17. Qin, C.A., Chen, G., Zhu, J., Tang, P.: A precursor of bedding rockslide: rock spalling in the key block triggered by tensile cracks. Bull. Eng. Geol. Environ. 79, 2513–2528 (2020) 18. Huang, D., Cen, D., Ma, G., Huang, R.: Step-path failure of rock slopes with intermittent joints. Landslides 12, 911–926 (2015) 19. Potyondy, D.O., Cundall, P.A.: A bonded-particle model for rock. Int. J. Rock Mech. Min. Sci. 41(8), 1329–1364 (2004) 20. Moës, N., Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng. 46(1), 131–150 (1999) 21. Zhou, Y., Ma, W., Tan, X., Chen, W., Yang, D., Su, Z., ..., Xu, F.: Numerical simulation of fracture propagation in freezing rocks using the extended finite element method (XFEM). Int. J. Rock Mech. Min. Sci. 148, 104963 (2021) 22. Bao, Y., Li, Y., Zhang, Y., Yan, J., Zhou, X., Zhang, X.: Investigation of the role of crown crack in cohesive soil slope and its effect on slope stability based on the extended finite element method. Natural Hazards, 1–20 (2021) 23. Ding, J., Yu, T., Bui, T.Q.: Modeling strong/weak discontinuities by local mesh refinement variable-node XFEM with object-oriented implementation. Theor. Appl. Fract. Mech. 106, 102434 (2020) 24. Camones, L.A.M., do Amaral Vargas Jr, E., de Figueiredo, R.P., Velloso, R.Q.: Application of the discrete element method for modeling of rock crack propagation and coalescence in the step-path failure mechanism. Eng. Geol. 153, 80–94 (2013) 25. Bonilla-Sierra, V., Scholtes, L., Donzé, F.V., Elmouttie, M.K.: Rock slope stability analysis using photogrammetric data and DFN-DEM modelling. Acta Geotechnica 10, 497–511 (2015) 26. Bonilla-Sierra, V., Scholtès, L., Donzé, F.V., Elmouttie, M.: DEM analysis of rock bridges and the contribution to rock slope stability in the case of translational sliding failures. Int. J. Rock Mech. Min. Sci. 80, 67–78 (2015) 27. Zhang, Z., Wang, T., Wu, S., Tang, H.: Rock toppling failure mode influenced by local response to earthquakes. Bull. Eng. Geol. Environ. 75, 1361–1375 (2016)

Phase-field Modeling of Brittle Failure in Rockslides

263

28. Zhou, X., Chen, J.: Extended finite element simulation of step-path brittle failure in rock slopes with non-persistent en-echelon joints. Eng. Geol. 250, 65–88 (2019) 29. Chen, G., Tang, P., Huang, R., Wang, D., Lin, Z., Huang, D.: Critical tension crack depth in rockslides that conform to the three-section mechanism. Landslides 18, 79–88 (2021) 30. Griffith, A.A.: VI. The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. Lond. Ser. A, Containing Papers of a Mathematical or Physical Character 221(582-593), 163–198 (1921) 31. Francfort, G.A., Marigo, J.J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46(8), 1319–1342 (1998) 32. Bourdin, B., Francfort, G.A., Marigo, J.J.: Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48(4), 797–826 (2000) 33. Miehe, C., Hofacker, M., Welschinger, F.: A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. Comput. Methods Appl. Mech. Eng. 199(45–48), 2765–2778 (2010) 34. Zhou, X.P., Zhang, J.Z., Qian, Q.H., Niu, Y.: Experimental investigation of progressive cracking processes in granite under uniaxial loading using digital imaging and AE techniques. J. Struct. Geol. 126, 129–145 (2019) 35. Ambati, M., Gerasimov, T., De Lorenzis, L.: A review on phase-field models of brittle fracture and a new fast hybrid formulation. Comput. Mech. 55, 383–405 (2015) 36. Wu, J.Y.: A unified phase-field theory for the mechanics of damage and quasi-brittle failure. J. Mech. Phys. Solids 103, 72–99 (2017) 37. Wu, J.Y., Nguyen, V.P., Nguyen, C.T., Sutula, D., Sinaie, S., Bordas, S.P.: Phase-field modeling of fracture. Adv. Appl. Mech. 53, 1–183 (2020) 38. Zhang, P., Hu, X., Bui, T.Q., Yao, W.: Phase field modeling of fracture in fiber reinforced composite laminate. Int. J. Mech. Sci. 161, 105008 (2019) 39. Quintanas-Corominas, A., Reinoso, J., Casoni, E., Turon, A., Mayugo, J.A.: A phase field approach to simulate intralaminar and translaminar fracture in long fiber composite materials. Compos. Struct. 220, 899–911 (2019) 40. Yang, Z.J., Li, B.B., Wu, J.Y.: X-ray computed tomography images based phase-field modeling of mesoscopic failure in concrete. Eng. Fract. Mech. 208, 151–170 (2019) 41. Saberi, H., Bui, T.Q., Furukawa, A., Rahai, A., Hirose, S.: FRP-confined concrete model based on damage-plasticity and phase-field approaches. Compos. Struct. 244, 112263 (2020) 42. Arguello, M.E., Gumulya, M., Derksen, J., Utikar, R., Calo, V.M.: Phase-field modeling of planar interface electrodeposition in lithium-metal batteries. J. Energy Storage 50, 104627 (2022) 43. Shen, B.T., Stephansson, O.: Modification of the G-criterion for crack propagation subjected to compression. Eng. Fract. Mech. 47(2), 177–189 (1994) 44. Zhang, X., Sloan, S.W., Vignes, C., Sheng, D.: A modification of the phase-field model for mixed mode crack propagation in rock-like materials. Comput. Methods Appl. Mech. Eng. 322, 123–136 (2017) 45. Bryant, E.C., Sun, W.: A mixed-mode phase field fracture model in anisotropic rocks with consistent kinematics. Comput. Methods Appl. Mech. Eng. 342, 561–584 (2018) 46. Zhou, S.W., Xia, C.C.: Propagation and coalescence of quasi-static cracks in Brazilian disks: an insight from a phase field model. Acta Geotechnica 14(4), 1195–1214 (2019) 47. Zhou, S., Zhuang, X., Rabczuk, T.: Phase field modeling of brittle compressive-shear fractures in rock-like materials: a new driving force and a hybrid formulation. Comput. Methods Appl. Mech. Eng. 355, 729–752 (2019) 48. Wang, Q., Feng, Y.T., Zhou, W., Cheng, Y., Ma, G.: A phase-field model for mixed-mode fracture based on a unified tensile fracture criterion. Comput. Methods Appl. Mech. Eng. 370, 113270 (2020) 49. Lepillier, B., Yoshioka, K., Parisio, F., Bakker, R., Bruhn, D.: Variational Phase-field modeling of hydraulic fracture interaction with natural fractures and application to enhanced geothermal systems. J. Geophys. Res. Solid Earth 125(7), e2020JB019856 (2020)

264

Y. Wang et al.

50. Zhou, S., Zhuang, X.: Phase field modeling of hydraulic fracture propagation in transversely isotropic poroelastic media. Acta Geotechnica 15(9), 2599–2618 (2020) 51. You, T., Zhu, Q.Z., Li, P.F., Shao, J.F.: Incorporation of tension-compression asymmetry into plastic damage phase-field modeling of quasi brittle geomaterials. Int. J. Plast. 124, 71–95 (2020) 52. Fei, F., Choo, J.: Double-phase-field formulation for mixed-mode fracture in rocks. Comput. Methods Appl. Mech. Eng. 376, 113655 (2021) 53. Wang, Y., Borja, R.I., Wu, W.: Dynamic strain localization into a compaction band via a phasefield approach. J. Mech. Phys. Solids 173, 105228 (2023) 54. Jia, Z.M., Zhou, X.P., Berto, F.: Compressive-shear fracture model of the phase-field method coupled with a modified Hoek-Brown criterion. Int. J. Fract. 229(2), 161–184 (2021) 55. Liu, S., Wang, Y., Peng, C., Wu, W.: A thermodynamically consistent phase field model for mixed-mode fracture in rock-like materials. Comput. Methods Appl. Mech. Eng. 392, 114642 (2022) 56. Ma, P.S., Ye, J.Y., Tian, K., Chen, X.H., Zhang, L.W.: Fracture phase field modeling of 3D stitched composite with optimized suture design. Comput. Methods Appl. Mech. Eng. 392, 114650 (2022) 57. Zhuang, X., Zhou, S.: An experimental and numerical study on the influence of filling materials on double-crack propagation. Rock Mech. Rock Eng. 53(12), 5571–5591 (2020) 58. Benzeggagh, M.L., Kenane, M.J.C.S.: Measurement of mixed-mode delamination fracture toughness of unidirectional glass/epoxy composites with mixed-mode bending apparatus. Compos. Sci. Technol. 56(4), 439–449 (1996) 59. Xu, J., Li, Z.: Crack propagation and coalescence of step-path failure in rocks. Rock Mech. Rock Eng. 52, 965–979 (2019) 60. Xu, Q., Fan, X., Huang, R., Yin, Y., Hou, S., Dong, X., Tang, M.: A catastrophic rockslidedebris flow in Wulong, Chongqing, China in 2009: background, characterization, and causes. Landslides 7, 75–87 (2010) 61. Li, W., Zhao, B., Xu, Q., Scaringi, G., Lu, H., Huang, R.: More frequent glacier-rock avalanches in Sedongpu gully are blocking the Yarlung Zangbo River in Eastern Tibet. Landslides 1–13 (2022) 62. Brideau, M.A., Yan, M., Stead, D.: The role of tectonic damage and brittle rock fracture in the development of large rock slope failures. Geomorphology 103(1), 30–49 (2009) 63. Zhang, Y., Jiang, Y., Asahina, D., Wang, Z.: Structural effect of en-echelon fractures on shear behavior of rock mass under constant normal load conditions: an experimental study. Rock Mech. Rock Eng. 54, 4825–4849 (2021) 64. Wu, L.Z., Shao, G.Q., Huang, R.Q., He, Q.: Overhanging rock: theoretical, physical and numerical modeling. Rock Mech. Rock Eng. 51, 3585–3597 (2018) 65. Paronuzzi, P., Bolla, A.: The prehistoric Vajont rockslide: an updated geological model. Geomorphology 169, 165–191 (2012) 66. Xing, A., Wang, G., Yin, Y., Tang, C., Xu, Z., Li, W.: Investigation and dynamic analysis of a catastrophic rock avalanche on September 23, 1991, Zhaotong, China. Landslides 13, 1035–1047 (2016) 67. Xu, Q., Shang, Y., van Asch, T., Wang, S., Zhang, Z., Dong, X.: Observations from the large, rapid Yigong rock slide-debris avalanche, Southeast Tibet. Can. Geotech. J. 49(5), 589–606 (2012) 68. Sturzenegger, M., Stead, D.: The Palliser Rockslide, Canadian Rocky Mountains: characterization and modeling of a stepped failure surface. Geomorphology 138(1), 145–161 (2012) 69. Elmo, D., Rogers, S., Stead, D., Eberhardt, E.: Discrete fracture network approach to characterise rock mass fragmentation and implications for geomechanical upscaling. Min. Technol. 123(3), 149–161 (2014) 70. Zhao, L., Huang, Y.: Insights into the dynamic and thermal characteristics of rockslide motion: a model experiment. Acta Geotechnica 17(1), 221–230 (2022) 71. Yang, B., Qin, S., Xue, L., Chen, H.: The reasonable range limit of the shape parameter in the Weibull distribution for describing the brittle failure behavior of rocks. Rock Mech. Rock Eng. 54, 3359–3367 (2021)

Triggering Mechanism and Mitigation Strategies of Freeze-Thaw Landslides for Engineering in Cold Regions: A Review Xi Xu, Xiuli Du, and Wei Wu

Abstract With the accelerated development of the engineering constructed in the frozen soil regions, the freeze-thaw landslides have caused huge financial repercussions, irreversible damage to the fragile environment and severely delay engineering progress. Due to the complicated failure mechanism and various triggering factors, the study of free-thaw landslides merits our great attention. This paper presents a review on engineering geological analysis of freeze-thaw landslide. The instability features of freeze-thaw landslide of freeze-thaw cycle effect and freezing-stagnatingslipping effect are discussed. Three main controlling factors affecting the stability of freeze-thaw landslides including global warming, earthquake and anthropogenic factors are investigated. Furthermore, the methods for freeze-thaw landslide prevention and mitigation are reviewed and discussed. This paper may have great value for landslide prevention in cold regions, and meet the urgent requirement of engineering constructions.

