Dynamic Mechanical Behaviors and Constitutive Model of Roller Compacted Concrete 9811989869, 9789811989865

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Hydroscience and Engineering

Sherong Zhang · Xiaohua Wang · Chao Wang

Dynamic Mechanical Behaviors and Constitutive Model of Roller Compacted Concrete

Hydroscience and Engineering Series Editors Feng Jin, Department of Hydraulic Engineering, Tsinghua University, Beijing, China Duruo Huang, Department of Hydraulic Engineering, Tsinghua University, Beijing, China

This Springer book series Hydroscience and Engineering addresses multidisciplinary advancements in water-related science and engineering practice. Main scope of the book series includes fundamental research and cutting-edge engineering technology in water resources and river/reservoir management in relation with hydraulic structures, erosion control, sediment transport, river basin management and planning, flood control, geological/hydrological risk assessment, offshore and costal engineering. The book series focuses on developing innovative, sustainable and environment-friendly materials and structures to improve infrastructural functionality and sustainability, and to enhance adaptability to changing environmental conditions. Recent advancements and successful insights obtained from engineering practice in China and worldwide will also be thoroughly introduced. The overarching goal of the book series is to provide combined knowledge on foremost research and solid engineering expertise on hydroscience and engineering from an interdisciplinary standpoint. The individual book volumes in the series are thematic, each specializing a focused topic, such as a new dam type and innovative dam construction materials. As a collection, the book series provides valuable resources targeting on a wide spectrum of audience, including researchers in academia, practicing engineers in engineering community and students who work on expanding their knowledge in the related areas.

Sherong Zhang · Xiaohua Wang · Chao Wang

Dynamic Mechanical Behaviors and Constitutive Model of Roller Compacted Concrete

Sherong Zhang State Key Laboratory of Hydraulic Engineering Simulation and Safety School of Civil Engineering Tianjin University Tianjin, China

Xiaohua Wang State Key Laboratory of Hydraulic Engineering Simulation and Safety School of Civil Engineering Tianjin University Tianjin, China

Chao Wang State Key Laboratory of Hydraulic Engineering Simulation and Safety School of Civil Engineering Tianjin University Tianjin, China

ISSN 2730-9002 ISSN 2730-9010 (electronic) Hydroscience and Engineering ISBN 978-981-19-8986-5 ISBN 978-981-19-8987-2 (eBook) https://doi.org/10.1007/978-981-19-8987-2 Jointly published with Science Press The print edition is not for sale in China mainland. Customers from China mainland please order the print book from: Science Press. ISBN of the Co-Publisher’s edition: 978-7-03-074226-1 © Science Press 2023 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 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 publishers, 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 publishers 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 publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

To our families, teachers and students

Foreword

The dynamic constitutive model of concrete-like materials is an important theoretical basis for the engineers and practitioners of civil and hydraulic engineering to analyze functional buildings and structures subjected to extreme dynamic loads. As a special kind of concrete material, RCC is widely used in constructions of hydraulic structures, transportation, and highway engineering facilities. The significant differences between RCC and normal concrete lie in the mix design and construction technique, which leads to the different mechanical properties. For a long time, however, RCC has always lacked a reasonable dynamic constitutive model to understand its mechanical behaviors under dynamic loads. Many engineers and practitioners, and even some researchers, have reservations about this topic. There is a mismatch between research and application. Especially in recent years, constitutive models aiming at refined simulations have been continuously developed, with increasing complexity and parameters. Therefore, this book is a welcome attempt to establish a reasonable dynamic constitutive model for RCC that can serve as a bridge between laboratory test and numerical simulation. I came to know Prof. Zhang since the years of his undergraduate period at Tianjin University. He has engaged in teaching and research works related to hydraulic structures for 40 years. He has been contributing to improve the static and dynamic analysis of hydraulic structures. I have known Dr. Xiaohua Wang since the years of his Ph.D. study at Tianjin University. His research project focused on blast-resistance analysis and protection design for dam. As one of his Ph.D. examiners, I was impressed by his enthusiasm in research and knowledge in this field. I came to know Dr. Chao Wang during the Spring of 2017. At that time, he was at Curtin University as a Visiting Scholar from Tianjin University. We began academic discussions and collaboration from then on. His rigorous research approach, deep understanding of concrete dynamic behavior, and solid analytical skills have left a deep impression on me. This book consists of a summary and a refinement of results obtained by the authors on the topic of dynamic behaviors and constitutive model of RCC. It gathers the authors’ original results in the effects of construction technique, aggregate size, strain rate, and specimen size on the dynamic mechanical properties of RCC, and then the constitutive modeling approach and engineering applications are clearly vii

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Foreword

presented based on rigorous analyses. In summary, this book provides an informative exposition of the current advances in the formulation of numerical simulation in blast-resistance analysis for RCC dams and will benefit both scientists and practitioners. Students will also find in this book a stimulating introduction to the dynamic mechanics of concrete-like materials. Hong Hao Fellow Australian Academy of Technological Science and Engineering (ATSE) President, International Association of Protective Structures Professor of School of Civil and Mechanical Engineering Curtin University Bentley, WA, Australia

Preface

The concrete as the research object of engineering material has very complex property especially under dynamic loads. How to correctly understand or describe the concrete physical and mechanical behaviors under dynamic loads is crucial to the development of the protection design for concrete structures. Thus, the dynamic mechanical constitutive model of concrete-like materials is an important theoretical basis for the graduate students of civil and hydraulic engineering to analyze functional buildings and structures subjected to extreme loads. In protective engineering, the laboratory tests are important ways to study and determine a certain physical property or parameters in concrete dynamic constitutive model. As a blend of concrete, RCC has been widely applied in the infrastructure construction such as hydraulic structures and pavements due to its advantages of cost-effectiveness and rapid construction. The special mixture design and construction technique of RCC evidently cause the physical and mechanical properties to be quite different from normal concrete. But for a long time, the constitutive models of RCC lack a simple and reliable mathematical model that can truly reflect the strainrate effect on its mechanical behaviors which is verified to be coupled with various factors such as specimen size and aggregate size. Based on a large amount of experiments and engineering practices, the researchers have proposed various concrete dynamic constitutive models for describing the rate-dependent behaviors. These models usually contain many unknown parameters that are generally determined using the regression analysis based on test data of normal concrete and evidently not suitable for RCC. Hence, research works in this book have been conducted to investigate the dynamic mechanical behaviors and constitutive model for RCC. This book begins with the most basic theoretical knowledge and combines recently concrete experiments to realize how advanced theory and direction can be applied to establish a dynamic constitutive model for RCC. Generally, this book solves the following four main concerns, i.e., special physical and dynamic mechanical properties, dynamic sizedependence, aggregate size effect on dynamic mechanical behaviors, and dynamic

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constitutive model modification for RCC. We also hope to enrich the developed constitutive model for RCC in the future by sharing with readers the latest scientific research results and to promote the application of this model to achieve the purpose of blast-resistance analysis and protective design for concrete dams. Tianjin, China September 2022

Sherong Zhang Xiaohua Wang Chao Wang

Acknowledgements

In the process of writing this book, we received much sincere guidance and help from experts and colleagues. Here, we would like to express our sincere gratitude to Dr. Mao Yu, Dr. Ran Song, and Dr. Peiyong Wei formerly from Tianjin University for their participation in the research projects and hard work in the process of organizing and reviewing the book. We are also grateful to Science Press and Springer for performing detailed work on the publication of this book. We would like to express our heartfelt thanks to National Natural Science Foundation of China (Grant No. 51779168, 51979188, and 52109163) and the Open Fund of State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University (Grant No. 2021SGG03) for their funding.

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About This Book

This book systematically illustrates the dynamic mechanical behaviors and discusses the fundamentals of the constitutive modeling of roller compacted concrete (RCC), influenced by the construction technique and mix design. Four typical problems are analyzed using laboratory tests, numerical simulation, and theoretical analysis, i.e., to illustrate the special dynamic mechanical behaviors of RCC, to reveal the dynamic size-dependence of mechanical properties, to discuss the aggregate size effect on dynamic mechanical properties, and to modify the dynamic constitutive model for RCC. Generally, the constitutive modeling of RCC needs a comprehensive understanding of dynamic size-dependence and aggregate size effect of concrete that coupled with the strain-rate sensitivity. So that, readers can master the modified dynamic constitutive model of RCC to analyze and solve the problems in blastresistance analysis and protective design of RCC dams. This book can be used as a postgraduate textbook for civil and hydraulic engineering in colleges and universities, and as an elective course for senior undergraduates. It can also be used as a reference for relevant professional scientific researchers and engineers in field of protective design of concrete structures.

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Contents

1 Constitutive Relations of RCC: An Overview . . . . . . . . . . . . . . . . . . . . . 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Dynamic Behaviors and Constitutive Models for Normal Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Special Physical and Mechanical Properties of RCC . . . . . . 1.2.3 Size-Dependence of Concrete Under Dynamic Loads . . . . . 1.2.4 Aggregate Effect on Mechanical Behaviors of Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Experimental Research on Dynamic Behaviors of RCC . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Experimental Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Material and RCC Mix Proportion . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Specimen Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Quasi-static Testing Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Reliability Analysis of SHPB Test . . . . . . . . . . . . . . . . . . . . . . 2.3 Effect of Construction Technique on Dynamic Behaviors . . . . . . . . . 2.3.1 Dynamic Mechanical Properties . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Stratification Effect of RCC on Dynamic Mechanical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Shock Wave Propagation Across Interlayers in RCC . . . . . . . . . . . . . 2.4.1 Experimental Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Incident and Transmitted Waveforms . . . . . . . . . . . . . . . . . . . 2.4.3 Reflection and Transmission of Shock Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Theoretical Analysis on the Shock Wave Propagation . . . . . . . . . . . . 2.5.1 Wave Propagation in Viscoelastic Medium . . . . . . . . . . . . . . . 2.5.2 Wave Attenuation During Propagation Across RCC . . . . . . . 2.5.3 Influence of Interlayers on Transmitted Wave . . . . . . . . . . . .

1 1 3 3 4 6 8 11 17 17 18 18 21 22 23 26 26 29 31 31 33 34 36 36 38 41

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2.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Meso-mechanic-Based Dynamic Behaviors of RCC . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Mesoscopic Simulation Method and Validation . . . . . . . . . . . . . . . . . 3.2.1 Meso-simulation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 The Constitutive Model and Parameters of Meso-components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Validation of Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Effect of Maximum Aggregate Size on Dynamic Mechanical Properties of RCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Two Dimensional Mesoscopic Model . . . . . . . . . . . . . . . . . . . 3.3.2 Effect of Aggregate Size on Dynamic Compressive Behaviors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Effect of Aggregate Size on Dynamic Tensile Behaviors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Influence of Layer Effect on Dynamic Mechanical Properties . . . . . 3.4.1 The Influence of Layer Effect on Dynamic Compressive Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Effect of Layer Effect on Dynamic Tensile Mechanical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Consturction-Induced Damage Effect on Dynamic Compressive Behaviors of RCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Specimen Preparation and Damage Quantification . . . . . . . . . . . . . . . 4.2.1 Specimen Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Quantification of the Initial Damage . . . . . . . . . . . . . . . . . . . . 4.3 Initial Damage Effect on the Dynamic Behaviors of RCC . . . . . . . . 4.3.1 Mechanical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Initial Damage Effect on Stress–Strain Curves . . . . . . . . . . . . 4.3.3 Initial Damage Effect on Dynamic Mechanical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Statistical Characteristics of Dynamic Compressive Behaviors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Assessment to the Initial Damage Effect on the Dynamic Behaviors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Correlation Between Initial Damage and Dynamic Behaviors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Evaluation on the Initial Damage from Improper Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50 53 55 56 57 58 61 61 62 63 64 67 67 68 68 69 71 71 72 73 73 76 76 78 80 81

Contents

5 Size-Dependence of Dynamic Behaviors for RCC Under High-Strain-Rate Loadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Dynamic Size Effect on Experimental Results . . . . . . . . . . . . . . . . . . 5.2.1 Schematic Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Failure Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Stress-Strain Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Dynamic Increase Factor for Compressive Stress . . . . . . . . . 5.3 Dynamic Size Effect on Compressive Behaviors of RCC . . . . . . . . . 5.3.1 Definitions of Dynamic Mechanical Properties . . . . . . . . . . . 5.3.2 Size Effect on Various Dynamic Mechanical Properties . . . . 5.3.3 Statistical Significance of the Dynamic Size Effect . . . . . . . . 5.3.4 Distribution Characteristic of Dynamic Compressive Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Modified Weibull Size Effect Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Fragmentation-Based Dynamic Size Effect of RCC Under Impact Loadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Fragment Characteristics of RCC Under Impact Loads . . . . . . . . . . . 6.2.1 Dynamic Fragmentation Process . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Fragment Size Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Relationship Between Fragment Size and Dynamic Behaviors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Dynamic Size Effect Depicted by Fractal Characteristics . . . . . . . . . 6.4 Fractal Mechanism of Dynamic Size Effect . . . . . . . . . . . . . . . . . . . . 6.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Dynamic Constitutive Model of RCC for Fully-Graded Dam . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Strength Surface Modification of Fully-Graded RCC . . . . . . . . . . . . 7.2.1 Experimental Study on RCC Triaxial Compression Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Meso-simulation of Triaxial Compressive Behavior of Fully-Graded RCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Strength Surface Modification for RCC Constitutive Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 True Strain-Rate Effect Model of Fully-Graded RCC . . . . . . . . . . . . 7.3.1 True Strain Rate Effect Decoupling Method . . . . . . . . . . . . . . 7.3.2 True Strain-Rate Effect on Dynamic Compressive Strength of RCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 True Strain-Rate Effect on Dynamic Tensile Strength of RCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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83 83 84 84 85 86 87 88 88 89 90 91 95 98 98 101 101 102 102 104 107 110 113 116 117 119 119 119 120 124 127 128 128 130 132

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7.4 Modification of the Damage Equation in the K&C Model . . . . . . . . 7.5 Validation of Modified Full-Graded RCC Constitutive Model . . . . . 7.5.1 Validation of Modified K&C Constitutive Model with Single Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Applicability of Modified K&C Constitutive Model in Slab Subjected To Air Explosion . . . . . . . . . . . . . . . . . . . . . 7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Constitutive Relations of RCC: An Overview

1.1 Background Since most dams are not designed to resist high-intensity blast-impact loads, these hydraulic structures may be accessible and vulnerable to the impacts of an explosive blast with potentially catastrophic consequences, such as loss of electricity, flood, injury and fatality. Nowadays, all over the world, terrorist attacks, accident explosions and military conflicts happen from time to time. Thus, the vulnerability assessment of dams to explosion effects is of utmost importance. Although no terrorist attacks have been successful in destroying dams, literatures show many successful and unsuccessful attempts to attack dams with explosives, for example Chingaza Dam in Columbia and Bhakra Dam in India. Moreover, cases about blasts acting on dams are not rare during the war time. In the World War II, the Möhne dam and the Edersee dam in German failed due to intensive bombings, killing thousands of people in such events. A clay-core-wall rockfill dam in Croatia, Peruca Dam, was attacked during the Balkan War on January 28, 1993. Although the explosions failed to breach the dam, turbid water flowed out from the shaft of the gallery, indicating that the cracks were formed in the core wall and the material of core wall was eroded. Excessive settlement was also observed on the dam crest. Lately, Turkey offended into the northeast Syria and bombed the local Mansour dam, cutting off millions of people from drinking water. Besides the cases during the war time, dams can also be influenced by blasting in engineering operations like rock and soil excavation. For example, it has been reported that in 1935, the upstream sand shell of the Swir III Dam in Russia liquefied due to the blasting operations at the 200 m upstream of the dam. For the occurrence of these events, agencies and dam owners are taking measures to reduce this possibility through risk analyses and security measures. Many risk and vulnerability assessments have focused on qualitative vulnerability assessments and security upgrades. These assessments rarely include computational analyses on the impacts of explosions, but rather focus on the accessibility of a structure, possible © Science Press 2023 S. Zhang et al., Dynamic Mechanical Behaviors and Constitutive Model of Roller Compacted Concrete, Hydroscience and Engineering, https://doi.org/10.1007/978-981-19-8987-2_1

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1 Constitutive Relations of RCC: An Overview

threat scenarios, increased security measures, etc. Computational analyses to assess the impacts of explosions involve complex, costly and time-consuming numerical modeling and simulations. As a result, such computational analyses may be beyond a dam owner’s limited resources, or may only be done for the highest-priority structure within a particular owner’s inventory of dams. The overall number, size, age and location of these types of structures can make such assessments a daunting task. Significant benefit to the profession could be realized if a consistent analysis method and simplified tools could be developed to facilitate how such vulnerabilities are assessed. Based on the structural forms and materials, dams can be generally categorized into gravity dam, arch dams and embankment dams. In dam construction, roller compacted concrete (RCC) began its initial development with the construction of the Alpe Gere Dam near Sondrio in North Italy between 1961 and 1964, and has been widely applied to construct concrete gravity dam and concrete arch dam. RCC is a special blend of concrete that has essentially the same ingredients as conventional concrete but in different ratios, and increasingly with partial substitution of fly ash for cement. The partial substitution of fly ash for cement is an important aspect of RCC dam construction because the heat generated by fly ash hydration is significantly less than the heat generated by cement hydration. This in turn reduces the thermal loads on the dam and reduces the potential for thermal cracking to occur. Moreover, RCC is placed in a manner similar to paving; the material is delivered by dump trucks or conveyors, spread by small bulldozers or specially modified asphalt pavers, and then compacted by vibratory rollers. This special rolling compaction technique improves the rapid construction capacity. The safety of RCC gravity dams under blast loadings is quite an intractable problem for dam operation and maintenance, because the basic understanding of the dynamic behaviors for RCC material is yet to be improved, especially for the effects of construction technique and mix design. Moreover, the strainrate sensitivity and size-dependence of RCC is seldom discussed, which may greatly influence the vulnerability assessments through numerical simulations. A proper assessment to the risk that an explosive impact could pose to dams usually involves a multistage process that consists of: (1) Blast- and post-blast analysis; (2) Threat/vulnerability/risk assessment; (3) Development of damage mitigation measures and preventive countermeasures.

1.2 Literature Review

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1.2 Literature Review 1.2.1 Dynamic Behaviors and Constitutive Models for Normal Concrete The properties of concrete-like materials under dynamic loadings are different from those under static loads. For the dynamic mechanical properties, concrete-like materials under high-strain- rate loadings are usually studied with split Hopkinson pressure bar (SHPB) [1–5]. The rate-dependence of dynamic compressive behaviors for concrete-like materials also can be investigated by other experimental methods [1, 6]. For example, under quasi-static or low-velocity impact, only one or two macrocracks can be observed, while several macro-cracks will develop and spread widely through higher resistance regions in the concrete specimen when loaded rapidly, which needs more energy to crack the specimen and corresponds to higher strength [4]. The ultimate strain defined as the strain at peak stress also addressed on the strain-rate effect but was independent on the water-to-cement ratio [7]. Hao et al. presented a state-of-the-art review about the influencing factors in dynamic tests on concrete-like materials including the lateral inertia effects, the contribution of coarse aggregates and the influence of moisture condition [8]. Generally, concrete material exhibits higher strength, modulus of elasticity and strain at maximum stress when the loading rate increases as is demonstrated by the laboratory test results. Together with the strain-rate effect, a multiaxial stress state due to stress wave propagation may be generated as concrete-like materials are subjected to dynamic loadings. In the local area that the impact or blast load is applied, high hydrostatic pressure might be generated. Moreover, as a heterogeneous and anisotropic composite material, initial defects such as meso–cracks and air voids always exist in concrete materials which affect the dynamic material properties. The formation, development and propagation of these natural defects, as well as macro-cracks under complex stress state, result in the significantly complicated nonlinear behaviors of the concrete materials. Nowadays, a variety of dynamic constitutive models for normal concrete have been developed and implemented within various theoretical frameworks under blastimpact loadings. Generally, five major categories can be divided [9]: (a) phenomenological models based on statistical regression analysis of test results. (b) Mesomechanical models associated with the concepts of the meso-scopic properties and damage accumulation. (c) Modified models introducing dynamic correction terms to existing theory. (d) Theoretical models on the basis of thermodynamic laws (e) combinations of the above methods. Numerical simulations have been widely used to study the problems of concrete structures exposed to impact and explosion with the constitutive models of HJC [10], K&C [11], RHT [12] and their modifications [13–15], which take expansion of destruction surface and dynamic characteristics of damage evolution into consideration to describe the dynamic properties of concrete.

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1 Constitutive Relations of RCC: An Overview

1.2.2 Special Physical and Mechanical Properties of RCC As a special kind of concrete material, RCC is widely used in constructions of hydraulic structures, transportation and highway engineering facilities. It has essentially the same material components as the conventional concrete, but in different mix proportion, such as much less water and relatively more fly ash to replace Portland cement. The mixture is much drier with no slump, spread by bulldozer, then compacted by vibratory roller, and this construction technique is conducive to rapid pouring of mass concrete. It was exciting that the revolutionary usage of RCC had promoted rapid progress of dam construction since the initial application in the construction of the Alpa Gera Dam [16]. To properly design the structures using RCC material, it is important to understand its mechanical properties under all types of loadings, especially dynamic loads arising from natural or man-made disasters such as earthquake and explosion, and there are still many problems to be solved. The significant differences between RCC and normal concrete in mechanical properties may come from the different mix proportion and construction technique. Recently, the experiments on RCC material pay more attention to the compaction effect and the material physical properties under quasi-static loadings, such as seepage [17], mechanical properties of interlayers [18, 19], temperature field and temperature control [20, 21], aggregate size [22], construction quality control [23, 24], and so on. Some scholars also began to investigate the mechanical properties from mesoscopic to macroscopic, and had achieved certain results, such as the effects of RCC components on the mesoscopic structure [25] and the compaction mechanism of RCC mixture [26]. In a word, the construction technique of thin-layer pouring and rolling compaction results in a layered structure in the horizontal direction. It is easy for interlayers to become relatively weak surfaces if processed improperly or time-interval is inappropriately controlled. It has been reported that the thickness of a interlayer usually is about 0.5–2.0 cm, which significantly influences the physical and mechanical behaviors of the RCC structures under static loads [18, 20]. For example, a gradual change in Young’s modulus and viscosity coefficient of interlayers for layered RCC was studied in detail based on the experimental investigation and theoretical analysis [26]. However, the influence of these interlayers on dynamic compressive behaviors of RCC cannot be found in previous works. Referring to the researches in geologic engineering, many experiments were carried out to study the effect of joints in rock masses on wave propagation using different experimental methods [27–29]. For example, Seinov [30] presented that the width, density and filling material of joints greatly affect the attenuation of shock wave during propagation. SHPB apparatus was adopted by Li [31] to analyze the influence of contact area and spatial geometry of contact surface on the dynamic mechanical properties of rock joint and wave propagation. Besides the experimental studies, much efforts have been paid to illustrate the propagation mechanism of stress waves propagating across interlayers using different theoretical methods. In order to make a continuum medium analysis for joints in rock masses, the equivalent medium method is proposed to treat the discontinuity as a whole, so that its

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mechanical behavior can be predicted with a representative elementary volume. In this way, an elastic continuum medium is commonly used to represent the rock mass containing joints, ignoring the nonlinear deformation behavior under different levels of stress waves [32, 33]. Compared with the existing elastic continuum medium methods, the equivalent viscoelastic medium method has been proved to be an effective approach to predict the stress waves propagating across jointed rock masses [28, 34–36]. Furthermore, the dispersion and attenuation of stress waves, as well as its frequency-dependence for jointed rock masses can be theoretically researched by solving a specific stress wave propagation equation [37, 38]. All of these can be used as a reference to understand the propagation rules of stress waves across interlayers in RCC masses. On the other hand, the compaction quality for RCC material involves the comprehensive effects of various compaction parameters (including the compaction passes, roller velocity, vibration frequency, and compacted thickness), which leads to the difficulty in construction control [39]. Moreover, the compaction mechanism of RCC still needs to be further studied and the construction quality of the whole storehouse is difficult to be controlled real-timely [40]. Improper compaction parameters will generate numerous air voids hiding in the hardened RCC, as well as the substandard interlayers. As a meso-structural feature, air voids and unbonded surfaces are widely known to affect the fracture process and mechanical properties of concretelike materials [41]. For example, 5% air void due to the improper compaction can result in a 30% loss of strength, whereas 20% air void may cause a strength loss of 80% [42]. Therefore, due to the difficulty in construction control and interlayer processing method, the hardened RCC in the field construction usually differs from that in the phase of laboratory mix design and suffers from the attenuation of mechanical behaviors, which is defined as the initial damage of RCC in this research. An inadequate consideration of damage effect will lead to the overestimation of structural safety and increase the risk level of concrete structures, which should be taken into consideration in the context of original material or engineering design. Recent researches about dynamic mechanical properties of concrete-like materials always do not take the initial damage state of concrete into consideration. The complicated interaction between dynamic mechanical properties and concrete damage needs further investigation. To better understand the concrete damage effect under impact loadings, the effect of air void on the dynamic compressive behaviors of concrete was reported by Huang [43]. The testing results showed that the dynamic compressive strength decreased with the increasing void content and more significant variability of dynamic compressive strength occurred at higher strain rates. Therefore, it is necessary to further understand the initial damage effect on dynamic mechanical properties of RCC material under impact loadings. To solve this problem above, the primary issue is to find a reasonable and feasible method to quantify the damage levels and feed structural computations with damaged material properties. As known, Nondestructive testing (NDT) provides a feasible approach to quantify the damage of concrete-like material, keeping rapid and inexpensive [44–47]. In addition, there is an inherent correlation between NDT variables and mechanical properties of concrete. In concrete structures after fire damage or

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environmental degradation, for example, the attenuation of ultrasonic pulse velocity (UPV) due to the creation of meso–cracks and the loss of stiffness reflects the variation in mechanical properties [44]. A lot of researches have tried to develop NDT or data processing to acquire a better assessment of building materials [48–52]. Generally, rebound hammer test and ultrasonic test are treated as the most useful and reliable tools for assessing materials’ mechanical properties [51, 53, 54]. The unbonded interfaces and pores in concrete at meso- scale, acting as the void inclusions, decrease the phase velocity and increase the attenuation of the propagating stress wave [44]. In practical engineering, UPV measurement is sensitive enough to the material damage and has been conducted to evaluate the concrete quality of ancient Frontenc Dam built in the early twentieth century [55].

1.2.3 Size-Dependence of Concrete Under Dynamic Loads It is well accepted, based on experimental and theoretical investigations, that the mechanical response of concrete-like materials under compression, shear, tension and torsion under quasi-static loading is significantly affected by the specimen size. Generally, a smaller specimen requires higher stress to fracture under quasi-static loading. The mechanism of the concrete size effect law for the quasi-static strength can be classified into three categories: (1) Weakest-link hypothesis [56, 57]: larger structures have a larger chance of containing a critical flaw that can cause complete collapse, and the structures will fail as soon as the first critical defect fails; (2) Energetic (deterministic) mechanism [58–61]: two fundamental causes of the size effect in concrete structures are the material heterogeneity and the stress discontinuities at the crack tips, which cause stress redistribution and stored energy release (i.e., strain energy dissipation) during the development of macro-cracks; (3) Fractal mechanism [62]: the roughness of the crack surfaces in concrete exhibits inherent fractal characteristics. When the meso-structural disorder and self-similar features (i.e., fractality) dominate the damage and fracturing process, the fractal mechanism permits better interpolation of experimental data than the energetic mechanism. However, when exposed to high-strain-rate loading, concrete-like materials have a higher dynamic compressive strength than their corresponding static compressive strength [4, 8, 10, 63]; the fracture energy is also increased [64]. The experiments conducted by Elfahal [65] and theoretical analysis by Qi [66] indicate that the size effect under impact loading is notably different from the well-known static size effect, in which the dynamic strength increases with the increasing sample size (in terms of the diameter) at a similar strain rate. Moreover, larger specimens display a more significant strain-rate effect. However, the interpretation and application of the dynamic size effect are still unclear, i.e., the mechanism of strength enhancement for larger structures under dynamic loading, and the application of laboratory testing results from small structures to real full-scale structures. Based on the concept of the size effect from Vliet [67, 68], the size effect can be considered a combination of the material size effect caused by material heterogeneity

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and the structural size effect induced by the boundary and shape of the specimen. Similarly, the dynamic strength enhancement of concrete under high-speed impact loading consists of contributions from the material strain-rate effect (which occurs due to the inherent meso–structure and crack propagation in aggregates and is considered part of the material size effect) and the structural effect (which occurs due to the lateral confinement and end friction and is considered part of the structural size effect) [69–71]. Under this view, the dynamic increase factor (DIF) obtained from the experimental tests can be expressed as D I F = f d / f c = ( f c +  f ε˙ +  f i )/ f c

(1.1)

where f d is the dynamic compressive strength; f c is the quasi-static strength;  f ε˙ is the dynamic strength increment due to the material strain-rate effect; and  f i is the dynamic strength increment due to the structural effect [71]. With respect to the material size effect, many types of meso–structure analysis have been conducted to further understand the macroscopic failure phenomena occurring under impact loading [72, 73]. The size effect in concrete has been investigated using Monte Carlo simulations of mesoscale finite element models in which the random inclusions (aggregates and pores) with the prescribed volume fractions, shapes and size distributions are considered [74]. It has been confirmed that the mesoscale heterogeneity, aggregate volume fraction and porosity should not be ignored in the size effect studies of concrete [57, 75, 76]. The structural size effect on the compressive strength enhancement of the concrete-like material in SHPB tests has gained the attention of many researchers. The factors responsible for the structural effects include the material parameters (i.e., hydrostatic dependence and dilation parameter), specimen geometry (i.e., diameter and aspect ratio), end interface friction and material inertia [77]. Specially, the structural size effect cannot reflect the true strain-rate effect of material, and an overestimation of dynamic mechanical properties can be resulted in when the structural size effect is taken into consideration [2]. Thus, it is important to understand the dynamic size effect of concrete and introduce its true strainrate effect into the vulnerability assessment to concrete structures under blasting or impacting. So far, many researches, mainly numerical based, have been carried out for a more comprehensive understanding of the structural effect influencing the results of impact tests, and some controversial conclusions have been drawn. One of the primary concerns and most intensively studied influencing factors in dynamic material tests is the lateral inertia confinement. Bertholf and karnes [78] performed numerical simulations and indicated that the lateral and axial inertia and friction could produce additional constraints and result in multi-axial stress states. They also suggested an optimal aspect ratio of 0.5 to design SHPB specimens, so that the axial inertia in specimens can be eliminated. Li and his co-workers performed a series of numerical and experimental studies of SHPB tests on the compressive strength of mortar or concrete-like materials and concluded that significant lateral inertia confinement occurs only when the strain rate exceeds 200/s [69, 70, 79]. Based on numerical simulation results, revised DIF relations of mortar matrix and

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aggregate were proposed in which the contribution of lateral inertia confinement was removed [71]. Another important structural effect in dynamic compressive tests is the end friction confinement. Similar to the lateral inertia confinement, the friction at specimen-apparatus interfaces constrains the lateral deformation of the specimen under rapid compression, contributing to the increase in material strength, when specimen is subjected to impact loadings. Parametric numerical simulations with mesoscale model were conducted by Hao et al. [80] to investigate the end friction confinement effect on concrete material DIF, considering the components in typical concrete material, i.e. mortar and aggregates. It was found that the L/D ratio, friction coefficient and strain rate strongly affect the stress and strain distributions and the failure patterns of concrete specimen, and their influences are often coupled with each other. An equation to remove the influence of end friction confinement on DIF considering the L/D ratio, friction coefficient and strain rate as variables was also proposed [80]. For a deeper understanding of the dynamic size effect for concrete, fractal geometry is an effective way to describe the self-similarity in irregular and chaotic phenomena [81]. In recent years, the fractal theory has already been used to link fractal geometry and concrete fragmentation, such as fracture surface and path, and acoustic emission characteristics [82, 83]. Focusing on the breaking behaviors of concrete under static loadings, an alternative explanation for the static size effect on strength and toughness can be obtained with the help of the fractal and multifractal approaches [62, 84, 85]. The results indicate that the fragmentation of concrete-like materials exhibits a fractal behavior in the condition of static loadings. When it comes to the high-speed impact loading, it also has been confirmed that a fractal dimension, calculated by mass-size relationship, can be statistically regarded as an ideal indicator to describe the fragment size distribution of concrete after impact fragmentation [86, 87]. The more serious concrete fragmentation at higher strain rates will lead to a larger fractal dimension. Therefore, the fractal theory is an effective way to investigate the typical non-linear process of concrete impact fragmentation with pronounced irregularity. Then, the failure mechanism hiding in concrete fragmentation (including fragment size distribution, energy dissipation law, and crack branching) can be further revealed.

1.2.4 Aggregate Effect on Mechanical Behaviors of Concrete Recent researches show that aggregate properties (aggregate type [88] and aggregate gradation [89]) have a certain impact on the dynamic mechanical properties of concrete. In order to meet the assumption of stress uniformity, it is necessary to scale the coarse aggregate in the laboratory tests. Therefore, the maximum particle size of coarse aggregate in the wet sieve specimens is usually less than 20 mm [5, 71, 80, 90–100]. However, in practical projects (especially hydraulic dams), three-grading proportion (maximum aggregate particle size d max = 80 mm) or four-grading proprotion (d max = 150 mm) is mostly used in the mix design of hydraulic engineering.

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Therefore, the results of laboratory tests based on the scaled concrete specimens are difficult to fully reflect the mechanical properties of fully graded wet-sieve concrete. Nowadays, mesoscopic numerical simulation has become another reliable way to explore the failure mechanism and macro dynamic response of concrete materials [101]. Through the meso-mechanical tests, the limitations in laboratory tests such as high cost and measuring error can be easily avoided, and the influence of internal nonuniform structure in concrete on the macro static and dynamic mechanical properties of concrete can be deeply analyzed. In the meso-mechanics research of concrete materials, the meso- structure of concrete is usually simplified into a multiphase composite structure composed of coarse aggregate, mortar and interface transition zone (ITZ) [102] (as shown in Fig. 1.1): (1) The mortar matrix is mainly composed of fine aggregate (d max ≤ 5 mm) and cement slurry filling the gap between particles. Because the volume of fine aggregate is much smaller than that of concrete specimen, mortar matrix is generally regarded as homogeneous and isotropic material in mesomechanical analysis. Some scholars also try to use material weakening [103], threedimensional random defect interface spring element [104], direct modeling [105] to reflect the initial defects (holes and meso-cracks, etc.) in the mortar. (2) Coarse aggregate is the skeleton structure of concrete materials. The lithology, stiffness, shape, gradation and volume content of coarse aggregate will influence the macromechanical behaviors of concrete in different degrees. (3) The ITZ is a transition thin layer formed on the surface of coarse aggregate particles due to sidewall effect (the actual thickness is about 20 ~ 50 µm). In order to take the calculation efficiency and result accuracy into account, the ITZ thickness in the meso-simulation can be set within the range of 0.5 ~ 2.0 mm [106, 107]. Although the basic material composition of ITZ is consistent with that of cement paste, the high porosity and meso-cracks make it become the weakest link in the meso-structure of concrete. Therefore, it is usually treated as a separate phase in the meso-mechanical analysis.

Fig. 1.1 Schematic diagram of meso structure composition of concrete materials

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Through the meso-simulation test, we can get rid of the test errors from equipment, laboratory conditions and other factors, and further reveal the essential material properties, so as to provide a basis for the concrete constitutive model [101]. Considering the influence of aggregate gradation, Tian [108] established two-dimensional mesoscopic models of two-graded (5 ~ 40 mm), three-graded (5 ~ 80 mm) and fourgraded (5 ~ 150 mm) concrete, and carried out static and dynamic tests (˙ε= 10−5 ~ 10−2 s−1 ). The results indicate that the aggregate gradation effect and size effect on the compressive strength of concrete. Three-dimensional mesoscopic models are also proposed by Fang et al. to investigate the aggregate gradation effect for fully-graded concrete [109]. On this basis, Zhang et al. [102] further studied the static mechanical properties of fully-graded concrete under uniaxial, biaxial and triaxial conditions, and verified the rationality of deriving the macro-mechanical properties of concrete through meso-simulations. Zhou et al. [110] established a two-dimensional aggregate database that can reflect the real aggregate shape by combining laser scanning and spatial random cutting technology, and the aggregate effects of shape and size (d = 16/12/8 mm) on uniaxial compressive strength were studied. The numerical simulation results show that the dispersion of the peak strength of large aggregate specimens is stronger, while the aggregate shape quantified by roundness and slenderness ratio has nearly no influence on the peak strength. Du et al. [97, 98] established three-dimensional mesoscopic models of concrete to consider the effects of aggregate gradation (d = 5 ~ 20 mm/5 ~ 30 mm/5 ~ 40 mm), aggregate volume content (20 and 40%) and specimen size (L = 150/300/450 mm) on the splitting tests at different strain rates (˙ε =10−3 ~ 10 s−1 ). It was verified that the splitting tensile strength of concrete increases with the increase of maximum aggregate size and aggregate content, and the splitting tensile strength does not show obvious size effect at high strain rate. Based on the meso-simulations, Liu et al. [99–101] discussed the influence of aggregate size (d = 10, 20, and 30 mm) on the failure behavior and concrete size effect at low strain rates. The results showed that the tensile strength increases with the increase of aggregate size, while the compressive strength increases firstly and then decreases with the increase of aggregate size. In conclusion, the shape of coarse aggregate particles (roundness, slenderness ratio, etc.) have limited influence on the dynamic strength of concrete materials, and the influence of aggregate volume content and aggregate gradation should be paid more attentions. As a special branch of concrete-like materials, the aggregate gradation of RCC usually reaches four-graded and the maximum aggregate size usually reaches 150 mm. Zhang et al. [111] conducted dynamic direct tensile tests at different strain rates (˙ε= 10–6 ~ 10−3 s−1 ) with fully-graded RCC. The results showed that the dynamic tensile strength of RCC has a logarithmic linear relationship with strain rate, and the ultimate tensile strain and energy absorption capacity also have strain rate effect. Thus, laboratory tests cannot reflect the essential material properties due to the scaled aggregate size as has done in many previous works, and it is necessary to reveal the dynamic mechanical properties with due consideration of the large aggregate size.

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References 1. Deng Z, Cheng H, Wang Z et al (2016) Compressive behavior of the cellular concrete utilizing millimeter-size spherical saturated SAP under high strain-rate loading [J]. Constr Build Mater 119:96–106 2. Hao Y, Hao H, Jiang GP et al (2013) Experimental confirmation of some factors influencing dynamic concrete compressive strengths in high-speed impact tests [J]. Cem Concr Res 52:63– 70 3. Tai YS (2009) Uniaxial compression tests at various loading rates for reactive powder concrete [J]. Theoret Appl Fract Mech 52(1):14–21 4. Xiao J, Li L, Shen L et al (2015) Compressive behaviour of recycled aggregate concrete under impact loading [J]. Cem Concr Res 71:46–55 5. Zhou J-K, Ding N (2014) Moisture effect on compressive behavior of concrete under dynamic loading [J]. J Centl South Univ 21(12):4714–4722 6. Ross CA, Tedesco JW, Kuennen ST (1995) Effects of strain rate on concrete strength [J]. ACI Mater J 92(1):37–47 7. Zhou J, Ge L (2015) Effect of strain rate and water-to-cement ratio on compressive mechanical behavior of cement mortar [J]. J Centl South Univ 22(3):1087–1095 8. Hao H, Hao Y, Li J et al (2016) Review of the current practices in blast-resistant analysis and design of concrete structures [J]. Adv Struct Eng 19(8):1193–1223 9. Li X, Liu Z (2015) Review of concrete dynamic constitutive model [J]. J Tianjin Univ (Sci Technol) 48(10):853–863 (in Chinese) 10. Holmquist TJ, Johnson GR, Cook WR (1993) A computational constitutive model for concrete subjected to large strains, high strain rates, and high pressures [C]. ADPA, Quebec Canada, pp 591–600 11. Malvar LJ, Crawford JE, Wesevich JW et al (1997) A plasticity concrete material model for DYNA3D [J]. Int J Impact Eng 19(9):847–873 12. Riedel W, Thoma K, Hiermaier S et al (1999) Penetration of reinforced concrete by BETA-B500 numerical analysis using a new macroscopic concrete model for hydrocodes [C]. ISIEMS, Berlin-Strausberg, pp 315–322 13. Xu H, Wen HM (2016) A computational constitutive model for concrete subjected to dynamic loadings [J]. Int J Impact Eng 91:116–125 14. Tu Z, Lu Y (2010) Modifications of RHT material model for improved numerical simulation of dynamic response of concrete [J]. Int J Impact Eng 37(10):1072–1082 15. Polanco-Loria M, Hopperstad OS, Børvik T et al (2008) Numerical predictions of ballistic limits for concrete slabs using a modified version of the HJC concrete model [J]. Int J Impact Eng 35(5):290–303 16. Wikipedia. Roller-compacted concrete [EB/OL]. Wikipedia, 2016-10-23 [2016-11-11]. https://en.wikipedia.org/wiki/Roller-compacted_concrete 17. Li M-C, Guo X-Y, Shi J et al (2015) Seepage and stress analysis of anti-seepage structures constructed with different concrete materials in an RCC gravity dam [J]. Water Sci Eng 8(4):326–334 18. Martin F-A, Rivard P (2012) Effects of freezing and thawing cycles on the shear resistance of concrete lift joints [J]. Can J Civ Eng 39(10):1089–1099 19. Liu YD (2015) Study on the effects of layer processing on the layer adhesion properties of RCC [D]. Zhejiang University, Hangzhou (in Chinese) 20. Jaafar MS, Bayagoob KH, Noorzaei J et al (2007) Development of finite element computer code for thermal analysis of roller compacted concrete dams [J]. Adv Eng Softw 38(11):886– 895 21. Gaspar A, Lopez-Caballero F, Modaressi-Farahmand-Razavi A et al (2014) Methodology for a probabilistic analysis of an RCC gravity dam construction. Modelling of temperature, hydration degree and ageing degree fields [J]. Eng Struct 65:99–110 22. Rao M, Yang H, Lin Y et al (2016) Influence of maximum aggregate sizes on the performance of RCC [J]. Constr Build Mater 115:42–47

12

1 Constitutive Relations of RCC: An Overview

23. Karimpour A (2010) Effect of time span between mixing and compacting on roller compacted concrete (RCC) containing ground granulated blast furnace slag (GGBFS) [J]. Constr Build Mater 24(11):2079–2083 24. Liu D, Li Z, Liu J (2015) Experimental study on real-time control of roller compacted concrete dam compaction quality using unit compaction energy indices [J]. Constr Build Mater 96:567– 575 25. Sun H (2010) Study on FA critical dosage of hydraulic concrete based on the meso-structure characteristic and deterioration mechanism [D]. Wuhan University, Wuhan (in Chinese) 26. Zhang N (2015) Three-dimensional numerical analysis of the compaction properties of RCC based on DEM [D]. Zhejiang University, Hangzhou (in Chinese) 27. Huang X, Qi S, Guo S et al (2014) Experimental study of ultrasonic waves propagating through a rock mass with a single joint and multiple parallel joints [J]. Rock Mech Rock Eng 47(2):549–559 28. Li JC, Li HB, Zhao J (2015) An improved equivalent viscoelastic medium method for wave propagation across layered rock masses [J]. Int J Rock Mech Min Sci 73:62–69 29. Zhang J, Zhang Y, Fang Q (2018) Numerical simulation of shock wave propagation in dry sand based on a 3D mesoscopic model [J]. Int J Impact Eng 117:102–112 30. Seinov NP, Chevkin AI (1968) Effect of fissure on the fragmentation of a medium by blasting [J]. Soviet Mining 4(3):254–259 31. Li JC, Li NN, Li HB et al (2017) An SHPB test study on wave propagation across rock masses with different contact area ratios of joint [J]. Int J Impact Eng 105:109–116 32. Pyrak-Nolte LJ, Myer LR, Cook NGW (1990) Anisotropy in seismic velocities and amplitudes from multiple parallel fractures [J]. J Geophys Res Solid Earth 95(B7):11345–11358 33. Pyrak-Nolte LJ, Myer LR, Cook NG (1990) Transmission of seismic waves across single natural fractures [J]. J Geophys Res Solid Earth 95(B6):8617–8638 34. Häussler-Combe U, Kühn T (2012) Modeling of strain rate effects for concrete with viscoelasticity and retarded damage [J]. Int J Impact Eng 50:17–28 35. Fan LF, Ma GW, Li JC (2012) Nonlinear viscoelastic medium equivalence for stress wave propagation in a jointed rock mass [J]. Int J Rock Mech Min Sci 50:11–18 36. Zhu JB, Zhao XB, Wu W et al (2012) Wave propagation across rock joints filled with viscoelastic medium using modified recursive method [J]. J Appl Geophys 86:82–87 37. Fan LF, Wu ZJ (2016) Evaluation of stress wave propagation through rock mass using a modified dominate frequency method [J]. J Appl Geophys 132:53–59 38. Li J (2013) Wave propagation across non-linear rock joints based on time-domain recursive method [J]. Geophys J Int 193(2):970–985 39. National Energy Administration of People’s Republic of China (2009) Construction specifications for hydraulic RCC (DL/T 5112–2009) [S]. Water Power Press, Beijing (in Chinese) 40. Liu Y, Zhong D, Cui B et al (2015) Study on real-time construction quality monitoring of storehouse surfaces for RCC dams [J]. Autom Constr 49:100–112 41. Liu SH, Li QL, Rao MJ et al (2017) Properties and meso-structure of roller compacted concrete with high volume low quality fly ash [J]. Mater Sci 23(3):273–279 42. United States Army Corps of Engineers (2000) Engineering and design of roller compacted concrete (Engineer Manual 1110-2-2006) [S]. HQUSACE, Washington 43. Huang YJ, Yang ZJ, Chen XW et al (2016) Monte Carlo simulations of meso-scale dynamic compressive behavior of concrete based on X-ray computed tomography images [J]. Int J Impact Eng 97:102–115 44. Yim HJ, Kim JH, Park S-J et al (2012) Characterization of thermally damaged concrete using a nonlinear ultrasonic method [J]. Cem Concr Res 42(11):1438–1446 45. Payan C, Garnier V, Moysan J (2010) Effect of water saturation and porosity on the nonlinear elastic response of concrete [J]. Cem Concr Res 40(3):473–476 46. Antonaci P, Bruno CLE, Gliozzi AS et al (2010) Monitoring evolution of compressive damage in concrete with linear and nonlinear ultrasonic methods [J]. Cem Concr Res 40(7):1106–1113

References

13

47. Shah AA, Ribakov Y (2009) Non-destructive evaluation of concrete in damaged and undamaged states [J]. Mater Des 30(9):3504–3511 48. Rosado LS, Santos TG, Piedade M et al (2010) Advanced technique for non-destructive testing of friction stir welding of metals [J]. Measurement 43(8):1021–1030 49. Amenabar I, Mendikute A, López-Arraiza A et al (2011) Comparison and analysis of nondestructive testing techniques suitable for delamination inspection in wind turbine blades [J]. Compos B Eng 42(5):1298–1305 50. Garnier C, Pastor M-L, Eyma F et al (2011) The detection of aeronautical defects in situ on composite structures using non destructive testing [J]. Compos Struct 93(5):1328–1336 51. Alwash M, Sbartaï ZM, Breysse D (2016) Non-destructive assessment of both mean strength and variability of concrete: a new bi-objective approach [J]. Constr Build Mater 113:880–889 52. Shah AA, Ribakov Y, Hirose S (2009) Nondestructive evaluation of damaged concrete using nonlinear ultrasonics [J]. Mater Des 30(3):775–782 53. Azari H, Nazarian S, Yuan D (2014) Assessing sensitivity of impact echo and ultrasonic surface waves methods for nondestructive evaluation of concrete structures [J]. Constr Build Mater 71:384–391 54. Yıldırım H, Sengul O (2011) Modulus of elasticity of substandard and normal concretes [J]. Constr Build Mater 25(4):1645–1652 55. Saint-Pierre F, Philibert A, Giroux B et al (2016) Concrete quality designation based on ultrasonic pulse velocity [J]. Constr Build Mater 125:1022–1027 56. Man HK, Van Mier JGM (2011) Damage distribution and size effect in numerical concrete from lattice analyses [J]. Cement Concr Compos 33(9):867–880 57. Lei W-S, Yu Z (2016) A statistical approach to scaling size effect on strength of concrete incorporating spatial distribution of flaws [J]. Constr Build Mater 122:702–713 58. Bažant ZP, Voˇrechovský M, Novák D (2007) Asymptotic prediction of energetic-statistical size effect from deterministic finite-element solutions [J]. J Eng Mech 133(2):153–162 59. Hoover CG, Bažant ZP (2014) Cohesive crack, size effect, crack band and work-of-fracture models compared to comprehensive concrete fracture tests [J]. Int J Fract 187(1):133–143 60. Bažant ZP (2004) Probability distribution of energetic-statistical size effect in quasibrittle fracture [J]. Probab Eng Mech 19(4):307–319 61. Morel S (2008) Size effect in quasibrittle fracture: derivation of the energetic size effect law from equivalent LEFM and asymptotic analysis [J]. Int J Fract 154(1):15–26 62. Carpinteri A, Puzzi S (2006) A fractal approach to indentation size effect [J]. Eng Fract Mech 73(15):2110–2122 63. Ren W, Xu J, Su H (2016) Dynamic compressive behaviour of concrete after exposure to elevated temperatures [J]. Mater Struct 49(8):3321–3334 64. Xu M, Wille K (2015) Fracture energy of UHP-FRC under direct tensile loading applied at low strain rates [J]. Compos B Eng 80:116–125 65. Elfahal MM, Krauthammer T, Ohno T et al (2005) Size effect for normal strength concrete cylinders subjected to axial impact [J]. Int J Impact Eng 31(4):461–481 66. Qi C, Wang M, Bai J et al (2014) Mechanism underlying dynamic size effect on rock mass strength [J]. Int J Impact Eng 68:1–7 67. Van Vliet MRA, Van Mier JGM (1999) Effect of strain gradients on the size effect of concrete in uniaxial tension [J]. Int J Fract 95(1):195–219 68. Van Vliet MRA, Van Mier JGM (2000) Experimental investigation of size effect in concrete and sandstone under uniaxial tension [J]. Eng Fract Mech 65(2):165–188 69. Li QM, Meng H (2003) About the dynamic strength enhancement of concrete-like materials in a split Hopkinson pressure bar test [J]. Int J Solids Struct 40(2):343–360 70. Li QM, Lu YB, Meng H (2009) Further investigation on the dynamic compressive strength enhancement of concrete-like materials based on split Hopkinson pressure bar tests. Part II: numerical simulations [J]. Int J Impact Eng 36(12):1335–1345 71. Hao Y, Hao H (2011) Numerical evaluation of the influence of aggregates on concrete compressive strength at high strain rate [J]. Int J Prot Struct 2(2):177–206

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72. Chen G, Hao Y, Hao H (2015) 3D meso-scale modelling of concrete material in spall tests [J]. Mater Struct 48(6):1887–1899 73. Su Y, Li J, Wu C et al (2016) Influences of nano-particles on dynamic strength of ultra-high performance concrete [J]. Compos B Eng 91:595–609 74. Wang X, Yang Z, Jivkov AP (2015) Monte Carlo simulations of mesoscale fracture of concrete with random aggregates and pores: a size effect study [J]. Constr Build Mater 80:262–272 75. Taha MMR, El-Dieb AS, El-Wahab MA et al (2008) Mechanical, fracture, and meso-structural investigations of rubber concrete [J]. J Mater Civ Eng 20(10):640–649 76. Van Mier JGM, Van Vliet MRA (2003) Influence of meso-structure of concrete on size/scale effects in tensile fracture [J]. Eng Fract Mech 70(16):2281–2306 77. Flores-Johnson EA, Li QM (2017) Structural effects on compressive strength enhancement of concrete-like materials in a split Hopkinson pressure bar test [J]. Int J Impact Eng 109:408–418 78. Bertholf LD, Karnes CH (1975) Two-dimensional analysis of the split Hopkinson pressure bar system [J]. J Mech Phys Solids 23(1):1–19 79. Zhang M, Wu HJ, Li QM et al (2009) Further investigation on the dynamic compressive strength enhancement of concrete-like materials based on split Hopkinson pressure bar tests. Part I: Experiments [J]. Int J Impact Eng 36(12):1327–1334 80. Hao Y, Hao H, Li ZX (2013) Influence of end friction confinement on impact tests of concrete material at high strain rate [J]. Int J Impact Eng 60:82–106 81. Carpinteri A, Pugno N (2002) A fractal comminution approach to evaluate the drilling energy dissipation [J]. Int J Numer Anal Meth Geomech 26(5):499–513 82. Vidya Sagar R, Raghu Prasad BK (2011) Fracture analysis of concrete using singular fractal functions with lattice beam network and confirmation with acoustic emission study [J]. Theoret Appl Fract Mech 55(3):192–205 83. Hossain MDS, Kruhl JH (2015) Fractal geometry-based quantification of shock-induced rock fragmentation in and around an impact crater [J]. Pure Appl Geophys 172(7):2009–2023 84. Carpinteri A, Corrado M (2009) An extended (fractal) overlapping crack model to describe crushing size-scale effects in compression [J]. Eng Fail Anal 16(8):2530–2540 85. Carpinteri A, Spagnoli A (2004) A fractal analysis of size effect on fatigue crack growth [J]. Int J Fatigue 26(2):125–133 86. Ren W, Xu J (2017) Fractal characteristics of concrete fragmentation under impact loading [J]. J Mater Civ Eng 29(4):04016244 87. Hou T-X, Xu Q, Zhou J-W (2015) Size distribution, morphology and fractal characteristics of brittle rock fragmentations by the impact loading effect [J]. Acta Mech 226(11):3623–3637 88. Wu K, Chen B, Yao W et al (2001) Effect of coarse aggregate type on mechanical properties of high-performance concrete [J]. Cem Concr Res 31(10):1421–1425 89. Fu S, Feng X, Lauke B et al (2008) Effects of particle size, particle/matrix interface adhesion and particle loading on mechanical properties of particulate–polymer composites [J]. Compos B Eng 39(06):933–961 90. Chen X, Wu S, Zhou J (2013) Experimental and modeling study of dynamic mechanical properties of cement paste, mortar and concrete [J]. Constr Build Mater 47:419–430 91. Yu S, Lu Y, Cai Y (2013) A correction methodology to determine the real strain-rate effect for rock-like materials based on SHPB testing [J]. J Wuhan Univ Technol 06:96–100 (in Chinese) 92. Hao Y, Hao H, Li Z-X (2010) Numerical analysis of lateral inertial confinement effects on impact test of concrete compressive material properties [J]. Int J Prot Struct 1(1):145–167 93. Wang X, Zhang S, Wang C et al (2018) Experimental investigation of the size effect of layered roller compacted concrete (RCC) under high-strain-rate loading [J]. Constr Build Mater 165:45–57 94. Jin L, Yu W, Du X et al (2019) Dynamic size effect of concrete under tension: a numerical study [J]. Int J Impact Eng 132:103318 95. Jin L, Yu W, Du X et al (2019) Mesoscopic numerical simulation of dynamic size effect on the splitting-tensile strength of concrete [J]. Eng Fract Mech 209:317–332 96. Jin L, Yu W, Du X et al (2019) Meso-scale modelling of the size effect on dynamic compressive failure of concrete under different strain rates [J]. Int J Impact Eng 125:01–12

References

15

97. Jin L, Yu W, Du X et al (2020) Meso-scale simulations of size effect on concrete dynamic splitting tensile strength: influence of aggregate content and maximum aggregate size [J]. Eng Fract Mech 230:106979 98. Du M, Jin L, Li D et al (2017) Mesoscopic simulation study of the influence of aggregate size on mechanical properties and specimen size effect of concrete subjected to splitting tensile loading [J]. Eng Mech 34(09):54–63 (in Chinese) 99. Jin L, Yang W, Yu W et al (2020) lnfluence of maximum aggregate size on dynamic size effect of concrete under low strain rates: meso-scale simulations [J]. Trans Nanjing Univ Aeronaut Astronaut 37(001):27–39 100. Yu W (2019) Meso-scale simulation in dynamic size effect on compressive and tensile failure of concrete materials[D]. School of Civil Engineering and Architecture, Beijing University of Technology, Beijing (in Chinese) 101. Fa MH, Chen HQ, Kun LB (2004) Review on micro-mechanics studies of concrete [J]. J China Inst Water 02(002):124–130 (in Chinese) 102. Zhang J, Fang Q, Gong Z et al (2012) Numerical simulation of static mechanical properties based on 3D mesoscale model of fully-graded concrete [J]. Chin J Comput Mech 29(06):927– 933+947. (in Chinese) 103. Zhao J (2009) 3D Meso-scale failure simulation of four-phase composite concrete [J]. J Civ Environ Eng 31(004):37–43. (in Chinese) 104. Wang J, Li Q, Qing L et al (2014) 3D Simulation of concrete strength under uniaxial compressive load [J]. Eng Mech 31(03):39–44. (in Chinese) 105. Stamati O, Roubin E, Andò E et al (2018) Phase segmentation of concrete x-ray tomographic images at meso-scale: validation with neutron tomography [J]. Cem Concr Compos 88:08–16 106. Maleki M, Rasoolan I, Khajehdezfuly A et al (2020) On the effect of ITZ thickness in mesoscale models of concrete [J]. Constr Build Mater 258:119639 107. Du X, Jin L, Ma G (2014) Numerical simulation of dynamic tensile-failure of concrete at meso-scale [J]. Int J Impact Eng 66:05–17 108. Ruijun T (2008) Study on the static, dynamic mechanical properties of (fully-graded) concrete based on mesomechanics [D]. School of Civil Engineering and Architecture, Beijing University of Technology, Beijing (in Chinese) 109. Fang Q, Zhang J, Huai Y et al (2013) The investigation into three-dimensional mesoscale modelling of fully-graded concrete [J]. Eng Mech 30(01):14–21+30 (in Chinese) 110. Zhou Y, Jin H, Wang B (2019) Modeling and mechanical influence of meso-scale concrete considering actual aggregate shapes [J]. Constr Build Mater 228:116785 111. Zhang K, Wang H, Tu J et al (2021) Dynamic tensile test of fully-graded roller compacted concrete [J]. J China Inst Water Resour Hydropower Res 19(03):290–300 (in Chinese)

Chapter 2

Experimental Research on Dynamic Behaviors of RCC

2.1 Introduction Layered RCC is a special concrete and has been widely applied in the infrastructure construction such as hydraulic structures and pavements due to its advantages of costeffectiveness and rapid construction. The mixture proportion of RCC, i.e., less water and more fly ash used to replace Portland cement, makes it different from normal concrete. Another significant difference between RCC and normal concrete lies in the construction technique. The RCC mixture is spread by bulldozer and compacted to a layered structure by vibratory roller, which leads to the numerous horizontal construction joints in the hardened RCC structure [1]. For a properly proportioned RCC mixture, the hardened performance of RCC primarily depends on the vibrating compaction control, including the compaction quality and the interlayer processing method. Recent researches on RCC material pay more attentions to the compaction effect and the material physical properties under quasi-static loadings, while its dynamic behaviors under high-strain-rate loadings are seldom involved in the previous studies. As a matter of fact, all kinds of factors, such as mix design and weak bonding interlayers, may lead to significant differences between RCC and normal concrete in mechanical properties. Dynamic loads, in the form of shock wave propagating across RCC masses, usually induce the concrete fragmentation [2, 3]. The discontinuities induced by the particular interlayers have great impact on stress wave propagation [4, 5]. It has been verified that when a shock wave propagates across the rock mass, it is attenuated and slowed due to the presence of interlayers. The amplitude and waveform of transmitted longitudinal wave can be observed to vary with the number and thickness of weak bonding interlayers. This can be a reference to investigate the stress wave propagation across interlayers in concrete structures, especially for the layered RCC mass.