1 Introduction Landslide is one of the most problematic natural disasters in the world and occurs worldwide with high frequency every year, threatening human life and influencing the social and economic development of many countries. In cold areas, the slope failure is closely related to the free-thaw effect characterized by the frozen soil. The distribution area of frozen soil in China is very wide (Fig. 1). The area of China’s permafrost is about 2.19.× 10.6 km.2 , accounting for about 22.83% of China’s territory. X. Xu (B) · X. Du Key Laboratory of Urban Security and Disaster Engineering of Ministry of Education, Beijing University of Technology, Beijing 100124, China e-mail: [email protected] W. Wu Institute of Geotechnical Engineering, University of Natural Resources and Life Sciences, 1180 Vienna, Austria e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 W. Wu and Y. Wang (eds.), Recent Geotechnical Research at BOKU, Springer Series in Geomechanics and Geoengineering, https://doi.org/10.1007/978-3-031-52159-1_17

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Fig. 1 Geocryological regionalization and classification map of the frozen soil in China (The dataset is provided by National Cryosphere Desert Data Center. http://www.ncdc.ac.cn)

Among them, the mountain permafrost is 0.42.× 10.6 km.2 , accounting for about 4.39% of China’s territory and the area of short time frozen ground is about 1.86.× 10.6 km.2 , accounting for 19.33% of China’s territory. The area of seasonally frozen ground is about 4.76.× 10.6 km.2 , accounting for 49.6% of China’s territory, where frost heave, ice split, solifluction, freeze-thaw landslides, thaw collapse and other disasters have often occurred, which have done great harm to road engineering, industrial civil construction, water conservancy and hydropower projects etc.

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With continuous development and gradual implementation of “Western Development”, “One Belt One Road” and “The revitalization of the Northeast” strategy in China, a large number of civil engineering projects related to people’s livelihood are under construction or planning to build under harsh conditions such as high altitude and severe cold, where the frozen soil is widely distributed. In the northeastern China, landslides occurred along high-speed railway line connecting Jilin and Hunchun, which are the result of drying-wetting-freeze-thaw cycles caused by the varying seasonal climate conditions [1]. In southeastern, the Sichuan-Tibet railway, which is an important transportation route to Tibet stretches from Chengdu to Lhasa and goes through the southeast of the Qinghai-Tibet Plateau, one of the world’s most geologically active areas [2]. The disasters such as landslides and debris flow along the railway are caused by the coupling of endogenic processes including the complex geological structure and active seismic activity, and the exogenic factors of regional arid climate and rainfall events [3]. At present, the research on the slope stability in the frozen soil region are mainly focused on the instability of the landslides under the natural influence of freeze-thaw cycles at home and abroad. Field surveys and related experiments have shown that freeze-thaw cycles and dry-wet processes can significantly change the physical and mechanical properties of rock and soil, lead to the damage to slopes and further affect the slope stability [4, 5]. Meanwhile, as the rock mass porosity of the slope increases, irreversible freeze-thaw damage occurs to the slope, and the closer the slope foot, the greater the damage. Moreover, when the freeze-thaw cycles and dry-wet alternation increase, the slope safety factor decreases, especially for geotechnical engineering construction impact by human beings [6]. In recent years, scientists have been extensively studying landslides related to freeze-thaw cycles under dynamic conditions. Wang et al. [7] used the data collected along Qinghai-Tibet Railway to analyze the dynamic response characteristics of frozen soil sites under different ground temperatures, and provided a scientific basis for the slope defense of permafrost. Wu et al. [8, 9] calculated the vibration acceleration of the slope subgrade by the summer and winter trainload of QinghaiTibet Railway, concluded that the energy attenuation of the slope in the summer was greater than that in the winter, and compared the advantages and disadvantages of the block roadbed and the soil subgrade. Li et al. [10] pointed out the main controlling factors of the catastrophic landslide triggered debris flow in the Zhamunong Gully were the long-term freeze-thaw and dry-wet cycles and a middle magnitude earthquake. However, the instability mechanism of such slopes under earthquake action and the effective support measures have been less considered. The problems that the landslides consisting of the frozen soil under earthquake and the related treatment methods are drawing more and more attention to the geotechnical engineers. However, the characteristics of freeze-thaw landslides and the latest treatment methods have not been reviewed systematically. In this paper, the instability features of freeze-thaw landslide of freeze-thaw cycle effect and freezing-stagnating-slipping effect are discussed based on the analysis of reviewed case studies. Furthermore, the controlling factors affecting the stability of freeze-thaw landslides including global warming, earthquake and human activities are investigated. Ultimately, the methods

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for freeze-thaw landslide treatment are presented, and the novel soil improvement by using nanoparticles are reviewed and discussed, which have great value for landslide prevention and control in cold regions.

2 Features of Freeze-Thaw Landslides In high-altitude mountain regions in western China where distributed seasonally frozen soil, freeze-thaw is a prominent weathering process [11, 12]. The landslides hazards in these regions (see Fig. 2 as the example of landslides on Qinghai-Tibet Plateau) are connected closely to the freeze-thaw cycles. In addition to the rainy season, a significant part of the landslides listed in Table 1 occurs in freeze-thaw season when it is fairly dry and cold, from November to January, during which the minimum air temperature could reach –30.◦ C and the deepest freezing depth could be up to 2.5 m. Then, it will not start to thaw until the following year from February to June. Meanwhile, the western region of China is not only the region with strong seasonal freeze-thaw action, but the region with the most landslide developed as well. Since the 1980s, a series of landslide disasters have successively occurred in the area, such as Saleshan landslide (1983 A.D.), Huangci landslide (1995 A.D.) and Heifangtai landslide (2015 A.D.) etc., which have caused huge losses. It is undeniable that the hysteresis recharge of precipitation supply (rainfall, snowmelt, etc.) causing landslides exists extensively. However, some freeze-thaw landslides still occur in some areas where there is no obvious precipitation. For example, the loess plateauHeifangtai, located in Yongjing County, Gansu Province, has frequent landslides since March, but the area belongs to a medium temperate and semi-arid climate with less precipitation, adequate sunshine, large evaporation and dry climate. Therefore, the frequent occurrence of landslides in freeze-thaw season is not only related to the hysteresis recharge of precipitation supply, but also to the seasonal freeze-thaw effect. The upper layer of the slope humidity significantly increases due to the temperature comes below zero. A large amount of ice in the fine soil of the frozen soil,

Fig. 2 Freeze-thaw landslides on the Qinghai-Tibet Plateau: a Shannan, Tibet; b Xining, Qinghai

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Table 1 Typical freeze-thaw landslides Location

Landslides

Date

Scale (.×10.4 m.3 )

Stratigraphic lithology

Likely causes

Gansu Province, China

Saleshan [16]

7 Mar. 1983

4000

Loess, mudstone

Rainfall, snowmelt in thaw season and unscientific water conservancy construction

Huangci [17]

30 Jan. 1995

360

Loess, sandy mudstone

Loose edge of the old landslide informed cracks, the irrigation water was easy to penetrate and the loess became saturated, forming a new slide body to push the old slide body as whole

Heifangtai [18]

19 Feb. 2017

26

Loess, Silty clay

Agricultural irrigation coupled with the declined shear strength of loess by freeze-thaw cycles

29 Apr. 2015

1400

25 Jan. 1986

150

Red clay, sand

Shear stress was concentrated in the foot area of the slope, and was located near the groundwater level. There was a weak moving surface between the viscous / sand layers where shear failure occurred first in the area

Longwuxishan 21 Nov. 1984 [20]

200

Loess, mudstone

Inappropriate human activities for toe excavation, coupled with the snowmelt and decreased shear strength of sliding soil, which triggered the developing new landslides based on old landslides

Sichuan Province, China

Zhaoma [21]

20 Mar. 2010

52.5

Siltstone, sandstone, slate

Freeze-thaw caused a sudden collapse of the upper part of the rock mass, and then transferred into debris flow, brought the loose material of slope body along the way and produced high-speed movement

Shaanxi Province, China

Jiangliu [22]

2 Dec. 1984

100

Loess

Abundance high underground water, coupled with rich irrigation water

Tibet Zhamunong Autonomous gully [10] Region, China

9 Apr. 2000

30000

Granite, limestone, slate, marble, gneiss, schist

The saturated rock mass after freeze-thaw cycle and dry-wet cycle, coupled with a middle magnitude earthquake

Northern British Columbia, Canada

Mink Creek [23]

Dec. 1993-Jan. 1994

2500

Sand (0-13%), silt (44-62%), clay (45-58%)

Warmer and wetter conditions and a warm wet fall lasting for a decade; bank erosion; earthquake; site loading

Southwestern Yukon Territory, Canada

Takhini river [24]

Headscarp retreat rates: 1971-1979: 3 m/a; 1979-1987: 16 m/a

4

Laminated silt and clay

Vegetation was removed by forest fire which accelerated thaw of icy material, and the river erosion-related thawing of ice-rich sediment caused the failure of fine-grained lacustrine terraces

Qinghai Province, China

Longxi [19]

(continued)

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Table 1 (continued) Location

Landslides

Date

Scale (.×10.4 m.3 )

Stratigraphic lithology

Likely causes

Northern Caucasus, Eastern Europe

KolkaKarmadon [25]

20 Sep. 2002

11000

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Thermal perturbation caused by steep glaciers on the underlying bedrock by warming the glacier ice due to warming temperature, and the geothermally active spots featured by the active tectonic and geothermal

South-central Alaska, US

Mt. Steller [26]

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Western Alps, Mt. Grivola Cogne, Italy [27]

during the thawing period, causes water content in thawing layer to exceed the liquid limit, and excess water is not able to be infiltrated or discharged in time, so that the strength of the soil and the stability of the slope are drastically reduced and the underlying frozen layer slides down eventually. The phase transitions between water and ice during the freeze-thaw process can significantly change the soil structure, affect its physical-mechanical properties and furtherly undermine the slope stability in high altitude mountain regions. The freeze-thaw landslide has the characteristics of a generally shallow stratigraphic landslide and a cohesive plastic landslide, which can be developed from small pieces of debris with a diameter of only a few meters to a large landslide of several hundred meters [13–15].

2.1 Freeze-Thaw Cycle Effect The freeze-thaw cycle is a special form of strong weathering that has a strong influence on the physical and mechanical properties of the soil, which acts as the direct factor affecting the stability of the slope. During the freeze-thaw cycle, the ice crystals and unfrozen water in porous media of permafrost migrate, diffuse and phase change with the positive and negative temperature fluctuations. Due to the different densities of water and ice, the volume of solid ice is larger than the mass of liquid water. When the liquid water is transformed into solid ice, the ice crystal grows in

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volume and crushes the surrounding agglutinate soil particles, so that not only the displacement even broken deformation of soil particles may occur, but will change the shape of the pores as well. More importantly, besides the phase transition of the water in the freeze-thaw cycle, there is also the migration of water. Moisture migration significantly changes the structural elements such as pore shape and particle arrangement of soil so that the influence of freeze-thaw cycle on soil structure becomes more complicated. The structure of undisturbed soil is significantly weakened by freeze-thaw cycles, which is embodied in the pre-consolidation pressure decrease of undisturbed soil and the gradual disappearance of peak intensity on the triaxial undrained shear stressstrain curve. Among them, the triaxial test results show that as the number of freezethaw cycle increases, the cohesion of soil decreases [28]. Meanwhile, for rock mass, with enlarge and penetration of microcracks and pores, the uniaxial compressive strength and the tensile strength decrease due to the freeze-thaw cycle. Hence, the changes of the internal structure of rock promote the deterioration of physical and mechanical properties. The failure of Heifangtai landslide [29] was caused by great reduction of the loess shear strength due to freeze-thaw cycles. In the period of 2010–2011, the soil within 40 cm from surface was monitored suffering more than 10 cycles of freezing and thawing. During the freezing period, the cohesive of surface loess, one of the shear strength indices, increased greatly and then declined rapidly during the melting period, so that the sharply change of soil strength was extremely unfavorable to the stability of slope.

2.2 Freezing-Stagnating-Slipping Effect As opposed to the landslide caused by freeze-thaw cycle, in freeze season, the landslide disasters may also occur due to freezing-stagnating-slipping effect. Freezing makes the groundwater enrich in the slope body, which can not only soften the soil, but produce a large hydrostatic pressure, float the slope, lower the effective pressure in the slope and reduce the stability of the slope overall as well. According to Niu et al. [13], the phenomenon of freezing and stagnating is the freezing potential (thermodynamic potential) in the frozen soil layer due to the freezing effect in winter, which makes the freezing layer have strong adsorption and condensate water capacity, so that aeration zone water, phreatic water, perched water and meteoric water are transported in the liquid or gaseous form to the frozen soil, forming seasonal solid groundwater and freezing layer. It is the shear strength reduction of soil that causes the thaw slumping of freeze-thaw landslide. Moreover, from a micro perspective on the other hand, it is vital that the addition of hydrogen bond adsorption energy, the formation of saturated vapor pressure difference in the soil layer and the influence of capillary membrane mechanism form frozen perched groundwater. That is, during the freezing period, the freezing water of the frozen soil layer in aeration zone is frozen, and the hydrogen bond reaches the perfect degree, resulting in a strong adsorption energy, so that groundwater in the frozen soil gathers to freeze. The failure

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of colluvial landslides in Qinghai-Tibet Plateau, as shown by Hu et al. [30], occurred because the water content of shallow deposits increased and deteriorated gradually under the action of moisture aggregation, and then the frozen soil melt quickly to be saturated in thawing period, ultimately the deformation accumulates to be shallow landslide and creep downwards continuously.

3 Controlling Factors Affecting the Stability of Freeze-Thaw Landslides 3.1 Global Warming The frozen soil is sensitive to the temperature fluctuation, as the structure and strength may change significantly when temperature increases [13]. In the context of the climate change which have raised global concerns nowadays, the ground temperature significant rises and the permafrost area is decreasing. The abnormal temperature change seriously affects the stability of slope under the freeze-thaw cycles. Taking the temperature on the Qinghai-Tibet Plateau as an example. It is observed from the historical meteorological data provided by China Meteorological Data Sharing Service System (Fig. 3a), with the rising trend of the annual average temperature from 1956 to 2022, there is an obvious decreasing trend of frozen days and an increasing of days with average temperature above 18.◦ C in Lhasa. Meanwhile, according to Luo et al. [31], there is a decreasing trend in maximum freeze depth and freezethaw duration occurred on the Qinghai-Tibet Plateau as statistics from 1960 to 2014. Specifically, the freeze start date has been later, and the thaw end date has been significantly earlier. If that continues, after 50 years and 100 years, it is assumed by Nan et al. [32] that when the temperature increment is 0.02.◦ C/a, the permafrost area decreases by approximately 8.8 and 13.4% respectively, and when the temperature increment is 0.052.◦ C/a, the permafrost area will decrease by approximately 13.5% and 46% respectively, which may intensify the freeze-thaw cycle and accelerate the instability of the landslides. Simultaneously, influenced by global warming, rainfall is more and more frequent with higher intensity by the rise of temperature. So, the seasonal freeze-thaw cycle and rainfall infiltration may make the physical and mechanical properties of deposits deteriorate continuously. The excessive water content of the soil will not only reduce the strength of the soil, but also increase the soil self-weight [33]. Large plastic or creeping deformation is easier to appear in slope driven by effects of gravity and groundwater, becoming the main source of landslide [30]. Indeed, differences in rainfall density resulted in different degree and processes of slope erosion. In thaw seasons, when the heavier rainfall is applied, the more rapid mudflows and debris flows become prominent [34]. For example, the rainfall gave rise to the mud flows after blocks suffering disaggregation due to wetting-drying and freeze-thaw cycles in the French Southern Alps [35]. As shown in Fig. 3b, with the increasing trend

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Fig. 3 a History curve of Lhasa average temperature and interdecadal variations of number of days with average temperature below 0.◦ C and above 18.◦ C from 1956 to 2022; b Interdecadal variations of Lhasa precipitation in the freeze season and thaw season from 1956 to 2022 (China Meteorological Data Sharing Service System. http://data.cma.cn/)

of the precipitation in recent two decades caused by climate change, the rainfall in freeze season and thaw season shows a significant increase. Meanwhile, in freeze season, warming and wetting conditions of the soil resulted in a significant decrease in maximum freeze depth and freeze-thaw duration in the most area of the Qinghai-Tibet Plateau [31], which may give rise to the enhancement of freezing-stagnating-slipping effect and the landslide occurrence. In addition, as the fastest rate of global warming, extensive forest fires removed the vegetation which could limit thaw by increasing shading and reducing solar radiation, and seriously caused deepening of the active layer and thawing of near-

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surface permafrost [36]. Burn [37] investigated that the 1958 Takhini Valley fire, which happened in Canada, had led to 2.4 m of permafrost degradation by 1997. Meanwhile, the ground temperatures throughout permafrost had been warmed to .≽ –0.2.◦ C, causing the thickness of active layer increased dramatically after fire, thus leading to the landslide occurred in 1971 [24]. Therefore, the global warming effect may promote the action of freeze-thaw cycle and the freezing-stagnating-slipping effect, which exacerbate disasters of landslides occurring for frozen soil region to some extent.

3.2 Earthquakes Slopes located at the seasonal frozen ground will have great changes in the mechanical properties after freezing and thawing cycles, and there is a huge hidden danger under the action of seismic loading. Therefore, it is a vital problem to study the development mechanism, formation process and dynamic change of low slope permafrost under earthquake action. Earthquakes can affect the stability of freeze-thaw slopes in a few ways theoretically. The stability of freeze-thaw slope will reduce under the influence of seismic acceleration evaluated by Niu et al. [14]. With the help of earthquake forces, the material within the freeze-thaw landslide will be loosened and the cracks along the surface of the slope will be generated. The moisture content at the ice-soil interface of freeze-thaw landslide may gradually increase during the vibration process, and the increase of water content in the weak layer of ice-soil will lead to the decrease of shear strength. Earthquake-induced liquefaction of the frozen soil foundation cannot be ignored as well: When liquefaction occurs, the frozen soil layer will restrict the discharge of pore water, which will aggravate the liquefaction of the foundation and show a more relative slip tendency along the contact interface. As analyzed by Li et al. [10], the middle magnitude earthquake is considered as a main controlling factors triggering the catastrophic landslides occurred in the Zhamunong gully located on the Qinghai-Tibet Plateau. Under seismic loading, the joint and crack development in the landslide, adding the saturated rock mass under the long-term freeze-thaw cycles, evolves into the catastrophic debris flow moving at a high speed.

3.3 Anthropogenic Factors Anthropogenic factors are also necessary to trigger destabilization of freeze-thaw landslide. Engineering construction changes the existing natural environment [38]. When conducting engineering construction, there is a great deal of excavation natural rock masses, which reduces the sliding resistance force and creates a high concentration of stress at the lower part of the slope, and thus changes the originally stable

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slope [2, 39]. In the thaw season, after excavation at the slope toe of the frozen soil slope, the ground ice is exposed to air and begins to melt, which leads to the soil lose support and collapse. When the ice-melting water accumulates on the surface of the ground ice, the thawed soil near the surface is tend to be saturated or extra-saturated, and the slope begins to slide [13]. Besides, in the freeze season, the exposed water outlet due to toe excavation in the slope is bunged up during freezing period, which causes the accumulation of groundwater within the slope, and slope fails consequently by increased pore water pressure and reduced soil strength. High hydraulic gradient may be generated by the release of groundwater during thawing period, leading slope stability reduced finally [40]. In addition, irrigation in winter leads water permeate into the bottom of the soil layer and thus form a saturated zone, reducing soil strength and pushing the old landslide whose slope body has already been dehiscent and unconsolidated during the dry freezing period. Water penetrates into the bedrock and softens the slip zone of the old landslide, prompting the revival of the old landslide partly. Water catchment measures such as the construction of reservoirs and other activities, raising the water level of the landslide area, increasing the extent and time of soaking soil in the groundwater to further reduce the stability of the slope.

4 Prevention and Mitigation Methods of Freeze-Thaw Landslide With the requirement of the national strategies and develop of economy, to ensure the safety of the important livelihood projects, it is necessary to prevent and mitigate the free-thaw landslide around. While, the traditional slope reinforcement methods employed in normal regions such as retaining wall, anchored supporting wall and framework structures are not available for the inharmonious deformation in the frozen soil [13]. The prevention and mitigation methods of freeze-thaw landslide should be fundamentally focus on draining ground water in a faster speed and controlling thermal energy entering into frozen soil.

4.1 Drainage Accelerated Measures According to the forming and developing conditions of the freeze-thaw landslide, the drainage accelerated measures of freeze-thaw landslide treatment can be summarized as follows: The drainage blind ditch can be built to discharge groundwater in time, thereby reducing the water content within the slope body, lessening the change range of rock and soil morphology during freezing and thawing, to reduce the extent of damage to rock and soil body.

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The layered geotextile installed into the slope is considered as an effective method with the characteristic of filter, that can discharge the excess water in a fast speed and maintain the stability of the slope during thawing period. Furthermore, partial soil replacement can be operated in the geotextile-treated slope with non-frost heave soil instead, which can play a significant part of filter and drainage, reduce soil moisture content and reduce the intensity of soil frost heave. Meanwhile, in the aspect of soil conservation property, the layered setting of geotextile can reinforce soil, enhance shear strength of soil, and reduce frost heave deformation.