© Science Press 2023 S. Zhang et al., Dynamic Mechanical Behaviors and Constitutive Model of Roller Compacted Concrete, Hydroscience and Engineering, https://doi.org/10.1007/978-981-19-8987-2_2

17

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2 Experimental Research on Dynamic Behaviors of RCC

This chapter extends previous studies and focuses on the fundamental research on the dynamic compressive behaviors of RCC under impact loadings. The experimental procedures include specimen preparation and dynamic tests on RCC specimens via SHPB device. Then, dynamic mechanical properties of RCC material at different strain rates are analyzed in terms of peak strength, Young’s modulus, and ultimate strain. Furthermore, the influence of weak bonding interlayers on reflection and transmission of stress wave propagation is discussed in detail. An equivalent viscoelastic medium model is proposed to further illustrate the ability for weak bonding interlayers in RCC to obstruct the propagation of shock waves under impact loadings.

2.2 Experimental Procedures 2.2.1 Material and RCC Mix Proportion RCC materials are made of mortar matrix, aggregates and additive. The RCC mix design in this study referred to that of Huangdeng hydropower station. As shown in Table 2.1, the water reducing agent (JM-II RCC) and air-entraining agent (HLAE) were used for mixing. The water-cement ratio (W/C) was set equal to 0.50 and the fly ash content reached 60% by mass. The sand ratio was 31% by mass according to the performance of the mixture and the strength of concrete. From Table 2.2, the moderate heat ordinary Portland cement (OPC) of 42.5 Grade was used, having a surface area of 325 m2 /kg and 62.97% CaO content. The fly ash of II Grade was produced in Guizhou with fineness modulus of 19.60% and water demand ratio of 101. The artificial medium sand with an apparent density of 2680 kg/m3 was selected as fine aggregate. To sum up, the physical and chemical properties of basic materials were met the code requirements [6, 7]. Based on the code for mix design of hydraulic concrete [8], the proportion of RCC mixture adopted in this study was prepared. According to Table 2.3, the maximum aggregate size of RCC casted in experimental site was no more than 19 mm, since the maximum size of coarse aggregate should not be allowed to exceed a quarter of the cylinder’s diameter. It is worthy to noting that the reduction of coarse aggregate size significantly increases the material heterogeneity, which not only influences the stress wave propagation [9] but also increases interfacial transition zones. It has been reported that interfacial transition zone is the weakest link of concrete and the fracture toughness or compressive strength decreases with the reduction of coarse aggregate size [10, 11]. The general rule of the aggregate size effect on dynamic behavors has been concluded in the CSC model in LS-DYNA based on the CEB-FIB model code [12].

0.50

31

0.8

0.05

88

70

106

672

1507

3.8

2453

W/C Sand ratio Water reducing agent (%) Air entraining agent (%) Material consumption (kg/m3 ) Air content Wet density (kg/m3 ) (%) Water Cement Fly ash Sand Aggregate (%)

Table 2.1 The mixture proportion and parameters for RCC

2.2 Experimental Procedures 19

Code requirement

Material property

Indexes

–

–

5d, where λ is the width of incident pulse, and d is the diameter of pressure bar. In this chapter, typical strain signals obtained from a SHPB test shown as Fig. 2.5a are used to verify the validation of SHPB tests. The propagation time of incident pulse (τe ) can be generally taken as 350 µs and the wave velocity of stress pulse (Ce ) propagating in pressure bars is detected to be 5400 m/s. Then, λ can be expressed as Eq. (2.8), indicating that the stress pulse in pressure bars satisfied the assumption of one-dimensional wave propagation theory. λ = Ce τe = 5400 × 350 µm = 1.890 m > 5d = 0.375 m

(2.8)

Fig. 2.5 Reliability of SHPB test on RCC: a Stress pulse in incident and transmitted bars; b System applicability of SHPB device to RCC

26

2 Experimental Research on Dynamic Behaviors of RCC

On the other hand, according to Ravichandran and Subhash [19], the homogeneity assumption of stress and strain in specimens needs the stress pulse to reflect three to four times before destruction. The velocity of stress-wave in specimens can be generalized as 3500 m/s, and then 2ls /Cs = 23 µs. So, the rising time of incident pulse should not be less than the time of stress pulse reflecting three to four times (T = 69 µs ∼ T = 92 µs). It is obviously in Fig. 2.5a that the rising time of incident pulse is about 120 µs, during which the homogeneity assumption of stress and strain can be realized easily. In order to further verify the Eq. (2.4), Fig. 2.5b shows that the gap between εi (t) + εr (t) and εt (t) decreases as the stress pulse reflects in the specimen, and the specimen almost reaches the dynamic stress equilibrium at the peak value in terms of strain. In addition, Fig. 2.5b indicates that this stress equilibrium state begins to disappear after peak stress pulse. The sum of incident and reflected strain begins to fluctuate for the reason of specimen destruction. Based on the tendencies of εi (t) + εr (t) and εt (t), the dynamic loading process can be divided into three stages: pulse reflection, stress equilibrium and material destruction. Moreover, according to the strain of specimen in Fig. 2.5b, the increase in strain seems to be linear during the stress equilibrium state, indicating a nearly constant strain-rate loading.

2.3 Effect of Construction Technique on Dynamic Behaviors 2.3.1 Dynamic Mechanical Properties After verifying the reliability of SHPB tests on RCC, a total of 35 specimens from 7 drilling columns are subjected to impact loadings with 4 different gas pressures, in which there are 3 testing results cannot meet the one-dimensional wave propagation theory and the homogeneity assumption of stress and strain. Based on the test results, Fig. 2.6a plots some typical stress–strain curves of RCC specimens subjected to various strain rates. As shown in Fig. 2.6a, it can be clearly noticed that the stress– strain responses for specimens apparently change with the increasing strain rates. In addition, the peak stress and initial elastic modulus increase with the increasing strain rate, which means RCC material is also strain-rate sensitive. The descending portions of stress–strain curves subjected to low strain-rate loadings (especially for specimen with strain rate of 40.67/s) seem much flatter than those subjected to relatively higher strain-rate loadings (i.e. specimens at strain rates of 92.27/s and 110.54/s). It is noted that the tail portion of specimen at the strain rate of 74.83/s has a second peak value. This can be illustrated as some hard coarse aggregates stripping from specimen are damaged further. As a matter of fact, the tail portions of stress–strain curves cannot express the real dynamic behaviors of RCC, instead, they are just dynamic responses of specimens after completely crushing, as the stress equilibrium cannot be maintained well in this state.

2.3 Effect of Construction Technique on Dynamic Behaviors

27

Fig. 2.6 Dynamic behaviors of RCC at various strain rates: a Stress–strain curves of RCC at various strain rates; b Comparison of DIFs from test results with different empirical models

It has been widely accepted that concrete is such a strain-rate sensitive material that the strength model or strength criterion should take the strain-rate effect into consideration. The strain-rate effect of concrete is usually described by the DIF, defined as the mechanical properties under dynamic loads divided by those under static loads. Various concretes and cement mortars classified into different strength grades have been studied with laboratory tests to quantify the strain-rate effects, and the polynomial fitting method is used to illustrate the empirical relationships between strain rates and DIFs [20–23]. Figure 2.6b compares the testing DIFs with the existing models proposed by other researchers. From this figure, the test results are distributed among these empirical models on the whole, and the RCC material seems more sensitive to strain rate. Many researchers have observed that concretes with higher static compressive strengths exhibit lower DIFs than weaker concretes at the intermediate-high strain-rate zone, and appropriate recommendations have been made by the CEB Committee [24]. As shown in Fig. 2.7a, the test results demonstrate that DIF of peak strength (D I Fσ ) increases rapidly with the increasing strain rate, and D I Fσ is close to 1 at the critical strain rate (about 40/s) where the specimen is not visibly damaged or nearly vertical splitting. Moreover, the discreteness of experimental data is obvious, especially at higher strain rates, which may be induced by the numerous meso– cracks in RCC. Therefore, at the critical strain rate, some DIFs are little lower than 1. These meso–cracks cannot expand immediately when loading rate do not exceed the critical strain rate. With the increasing loading rate, the impact energy gradually interconnects these meso–cracks, leading the specimen failure at last. Based on the experimental data, the DIF regression equation for dynamic compressive strength is suggested as Eq. (2.9), which describe the general rules of the strain-rate effect for RCC and can be used for further discussion on the discreteness of RCC. D I Fσ (˙ε) = 14.473(lg ε˙ )2 − 47.348(lg˙ε) + 39.624 when ε˙ ≥ 40/s

(2.9)

28

2 Experimental Research on Dynamic Behaviors of RCC

Fig. 2.7 Dynamic increase factor of RCC at different strain rates: a Dynamic compressive strength; b Dynamic elastic modulus; c Dynamic ultimate strain

The elastic modulus of RCC under impact loadings can be obtained from the stress–strain curve with the following equation [18]: E d = (σb − σa )/(εb − εa )

(2.10)

where σa and σb denote the stress values corresponding to 5 and 30% peak stress respectively; εa and εb denote the strain values corresponding to σa and σb respectively. In this chapter, the DIF for elastic modulus is expressed as D I FE , whose calculation method is similar to that for strength. As shown in Fig. 2.7b, although the strain-rate effect of elastic modulus is particularly vague compared to those of peak strength and ultimate strain, a higher elastic modulus approximately appears at higher strain rates. Based on the test results, the strain-rate effect of elastic modulus can be generalized as the following equation: D I FE (˙ε ) = 3.190(lg˙ε ) − 3.925 when ε˙ ≥ 40/s

(2.11)

2.3 Effect of Construction Technique on Dynamic Behaviors

29

D I Fε (˙ε) = 3.508(lg ε˙ )2 − 10.607(lg ε˙ ) + 8.842 when ε˙ ≥ 40/s

(2.12)

From Fig. 2.7c, it is also obvious that the ultimate strain is also strain-rate sensitive, while the strain rate effect on ultimate strain is less significant than that on peak strength. The trend in Fig. 2.7c can be generalized as Eq. (2.12).

2.3.2 Stratification Effect of RCC on Dynamic Mechanical Properties The dynamic mechanical properties of RCC are intensively influenced by compaction construction quality, and improper construction control will lead to numerous voids in RCC material and further influence its mechanical properties. As shown in Fig. 2.7, the experimental DIFs for dynamic mechanical properties, containing peak strength, Young’s modulus and ultimate strain, show strong discreteness that may be caused by the initial defects in RCC, and the influencing factors of DIF may include the porosity, strain rate, distribution of aggregates, saturation, and so on, which can be described as the following equations: X d = X d (φ, ε˙ , sr , · · · )

(2.13)

where X d is the dynamic compressive properties, which can be dynamic compressive strength (σd ), dynamic ultimate strain at peak strength (εd ) and dynamic Young’s modulus (E d ). Besides, φ, ε˙ , sr represent the influencing factors: porosity, strain rate and saturation respectively. d Xd =

∂ Xd dφ ∂φ

+

∂ Xd d ε˙ ∂ ε˙

+

∂ Xd dsr ∂sr

+ ···

(2.14)

In the same testing conditions. Equation (2.13) can be expressed as Eq. (2.14). Based on the strain-rate effect on dynamic mechanical properties, demonstrated as Eqs. (2.9), (2.11) and (2.12), the strain rate will be unified to the critical strain rate (taking as 40/s). Then, the strain-rate effect can be eliminated, such that ∂ X d /∂ ε˙ = 0. Based on the proposed method above, we defined the normalized dynamic mechanical properties as follows: X d∗ =

X d0 Xs

=

Xd Xs

×

D I FX (40) D I FX (˙ε )

=

D I FX∗ ×D I FX (40) D I FX (˙ε )

(2.15)

where X d∗ is the normalized dynamic mechanical properties, in which X can be σ , ε and E; X d0 and X s represent mechanical properties at the critical strain rate of 40/s and quasi-static loading tests; X d denotes the dynamic mechanical properties in practical SHPB tests at various strain rates, and D I FX∗ is the corresponding dynamic increase factor. It is worth noting that different from above parameters obtained from experimental data, D I FX (˙ε) is derived from Eqs. (2.9), (2.11) and (2.12), denoting

30

2 Experimental Research on Dynamic Behaviors of RCC

the empirical dynamic increase factor at the strain rate of ε˙ . Therefore, D I FX (40) is the empirical dynamic increase factor at the critical strain rate. In order to eliminate the strain rate effect, the tested dynamic strength is converted to the strength at the critical strain rate (σd0 ) with Eq. (2.15), named normalized critical strength. In this study, five drilling columns from which the specimens are all tested successfully are selected to analyze the stratification effect on mechanical properties of RCC. Table 2.4 lists all the tested results from the drilling columns, as well as the statistical characteristics for specimens from each drilling column. From Table 2.4, it is clear that there is a striking difference in the test results for five specimens from one drilling column, while the mean values for different drilling columns are quite similar. This rule is shown more intuitively in Fig. 2.8a. During the test, the RCC mixture preparation for each layer is strictly in accordance with the specified requirements. Thus, the existing significant strength difference in vertical direction is mainly induced by the vertical roller compaction, rather than the RCC mixture preparation for different layers. On the other hand, we use the mean normalized critical strength of every five specimens from one drilling column to represent the compressive behavior of this drilling column. Then, the standard deviation of these five mean values can present the horizontal construction characteristic of RCC. From Fig. 2.8b, it is obvious that the normalized standard deviations of five drilling columns representing the vertical construction characteristic are quite higher than that in horizontal direction. Therefore, the vertical stratification is one of main concerns in the analysis of RCC structures, and the bonding quality of interlayers is the key to RCC structure construction. In practical construction, the compaction of RCC related to the comprehensive effect of all compaction parameters usually leads to the bonding quality of interlayers cannot be guaranteed strictly in the whole process or compaction zone. Previous works [25–27, 28] have confirmed that some certain important construction factors can positively affect the bonding quality, such as compactness of upper Table 2.4 Normalized dynamic compressive strengths for specimens from each drilling column Number of specimen

Number of drilling column

1

13.81 13.73 10.16 11.09 11.52

2

9.48

12.37 14.18 12.64 12.45

3

6.9

8.8

4

6.8

10.11 6.5

5

9.65

11.06 11.74 8.74

E1

Vertical 9.33 StDev construction 2.85 characteristic

Mean

E2

G1

G4

J5

Distribution of drilling columns

15.86 13.82 8.59 10.68 12.23 7.48

11.21 11.69 11.40 10.45 Horizontal 1.92 3.64 1.94 2.27 construction characteristic

10.82 0.95

2.4 Shock Wave Propagation Across Interlayers in RCC

31

Fig. 2.8 RCC construction characteristic considering vertical stratification: a Normalized strength of each drilling column; b Normalized strength and standard deviation

RCC layers, concrete consistency, interval construction time and interfacial situation. Therefore, there exists more uncertainty in vertical construction than that in horizontal construction.

2.4 Shock Wave Propagation Across Interlayers in RCC 2.4.1 Experimental Scheme The preparation procedures for RCC masses have been described in Sect. 2.2.1.1, and the effect of construction technique on the dynamic mechanical behaviors of RCC can be further investigated below. In this Section, as shown in Fig. 2.9, two categories of RCC specimens were prepared for SHPB tests, judging whether the RCC specimen contains an interlayer or not.

Fig. 2.9 Diagrammatic sketches of specimen preparation

32

2 Experimental Research on Dynamic Behaviors of RCC

Table 2.5 Experimental scheme for SHPB tests Specimen category

Specimen dimensions (D × Hs : mm × mm)

Hs /D

Group

Number

Gas pressure (MPa)

Without interlayer

75 × 37.5

0.5

A-a

3

0.40

A-b

3

0.50

A-c

3

0.60

A-d

3

0.80

B-a

5

0.40

B-b

5

0.50

B-c

5

0.60

B-d

5

0.80

With interlayer

75 × 37.5

0.5

High-speed impact tests were carried out by the SHPB apparatus with the diameter of 75 mm. Prior to the SHPB tests, a set of density and ultrasonic tests were conducted on RCC specimens. As listed in Table 2.5, the RCC specimens in each category were further divided into four groups according to the predetermined gas pressure. In this laboratory test, the half-sine stress wave was achieved to load the RCC specimens by the various cross-sectional striker. A total of 32 RCC specimens (i.e., 12 specimens without interlayer and 20 specimens with interlayer) were loaded successfully by SHPB apparatus. The procedures of the density and ultrasonic pulse velocity tests in this study were described in detail as below: (1) Density test: The mass measurement was conducted at a natural moisture and the volume was averaged from a total of threetime dimension measurements for each cylindrical specimen, including diameter and length; (2) Ultrasonic tests: Owing to the significant wave attenuation in layered RCC, it is hard to exactly measure the velocity of longitudinal ultrasonic wave propagating in RCC. To perform ultrasonic tests more accurately, a total of 4 acquired signals in a frequency range of 200–500 kHz were applied to measure the wave velocity propagating in RCC [29]. Then, the ultrasonic wave velocity for each specimen was averaged from the propagation velocities of these 4 signals. Based on the density and ultrasonic testing results, the wave impedance for each RCC specimen can be calculated by Eq. (2.16). At the same time, the wave impedance of incident and transmission bars can be set as 45.19 × 106 kg/(m2 · s). Z s = ρC

(2.16)

where ρ is the medium density and C is the ultrasonic wave velocity in the RCC specimen.

2.4 Shock Wave Propagation Across Interlayers in RCC

33

2.4.2 Incident and Transmitted Waveforms In Fig. 2.10, the waveform characteristics for RCC specimens without interlayers are compared under different gas pressures, in terms of the incident and transmitted waveforms and the corresponding normalized frequency spectrums. From Fig. 2.10a, both the incident and transmitted waves are approximately half-sinusoidal owing to the usage of various cross-sectional striker. A higher gas pressure leads to higher incident and transmitted waves, while their durations are approximately identical and insensitive to the gas pressure. Figure 2.10b shows the corresponding normalized frequency spectrums for incident and transmitted waves under different gas pressure, which come from the Fast Fourier Transform (FFT) of stress waves in Fig. 2.10a. It is obvious that the normalized frequency spectrums for incident waves are also insensitive to the gas pressure and the RCC specimens in the propagation path almost have no influence on the normalized frequency spectrums for transmitted waves. As shown in Fig. 2.11, the rules of stress wave propagation in Fig. 2.10 are also applicable to the stress wave propagation across RCC specimen with interlayer, even though a relatively higher stress attenuation exists in Fig. 2.11. Therefore, it is reasonable to ignore the change in frequency of transmitted waves in SHPB tests, no matter whether the specimen contains a weak bonding interlayer. Focusing on the transmitted waves after the incident waves propagating across the RCC specimens, it can be observed from Figs. 2.10a and 2.11a that for both RCC specimens with and without an interlayer, the transmitted waves gradually deviate from the half-sinusoidal waveform and get earlier to reach the peak stresses, when the gas pressure reaches 0.8 MPa.

Fig. 2.10 The waveform characteristics for RCC specimen without interlayer: a Incident and transmitted waveforms under different gas pressure; b Corresponding normalized frequency spectrums

34

2 Experimental Research on Dynamic Behaviors of RCC

Fig. 2.11 The waveform characteristics for RCC specimen with interlayer: a Incident and transmitted waveforms under different gas pressure; b Corresponding normalized frequency spectrums

2.4.3 Reflection and Transmission of Shock Wave Propagation Once the incident wave reaches the front end of the RCC specimen in a SHPB test, the reflection and transmission of stress wave will occur, which is tightly related to the difference of wave impedance between steel bars and RCC specimens. The wave impedance is an important measure of how much a material resists motion when subjected to a stress wave. As explained in Sect. 2.3.1.1, a set of density and ultrasonic tests were conducted on RCC specimens before SHPB tests, so that the wave impedance for each RCC specimen could be estimated. The testing results have been listed in Table 2.6, characterized by the mean and standard deviation (StDev). It can be observed that the RCC interlayers from rolling compaction technique exist an obvious influence on physical properties of RCC, in terms of density, ultrasonic pulse velocity and wave impedance. Compared to the specimen without interlayer, the specimen containing interlayer has relatively lower mean values of density and ultrasonic pulse velocity, but higher StDevs, indicating much more significant variability of material attributes. According to the Eq. (2.16), the wave impedance for RCC specimen can be easily obtained based on the density and ultrasonic tests. As a result, the mean wave impedance is 9.20 × 106 kg/(m2 · s) for RCC specimen without an interlayer, but 7.11 × 106 kg/(m2 ·s) for RCC specimen containing an interlayer. Therefore, the specimen with interlayer is easier to deform and induce a higher wave attenuation under impact loadings. Tc = max|εt (t)|/max|εi (t)|

(2.17)

Rc = max|εr (t)|/max|εi (t)|

(2.18)

2.4 Shock Wave Propagation Across Interlayers in RCC

35

Table 2.6 Basic physical properties of the prepared RCC specimens Specimen category

Density (kg/m3 ) Mean

StDev

Ultrasonic pulse velocity (m/s)

Wave impedance (kg/(m2 · s))

Mean

Mean

StDev

StDev

Specimen without interlayer

2381.23

25.80

3863.04

153.08

9.20 × 106

0.40 × 106

Specimen with interlayer

2357.52

90.33

3018.29

286.21

7.11 × 106

0.63 × 106

To analyze effect of weak bonding interlayers on wave propagation, transmission and reflection coefficients are usually adopted. For example, based on the Hertzian contact theory, the issue of reflection and transmission for plane waves propagating across fractures was discussed by Misra and Marangos [30]. Here, the transmission coefficient is denoted as Tc and the reflection coefficient is denoted as Rc , which are the ratio of the peak transmitted (reflected) stress wave to the peak incident stress wave. Hence, Tc and Rc can be expressed as Eqs. (2.17) and (2.18). Figure 2.12 illustrates the variation of reflection and transmission coefficients with the representative strain rate, which is defined as the instantaneous strain rate at peak stress for specimens under impact loadings. Figure 2.12 demonstrates that for RCC specimen without an interlayer, the reflection coefficient decreases with the increasing strain rate, while its transmission coefficient shows a positive correlation with the increasing strain rate. On the other hand, the reflection and transmission coefficients for RCC specimen containing an interlayer exhibit similar correlation with the strain rate, even though the strain-rate effect is relatively less remarkable. Moreover, the reflection and transmission coefficients are also dependent to the material attributes (i.e., wave impedance), since the RCC specimen containing interlayer usually has higher reflection coefficient or lower transmission coefficient. The sum of reflection coefficient and transmission coefficient for each RCC specimen exposed to impact loadings is approximately equal to 1, no matter whether the RCC specimen contains an interlayer.

Fig. 2.12 The variation of reflection and transmission coefficient with the strain rate: a reflection coefficient; b transmission coefficient

36

2 Experimental Research on Dynamic Behaviors of RCC

In a word, the reflection and transmission will occur once the stress wave propagates from one medium to another medium. As shown in Eq. (2.17) and Fig. 2.12, the transmitted stress is tightly related to the incident stress, strain rate, and material attribute difference of these two mediums. As a conjugate term of transmission coefficient Tc , the reflection coefficient of RCC specimen exposed to impact loadings can be preliminarily evaluated by Rc = 1 − Tc based on the one-dimensional wave propagation theory, which describes the magnitude of stress wave reflected back into the incident bar. Therefore, the primary issue is to evaluate the influence of RCC noumenon and interlayers on the transmitted wave more rationally.

2.5 Theoretical Analysis on the Shock Wave Propagation 2.5.1 Wave Propagation in Viscoelastic Medium The equivalent medium method has been regarded as a useful approach to describe the macroscopic mechanical properties of a discontinuous material with the common continuum mechanics, which has been widely used in the simulation of finite element method. In finite element numerical simulation, the constitutive models of a representative elementary volume are derived from the average stress–strain relationship for all kinds of materials, taking the meso–structures without consideration. For example, concrete-like materials, consisting of coarse aggregates, mortar and meso-defects, are usually regarded as the continuous medium with an isotropic continuum damage model in the damage prediction of concrete gravity dams subjected to underwater explosion shock loadings [31, 32]. On the other hand, the viscoelastic constitutive model has been widely used to describe the mechanical and deformation properties of concrete-like materials [33, 34]. As for the RCC material, the viscoelastic constitutive model for interlayers and RCC noumenon is used by Gu et al. [35] to consider the effect of weak bonding interlayers on the displacement of RCC gravity dam with finite element method. This subsection aims to propose a methodology to describe the nonlinear mechanical behaviors of RCC under impact loadings with the equivalent viscoelastic medium model, so that the mechanism of stress wave propagation across the layered RCC can be further revealed. Figure 2.13 shows the representative elementary volume of RCC, whose nonlinear mechanical behaviors under all kinds of loadings are described with an equivalent viscoelastic medium model. When the stress wave transmits into the RCC specimen, the attenuation and dispersion of stress wave are displayed. Based on the equivalent viscoelastic model in Fig. 2.13b, derivation for the equation of wave propagation across RCC specimens is addressed in detail as following. As shown in Fig. 2.13b, a linear spring with a modulus of E υ is in parallel with a linear dashpot ηυ , then in series with another linear spring E s . The constitutive equation of the nonlinear model is given as

2.5 Theoretical Analysis on the Shock Wave Propagation

37

Fig. 2.13 The simplified continuum medium model for layered RCC: a Diagram of the representative elementary volume; b Equivalent viscoelastic medium model

(E s + E υ )σ + ηυ ∂σ/∂t − ηυ E s ∂ε/∂t − E s E υ ε = 0

(2.19)

where σ and ε denote the stress and the corresponding strain, respectively; t denotes the time. The parameters in the nonlinear model (i.e., E υ , E s and ηυ ) vary with the material attributes. On the other hand, as explained in Sect. 2.2.1, the stress wave propagating along the bars and RCC specimens in SHPB tests approximately reaches to the one-dimensional propagation theory. Therefore, for simplicity, we only consider the longitudinal wave propagation and the corresponding motion equation of the RCC specimen can be expressed as ρ∂v p /∂t = ∂σ/∂ x

(2.20)

where v p denotes the particle velocity, and x is the normal direction of stress wave propagation. Differentiating Eq. (2.19) with respect to x and taking Eq. (2.20) into consideration, the wave propagation equation across RCC specimens described by the viscoelastic medium can be rewritten as ρηυ

∂2vp ∂t 2

+ ρ(E s + E υ )

∂v p ∂t

− ηυ E s

∂2vp ∂x2

− Es Eυ



∂2vp dt ∂x2

=0

(2.21)

For simplification, the wave propagation equation can be further generalized as v p (x, t) = Aex p(−αx)exp[iω(t − β x)]  α=  β=

ρω2 2Er E s

ρ 2Er E s

 

E s 2 +Er 2 ω2 τ 2 1+ω2 τ 2

E s 2 +Er 2 ω2 τ 2 1+ω2 τ 2

1/2 1/2

−

E s +Er ω2 τ 2 1+ω2 τ 2

+

E s +Er ω2 τ 2 1+ω2 τ 2

(2.22a)

1/2 (2.22b) 1/2 (2.22c)

38

2 Experimental Research on Dynamic Behaviors of RCC

where A is the peak velocity of the incident wave; 1/Er = 1/E s + 1/E υ and τ = ηυ /E υ ; ω is the velocity of phase shift and ω = 2π f , where f is the frequency of the incident wave. α is the wave attenuation coefficient characterizing the wave attenuation during propagation; β is another coefficient denoting the velocity of phase shift, which has no influence on the amplitude of the transmitted wave. In addition, by setting η to zero, the wave attenuation coefficient also becomes zero (α = 0). That means the wave attenuation lies in the nonlinear deformation behavior of propagation medium. Many researches have confirmed that the wave attenuation and phase shift are tightly related to the characteristics of the incident wave (such as frequency) and the material attributes (such as density, elastic modulus, wave impedance, and so on) [36, 37]. As for the SHPB tests in this chapter, it has been confirmed in Sect. 2.3.1 that the frequency spectrums almost remain constant after propagation across RCC specimens under different levels of gas pressure. Therefore, in the present study, the wave attenuation mainly lies in the contributions of material attributes, i.e. the existence of weak bonding interlayers and the intrinsic material inhomogeneity.

2.5.2 Wave Attenuation During Propagation Across RCC It has been confirmed in Refs. [38, 39] that when exposed to impact loadings, the RCC specimen with interlayer shares a similar trend of stress–strain curve with that of specimen without interlayer. However, the dynamic compressive strength for specimen with interlayer is relatively lower than that for specimen without interlayer at a similar strain rate, indicating more wave attenuation as the stress wave propagates across interlayers [39]. In this subsection, wave attenuation during propagation across RCC is further discussed in the aspect of wave propagation. In SHPB tests, it is necessary to consider the interfacial effect for concrete-like materials. Liu et al. [40, 41] conducted the sensitivity and uncertainty analysis of the parameters for interfaces, where the reflection and transmission of wave take place. Based on the Hertzian contact theory, the stress transmission coefficients Ti−s (from the incident bar to the RCC specimen) and Ts−t (from the RCC specimen to the transmission bar) can be calculated from the following equations respectively [42]. Ti−s = 2Z s Ab /(Z b Ab + Z s As )

(2.23a)

Ts−t = 2Z b As /(Z b Ab + Z s As )

(2.23b)

where Z s is the wave impedance of RCC specimens, which can be estimated with Eq. (2.16) by density and ultrasonic tests. Z b denotes the wave impedance of bars and is set to be 45.19 × 106 kg/(m2 · s) in this chapter. In this experimental study, the end surfaces of prepared RCC specimens are applicable to those of incident and transmission bars, i.e., As = Ab . Therefore, considering

2.5 Theoretical Analysis on the Shock Wave Propagation

39

the attenuation of stress wave propagating in RCC specimen with the length of Hs , the transmission coefficient from the incident bar to the transmission bar can be given by the following equation derived as Eq. (2.24), since phase shift of stress wave has no influence on the amplitude of the transmitted wave. Tc = Ti−s Ts−t

maxv p (Hs ,t) maxv p (0,t)

=

4Z s /Z b exp(−α Hs ) (1+Z s /Z b )2

(2.24)

Then, the wave attenuation coefficient (α) during propagation across RCC, can be estimated as α = −[lnTc + 2ln(Z s + Z b ) − ln(4Z s Z b )]/Hs

(2.25)

In this experimental study, the wave attenuation coefficients for layered RCC under impact loadings have been illustrated in Fig. 2.14 graphically. It is obvious that the wave attenuation coefficient is sensitive to the strain rate, which may be related to the slight strain-rate effect of elastic modulus [43]. Much more serious lateral confinement will restrict the deformation of RCC specimen at a higher strain rate. In this condition, a higher elastic modulus will be obtained, so that less wave attenuation will occur at a higher strain rate based on Eq. (2.25). Moreover, as shown in Fig. 2.14, more wave attenuation occurs when the stress wave propagates across an interlayer in RCC material, accompanied by a more invisible strain-rate effect on the wave attenuation coefficient and apparently different from that for specimen without interlayer. The reasom may also relate to a lower elastical behavior induced by the interlayer. As shown in Fig. 2.14, the fitting curves of wave attenuation coefficients exhibit the influence of interlayers on stress wave propagation across RCC specimen much more graphically. It is also obvious from Fig. 2.14 that the wave attenuation coefficient is accompanied by a significant variability due to the inhomogeneity of RCC material, especially for those with interlayers. The influence of material inhomogeneity on Fig. 2.14 Variability of wave attenuation coefficient with strain rate for layered RCC

40

2 Experimental Research on Dynamic Behaviors of RCC

Fig. 2.15 Characteristics of wave propagation across RCC specimens: a The correlation among strain rate, relative wave impedance, and wave attenuation coefficient; b Theoretical transmission and reflection coefficients

wave propagation can be generalized by the variability of relative wave impedance, defined as Z s /Z b . Therefore, the wave attenuation coefficient is dependent on the strain rate and the wave impedance. Figure 2.15a illustrates the correlation among strain rate, relative wave impedance, and wave attenuation coefficient. As shown in Fig. 2.15a, the wave attenuation coefficient exhibits an obvious negative correlation with the relative wave impedance. No matter whether the RCC specimen containing an interlayer, the two kinds of RCC specimens obey a similar trend as the relative wave impedance and strain rate change. This trend has been generalized with the multiple linear regression method, shown as below: α = 111.50 − 161.70 × Z s /Z b − 17.37 × log(˙ε)

(2.26)

Based on Eq. (2.26), the wave attenuation coefficient for layered RCC at different strain rates can be easily estimated by the conventional density and ultrasonic tests. Introducing Eqs. (2.26) into (2.24), the transmission coefficient for stress wave propagating across RCC specimens can be easily obtained, which is determined by the relative wave impedance, strain rate, and specimen’s length. Furthermore, based on the one-dimensional wave propagation theory in SHPB tests, the corresponding reflection coefficient can be evaluated by Rc = 1 − Tc . As shown in Fig. 2.15b, the transmission and reflection coefficients are both strain-rate sensitive and their strain-rate effect becomes more significant at a higher relative wave impedance. Moreover, the relative wave impedance is such an important parameter for wave propagation that the strength enhancement of transmitted wave increases nonlinearly with the increasing relative wave impedance at the same strain rate, leading to a lower reflected wave.

2.5 Theoretical Analysis on the Shock Wave Propagation

41

2.5.3 Influence of Interlayers on Transmitted Wave As explained in Sect. 2.3.1, the incident wave generated by the various cross-sectional striker in this experimental study is a typical half-sine stress wave. Moreover, the frequency spectrums for incident and transmitted waves have been confirmed to be independent on the gas pressure. Then, the incident wave can be generalized as σi (t) = σm sin(ωt)

(2.27)

where σm is the peak stress of the incident wave. Figures 2.10b and 2.11b also illustrate that the steel bars and RCC specimens in the propagation path of stress wave almost have no influence on the frequency spectrums for stress waves. Thus, in SHPB tests, the influence of frequency spectrum on the transmitted waves can be ignored for all RCC specimens, no matter whether the specimen contains a weak bonding interlayer. Combining Eq. (2.24) with Eq. (2.27), the transmitted wave can be further rewritten as σt (t) = Tc σm sin(ωt) =

4Z s /Z b exp(−α Hs )σm sin(ωt) (1+Z s /Z b )2

(2.28)

where the transmitted wave at a specific moment is determined by the peak stress of incident wave, relative wave impedance, wave attenuation coefficient, specimen’s length and the velocity of phase shift. Meanwhile, the wave attenuation coefficient is dependent on the peak stress of incident wave and the relative wave impedance, ignoring the influence of velocity of phase shift in SHPB test. In this subsection, we focus on the transmitted waves experiencing the stress attenuation from RCC specimens in SHPB tests, which can be evaluated by the theoretical model of Eq. (2.28). Based on the results of density and ultrasonic tests, the wave impedance for each RCC specimen can be obtained by Eq. (2.16). Then, the wave attenuation coefficients at different strain rates can be easily obtained by Eq. (2.26). In this chapter, the constant frequency of the incident waves is about 1818 Hz and the corresponding velocity of phase shift is about 11,424 rad/s, which almost remain constant after propagating across RCC specimens. As shown in Fig. 2.16, the experimental transmitted waves for each group characterized by the mean and StDev can be described well by Eq. (2.28), where the parameters for simulating transmitted waves of each group are averaged from the testing results of specimens in corresponding group. It is obvious that the proposed theoretical transmitted wave, illustrated as Eq. (2.28), can well describe the experimental transmitted wave under different gas pressures, no matter whether an interlayer exists in the RCC specimen. As shown in Fig. 2.16, the theoretical transmitted wave increases with the increasing gas pressure, while a convincing wave attenuation occurs due to the existence of an interlayer in RCC specimen. Moreover, much wave attenuation will occur at a higher gas pressure and the theoretical transmitted wave can well describe this phenomenon. For example, at a gas pressure of 0.40 MPa, the peak stress of theoretical transmitted

42

2 Experimental Research on Dynamic Behaviors of RCC

Fig. 2.16 Comparison between simulating and experimental transmitted waves: a RCC specimens without interlayer; b RCC specimens with an interlayer

wave decreases 4.39 MPa due to the existing interlayer, while the maximum attenuation of transmitted wave reaches 16.06 MPa at a gas pressure of 0.80 MPa. Therefore, the existence of weak interlayers will effectively obstruct the propagation of stress waves. This provides a new way to protect concrete structures from blast-impact loadings, as a supplement of the one-sidedness of using reinforced concrete.

2.6 Summary and Conclusions The dynamic compressive behaviors of RCC under various strain rates were experimentally studied by SHPB device. Based on the experimental results, the strain-rate sensitivity is the most distinct characteristic of various dynamic mechanical properties for RCC material. Especially, the stratification effect and the weak bonding interlayers induced by rolling compaction technique can be identified as the main differences between RCC and normal concrete. This chapter focuses on the stratification effect on the dynamic mechanical properties of RCC and the shock wave propagation across the weak bonding interlayers. Herein, the essential conclusions are listed as following: (1) The reliability of SHPB test on RCC material has been verified with typical experimental signals. Although the inhomogeneity of RCC might be particularly remarkable due to the aggregate distribution and construction technique, the one-dimensional wave propagation theory and the homogeneity assumption of stress and strain still can be met in SHPB tests. (2) The empirical equations for the strain-rate effects on mechanical properties are concluded, including dynamic compressive strength, Young’s modulus and ultimate strain at peak stress. All of them increase with the increasing strain rate, and the polynomial fitting method is suggested to describe the empirical relationships between dynamic mechanical properties of RCC and loading rate. In addition, when the discreteness of dynamic compressive strength is decoupled

References

43

from the strain rate, it can be found that the dynamic compressive strength of RCC in vertical direction is affected by rolling compaction technique more significantly than that in horizontal direction. (3) The shock wave propagation across RCC specimen is analyzed in terms of incident and transmitted waves and corresponding frequency spectrums, in which the effect of weak bonding interlayer is also taken into consideration. The normalized frequency spectrums of incident and transmitted waves are approximately identical and insensitive to the loading rate, no matter whether a weak bonding interlayer exists in the RCC specimen. Therefore, the frequency change of stress wave is reasonable to be ignored in the theoretical analysis on the shock wave propagation across RCC specimen in SHPB test. (4) Reflection and transmission coefficients, observed to be strain-rate sensitive, are proposed to characterize the stress wave propagation across RCC. The reflection coefficient decreases with the increasing strain rate, while the transmission coefficient shows a positive correlation with the strain rate. However, comparing to the RCC specimens without interlayer, the strain-rate effect on the reflection and transmission coefficients becomes less significant when interlayers exist in RCC specimens. (5) The nonlinear deformation behavior of RCC during stress wave propagation is described by the suggested equivalent viscoelastic medium model in the present study, which is verified to be the main reason for the wave attenuation through the theoretical deduction. The theoretical transmission coefficient of shock wave in SHPB test is negatively related to relative wave impedance, wave attenuation coefficient and specimen length. Besides, much less wave attenuation will occur when RCC specimens are exposed to higher loading rates.

References 1. Wikipedia. Roller-compacted concrete (EB/OL). Wikipedia, 2016-10-23 [2016-11-11]. https:// en.wikipedia.org/wiki/Roller-compacted_concrete 2. Tai YS (2009) Uniaxial compression tests at various loading rates for reactive powder concrete [J]. Theoret Appl Fract Mech 52(1):14–21 3. Xiao J, Li L, Shen L et al (2015) Compressive behaviour of recycled aggregate concrete under impact loading [J]. Cem Concr Res 71:46–55 4. Zhou J-K, Ding N (2014) Moisture effect on compressive behavior of concrete under dynamic loading [J]. J CentL South Univ 21(12):4714–4722 5. Ross CA, Tedesco JW, Kuennen ST (1995) Effects of strain rate on concrete strength [J]. ACI Mater J 92(1):37–47 6. China’s General Administration of Quality Supervision, Inspection and Quarantine (2003) Moderate heat portland cement, low heat portland cement, low heat portland slag cement (GB200-2003) [S]. China Standards Press, Beijing (in Chinese) 7. Development N, Commission R (2007) Technical specification of fly ash for use in hydraulic concrete (DL/T5055-2007) [S]. China Electric Power Press, Beijing (in Chinese) 8. National Energy Administration (2015) Code for mix design of hydraulic concrete (DL/T53302015) [S]. China Electric Power Press, Beijing (in Chinese)

44

2 Experimental Research on Dynamic Behaviors of RCC

9. Chen G, Hao Y, Hao H (2015) 3D meso-scale modelling of concrete material in spall tests [J]. Mater Struct 48(6):1887–1899 10. Akçao˘glu T, Tokyay M, Celik T (2004) Effect of coarse aggregate size and matrix quality on ITZ and failure behavior of concrete under uniaxial compression [J]. Cement Concr Compos 26(6):633–638 11. Karamloo M, Mazloom M, Payganeh G (2016) Effects of maximum aggregate size on fracture behaviors of self-compacting lightweight concrete [J]. Constr Build Mater 123:508–515 12. Cui J, Hao H, Shi Y (2017) Discussion on the suitability of concrete constitutive models for high-rate response predictions of RC structures [J]. Int J Impact Eng 106:202–216 13. Ministry of Water Resources of the People’s Republic of China (2006) Test code for hydraulic concrete (SL352-2006) [S]. China Electric Power Press, Beijing (in Chinese) 14. China National Energy Ministry (2009) Construction specification for hydraulic roller compacted concrete (DL/T5112-2009) [S]. China Electric Power Press, Beijing (in Chinese) 15. Chen X, Ge L, Zhou J et al (2016) Experimental study on split Hopkinson pressure bar pulseshaping techniques for concrete [J]. J Mater Civ Eng 28(5):04015196 16. Lindholm US (1964) Some experiments with the split Hopkinson pressure bar* [J]. J Mech Phys Solids 12(5):317–335 17. Xie Y-J, Fu Q, Zheng K-R et al (2014) Dynamic mechanical properties of cement and asphalt mortar based on SHPB test [J]. Constr Build Mater 70:217–225 18. Deng Z, Cheng H, Wang Z et al (2016) Compressive behavior of the cellular concrete utilizing millimeter-size spherical saturated SAP under high strain-rate loading [J]. Constr Build Mater 119:96–106 19. Ravichandran G, Subhash G (1994) Critical appraisal of limiting strain rates for compression testing of ceramics in a split Hopkinson pressure bar [J]. J Am Ceram Soc 77(1):263–267 20. Dilger WH, Koch R, Kowalczyk R (1984) Ductility of plain and confined concrete under different strain rates [J]. ACI Journal Proceedings 81(1):73–81 21. Parviz Soroushian K-BC, Abdulaziz A (1986) Dynamic constitutive behavior of concrete [J]. ACI J Proc 83(2):251–259 22. Li QM, Ye ZQ, Ma GW et al (2007) Influence of overall structural response on perforation of concrete targets [J]. Int J Impact Eng 34(5):926–941 23. Zhou XQ, Hao H (2008) Modelling of compressive behaviour of concrete-like materials at high strain rate [J]. Int J Solids Struct 45(17):4648–4661 24. Bischoff PH, Perry SH (1991) Compressive behaviour of concrete at high strain rates [J]. Mater Struct 24(6):425–450 25. Martin F-A, Rivard P (2012) Effects of freezing and thawing cycles on the shear resistance of concrete lift joints [J]. Can J Civ Eng 39(10):1089–1099 26. Liu YD (2015) Study on the effects of layer processing on the layer adhesion properties of RCC [D]. Zhejiang University, Hangzhou (in Chinese) 27. Liu D, Li Z, Liu J (2015) Experimental study on real-time control of roller compacted concrete dam compaction quality using unit compaction energy indices [J]. Constr Build Mater 96:567– 575 28. Schrader E (1999) Shear strength and lift joint quality of RCC [J]. Int J Hydropower Dams 6(1):46–55 29. Misra A, Marangos O (2011) Rock-joint meso-mechanics: relationship of roughness to closure and wave propagation [J]. Int J Geomech 11(6):431–439 30. Wang G, Zhang S (2014) Damage prediction of concrete gravity dams subjected to underwater explosion shock loading [J]. Eng Fail Anal 39:72–91 31. Zhang S, Wang G, Wang C et al (2014) Numerical simulation of failure modes of concrete gravity dams subjected to underwater explosion [J]. Eng Fail Anal 36:49–64 32. Scheiner S, Hellmich C (2009) Continuum meso-viscoelasticity model for aging basic creep of early-age concrete [J]. J Eng Mech 135(4):307–323 33. Lee H-J, Kim YR (1998) Viscoelastic constitutive model for asphalt concrete under cyclic loading [J]. J Eng Mech 124(1):32–40

References

45

34. Gu C-S, Song J-X, Fang H-T (2006) Analysis model on gradual change principle of effect zones of layer face for rolled control concrete dam [J]. Appl Math Mech 27(11):1523–1529 35. Zhu JB, Zhao XB, Wu W et al (2012) Wave propagation across rock joints filled with viscoelastic medium using modified recursive method [J]. J Appl Geophys 86:82–87 36. Zou Y, Li J, Laloui L et al (2017) Analytical time-domain solution of plane wave propagation across a viscoelastic rock joint [J]. Rock Mech Rock Eng 50(10):2731–2747 37. Zhang S, Song R, Wang C et al (2018) Experimental investigation of the compressive behavior of RCC under high strain rates: considering the rolling technique and layered structure [J]. J Mater Civ Eng 30(4):04018057 38. Wang C, Chen W, Hao H et al (2018) Experimental investigations of dynamic compressive properties of roller compacted concrete (RCC) [J]. Constr Build Mater 168:671–682 39. Liu P, Hu D, Wu Q et al (2018) Sensitivity and uncertainty analysis of interfacial effect in SHPB tests for concrete-like materials [J]. Constr Build Mater 163:414–427 40. Liu P, Han X, Hu D et al (2016) Sensitivity and uncertainty analysis of SHPB tests for concrete materials [J]. Int J Appl Mech 08(08):1650088 41. Ahmad Iram R, Shu Dong W (2011) Effect of specimen diameter in compression at high strain rates [J]. J Eng Mech 137(3):169–174 42. Hao H, Hao Y, Li J et al (2016) Review of the current practices in blast-resistant analysis and design of concrete structures [J]. Adv Struct Eng 19(8):1193–1223 43. Zhang S-R, Wang X-H, Wang C et al (2017) Compressive behavior and constitutive model for roller compacted concrete under impact loading: considering vertical stratification [J]. Constr Build Mater 151:428–440

Chapter 3

Meso-mechanic-Based Dynamic Behaviors of RCC

3.1 Introduction Both laboratory test and numerical simulation studies show that the rate dependence of concrete is mainly attributed to lateral confinement effect [1], meso-crack growth effect [2] and viscous effect [2]. Compared with the normal concrete in civil and industrial engineering, the dam concrete such as RCC is a kind of concrete material with low strength, large aggregate size (d max ≥ 80 mm), and lean cement. However, due to the limited scale of specimen in laboratory tests, it is difficult to carry out dynamic mechanical test on fully-graded RCC. With the development of numerical simulation, domestic and foreign scholars have carried out various meso-mechanical studies on the influences of ITZ thickness [3, 4], aggregate characteristics (lithology, particle gradation, content, etc.) [5–9] and specimen size [9] on dynamic mechanical properties of concrete. But the understanding of such large aggregate effect on dynamic mechanical properties is still ambiguous. Therefore, a series of uniaxial compression/tensile mesoscopic mechanics tests are conducted in this chapter based on the concrete mesoscopic model with due consideration of rate sensitive components. The effects of interlayer bonding surface and the maximum aggregate size on the dynamic compressive/tensile mechanical properties of RCC are analyzed.

3.2 Mesoscopic Simulation Method and Validation 3.2.1 Meso-simulation Method In this chapter, RCC is simplified into two parts: noumenon and interlayer bonding surface. The RCC noumenon is composed of coarse aggregate, mortar matrix and the interfacial transition layer between them. Considering the modelling and computation © Science Press 2023 S. Zhang et al., Dynamic Mechanical Behaviors and Constitutive Model of Roller Compacted Concrete, Hydroscience and Engineering, https://doi.org/10.1007/978-981-19-8987-2_3

47

48

3 Meso-mechanic-Based Dynamic Behaviors of RCC

efficiency, the two-dimensional circular random aggregate is applied herein, as has done in many literatures [5–14]. Moreover, the volume content of coarse aggregate is set to be 50%, and four different aggregate gradations (5–20/5–40/5–60/5–80 mm) are considered. The coarse aggregate particle distribution is described by a twodimensional grading curve [15], as illustrated by Eq. (3.1). In concrete, the actual thickness of the interface bonding surface is generally within the range of 20–50 µm, while the thickness is usually set to about 0.5–2.0 mm to simplify calculation [5– 9, 12–14] in numerical simulation. Song and Lu [16] studied the influence of the thickness of interfacial transition zone (ITZ), and their results showed that when the thickness of ITZ increased from 0.5 to 2.0 mm, it only had a slight influence on the post-peak section of stress–strain curve, but had no obvious influence on the macroscopic strength of concrete. Therefore, the thickness of the interface transition zone in this chapter is set to be 0.5 mm. Pc (d < d0 ) = ) −0.5 − 0.053d 4 d −4 − 0.012d 6 d −6 − 0.0045d 8 d −8 − 0.0025d 10 d −10 Pk 1.065d00.5 dmax 0 max 0 max 0 max 0 max (

(3.1) where Pc (d < d0 ) represents the percentage of two-dimensional aggregate with particle size less than d 0 in the cross section; Pk represents the proportion of aggregate volume. The interlayer bonding surface is a unique structure of RCC material, which is usually seen as a weak link in shear and tensile resistance. The thickness of interlayer bonding surface in actual engineering is generally 0.5–2.0 cm [17, 18]. The quasi-static meso-mechanical tests of RCC with consideration of interlayer bonding surface indicated that the interlayer joint surface with a uniform thickness (1.0 cm) can accurately reflect the real mechanical behavior of RCC [19–21]. Therefore, the thickness of interlayer bonding surface of RCC was set as 1.0 cm in this chapter. Based on “background grid projection method” [22, 23], the uniaxial compression/tension meso-models of RCC are established, h/b of which are 1 and 2 respectively as shown in Fig. 3.1. For the uniaxial dynamic compression test, vertical constraints are imposed on the bottom boundary of the specimen model, and the compression load is applied on the top boundary of the model with a constant vertical velocity boundary of v = ε˙ ×h. Similarly, normal velocity boundaries (v = ε˙ × h/2) are applied at the top and bottom of the specimen in uniaxial dynamic tensile tests. In order to avoid the failure of loading end due to stress concentration under dynamic tensile load, the initial velocity [24] is applied to all nodes as shown in Eq. (3.2). vy =

2v y b

where y = 0 represents the middle part of the specimen.

(3.2)

3.2 Mesoscopic Simulation Method and Validation

Fig. 3.1 Schematic diagram of RCC meso model

49

50

3 Meso-mechanic-Based Dynamic Behaviors of RCC

3.2.2 The Constitutive Model and Parameters of Meso-components 3.2.2.1

Description of K&C Constitutive Model

In this chapter, the dynamic mechanical behaviors of each meso-component are described by K&C dynamic constitutive model. In the K&C dynamic constitutive model, the deviatoric stress on the failure strength surface ∆σ can be obtained by linear interpolation of Eq. (3.3). As shown in Fig. 3.2, when the current strength surface ∆σ is between the initial yield surface ∆σ y and the ultimate strength surface ∆σm , it can be obtained by interpolation from the first equation, namely strain hardening state; otherwise, when the current strength surface ∆σ moves between ∆σm and ∆σr , it is in strain softening stage. The three strength surfaces are shown in Eqs. (3.5)–(3.7). ∆σ =

) { ( √ η ∆σm − ∆σ y + ∆σ y 3J2 = η(∆σm − ∆σr ) + ∆σr

(3.3)

| | where J2 = (σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ1 − σ3 )2 /6 is the second invariant of stress; σ1 , σ2 and σ3 are the principal stresses in three directions; η is the scale factor of strength surface related to the modified effective plastic strain; ∆σ y , ∆σm and ∆σr are the initial yield surface, ultimate strength surface and residual strength surface, respectively. ( ) ⎧ ⎨ a0y + P/ a1y( + a2y P ) ∆σ y = 1.35 f t + 3P 1 − 1.35 f t / f yc ⎩ 1.35(P + f t )

P ≥ f yc /3 0 ≤ P < f yc /3 P λm

∆σr = a0r + P/(a1r + a2r P)

(3.5)

(3.6)

where aim , ai y and air (i = 0, 1, 2) are strength surface parameters determined by fitting triaxial test data, respectively; P = (σ1 + σ2 + σ3 )/3 is hydrostatic pressure; ψ is the meridian ratio between tension and compression; f yc , f c' and f t are the initial yield strength, uniaxial compressive strength and uniaxial tensile strength of the material, respectively; λm is the modified effective strain; λ is the damage parameter, which is defined as follows: {{ε λ=

p

{0ε p 0

| ( )|−b1 γ f−1 1 + P/ γ f f t dε p P ≥ 0 | ( )|−b2 γ f−1 1 + P/ γ f f t dε p P < 0

(3.7)

where ε p is the cumulative effective plastic strain; dε p is the effective plastic strain increment at each integral step; r f is the dynamic growth factor of material strength, and γ f = 1 represents the quasi-static load condition; b1 and b2 are the control parameters for the damage evolution rate under compression and tension loading in the strain-softening phase, respectively. The relationship between η and λ is usually entered by the users. In general, as ∆σ changes from ∆σ y to ∆σm , η changes from 0 to 1. While, as ∆σ changes from ∆σm to ∆σr , η decreases from 1 to 0. When λ = λm , η = 1. As shown in Eq. (3.8), volume damage is introduced into the effective plastic strain. ( ) ∆λ = b3 f d kd εv − εvy

(3.8)

where b3 is the control parameter for the damage evolution rate of the material in the strain softening stage; kd is the internal scalar multiplier; εv and εvy are the volumetric strain and the yield strain, respectively; f d is the factor limiting the strength path under triaxial stretching, and its definition is shown in Eq. (3.9). { fd =

| |√ 1 − | 3J2 /P |/0.1 0

|√ | 0| < | 3J|2 /P | < 0.1 √ | 3J2 /P | > 0.1

(3.9)

In the K&C dynamic constitutive model, the relationship between volume strain and hydrostatic pressure is described by equation of state (EOS), which corresponds to *EOS_TABULATED_COMPACTION keywords in LS-DYNA. The curve of hydrostatic pressure changing with volumetric strain is shown in Fig. 3.3 and Eq. (3.10). P = C(εv ) + γt T (εv )E i

(3.10)

52

3 Meso-mechanic-Based Dynamic Behaviors of RCC

Fig. 3.3 Schematic diagram of equation of state in K&C model

where the equation C(εv ) is defined by ten sets of external input data (εv , P); γt is specific heat ratio; T (εv ) is the thermodynamic coefficient related to volumetric strain; E i is the initial internal energy per unit volume. It should be noted that the internal energy in the equation of state under quasi-static load can be ignored.

3.2.2.2

Strain Rate Effect

In the K&C dynamic constitutive model, the strain rate effect is reflected not only in the damage equation, but also in the failure surface. As shown in Eq. (3.11), the strain-rate enhancement coefficient γ f is introduced into the ultimate strength surface to consider the strain-rate effect. ( ) ∆σme = γ f × ∆σm P/γ f

(3.11)

Therefore, accurately describing the strain-rate effect of materials determines the rationality of the simulation results. The strain-rate effects of mortar matrix, interlayer bonding surface and ITZ are simplified as the same in this study. The CDIF relationship proposed by Hao [6] is used herein to consider the strain-rate effect on dynamic compressive strength after eliminating the inertia effect, as shown in Eq. (3.12). The TDIF relationship proposed by Hao and Zhou [25] is adopted to consider the strain-rate effect on dynamic tensile strength, as shown in Eq. (3.13). { CDIF =

0.0419 lg ε˙ + 1.2165 ε˙ ≤ 30 s−1 (3.12) 2 0.8988(lg ε˙ ) − 2.8255 lg ε˙ + 3.4907 30 s−1 < ε˙ < 1000 s−1 { 0.26 lg ε˙ + 2.06 10−4 s−1 ≤ ε˙ ≤ 1 s−1 TDIF = (3.13) 2.00 lg ε˙ + 2.06 1 s−1 < ε˙ < 100 s−1

As shown in Fig. 3.4, the strain-rate effect of coarse aggregate is fitted by the test data in the literatures [26–30], and then the DIF relationships are obtained by

3.2 Mesoscopic Simulation Method and Validation

53

Fig. 3.4 DIF relationship of dynamic strength of coarse aggregate

Eqs. (3.14) and (3.15). In order to avoid overestimating its dynamic tensile strength, the TDIF remains constant when the strain rate ε˙ ≥ 50 s−1 due to the lack of dynamic tensile test data of coarse aggregate under high strain rates. { CDIF = { TDIF =

0.0523 lg ε˙ + 1.3138 2.6475(lg ε˙ )2 − 11.7664 lg ε˙ + 14.4712

0.0598 lg ε˙ + 1.3588 0.5605(lg ε˙ )2 + 1.3871 lg ε˙ + 2.1256

10−6 s−1 ≤ ε˙ < 220 s−1 220 s−1 ≤ ε˙ ≤ 1000 s−1 (3.14)

10−6 s−1 ≤ ε˙ ≤ 0.1 s−1 (3.15) 0.1 s−1 ≤ ε˙ ≤ 50 s−1

3.2.3 Validation of Numerical Model The rationality of the meso-model is verified by the results of dynamic compressive/tensile tests on RCC [19, 31–34]. The ITZ is regarded as the mortar matrix with the weakened mechanical parameters [17]. At the same time, this study refers to the meso-simulation carried out by Peng et al. [20, 35], and the mechanical properties of interlayer bonding surface are consistent with the ITZ. Then, the K&C constitutive parameters of each meso-component were finally determined by repeated trial calculations. Based on the laboratory tests conducted by Liu [33], two- graded concrete specimens with dimensions of 450 × 450 mm were established, and the parameters of each mesoscopic component are determined by trial calculation as shown in Tables 3.1, 3.2 and 3.3. Figure 3.5a shows that the peak strength obtained by numerical simulation is in good agreement with that of laboratory test, but there is some differences in the post-peak curves. In order to further study the applicability of the model at high strain rates, the above parameters are used to carry out the uniaxial dynamic compression tests at three strain rates (˙ε = 30/50/70/100 s−1 ),

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3 Meso-mechanic-Based Dynamic Behaviors of RCC

Table 3.1 Parameters of K&C constitutive model of mortar matrix Basic parameters

Strength surface parameters

Damage parameters

ρ/kg·m−3

a0y /Pa

b1

2000

5.58 × 106

1.60

v

0.18

a1y

0.625

b2

1.35

G/GPa

10.63

a2y /Pa−1

1.03 × 10–9

b3

1.15

f ' c /MPa

21

a0m /Pa

1.60 × 106

I f rac /mm

10

f t /MPa

2.57

a1m

1.35

n

100

dε p

0.5

a2m /Pa−1

1.15 × 10–9

α

2.60

λt

8.7 × 10–3

a0r /Pa

0

αc

0.283

a1r

0.4417

αd

1.74

a2r /Pa−1

4.732 × 10–9

Table 3.2 Parameters of K&C constitutive model of ITZ interface Basic parameters

Strength surface parameters

ρ/kg·m−3

a0y /Pa

1800

5.58 × 106

Damage parameters b1

1.60

v

0.20

a1r

0.625

b2

1.35

G/GPa

8.65

a2y /Pa−1

1.03 × 10–9

b3

1.15

f ' c /MPa

16

a0m /Pa

1.60 × 106

I f rac /mm

10

f' λt

c /MPa

2.0

a1m

1.35

n

100

8.7 × 10–3

a2m /Pa−1

1.15 × 10–9

α

2.60

a0r /Pa

0

αc

0.283

a1r

0.4417

αd

1.74

a2r /Pa−1

4.732 × 10–9

and the CDIFs obtained by the mesoscopic simulation fall within the variability range of SHPB test results, as shown in Fig. 3.5b. Furthermore, according to the laboratory uniaxial dynamic tensile tests by Zhang et al. [34], a three-graded RCC model (5–80 mm) with the size of 450 × 1350 mm was established, and the results of uniaxial tensile tests at different strain rates (˙ε = 1 × 10−5 and 1 × 10−3 s−1 ) are shown in Fig. 3.5c. It is obvious that when the strain rate increases from 1 × 10−5 to 1 × 10−3 s−1 , the tensile strength from mesoscopic simulation increases from 2.32 to 3.09 MPa, which is consistent with the results (increasing from 2.54 MPa to 3.24 MPa) of laboratory tests by Zhang et. al. [34]. To sum up, the mesosimulation proposed in this chapter and the selected material parameters can well describe the dynamic compressive/tensile behaviors of RCC, and can be used for further meso-mechanics research.