4.2 Thermal Controlling Measures The measures aiming at controlling the thermal energy entering into frozen soil include controlling the solar radiation, heat convection and conduction [15]. Vegetation plating is one of effectively cooled measures that can be applied into engineering conduction, which can give full play to the insulating effect of the surface organic mat whose surface albedo would prevent temperature increase [41, 42]. The embankment slope with turf revetments (Fig. 4a) can be constructed in order to increase the leaf area where can reduce the amount of solar radiation, hence protect permafrost [36]. For controlling heat convection and conduction, the crushed-rock embankment, with low fines content and highly porous, which can exchange heat in the outward direction at the bottom of the crushed-rock layer by outward air convection in sum-

Fig. 4 Mitigation strategies for slope engineering disease of constructions on the frozen soil-based ground: a Embankment slope with turf revetments on the Qinghai-Tibet Plateau; b Embankment slope with thermosiphons (image adapted from Zhi et al. [45])

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mer, can be constructed to effectively remove heat from underlying soil and decrease the temperatures of the underlying soil layers thus protect the frozen soil [43, 44]. Thermosiphon (Fig. 4b) is a cooling technique free of power which is considered as an effective strategy for preventing settlement and heave especially for artificial permafrost slope [46]. The upside of the thermosiphon with a radiator is the condensation section, and the downside is the evaporation section. When the temperature of the condensation section is lower than the temperature of the evaporation section in cold season, the liquid inside vaporizes and rises to the condensation section, then cools into a liquid state, and flows back to the evaporation section. The whole cycle can take the heat from permafrost away and protect permafrost in case of melt [45].

4.3 Novel Soil Improvement by Using Nanoparticles With the advance of the nanotechnology, nanoparticles, as a novel stabilizer, have been widely used in various applications and the usage for soil improvement has developed rapidly [47]. Nanocomposite soil, with characteristics of special rheological properties, is compounded of nano suspension in gel state and soil (Fig. 5), which can alter the elastic threshold strain and permeability coefficient of the soil, thus delay the generation and propagation of excess pore pressure [48]. Meanwhile, Wang et al. [49] approved a dynamic centrifuge test conducted with laponite (a kind of nanoparticles)-treated soil, and observed that the amplitude of excess porewater pressure ratio was decreased, and both lateral and vertical displacements were effectively controlled. In the context of free-thaw cycle, Kalhor et al. [50] investigated the effects of freethaw cycles on the fine soil specimens stabilized by nanoparticles, and discovered

Fig. 5 The microstructure and scale of the nanocomposite soil (image adapted from Huang and Wang [47])

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that the unconfined compressive strength reduction of the soil treated with nanoparticles is less than that of non-stabilized soil after 9 freeze-thaw cycles, and exhibited a more ductile behavior as the number of freeze-thaw cycles increase. Therefore, the nanoparticles can be considered as an effective method in the prevention and treatment of freeze-thaw landslides, especially for its delay of pore pressure generation and the resistance of soil liquefaction under seismic loading (Fig. 5). From the above research, there are still limited methods in the prevention and mitigation of free-thaw landslides. The works are mostly dominated by empirical analysis, theoretical analysis and laboratory tests. In view of the complex and fragile geological environment of frozen soil regions, the cost of in-situ experiments is too high. Geotechnical centrifuge test is an effective method to study geotechnical seismic problems. Compared with 1g model test, the centrifuge model test uses centrifugal force to simulate gravity, so that the self-weight of slope can be increased to the prototype state to simulate the true stress field [51]. Furthermore, the thermal transfer models, pore-pressure models and soil-structure interactions need to be further studied by numerical simulations and model tests. With the development of the engineering constructed on the frozen soil-based ground, the novel landslide treatment fully considered the variation of frozen soil properties, temperature and ground water caused by free-thaw cycles is still regarded as key scientific issues of future research.

5 Conclusions This paper reviewed the features and controlling factors affecting the stability of the freeze-thaw landslides. On the basis, the prevention and mitigation methods are reviewed and discussed. The following conclusions can be drawn. The freeze-thaw cycle acts as the direct factor affecting the stability of the slope. The internal structure in undisturbed soil and rocks can destroy because of freezingthawing cycle effect. The ice crystals and unfrozen water in permafrost migrate, diffuse and phase change with the positive and negative temperature fluctuations is a negative factor that undermines the stability of freezing-thawing landslide. The landslide disasters occurred in freeze season is related to freezing-stagnatingslipping effect. The water content in the upper soil layer of the freeze-thaw landslide significantly increases during the freezing period. When the water content exceeds the liquid limit during thawing period, excess water cannot be infiltrated or discharged in time, causing the strength of the soil and the slope stability to drop sharply. The underlying frozen layer slides down ultimately. The gradually increment of temperature and precipitation caused by global warming is the controlling factor of the freeze-thaw landslide. The abnormal temperature change seriously affects the intensify the freeze-thaw cycle. On the other hand, the higher intensity of the rainfall leads to the excessive water content which deteriorates the physical and mechanical properties of deposits. Meanwhile, the removal of vegetation caused by severe forest fires may seriously cause the deepening of the active

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layer and thawing of near-surface permafrost, thus promote the action of freeze-thaw cycle and accelerate the landslide failure. Under seismic loading, the sliding force may increase due to seismic force, and the slope materials may loose and generate cracks which leads to the shear strength decline. Meanwhile, under the influence of earthquake-induced liquefaction, the frozen soil layer will restrict the discharge of pore water, and thus aggravate the liquefaction of the foundation and a more relative slip tendency occur along the contact interface. Anthropogenic factors including toe excavation when conducting engineering construction reduces the sliding resistance force and exposed water outlet is bunged up during freezing period, causing the accumulation of groundwater and the failure consequently. In addition, irrigation and water catchment measures raise the water level may reduce the soil strength and slope stability, and even softens the slip zone and prompt the old landslide. Conventional retaining structures in landslide reinforcement engineering is not applicable for the inharmonious deformation in the frozen soil. Appropriate prevention of the freeze-thaw landslide failure should be taken into consideration, and it can be summarized as the drainage accelerated measures and thermal controlling measures. Besides, the novel soil improvement by using nanoparticles takes fully consideration of the variation of frozen soil properties, temperature and ground water caused by free-thaw cycles and earthquakes, which may be seen as an effective method in the prevention and treatment of freeze-thaw landslides. Acknowledgements We sincerely appreciate the China Postdoctoral Science Foundation [grant number 2022M710285] and the Beijing Postdoctoral International Exchange Funding [grant number 2022-PC-02] for the financial support of this work.

References 1. Kong, L.W., Zeng, Z.X., Bai, W., Wang, M.: Engineering geological properties of weathered swelling mudstones and their effects on the landslides occurrence in the Yanji section of the Jilin-Hunchun high-speed railway. Bull. Eng. Geol. Environ. 7, 1491–1503 (2018) 2. Wang, W.D., Li, J.Y., Han, Z.: Comprehensive assessment of geological hazard safety along railway engineering using a novel method: a case study of the Sichuan-Tibet railway. China. Geomat. Nat. Hazards Risk. 11, 1–21 (2020) 3. Hu, G., Zhao, C., Chen, N., Chen, K., Wang, T.: Characteristics, mechanisms and prevention modes of debris flows in an arid seismically active region along the Sichuan-Tibet railway route, China: a case study of the Basu-Ranwu section, southeastern Tibet. Environ. Earth Sci. 78, 564 (2019) 4. Qu, Y.L., Chen, G.L., Niu, F.J., Ni, W.K., Mu, Y.H., Luo, J.: Effect of freeze-thaw cycles on uniaxial mechanical properties of cohesive coarse-grained soils. J. Mt. Sci. 16, 2159–2170 (2019) 5. Xu, J., Wang, Z., Ren, J., Wang, S., Jin, L.: Mechanism of slope failure in loess terrains during spring thawing. J. Mt. Sci. 15, 845–858 (2018)

280

X. Xu et al.

6. Qu, D., Luo, Y., Li, X., Wang, G., Zhang, G., Xu, K.: Study on the stability of rock slope under the coupling of stress field, seepage field, temperature field and chemical field. Arab. J. Sci. Eng. 45, 8315–8329 (2020) 7. Wang, L.M., Zhang, D.L., Wu, Z.J., Ma, W.: The influence of earth temperature on dynamic characteristics of frozen soil and the parameters of ground motion on the sites of frozen soil. Earthq. Res. China. 19, 23–27 (2003) (in Chinese) 8. Wu, Z.J., Sun, J.J., Wang, L.M., Cheng, G.D., Xu, S.H.: Study on characteristics of ground motion at permafrost sites along Qinghai-Tibet Railway. Chin. J. Rock Mech. Eng. 26, 2466– 2472 (2007) (in Chinese) 9. Wu, Z.J., Chen T, Ma, W., Wang, L.M.: Characteristics of vibration response of typical structure embankments at permafrost regions along the Qinghai-Tibet Railroad. J. Shanghai Jiaotong Univ. 45, 716–721 (2011) (in Chinese) 10. Li, J., Chen, N., Zhao, Y., Liu, M., Wang, W.: A catastrophic landslide triggered debris flow in China’s Yigong: factors, dynamic processes, and tendency. Earth Sci. Res. J. 24, 71–82 (2020) 11. Kang, S., Xu, Y., You, Q., Flügel, W.A., Pepin, N., Yao, T.: Review of climate and cryospheric change in the Tibetan Plateau. Environ. Res. Lett. 5, 015101 (2010) 12. Li, X., Jin, R., Pan, X., Zhang, T., Guo, J.: Changes in the near-surface soil freeze-thaw cycle on the Qinghai-Tibetan Plateau. Int. J. Appl. Earth Obs. Geoinformation. 17, 33–42 (2012) 13. Niu, F., Cheng, G., Ni, W., Jin, D.: Engineering-related slope failure in permafrost regions of the Qinghai-Tibet Plateau. Cold Reg. Sci. Technol. 42, 215–225 (2005) 14. Niu, F., Luo, J., Lin, Z., Fang, J., Liu, M.: Thaw-induced slope failures and stability analyses in permafrost regions of the Qinghai-Tibet Plateau. China. Landslides. 13, 55–65 (2016) 15. Wei, M., Guodong, C., Qingbai, W.: Construction on permafrost foundations: Lessons learned from the Qinghai-Tibet railroad. Cold Reg. Sci. Technol. 59, 3–11 (2009) 16. Su, B.L.: High speed ultra large landslide-Saleshan landslide and its research. Typ. Chin. Landslide (1986) (in Chinese) 17. Wang, G.X.: Sliding mechanism and prediction of critical sliding of Huangci landslide in Yongjing County. J. Catastrophology 3, 23–27 (1997) (in Chinese) 18. Xu, L., Dai, F., Kwong, A.K.L., Min, H.: Types and characteristics of loess landslides at Heifangtai loess plateau. China. J. Mt. Sci. (2008) 19. Guo, Z.M.: Analysis of slope instability in the lower part of Longxi steep slope on the bank of the Longyangxia Dam near the Yellow River. Northwest Hydropower 4, 3–8 (1986) (in Chinese) 20. Li, T.C., Liu, X.M., Wu, Q.W.: The hidden dangers and emergency measures of Longwuxishan landslide. Bull. Soil Water Conserv. 2, 19–23 (1985) (in Chinese) 21. Li, P.Y., Tie, Y.B., Wang, D.H., Tang, Y.Q.: Characteristics and cause analysis of FreezeThaw landslide on Western Sichuan Plateau. Bull. Soil Water Conserv. 34, 173–177 (2014) (in Chinese) 22. Wu, W.J.: Seasonal Freeze-Thaw action and the entire deformation, Failure of Slope. Chin. J. Geol. Hazard Control 7, 59–64 (1996) (in Chinese) 23. Geertsema, M., Clague, J.J., Schwab, J.W., Evans, S.G.: An overview of recent large catastrophic landslides in northern British Columbia. Canada. Eng. Geol. 83, 120–143 (2006) 24. Huscroft, C.A., Lipovsky, P.S., Bond, J.D.: Permafrost and landslide activity: case studies from southwestern Yukon Territory, vol. 15 (2003) 25. Huggel, C.: Recent extreme slope failures in glacial environments: effects of thermal perturbation. Quat. Sci. Rev. 28, 1119–1130 (2009) 26. Huggel, C., Salzmann, N., Allen, S., Caplan-Auerbach, J., Fischer, L., Haeberli, W., Larsen, C., Schneider, D., Wessels, R.: Recent and future warm extreme events and high-mountain slope stability. Philos. Trans. R. Soc. Math. Phys. Eng. Sci. 368, 2435–2459 (2010) 27. Frasca, M., Vacha, D., Chicco, J., Troilo, F., Bertolo, D.: Landslide on glaciers: an example from Western Alps (Cogne-Italy). J. Mt. Sci. 17, 1161–1171 (2020) 28. Guo, Y., Shan, W.: Monitoring and experiment on the effect of freeze-thaw on soil cutting slope stability. Procedia Environ. Sci. 10, 1115–1121 (2011)

Triggering Mechanism and Mitigation Strategies …

281

29. Cheng, X.J., Zhang, M.S., Zhu, L.F., Pei, Y., Li, Z.G., Hu, W.: Seasonal freeze-thaw action and its effect on the slope soil strength in Heifangtai area, Gansu Province. Geol. Bull. China. 32, 904–909 (2013) (in Chinese) 30. Hu, T., Liu, J., Zhu, B., Zheng, J.: Study on sliding characteristics and controlling measures of colluvial landslides in Qinghai-Tibet Plateau. Procedia Eng. 143, 1477–1484 (2016) 31. Luo, S.Q., Wang, J.Y., Pomeroy, J.W., Lyu, S.H.: Freeze-Thaw changes of seasonally frozen ground on the Tibetan Plateau from 1960 to 2014. J. Clim. 33, 9427–9446 (2020) 32. Nan, Z., Li, S., Cheng, G.: Prediction of permafrost distribution on the Qinghai-Tibet Plateau in the next 50 and 100 years. Sci. China Ser. Earth Sci. 48, 797–804 (2005) 33. Shan, W., Zhang, C.C., Guo, Y.: Mechanism of shallow slide on soil road cutting slope during spring in seasonal frozen region. Appl. Mech. Mater. 178–181, 1258–1263 (2012) 34. Vedie, E., Lagarde, J.L., Font, M.: Physical modelling of rainfall- and snowmelt-induced erosion of stony slope underlain by permafrost. Earth Surf. Process. Landf. 36, 395–407 (2011) 35. Descroix, L., Mathys, N.: Processes, spatio-temporal factors and measurements of current erosion in the French Southern Alps: a review. Earth Surf. Process. Landf. 28, 993–1011 (2003) 36. Fisher, J.P., Estop-Aragonés, C., Thierry, A., Charman, D.J., Wolfe, S.A., Hartley, I.P., Murton, J.B., Williams, M., Phoenix, G.K.: The influence of vegetation and soil characteristics on active-layer thickness of permafrost soils in boreal forest. Glob. Change Biol. 22, 3127–3140 (2016) 37. Burn, C.R.: The response (1958–1997) of permafrost and near-surface ground temperatures to forest fire, Takhini River valley, southern Yukon Territory. Can. J. Earth Sci. 35, 184–199 (1998) 38. Khak, V.A., Kozyreva, E.A.: Changes of geological environment due to the anthropogenic impacts: a case study of south of East Siberia. Russia. Z. Für Geomorphol. 56, 183–199 (2012) 39. Huang, Y., Fan, G.: Engineering geological analysis of municipal solid waste landfill stability. Nat. Hazards 84, 93–107 (2016) 40. Liu, W., Li, T.: Forming mechanism of the loess landslides triggered by irrigation and seasonal freezing. In: Presented at the 5th International Conference on Civil Engineering and Transportation November (2015) 41. Wang, B., Nichol, S., Su, X.: Geotechnical field observations of landslides in fine-grained permafrost soils in the Mackenzie Valley, Canada. In: Sassa, K., Fukuoka, H., Wang, F., Wang, G. (eds.) Landslides: Risk Analysis and Sustainable Disaster Management, pp. 203–212. Springer, Berlin, Heidelberg (2005) 42. Marsh, P., Bartlett, P., MacKay, M., Pohl, S., Lantz, T.: Snowmelt energetics at a shrub tundra site in the western Canadian Arctic. Hydrol. Process. 24, 3603–3620 (2010) 43. Zhang, M., Lai, Y., Gao, Z., Yu, W.: Influence of boundary conditions on the cooling effect of crushed-rock embankment in permafrost regions of Qinghai-Tibetan Plateau. Cold Reg. Sci. Technol. 44, 225–239 (2006) 44. Zhang, M., Lai, Y., Li, S., Zhang, S.: Laboratory investigation on cooling effect of sloped crushed-rock revetment in permafrost regions. Cold Reg. Sci. Technol. 46, 27–35 (2006) 45. Wen, Z., Sheng, Y., Ma, W., Qi, J., Wu, J.: Analysis on effect of permafrost protection by two-phase closed thermosyphon and insulation jointly in permafrost regions. Cold Reg. Sci. Technol. 43, 150–163 (2005) 46. Wen, Z., Sheng, Y., Ma, W., Qi, J.: Evaluation of EPS application to embankment of QinghaiTibetan railway. Cold Reg. Sci. Technol. 41, 235–247 (2005) 47. Huang, Y., Wang, L.: Experimental studies on nanomaterials for soil improvement: a review. Environ. Earth Sci. 75, 497 (2016) 48. Huang, Y., Wen, Z., Wang, L., Zhu, C.: Centrifuge testing of liquefaction mitigation effectiveness on sand foundations treated with nanoparticles. Eng. Geol. 249, 249–256 (2019) 49. Wang, L., Huang, Y., Wen, Z.Q.: Centrifuge modeling of a constructed reservoir embankment: antiliquefaction performance improvement using nanoparticles. J. Perform. Constr. Facil. 32, 06018001 (2018)

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50. Kalhor, A., Ghazavi, M., Roustaei, M., Mirhosseini, S.M.: Influence of nano-SiO2 on geotechnical properties of fine soils subjected to freeze-thaw cycles. Cold Reg. Sci. Technol. 161, 129–136 (2019) 51. Huang, Y., Xu, X., Liu, J., Mao, W.: Centrifuge modeling of seismic response and failure mode of a slope reinforced by a pile-anchor structure. Soil Dyn. Earthq. Eng. 131, 106037 (2020)

SPH Modeling of Water-Soil Coupling Dynamic Problems Chengwei Zhu, Chong Peng, and Wei Wu

Abstract This study introduces an SPH (Smoothed Particle Hydrodynamics) model that utilizes mixture theory to address the complex interactions between soil and water dynamics. The model incorporates an intrinsic density formulation for the fluid phase and accurately accounts for the spatial and temporal variations in soil porosity. To discretize the fluid and solid phases independently, it employs two layers of SPH particles. This model successfully reproduces the results of three benchmark problems, illustrating the effectiveness of the proposed SPH model for simulating water-soil dynamics.

1 Introduction Dynamic interactions between soil and water are prevalent in both natural occurrences and engineering applications, such as sediment transport, debris flows, and landslides. While techniques like the combination of smoothed particle hydrodynamics and DEM (SPH-DEM) [2] or the coupled discrete element and lattice Boltzmann method (LBM-DEM) [1] provide detailed insights at the microscale, they can be impractical for large-scale engineering simulations. A more pragmatic approach is to address water-soil coupling on a macroscale using a continuum framework. In this study, the mixture theory is adopted, which involves averaging actual flows and interC. Zhu Research Center of Coastal and Urban Geotechnical Engineering, Zhejiang University, 310058, Hangzhou, China e-mail: [email protected] C. Peng (B) ESS Engineering Software Steyr GmbH, Berggasse 35, 4400 Steyr, Austria e-mail: [email protected] W. Wu Institute of Geotechnical Engineering, University of Natural Resources and Life Sciences, Vienna, Feistmantelstrasse 4, 1180 Vienna, Austria e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 W. Wu and Y. Wang (eds.), Recent Geotechnical Research at BOKU, Springer Series in Geomechanics and Geoengineering, https://doi.org/10.1007/978-3-031-52159-1_18

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action forces within a representative volume. This approach allows us to describe the fluid solid interaction from a macro-scale perspective. One notable advantage of the mixture theory is its ability to analyze both water-soil mixtures and the pure water or soil phase within a single theoretical framework. This theory is useful and is adopted in plenty of water-soil coupling problems, including geophysical flows [3], submerged granular flows [4], and interactions with porous media in wave scenarios [5]. The previous numerical studies of water-soil coupling were predominantly based on depth-integration models. It is of challenge for The conventional grid-based methods such as the finite element method (FEM) and finite difference method (FDM) to numerically reproducing problems involving large deformations and free-surface flow. In this context, the smoothed particle hydrodynamics (SPH), as a typical example of Lagrangian particle-based meshfree methods, emerges as suitable candidate. The reason is twofold: (1) The numerical instability caused by mesh distortion is avoid as SPH is a particle-based method; (2) The free surfaces and interfaces can be tracked naturally due to the Lagrangian nature. In this study, the comprehensive mathematical framework is first developed for the dynamic water-soil coupling issues based on the mixture theory. As proceeds, a three-dimensional multi-layer smoothed particle hydrodynamics (SPH) method is proposed for the numerical solution of the governing equations. At last, validation is implemented to show the performance of the developed SPH model. Note that, the coding work of this study is based on our in-house SPH solver, LOQUAT [9].

2 Methodology Figure 1 presents the schematic diagram for the porous media within the mixture theory. In the region saturated by water, the following unity condition holds φ + φf = 1

. s

(1)

where .φs and .φ f are the volume fractions for the soil and fluid phases, respectively.

Fig. 1 Schematic diagram of the water-soil mixture based on mixture theory

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2.1 Governing Equations The mass conservation equation for the solid phase in Lagrangian form is expressed as follows ds ρ˜s φs . (2) = −ρ˜s φs ∇ · us , dt where .ds (·)/dt = ∂(·)/∂t + us · ∇(·) denotes the material derivative to the soil phase; .ρ˜s and .us denote the intrinsic density and the velocity of the soil phase, correspondingly. Note that the soil grains are usually assumed imcompressible, thus Eq. 2 reduces as d s φs = −φs ∇ · us . . (3) dt The conservation of linear momentum for the soil phase is as follows ρ˜ φs

. s

d s us = ∇ · σ 's + ρ˜s φs g − φs ∇ p f + f d , dt

(4)

in which .σ 's denotes the soil’s effective stress, . p f is the water pressure, and . f d refers to the inter-phase drag force. In the case of Darcy flow, . f d owns the following expression φfρf g . fd = (u f − us ) (5) Ks where . K s is the permeability coefficient and .u f denotes the fluid velocity. In numerous scenarios involving dynamic water-soil coupling, the porous flows exhibit high Reynolds numbers, placing them in a turbulent regime. Consequently, the linear drag force relationship mentioned above must be revised with .

f d = αd

|| μ(1 − φ f )2 ρ˜ f (1 − φ f ) || ||u f − us || (u f − us ). (u − u ) + β f s d φ f Dc2 Dc

(6)

Herein, . Dc is the characteristic length of the soil grains; .αd and .βd are two constants whose values are taken as .αd = 150 and .βd = 1.75, respectively. The governing equations for the water phase include the mass conservation equation ρ˜ f d f ρ˜ f = − ∇ · (φ f u f + φs us ) . (7) dt φf with .ρ˜ f denoting the intrinsic density for the water, and the linear momentum conservation equation φ f ρ˜ f

.

df uf = −φ f ∇ p f + φ f ρ˜ f g − f d . dt

(8)

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2.2 Constitutive Models The weakly compressible SPH is adopted to model the fluid phase by introducing the equation of state ) [( ] ρ˜ f γ .p f = B −1 . (9) ρ˜ f 0 The elastoplastic model with Drucker-Prager yield surface is employed to describe soil’s behavior with the following yield and plastic potential functions, .

f (I1 , J2 ) = .