3.3 Effect of Maximum Aggregate Size on Dynamic Mechanical Properties …

55

Table 3.3 K&C constitutive model parameters of coarse aggregate Strength surface parameters

Basic parameters ρ/kg·m−3

2650

a0y /Pa

7.70 × 107

Damage parameters b1

0.75

v

0.25

a1y

0.513

b2

3.21

G/GPa

20.65

a2y /Pa−1

7.70 × 10–10

b3

0.50

f ' c /MPa

80

a0m /Pa

7.90 × 107

L w /mm

1.35

f t /MPa

5.20

a1m

0.542

n

100

ω

0.90

a2m /Pa−1

1.50 × 10–10

α

2.60

a0r /Pa

0.0

αc

0.283

a1r

0.47

αd

1.74

a2r /Pa−1

2.00 × 10–10

Fig. 3.5 Validation of mesoscopic numerical model

3.3 Effect of Maximum Aggregate Size on Dynamic Mechanical Properties of RCC Due to the limitation of specimen size and numerical simulation efficiency of high strain rate impact test, laboratory tests and numerical simulations mostly regard concrete materials as homogeneous materials to study their mechanical properties. However, concrete materials are composed of meso components with different mechanical properties, so it is not accurate to assume them as homogeneous materials. Erzar et al. [36] studied the influence of maximum aggregate size (Dmax = 8 and 2 mm) on dynamic failure modes of specimens. Grote et al. [37] compared the dynamic characteristics of mortar specimens and concrete specimens through laboratory tests, and the addition of coarse aggregate intensified the internal heterogeneity of specimens and contributed to CDIF. The numerical simulation study of Hao et al. [1, 6, 11, 38] shows that the influence of aggregate size on CDIF of concrete materials under the impact load; Jin Liu et al. [8, 39] ignored the influence of continuous aggregate gradation and selected three aggregate particle sizes (d = 10/20/30 mm) to conduct a mesoscopic simulation study on the dynamic strength of concrete. At low strain rate, the dynamic tensile strength increases with the increase of aggregate particle size, while the compressive strength first increases and then decreases with

56

3 Meso-mechanic-Based Dynamic Behaviors of RCC

the increase of aggregate particle size. At high strain rates, the effect of aggregate size on dynamic tensile/compressive strength can be ignored. Similarly, the effect of four different aggregate particle sizes (d = 4/8/12/16/20 mm) on dynamic mechanical properties selected by Lei Guangyu et al. [38] selected the influence of four different aggregate particle sizes (d = 4/8/12/16/20 mm) on dynamic mechanical properties. The results show that the influence of aggregate size on the compressive strength and tensile strength of concrete under dynamic load is opposite. The dynamic compressive strength decreases with the increase of aggregate size, and the dynamic tensile strength increases with the increase of aggregate size, but the increase range is small. It can be seen from the above that due to the limitation of test conditions, it is difficult to study the dynamic performance of large aggregate concrete materials in the laboratory test. The influence of continuous aggregate gradation is mostly ignored in relevant numerical simulation. The influence of aggregate size under continuous aggregate gradation needs to be further studied.

3.3.1 Two Dimensional Mesoscopic Model In order to study the effect of coarse aggregate size on the dynamic tensile and compressive mechanical properties of RCC, the above meso-model is used to generate RCC specimens with four aggregate gradations (5–20, 5–40, 5–60 and 5–80 mm). Figure 3.6 shows the mesoscopic finite element models for uniaxial compression test, where the specimen width b is set to be 600 mm and the aspect ratio h:b is set to be 1:1, and the mesoscopic numerical models for dynamic tension test are established in the same way but with an aspect ratio of 2:1.

Fig. 3.6 Meso-model of different aggregate sizes

3.3 Effect of Maximum Aggregate Size on Dynamic Mechanical Properties …

57

3.3.2 Effect of Aggregate Size on Dynamic Compressive Behaviors Figure 3.7 shows the failure mode diagrams of RCC specimens with different coarse aggregate sizes under different loading rates. It can be seen from the figure that the failure modes for specimens with different aggregate sizes are basically the same and the internal damage distribution is non-uniform. When the strain rate is low (˙ε = 0.1 s−1 ), the crack firstly occurs in the mortar matrix and ITZ area. Then the crack gradually expands and forms an oblique penetrating crack. When the crack encounters the coarse aggregate component in the propagation process, the crack mainly develops around the coarse aggregate along the weak ITZ. When the strain rate increases to ε˙ = 50 s−1 , the crack number and damage area increase obviously. The reason can be explained that when the loading rate becomes faster, the internal stress distribution in the specimen is difficult to reach the equilibrium state in a short time, and the aggregate begin to crush in this case. Figure 3.8 shows the correlation between the maximum aggregate size and the dynamic compressive strength of RCC at different strain rates. It can be seen from the figure that the dynamic compressive strength shows significant strain-rate effect no matter what the aggregate gradation is. The dynamic compressive strength increases with the increase of strain rate, and the amplitude of strength enhancement becomes more and more significant at a higher strain rate. However, under the same loading rate, the dynamic compressive strength decreases with the increase of the maximum aggregate size. From the mesoscopic view, the existence of coarse aggregate can significantly change the crack propagation direction, and the larger the aggregate size is, the more obvious the crack arrest effect is. The total number and specific surface area of coarse aggregate decrease with the increase of the maximum aggregate size, indicating the more uneven damage distribution of specimens.

Fig. 3.7 Failure modes of RCC with different aggregate sizes under dynamic compression

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3 Meso-mechanic-Based Dynamic Behaviors of RCC

Fig. 3.8 Relationship between peak strength and maximum aggregate particle size of RCC

Figure 3.9 shows the relationship between CDIFs of dynamic compressive strength for specimens with different aggregate gradations. It can be seen that the CDIFs of RCC with large aggregate size is stronger than the results of normal concrete conducted by Hao et al. [10]. Taking the maximum aggregate size into consideration, the traditional formulae for strain-rate effect can be modified by Eqs. (3.16) and (3.17). ⎧ )γ1C ( ⎨ (a1C + b1C × lg ε˙ ) dmax ε˙ ≤ ε˙ cr d0 )γ2C ( CDIF = ( ) ⎩ a2C + b2C × lg ε˙ + c2C (lg ε˙ )2 dmax ε˙ > ε˙ cr d0

(3.16)

⎧ )0.04094 ( ⎨ (1.16085 + 0.03481 lg ε˙ ) dmax ε˙ ≤ 10 s−1 d0 ) ( CDIF = ( ⎩ 2.75678 − 3.21005 lg ε˙ + 1.58079(lg ε˙ )2 ) dmax 0.02291 ε˙ > 10 s−1 d0 (3.17) where a1C , a2C , b1C , b2C , c2C , γ1C , γ2C are the fitting parameter; d0 = 20 mm.

3.3.3 Effect of Aggregate Size on Dynamic Tensile Behaviors Figure 3.10 compares the tensile failure modes of RCC specimens with different aggregate sizes at different loading rates. It can be seen from the figure that when the strain rate is low (˙ε = 1 × 10−3 s−1 ), the crack mainly appears at the middle of the specimen and gradually develops to form a penetrating crack along the non-aggregate area with poor mechanical properties. However, the damage area increase obviously as the strain rate increases. Moreover, as the aggregate size increases, the damage area also get larger, especially at the low loading rate. The increase of aggregate size generally leads to a more obvious damage inhomogeneity.

3.3 Effect of Maximum Aggregate Size on Dynamic Mechanical Properties …

59

Fig. 3.9 Comparison between fitted CDIF relationship and existing results [10]

Fig. 3.10 Failure modes of RCC with different aggregate sizes under dynamic tension

Figure 3.11 shows the effect of aggregate size on the dynamic tensile strength of RCC at different strain rates. It can be seen from the figure that the dynamic tensile strength generally increases with the increase of strain rate, and the strength

60

3 Meso-mechanic-Based Dynamic Behaviors of RCC

Fig. 3.11 Relationship between dynamic tensile strength and aggregate size at different strain rates

enhancement range becomes larger as strain rate increases. On the other hand, the tensile strength decreases with the increase of the maximum aggregate size no matter with the loading rate, and the weakening trend is more obvious at higher loading rate. This can be mainly attributed to that the existence of larger aggregate restricts the crack propagate through the shortest path, even crushing the aggregates at enough high strain rates. Figure 3.12 shows the relationship between TDIF and strain rate for RCC specimens with different aggregate gradation. It can be seen that under dynamic tension, the strain rate effect of RCC obtained by meso-simulation is more significant than that recommended by CEB code, which is close to the research results of Hao et al. [10]. The general rule can be illustrated by Eqs. (3.18) and (3.19) after parameter fitting based on simulation results of this study. TDIF =

⎧ ⎨

(

)γ1T

ε˙ ≤ ε˙ cr )γ2T ( ( ) d ⎩ a2T + b2T × lg ε˙ + c2T (lg ε˙ )2 max ε˙ > ε˙ cr d0 (a1T + b1T × lg ε˙ )

dmax d0

(3.18)

where a1T , a2T , b1T , b2T , c2T , γ1T , γ2T are the fitting parameters; d0 = 20 mm. ⎧ ( )0.04007 ⎨ (1.93024 + 0.19467 lg ε˙ ) dmax ε˙ ≤ 1 s−1 d0 ) ( TDIF = ( ⎩ 1.82604 − 0.07062 lg ε˙ + 0.63942(lg ε˙ )2 ) dmax 0.11918 ε˙ > 1 s−1 d0 (3.19)

3.4 Influence of Layer Effect on Dynamic Mechanical Properties

61

Fig. 3.12 TDIF fitting results and comparison with existing results [10]

3.4 Influence of Layer Effect on Dynamic Mechanical Properties 3.4.1 The Influence of Layer Effect on Dynamic Compressive Properties In order to study the influence of interlayer bonding surface on the dynamic compressive properties of RCC, the dynamic compressive behaviors of layered specimens (b = 300 mm) at different loading rates (˙ε = 1 × 10−5 , 1 × 10−3 , 0.1, 1, 10, 30, 50, and 100 s−1 ) is numerically simulated based on the meso-models with an interlayer of 1.0 cm and a coarse aggregate gradation of 5–40 mm. Figure 3.13 compares the influence of interlayer on the failure modes of RCC specimens at different compressive loading rates. It can be seen that the interlayer will also be damaged as the cracks propagate in the specimens. However, although the interlayer is damaged under different loading rates, the controlling cracks still exist in the concrete matrix rather than the interlayers.

Fig. 3.13 Influence of layer effect on RCC dynamic compression failure mode

62

3 Meso-mechanic-Based Dynamic Behaviors of RCC

Fig. 3.14 Influence of layer effect on dynamic compressive strength of RCC

Figure 3.14 shows the effect of interlayer on the dynamic compressive strength of RCC. It is obvious that the compressive strength of RCC specimens with interface also shows a significant strain-rate effect. Under quasi-static loading (˙ε = 1 × 10−5 s−1 ), the compressive strength of RCC specimens with interface is slightly lower than that of RCC specimens without interface. The interlayer effect coefficient of compressive strength is nearly 0.94, which is defined by f c,w /f c,wo . The interlayer effect coefficient increases gradually as the strain rate increases, and when ε˙ ≥ 1 s−1 , the layer effect coefficient basically tends to be stable (close to 1). The results show that the interlayer nearly has no influence on the dynamic compressive strength at high loading rates. The reason maybe lie in that the stress distribute in the specimen is relatively uniform under the low loading rates and the interlayer will be damaged at first in this case. However, as the strain rate gradually increases, the failure of RCC materials often occurs in a very short time, and the stress has no time to propogate into the weak bonded interlayer. Thus, the influence of interlayer on dynamic compressive strength can be ignored at high loading rates.

3.4.2 Effect of Layer Effect on Dynamic Tensile Mechanical Properties In order to study the influence of interlayer bonding surface on the dynamic tensile properties of RCC, the prepared meso-models and loading conditions are the same to those for the study of dynamic compressive properties of RCC. Generally, the interlayer will be the weakest link of the specimens at various tensile loading rates, which is quite different from that under compression. Figure 3.15 compares the influence of interlayer on the dynamic tensile strength of RCC at different tensile loading rates. It can be seen from the figure that, similar to the

3.5 Summary

63

Fig. 3.15 Influence of layer on tensile strength of RCC with indirect joint

non-layered RCC specimen, the tensile strength of the layered specimen also shows a significant strain-rate effect. The strength attenuation due to the interlayer is obvious at the quasi-static loading (˙ε = 1 × 10−5 s−1 ), and the interlayer effect coefficient of tensile strength is 0.78 in this case, which is defined by f t,w /f t,wo . Similarly, the interlayer effect coefficinet of ultimate tensile strength of RCC in laboratory tensile test is usually between 0.70 and 0.82 [40], which further proves the rationality of numerical simulation results. However, the TDIFs of layered RCC specimens are slightly greater than those of the non-layered RCC specimen, indicating a more significant strain-rate effect. With the increase of strain rate, the interlayer effect coefficient increases gradually, getting to be higher than 0.94 and approaching to 1.0 when ε˙ ≥ 1 s−1 . Thus, the influence of interlayer becomes quite slight at the enough high strain rates. The reason also can be explained by the velocity of stress propagation as that in dynamic loading condition.

3.5 Summary This chapter discusses the effects of aggregate and interlayer on the dynamic compressive and tensile behaviors of RCC based on the proposed mesoscopic numerical models. The main conclusions are as follows: (1) The dynamic compressive/tensile strength shows significant strain-rate effect no matter what the aggregate gradation is. However, under the same loading rate, both the dynamic compressive and tensile strength decrease with the increase of the maximum aggregate size. From the mesoscopic view, the existence of coarse aggregate can significantly change the crack propagation direction, and the larger the aggregate size is, the more obvious the crack arrest effect is. This can be mainly attributed to that the existence of larger aggregate restricts the crack propagate through the shortest path, even crushing the aggregates

64

3 Meso-mechanic-Based Dynamic Behaviors of RCC

at enough high strain rates. Moreover, the strain-rate effect models for both compression and tension are proposed based on the meso-simulation results, taking the aggregate gradation into consideration. (2) The compressive/tensile strength attenuation due to the interlayer is obvious at the quasi-static loading, and the interlayer effect coefficients of material strength are 0.94 and 0.78 for compression and tension, respectively. Moreover, the interlayer effect coefficient increases gradually as the strain rate increases, and the interlayer nearly has no influence on the dynamic compressive/tensile strength at enough high loading rates. The reason maybe lie in that the stress distribute in the specimen is relatively uniform under the low loading rates and the interlayer will be damaged at first in this case. However, as the strain rate gradually increases, the failure of RCC materials often occurs in a very short time, and the stress has no time to propagate into the weak bonded interlayer.

References 1. Hao Y, Hao H, Jiang G et al (2013) Experimental confirmation of some factors influencing dynamic concrete compressive strengths in high-speed impact tests [J]. Cem Concr Res 52:63– 70 2. Hao H, Hao Y, Li J et al (2016) Review of the current practices in blast-resistant analysis and design of concrete structures [J]. Adv Struct Eng 19(08):1193–1223 3. Maleki M, Rasoolan I, Khajehdezfuly A et al (2020) On the effect of ITZ thickness in meso-scale models of concrete [J]. Constr Build Mater 258:119639 4. Du X, Jin L, Ma G (2014) Numerical simulation of dynamic tensile-failure of concrete at meso-scale [J]. Int J Impact Eng 66:05–17 5. Jin L, Yu W, Du X et al (2020) Meso-scale simulations of size effect on concrete dynamic splitting tensile strength: Influence of aggregate content and maximum aggregate size [J]. Eng Fract Mech 230:106979 6. Hao Y, Hao H (2011) Numerical evaluation of the influence of aggregates on concrete compressive strength at high strain rate[J]. Int J Prot Struct 02(02):177–206 7. Du M, Jin L, Li D et al (2017) Mesoscopic simulation study of the influence of aggregate size on mechanical properties and specimen size effect of concrete subjected to splitting tensile loading [J]. Eng Mech 34(09):54–63. (in Chinese) 8. Jin L, Yang W, Yu W et al (2020) lnfluence of maximum aggregate size on dynamic size effect of concrete under low strain rates: meso-scale simulations [J]. Trans Nanjing Univ Aeronaut Astronaut 37(001):27–39 9. Yu W (2019) Meso-scale simulation in dynamic size effect on compressive and tensile failure of concrete materials[D]. School of Civil Engineering and Architecture, Beijing University of Technology, Beijing. (in Chinese) 10. Hao Y, Hao H, Li Z (2013) Influence of end friction confinement on impact tests of concrete material at high strain rate [J]. Int J Impact Eng 60:82–106 11. Hao Y, Hao H, Li Z (2010) Numerical analysis of lateral inertial confinement effects on impact test of concrete compressive material properties [J]. Int J Prot Struct 01(01):145–167 12. Jin L, Yu W, Du X et al (2019) Dynamic size effect of concrete under tension: a numerical study [J]. Int J Impact Eng 132:103318 13. Jin L, Yu W, Du X et al (2019) Mesoscopic numerical simulation of dynamic size effect on the splitting-tensile strength of concrete [J]. Eng Fract Mech 209:317–332

References

65

14. Jin L, Yu W, Du X et al (2019) Meso-scale modelling of the size effect on dynamic compressive failure of concrete under different strain rates [J]. Int J Impact Eng 125:01–12 15. Walraven JC (1981) Fundamental analysis of aggregate interlock [J]. J Struct Div 107(11):2245–2270 16. Song Z, Lu Y (2012) Mesoscopic analysis of concrete under excessively high strain rate compression and implications on interpretation of test data [J]. Int J Impact Eng 46:41–55 17. Liu G, Hao J (1996) Simplified numerical method in stress analysis of RCC arch dam with layered structure [J]. J Tsinghua Univ (Sci Technol) 36(01):27–33. (in Chinese) 18. Gu C, Song J, Fang H (2006) Analysis model on gradual change principle of effect zones of layer face for RCCD [j]. Appl Math Mech 27(11):1335–1340. (in Chinese) 19. Zhang S, Wang X, Wang C et al (2017) Compressive behavior and constitutive model for roller compacted concrete under impact loading: considering vertical stratification [J]. Constr Build Mater 151:428–440 20. Peng Y, Li B, Liu B (2001) Numerical simulation of meso-level mechanical properties of roller compacted concrete [J]. J Hydraul Eng (06):19–22. (in Chinese) 21. Yan-ling QU, Yi-jiang PENG, Li-feng DU (2007) Numerical simulation for size effect on shear strength of roller compacted concrete specimen [J]. J Water Resour Arch Eng 05(03):22–24 (in Chinese) 22. Liu JIN, Wen-xuan YU, Xiu-li DU et al (2019) Reserch on size effect of dynamic compressive strength of concrete based on meso-scale simulation [J]. Eng Mech 36(11):50–61 (in Chinese) 23. Huai-fa MA, Hou-qun CHEN, Bao-kun LI (2004) Meso-structure numerical simulation of concrete specimens [J]. J Hydraul Eng 35(10):0027–0035 (in Chinese) 24. Naderi S, Zhang M (2021) Meso-scale modelling of static and dynamic tensile fracture of concrete accounting for real-shape aggregates [J]. Cement Concr Compos 116:103889 25. Hao H, Zhou X (2007) Concrete material model for high rate dynamic analysis[C]. Proceedings of the 7th international conference on shock and impact loads on structures. Lok Ts, Beijing, pp 753–768 26. Li X, Lok T, Zhao J (2005) Dynamic characteristics of granite subjected to intermediate loading rate [J]. Rock Mech Rock Eng 38(01):21–39 27. Li Y, Xia C (2000) Time-dependent tests on intact rocks in uniaxial compression [J]. Int J Rock Mech Min Sci 37(03):467–475 28. Lindholm U, Yeakley L, Nagy A (1974) The dynamic strength and fracture properties of dresser basalt [J]. Int J Rock Mech Min Sci Geomech Abstr 11(02):181–191 29. Olsson W (1991) The compressive strength of tuff as a function of strain rate from 10–6 to 103/sec [J]. Int J Rock Mech Min Sci Geomech Abstr 28(01):115–118 30. Li H, Zhao J, Li J et al (2004) Experimental studies on the strength of different rock types under dynamic compression [J]. Int J Rock Mech Min Sci 41(03):01–06 31. Wang X, Zhang S, Wang C et al (2018) Experimental investigation of the size effect of layered roller compacted concrete (RCC) under high-strain-rate loading [J]. Constr Build Mater 165:45–57 32. Ran S (2018) Study on dynamic mechanical properties and damage constitutive model of compacted concrete at high strain rate [D]. School of Civil Engineering and Architecture, Tianjin University, Tianjin (in Chinese) 33. Peng LIU (2011) Experimental research and numerical analysis on dynamic mechanical properties of concrete [D]. School of Civil Engineering and Architecture, Dalian University, Dalian (in Chinese) 34. Zhang K, Wang H, Tu J et al (2021) Dynamic tensile test of fully-graded roller compacted concrete [J]. J China Inst Water Resour Hydropower Res 19(03):s290–300. (in Chinese) 35. Peng Y, Li B, Qu Y (2003) Numerical simulation of shear test of specimen roller compacted concrete on meso-level [J]. J Univ Hydraul Electr Eng/Yichang 25(06):492–494+561. (in Chinese) 36. Erzar B, Forquin P, Pontiroli C et al (2010) Influence of aggregate size and free water on the dynamic behaviour of concrete subjected to impact loading [C]. Proceedings of 14th international conference on experimental mechanics, EPJ Web of Conferences. EDP Sciences, Washington, pp 01–08

66

3 Meso-mechanic-Based Dynamic Behaviors of RCC

37. Grote D, Park S, Zhou M (2001) Dynamic behavior of concrete at high strain rates and pressures: I. experimental characterization [J]. Int J Impact Eng 25(09):869–886 38. Hao Y, Hao H, Zhang X (2012) Numerical analysis of concrete material properties at high strain rate under direct tension [J]. Int J Impact Eng 39(01):51–62 39. Jin L, Yang W, Yu W et al (2020) Influence of aggregate size on the dynamic tensile strength and size effect of concrete [J]. J Vib Shock 39(09):24–34. (in Chinese) 40. Rongmei J, Wei F (2007) Impact of interlayer and size effect on the mechanical performances of fully graded RCC [J]. Water Power 33(04):20–22 (in Chinese)

Chapter 4

Consturction-Induced Damage Effect on Dynamic Compressive Behaviors of RCC

4.1 Introduction The technical parameters and construction control are very important in the construction process of RCC, which significantly determines the performance of the hardened RCC. Although real-time monitoring systems for concrete rolling-compaction quality have been developed and widely used in large dam engineerings to guarantee the construction quality of RCC, much more medium-small dam engineerings still remain the traditional construction control methods. Moreover, quality of hydropower project still suffers from poor quality awareness, imperfect quality control system, poor workers’ technical level, and inadequate implementation of responsibility. Referring to the results in Chap. 2, the remarkable variability of dynamic compressive behaviors for RCC in the vertical direction may relate to the initial damage from the construction technique of thin-layer pouring and vibration rolling. Thus, it is necessary to further understand the initial damage effect on dynamic behaviors of RCC under impact loadings. In this chapter, two types of RCC were constructed with the relative compaction degree of 99.04 and 97.15%, and the initial damage of RCC specimens was quantified by NDT methods before SHPB tests. Based on the experimental results, the initial damage effect on the dynamic compressive behaviors of RCC is investigated in detail, as well as the tight correlation between them.

© Science Press 2023 S. Zhang et al., Dynamic Mechanical Behaviors and Constitutive Model of Roller Compacted Concrete, Hydroscience and Engineering, https://doi.org/10.1007/978-981-19-8987-2_4

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4.2 Specimen Preparation and Damage Quantification 4.2.1 Specimen Preparation For a properly proportioned RCC mixture, the hardened performance primarily relies on the compaction quality, characterized by the relative compaction degree according to the test code for hydraulic RCC [1]. In this experiment, the relative compaction degrees of RCC with 8 vibration compaction passes and 12 vibration compaction passes were tested as 97.15% (regarded as the damaged RCC) and 99.04% (regarded as the standard RCC) respectively, even though they both met the control criterion for internal concrete (i.e., the relative compaction degree of 97%). It is noted that although the prepared RCC with a relative compaction degree of 97.15% is defined as the initial damaged RCC here, it still meets the demands of construction control for internal concrete in dam engineering. A set of drilling cores with the diameters of 75 and 100 mm were taken after curing for 90 days in a moist environment. The drilling cores were cut into several cylinders. The dimensions of cylindrical specimens for SHPB tests were diameter × length = 75 mm × 37.5 mm, and those for quasi-static tests were diameter × length = 100 mm × 200 mm. To minimize the testing errors from the specimens, the surfaces of the cylindrical specimens should be ground smoothly. It is worthy of being pointed out that the influence of drilling and grinding on testing results is limited as the diameters of drilling cores are larger enough (i.e. 75 and 100 mm in diameter). Moreover, we monitored the damage of RCC before and after drilling with UPV, so that some drilling cores with too much attenuation of UPV had been abandoned (Fig. 4.1). Vibrating compaction is a marked feature distinguishing roller compacted concrete (RCC) from conventional concrete. The performance of RCC mainly depends on the compaction quality. A vibratory roller was used in the laboratory experiments to simulate the practical construction of RCC. Evaluation of the interlayer bonding quality is the key way for controlling the RCC construction. It is obvious from Fig. 3.1 that there are several honeycombed voids at the interlayers of initial damaged RCC, while the interlayers of standard RCC seem relatively uniform and compacted. This indicates that with the increase of compaction passes, the coarse and fine aggregates of upper lifts are gradually embedded into the lower lifts, so that the interlayers become a dense and stable union. However, due to the numerous construction control parameters and the processing methods of interlayers during the large-scale successive construction of RCC, the significant uncertainty of construction quality and initial defects will exist in the RCC inevitably.

4.2 Specimen Preparation and Damage Quantification

69

Fig. 4.1 Interlayer bonding quality under different compaction passes

4.2.2 Quantification of the Initial Damage The damage from construction in RCC inherently accompanies amount of weak bonding interlayers and meso-pores. Characterization of damage from construction is important to indicate the initial degradation of RCC performance, which requires quantitative measurements of NDT variables related to the damage. The ultrasonic waves can penetrate deeply into the RCC matrix and monitor the internal deterioration and meso–cracks. According to the RCC construction specification in China [2], the density (or relative compaction degrees) is usually identified as the compaction quality index of RCC matrix. Moreover, the UPV and elasticity modulus are considered to quantitatively measure the damage degree of concrete, and the elasticity modulus is more sensitive to the concrete damage than UPV and density [3]. Therefore, the initial damage of RCC from construction can be quantified with density, UPV and elasticity modulus in this study. The elasticity modulus of RCC can be estimated with density and UPV, shown as below [4]: E d = cl2 ρ(1 + υ)(1 − 2υ)/(1 − υ)

(4.1)

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4 Consturction-Induced Damage Effect on Dynamic Compressive …

where E d is the elasticity modulus; cl is the propagation speed of longitudinal waves and υ represents the dynamic Poisson’s ratio. Since the dynamic Poisson’s ratio is assumed to be constant (i.e., 0.20), a decrease in elasticity modulus (E d ) is associated to the decrease of UPV (cl ) and density (ρ) for specimens with higher damage. A set of density and ultrasonic tests were conducted on RCC specimens before SHPB tests. The procedures of the density and UPV measurements in this study were described in detail as below: (1) Density measurement: The mass measurement was conducted at a natural moisture and the volume was averaged from a total of threetime dimension measurements for each cylindrical specimen, including diameter and length; (2) Longitudinal UPV measurements: Due to the significant wave attenuation in damaged RCC, it is hard to exactly measure the UPV in the high frequency (higher than 500 kHz). To perform UPV measurement more accurately, a total of 4 acquired signals were stacked for averaging at a frequency range of 200 to 500 kHz [3]. Then, the UPV for each specimen was averaged from the propagation velocity of these 4 signals. To investigate the initial damage effect on dynamic mechanical properties of RCC, we quantify the damage levels of specimens by means of three damage measurements (i.e., density, UPV and elasticity modulus). The statistical properties of damage measurements for RCC have been listed in Table 4.1, comparing the damage levels of standard RCC with those of initial damaged RCC. According to Table 4.1, standard RCC with the higher compaction quality has higher density and UPV, representing the more compacted and more homogeneous meso–structures. Similarly, a higher elasticity modulus can be acquired for standard RCC. On the other hand, the elasticity modulus for damaged RCC shows more significant variability than that for standard RCC, as well as the density and UPV. Therefore, construction quality control of RCC is the key to reduce the material variability. In this study, the elasticity modulus has been estimated by the density and UPV measurements to quantify the initial damage of RCC specimens. In continuum damage mechanics, material damage is defined as a relative change in the elasticity modulus. Thus, the initial damage parameter can be denoted as D 0E = 1 − E d0 /E d∗

(4.2)

Table 4.1 Statistical properties of damage measurements for RCC Specimen type

Compaction pass

Standard RCC

N = 12

Statistical indexes

Density (kg/m3)

UPV (m/s)

Elasticity modulus (GPa)

Mean

2383.44

3884.44

32.42

23.82

147.27

2.49

2326.84

3067.89

19.83

39.95

238.32

3.04

Standard deviation Damaged RCC

N=8

Mean Standard deviation

4.3 Initial Damage Effect on the Dynamic Behaviors of RCC

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where E d0 is the initial elasticity modulus for damaged RCC; the standard elasticity modulus (E d∗ ) is averaged from standard RCC specimens, set as 32.42 GPa. Therefore, the average initial damage parameter for damaged RCC is about 0.39.