√

J2 + αϕ I1 − kc ,

√

g(I1 , J2 ) =

J2 + 3I1 sin ψ.

(10) (11)

In the equations, . I1 represents the first invariant of the stress tensor .σ 's ; . J2 is the second invariant of the deviatoric stress tensor . s's = σ 's − (I1 /3)I; .αϕ and .kc are Drucker-Prager’s parameters; .ψ is the dilatancy angle.

2.3 SPH Formulations Within the SPH framework, function and its spatial derivative can be approximated with ∑ . fi = f j Wi j V j , (12) j

∇ fi =

∑

.

f j ∇i Wi j V j ,

(13)

j

where . f i and . f j represent abbreviated forms of . f (x i ) and . f (x j ), respectively, with x and . x j denoting the position vectors of particle .i and . j. The index . j iterates through particles within the support domain of particle .i. The kernel function takes the form of a bell-shaped weighting function, defined as.Wi j = W (x i − x j , h), where .h represents the smoothing length that determines the support domain size. In this study, we employ the Wendland C2 function as the kernel. The term .V j represents the volume of particle . j. .∇i Wi j stands for the gradient of the kernel function regarding the position of particle .i. In this work, we determine the soil’s volume fraction using Eq. 3, while the fluid volume fraction is calculated based on the saturation assumption, as expressed in Eq. 1 with . i

φ = 1 − φs (x i ) = 1 −

∑

. i

a

φa Wia Va .

(14)

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Here, the subscripts .i and .a are used to represent the fluid and soil particles, respectively. The soil governing equations in SPH discretized form are outlined as follows ∑ ds φa = φa (ua − ub ) · ∇a Wab Vb dt b

(15)

d s ua 1 ∑ ' (σ a + σ 'b + πab I)∇a Wab Vb − = dt ρa b . φa ∑ 1 ∑ ( pi − pa )∇a Wai Vi + f ai Wai Vi + g a ρa i ρa i

(16)

.

where . pa is the water pressure interpolated at the soil particle with ∑ .

j

pa = ∑

p j Wa j ΔV j j

Wa j ΔV j

,

(17)

and .πab is the artificial viscosity adopted for numerical stability. The following SPH form of mass conservation equation for the fluid phase is employed in this work

.

⎡ ⎤ ∑ ) ρ˜i ⎣∑ ( d f ρ˜i s =− φ j u j − φi ui · ∇i Wi j V j + (1 − φ j )u j · ∇i Wi j V j ⎦ + dt φi j j ∑ Ψ i j · ∇i Wi j V j δhc f

(18)

j

In this study, we utilize the .δ-SPH method, a crucial component in the WCSPH scheme for achieving smooth pressure fields [6]. The constant .δ is typically set at a value of 0.1 [7]. We use the symbol .usj to represent the interpolated soil velocity at the water particle as follows s . ui

∑ a ua Wia ΔVa = ∑ . a Wia ΔVa

(19)

The numerical density diffusion term .Ψ i j is defined as ) ( xi j Ψ i j = 2 ρ˜i − ρ˜ j || ||2 . || x i j || + η2

.

(20)

The SPH approximation to the momentum conservation is outlined as follows .

d f ui 1 ∑ 1 ∑ ( pi + p j − πi j )∇i Wi j V j − f Wia Va + g i . =− dt ρ˜i j φi ρ˜i a ia

(21)

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2.4 Boundary Condition Properly addressing boundary conditions is crucial when working with SPH. In this study, we manage solid boundaries through the application of a generalized boundary particle method, inspired by the approach introduced by Adami et al. [8]. As illustrated in Fig. 2, the solid boundary is represented by multiple layers of fictitious particles, strategically arranged in regular lattices. Since only a single layer of boundary particles is utilized, these particles are treated as distinct particle types (either water or soil) when calculating various interactions. The soil stress and water pressure are evaluated with the following equations, ∑ ∑ ' ∑ pi Wwi Vi + g w · i ρ˜i x wi Wwi Vi a σ a Wwa Va ∑ σ 'sw = ∑ , pwf = i a Wwa Va i Wwi Vi

.

(22)

where the subscript.w is assigned to denote boundary particles. In addition to addressing stress and pressure, it’s imperative to consider velocity extrapolation for accurate simulation results. However, it’s worth noting that the approach to velocity extrapolation varies based on the specific boundary conditions employed. In line with the concept of dummy particles introduced by [8], the calculation of boundary velocity under non-slip conditions is performed as follows ∑ ∑ ui Wwi Vi a ua Wwa Va us = 2u0w − ∑ , uwf = 2u0w − ∑i a Wwa Va i Wwi Vi

. w

(23)

in which .u0w represents the specified velocity of the boundary. In cases where the free-slip condition is in use, the normal component continues to be determined using Eq. 23, while the update for the tangential part is as follows

Fig. 2 Sketch of the dummy particle scheme for boundary condition

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∑ τ ∑ τ u Wwa Va i u i Wwi Vi u sτ = ∑a a , u wf τ = ∑ W V wa a a i Wwi Vi

. w

289

(24)

where the superscript .τ is used to signify the tangential direction. It’s worth noting that the pressure extrapolation accounts for the effects of gravity. These gravitational effects are also taken into consideration when extrapolating the vertical stress component of .σ 'sw .

3 Validation In this section, we introduce a series of numerical examples aimed at validating the multi-layer SPH method we propose. For these cases, we employ analytical solutions for examination. Additionally, we include comparisons between the results obtained in this study and findings from other sources. It’s important to note that if one phase is absent in the problem under consideration, the formulations deteriorate to the case with a single phase. Thus, we do not present validations for single-phase water and soil dynamics here, as they are readily accessible in the literature, such as [8] for water and [10, 11] for soil.

3.1 Volume Conservation Test In this section, we construct a straightforward 3D test scenario to assess if the multilayer SPH model we’ve introduced preserves fluid volume. Let’s consider the scenario depicted in Fig. 3, where a body of water within a tank collapses into the pure soil matrix on the right side. The width of the tank is set as 0.3 m for this test. We assume the soil non-deformable, with the volume fraction .φs kept constant. When the water restores static, the final free surface elevation is .

Fig. 3 Illustration of the experiment for testing volume conservation

H=

H0 L w L − L s φs

(25)

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Fig. 4 Profiles of the water body at three time instants Fig. 5 Time history of free-surfaces at three locations

We numerically simulate the aforementioned scenario using the proposed SPH model. The problem dimensions are as follows: . L = 1 m, . L w = 0.3 m, . L s = 0.4 m, and . H0 = 0.8 m. The soil volume fraction is assumed as .φs = 0.3 and the mean grain size for the soil as . D50 = 0.016 m. Accordingly, the analytical solution shows that the final water level should be . H = 0.5 m. In the simulations, we begin with an initial particle resolution of .Δr = 20 mm. Figure 4 presents three snapshots of the the pressure contours. The time history of water level at .x = 0.3, 0.6 and 0.9 m on the symmetric section is drawn in Fig. 5. The former two points lie in the pure water region, while the other one is located in the porous region. As anticipated, the simulation results reveal that the final water elevations at these three points are consistent and precisely match the analytical solution.

3.2 U-Tube Seepage Test In this section, a permeability test conducted in a U-tube configuration is proposed and numerically reproduced with the multi-layer SPH method. The fundamental setup is depicted in Fig. 6, which comprises a square porous medium in the middle with a length of 1.0 m, a left water domain with an initial water elevation of

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Fig. 6 Sketch of U-tube seepage test

1.0 m

A

B

C

D

1.0 m

2.0 m

3.35 m

ΔH

1.0 m

3.5 m, a right water domain with a lower water level of 2.0 m. In this case, seepage is so slow that Darcy flow is applicable. Therefore, Eq. 5 or the linear term in Eq. 6 is adopted when evaluating the inter-phase drag force. The analytical solution to the hydraulic elevation difference evolution is derivable, reading ΔH =

.

ΔH0 ), exp 2K eq t/L (

(26)

where .ΔH0 represents the initial difference in head, . L is the length of the seepage path, and . K eq refers to the equivalent hydraulic conductivity of the porous medium. With Eq. 26, the Darcy velocity can be obtained as .

u Df =

ΔH0 K eq ( ) . exp 2K eq t/L /L

(27)

The intrinsic velocity of the fluid, .u f , can be expressed as .

u Df = φ f (u f − us )

(28)

The initial SPH particle spacing is set as 5 cm. The soil volume fraction has a linear increase from 0.6 at the left end to 0.8 at the right end. As a result, the permeability of the porous media is spatially different. The water pressure on both sides of the soil matrix is set hydrostatic, while the porous medium region is assumed to have a linear distribution from left to right, as depicted in Fig. 7a. The fluid particles can be categorized into four groups in Fig. 7b. In Group I and III, the particle spacing is almost the same. In comparison, SPH particles from Group II become sparse while those from Group IV get compressed. Each group of SPH particles show different particle spacing as seepage develops, which is caused by the varying volume fraction. Figure 8 illustrates the time history of the water head difference and intrinsic velocity. In total, four tests are conducted with various permeability coefficients, as

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Fig. 7 Pressure contour with . K eq = 0.01 m/s

(a) Temporal curves of the head difference

(b) Temporal curves of the intrinsic velocity

Fig. 8 Evolution of head difference and intrinsic velocity. The lines and markers denote the analytical and numerical solutions, respectively

shown in Fig. 8a. In Fig. 8b, the temporal curves of the intrinsic velocity is evaluated with . K eq = 0.01 m/s. In general, good consistency is observed between the results from the SPH simulations and the analytical solution for both water elevation and intrinsic velocity evolution. This demonstrates, again, the high performance of the proposed multi-layer SPH model.

3.3 Submerged Soil Stress Initialization Under Gravity In this section, we validate a dynamic and fully coupled problem against an analytical solution. As depicted in Fig. 9, a soil body with a height of . Hs = 0.5 m is completely submerged in a water body with a depth of . Hw = 1.0 m. The soil is characterized

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Fig. 9 Sketch of the problem of submerged soil

by a particle density of .ρ˜s0 = 2750 kg/m.3 and a solid volume fraction of .φs = 0.5. Additionally, the mean grain size is assumed to be . D50 = 0.05 m. The soil’s material parameters include a Young’s modulus of . E = 1.5 × 106 Pa, a Poisson’s ratio of ◦ ◦ .ν = 0.2, a frictional angle of .ϕ = 25 , a dilatancy angle of .ψ = 0 , and cohesion of .c = 0 Pa. Initially, all stress and pressure for both the water and soil phases are set to zero. Subsequently, a gravitational load is applied in a single step at the beginning of the simulation. At the final stage, the water pressure and soil stress will balance the gravity, and have the following expressions .

p = (Hw − z)ρ˜ f 0 g,

( ) σzz' = (Hs − z)φs ρ˜s0 − ρ˜ f 0 g

(29)

In this case, the length, width and height are all set as 1.0 m, thus a threedimensional problem. At the initial stage, there will be a strong stress fluctuation due to the sudden application of the gravitational load. Following the practice from [12], numerical damping techniques is adopted herein. Note that in this simulation, all information of both water and soil phases is updated according to the governing equations. Besides, the inter-phase drag force in the current simulation is activated. The initial particle spacing is set as .Δr = 0.01 m, and the physical duration lasts for 2 s. As the gravitational load is activated, both water pressure and soil stress began to increase, as shown in Fig. 10. Under the unified action of the numerical damping, the inter-phase drag force and the viscous effects, stable water pressure and soil stress are observed in approximately 0.35 s. When compared to the analytical values, high consistency is found. Both the water pressure and soil vertical stress are presented in Fig. 11. In general, proportional relationships with elevation can be found both inside the porous media and on the boundary surfaces, demonstrating that the result is reasonable and the proposed free-slip boundary method works well. To give a quantitative analysis, the depth distribution of water pressure and soil vertical stress along the central line are presented in Fig. 12, from which a good agreement between the numerical and analytical results is observed. Particularly, error analysis shows that the normalized

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Fig. 10 Temporal evolution of vertical effective stress and water pressure at the point .x = 1.0 and = 0.2 m. Effective stress is negative and water pressure is positive

.z

Fig. 11 Contours of water pressure and vertical soil effective stress at .t = 2 s

root mean square error is 1.1% for soil vertical stress and only 0.5% for water pressure. This once again demonstrates the satisfactory performance of the proposed multi-layer SPH method on modeling the dynamic water-soil coupling problems.

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Fig. 12 The water pressure soil vertical effective stress along the central line at .t = 2 s. Pressure is positive, and effective stress is negative

4 Conclusions In this paper, we introduce an innovative SPH model rooted in the mixture theory to address general dynamic problems involving the interaction of water and soil. We derive the SPH formulation based on intrinsic fluid densities, and we propose a novel technique to emulate the free-slip boundary condition. Three numerical cases including the volume conservation of the fluid phase, Utube seepage test and submerged soil stress initialization under gravity are conducted and good agreement is observed between the results from numerical simulations and analytical solutions. The satisfactory performance demonstrates the model’s ability to accurately simulate water-soil coupling problems, indicating its significant potential for practical engineering applications. Acknowledgements The first author wishes to thank the Otto Pregl Foundation for financial support in Austria.

References 1. Han, K., Feng, Y.T., Owen, D.R.J.: Coupled lattice Boltzmann and discrete element modelling of fluid-particle interaction problems. Comput. Struct. 85, 1080–1088 (2007) 2. Peng, C., Zhan, L., Wu, W., et al.: A fully resolved SPH-DEM method for heterogeneous suspensions with arbitrary particle shape. Powder Technol. 387, 509–526 (2021) 3. Morland, L.W., Kelly, R.J., Morris, E.M.: A mixture theory for a phase-changing snowpack. Cold Reg. Sci. Technol. 17, 271–285 (1990) 4. Meruane, C., Tamburrino, A., Or, Roche: On the role of the ambient fluid on gravitational granular flow dynamics. J. Fluid Mech. 648, 381–404 (2010)

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5. Akbari, H., Taherkhani, A.: Numerical study of wave interaction with a composite breakwater located on permeable bed. Coast. Eng. 146, 1–13 (2019) 6. Antuono, M., Colagrossi, A., Marrone, S.: Numerical diffusive terms in weakly-compressible SPH schemes. Comput. Phys. Commun. 183, 2570–2580 (2012) 7. Meringolo, D.D., Marrone, S., Colagrossi, A., et al.: A dynamic .δ-SPH model: how to get rid of diffusive parameter tuning. Comput. Fluids 179, 334–355 (2019) 8. Adami, S., Hu, X.Y., Adams, N.A.: A generalized wall boundary condition for smoothed particle hydrodynamics. J. Comput. Phys. 231, 7057–7075 (2012) 9. Peng, C., Wang, S., Wu, W., et al.: LOQUAT: an open-source GPU-accelerated SPH solver for geotechnical modeling. Acta Geotech. 14(5), 1269–1287 (2019) 10. Bui, H.H., Fukagawa, R., Sako, K., et al.: Lagrangian meshfree particles method (SPH) for large deformation and failure flows of geomaterial using elastic-plastic soil constitutive model. Int. J. Numer. Anal. Meth. Geomech. 32, 1537–1570 (2008) 11. Peng, C., Wu, W., Yu, H.S., et al.: A SPH approach for large deformation analysis with hypoplastic constitutive model. Acta Geotech. 10, 703–717 (2015) 12. Bui, H.H., Fukagawa, R.: An improved SPH method for saturated soils and its application to investigate the mechanisms of embankment failure: Case of hydrostatic pore-water pressure. Int. J. Numer. Anal. Meth. Geomech. 37, 31–50 (2013) 13. Bui, H.H., Nguyen, G.D.: A coupled fluid-solid SPH approach to modelling flow through deformable porous media. Int. J. Solids Struct. 125, 244–264 (2017)

Assessing Slope Stability Based on Measured Data Coupled with PSO Jiaqiang Zou, Wei Zhang, and Aihua Liu

Abstract Real-time accessing slope stability has always been one of the research hotspots in landslide prevention and control. Coupled with Particle Swarm Optimization (PSO), a systematic method based on measured data assessing slope stability is developed in this paper. Through the numerical simulation of two case studies, the results show that: (1) using the monitoring dataset, PSO can be applied to effectively deduce the mechanical properties of the slope; (2) employing the ascertained mechanical parameters, the safety factor of the slope can be swiftly computed through numerical simulation. Therefore, with the help of measured data of the slope, the developed assessing method of slope stability based on PSO can fully consider the evolution of mechanical parameters of the slope, and implement corresponding preventive measures for slope deformation or even instability. To this end, a more reasonable and efficient tool for practical slope engineering is also provided.

J. Zou Construction Engineering Quality Supervision Station of Nanhai District, Foshan 528200, China e-mail: [email protected] J. Zou · W. Zhang · A. Liu (B) College of Water Conservancy and Civil Engineering, South China Agricultural University, Guangzhou 510642, China e-mail: [email protected] W. Zhang e-mail: [email protected] J. Zou Institute of Geotechnical Engineering, University of Natural Resources and Life Sciences, Vienna, Feistmantelstrasse 4, 1180 Vienna, Austria © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 W. Wu and Y. Wang (eds.), Recent Geotechnical Research at BOKU, Springer Series in Geomechanics and Geoengineering, https://doi.org/10.1007/978-3-031-52159-1_19

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1 Introduction In the long geological evolution process, soil become a combination of complex geological bodies, and they are very difficult to obtain accurate geotechnical mechanical parameters. The complex composition of the rock and soil on the slope frequently leads to substantial discrepancies between the mechanical parameters derived from laboratory tests or field assessments and the actual values. Consequently, the exploration of geotechnical parameter determination for slopes through real-time monitoring data and backward analysis has piqued significant interest among researchers [1–5]. The primary objective of back-analysis is to obtain soil properties that closely match the exact values at the site. Through the process of back analysis, it becomes feasible to directly extract the necessary soil parameters for geotechnical assessments from the specific data gathered during practical engineering monitoring. A variety of optimization algorithms are currently prevalent for conducting back analysis, genetic algorithm (GA) [6, 7], neural network (NN) [2, 8], ant colony optimization (ACO) [9, 10], and particle swarm optimization (PSO) [11–14], and more. Among this array of optimization techniques, PSO has showcased its exceptional efficiency and speed, outperforming most other methods in various scenarios [11], so PSO is used for conducting back analysis in this paper. Since the previous researches mainly use empirical methods or mathematical statistics methods, which inevitably weakens the influence of deformation mechanism of slope, the assessment results of the slope based on monitoring information would be more accurate. In general, when conducting numerical back analysis, the more parameters to be solved, the more complex the calculation will be, and the test error will have a transfer effect on the calculation results. Hence, several values of slope soil mass are selected as known quantities according to engineering survey, such as Poisson’s ratio, elastic modulus, unit weight, etc., which will help eliminate the transmission of measurement errors and obtain good results. In the back analysis, while the finite element method (FEM) remains the prevailing approach [8, 12, 15, 16], it is susceptible to mesh deformation, leading to certain constraints in its utility for large deformation back analysis. In light of this concern, this paper opts for the Particle Finite Element Method (PFEM), known for its significant advantages in geotechnical engineering large deformation simulations [17–21], as the numerical analysis and back analysis method. Therefore, in this study, PSO incorporated to inversely ascertain the geotechnical parameters of the rock and soil mass. Subsequently, numerical experiments were conducted to dynamically assess the stability of the slope, utilizing the determined geotechnical parameters in conjunction with real-time monitoring data. What’s more, this paper serves the purpose to provide a solution to the key problems in real-time early warning of slope.

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2 Developed Assessing Method for Slope Stability 2.1 Basic Theory of PSO Particle Swarm Optimization (PSO) was proposed by J. Kennedy and R. C. Eberhart in 1995. Within the PSO algorithm, every particle implies a prospective solution to the problem. The calculation process can be summarized as follows: Initially, the optimization problem’s feasible solutions are randomized and initialized using a swarm of particles, and the spatially optimal solution is determined by specifying a random speed of each particle in the population. Subsequently, during each iterative calculation phase, the optimal results for both the global and individual particles in the current step are acquired and compared to the target results to calculate the error accuracy. Finally, through continuous iterations, each particle continuously enhances its flight direction within the defined constraints, ensuring a consistent approach towards the global optimal value (.gbest ) and the local optimal value (. pbest ) in the feasible solution space. In mathematical terms, the search space is a .d-dimensional domain that encompasses .n particles. For an .ith particle, the updating velocity and position can be calculated by .

Vid (k + 1) = Vid (k) + c1r1 (Pid (k) − X id (k)) + c2 r2 (Pid (k) − X id (k))

(1)

X id (k + 1) = X id (k) + Vid (k + 1)

(2)

.

where .Vid is the velocity; . X id is the position of the particle; .k is the current iteration step; .c1 and .c2 are both the acceleration coefficient, where .c1 is mainly used to represent the ability of a particle to approach its own optimal position, and .c2 represents the ability of a particle to approach the global optimal position; .r1 ∼ U (0, 1) and .r 2 ∼ U (0, 1) are two independent random numbers.

2.2 Real-Time Updating of Mechanical Parameters Based on the basic idea of PSO outlined in Sect. 2.1, the procedure for calculating real-time updates of geotechnical parameters using PSO unfolds as follows: 1. Utilizing available engineering geological data, identify those needs for inversely determining the mechanical properties of rock and/or soil, and establish the upper and lower limits for these parameters as the search boundaries for PSO. Meanwhile, select suitable real-time monitoring data to serve as target results.

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2. Commence the PSO process by configuring the various parameters for the population and initiating random initialization for each particle, generating a total of .n initial values for the mechanical parameters requiring simultaneous inversion in the same computational step as the population size. 3. Upon completion of the PSO process, ascertain the global optimal value to acquire the optimal solution for the mechanical properties of rock and soil that require inversion.