4.3 Initial Damage Effect on the Dynamic Behaviors of RCC 4.3.1 Mechanical Tests Quasi-static uniaxial compressive tests were conducted by the computer-controlled electro-hydraulic servo loading test machine (PLS-500 T) at the 1 mm/min loading rate. High rate impact tests were carried out on SHPB system. Specimens prepared for mechanical tests have been listed in Table 4.2. It is obvious from Table 4.2 that specimens for SHPB test have been classified into two groups, identified as the standard RCC with 12 compaction passes and damaged RCC with 8 compaction passes. Moreover, specimens for quasi-static tests also were standard RCC to characterize the general quasi-static mechanical properties. By varying the gas pressure of gas gun (0.4−0.8 MPa), different loading rates can be acquired. To avoid oscillation of stress–strain curves, half-sine stress waveform was chosen as the most ideal loading method to carry out SHPB tests on quasibrittle materials [5]. In addition, the incident waves also must have a certain rising time to avoid the destruction of the specimen before the stress balance between the two surfaces of specimens. Vaseline was daubed on the two specimen/bar contact surfaces uniformly to reduce friction. The strain gauges pasted on the incident and transmitted bars could record the incident, reflected and transmitted pulses during the whole impact process. Table 4.2 Specimens size and shape for mechanical tests Test type

Dimensions Compaction L/D Cross-sectional passes shape (D ×L: mm)

Number Gas Comment pressure (MPa)

Quasi-static 100 × 200 tests

12

2.0

Cylindrical

3

–

Standard RCC

75 × 37.5

12

0.5

Cylindrical

4

0.40

4

0.50

Standard RCC

4

0.60

4

0.80

6

0.40

6

0.50

6

0.60

6

0.80

SHPB tests

75 × 37.5

8

0.5

Cylindrical

Initial damaged RCC

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4.3.2 Initial Damage Effect on Stress–Strain Curves The stress–strain curves of standard RCC specimens are compared with those of damaged RCC specimens, as shown in Fig. 4.2. The UPVs for these three damaged RCC specimens in Fig. 4.2 are very close to 3200 m/s, as well as the UPVs of 3900 m/s for standard RCC specimens. Therefore, the initial damage gap between standard RCC and damage RCC is approximately constant in Fig. 4.2. However, as shown in Fig. 4.2, at a similar damage gap, the attenuation of dynamic compressive strength for damaged RCC becomes more significant when the strain rate gets higher, compared with the dynamic compressive strength of standard RCC. For example, the attenuation of dynamic compressive strength is relatively low (i.e., 1.90 MPa) at the strain rate of 40/s, while a more prominent attenuation of dynamic compressive strength occurs at the strain rate of 110/s, reaching 15.59 MPa (or 33.05% dynamic compressive strength for standard RCC). This suggests that the dynamic compressive strength is more sensitive to the initial damage (or internal defects) at higher strain rates, which can be generized as the strain-rate sensitivity of damage-induced strength attenuation. Thus, this phenomenon emphasizes the importance of damage evaluation for concrete, and the internal defects should be minimized for optimal material design and construction, especially in the conditions of blast-impact loads. Moreover, it is obvious that the failure patterns of damaged RCC show a great strain-rate sensitivity. As shown in Fig. 4.2, the fracture status of the specimen changes from large blocks to fine fragments when the strain rate increases from 40.67/s to 112.51/s. It also proves that cracks under impact loading usually propagate through the coarse aggregates, rather than the weakest aggregate-mortar interfaces in concrete.

Fig. 4.2 Initial damage effect on dynamic stress–strain curves of RCC at a similar damage gap (the UPVs of 3900 m/s for standard RCC specimens and the UPVs of 3200 m/s for damaged RCC specimens)

4.3 Initial Damage Effect on the Dynamic Behaviors of RCC

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4.3.3 Initial Damage Effect on Dynamic Mechanical Properties Under quasi-static loadings, the 90-day uniaxial compressive strength of standard RCC was obtained as 11.97 MPa, and the ultimate strain corresponding to peak strength was 0.67%, averaging from three specimens. Moreover, the toughness related to the ductility and strength can be determined by the area under the full stress–strain curve, which is the amount of absorbed energy per unit volume. In this study, the toughness for standard RCC was tested as 0.13 MJ/m3 . Based on the testing results of quasi-static loading tests, the dynamic mechanical properties of RCC can be written into the form of dynamic increase factor (DIF). So far, the polynomial fitting method has been widely used to illustrate the empirical relationship between the strain rates and the DIFs. To further investigate the initial damage effect, Fig. 4.3 compares the dynamic compressive behaviors of standard RCC with those of damaged RCC at different strain rates, in terms of dynamic compressive strength, ultimate strain and toughness. It is obvious that the initial damage of RCC has a significant effect on its dynamic compressive strength and toughness. In this study, the polynomial fitting method has been used to illustrate the empirical relationship between the DIFs and the strain rates, in term of the base-10 logarithm (donated as lg˙ε). The fitting dynamic compressive strength for standard RCC is more sensitive to the strain rate than that for damage RCC, as well as the fitting toughness. The attenuation of dynamic compressive strength and toughness is also dependent on the strain rate, and more remarkable attenuation will occur at a higher strain rate. More specifically, the attenuation of dynamic compressive strength for damaged RCC reaches 28.17% at the strain rate of 40/s, while it gets to be 35.24% at the strain rate of 120/s. However, different from the dynamic compressive strength and toughness, the ultimate strain seems to be insensitive to the initial damage. This phenomenon is more intuitively in the form of stress–strain curve, as shown in Fig. 4.2. Moreover, the dynamic mechanical properties of damaged RCC shows more significant discreteness, corresponding to higher variability of damage measurements in Table 4.1.

4.3.4 Statistical Characteristics of Dynamic Compressive Behaviors It has been confirmed that the due to the interlayers and fast construction, most dynamic mechanical properties of damaged RCC exhibit more significant discreteness than those of standard RCC, which emphasize the usage of statistical analysis. However, different from the statistical analysis of static mechanical properties, the consideration of strain-rate effect is the primary and difficult issue in the dynamic

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Fig. 4.3 Initial damage effect on dynamic increase factors of RCC at different strain rates: a dynamic compressive strength; b ultimate strain; c toughness

condition. As known, a higher strain rate will cause the enhancement of dynamic mechanical properties, accompanied with more significant variability [6, 7]. The variability of dynamic mechanical properties, such as dynamic compressive strength and toughness, usually relates to the initial damage of RCC. However, based on the analysis in Subsection 4.3.3, the attenuation of ultimate strain due to initial damage has been confirmed to be indistinct, that means there is no relationship between initial damage and ultimate strain. Figure 4.2 shows this phenomenon more graphically. Thus, this study focuses on the initial damage effect on dynamic compressive strength and toughness. A statistical model is proposed in this chapter to describe the strain-rate dependent strength and toughness for damaged RCC, accompanied with the variability: ⎧ ⎪ ⎨

yi = μ0 − ∆μ + εi | | μ0 = A(lg˙ε )2 + B(lg˙ε) + C μs ⎪ ⎩ εi = εi /∆μ ∼ D

(4.3)

where the real dynamic mechanical property of damaged RCC (yi ) is determined by the mean value (μ0 ) of standard RCC at a certain strain rate, the gap between the mean values of standard and damaged RCC (∆μ), and the residual error (εi ). Moreover,

4.3 Initial Damage Effect on the Dynamic Behaviors of RCC

75

the standardized residual error (εi ) obeys any theoretical or empirical distribution (D). So that, the average dynamic mechanical properties (μ0 ) for standard RCC can be usually quantified by the product of empirical DIF equations and corresponding static mechanical properties (μs ). The damage gap (∆μ) is the key of statistical model, which describe the strain-rate dependent attenuation of dynamic mechanical properties. So that, the residual error enhancement of dynamic mechanical properties due to strain-rate effect for damaged RCC (i.e., more significant attenuation at a higher strain rate) can be eliminated with εi /∆μ. It has been confirmed that the damage gap for dynamic compressive strength (or toughness) is strain-rate sensitive and increases with the increasing strain rates. According to the Fig. 4.3, the damage gaps of dynamic compressive strength and toughness can be respectively illustrated as | | ∆μσ = 4.35(lg˙ε )2 − 13.34(lg˙ε ) + 10.46 × 11.97

(4.4)

| | ∆μT = 27.17(lg˙ε)2 − 92.66(lg˙ε ) + 79.23 × 0.13

(4.5)

where ∆μσ is the damage gap of dynamic compressive strength; ∆μT is the damage gap of toughness; and ε˙ denotes the strain rate. Then, the standardized residual errors for dynamic compressive strength and toughness can be obtained. | ( ( ) ) | x −c b b x − c b−1 exp − P(x; a, b, c) = x ≥c a a a

(4.6)

where a is scale parameter; b is shape parameter; c is position parameter; and P(x; a, b, c) denotes the probability density when the random variable X equals to x. To further investigate the statistical characteristics of dynamic mechanical properties for damaged RCC, Fig. 4.4 shows the histograms of their standardized residual errors. According to Table 4.3, the residual errors of dynamic compressive strength and toughness cannot strictly obey the Normal distribution, whose skewness and kurtosis are equal to 0 and 3 respectively. The three-parameter Weibull distribution, as illustrated in Eq. (4.6), has been used to describe their statistical characteristics. The distribution parameters are estimated with maximum likelihood method, as shown in Table 4.3. In addition, Kolmogorov–Smirnov test (K-S test) has been used to judge that whether the experimental data can be drawn from the proposed three-parameter Weibull distribution. Using the standardized residual strength as an example, it is obvious from Fig. 4.4a that the fitted three-parameter Weibull distribution can well describe the variability of dynamic compressive strength for damaged RCC, compared with the histogram of standardized residual strength. In K-S test, the p-value and test statistic can be used to estimate whether the null hypothesis can be rejected. Moreover, a larger p-value or a smaller test statistic means that it is more valid to accept the null hypothesis. In this study, the critical value for K-S test equals 0.27 at the 5% significance level (α =

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4 Consturction-Induced Damage Effect on Dynamic Compressive …

Fig. 4.4 Distribution characteristics of dynamic mechanical properties for damaged RCC: a dynamic compressive strength; b toughness

Table 4.3 Distribution parameters of standardized residual errors and probability distribution tests Dynamic mechanical properties

Statistical characteristics

Distribution parameters K−S testa

Skewness

Kurtosis

a

b

c

p-value

K−S statistics

Dynamic compressive strength

−0.08

2.50

1.17

3.51

−1.06

0.97

0.09

Toughness

−0.43

2.93

3.38

6.78

−3.14

0.99

0.08

a

The critical value for K-S test equal to 0.27 at the 5% significance level

0.05). Therefore, the null hypothesis, explained as the standardized residual strength obeys the three-parameter Weibull distribution, can be accepted since p-value = 0.97 > 0.05 and test statistic = 0.09 < 0.23 for K-S test. The statistical model is also applied to analyze the distribution characteristic of toughness. In general, the fitted three-parameter Weibull distribution can well describe the statistical characteristics of dynamic compressive strength and toughness for damaged RCC.

4.4 Assessment to the Initial Damage Effect on the Dynamic Behaviors 4.4.1 Correlation Between Initial Damage and Dynamic Behaviors The correlation between initial damage and dynamic compressive behaviors are further investigated in this subsection. A bivariate probability distribution function (PDF) is introduced in this chapter using kernel density estimation, which is a

4.4 Assessment to the Initial Damage Effect on the Dynamic Behaviors

77

Fig. 4.5 Influence of initial damage on dynamic compressive behaviors: a dynamic compressive strength; b toughness

nonparametric probabilistic method with great flexibility and without any deviations from the experimental data [8]. The correlation between initial damage and dynamic compressive behaviors has been illustrated with the contour lines of empirical probability density against the experimental data in Fig. 4.5. It is obvious from Fig. 4.5 that both dynamic compressive strength and toughness are sensitive to the initial damage and get lower as the initial damage increases. More intensive contour lines show more notable relationship. For example, the initial damage parameter is mainly concentrated in the range of 0.3 to 0.5 with a decreasing standardized residual strength from 0.4 to –0.4. In general, the Pearson correlation (r), i.e., linear correlation, is the most popular method to measure the dependence structure of initial damage and dynamic compressive behaviors. However, the Pearson correlation is not invariant under nonlinear transformations. In this chapter, the proposed statistical model is a typical nonlinear transformation, which has been used to turn dynamic compressive behaviors into their standardized residual forms. Here, their dependence structure is improved with the Kendall rank correlation (τ ), which is a nonlinear method to measure the variability concordance of initial damage and dynamic compressive behaviors [9]. The testing results have been listed in Table 4.4, where both dynamic compressive strength and toughness show negative correlation with the initial damage. Moreover, |r | (or |τ |) for dynamic compressive strength is higher than that of toughness. This suggests a higher level of correlation between initial damage and dynamic compressive strength. Table 4.4 also lists the coefficients α1 and α2 of the straight lines ε = α1 + α2 D 0E from least-square curve-fitting based on the non-dimensional and strain-rate independent relationship between the dynamic compressive behaviors and the initial damage parameters. The determination coefficient R2 indicates the validation of the fitting curves. It can be noted that the coefficient α2 for toughness is lower than that of dynamic compressive strength, indicating a higher attenuation rate with the increasing initial damage for toughness.

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4 Consturction-Induced Damage Effect on Dynamic Compressive …

Table 4.4 Correlation and curve-fitting parameters between initial damage and dynamic mechanical properties Correlation coefficients

Curve-fitting parameters

Pearson (r)

Kendall (τ )

α1

α2

Determination coefficient (R2 )

Dynamic compressive strength

−0.86

−0.67

1.18

−3.06

0.73

Toughness

−0.80

−0.56

1.82

−4.64

0.62

Dynamic mechanical properties

4.4.2 Evaluation on the Initial Damage from Improper Construction The structural safety assessment for concrete structures becomes more important for engineers and designers, as more attenuation of dynamic compressive strength occurs at higher strain rates. Therefore, the structural safety assessment without considering the damage state, feeding the designed compressive strength and empirical equation of strain-rate effect for standard RCC into the structural calculation directly, will result in a serious overestimation of structural safety. In this study, the relationship between initial damage parameter (D 0E ) and standardized dynamic compressive strength (εσ ) has been investigated in Fig. 4.5a. Then, the dynamic compressive strength of RCC specimens with different damage levels can be generalized with the Eq. (4.7), which has been illustrated graphically in Fig. 4.6a. As shown in Fig. 4.6a, the damage levels of RCC materials have been classified into three grades, including slight damage (D 0E ∈ (0, 0.2)), moderate damage (D 0E ∈ [0.2, 0.4)) and severe damage (D 0E ∈ [0.4, 1.0)). Referring to the damage graph, the damage grade of RCC under impact loadings can be easily determined after a set of SHPB tests. For example, the average dynamic compressive strength of standard RCC, estimated from the empirical formula in Fig. 4.3a, indicates that the standard RCC belongs to the slight damage grade. Therefore, it is a convenient method to evaluate the initial damage of RCC based on the damage graph. ⎧ ⎪ ⎪ ⎨

yσ =| μσ 0 − ∆μσ + εσ = μσ 0 − (1 − ε|σ )∆μσ μσ 0 = 12.23(lg˙ ε )2 − 37.82(lg˙ε ) + 30.15| × 11.97 | ⎪ ∆μσ = 4.35(lg˙ε )2 − 13.34(lg˙ε ) + 10.46 × 11.97 ⎪ ⎩ εσ = 1.18 − 3.06D 0E

(4.7)

However, the damage graph is a deterministic method without considering the joint distribution characteristics of initial damage and dynamic compressive strength. In this study, the kernel density estimation has been introduced to investigate the correlation between initial damage and dynamic compressive strength in Fig. 4.5a, suggesting a non-absolute linear relationship. Moreover, the determination coefficient (R2 ) of the linear regression equals to 0.73. Therefore, the usage of linear

4.4 Assessment to the Initial Damage Effect on the Dynamic Behaviors

79

Fig. 4.6 Initial damage evaluation of RCC: a based on the damage graph; b based on the strength assurance rate

regression to describe the correlation between initial damage and dynamic compressive strength will result in an incomplete initial damage evaluation of RCC. For example, the average dynamic compressive strength of damaged RCC in Fig. 4.6a is tightly close to boundary between the moderate damage and the severe damage, which necessitates a further damage evaluation. The designed compressive strength of RCC in this study is 10 MPa for the standard 150 mm cubic specimen, usually used in the construction of RCC dams. Neville [10] considered the conversion factor of concrete compressive strength as Eq. (4.8), which is a function of the volume V, lateral dimension d, and height h. Then, the compressive strength of specimens with various shapes and sizes (fc) can be converted to the strength of 150 mm cubic specimen. Therefore, the designed compressive strength for 100 mm × 200 mm (diameter × length) cylindrical specimens can be taken as 8.37 MPa for the standard 150 mm cubic specimen, since the corresponding conversion factor is 0.837 in this study. Based on the fitting results of the average dynamic compressive strength of standard RCC (μσ 0 = 11.47 MPa) and the damage gap of dynamic compressive strength (∆μσ = 3.23 MPa) at the strain rate of 40/s, the designed value of standardized residual strength (εσ r ) is taken as 0.04. f c / f cu15 = 0.56 + 0.697/(V /6hd + h/d)

(4.8)

where the unit of dimension is inch, and 1-inch equals to 25.4-mm. | ( ) | x −c b P(x; a, b, c) = 1 − exp − x ≥c a

(4.9)

According to the analysis in Subsection 3.3.4, the three-parameter Weibull distribution has been verified to be able to describe the variability of dynamic compressive strength well, whose cumulative probability distribution (P) can be describe as Eq. (4.9). Here, we define the strength assurance rate (1-P) as the probability that the standardized residual strength is higher than the designed value (εσ > ε σ r ). In

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4 Consturction-Induced Damage Effect on Dynamic Compressive …

this study, the strength assurance rate sof damaged RCC is 44.24%, as shown in Fig. 4.6b, and it is more reasonable to classify the damaged RCC into the severe damage grade when 1-P < 50%. In a word, the method of damage graph accompanied with the strength assurance rate shows a practical value for guiding the material or engineering design.

4.5 Summary and Conclusions RCC material usually remains in the initial damage state inevitably due to the improper rolling compaction control, which is usually ignored in original design. Focusing on the construction-induced initial damage effect, three damage measurements (i.e., density, UPV and elasticity modulus) were conducted to quantify the damage states of RCC specimens before SHPB tests. Then, the initial damage effect on dynamic compressive behaviors of RCC under impact loadings was investigated and illustrated. The main contributions and findings can be drawn as follows: (1) The rolling compaction control has a significant influence on the dynamic compressive strength and toughness, while the ultimate strain seems to be insensitive to the construction-induced initial damage. Moreover, the attenuation of dynamic behaviors from initial damage is strain-rate sensitive under impact loadings, and the variability of dynamic compressive behaviors can be described by a proposed statistical model. After eliminating the strain-rate effect, the distributions of dynamic compressive behaviors can be illustrated with the three-parameter Weibull distribution. (2) The correlation between initial damage and dynamic compressive behaviors is revealed in this study, which can be described by the linear regression between initial damage parameter and standardized residual items of dynamic compressive behaviors. Then, the dynamic compressive strength at high strain rates can be predicted more accurately after the initial damage state is detected by various damage measurements. (3) A damage graph accompanied with the strength assurance rate is proposed to determine the material damage grade of RCC. Based on the damage graph, the damage state of RCC can be classified into three grades preliminarily. To gain a more reasonable damage grade, the strength assurance rate is used to consider the high discreteness of dynamic compressive strength induced by the construction technique.

References

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References 1. National Energy Administration of People’s Republic of China (2009) Test code for hydraulic roller compacted concrete (DL/T5433-2009) [S]. China Electric Power Press, Beijing 2. National Energy Administration of People’s Republic of China (2009) Construction specifications for hydraulic RCC (DL/T 5112–2009) [S]. Water Power Press, Beijing (in Chinese) 3. Yim HJ, Kim JH, Park S-J et al (2012) Characterization of thermally damaged concrete using a nonlinear ultrasonic method [J]. Cem Concr Res 42(11):1438–1446 4. Antonaci P, Bruno CLE, Gliozzi AS et al (2010) Monitoring evolution of compressive damage in concrete with linear and nonlinear ultrasonic methods [J]. Cem Concr Res 40(7):1106–1113 5. Chen X, Ge L, Zhou J et al (2016) Experimental study on split Hopkinson pressure bar pulseshaping techniques for concrete [J]. J Mater Civ Eng 28(5):04015196 6. Huang YJ, Yang ZJ, Chen XW et al (2016) Monte carlo simulations of meso-scale ssdynamic compressive behavior of concrete based on X-ray computed tomography images [J]. Int J Impact Eng 97:102–115 7. Wang X-H, Zhang S-R, Wang C et al (2018) Experimental investigation of the size effect of layered roller compacted concrete (RCC) under high-strain-rate loading [J]. Constr Build Mater 165:45–57 8. Wang Y, Hao Y, Hao H et al (2016) An efficient method to derive statistical mechanical properties of concrete reinforced with spiral-shaped steel fibres in dynamic tension [J]. Constr Build Mater 124:732–745 9. Li D-Q, Tang X-S, Phoon K-K et al (2013) Bivariate simulation using copula and its application to probabilistic pile settlement analysis [J]. Int J Numer Anal Meth Geomech 37(6):597–617 10. Sabnis GM, Mirza SM (1979) Size effect in model concretes [J]. J Struct Div 105(6):1007–1020

Chapter 5

Size-Dependence of Dynamic Behaviors for RCC Under High-Strain-Rate Loadings

5.1 Introduction A series of experiments [1–3] and theoretical analysis [4, 5] indicate that the wellaccepted concrete size effect laws for static loads are incapable to reflect the size effect on dynamic behaviors of RCC. The dynamic compressive strength usually is sensitive to both strain rate and specimen size, and larger specimens display a much more significant strain-rate effect. So far, much effort has been devoted to explain the relationship among the strength, strain rate and specimen size in the context of the complex meso--structural hierarchy and finiteness of the crack propagation speed [2, 6]. Although the material strength enhancement under impact loading has been proven to be size-dependent, the size effect on other dynamic material properties at different strain rates remains unclear. Moreover, the size effect law for concrete-like materials is not fully understood under impact loading, resulting in an urgent need to extend size effect law to the full range of strain rates, applicable to both static and dynamic loadings. This chapter focuses on the dynamic size effect of the RCC under high-strainrate loading. The actual construction technique was replicated in the laboratory, and different dimensional RCC specimens were prepared for SHPB tests. Taking the natural material heterogeneity from rolling construction technique into consideration, which may lead to high scatters of dynamic behaviors for RCC, the usage of statistical methods is emphasized here to estimate the size effect on the strength of concrete and on the damage/fracture process in general. Then, the exposed size effect on the dynamic mechanical properties is further analyzed and discussed to gain a deeper understanding of the physical mechanisms, and a strain-rate-dependent size effect law is developed to model the dynamic behavior of geometrically similar specimens.

© Science Press 2023 S. Zhang et al., Dynamic Mechanical Behaviors and Constitutive Model of Roller Compacted Concrete, Hydroscience and Engineering, https://doi.org/10.1007/978-981-19-8987-2_5

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5.2 Dynamic Size Effect on Experimental Results 5.2.1 Schematic Design The RCC prepared in Chap. 2 was used here whose compaction degree was 97.15% meeting the demands of construction control for internal concrete. Regarding the dynamic mechanical properties, various cross-sectional strikers or thin copper pulse shapers were used in the tests to achieve the half-sine stress wave loading. By varying the gas pressure of the gas gun (1.5–8 bar) and diameters of the specimens (D50, D75, and D100mm), we caused specimens with different sizes to undergo various strain rates. The schematic design of the experimental tests is illustrated as Table 5.1. The specimens prepared for the SHPB tests can be classified into three categories based on the specimen diameter. For each category with identical diameter, the specimens can be divided into four groups based on the gas pressure in the SHPB tests. We prepared 10 specimens for each group (i.e., 120 specimens for the SHPB tests in total) at first. However, some specimens for the SHPB tests failed due to incorrect operation or other reasons, and only 101 specimens successfully satisfied the onedimensional wave propagation assumption and homogeneity assumption of stress and strain. The 101 successfully-tested specimens were not uniformly distributed. The specific number for each group has been listed in Table 5.1. Table 5.1 Specimen size and shape for the dynamic loading tests Test type

Specimen dimensions (D × L: mm)

L/D

Group

Quasi-static tests

100 × 200

2.0

A

SHPB tests

50 × 25

0.5

75 × 37.5

100 × 50

a

0.5

0.5

Number

Gas pressure (MPa)

Average strain ratea (/s)

5

–

8.30×10-5

D50–A

8

0.30

70.15

D50–B

8

0.37

88.54

D50–C

8

0.45

114.45

D50–D

8

0.55

143.78

D75–A

7

0.40

46.68

D75–B

9

0.50

75.84

D75–C

8

0.60

90.47

D75–D

8

0.80

111.20

D100–A

8

0.15

33.11

D100–B

10

0.22

47.10

D100–C

10

0.30

59.83

D100–D

9

0.37

74.08

The average strain rate for each group is the mean value of the representative strain rates for the specimens in this group, not the real strain rate for each specimen

5.2 Dynamic Size Effect on Experimental Results

85

5.2.2 Failure Patterns The RCC specimens with various sizes were prepared and tested under different loading rates by varying the gas gun pressure. The failure patterns at different strain rates, as was concluded in Ref. [7], are notably similar in specimens of different size. Figure 5.1 shows the schematic failure patterns of the RCC at various strain rates, which can be summarized as follows. Visible cracks form in the ITZs and propagate along the interfaces under static loading, as shown in Fig. 5.1a. At the critical strain rate in Fig. 5.1b, the propagating path becomes straighter, and the fracture surface is less ragged. At a high strain rate in Fig. 5.1c, the cracks propagate along several direct paths with more fractured aggregates and the specimens ultimately are crushed into several fragments. With increasing strain rate, the specimens can be further crushed into finer granularities, which dissipate more energy, as shown in Fig. 5.1d. To sum up, stress increases so rapidly at higher strain rates that the cracks do not have sufficient time to propagate along the path of least resistance and propagate in aggregates instead, resulting in smaller fragments. Notably, the critical strain rate can be reduced by increasing the specimen size. Therefore, the failure pattern or other dynamic mechanical properties under impact loading may be significantly affected by both strain rate and specimen size. For example, the fracture status of the samples changes from fine fragments to large blocks when the specimen diameter is decreased from 100 to 50 mm at a similar strain rate of approximately 70/s in the SHPB tests, as shown in Fig. 5.1e.

Fig. 5.1 Failure patterns of the RCC specimens: a–d schematic failure patterns at various strain rates; e failure patterns of specimens with various diameters

86

5 Size-Dependence of Dynamic Behaviors for RCC Under …

5.2.3 Stress-Strain Curves Considering the significant discreteness of the RCC from the construction technique, the stress-strain responses of the RCC material for each group are represented by the means and standard deviations. For the specimens with identical sizes, the stressstrain responses of four groups characterized by different average strain rates are compared in Fig. 5.2a–c to illustrate the strain-rate effect along with significant discreteness. The average stress-strain responses of the RCC specimens with different sizes share common characteristics: they change significantly with increasing strain rates. The peak stress increases at a higher strain rate, which we refer to as strain-rate dependence resulting from the comprehensive effects of the inertial effect, crack propagation effect and viscosity effect. The slopes of the ascending and descending parts of the stress-strain curves tend to be steeper when the strain rate increases. The average stress–strain curves for specimens of different dimensions at a similar strain rate (approximately 70/s) are compared in Fig. 5.2d. A notable change in these curves demonstrates the dynamic size effect. The stress-strain curves of the larger specimens are much steeper than those of smaller specimens, i.e., the peak stress increases when the specimen size increases.