2.3 Real-Time Assessment Slope Stability According to the updated inversion results of parameters of the slope from Sect. 2.2, the acquired inverted mechanical properties are used for numerical simulations to calculate the factor of safety (FOS). This paper combines the strength reduction technique and PFEM to further evaluate the safety of the slope body. Leveraging GPU acceleration for computation allows for a swift and efficient evaluation of slope stability. More information about GPU acceleration technique can be found in Ref. [19]. In this paper, the Mohr-Coulomb model is employed and the total simulation time is set to be 20 s. The numerical model is established by focusing on the weakest section within a sufficiently large range, within which feature points are identified based on monitoring data. Subsequently, boundary conditions are applied, with horizontal restraints at the lateral boundaries and fixed constraints in both directions at the bottom boundary. In addition, the self-weight gradually increases linearly from 0 to its actual value in the first 5 s, after which it remains constant. The real-time assessment is calculated by the following equations: c SRF ) ( tanφ .φt = arctan SRF c =

. t

(3)

(4)

where SRF is the Strength Reduction Factor. During the simulation process, the SRF value increases incrementally from 1.0 to 2.0. By the end of each simulation, the maximum displacement of the slope will be recorded to further help distinguish a sudden change in displacement, indicating the corresponding SRF is regarded as FOS of the slope.

Assessing Slope Stability Based on Measured Data Coupled …

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3 Case Study 3.1 Landslide of Longjiang Hydropower Project 3.1.1

Real-Time Back Analysis

To gain a comprehensive understanding of the deformation and pressure on the left bank slope of the Longjiang Project, various kinds of data monitoring have been employed, such as surface displacement, internal deformation, anchor bolt stress, and seepage pressure, were equipped. According to previous geological survey results, a landslide occurred in 2013. Therefore, in order to monitor the safety of the slope, GNSS equipment was installed, as illustrated in Fig. 1. According to the GNSS monitoring database of the slope, the measuring point positioned at an elevation of 905 m showed the maximum displacement change, as shown in Fig. 2. As such, the section containing the GNSS point is located is chosen as the analysis section, and the displacement data is taken as the target value. Based on the geological survey, three distinct soil types are identified in this area, namely clay, completely weathered gneiss (CWG), and strongly weathered gneiss (SWG). So, it is necessary to distinguish the determining mechanical parameters among various layers. Given the large buried depth of SWG, rainfall infiltration has little impact on its mechanical properties, the cohesion and internal friction angle of clay and CWG can be obtained through back analysis using real-time monitoring

GNSS monitoring points

Back analysis reference point

Fig. 1 Distribution of GNSS monitoring points

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07/17/20

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Fig. 2 Displacement versus time monitored by GNSS at the elevation of 905 m Table 1 Mechanical parameters of longjiang project Soil layer Clay Completely weathered Strongly weathered gneiss (CWG) gneiss (SWG) Saturated unit weight m.3 Deformation modulus E/MPa Poisson’s ratio .ν Cohesion c.' /kPa Internal friction angle .φ/.◦

17.8

18.8

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50.0

50.0

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0.3 To be inversed To be inversed

0.3 50.0 28.0

.γsat /kN.·

data. Meanwhile, the remaining mechanical properties of the slope are suggested by the geological survey and are detailed in Table 1. As depicted in Fig. 3b, The numerical model is established based on the section where the GNSS is equipped at the elevation of 905 m, as illustrated in Fig. 3a. The model encompasses a range from –116.379 m to 30.929 m in the X direction and 831.049 m to 927.500 m in the Y direction. Consequently, a mesh with 23882 elements and 12190 nodes is divided. The applied loads only include the self-weight of soil. Boundary conditions impose full constraints at the bottom and normal constraints on both sides. Before conducting the inversion analysis, a parametric examination of the various mechanical properties of each soil layer is undertaken. The results show that the slope deformation caused by the reduction of soil cohesion and internal friction angle is the most significant, which would be the focus of inversion analysis. This is also in line with the existing geological survey. Therefore, as depicted in Fig. 2, March 18, 2021, is selected as the representative date for back analysis, and the

Assessing Slope Stability Based on Measured Data Coupled … Fig. 3 a Design drawing of monitoring section; b Mesh of monitoring section

m 934

927.50

924

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914

CWG

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Ground surface GNSS Monitoring point 905.00

904 SWG

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864 854 844

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error precisions

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measured GNSS value on that date was 48.76 mm. The results of the PSO-based back analysis at this specific date are presented in Fig. 4. Furthermore, the numerical displacement of the monitoring point matches the GNSS data closely, measuring 48.7 mm. This indicates that the parameters obtained from the back analysis are reliable. The inversely obtained parameters are as follows: cohesion and internal friction angle of clay is 36.86 kPa and 14.06.◦ , respectively; while those of CWG are 37.13 kPa and 18.54.◦ , respectively.

3.1.2

Numerical Example

With the mechanical properties inversely determined, the slope’s safety can now be assessed numerically. The maximum displacement of the slope after 20 s under different SRFs is demonstrated in Fig. 5. It can be seen that as the SRF increases, the maximum displacement goes up gradually from 0.27 m to more than 2.0 m.

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Fig. 5 Maximum displacement of the slope after 20 s under different SRFs Displacement / m

2

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0.5

0

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1.1

1.15

1.2

1.25

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When the SRF reaches 1.23, there is a sudden change in the maximum displacement, implying a landslide occurrence at that point. Consequently, the FOS for the slope is determined to be 1.23. Figure 6 presents the distribution of slope plastic zone and displacement field under different SRFs are given in. It’s observable that when SRF reaches 1.23, the plastic zone of the slope nearly connects to form a sliding surface, situated in the middle and upper portion of the slope. This indicates a critical state for the slope, and as a result, the maximum displacement is recorded to be 0.60 m. However, when SRF reaches 1.50, the plastic zone of the slope extends through to form a continuous sliding surface, leading to substantial overall deformation and sliding of the slope, with a maximum displacement of 18.0 m. There is no doubt that a landslide occurs.

4 Conclusions To sum up, it is a trend of great significance to study slope instability based on the assessing method of parametric inversion. Based on the measured data of slope, coupled with Particle Swarm Optimization (PSO), a systematic assessment method of slope stability is developed in this paper. Through a case study, the developed method is verified to be effective for real-time monitoring and assessing slope safety. The main conclusions are as follows: 1. Utilizing PSO, the back analysis of soil mechanical parameters can be conducted with remarkable speed, facilitating a more efficient assessment of slope safety. 2. The inverse analysis of the geotechnical parameters for the slope, based on measured data, proves to be reasonable and serves as a solid foundation for subsequent safety analysis.

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0.3 0.2 0.1 0.0

Disp / m 18 16 14 12 10 8 6 4 2 0

(b)

Fig. 6 Contour plot of the slope after 20s under different SRFs: a incremental deviatoric plastic strain invariant distribution; b magnitude of displacement

3. The methodology developed in this paper helps address the technical challenges associated with dynamically evaluating slope safety and implementing real-time landslide warnings. Acknowledgements This research is supported by the Water Conservancy Science and Technology Innovation Project of Guangdong Province (grant No. 2017-30), and the International Training Program for Outstanding Young Scientific Researchers in Colleges and Universities of Guangdong Province (grant No. HT202002314-4).

References 1. Cai, M., Morioka, H., Kaiser, P.K., et al.: Back-analysis of rock mass strength parameters using AE monitoring data. Int. J. Rock. Mech. Min. 44(4), 538–549 (2007) 2. You, K.: A case study on the utilization of tunnel face mapping data for a back analysis based on artificial neural network. KSCE J. Civ. Eng. 18, 751–759 (2014) 3. Wang, M., Chi, S., Xie, Y., et al.: Dynamic parameters inversion analysis of rockfill materials considering interaction effects based on weak earthquakes. Soil Dyn. Earthq. Eng. 130, 105968 (2020)

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4. Guo, C., Ma, G., Xiao, H., et al.: Displacement back analysis of reservoir landslide based on multi-source monitoring data: a case study of the cheyiping landslide in the Lancang river basin. China. Remote Sens. 14(11), 2683 (2022) 5. Stuyts, B., Weijtjens, W., Devriendt, C.: Development of a semi-structured database for backanalysis of the foundation stiffness of offshore wind monopiles. Acta Geotech. (2022). https:// doi.org/10.1007/s11440-022-01551-3 6. Jurado-Piña, R., Jimenez, R.: A genetic algorithm for slope stability analyses with concave slip surfaces using custom operators. Eng. Optim. 47(4), 453–472 (2015) 7. Palazzolo, N., Peres, D.J., Bordoni, M., et al.: Improving spatial landslide prediction with 3D slope stability analysis and genetic algorithm optimization: application to the Oltrepò Pavese. Water 13(6), 801 (2021) 8. Zhang, J., Li, P., Yin, X., et al.: Back analysis of surrounding rock parameters in Pingdingshan mine based on bp neural network integrated mind evolutionary algorithm. Mathematics 10(10), 1746 (2022) 9. Gao, W.: Displacement back analysis for underground engineering based on immunized continuous ant colony optimization. Eng. Optim. 48(5), 868–882 (2015) 10. Ebrahim, G., Mahdi, M., Mohammad Farouq, H., et al.: Determination of initial stress state and rock mass deformation modulus at Lavarak HEPP by back analysis using ant colony optimization and multivariable regression analysis. Bull. Eng. Geol. Environ. 80, 429–442 (2021) 11. Cheng, Y.M., Li, L., Chi, S.C.: Performance studies on six heuristic global optimization methods in the location of critical slip surface. Comput. Geotech. 34(6), 462–484 (2007) 12. Jia, Y., Chi, S.: Back-analysis of soil parameters of the Malutang II concrete face rockfill dam using parallel mutation particle swarm optimization. Comput. Geotech. 65, 87–96 (2015) 13. Huang, F., Huang, J., Jiang, S., et al.: Landslide displacement prediction based on multivariate chaotic model and extreme learning machine. Eng. Geol. 218, 173–186 (2017) 14. Pallero, J.L.G., Fernández-Martínez, J.L., Fernández-uñiz, Z., et al.: GravPSO2D: a Matlab package for 2D gravity inversion in sedimentary basins using the particle swarm optimization algorithm. Comput. Geosci. 146, 104653 (2021) 15. Schädler, W., Borgatti, L., Corsini, A., et al.: Geomechanical assessment of the Corvara earthflow through numerical modelling and inverse analysis. Landslides 12, 495–510 (2015) 16. Liu, Y., Ren, W., Liu, C., et al.: Displacement-based back-analysis frameworks for soil parameters of a slope: using frequentist inference and Bayesian inference. Int. J. Geomech. 22(4), 04022026 (2022) 17. Zhang, W., Yuan, W., Dai, B.: Smoothed particle finite-element method for large-deformation problems in geomechanics. Int. J. Geomech. 18(4), 04018010 (2018) 18. Yuan, W., Liu, K., Zhang, W., et al.: Dynamic modeling of large deformation slope failure using smoothed particle finite element method. Landslides 17(7), 1591–1603 (2020) 19. Zhang, W., Zhong, Z., Peng, C., et al.: GPU-accelerated smoothed particle finite element method for large deformation analysis in geomechanics. Comput. Geotech. 129, 103856 (2021) 20. Zhang, W., Zou, J., Zhang, X., et al.: Interpretation of cone penetration test in clay with smoothed particle finite element method. Acta Geotech. 16(8), 2593–2607 (2021) 21. Zou, J., Yang, F., Yuan, W., et al.: A kinetic energy-based failure criterion for defining slope stability by PFEM strength reduction. Eng. Fail. Anal. 145, 107040 (2023)