Fig. 5.2 Average stress-strain curves of the RCC specimens at various strain rates: a–c for specimens D50, D75, and D100; d for specimens of various dimensions at similar strain rates (approximately 70/s)

5.2 Dynamic Size Effect on Experimental Results

87

5.2.4 Dynamic Increase Factor for Compressive Stress Using the D50 specimens as an example, Table 5.2 summarizes the testing results of dynamic compressive strength and corresponding strain rate for the D50 specimens, which have been classified into four groups according to the gas gun pressure as shown in Table 5.1. It is obvious from Table 5.2 that the mean values of dynamic compressive strength for each RCC group increases with the increasing average strain rate. The same rules have been seen in specimens with diameters of 75 and 100 mm as well. Various concrete-like materials have been studied using laboratory tests to quantify the strain-rate effects, and the polynomial fitting method has been widely used to illustrate the empirical relationship between the strain rates and the DIFs [8–13]. Figure 5.3 compares the obtained DIFs with the existing empirical models from other studies. In general, the test results are distributed among these empirical models. Moreover, the DIF increases with increasing strain rate, and the DIFs of the RCC appear more sensitive to the strain rate than the DIFs of normal concrete. Figure 5.3 also shows that the DIFs tend to be more sensitive to the strain rate for larger specimens. The test results are consistent with the observations of previous studies [14] that the DIFs obtained from the impacting tests are directly composed of contributions from the inherent strain-rate effect and structural effect. With increasing strain rate, the difference in DIFs among specimens of various dimensions tends to be more significant, which indicates the strain rate sensitivity of the size effect. Table 5.2 Dynamic compressive strength of D50 RCC specimens (D × L: 50 × 25 mm) Group Indexes

Testing results Test 1

D50-A Strain rate (/s) Strength (MPa) D50-B Strain rate (/s) Strength (MPa) D50-C Strain rate (/s) Strength (MPa) D50-D Strain rate (/s) Strength (MPa)

Test 2

Test 3

Test 4

Test 5

Test 6

Test 7

Test 8

Average

64.27

69.26

70.58

69.28

70.95

71.51

74.31

71.01

70.15

12.32

8.59

9.12

10.44

8.57

10.74

9.94

10.92

10.08

80.75

98.75

89.75

94.25

92.00

81.68

86.84

84.26

88.54

12.17

13.36

10.74

12.05

11.51

13.96

12.36

10.21

12.05

111.43 106.10 105.29 123.35 106.36 124.15 123.23 115.72 114.45 15.30

16.64

18.65

17.11

17.83

15.66

12.59

17.22

16.38

139.45 140.46 143.38 145.75 145.44 142.09 144.68 148.95 143.78 17.23

23.33

22.32

21.19

22.09

19.17

25.43

27.47

22.28

88

5 Size-Dependence of Dynamic Behaviors for RCC Under …

Fig. 5.3 Comparison of the DIFs from the laboratory tests with that of existing empirical models

5.3 Dynamic Size Effect on Compressive Behaviors of RCC 5.3.1 Definitions of Dynamic Mechanical Properties Figure 5.4 graphically illustrates the definitions of key mechanical properties in this study: the peak strength, ultimate strain, maximum strain and toughness. The peak value of the stress time history is considered the material strength. The ultimate strain is taken as the strain at peak stress, and the maximum strain of the stress-strain curve is the strain at the end of the softening stage. Moreover, the toughness related to the ductility and strength can be expressed as the specific energy absorption, which is the capacity to absorb the energy of the stress wave for the RCC per unit volume. It is also observed that the material strength occurs nearly at the peak strain rate duration and that the breakage process of the specimen maintains approximately constant strain rate loading since the variation in strain rate with time near the failure point is minimal. Because the strain rates obtained in the SHPB tests are not constant, the representative strain rate for each specimen can be defined in different ways. However, the strain rate at failure cannot be used to characterize the strain rate during the entire loading process [13], and the mean strain rate, defined as the maximum strain divided by the entire time duration, is notably lower than the instantaneous strain rate at specimen failure [15]. Therefore, the ultimate strain divided by the time duration to reach the peak stress is used as the representative strain rate in this study. Moreover, the mean value of the representative strain rates for specimens in each group is calculated to present the loading condition of this group, denoted as the average strain rate.

5.3 Dynamic Size Effect on Compressive Behaviors of RCC

89

Fig. 5.4 Diagrammatic sketch of the definitions of characteristic values in the stress-strain curve

5.3.2 Size Effect on Various Dynamic Mechanical Properties Figure 5.5 compares the experimental results of the dynamic compressive properties, which are represented by the mean values and standard deviations, for specimens with various sizes and strain rates. For specimens with 50 mm in diameter (D50mm), when the strain rate increases from 70.15 to 143.78/s, the average peak strength continuously increases from 10.08 to 22.28 MPa. The ultimate strain and maximum strain continually increase from 0.0070 to 0.0113 and from 0.0205 to 0.0278, respectively. In addition, the average toughness slightly increases by as much as 0.19 MJ/m3. The same trend is observed in the D75mm specimens. Immediately after the average strain rate increases from 46.68 to 111.20/s, the average peak strength increases from 10.75 to 36.71 MPa, as much as 241%. In addition, the average ultimate strain consistently increases from 0.0065 to 0.0131. Because of the increasing ductility and strength, the average toughness increases from 0.11 to 0.98 MJ/m3 . The average peak strength of the D100mm specimens increases from 15.31 to 30.44 MPa when the strain rate increases from 33.11 to 74.08 /s, as well as ultimate strain, maximum strains, and toughness all increase with the strain rate. All experimental results of the dynamic mechanical properties for the RCC material share significant variability at different strain rates. Moreover, the experimental results suggest that the specimen size is another key factor for the increase in dynamic mechanical properties, e.g., the peak strength, ultimate strain and toughness, as shown in Fig. 5.5. When the specimen size is larger, the lateral inertia confinement becomes more significant, which ultimately leads to a higher peak strength. For example, the testing results of different dimensional specimens at a similar strain rate (approximately 70/s) show that when the diameter of the specimen increases from 50 to 75 to 100 mm, the average peak strength monotonically increases from 10.08 to 20.33 to 30.44 MPa. This phenomenon is more prominent in Fig. 5.5a. Similar trends are observed for other dynamic mechanical properties, such as the ultimate strain and toughness. For the ductility, the maximum strain increases with increasing strain rate for same-size specimens, whereas the size dependence of the maximum strain

90

5 Size-Dependence of Dynamic Behaviors for RCC Under …

Fig. 5.5 Dynamic mechanical properties of the RCC specimens at various strain rates: a peak strength; b ultimate strain; c maximum strain; d toughness

is indistinct for specimens at a similar strain rate. This phenomenon is graphically illustrated in Fig. 5.5c. Because of the increasing ductility and strength, the specimens at a high strain rate exhibit a greater toughness than those at a relative low strain rate.

5.3.3 Statistical Significance of the Dynamic Size Effect To further investigate the statistical significance of the strain-rate sensitivity and size dependence on the dynamic mechanical parameters of RCC at high strain rates, the widely accepted method of ANOVA was performed [16]. The F-distribution was used in the ANOVA to evaluate the equality of three or more populations. F 0 is defined as the ratio of two mean squares to estimate whether the null hypothesis can be rejected. In this study, referring to the F-distribution table, the critical value (F 0 ) can be taken as F 0.05,3,28 = 2.95, representing the critical value with corresponding DOFs of 3 and 28 at a 5% significance level. Therefore, the null hypothesis (μ1 = μ2 = μ3 = μ4 ) (i.e., the strain rate has no effect on the dynamic mechanical properties) can be rejected if the P-value is smaller than 0.05 or F 0 > 2.95.

5.3 Dynamic Size Effect on Compressive Behaviors of RCC

91

Table 5.3 Summary of the ANOVA for the strain-rate effect and size effect of the RCC Property

Strain-rate effect D50

Size effect D75

F0

P-value

Peak strength

53.80

9.58 × 10−12

Ultimate strain

5.12

Maximum strain Toughness

F0

D100

At about 70/s

P-value

F0

P-value

F0

P-value

16.84

1.91 × 10−6

13.76

5.47 × 10−6

39.37

3.74 × 10−8

6.00 × 10−3

48.69

3.10 × 10−11

0.90

0.45

16.77

3.21 × 10−5

5.39

4.70 × 10−3

109.30

1.45 × 10−15

4.59

8.60 × 10−6

1.27

11.94

5.55 × 10−5

33.55

2.07 × 10−9

7.86

4.32 × 10−4

25.41

0.30 3.22 × 10−6

Table 5.3 summarizes all ANOVA results of specimens of various dimensions for the RCC material. The peak strength, maximum strain, and toughness can be significantly affected by the strain rates for specimens of different dimensions. A smaller P-value provides stronger justification to reject the null hypothesis. The strain-rate effect is notable for almost all of the dynamic mechanical properties, whereas the ultimate strain for the D100mm specimens is less sensitive to the strain rates. The size effect is detectable for most of the dynamic mechanical properties, apart from the maximum strain. The P-value of the maximum strain, recorded as 0.30 > 0.05, indicates that specimens of different sizes have similar maximum strains. Thus, the size dependence of the maximum strain for the RCC is insignificant.

5.3.4 Distribution Characteristic of Dynamic Compressive Strength Zhang suggested that experimental results of the dynamic mechanical properties for RCC share more significant discreteness than that of normal concrete, particularly in the vertical direction, as a result of different mix proportions and construction techniques [7]. Therefore, it is necessary to address the discreteness of the peak strength for the specimens of different dimensions. The weakest-link assumption based on Weibull statistics has been widely used to describe the scatter of material strengths, i.e., that the strength of a structure depends on the weakest volume element [17–19]. The Weibull analysis is used with a two-parameter form in the present study, as shown in Eq. (5.1). In addition, by taking the logarithm twice, the Weibull distribution can be rewritten in linear form as Eq. (5.2). P(σ ) = 1 − exp[−(σ/σ0 )m ]

(5.1)

ln[− ln(1 − P)] = m(ln σ − ln σ0 )

(5.2)

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5 Size-Dependence of Dynamic Behaviors for RCC Under …

where σ is the peak strength, σ0 is the scaling value concerned with the mean, and m is the shape parameter or Weibull modulus. Then, the mean and standard deviation can be derived as follows: σ¯ = σ0 Γ(1 + 1/m)

(5.3)

√ s = σ0 Γ(1 + 2/m) − Γ 2 (1 + 1/m)

(5.4)

where σ¯ is the mean strength, s is the standard deviation, and Γ(∗) is the gamma function. The cumulative probability density P can be estimated as P = i /(N + 1)

(5.5)

where N is the total number of tests and i is the current test number. The cumulative distribution plots of the experimental results show satisfactory qualitative agreement. Figure 5.6 compares the fitted cumulative distribution functions (CDFs) for each group in Table 5.1, and the Weibull distribution is shown to accurately describe the discrete material strength. Figure 5.6 also shows that the fitted cumulative probability curve shifts toward a higher stress value when the strain rate increases for same-size specimens, which indicates a positive effect of the strain rate on material strength. However, the peak strength shows more significant discreteness at higher strain rates since the fitted cumulative probability curve tends to be less steep. Table 5.4 summarizes the calculated parameters of the fitted Weibull distributions. For the dynamic experimental data, σ0 increases with increasing strain rate for same-size specimens. However, the statistical results show that the strain-rate effect on the shape parameter is not pronounced, without notable increasing or decreasing trend as the strain rate increases, which indicates an approximately uniform distribution at the selected strain rates. The average shape parameters for the D50, D75, and D100mm specimens are 10.15, 4.28, and 8.07, respectively, indicating no strong relationship with the specimen size. Table 5.4 also shows the effect of the strain rate and specimen size on the mean strength (σ¯ ) and standard deviation (s). Both the mean strength and standard deviation increase somewhat with increasing strain rate and specimen size. Figure 5.7 graphically presents the mean and standard deviation of strength from Weibull analysis. Relative to the results shown in Fig. 5.5a, the means and standard deviations of strength derived from the Weibull distribution are notably near to those of the experimental data. Based on the analysis results in this study, empirical formulae in terms of the dynamic compressive strength for the RCC material under impact loading are suggested as follows: for D50 specimens at 70/s < ε˙ < 150/s | | σ¯ = 8.36(lg˙ε )2 − 29.94(lg˙ε ) + 27.70 f c

(5.6)

5.3 Dynamic Size Effect on Compressive Behaviors of RCC

93

Fig. 5.6 CDFs of the dynamic compressive strength for specimens with various diameters: a D50; b D75; and c D100 Table 5.4 Statistical Weibull parameters for specimens of various dimensions Specimen dimensions (D×L: mm)

Statistical parameters

50×25

D50-A

75×25

100×25

Group

ε˙ (/s) 70.15

σ0 (MPa) 10.63

m 8.79

σ¯ (MPa)

s (MPa)

10.06

1.37

D50-B

88.54

12.58

11.37

12.03

1.28

D50-C

114.45

17.11

12.39

16.42

1.61

D50-D

143.78

23.62

8.06

22.25

3.28

D75-A

46.68

11.71

4.88

10.74

2.51

D75-B

75.84

21.72

4.10

19.71

5.41

D75-C

90.47

30.29

3.20

27.13

9.31

D75-D

111.20

40.03

4.95

36.73

8.49

D100-A

33.11

16.34

8.28

15.41

2.21

D100-B

47.10

20.28

7.69

19.06

2.93

D100-C

59.83

25.46

9.81

24.20

2.96

D100-D

74.08

32.62

6.51

30.40

5.46

94

5 Size-Dependence of Dynamic Behaviors for RCC Under …

Fig. 5.7 Strain-rate effect on the dynamic strength using weibull analysis

for D75 specimens at 40/s 0 always stands when 2 < Df < 3 and dmax  dmin . Therefore, the specific surface area of fragments always increases with the increasing fractal dimension in this condition. That also indicates a positive association between dynamic behaviors and fractal dimension. As shown in Fig. 6.8, the size-independent relationships between dynamic behaviors of RCC and fractal dimension can be generalized as   DIFσ = 1 + 7.69 × 10−7 exp 5.74Df

(6.18)

  DIFe = 1 + 1.96 × 10−7 exp 6.87Df

(6.19)

It has been confirmed that the fractal dimension is size-dependent and strain-rate sensitive, describing the variability of dynamic behaviors for various dimensional RCC specimens at different strain rate, as shown in Fig. 6.6. However, from Fig. 6.8, the influence of specimen size on dynamic behaviors of RCC becomes obscure when expressed by fractal dimension, since the fractal dimension is such a scientific indicator that the size-dependence and strain-rate effect on the dynamic behaviors of RCC both can be represented by it. The tight correlations between dynamic behaviors of RCC and fractal dimension can be generalized with the exponential growth model, illustrated as Eqs. (5.18) and (5.19). Therefore, the failure pattern (or fragment size distribution), determined by the crack propagation (strain-rate effect) and the multiaxial stress state (lateral inertia confinement) in the specimens, can be used to evaluate the dynamic behaviors of RCC under impact loadings with the specific exponential growth model. In practical engineering, the fragmentation prediction, blasting effectiveness, and damage degree evaluation of a concrete structure under impact loadings can be estimated through the characteristics of fragments including size distribution and fractal dimension. This macroscopic indicator of concrete fragmentation provides a relatively convenient approach to quantitatively analyze the dynamic behaviors of structures to avoid the secondary injury caused by the flying fragments. It is more

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6 Fragmentation-Based Dynamic Size Effect of RCC Under Impact Loadings

important that the fragmentation-based fractal dimension is another potential indicator to describe the strain-rate effect in structural analysis, rather than the traditional DIF indicator, so as to solve the overestimation of strain-rate effect induced by the inertia-related structural effect on dynamic behaviors.

6.5 Summary and Conclusions This chapter investigates the impact fragmentation of different dimensional RCC specimens in SHPB tests implementing various loading rates. The dynamic fragmentation process detected by high speed camera verifies the existing of lateral inertia confinement under impact loadings. The sieving analysis has been carried out to investigate the strain-rate effect and dynamic size effect on fragment size distribution, as well as its correlation with dynamic behaviors. Furthermore, the mechanism for dynamic behaviors of RCC is discussed based on the fractal characteristics of concrete fragmentation under impact loadings. The main contributions of this chapter can be listed as below: (1) Impact fragmentation process and the corresponding fragment size distribution of RCC specimens in SHPB tests are discussed in this chapter. The dynamic fragmentation process of RCC specimens is highly related to the multiaxial stress state in the specimens, which leads to the concrete failure of external surface at first. Moreover, the fragment size distribution is both strain-rate sensitive and size-dependent, which means the specimens exposed to higher loading rates or with a larger dimension have a higher content of smaller fragments. (2) The dynamic compressive strength and toughness of RCC are proved to be sensitive to the fragment size distribution, which get down exponentially with the increasing average grain diameter. Therefore, the fragment size distribution under impact loadings can be used to depicture the dynamic behaviors of concrete, indicating the tight correlation among crack propagation, energy dissipation and failure strength. (3) The fractal dimension, calculated by the mass-size relationship, is revealed as an ideal indicator to describe the size-dependence and strain-rate sensitivity of the dynamic behaviors for RCC, so that the influence of specimen size on dynamic behaviors of RCC becomes obscure when expressed by fractal dimension. The mechanism of the difference between static size effect and dynamic size effect can be interpreted by the specific surface area of fragments. The exponential growth models in terms of fractal dimension provide a meaningful and convenient way to evaluate the dynamic compressive strength and toughness of RCC with the macroscopic concrete fragmentation.

References

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References 1. Carpinteri A, Pugno N (2002) A fractal comminution approach to evaluate the drilling energy dissipation. Int J Numer Anal Meth Geomech 26(5):499–513 2. Carpinteri A, Lacidogna G, Pugno N (2004) Scaling of energy dissipation in crushing and fragmentation: a fractal and statistical analysis based on particle size distribution. Int J Fract 129(2):131–139 3. Grady DE (2008) Fragment size distributions from the dynamic fragmentation of brittle solids. Int J Impact Eng 35(12):1557–1562 4. Wu C, Nurwidayati R, Oehlers DJ (2009) Fragmentation from spallation of RC slabs due to airblast loads. Int J Impact Eng 36(12):1371–1376 5. Ren W, Xu J (2017) Fractal characteristics of concrete fragmentation under impact loading. J Mater Civ Eng 29(4):04016244 6. Hou T-X, Xu Q, Zhou J-W (2015) Size distribution, morphology and fractal characteristics of brittle rock fragmentations by the impact loading effect. Acta Mech 226(11):3623–3637 7. Zhang S-R, Wang X-H, Wang C et al (2017) Compressive behavior and constitutive model for roller compacted concrete under impact loading: Considering vertical stratification. Constr Build Mater 151:428–440 8. Wang M, Hao H, Ding Y et al (2009) Prediction of fragment size and ejection distance of masonry wall under blast load using homogenized masonry material properties. Int J Impact Eng 36(6):808–820 9. Shi Y, Xiong W, Li Z-X et al (2016) Experimental studies on the local damage and fragments of unreinforced masonry walls under close-in explosions. Int J Impact Eng 90:122–131 10. Li QM, Lu YB, Meng H (2009) Further investigation on the dynamic compressive strength enhancement of concrete-like materials based on split Hopkinson pressure bar tests. Part II: numerical simulations. Int J Impact Eng, 36(12):1335–1345 11. Zhang M, Wu HJ, Li QM, et al. (2009) Further investigation on the dynamic compressive strength enhancement of concrete-like materials based on split Hopkinson pressure bar tests. Part I: experiments. Int J Impact Eng, 36(12):1327–1334 12. Forrestal MJ, Wright TW, Chen W (2007) The effect of radial inertia on brittle samples during the split Hopkinson pressure bar test. Int J Impact Eng 34(3):405–411 13. Zhang M, Li QM, Huang FL et al (2010) Inertia-induced radial confinement in an elastic tubular specimen subjected to axial strain acceleration. Int J Impact Eng 37(4):459–464 14. Wang X-H, Zhang S-R, Wang C et al (2018) Experimental investigation of the size effect of layered roller compacted concrete (RCC) under high-strain-rate loading. Constr Build Mater 165:45–57 15. Qi C, Wang M, Bai J et al (2016) Investigation into size and strain rate effects on the strength of rock-like materials. Int J Rock Mech Min Sci 86:132–140 16. Chen X, Wu S, Zhou J (2013) Experimental and modeling study of dynamic mechanical properties of cement paste, mortar and concrete. Constr Build Mater 47:419–430 17. Hao Y, Hao H (2013) Dynamic compressive behaviour of spiral steel fibre reinforced concrete in split Hopkinson pressure bar tests. Constr Build Mater 48:521–532 18. Hao Y, Hao H, Jiang GP et al (2013) Experimental confirmation of some factors influencing dynamic concrete compressive strengths in high-speed impact tests. Cem Concr Res 52:63–70 19. Elfahal MM, Krauthammer T, Ohno T et al (2005) Size effect for normal strength concrete cylinders subjected to axial impact. Int J Impact Eng 31(4):461–481 20. Qi C, Wang M, Bai J et al (2014) Mechanism underlying dynamic size effect on rock mass strength. Int J Impact Eng 68:1–7 21. Krauthammer T, Elfahal MM, Lim J et al (2003) Size effect for high-strength concrete cylinders subjected to axial impact. Int J Impact Eng 28(9):1001–1016 22. Zou C, Wong LNY (2016) Size and geometry effects on the mechanical properties of carrara marble under dynamic loadings. Rock Mech Rock Eng 49(5):1695–1708

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23. Carpinteri A, Puzzi S (2006) A fractal approach to indentation size effect. Eng Fract Mech 73(15):2110–2122 24. Carpinteri A, Corrado M (2009) An extended (fractal) overlapping crack model to describe crushing size-scale effects in compression. Eng Fail Anal 16(8):2530–2540 25. Carpinteri A, Spagnoli A (2004) A fractal analysis of size effect on fatigue crack growth. Int J Fatigue 26(2):125–133

Chapter 7

Dynamic Constitutive Model of RCC for Fully-Graded Dam

7.1 Introduction Based on the discuss in the above chapters, the special characteristics of RCC such as low strength and large aggregate gradation will significantly influence its dynamic mechanical properties, and it is verified that the stratified structure of RCC has little impact on the dynamic compressive and tensile behaviors at high loading rates. However, due to the limitation of laboratory test equipments, the scaled aggregates are used in the tests that cannot reflect the true large aggregate effect on the dynamic behaviors of RCC and the inertia effect manifesting as the size dependence also cannot be avoided in the dynamic tests. Therefore, it is necessary to establish a modified concrete dynamic constitutive model to describe the true dynamic behaviors for fully-graded RCC. The K&C constitutive model can accurately describe the dynamic mechanical behaviors of normal concrete, and it has been modified to describe the dynamic behaviors for different concrete materials at home and abroad. This chapter tries to discuss the modification approach for full-graded RCC dynamic constitutive model based on the framework of K&C constitutive theory.

7.2 Strength Surface Modification of Fully-Graded RCC A well understanding of the multi-axial mechanical behaviors of RCC is the foundation of strength surface modification. So far, there are limited experiments for RCC at present. In this chapter, the triaxial mechanical properties of RCC are firstly studied via laboratory tests and then the triaxial mechanical properties under medium and high confining pressures of RCC with different aggregate gradations are further studied through the verified meso-simulation method. The results are supplemented to provide a basis for the modification of strength riterion for RCC. © Science Press 2023 S. Zhang et al., Dynamic Mechanical Behaviors and Constitutive Model of Roller Compacted Concrete, Hydroscience and Engineering, https://doi.org/10.1007/978-981-19-8987-2_7

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120

7 Dynamic Constitutive Model of RCC for Fully-Graded Dam

7.2.1 Experimental Study on RCC Triaxial Compression Behavior The specimen preparation is shown in Chap. 2, and a series of RCC cores in a diameter of 100 mm were drilled. Then, the cores were cut to lengths of 200 mm and ground in both ends for tests. The uniaxial and triaxial compression tests were performed with a triaxial system as shown in Fig. 7.1. The confining pressure of this cell could be up to 100 MPa. The axial load F d and confining pressure σ3 can be directly measured by the transducer in the system. In this study, axial extensometer and circumferential extensometer were employed to measure axial displacement and circumferential deformation, respectively. In this way, the total circumferential deformation along the whole perimeter of the cylinder can be obtained, which was utilized to derive the average lateral strain ε3 . Under uniaxial compression, a displacement control mode was used to obtain the stress– strain curve, with a rate of 0.002 mm/s. As for triaxial compression tests, the load path is given in Fig. 7.2. As shown, the confining pressure was firstly applied to the target value in load control mode and then kept constant. After the target σ3 was obtained, the control mode was switched to displacement control and deviatoric was applied all the way up to the fracture of the RCC specimens at a loading rate of 0.002 mm/s. The penetration of confining fluid into the concrete has a significant influence on results of triaxial tests, so concrete specimens were jacketed with heat shrinkable tube (1.5 mm thick) to prevent penetration of oil. A total of seven levels of confining pressure (σ3 ) were employed to cover the response under both moderate and high confinement: 0, 5, 10, 15, 20, 25, and 30 MPa, corresponding to confinement ratios

Fig. 7.1 Computer-controlled electro-hydraulic servo triaxial testing machine

7.2 Strength Surface Modification of Fully-Graded RCC

121

Fig. 7.2 Schematic diagram of loading path in triaxial test

of about 0.00, 0.34, 0.67, 1.01, 1.34, 1.68, and 2.01, respectively. Mention that at least three reasonable test results were obtained at each confinement level. According to the triaxial test results, the historical curves of deviatoric stress and volumetric strain of RCC under different confining pressures are shown in Figs. 7.3 and 7.4. It should be noted that the volumetric strain εv is obtained from the axial strain ε1 and radial strain ε3 , as shown in Eq. (7.2). σ1 = ∆σ + σ3 = N /A + σ3

(7.1)

εv = ε1 − 2ε3

(7.2)

where σ1 is the axial stress of specimens under different confining pressures σ3 ; ∆σ is the deviatoric stress measured in the triaxial test. Fig. 7.3 Deviatoric stress–strain curves under different confining pressures

122

7 Dynamic Constitutive Model of RCC for Fully-Graded Dam

Fig. 7.4 Volumetric strain history curve

The experimental results of RCC under various confining pressures are plotted in Figs. 7.3 and 7.4. The axial deviatoric stress–strain curves and the volumetric strain history are plotted by different level of confining pressures. It’s obvious that the deviatoric axial stress (∆σ = σ1 − σ3 ) of RCC specimens increases significantly with increasing confining pressure at given strain. Furthermore, RCC sample shows ductile and plastic behavior in the presence of higher confinement. During the increase of axial load, RCC specimens contract axially. Meanwhile, due to Poisson’s law, lateral dilation can be observed accompanied by axial contraction. The above process is accompanied by the generation and propagation of cracks, while the confining pressure places a restriction on the development of these cracks. According to Figs. 7.3 and 7.4, softening behavior can be observed for RCC specimens under all confining pressures (0–30 MPa). It can be found that the peak strength moves towards higher axial strain with the increasing confining pressures. As the cracks develop, the volumetric strain follows the tendency from contraction to dilation. However, when subjected to relatively higher confining pressures (15–30 MPa), the RCC specimens show more plastic behavior. The slope of post-peak curves decreases with the enhancement of confinement, which leads to horizontal plateaus. The reduction in crack developments induced by increasing confining pressures contributes to this phenomenon. Table 7.1 shows the peak strength ∆σp , peak axial strain ε1p and transverse strain ε3p of RCC under different confining pressures. It can be seen that both the peak axial strain and transverse strain increase significantly with the increase of confining pressure. Figure 7.5 gives the relationship between normalized peak axial train ratio (ε1p /εcp ) and confinement ratio (σ3 /fc' ), in which εcp denotes the peak axial strain of RCC under uniaxial compression, and a linear formula can describe their relationship as shown in Eq. (7.3). Similarly, the linear trend above was also found in high

7.2 Strength Surface Modification of Fully-Graded RCC

123

strength concrete (HSC) [1–4], cement mortar [5], ultra-high performance cementbased composite material (UHPCC) [6] with a range of slope from 6.24 to 20. It indicates that the effect of confinement on the enhancement of peak axial strain was not pronounced for RCC, compared with other concrete-like materials. ε1p /εcp = 1 + 4.74σ3 / fc'

(7.3)

The Mohr–Coulomb failure criterion model is a kind of pressure-dependent failure criteria and assumes only two different failure modes: the sliding failure and the separation failure. The Mohr–Coulomb failure criterion can be expressed by Eq. (7.4). For the experimental data, the fitting parameter of k is 3.65 for RCC as shown in Fig. 7.6 and Table 7.2, compared with the available triaxial compressive data of RCC (the maximum of confining pressure is limited to 10 MPa) and other concrete-like materials in literatures. This difference also indicates that the effect of confining pressure on the peak axial strength of RCC specimens is less significant than that of other concrete materials. Tabel 7.1 Peak strain of RCC under various confining pressures σ3 (MPa)

σ3 /fc'

∆σp (MPa)

ε1p (%)

ε3p (%)

0

0

14.9

0.411

−0.237

5

0.34

26.1

0.625

−0.490

10

0.67

37.3

1.578

−0.760

15

1.01

41.9

2.270

−1.279

20

1.34

50.3

3.078

−2.228

25

1.68

54.1

3.630

−2.849

30

2.01

60.1

4.516

−3.478

Fig. 7.5 Relationship between peak strain and confining pressure [1–6]

124

7 Dynamic Constitutive Model of RCC for Fully-Graded Dam

Fig. 7.6 Relationship between peak strength and confining pressure [1–8]

Table 7.2 Fitting results of different concrete material parameters k Material Farnam et al. [7]

Ansari and Li [1, 8]

Lu and Hsu [3] Candappa et al. [2]

Results of this study

fc' (MPa)

σ3 /fc'

k

HSC

76

0.28

4.46

HPFRC

87

0.25

4.82

SIFCON

146

0.15

4.82

HSC

47

0.90

3.00

71

0.90

2.60

107

0.80

2.60

HSC

67

0.80

4.00

SFHSC

69

1.00

3.95

HSC

RCC

61

0.20

5.30

73

0.20

5.30

103

0.10

5.30

14.9

2.01

2.64

σ1 /fc' = 1 + kσ3 /fc'

(7.4)

where k = (1 + sinϕ)/(1 − sinϕ) and ϕ denotes the internal-friction angle.

7.2.2 Meso-simulation of Triaxial Compressive Behavior of Fully-Graded RCC In order to further accurately reflect the influence of confining pressure on the triaxial compressive behaviors of fully-graded RCC, this section adopts a three-dimensional meso-models to consider the aggregate gradation [9]. As shown in Fig. 7.7, the size

7.2 Strength Surface Modification of Fully-Graded RCC

125

Fig. 7.7 Schematic diagram of triaxial compression test based on mesoscopic numerical model

of the mesoscopic model is F 100 × 200 mm, which is consistent with the laboratory test. It is assumed that coarse aggregate particles are spheres, and meso-models with four different aggregate gradations of 5–20 mm, 5–40 mm, 5–60 mm and 5–80 mm are generated. Vertical constraints were set at the bottom of the model, and the load procedure was consistent with the laboratory test as shown in Fig. 7.2. In this section, the K&C constitutive model is still used to describe the mechanical properties of mortar matrix, coarse aggregate particles and ITZ. Referreing to the constitutive parameters of each meso-component in Chap. 3, the model parameters are generated by the parameter automation generation of KCC model in LS-DYNA. The only input parameters in Table 7.3 are determined through repeated trial calculations. Figure 7.8 compares the results of meso-simulation with those of laboratory triaxial compressive tests, where the maximum diameter of coarse aggregate is 20 mm. It can be found that the numerical stress–strain curves under different confining pressures are close to the results of laboratory tests, indicating that the established meso-model and the selected mechanical parameters are reasonable. Table 7.3 Parameters for K&C constitutive models of concrete meso-components

Parameters

Mortar

ITZ

Coarse aggregate

Densityρ/kg·m−3

2000

1800

2600

Poisson’s ratio v

0.167

0.167

0.20

Shear modulus G/GPa

10.63

8.65

18.50

Compressive strength fc' /MPa

15

10

50

Tensile strength ft' /MPa

2.10

1.80

5.20

126

7 Dynamic Constitutive Model of RCC for Fully-Graded Dam

Fig. 7.8 Validation of mesoscopic numerical model and model parameters for RCC triaxial tests

In order to explore the influence of aggregate gradation, the peak compressive strength of RCC under confining pressures at all levels was obtained through mesotriaxial test. Among them, seven groups of confinement ratios were set (σ3 /fc' = 0, 0.34, 0.67, 1.01, 1.34, 1.68, 2.01). Table 7.4 lists the results of laboratory tests and numerical simulations. It is obvious that the aggregate size effect on the compressive strength is relatively limited at the low confining state, while the concrete strength will increase as the aggregate size increases especially at the high confining state. The reason maybe lie in that specimens with a smaller coarse aggregates usually have more ITZs, which leads to the strength attenuation. Moreover, the strength attenuation will be enlarged by the confining pressure. On the basis of Eq. (7.4), the maximum aggregate size factor is introduced into the Mohr-Coulomb failure criterion so as to illustrate the triaxial compressive strength of RCC influenced by the aggregate gradation, as shown in Eq. (7.5) and Fig. 7.9. Table 7.4 Peak strength of RCC under various confining pressures σ3 /fc'

σ1 (MPa) Mesoscopic simulation Laboratory test 5–20 mm

5–20 mm

5–40 mm

5–60 mm

5–80 mm

0

14.9

14.8

13.9

13.3

12.8

0.34

31.1

30.5

32.4

32.8

32.3

0.67

47.3

47.4

43.7

45.8

48.1

1.01

56.9

55.6

57.8

58.3

57.9

1.34

70.3

69.6

69.7

71.2

70.8

1.68

79.1

78.7

80.5

80.7

82.4

2.01

90.1

89.7

88.2

90.4

95.8

7.2 Strength Surface Modification of Fully-Graded RCC

127

Fig. 7.9 Triaxial compression test results of RCC specimens with different aggregate gradations

( ) σ1 σ3 dmax 0.1605 = 1 + 2.6353 fc' fc' d0

(7.5)

where d0 = 20mm and dmax are the reference aggregate size and maximum aggregate size in the mix design, respectively.

7.2.3 Strength Surface Modification for RCC Constitutive Model Through the indoor triaxial compressive tests and three-dimensional mesosimulations, the maximum aggregate size factor is introduced to modify the concrete strength surface in K&C model, as shown in Eq. (7.6). The corresponding model parameters fitted by the least square method are shown in Fig. 7.10 and Eq. (7.7). For comparation, Fig. 7.10 also presents the original ultimate strength of normal concrete, whose model parameters are automatically generated in the programme LS-DYNA. It is obvious that the parameters of the original K&C constitutive model cannot accurately describe the ultimate strength surface of RCC materials with different aggregate gradation. ( ∆σm = a0m +

P a1m + a2m P

)(

dmax d0

)γm (7.6)

a0m = 0.4490fc' ; a1m = 0.6780; a2m = 0.0760/fc' ; γm = 0.1240; d0 = 20mm (7.7) On the other hand, the initial yield surface is usually generated as 0.45 times ultimate strength surface. The relationship between the ultimate strength surface and the initial yield strength surface can be expressed ∆σ y = 0.45∆σ m . In( this case, ) a point (P, ∆σ m ) on the ultimate strength surface corresponds to the point P ' , ∆σy on

128

7 Dynamic Constitutive Model of RCC for Fully-Graded Dam

Fig. 7.10 Modified ultimate strength surface of RCC with different aggregate gradations

the yield surface, as expressed by Eq. (7.8). As for the residual strength surface, Kong et al. [10] assumed that the residual strength surface ∆σr was parallel to the ultimate strength surface and this assumption is also used herein. Then, the strength surface of RCC with different aggregate gradation can finally be determined. Figure 7.11 shows the modified strength surface for RCC specimens under the coarse aggregate gradation of 5–20 mm and 5–80 mm based on the K&C constitutive concept. ∆σy = 0.45∆σm , P ' = P − 0.55∆σm /3

(7.8)

7.3 True Strain-Rate Effect Model of Fully-Graded RCC 7.3.1 True Strain Rate Effect Decoupling Method In general, the apparent DIFs directly obtained from laboratory tests or numerical simulation results are usually composited of the true strain-rate effect, inertia enhancement and end friction enhancement, and the enhancements of inertia and

7.3 True Strain-Rate Effect Model of Fully-Graded RCC

129

Fig. 7.11 Modified strength surfaces in K&C constitutive model for RCC

end friction are commonly seen as the structural effect. Thus, an overestimation of structural resistance will occur once the apparent DIFs directly from laboratory tests are used in analysis. In order to obtain the true material strain-rate effect, many works have been done to eliminate the structural effect [11]. Taking dynamic compressive strength as an example, the apparent CDIF can be defined as Eq. (7.9). ( ) CDIFA = fc /fc' = fc' + ∆fε˙ + ∆fi + ∆fμ /fc'

(7.9)

where CDIFA represents the apparent CDIF directly obtained from laboratory tests or meso-mechanical tests; fc and fc' are dynamic compressive strength and quasi-static compressive strength, respectively; ∆fε˙ , ∆fi , and ∆fμ are the dynamic compressive strength enhancement caused by the real strain-rate effect, inertia effect and end friction effect of the material, respectively. According to Eq. (7.9), when ∆fε˙ = 0 and ∆fμ = 0, that is, the DIF value in the constitutive model of concrete material is set to 1 in numerical simulation and the friction coefficient of the loading face is set to 0 at the same time, the CDIF increment caused only by the inertia effect can be genelized as Eq. (7.10): ) ( ∆CDIFi = fc' + ∆fi /fc' − 1

(7.10)

Similarly, when ∆fε˙ = 0 and ∆fi = 0, the CDIF increment caused by the end can be defined as: ) ( ∆CDIFμ = fc' + ∆fμ /fc' − 1

(7.11)

At this point, the true strain-rate effect of the material can be obtained by Eq. (7.12). In the meso-mechanical tests of this chapter, the velocity boundary is directly applied to the end face of the specimen, so the dynamic strength enhancement in the numerical

130

7 Dynamic Constitutive Model of RCC for Fully-Graded Dam

results does not include the strength enhancement of end friction, and the true strainrate effect of the material can be further simplified as Eq. (7.13). ) ( CDIFε˙ = CDIFA − ∆CDIFμ − ∆CDIFi = fd − ∆fμ − ∆fi /fc' CDIFε˙ = CDIFA − CDIFi = (fd − ∆fi )/fc'

(7.12) (7.13)

7.3.2 True Strain-Rate Effect on Dynamic Compressive Strength of RCC 7.3.2.1

Inertia Effect on Dynamic Compressive Strength

Chapter 3 discusses the strain-rate effect on dynamic compressive strength for RCC with different aggregate gradation, and an empirical formula with due consideration of the aggregate gradation is proposed to quantify the apparent strength enhancement from direct numerical results. To obtain the true material strain-rate effect, it is necessary to quantify the strength enhancement induced by the inertia effect at first. In this section, the strength enhancement induced by the inertia effect is further studied with two-dimensional mesoscopic model in Chap. 3 by assuming the DIFs of each concrete component to be 1. Then, the contribution of compressive strength from inertia effect can be calculated. Here, the contribution ratio of inertia effect to the dynamic compressive strength is defined as Eq. (7.14), and those for RCC with different coarse aggregate gradation under various loading rates are illustrated in Fig. 7.12. τ=

Fig. 7.12 Contribution ratio of lateral inertia to dynamic compressive strength

∆fi ∆fi + ∆fε˙

(7.14)

7.3 True Strain-Rate Effect Model of Fully-Graded RCC

131

It can be seen from Fig. 7.12 that the coarse aggregate gradation has extremely limited influence on the contribution rate of inertia effect to the dynamic compressive strength at any loading rates. Before critical strain rate ε˙ cr = 1 s−1 , the contribution ratio of inertia effect is small, while it becomes more significant at high loading rates, which is basically consistent with the results in existing literature [12, 13]. This is mainly because the compressive strength enhancement under a low loading rate is mainly affected by free water effect. When the strain rate continues to increase, the influence of inertia effect on dynamic compressive strength gradually appears [14].

7.3.2.2

True Strain-Rate Effect of Dynamic Compressive Strength

Based on Eq. (7.13), the dynamic compressive strength increment caused by inertia effect is quantified and removed from the apparent CDIFs. Then, the true strain-rate effect on dynamic compressive strength for RCC with different coarse aggregate gradation can be obtained, as shown in Fig. 7.13a. Based on the least square method, the true strain-rate effect on dynamic compressive strength for RCC with different coarse aggregate gradation can be generalized by Eq. (7.15).

CDIFε˙ =

⎧ ⎪ ⎨

(1.14024 + 0.02911 lg ε˙ )

(

dmax d0

)0.04061

( ) ⎪ ⎩ 2.21977 − 2.35299 lg ε˙ + 1.20956(lg ε˙ )2

(

dmax d0

)0.09275

ε˙ ≤ 10 s−1 ε˙ > 10 s−1 (7.15)

Figure 7.13b compares the true strain-rate effect of fully-graded RCC (5–80 mm) on dynamic compressive strength with other CDIF relations in literatures [15]. It can be seen that under the condition of low strain rate (˙ε < 10 s−1 ), the true strain-rate effect on dynamic compressive strength for RCC with full gradation (5–80 mm) is close to the formula proposed by Hao et al. [15], but it is lower than CEB

Fig. 7.13 True strain rate effect on dynamic compressive strength of RCC

132

7 Dynamic Constitutive Model of RCC for Fully-Graded Dam

empirical formula. However, at a higher strain rate (˙ε > 10 s−1 ), the true strainrate effect on dynamic compressive strength for RCC with full aggregation is close to CEB empirical formula, which is significantly stronger than the results obtained by Hao et al. [15].

7.3.3 True Strain-Rate Effect on Dynamic Tensile Strength of RCC 7.3.3.1

Dynamic Tensile Stress–Strain Relationship

The dynamic tensile behaviors of RCC with various specimen sizes (b = 150 mm, 300 mm, 450 mm and 600 mm) were studied by meso-simulation at different loading rates (˙ε = 10−5 s−1 , 10−3 s−1 , 10−1 s−1 , 1 s−1 , 10 s−1 , 30 s−1 , 50 s−1 , 100 s−1 , 200 s−1 and 300 s−1 ). Two-dimensional mesoscopic model in Chap. 3 was used herein with a constant length-to-width ratio of 2.0 and the coarse aggregate gradation is set to be 5–40 mm. In the numerical simulation, the tensile stress is calculated by the reaction force at the loaded boundary divided by the transverse length of the specimen (ft = Ft /b), and the tensile strain calculated by the vertical displacement of the specimen loaded boundary divided by the specimen height in the loading direction (εt = ut /l). Figure 7.14 shows the dynamic tensile stress–strain curves at different loading rates for RCC with different specimen sizes, and Fig. 7.15 shows the rate sensitivity of dynamic tensile strength in term of TDIFs. It can be seen from the figures that the tensile strength of RCC also shows a significant strain-rate effect, that is, when the specimen size remains the same, the dynamic tensile strength of RCC exponentially increases with the increase of strain rate. Moreover, under the quasi-static loads (˙ε = 1 × 10−5 s−1 ) or the low loading rates, the tensile strength of RCC decreases with the increasing specimen size. However, the size dependence of dynamic tensile strength effect gradually weakens as the strain rate increases, and it can be ignored when the loading rate is higher than 1 s−1 . The result is consistent with relevant research of normal concrete materials [16].

7.3.3.2

Inertia Effect on Dynamic Tensile Strength

In order to further quantify the strength enhancement under tension induced by the inertia effect, the DIFs of each component in meso-model was set to 1. The contribution ratio of inertia effect defined by Eq. (7.14) is also applied to measure the tensile strength enhancement induced by inertia effect. Figure 7.16 shows the contribution ratio of inertia effect to dynamic tensile strength for RCC with various coarse aggregate gradation.

7.3 True Strain-Rate Effect Model of Fully-Graded RCC

133

Fig. 7.14 Tensile stress–strain relationship curves of specimens with different sizes under different loading rates Fig. 7.15 Size effect on the dynamic tensile strength of RCC

134

7 Dynamic Constitutive Model of RCC for Fully-Graded Dam

Fig. 7.16 Contribution ratio of lateral inertia to dynamic tensile strength

It can be seen from Fig. 7.16 that the contribution ratios of inertia effect to TDIF under high loading rates are more obvious than those under low loading rates, indicating a certain strain-rate sensitivity. At the same time, the contribution ratio of inertia effect on TDIFs is not sensitive to the aggregate gradation. When strain rate reaches 100 s−1 , the contribution ratio of inertia effect on TDIFs is about 6%. Generally, the analysis results by meso-simulation in this study are consistent with the research conclusions from Hao et al. [17]. However, the numerical simulation results of Zhang et al. [18] show that the inertia effect has almost no influence on the dynamic tensile strength enhancement. This contrary conclusion needs more systematic and in-depth studies through a large number of laboratory tests and numerical simulations.

7.3.3.3

True Strain-Rate Effect of Tensile Strength

After the removal of the tensile strength enhancement induced by the inertial effect, the true strain-rate effect of RCC material under tension are shown in Fig. 7.17a. It can be found that the coarse aggregate gradation has a relatively significant influence on the dynamic tensile strength of RCC. Larger coarse aggregate usually indicates a more sensitive strain-rate effect. Moreover, based on the analysis data, the true strain-rate effect of tensile strength for RCC with different aggregate gradation can be generalized by Eq. (7.17).

TDIFε˙ =

⎧ ⎪ ⎨

(1.92725 + 0.19065 lg ε˙ )

(

dmax d0

)0.02475

ε˙ ≤ 1 s−1

)0.11691 ( )( ⎪ ⎩ 1.82579 + 0.08164 lg ε˙ + 0.47383(lg ε˙ )2 dmax ε˙ > 1 s−1 d0 (7.17)

Figure 7.17b compares the obtained true strain-rate effect on tensile strength for RCC with the coarse aggregate of 5–80 mm with other TDIF relation formulae in the

7.4 Modification of the Damage Equation in the K&C Model

135

Fig. 7.17 True strain rate effect on dynamic tensile strength of RCC

literatures. It is obvious that under the condition of low loading rates (˙ε < 1 s−1 ), the obtained true strain-rate effect on tensile strength for full-graded RCC in this study is close to the formula proposed by Hao et al. [15] for normal concrete, relatively higher than the empirical formulae proposed by CEB, Malvar and Crawford [19]. However, at a higher loading rate (˙ε > 1 s−1 ), the true strain-rate effect on tensile strength for full-graded RCC falls between the empirical formula proposed by CEB and Hao et al. [15].

7.4 Modification of the Damage Equation in the K&C Model In the K&C constitutive model introduced in Chap. 3, the deviatoric stress on the failure strength surface ∆σ is obtained by linear interpolation between the two strength surfaces, and its state is determined by the damage evolution function η(λ). The relationship between the damage proportional factor η and effective plastic strain λ describes the stress–strain curve of the material in the whole process. Therefore, in order to accurately characterize the damage evolution process of RCC materials, it is necessary to modify the damage evolution equation. Attard and Setunge define the relationship between damage scaling factor η and effective plastic strain λ as follows: ⎧ ( ) ( )2 ( )3 ⎨ α λλm + (3 − 2α) λλm + (α − 2) λλm λ < λm η(λ) = λ/λm ⎩ λ ≥ λm αc (λ/λm −1)αd +λ/λm

(7.18)

where α, αc and αd are constants. The former mainly controls the strain-hardening stage, while the latter two mainly control the strain-softening stage. λm is the peak value of effective plastic strain.

136

7 Dynamic Constitutive Model of RCC for Fully-Graded Dam

To determine the values of α, α c , α d and λm , the single-element method was used to simulate the RCC uniaxial compressive test. The single-element model size was 2 mm [20, 21] as shown in Fig. 7.18. The simulated stress–strain relationship curve was consistent with the test results through trial calculation to determine the parameter values. Finally, the values of the parameters are determined as follows: α = 2.70, α c = 0.281, α d = 1.84, λm = 8.60 × 10–5 . The modified damage evolution equation is shown in Fig. 7.19. Fig. 7.18 Single element model of compression test

Fig. 7.19 Modified damage evolution equation

7.5 Validation of Modified Full-Graded RCC Constitutive Model

137

7.5 Validation of Modified Full-Graded RCC Constitutive Model 7.5.1 Validation of Modified K&C Constitutive Model with Single Element Method The basic mechanical parameters including density ρ, elastic modulus E and quasistatic compressive strength fc' for the modified K&C consititutive model of fullgraded RCC can be obtained from uniaxial compressive test. The shear modulus G and volume modulus Ke can be obtained from the relation equation E = 2G(1 + ν) and Ke = E/3(1 − 2ν) respectively, and v is Poisson’s ratio. The basical physical and mechanical parameters are obtained according to the laboratory uniaxial tests of RCC in Chap. 2. The modified strength surface model in Sect. 7.2, modified strainrate effect model in Sect. 7.3, and modified damage evaluation equation in Sect. 7.4 are all used herein to redefine the dynamic mechanical behaviors for the proposed RCC constitutive model. The parameters for RCC constitutive model has been listed in Table 7.8. In order to verify the rationality of the modified K&C constitutive model for fully-graded RCC, the results obtained through single-element simulation are shown in Fig. 7.20, compared with the laboratory test data. It can be seen that the stress– strain curves of the proposed RCC constitutive model are in good agreement with the laboratory test data under both compression and tension at different loading rates, which verifies the rationality of the proposed RCC constitutive model. Table 7.8 Parameters for modified K&C constitutive model of fully-graded RCC Basic parameter ρ/kg · m−3

2400

Strength surface parameter

Damage parameter

a0m /MPa

b1

7.184

1.6

ν

0.17

a1m

0.6780

b2

1.35

G/GPa

8.65

a2m /MPa−1

4.75 × 10–3

b3

1.15

fc' /MPa

16

a0y /MPa

3.2328

Ifrac /mm

10

ft /MPa

2.0

a1y

1.5067

els

1.15

d εp

0.5

a2y /MPa−1

10.56 × 10–3

λm

8.6 × 10–5

λt

8.7 × 10–3

a0r /MPa

0

n

100

dmax /mm

20

a1r

0.4898

a

2.70

a2r /MPa−1

7.50 × 10–3

ac

0.281

γm

0.1240

ad

1.84

138

7 Dynamic Constitutive Model of RCC for Fully-Graded Dam

Fig. 7.20 Validation of modified K&C constitutive model for dam RCC

7.5.2 Applicability of Modified K&C Constitutive Model in Slab Subjected To Air Explosion Due to the lack of field experimental study on RCC structure exposed to explosion loads, this study first establishes a fully coupled numerical simulation model based on the laboratory air explosion test of normal concrete, and verifies the rationality of the fully coupled model with the mechanical parameters of normal concrete. Then, the modified constitutive model for full gradation (d = 5–80 mm) and wet-screened RCC (d = 5–20 mm) were used to analyze the failure modes of RCC slabs with different aggregate gradations.

7.5.2.1

Validation of Model Rationality

At first, the air explosion test of concrete slab in Reference [22] was selected as the verification test. The numerical model was established strictly according to the test conditions, and the explosive equivalent was 0.31 kg. Considering the model symmetry, only the 1/4 model shown in Fig. 7.21 is established for analysis, in which the symmetric constraint boundary is applied to the symmetric plane, and the nonreflective boundary condition is applied to the rest of the plane. In order to simulate the failure mode of reinforced concrete slabs subjected to air explosion and avoid unexpected termination of calculation due to severe distortion of large deformed elements, the erosion failure criterion *MAT_ADD_EROSION was introduced in the calculation, and the failure principal strain MXEPS = 0.018 was selected as the basis for judging the failure of the elements [23]. In numerical verification, the constitutive models and corresponding parameters of other materials refer to existing literature [24]. Among them, the original K&C constitutive model parameters are generated by the parameter automation generation algorithm in LS-DYNA, and the necessary inputs are ρ = 2400 kg/m3 , fc' = 39.5 MPa.

7.5 Validation of Modified Full-Graded RCC Constitutive Model

139

Fig. 7.21 Model of reinforced RCC slab

As shown in Fig. 7.22, the failure mode of concrete slab subjected to air explosion of 0.31 kg TNT equivalent explosive was obtained based on numerical simulation, in which the fluid–solid interaction was solved by ALE algorithm. It can be seen from the figure that, under the air explosion, circumferential cracks appear on the facing surface of the concrete slab, while the back surface appears shock collapse and crack, which is basically the same as the observation result of outdoor test. Compared with the original test result [22], it was verified that the numerical method and selected constitutive models can successfully reflect the dynamic structural responses of concrete structures under blasting, indicating the rationality of numerical simulation.

7.5.2.2

Effectiveness of Modified RCC Constitutive Model

In order to study the effectiveness of modified RCC constitutive model, the fully coupled numerical model described in the above section is adopted. The only difference in the study cases lies in the simulated RCC with the coarse gradation of 5–20 mm and 5–80 mm. The results of numerical simulation are shown in Fig. 7.23. It can be seen that the failure mode of concrete slab using the modified RCC constitutive model is similar to that of the outdoor test, but the damage degree and damage range are significantly increased due to the low strength characteristic of RCC. More importantly, the damage range of full-graded RCC slab with aggregate gradation of 5–80 mm is significantly smaller than that of wet sieving RCC slab with aggregate gradation of 5–20 mm. Thus, the special large aggregate gradation can also significantly affect the structural dynamic responses when RCC structures are exposed to blast-impact loads.

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7 Dynamic Constitutive Model of RCC for Fully-Graded Dam

Fig. 7.22 Numerical model validation of concrete slab subjected to air explosion

7.6 Summary The existing concrete dynamic constitutive models are difficult to accurately describe the dynamic mechanical behaviors of RCC due to its special mix design and construction technique. Based on the results of the laboratory tests and meso-simulation, the strength surface, strain rate effect and damage evaluation equation of the K&C constitutive model were modified, respectively, so as to better describe the structural dynamic responses of RCC structures subjected to blast-impact loads. The main conclusions are summarized as below: (1) Triaxial mechanical properties of RCC with different aggregate gradations were studied by laboratory test and mesoscale simulation. The results show that the peak strength, peak strain and initial stiffness of RCC increase significantly with the increase of confining pressure, and the slope of stress–strain curve decreases after the peak strength, indicating that the ductility of RCC becomes

7.6 Summary

141

Fig. 7.23 Modified RCC constitutive model for slab subjected to air explosion

stronger under triaxial compression. The confining effect on the compressive strength of RCC with full gradation (5–80 mm) is more significant, and the maximum aggregate size factor is introduced to modify the concrete strength surface in K&C model, which can better describe the strength characteristics of fully-graded RCC. (2) Based on the results of meso-simulation, the true strain rate effect of compressive/tensile strength for RCC is deduced. The contribution ratio of inertia effect to compressive/tensile strength enhancement was quantified by numerical simulation, which were verified to be insensitive to the aggregate gradation. The modified strain rate effect model taking the aggregate gradation effect into account was more consistent with the dynamic mechanical behaviors of RCC with large aggregates in engineering. (3) The K&C constitutive model was modified in terms of strength criterion, damage evolution equation and strain-rate effect model for better describing the dynamic behaviors of fully-graded RCC. The failure strength surface, damage equation and true strain-rate effect were modified, and the modified RCC constitutive model was verified to better describe the structural dynamic responses of fullygraded RCC structure. Based on the modified RCC constitutive model, the failure modes of RCC slabs with wet-sieving aggregate (5–20 mm) and full

142

7 Dynamic Constitutive Model of RCC for Fully-Graded Dam

gradation (5–80 mm) are compared to further illustrate the large aggregate effect on the structural dynamic responses.

References 1. Ansari F, Li Q (1998) High-strength concrete subjected to triaxial compression. Mater J 95(06):747–755 2. Candappa D, Sanjayan J, Setunge S (2001) Complete triaxial stress–strain curves of highstrength concrete. J Mater Civ Eng 13(03):209–215 3. Lu X, Hsu C (2006) Behavior of high strength concrete with and without steel fiber reinforcement in triaxial compression. Cem Concr Res 36(09):1679–1685 4. Lu X, Hsu C (2007) Stress–strain relations of high-strength concrete under triaxial compression. J Mater Civ Eng 19(03):261–268 5. Kohees M, Sanjayan J, Rajeev P (2019) Stress–strain relationship of cement mortar under triaxial compression. Constr Build Mater 220:456–463 6. Ren G, Wu H, Fang Q et al (2016) Triaxial compressive behavior of UHPCC and applications in the projectile impact analyses. Constr Build Mater 113:01–14 7. Farnam Y, Moosavi M, Shekarchi M et al (2010) Behaviour of slurry infiltrated fibre concrete (SIFCON) under triaxial compression. Cem Concr Res 40(11):1571–1581 8. Deng RG (2002) Mechanical properties of roller compacted concrete. J Southwest Jiaotong Univ 15(1):15–19 (in Chinese) 9. Zhang JH, Fang Q, Gong ZM, et al. (2012) Numerical simulation of static mechanical properties based on 3D mesoscale model of fully-graded concrete. Chin J Comput Mech, 29(6):927– 933+947 (in Chinese) 10. Kong X, Fang Q, Chen L et al (2018) A new material model for concrete subjected to intense dynamic loadings. Int J Impact Eng 120:60–78 11. Shui-Sheng YU, Yu-Bin LU, Yong C (2013) A correction methodology to determine the real strain-rate effect for rock-like materials based on SHPB testing. J Wuhan Univ Technol 06:96– 100 (in Chinese) 12. Hao Y, Hao H, Li Z (2010) Numerical analysis of lateral inertial confinement effects on impact test of concrete compressive material properties. Int J Prot Struct 01(01):145–167 13. Wang X, Zhang S, Wang C et al (2018) Fragmentation-based dynamic size effect of layered roller compacted concrete (RCC) under impact loadings. Constr Build Mater 192:58–69 14. Liu ZG (2012) Research on fracture simulation and dynamic behavior of concrete. School of Civil Engineering and Water Conservancy, Dalian University of Technology, Dalian (in Chinese) 15. Hao Y, Hao H, Li Z (2013) Influence of end friction confinement on impact tests of concrete material at high strain rate. Int J Impact Eng 60:82–106 16. Liu JIN, Wangxian YANG, Wenxuan YU et al (2020) Influence of aggregate size on the dynamic tensile strength and size effect of concrete. J Vib Shock 39(09):24–34 (in Chinese) 17. Hao Y, Hao H, Zhang X (2012) Numerical analysis of concrete material properties at high strain rate under direct tension. Int J Impact Eng 39(01):51–62 18. Shu Z, Yubin L, Dongmei Z, et al. (2017) Inertia effect in dynamic tensile tests of concrete-like materials. J Syst Simul, 29(07):1531–1537+1545. (in Chinese) 19. Malvar L, Crawford J (1998) Dynamic increase factors for concrete. Proceedings of 28th department of defense explosives safety seminar, Orlando, FL, pp 01–17 20. Shilang X, Ping W, Qinghua L, et al. (2021) Determination of K&C model parameters for ultra-high toughnessb cementitious composites. J Build Struct, 1–16. (in Chinese) 21. Huawei Y, Ke J, Liao Z, et al. (2020) K&C model of steel fiber reinforced concrete plate under impact and blast load. Chin J High Press Phys, 34(3):024201. (in Chinese)

References

143

22. Wang W, Zhang D, Lu F et al (2012) Experimental study on scaling the explosion resistance of a one-way square reinforced concrete slab under a close-in blast loading. Int J Impact Eng 49:158–164 23. Rao B, Chen L, Fang Q et al (2018) Dynamic responses of reinforced concrete beams under double-end-initiated close-in explosion. Def Technol 14(05):527–539 24. Wang GH (2014) Dynamic response and damage mechanism of concrete gravity dams under extreme loadings. School of Civil Engineering and Architecture, Tianjin University, Tianjin (in Chinese)